Optical properties of materials and their applications [Second edition] 9781119506003, 111950600X, 9781119506058, 1119506050, 9781119506065, 1119506069, 9781119506317

Table of contents :
Content: List of Contributors xv Series Preface xvii Preface xix 1 Fundamental Optical Properties of Materials I 1S.O. Kasap, W.C. Tan, Jai Singh, and Asim K. Ray 1.1 Introduction 1 1.2 Optical Constants n and K 2 1.2.1 Refractive Index and Extinction Coefficient 2 1.2.2 n and K, and Kramers-Kronig Relations 5 1.3 Refractive Index and Dispersion 7 1.3.1 Cauchy Dispersion Relation 7 1.3.2 Sellmeier Equation 8 1.3.3 Refractive Index of Semiconductors 10 1.3.3.1 Refractive Index of Crystalline Semiconductors 10 1.3.3.2 Bandgap and Temperature Dependence 11 1.3.4 Refractive Index of Glasses 11 1.3.5 Wemple-DiDomenico Dispersion Relation 14 1.3.6 Group Index 15 1.4 The Swanepoel Technique: Measurement of n and

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Optical Properties of Materials and Their Applications

Wiley Series in Materials for Electronic and Optoelectronic Applications www.wiley.com/go/meoa Series Editors Professor Arthur Willoughby, University of Southampton, Southampton, UK Dr Peter Capper, Ex-Leonardo MW Ltd, Southampton, UK Professor Sofa Kasap, University of Saskatchewan, Saskatoon, Canada Published Titles Bulk Crystal Growth of Electronic, Optical and Optoelectronic Materials, Edited by P. Capper Properties of Group-IV, III—V and II—VI Semiconductors, S. Adachi Charge Transport in Disordered Solids with Applications in Electronics, Edited by S. Baranovski Optical Properties of Condensed Matter and Applications, Edited by J. Singh Thin Film Solar Cells: Fabrication, Characterization, and Applications, Edited by J. Poortmans and V. Arkhipov Dielectric Films for Advanced Microelectronics, Edited by M. R. Baklanov, M. Green, and K. Maex Liquid Phase Epitaxy of Electronic, Optical and Optoelectronic Materials, Edited by P. Capper and M. Mauk Molecular Electronics: From Principles to Practice, M. Petty Luminescent Materials and Applications, A. Kitai CVD Diamond for Electronic Devices and Sensors, Edited by R. S. Sussmann Properties of Semiconductor Alloys: Group-IV, III—V and II—VI Semiconductors, S. Adachi Mercury Cadmium Telluride, Edited by P. Capper and J. Garland Zinc Oxide Materials for Electronic and Optoelectronic Device Applications, Edited by C. Litton, D. C. Reynolds, and T. C. Collins Lead-Free Solders: Materials Reliability for Electronics, Edited by K. N. Subramunian Silicon Photonics: Fundamentals and Devices, M. Jamal Deen and P. K. Basu Nanostructured and Subwavelength Waveguides: Fundamentals and Applications, M. Skorobogatiy Photovoltaic Materials: From Crystalline Silicon to Third-Generation Approaches, Edited by G. Conibeer and A. Willoughby Glancing Angle Deposition of Thin Films: Engineering the Nanoscale, Matthew M. Hawkeye, Michael T. Taschuk, and Michael J. Brett Physical Properties of High-Temperature Superconductors, R. Wesche Spintronics for Next Generation Innovative Devices, Edited by Katsuaki Sato and Eiji Saitoh Inorganic Glasses for Photonics: Fundamentals, Engineering and Applications, Animesh Jha Amorphous Semiconductors: Structural, Optical and Electronic Properties, Kazuo Morigaki, Sandor Kugler, and Koichi Shimakawa Microwave Materials and Applications, Two volume set, Edited by Mailadil T. Sebastian, Rick Ubic, and Heli Jantunen Molecular Beam Epitaxy: Materials and Applications for Electronics and Optoelectronics, Edited by Hajime Asahi and Yoshiji Korikoshi Metalorganic Vapor Phase Epitaxy (MOVPE): Growth, Materials Properties, and Applications, Edited by Stuart Irvine and Peter Capper

Optical Properties of Materials and Their Applications

Edited by Jai Singh College of Engineering, IT and Environment Charles Darwin University, Darwin, Australia

Second Edition

This edition first published 2020 © 2020 John Wiley & Sons Ltd Edition History John Wiley & Sons Inc. (1e, 2006) All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/ permissions. The right of Jai Singh to be identified as the author of the editorial material in this work has been asserted in accordance with law. Registered Offices John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK Editorial Office The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats. Limit of Liability/Disclaimer of Warranty In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Library of Congress Cataloging-in-Publication Data Names: Singh, Jai, editor. Title: Optical properties of materials and their applications / edited by Jai Singh (College of Engineering, IT, and Environment, Charles Darwin University, Darwin, Australia) Other titles: Optical properties of condensed matter and applications. | Optical properties of condensed matter and applications. Description: Second edition. | Hoboken, NJ : John Wiley & Sons, 2020. | Series: Wiley series in materials for electronic and optoelectronic applications | Previous edition: Optical properties of condensed matter and applications, 2006. | Includes bibliographical references and index. Identifiers: LCCN 2019023895 (print) | LCCN 2019023896 (ebook) | ISBN 9781119506317 (cloth) | ISBN 9781119506065 (adobe pdf ) | ISBN 9781119506058 (epub) Subjects: LCSH: Condensed matter–Optical properties. | Materials–Optical properties. | Electrooptics–Materials. Classification: LCC QC173.458.O66 O68 2020 (print) | LCC QC173.458.O66 (ebook) | DDC 530.4/12–dc23 LC record available at https://lccn.loc.gov/2019023895 LC ebook record available at https://lccn.loc.gov/2019023896 Cover Design: Wiley Cover Images: © mitchFOTO / Shutterstock Set in 10/12pt WarnockPro by SPi Global, Chennai, India 10 9 8 7 6 5 4 3 2 1

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Contents List of Contributors Series Preface Preface 1 Fundamental Optical Properties of Materials I S.O. Kasap, W.C. Tan, Jai Singh, and Asim K. Ray

1.1 1.2

Introduction Optical Constants n and K 1.2.1 Refractive Index and Extinction Coefficient 1.2.2 n and K, and Kramers–Kronig Relations 1.3 Refractive Index and Dispersion 1.3.1 Cauchy Dispersion Relation 1.3.2 Sellmeier Equation 1.3.3 Refractive Index of Semiconductors 1.3.3.1 Refractive Index of Crystalline Semiconductors 1.3.3.2 Bandgap and Temperature Dependence 1.3.4 Refractive Index of Glasses 1.3.5 Wemple–DiDomenico Dispersion Relation 1.3.6 Group Index 1.4 The Swanepoel Technique: Measurement of n and 𝛼 for Thin Films on Substrates 1.4.1 Uniform Thickness Films 1.4.2 Thin Films with Non-uniform Thickness 1.5 Transmittance and Reflectance of a Partially Transparent Plate 1.6 Optical Properties and Diffuse Reflection: Schuster–Kubelka–Munk Theory 1.7 Conclusions Acknowledgments References 2 Fundamental Optical Properties of Materials II S.O. Kasap, K. Koughia, Jai Singh, Harry E. Ruda, and Asim K. Ray

2.1 2.2 2.3

Introduction Lattice or Reststrahlen Absorption and Infrared Reflection Free Carrier Absorption (FCA)

xv xvii xix 1

1 2 2 5 7 7 8 10 10 11 11 14 15 16 16 22 25 27 31 31 32 37

37 40 42

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2.4 2.5 2.6

Band-to-Band or Fundamental Absorption (Crystalline Solids) Impurity Absorption and Rare-Earth Ions Effect of External Fields 2.6.1 Electro-Optic Effects 2.6.2 Electro-Absorption and Franz–Keldysh Effect 2.6.3 Faraday Effect 2.7 Effective Medium Approximations 2.8 Conclusions Acknowledgments References 3 Optical Properties of Disordered Condensed Matter Koichi Shimakawa, Jai Singh, and S.K. O’Leary

3.1 3.2

Introduction Fundamental Optical Absorption (Experimental) 3.2.1 Amorphous Chalcogenides 3.2.2 Hydrogenated Nano-Crystalline Silicon (nc-Si:H) 3.3 Absorption Coefficient (Theory) 3.4 Compositional Variation of the Optical Bandgap 3.4.1 In Amorphous Chalcogenides 3.5 Conclusions References 4 Optical Properties of Glasses Andrew Edgar

4.1 4.2 4.3 4.4 4.5

Introduction The Refractive Index Glass Interfaces Dispersion Sensitivity of the Refractive Index 4.5.1 Temperature Dependence 4.5.2 Stress Dependence 4.5.3 Magnetic Field Dependence—The Faraday Effect 4.5.4 Chemical Perturbations—Molar Refractivity 4.6 Glass Color 4.6.1 Coloration by Colloidal Metals and Semiconductors 4.6.2 Optical Absorption in Rare-Earth-Doped Glass 4.6.3 Absorption by 3d Metal Ions 4.7 Fluorescence in Rare-Earth-Doped Glass 4.8 Glasses for Fiber Optics 4.9 Refractive Index Engineering 4.10 Glass and Glass–Fiber Lasers and Amplifiers 4.11 Valence Change Glasses 4.12 Transparent Glass Ceramics 4.12.1 Introduction 4.12.2 Theoretical Basis for Transparency

45 48 54 54 55 56 58 61 61 62 67

67 69 69 72 74 79 79 80 80 83

83 84 86 88 90 90 91 92 94 95 95 96 99 102 104 106 109 111 114 114 116

Contents

4.12.3 Rare-Earth-Doped Transparent Glass Ceramics for Active Photonics 4.12.4 Ferroelectric Transparent Glass Ceramics 4.12.5 Transparent Glass Ceramics for X-ray Storage Phosphors 4.13 Conclusions References 5 Concept of Excitons Jai Singh, Harry E. Ruda, M.R. Narayan, and D. Ompong

5.1 5.2

Introduction Excitons in Crystalline Solids 5.2.1 Excitonic Absorption in Crystalline Solids 5.3 Excitons in Amorphous Semiconductors 5.3.1 Excitonic Absorption in Amorphous Solids 5.4 Excitons in Organic Semiconductors 5.4.1 Photoexcitation and Formation of Excitons 5.4.1.1 Photoexcitation of Singlet Excitons Due to Exciton–Photon Interaction 5.4.1.2 Excitation of Triplet Excitons 5.4.2 Exciton Up-Conversion 5.4.3 Exciton Dissociation 5.4.3.1 Conversion from Frenkel to CT Excitons 5.4.3.2 Dissociation of CT Excitons 5.5 Conclusions References 6 Photoluminescence Takeshi Aoki

6.1 6.2

6.3

6.4

Introduction Fundamental Aspects of Photoluminescence (PL) in Materials 6.2.1 Intrinsic Photoluminescence 6.2.2 Extrinsic Photoluminescence 6.2.3 Up-Conversion Photoluminescence (UCPL) 6.2.4 Other Related Optical Transitions Experimental Aspects 6.3.1 Static PL Spectroscopy 6.3.2 Photoluminescence Excitation Spectroscopy (PLE) and Photoluminescence Absorption Spectroscopy (PLAS) 6.3.3 Time Resolved Spectroscopy (TRS) 6.3.4 Time-Correlated Single Photon Counting (TCSPC) 6.3.5 Frequency-Resolved Spectroscopy (FRS) 6.3.6 Quadrature Frequency Resolved Spectroscopy (QFRS) Photoluminescence Lifetime Spectroscopy of Amorphous Semiconductors by QFRS Technique 6.4.1 Overview 6.4.2 Dual-Phase Double Lock-in (DPDL) QFRS Technique

120 121 121 124 124 129

129 130 133 135 137 139 140 141 142 147 148 151 152 153 154 157

157 158 159 160 162 163 164 164 167 168 171 172 173 175 175 176

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6.4.3

Exploring Broad PL Lifetime Distribution in a-Si:H by Wideband QFRS 6.4.3.1 Effects of Excitation Intensity, Excitation, and Emission Energies 6.4.3.2 Temperature Dependence 6.4.3.3 Effect of Electric and Magnetic Fields 6.4.4 Residual PL Decay of a-Si:H 6.5 QFRS on Up-Conversion Photoluminescence (UCPL) of RE-Doped Materials 6.6 Conclusions Acknowledgments References 7 Photoluminescence, Photoinduced Changes, and Electroluminescence in Noncrystalline Semiconductors Jai Singh

7.1 7.2

Introduction Photoluminescence 7.2.1 Radiative Recombination Operator and Transition Matrix Element 7.2.2 Rates of Spontaneous Emission 7.2.2.1 At Nonthermal Equilibrium 7.2.2.2 At Thermal Equilibrium 7.2.2.3 Determining E0 7.2.3 Results of Spontaneous Emission and Radiative Lifetime 7.2.4 Temperature Dependence of PL 7.2.5 Excitonic Concept 7.3 Photoinduced Changes in Amorphous Chalcogenides 7.3.1 Effect of Photo-Excitation and Phonon Interaction 7.3.2 Excitation of a Single Electron–Hole Pair 7.3.3 Pairing of Like Excited Charge Carriers 7.4 Radiative Recombination of Excitons in Organic Semiconductors 7.4.1 Rate of Fluorescence 7.4.2 Rate of Phosphorescence 7.4.3 Organic Light Emitting Diodes (OLEDs) 7.4.3.1 Second- and Third-Generation OLEDs: TADF 7.5 Conclusions Acknowledgments References

8 Photoinduced Bond Breaking and Volume Change in Chalcogenide Glasses Sandor Kugler, Rozália Lukács, and Koichi Shimakawa

8.1 8.2 8.3 8.4

Introduction Atomic-Scale Computer Simulations of Photoinduced Volume Changes Effect of Illumination Kinetics of Volume Change

178 179 184 185 189 192 197 198 198

203

203 205 206 211 212 214 215 216 222 223 225 226 228 229 232 233 233 234 235 236 236 237 241

241 243 244 245

Contents

8.4.1 a-Se 8.4.2 a-As2 Se3 8.5 Additional Remarks 8.6 Conclusions References 9 Properties and Applications of Photonic Crystals Harry E. Ruda and Naomi Matsuura

9.1 9.2

Introduction PC Overview 9.2.1 Introduction to PCs 9.2.2 Nanoengineering of PC Architectures 9.2.3 Materials Selection for PCs 9.3 Tunable PCs 9.3.1 Tuning PC Response by Changing the Refractive Index of Constituent Materials 9.3.1.1 PC Refractive Index Tuning Using Light 9.3.1.2 PC Refractive Index Tuning Using an Applied Electric Field 9.3.1.3 Refractive Index Tuning of Infiltrated PCs 9.3.1.4 PC Refractive Index Tuning by Altering the Concentration of Free Carriers (Using Electric Field or Temperature) in Semiconductor-Based PCs 9.3.2 Tuning PC Response by Altering the Physical Structure of the PC 9.3.2.1 Tuning PC Response Using Temperature 9.3.2.2 Tuning PC Response Using Magnetism 9.3.2.3 Tuning PC Response Using Strain 9.3.2.4 Tuning PC Response Using Piezoelectric Effects 9.3.2.5 Tuning PC Response Using MEMS Actuation 9.4 Selected Applications of PC 9.4.1 Waveguide Devices 9.4.2 Dispersive Devices 9.4.3 Add/Drop Multiplexing Devices 9.4.4 Applications of PCs for Light-Emitting Diodes (LEDs) and Lasers 9.5 Conclusions Acknowledgments References 10 Nonlinear Optical Properties of Photonic Glasses Keiji Tanaka

10.1 Introduction 10.2 Photonic Glass 10.3 Nonlinear Absorption and Refractivity 10.3.1 Fundamentals 10.3.2 Two-Photon Absorption

245 246 248 249 249 251

251 252 252 253 255 255 256 256 256 257

257 258 258 258 258 259 260 260 261 262 262 263 265 265 265 269

269 271 272 272 275

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10.3.3 Nonlinear Refractivity 10.4 Nonlinear Excitation-Induced Structural Changes 10.4.1 Fundamentals 10.4.2 Oxides 10.4.3 Chalcogenides 10.5 Conclusions 10.A Addendum: Perspectives on Optical Devices References 11 Optical Properties of Organic Semiconductors Takashi Kobayashi and Hiroyoshi Naito

11.1 Introduction 11.2 Molecular Structure of π-Conjugated Polymers 11.3 Theoretical Models 11.4 Absorption Spectrum 11.5 Photoluminescence 11.6 Non-Emissive Excited States 11.7 Electron–Electron Interaction 11.8 Interchain Interaction 11.9 Conclusions References 12 Organic Semiconductors and Applications Furong Zhu

12.1 Introduction 12.1.1 Device Architecture and Operation Principle 12.1.2 Technical Challenges and Process Integration 12.2 Anode Modification for Enhanced OLED Performance 12.2.1 Low-Temperature High-Performance ITO 12.2.1.1 Experimental Methods 12.2.1.2 Morphological Properties 12.2.1.3 Electrical Properties 12.2.1.4 Optical Properties 12.2.1.5 Compositional Analysis 12.2.2 Anode Modification 12.2.3 Electroluminescence Performance of OLEDs 12.3 Flexible OLEDs 12.3.1 Flexible OLEDs on Ultrathin Glass Substrate 12.3.2 Flexible Top-Emitting OLEDs on Plastic Foils 12.3.2.1 Top-Emitting OLEDs 12.3.2.2 Flexible TOLEDs on Plastic Foils 12.4 Solution-Processable High-Performing OLEDs 12.4.1 Performance of OLEDs with a Hybrid MoO3 -PEDOT:PSS Hole Injection Layer (HIL) 12.4.2 Morphological Properties of the MoO3 -PEDOT:PSS HIL

278 280 280 281 283 285 286 288 295

295 296 298 300 304 306 309 314 320 321 323

323 324 325 327 327 328 329 331 333 336 339 340 345 346 347 348 350 353 353 361

Contents

12.4.3 Surface Electronic Properties of MoO3 -PEDOT:PSS HIL 12.5 Conclusions References 13 Transparent White OLEDs Choi Wing Hong and Furong Zhu

13.1 Introduction—Progress in Transparent WOLEDs 13.2 Performance of WOLEDs 13.2.1 Optimization of Dichromatic WOLEDs 13.2.2 J-L-V Characteristics of WOLEDs 13.2.3 Electron-Hole Current Balance in Transparent WOLEDs 13.3 Emission Behavior of Transparent WOLEDs 13.3.1 Visible-Light Transparency of WOLEDs 13.3.2 L-J Characteristics of Transparent WOLEDs 13.3.3 Angular-Dependent Color Stability of Transparent WOLEDs 13.4 Conclusions References 14 Optical Properties of Thin Films V.-V. Truong, S. Tanemura, A. Haché, and L. Miao

14.1 Introduction 14.2 Optics of Thin Films 14.2.1 An Isotropic Film on a Substrate 14.2.2 Matrix Methods for Multi-Layered Structures 14.2.3 Anisotropic Films 14.3 Reflection-Transmission Photoellipsometry for Determination of Optical Constants 14.3.1 Photoellipsometry of a Thick or a Thin Film 14.3.2 Photoellipsometry for a Stack of Thick and Thin Films 14.3.3 Remarks on the Reflection-Transmission Photoellipsometry Method 14.4 Application of Thin Films to Energy Management and Renewable-Energy Technologies 14.4.1 Electrochromic Thin Films 14.4.2 Pure and Metal-Doped VO2 Thermochromic Thin Films 14.4.3 Temperature-Stabilized V1-x Wx O2 Sky Radiator Films 14.4.4 Optical Functional TiO2 Thin Film for Environmentally Friendly Technologies 14.5 Application of Tunable Thin Films to Phase and Polarization Modulation 14.6 Conclusions References 15 Optical Characterization of Materials by Spectroscopic Ellipsometry J. Mistrík

15.1 Introduction 15.2 Notions of Light Polarization

363 368 369 373

373 374 374 377 384 386 386 390 395 400 400 403

403 404 404 406 407 408 408 410 412 412 413 414 417 420 424 430 430 435

435 436

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15.3 Measureable Quantities 15.4 Instrumentation 15.5 Single Interface 15.6 Single Layer 15.7 Multilayer 15.8 Linear Grating 15.9 Conclusions Acknowledgments References 16 Excitonic Processes in Quantum Wells Jai Singh and I.-K. Oh

16.1 16.2 16.3 16.4

438 441 442 448 454 458 462 463 463 465

Introduction Exciton–Phonon Interaction Exciton Formation in QWs Assisted by Phonons Nonradiative Relaxation of Free Excitons 16.4.1 Intraband Processes 16.4.2 Interband Processes 16.5 Quasi-2D Free-Exciton Linewidth 16.6 Localization of Free Excitons 16.7 Conclusions References

465 466 467 474 475 479 485 491 499 500

17 Optoelectronic Properties and Applications of Quantum Dots Jørn M. Hvam

503

17.1 Introduction 17.2 Epitaxial Growth and Structure of Quantum Dots 17.2.1 Self-Assembled Quantum Dots 17.2.2 Site-Controlled Growth on Patterned Substrates 17.2.3 Natural or Interface Quantum Dots 17.2.4 Quantum Dots in Nanowires 17.3 Excitons in Quantum Dots 17.3.1 Quantum-Dot Bandgap 17.3.2 Optical Transitions 17.4 Optical Properties 17.4.1 Radiative Lifetime, Oscillator Strength, and Internal Quantum Efficiency 17.4.2 Linewidth, Coherence, and Dephasing 17.4.3 Transient Four-Wave Mixing 17.5 Quantum Dot Applications 17.5.1 Quantum Dot Lasers and Optical Amplifiers 17.5.1.1 Gain Dynamics 17.5.1.2 Homogeneous Broadening and Dephasing 17.5.1.3 Long-Wavelength Lasers 17.5.1.4 Nano Lasers 17.5.2 Single-Photon Emitters 17.5.2.1 Micropillars and Nanowires

503 504 504 505 506 507 508 509 510 513 514 516 517 520 520 522 524 526 527 527 530

Contents

17.5.2.2 Photonic Crystal Waveguide 17.6 Conclusions Acknowledgments References 18 Perovskites – Revisiting the Venerable ABX3 Family with Organic Flexibility and New Applications Junwei Xu, D.L. Carroll, K. Biswas, F. Moretti, S. Gridin, and R.T. Williams

18.1 Introduction 18.1.1 Review 18.1.2 The Structures 18.1.2.1 Simple Cubic Frameworks 18.1.2.2 The Multiplicity of Hybrids 18.1.2.3 Structural Variation 18.2 Hybrid Perovskites in Photovoltaics 18.2.1 Review 18.2.2 The Phenomena Characterized as “Defect Tolerance” 18.3 Light-Emitting Diodes Using Solution-Processed Lead Halide Perovskites 18.3.1 Review 18.3.2 Construction and Characterization of LEDs Utilizing CsPbBr3 Nano-Inclusions in Cs4 PbBr6 as the Electroluminescent Medium 18.4 Ionizing Radiation Detectors Using Lead Halide Perovskite Materials: Basics, Progress, and Prospects 18.5 Conclusions Acknowledgments References

531 533 534 534

537

537 537 538 538 539 540 544 544 548 549 549

553 562 582 583 583

19 Optical Properties and Spin Dynamics of Diluted Magnetic Semiconductor Nanostructures Akihiro Murayama and Yasuo Oka

589

19.1 Introduction 19.2 Quantum Wells 19.2.1 Spin Injection 19.2.2 Study of Spin Dynamics by Pump-Probe Spectroscopy 19.3 Fabrication of Nanostructures by Electron-Beam Lithography 19.4 Self-Assembled Quantum Dots 19.5 Hybrid Nanostructures with Ferromagnetic Materials 19.6 Conclusions Acknowledgments References

589 591 591 594 596 599 604 607 608 609

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20 Kinetics of the Persistent Photoconductivity in Crystalline III-V Semiconductors Ruben Jeronimo Freitas and Koichi Shimakawa

611

20.1 Introduction 20.2 A Review of PPC in III-V Semiconductors 20.3 Key Physical Terms Related to PPC 20.3.1 Dispersive Reaction 20.3.2 SEF and Power Law 20.3.3 Waiting Time Distribution 20.4 Kinetics of PPC in III-V Semiconductors 20.5 Conclusions Acknowledgments 20.A On the Reaction Rate Under the Uniform Distribution References

611 613 615 615 616 617 617 623 623 623 625

Index

627

xv

List of Contributors Takeshi Aoki Joint Research Center of High-technology, Department of Electronics and

Information Technology, Tokyo Polytechnic University, Atsugi, Japan K. Biswas Department of Chemistry and Physics, Arkansas State University, Jonesboro,

USA D.L. Carroll Department of Physics and Nanotechnology Center, Wake Forest University,

Winston-Salem, North Carolina, USA Andrew Edgar School of Chemical and Physical Sciences, Victoria University of

Wellington, New Zealand Ruben Jeronimo Freitas Department of Electrical and Electronic Engineering, National

University of Timor Lorosae, Díli, East Timor S. Gridin Department of Physics and Nanotechnology Center, Wake Forest University,

Winston-Salem, North Carolina, USA A. Haché Département de physique et d’astronomie, Université de Moncton, New

Brunswick, Canada Jørn M. Hvam Department of Photonics Engineering, Technical University of Denmark,

Kongens Lyngby, Denmark S.O. Kasap Department of Electrical and Computer Engineering, University of

Saskatchewan, Saskatoon, Canada Takashi Kobayashi Department of Physics and Electronics, Osaka Prefecture University,

Sakai, Japan K. Koughia Department of Electrical and Computer Engineering, University of

Saskatchewan, Saskatoon, Canada Sandor Kugler Department of Theoretical Physics, Budapest University of Technology

and Economics, Hungary Rozália Lukács Norwegian University of Life Sciences, Ås, Akershus, Norway Naomi Matsuura Centre for Nanotechnology, University of Toronto, Canada L. Miao Guilin University of Electronic Technology, Guangxi, P.R. China

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List of Contributors

J. Mistrík Center of Materials and Nanotechnologies, Faculty of Chemical Technology,

University of Pardubice, Czech Republic F. Moretti Lawrence Berkeley National Laboratory, Berkeley, California, USA Akihiro Murayama Graduate School of Information Science and Technology, Hokkaido

University, Sapporo, Japan Hiroyoshi Naito The Research Institute for Molecular Electronic Devices, Osaka

Prefecture University, Sakai, Japan M.R. Narayan College of Engineering, Information Technology and Environment,

Charles Darwin University, Darwin, Australia S.K. O’Leary School of Engineering, The University of British Columbia, Kelowna,

Canada I.-K. Oh College of Engineering, Information Technology and Environment, Charles

Darwin University, Darwin, Australia Yasuo Oka Institute of Multidisciplinary Research for Advanced Materials, Tohoku

University, Sendai, Miyagi, Japan D. Ompong College of Engineering, Information Technology and Environment, Charles

Darwin University, Darwin, Australia Asim K. Ray Department of Electrical & Computer Engineering, Brunel University

London, Uxbridge, UK Harry E. Ruda Centre for Nanotechnology and Electronic and Photonic Materials Group,

Department of Materials Science, University of Toronto, Ontario, Canada Koichi Shimakawa Department of Electrical and Electronic Engineering, Gifu University,

Japan Jai Singh College of Engineering, Information Technology and Environment, Charles

Darwin University, Darwin, Australia W.C. Tan Department of Electrical & Computer Engineering, National University of

Singapore, Kent Ridge, Singapore Keiji Tanaka Department of Applied Physics, Graduate School of Engineering, Hokkaido

University, Sapporo, Japan S. Tanemura Japan Fine Ceramics Centre, Mutsuno, Atsuta-ku, Nagoya, Japan V.-V. Truong Physics Department, Concordia University, Montreal, Quebec, Canada R.T. Williams Department of Physics and Nanotechnology Center, Wake Forest

University, Winston-Salem, North Carolina, USA Choi Wing Hong, Department of Physics, Hong Kong Baptist University, Kowloon Tong,

China Junwei Xu Department of Physics and Nanotechnology Center, Wake Forest University,

Winston-Salem, North Carolina, USA Furong Zhu Department of Physics, Hong Kong Baptist University, Kowloon Tong, China

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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 electronic and optoelectronic applications. It is designed to provide much-needed information on the fundamental scientific principles of these materials, together with how these are employed in technological applications. These books are aimed at (postgraduate) students, researchers, and technologists engaged in research, development, and the study of materials in electronics and photonics, and at 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 on materials engineering at a practical level, but also on a clear understanding of the properties of materials and the fundamental science behind these properties. It is the properties of a material that eventually determine its usefulness in an application. The series therefore also includes 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 as much as it is impossible to develop new materials without the proper characterization of their structure and properties. Structure–property relationships have always been fundamentally and intrinsically important to materials science and engineering. Materials science is well known for being one of the most interdisciplinary sciences. It is the 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 chemical engineering background working on materials projects with applications in electronics. In selecting titles for the series, we have tried to maintain the interdisciplinary aspect of the field, and hence its excitement to researchers in this field. Arthur Willoughby Peter Capper Safa Kasap

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Preface The second edition, being published more than 10 years after the first edition, presents state-of-the-art developments in almost all topics related to the optical properties of materials and their applications presented in the first edition. Since the publication of the first edition in 2006, many advances have been made in fields such as the optical properties of materials, electroluminescence in organic light-emitting devices, organic solar cells, opto-electronic devices, etc. It is hence very timely to update all the chapters in the first edition by adding developments since 2006 to produce the second edition. This second edition contains 15 of the original 16 chapters, all of which have been updated, as well as 5 brand new chapters, contributed by very experienced and well-known scientists and groups available on different aspects of the optical properties of materials. The study of optical properties of materials has now become an interdisciplinary field, and scientists of physical, chemical, and biological sciences; nanotechnology engineers; and industry researchers have strong interests in this field. The field offers one of the fastest-growing research platforms in material sciences. The second edition covers many examples and applications in the field of electronic and optoelectronic properties of materials, and in photonics. Most chapters are presented to be relatively independent with minimal cross-referencing, and chapters with complementary contents are arranged together to facilitate a reader with cross-referencing. Books written in this field mostly follow one of the two pedagogies: chapters are either based on (i) physical processes, or (ii) the various classes of materials. This book combines the two approaches by first identifying the processes that should be described in detail, and then introducing the relevant classes of materials. Many books also miss the details of how various optical properties are measured. This book presents a comprehensive review of experimental techniques, including recent advances in ultrafast (femtosecond) spectroscopy of materials. Not many books are currently available with such a wide coverage of the field with clarity and levels of readership in a single volume as this book. In Chapters 1 and 2 by Kasap et al., the fundamental optical properties of materials are reviewed, and as such these chapters are expected to refresh the readers with the basics by providing useful optical relations. In Chapter 3, Shimakawa et al. present an up-to-date review of the optical properties of disordered inorganic solids, and Chapter 4 by Edgar presents an extensive discussion on the optical properties of glasses. Chapter 5 by Singh and co-workers presents the concept of excitons in inorganic and organic semiconductors, both crystalline and non-crystalline variants. In Chapter 6, Aoki has presented a comprehensive review of the experimental advances in the techniques of

xx

Preface

measuring photoluminescence together with updates in luminescence results in amorphous semiconductors, and Chapter 7 by Singh complements the theoretical advances in the field of photoluminescence and photoinduced changes in non-crystalline semiconductors. In Chapter 8 by Kugler et al., recent advances in the simulation of photoinduced bond breaking and volume changes in chalcogenide glasses are presented. In Chapter 9, Ruda and Matsuura present a comprehensive review of the properties and applications of photonic crystals. In Chapter 10, Tanaka has presented an up-to-date review of the nonlinear optical properties of photonic glasses. Chapter 11 by Kobayashi and Naito discusses the fundamental optical properties of organic semiconductors. In Chapter 12, Zhu has presented a comprehensive review of the applications of organic semiconductors, in particular, in developing organic light-emitting diodes (OLEDs). In Chapter 13, Hong and Zhu have reviewed the recent developments in the fabrication of transparent white light-emitting diodes (WOLEDs). This is a new chapter added in the second edition. In Chapter 14, Truong and Tanemura have presented an up-to-date review of the optical properties of thin films and their applications, and Chapter 15 by Mistrik deals with the optical characterization of materials by spectroscopic ellipsometry. This is the second new chapter in the second edition. In Chapter 16, Singh and Oh have discussed the excitonic processes in quantum wells. In Chapter 17, the third new chapter in this edition, Hvam has presented an up-to-date comprehensive review of the optoelectronic properties and applications of quantum dots. Chapter 18 by Xu et al. presents up-to-date developments in the applications of perovskites. This is the fourth new chapter in the second edition. In Chapter 19, Murayama and Oka have presented the optical properties and spin dynamics of diluted magnetic semiconductor nanostructures. In the final Chapter 20, the fifth new chapter in this edition, Freitas and Shimakawa have discussed the kinetics of the persistent photoconductivity in Crystalline III–V semiconductors. Thus, the addition of the five new chapters on transparent WOLELDs, ellipsometry, quantum dots, perovskites, and persistent photoconductivity widens the scope of the second edition to a new level. One of the chapters on the negative index of refraction in the first edition has not been included in the second edition at the request of the authors. The readership of the book is expected to be the senior undergraduate and postgraduate students, and teaching and research professionals in the field. In conclusion, I am very grateful to all the contributing authors of the second edition for their utmost co-operation in meeting the deadlines, without which this project would not have concluded. I also would like to acknowledge the technical support from Drs Stefanija Klaric and Luis Herrera Diaz in preparing my chapters. I would also like to thank my friend Beth Woof for her support throughout the course of preparation of this volume. Darwin, Australia

Jai Singh

1

1 Fundamental Optical Properties of Materials I S.O. Kasap 1 , W.C. Tan 2 , Jai Singh 3 , and Asim K. Ray 4 1

Department of Electrical and Computer Engineering, University of Saskatchewan, 57 Campus Drive, Saskatoon, Canada Department of Electrical & Computer Engineering, National University of Singapore, Kent Ridge, Singapore College of Engineering, IT and Environment, Purple 12, Charles Darwin University, Ellengowan Drive, Darwin, Australia 4 Department of Electrical & Computer Engineering, Brunel University London, Kingston Lane, Uxbridge, UK 2 3

CHAPTER MENU Introduction, 1 Optical Constants n and K, 2 Refractive Index and Dispersion, 7 The Swanepoel Technique: Measurement of n and 𝛼 for Thin Films on Substrates, 16 Transmittance and Reflectance of a Partially Transparent Plate, 25 Optical Properties and Diffuse Reflection: Schuster–Kubelka–Munk Theory, 27 Conclusions, 31 References, 32

1.1 Introduction Optical properties of a material change or affect the characteristics of light passing through it by modifying its propagation vector or intensity. Two of the most important optical parameters are the refractive index n and the extinction coefficient K, which are generically called optical constants, although some authors include other optical coefficients within this terminology. The latter is related to the attenuation or absorption coefficient 𝛼. In Part I, in this chapter, we present the complex refractive index, the frequency or wavelength dependence of n and K, so-called dispersion relations, how n and K are inter-related, and how n and K can be determined by studying the transmission as a function of wavelength through a thin film of the material. Physical insights into n and K are provided in Part II (Chapter 2). In addition, there has been a strong research interest in characterizing the optical properties of inhomogeneous media, such as porous media, in which both light absorption and scattering take place so that the reflectance is not specular but diffuse. The latter problem is now included in this second edition. The optical properties of various materials, with n and K being the most important, are available in the literature in one form or another, either published in journals, books, and handbooks, or posted on websites of various researchers, organizations (e.g. NIST), or companies (e.g. Schott Glass). Nonetheless, the reader is referred to the Optical Properties of Materials and Their Applications, Second Edition. Edited by Jai Singh. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

2

1 Fundamental Optical Properties of Materials I

works of Greenway and Harbeke [1], Wolfe [2], Klocek [3], Palik [4, 5], Ward [6], Efimov [7], Palik and Ghosh [8], Nikogosyan [9], and Weaver and Frederikse [10] for the optical properties of a wide range of materials. Adachi’s books on the optical constants of semiconductors are highly recommended [11–13], along with Madelung’s third edition of Semiconductors: Data Handbook [14]. There are, of course, other books and handbooks that also contain optical constants in various chapters; see, for example, references [15–20]. There are also various books that describe optical properties of solids at the senior undergraduate and introductory graduate levels, such as those by Tanner [21], Jimenez and Tomm [22], Stenzel [23], Fox [24], Simmons and Potter [25], Toyozawa [26], Wooten [27], and Abeles [28], which are highly recommended. A number of experimental techniques are available for measuring n and K, some of which have been summarized by Simmons and Potter [25]. For example, ellipsometry measures changes in the polarization of light incident on a sample to sensitively characterize surfaces and thin films (see Chapter 23 in this volume). The interaction of incident polarized light with the sample causes a polarization change in the light, which may then be measured by analyzing the light reflected from the sample. Collins has also provided an extensive in-depth review of ellipsometry for optical measurements [29]. One of the most popular and convenient optical experiments involves a monochromatic light passing through a thin sample, and measuring the transmitted intensity as a function of wavelength, T(𝜆), using a simple spectrophotometer. For thin samples on a thick transparent substrate, the transmission spectrum shows oscillations in T(𝜆) with the wavelength due to interferences within the thin film. Swanepoel’s technique uses the T(𝜆) measurement to determine n and K, as described in Section 1.4.

1.2 Optical Constants n and K One of the most important optical constants of a material is its refractive index, which in general depends on the wavelength of the electromagnetic (EM) wave, through a relationship called dispersion. In materials where an EM wave loses its energy during its propagation, the refractive index becomes complex. The real part is usually the refractive index, n, and the imaginary part is called the extinction coefficient, K. In this section, the refractive index and extinction coefficient will be presented in detail, along with some common dispersion relations. A more practical and a semiquantitative approach is taken along the lines in [30] rather than a full dedication to rigor and mathematical derivations. More analytical approaches can be found in other texts, such as [25, 26]. 1.2.1

Refractive Index and Extinction Coefficient

The refractive index of an optical or dielectric medium, n, is the ratio of the velocity of light c in vacuum to its velocity v in the medium; n = c/v. Using this and Maxwell’s equations, one obtains the well-known Maxwell’s formula for the refractive index of a √ substance as n = 𝜀r 𝜇r , where 𝜀r is the static dielectric constant or relative permittivity and 𝜇r the relative magnetic permeability of the medium. As 𝜇r = 1 for nonmagnetic √ substances, one gets n = 𝜀r , which is very useful in relating the dielectric properties to optical properties of materials at any particular frequency of interest. As 𝜀r depends on the wavelength of light, the refractive index also depends on the wavelength of light, and

1.2 Optical Constants n and K

this dependence is called dispersion. In addition to dispersion, an EM wave propagating through a lossy medium experiences attenuation, which means it loses its energy, due to various loss mechanisms such as the generation of phonons (lattice waves), photogeneration, free carrier absorption, scattering, etc. In such materials, the refractive index becomes a complex function of the frequency of the light wave. The complex refractive index in this chapter is denoted by n*, with real part n, and imaginary part K, called the extinction coefficient, is related to the complex relative permittivity, 𝜀r = 𝜀′r + i𝜀′′r , by, √ √ (1.1a) n∗ = n + iK = 𝜀r = 𝜀′r + i𝜀′′r where 𝜀′r and 𝜀′′r are, respectively, the real and imaginary parts of 𝜀r . Eq. (1.1a) gives: n2 − K 2 = 𝜀′r and 2nK = 𝜀′′r . In explicit terms, n and K can be obtained as √ ′′2 1∕2 n = (1∕ 2)[(𝜀′2 + 𝜀′r ]1∕2 r + 𝜀r ) √ ′′2 1∕2 K = (1∕ 2)[(𝜀′2 − 𝜀′r ]1∕2 r + 𝜀r )

(1.1b)

(1.2a) (1.2b)

Some books (particularly in electrical engineering) use 𝜀r = 𝜀′r − i𝜀′′r and n* = n − iK instead of 𝜀r = 𝜀′r + i𝜀′′r and n* = n + iK. The preference lies in what was assumed for the propagating electric field, whether it is represented by expi(𝜔t − kx) or expi(kx − 𝜔t), where k is the propagation constant. In a lossy medium, the imaginary part of√ n* must ′′ ′ lead to a traveling wave whose amplitude decays. Notice that, for 𝜀r ≪ 𝜀r , n = 𝜀′r and K = 𝜀′′r ∕2n—that is, the refractive index is essentially determined by the real part of 𝜀r and K is determined by the imaginary part of 𝜀r , which is known to represent losses in a dielectric medium. The extinction coefficient K represents loss from the energy carried by the propagating EM wave by conveniently including this loss as the imaginary part in the complex refractive index. The optical attenuation coefficient 𝛼 gauges the rate of this loss from the propagating EM wave. In the absence of scattering, the attenuation would be due to absorption within the medium. For an EM wave that is propagating along x with an intensity I, 𝛼 is defined by dI (1.3) 𝛼=− Idx We can relate 𝛼 and K quite easily by taking a plane wave traveling along x for which the electric field in the wave propagates as E = Eo expi(kx − 𝜔t), where Eo is a constant, 𝜔 is the angular frequency and k is the complex propagation constant in the medium, related to n* by its definition k = n*𝜔/c = (n + iK)(𝜔/c). In free space k = k o = 𝜔/c = 2𝜋/𝜆, where 𝜆 is the free space wavelength. We can substitute for n* and then use I is proportional to |E|2 to find I ∝ exp[−2(𝜔/c)Kx)]—that is, I decays exponentially with the distance propagated. We can substitute for I in (1.3) to find 2𝜔 K (1.4) 𝛼= c The optical constants n and K can be determined by measuring the reflectance from the surface of a material as a function of polarization and the angle of incidence. For normal incidence, the reflection coefficient, r, is obtained as 1 − n∗ 1 − n − iK r= = (1.5) 1 + n∗ 1 + n + iK

3

4

1 Fundamental Optical Properties of Materials I

The reflectance R is then defined by: | 1 − n − iK |2 (1 − n)2 + K 2 | = R = |r|2 = || . (1.6) | (1 + n)2 + K 2 | 1 + n + iK | Notice that whenever K is large, for example, over a range of wavelengths, the absorption is strong, and the reflectance is almost unity. The light is then reflected, and any light in the medium is highly attenuated (typical sample calculations may be found in [24, 30]). Optical properties of materials are typically presented either by showing the frequency dependences (dispersion relations) of n and K or 𝜀′r and 𝜀′′r . An intuitive guide to explaining dispersion in insulators is based on a single oscillator model in which the electric field in the light induces forced dipole oscillations in the material (displaces the electron shells in an atom to oscillate about the positive nucleus) with a single resonant frequency 𝜔o . The frequency dependences of 𝜀′r and 𝜀′′r are then obtained as: 𝜀′r = 1 +

Nat ′ N 𝛼 and 𝜀′′r = 1 + at 𝛼e′′ , 𝜀o e 𝜀o

(1.7)

where N at is the number of atoms per unit volume, 𝜀o is the vacuum permittivity, and 𝛼e′ and 𝛼e′′ are, respectively, the real and imaginary parts of the electronic polarizability, given respectively by: 𝛼e′ = 𝛼eo

1 − (𝜔∕𝜔o )2 [1 − (𝜔∕𝜔o )2 ]2 + (𝛾∕𝜔o )2 (𝜔∕𝜔o )2

(1.8a)

(𝛾∕𝜔o )(𝜔∕𝜔o ) [1 − (𝜔∕𝜔o )2 ]2 + (𝛾∕𝜔o )2 (𝜔∕𝜔o )2

(1.8b)

and 𝛼e′′ = 𝛼eo

where 𝛼 eo is the DC polarizability corresponding to 𝜔 = 0 and 𝛾 is the loss coefficient that characterizes the EM wave losses within the material system. Using Eqs. (1.1)–(1.2) and (1.7)–(1.8), the frequency dependence of n and K can be studied. Figure 1.1a shows the dependence of n and K on the normalized frequency 𝜔/𝜔o for a simple single electronic dipole oscillator of resonance frequency 𝜔o . (a)

n, K Complex refractive index 8

Figure 1.1 Refractive index n and extinction coefficient K obtained from a single electronic dipole oscillator model. (a) n and K versus normalized frequency, and (b) reflectance versus normalized frequency.

6 4

γ/ωo = 0.1

n K

2 0 0 R

(b)

n

K

1

1

εDC1/2 = 3

2

3

2

3

ω/ωo

Reflectance

0.5 0 0

1

ω/ωo

1.2 Optical Constants n and K

It is seen from Figure 1.1 that √ n and K peak close to 𝜔 = 𝜔o . If a material has a 𝜀′′r ′ ′′ ≫ 𝜀r , then 𝜀r ≈ i𝜀r , and n ≈ K ≈ 𝜀′′r ∕2 is obtained from Eq. (1.1b). Figure 1.1b shows the dependence of the reflectance R on the frequency. It is observed that R reaches its maximum value at a frequency slightly above 𝜔 = 𝜔o , and then remains high until 𝜔 reaches nearly 3𝜔o ; thus, the reflectance is substantial while absorption is strong. The normal dispersion region is the frequency range below 𝜔o , where n falls as the frequency decreases; that is, n decreases as the wavelength 𝜆 increases. Anomalous dispersion region is the frequency range above 𝜔o where n decreases as 𝜔 increases. Below 𝜔o , K is small and, if 𝜀DC is 𝜀r (0), the DC permittivity, then 𝜔2o

; 𝜔 < 𝜔o . (1.9) 𝜔2o − 𝜔2 Since, 𝜆 = 2𝜋c/𝜔, defining 𝜆o = 2𝜋c/𝜔o as the resonance wavelength, one gets: n2 ≈ 1 + (𝜀DC − 1)

𝜆2 ; 𝜆 > 𝜆o . (1.10) − 𝜆2o While intuitively useful, the dispersion relations in Eq. (1.8) are far too simple. More rigorously, we have to consider the dipole oscillator quantum mechanically, which means a photon excites the oscillator to a higher energy level—see, for example, Fox [24] or Simmons and Potter [25]. The result is that we would have a series of 𝜆2 /(𝜆2 − 𝜆i 2 ) terms with various weighting factors Ai that add to unity, where 𝜆i represent different resonance wavelengths. The weighting factors Ai involve quantum mechanical matrix elements. Figure 1.2 shows the complex relative permittivity and the complex refractive index of crystalline silicon in terms of photon energy h𝜈 [31, 32]. For photon energies below the bandgap energy (1.1 eV), both 𝜀′′r and K are negligible and n is close to 3.7. Both 𝜀′′r and K increase and change strongly as the photon energy becomes greater than 3 eV, far beyond the bandgap energy. Notice that both 𝜀′′r and K peak at h𝜈 ≈ 3.5 eV, which corresponds to a direct photoexcitation processes, electrons excited directly from the valence band to the conduction band, as discussed in Chapter 2. n2 ≈ 1 + (𝜀DC − 1)

1.2.2

𝜆2

n and K, and Kramers–Kronig Relations

If we know the frequency dependence of the real part, 𝜀′r , of the relative permittivity of a material, we can, using the Kramers–Kronig relations between the real and the imaginary parts, determine the frequency dependence of the imaginary part 𝜀′′r , and vice versa. The transform requires that we know the frequency dependence of either the real or imaginary part over as wide a range of frequencies as possible, ideally from zero (DC) to infinity, and that the material has linear behavior, that is, it has a relative permittivity that is independent of the applied field. The Kramers–Kronig relations for the relative permittivity 𝜀r = 𝜀′r + i𝜀′′r are given by [33–35] (see also Appendix 1C in [25] as well as [27]) ∞ 𝜔′ 𝜀′′ r (𝜔′ ) ′ 2 d𝜔 (1.11a) 𝜀′r (𝜔) = 1 + P 𝜋 ∫ 0 𝜔′ 2 − 𝜔2 and 𝜀′′r (𝜔) = −

2𝜔 P 𝜋 ∫0

∞

𝜀′r (𝜔′ ) − 1 𝜔′ 2 − 𝜔2

d𝜔′

(1.11b)

5

6

1 Fundamental Optical Properties of Materials I

50 40

8

ε r″

30

Real

Imaginary

7 6

20

εr′

10

n

4

εr″

0

Real

5 Imaginary

3 εr′

–10

2

K

1

–20

K n

0 1.5

2

3 4 5 Photon energy (ћω) (a)

6

0

2

4 6 8 Photon energy (ћω) (b)

10

Figure 1.2 (a) Complex relative permittivity of a silicon crystal as a function of photon energy plotted in terms of real (𝜀′r ) and imaginary (𝜀′′r ) parts. (b) Optical properties of a silicon crystal vs. photon energy in terms of real (n) and imaginary (K) parts of the complex refractive index. Source: Adapted from D. E. Aspnes and A. A. Studna, 1983 [32] and H.R. Philipp and E.A. Taft, 1960 [31].

where 𝜔′ is the integration variable, P represents the Cauchy principal value of the integral, and the singularity at 𝜔 = 𝜔′ is avoided. Similarly, one can relate the real and imaginary parts of the polarizability, 𝛼 ′ (𝜔) and ′′ 𝛼 (𝜔), and those of the complex refractive index, n(𝜔) and K(𝜔), as well. For a complex refractive index written as n* = n(𝜔) + iK(𝜔), n(𝜔) = 1 +

2 P 𝜋 ∫0

∞

∞ 𝜔′ K(𝜔′ ) ′ n(𝜔′ ) − 1 ′ 2 d𝜔 and K(𝜔) = − d𝜔 P 2 ′ 2 𝜋 ∫ 0 𝜔′ 2 − 𝜔2 𝜔 −𝜔

(1.12)

Although it appears, in theory, that one needs to integrate the spectrum of n or K from DC to infinite frequencies, this is obviously not feasible, and is unnecessary. It should be noted that the experimental setup usually has low- and high-frequency limitations that truncate the preceding integrations. Moreover, in many cases, we are interested in the spectrum of n and K in and around an absorption band. Thus, before and after the absorption frequency range, K would be negligibly small, and we can use this absorption frequency range in the preceding integrals in Eq. (1.12). There are numerous studies in the literature that use the preceding Kramers–Kronig relations in extracting the wavelength dependence of n from that of K, and vice versa, especially around clear absorption bands; a few selected examples can be found in [36–40], and there are many others in the literature. There are also several useful approaches in which the absorption spectrum, or K(𝜔), is described in terms of a particular physical model with a particular expression, and the corresponding refractive index n(𝜔) is derived from the Kramers–Kronig transformation for both amorphous and crystalline solids—for examples, see [41, 42]. It should be emphasized that the optical constants n and K have to obey what are called f-sum rules [43]. For example, the integration of [n(𝜔) – 1] over all frequencies must be zero, and the integration of 𝜔K(𝜔) over all frequencies gives (𝜋/2)𝜔p 2 , where 𝜔p = ℏ(4𝜋NZe2 /me )1/2 is the free electron plasma frequency in which N is the atomic

1.3 Refractive Index and Dispersion

concentration, Z is the total number of electrons per atom, and e and me are the charge and mass of the electron, respectively. The f -sum rules provide a consistency check and enable various constants to be interrelated.

1.3 Refractive Index and Dispersion There are several popular models describing the spectral dependence of refractive index n in a material. Most of these are described in the following text, although some, such as the infrared refractive index, is covered in the discussion on Reststrahlen absorption in Part II, since it is closely related to the coupling of the EM wave to lattice vibrations. The most popular dispersion relation in optical materials is probably the Sellmeier relationship, since one can sum any number of resonance-type terms to get as wide a range of wavelength dependence as possible. However, its main drawback is that it does not accurately represent the refractive index when there is a contribution arising from free carriers in narrow bandgap or doped semiconductors. There are many handbooks, books, and websites that now provide empirical equations for the refractive index of a wide range of solids, for example as in references [1–19, 44]. 1.3.1

Cauchy Dispersion Relation

In the Cauchy relationship, the dispersion relationship between the refractive index (n) and the wavelength of light (𝜆) is commonly stated in the following form: C B + 4 (1.13) 2 𝜆 𝜆 where A, B, and C are material-dependent specific constants. Equation (1.13) is known as Cauchy’s formula; it is typically used in the visible spectrum region for various optical glasses, and it applies to normal dispersion, when n decreases with increasing 𝜆 [45, 46]. The third term is sometimes dropped for a simpler representation of n versus 𝜆 behavior. The original expression was a series in terms of the wavelength, 𝜆, or frequency, 𝜔, or photon energy ℏ𝜔 of light as: n=A+

n = a0 + a2 𝜆−2 + a4 𝜆−4 + a6 𝜆−6 + … 𝜆 > 𝜆th ,

(1.14a)

n = n0 + n2 (ℏ𝜔)2 + n4 (ℏ𝜔)4 + n6 (ℏ𝜔)6 + … ℏ𝜔 < ℏ𝜔th ,

(1.14b)

or where ℏ𝜔 is the photon energy; ℏ𝜔th = hc/𝜆th is the optical excitation threshold (e.g. bandgap energy); and a0 , a2 ,… and n0 , n2 ,… are constants. It has been found that a Cauchy relation in the following form [47]: n = n−2 (ℏ𝜔)−2 + n0 + n2 (ℏ𝜔)2 + n4 (ℏ𝜔)4 ,

(1.15)

can be used satisfactorily over a wide range of photon energies. The dispersion parameters of Eq. (1.15) are listed in Table 1.1 for a few selected materials over specific photon energy ranges. Cauchy’s dispersion relations given in Eqs. (1.13)–(1.14) were originally called the elastic ether theory of the refractive index. It has been widely used for many materials,

7

8

1 Fundamental Optical Properties of Materials I

Table 1.1 Cauchy’s dispersion parameters of Eq. (1.15) for Ge, Si, and Diamond from [43]. n2 (eV−2 )

n4 (eV−4 )

2.378

0.00801

0.000104

3.4189

0.0815

0.0125

4.0030

0.220

0.140

Material

ℏ𝝎(eV) Min

ℏ𝝎(eV) Max

n−2 (eV2 )

n0

Diamond

0.0500

5.4700

−1.07 × 10−5

Si

0.0020

1.08

−2.04 × 10−8

0.75

−8

Ge

0.0020

−1.00 × 10

although, in recent years, many researchers have preferred to use the Sellmeier equation, described in the following text. 1.3.2

Sellmeier Equation

The Sellmeier equation [48] is an empirical relation between the refractive index n of a substance and wavelength 𝜆 of light in the form of a series of single dipole oscillator terms, each of which has the usual 𝜆2 /(𝜆2 − 𝜆i 2 ) dependence as in n2 = 1 +

A1 𝜆2 𝜆2 − 𝜆21

+

A2 𝜆2 𝜆2 − 𝜆22

+

A3 𝜆2

(1.16)

𝜆2 − 𝜆23

where A1 , A2 , A3 and 𝜆1 , 𝜆2 , and 𝜆3 are constants, called Sellmeier coefficients, which are determined by fitting this expression to the experimental data. The actual Sellmeier formula is more complicated. It has more terms of similar form, such as Ai 𝜆2 /(𝜆2 – 𝜆i 2 ), where i = 4, 5, ..., but these can generally be neglected in representing n vs. 𝜆 behavior over typical wavelengths of interest and by ensuring that the three terms included in Eq. (1.16) correspond to the most important or relevant terms in the summation [49]. The Sellmeier coefficients for some materials, including pure Silica (SiO2 ) and 86.5 mol% SiO2 –13.5 mol% GeO2 , are given in Table 1.2 as examples. A quantitative analysis of the application of the Sellmeier dispersion relation to a range of materials, from glasses to semiconductors, has been discussed by Tatian [49]. There are two methods for determining the refractive index of silica–germania glass (SiO2 )1-x (GeO2 )x . The first is a simple, but approximate, linear interpolation of the refractive index between known compositions, for example, n(x) − n(0.135) = (x − 0.235)[n(0.135) − n(0)]/0.135, where n(x) is for (SiO2 )1−x (GeO2 )x ; n(0.135) is for 86.5 mol% SiO2 –13.5 mol% GeO2 ; and n(0) is for SiO2 . The second is an interpolation for coefficients Ai and 𝜆i between SiO2 and GeO2 as [50]: n2 − 1 =

{A1 (S) + X[A1 (G) − A1 (S)]}𝜆2 𝜆2 − {𝜆1 (S) + X[𝜆1 (G) − 𝜆1 (S)]}𝜆21

+ …,

(1.17)

where S and G in parentheses refer to silica and germania, respectively. The theoretical basis of the Sellmeier equation lies in representing the solid as a sum of N lossless (frictionless) Lorentz oscillators such that each has the usual form of 𝜆2 /(𝜆2 – 𝜆i 2 ) with different 𝜆i and each has a different strength, or weighting factor; Ai , i = 1 to N [51, 52]. Such dispersion relationships are essential in designing photonic devices such as waveguides. (Note that although Ai weighs different Lorentz contributions, they do not sum to 1 since they include other parameters besides the oscillator strength f i .) The refractive indices of most optical glasses have been extensively modeled by the Sellmeier equation.

1.3 Refractive Index and Dispersion

Table 1.2 Sellmeier coefficients of a few materials, where 𝜆1 , 𝜆2 , 𝜆3 are in μm. Material

A1

A2

A3

𝝀1

𝝀2

𝝀3

SiO2 (fused silica)

0.696749

0.408218

0.890815

0.0690660

0.115662

9.900559

86.5% SiO2 –13.5% GeO2

0.711040

0.451885

0.704048

0.0642700

0.129408

9.425478

GeO2

0.80686642 0.71815848 0.85416831 0.068972606

0.15396605 11.841931

Barium fluoride

0.3356

0.506762

3.8261

0.057789

0.109681

46.38642

Sapphire

1.023798

1.058264

5.280792

0.0614482

0.110700

17.92656

Diamond

0.3306

4.3356

0.175

0.106

Quartz, no

1.35400

0.010

0.9994

0.092612

10.700 11.310

Quartz, ne

1.38100

0.0100

0.9992

0.093505

KTP, no

1.2540

0.0100

0.0992

0.09646

6.9777

KTP, ne

1.13000

0.0001

0.9999

0.09351

7.6710

9.8500 9.5280 5.9848 12.170

Source: From various sources.

Various optical glass manufacturers such as Schott Glass normally provide the Sellmeier coefficients for their glasses [53]. The optical dispersion relations for glasses have been discussed by a number of authors [7, 25, 54]. There are other Sellmeier–Cauchy-like dispersion relationships that inherently take account of various contributions to the optical properties, such as the electronic and ionic polarization and the interaction of photons with free electrons. For example, for many semiconductors and ionic crystals, two useful dispersion relations are, n2 = A +

B𝜆2 D𝜆2 + 2 , −C 𝜆 −E

(1.18)

𝜆2

and n2 = A +

𝜆2

B C + + D𝜆2 + E𝜆4 , 2 2 − 𝜆o (𝜆 − 𝜆2o )2

(1.19)

where A, B, C, D, E, and 𝜆o are constants particular to a given material. Eq. (1.18) is equivalent to the Sellmeier equation. Eq. (1.19) is known as the Herzberger dispersion relation [52]. Table 1.3 provides a few examples. Both Cauchy and Sellmeier equations are strictly applicable in wavelength regions where the material is transparent, that is, the extinction coefficient is relatively small. The refractive index dispersion relations Table 1.3 Parameters of Eq. (1.19) for some selected materials. D (𝛍m)−2

Material

𝝀o (𝛍m)

A

B (𝛍m)2

C (𝛍m)4

Silicon

0.028

3.41983

0.159906

−0.123109

MgO LiF AgCl

0.11951 0.16733 0.21413

2.95636 1.38761 4.00804

0.021958 0.001796 0.079009

0

1.269 × 10−6

−4.1 × 10

−1.951 × 10−9

−2

−2.05 × 10−5

−3

−5.57 × 10−6

−4

−1.976 × 10−7

−1.0624 × 10 −5

0

E (𝛍m)−4

−2.3045 × 10

−8.5111 × 10

Source: Si data from D.F. Edwards and E. Ochoa, Appl. Optics 19, 4130 (1980), others from W. L. Wolfe, The Handbook of Optics, W.G. Driscoll and W. Vaughan, McGraw-Hill, New York, 1978.

9

10

1 Fundamental Optical Properties of Materials I

for a wide range of semiconductors have been compiled by Madelung in [14]. There are many application-based articles in the literature that provide empirical dispersion relations for a variety of materials; a recent example on far infrared substrates (Ge, Si, ZnSe, ZnS, ZnTe) is given in reference [55]. There are both websites and various journal articles in the literature that give the refractive index of numerous materials as a function of wavelength.

1.3.3 1.3.3.1

Refractive Index of Semiconductors Refractive Index of Crystalline Semiconductors

A particular interest in the case of semiconductors is in n and K for photon energies greater than the bandgap Eg for optoelectronics applications. Due to various features and singularities in the E-k diagrams of crystalline semiconductors, the optical constants n and K for ℏ𝜔 > Eg are not readily expressible in simple terms. Various authors, for example, Forouhi and Bloomer [42, 56] and Chen et al. [57], have nonetheless provided useful and tractable expressions for modeling n and K in this regime. In particular, Forouhi–Bloomer (FB) equations express n and K in terms of the photon energy ℏ𝜔 in a consistent way that obey the Kramers–Kronig relations [42], that is K=

q ∑

Ai (ℏ𝜔 − Eg )2

i=1

(ℏ𝜔)2 − Bi (ℏ𝜔) + Ci

and n = n(∞) +

q ∑ i=1

Boi (ℏ𝜔) + Coi , (ℏ𝜔)2 − Bi (ℏ𝜔) + Ci

(1.20)

where (ℏ𝜔) is the photon energy; q is an integer that represents the number of terms needed to suitably model experimental n, K; Eg is the bandgap and Ai , Bi , C i , Boi , C oi are constants; Boi and C oi depend on Ai , Bi , C i , and Eg —only the latter four are independent parameters; and Boi = (Ai /Qi )[−(1/2)Bi 2 + Eg Bi – Eg 2 + C i ], C oi = (Ai /Qi )[(1/2)(Eg 2 + C i ) Bi − 2Eg C i ], and Qi = (1/2)(4C i − Bi 2 )1/2 . Forouhi and Bloomer provide a table of FB coefficients, Ai , Bi , C i , and Eg for four terms in the summation in Eq. (1.20) [42] for a number of semiconductors; an example that shows an excellent agreement between the FB dispersion relation and the experimental data is shown in Figure 1.3. Table 1.4 provides the FB coefficients for a few selected semiconductors. Other useful theoretical or somewhat semiempirical dispersion relationships have also been proposed, for example, by Afromowitz [58], Adachi [59–63], Campi and Papuzza [64], and others [65]. These models have been applied to various semiconductors and their alloys with relative success over certain photon energy ranges. One of the useful and straightforward approaches to modeling the dispersion has been based on writing the complex relative permittivity 𝜀r (ℏ𝜔) as a finite sum of a number of damped harmonic oscillators (the so-called harmonic oscillator approximation), and fitting this expression to the experimental data as in references [66, 67], even though many terms may be needed and the curve fit process has to be carefully chosen to ensure a reliable representation of the data. One of best models considered so far, however, has involved parametric modeling [68–70], in which not only a sum of harmonic oscillators are used but also Gaussian broadened polynomials to represent the dispersion of the complex relative permittivity, and hence n and K.

1.3 Refractive Index and Dispersion

5 SiC 4 n

3 n, K 2

1

K

0 0

2

4

6 8 10 Photon energy (eV)

12

14

Figure 1.3 n and K versus photon energy for crystalline SiC. The solid line is obtained from the FB equation with four terms with appropriate parameters, and the points represent the experimental data. See original reference [42] for the data and details. Source: Reprinted with permission, from Figure 2c, A.R. Forouhi and I. Bloomer, Phys. Rev. B, 38, 1865. Copyright (1988) by the American Physical Society.

1.3.3.2

Bandgap and Temperature Dependence

The refractive index of a semiconductor (typically for ℏ𝜔 < Eg ) typically decreases with increasing bandgap Eg . There are various empirical and semi-empirical rules and expressions that relate n to Eg . Based on an atomic model, Moss has suggested that n and Eg are related by n4 Eg = K = constant [71, 72] (K is about ∼100 eV). In the Hervé–Vandamme relationship [73], ( )2 A 2 , (1.21) n =1+ Eg + B where A and B are constants as A ≈ 13.6 eV and B ≈ 3.4 eV. The temperature dependence of n arises from the variation of Eg with the temperature T, and it typically increases with increasing temperature. The temperature coefficient of refractive index (TCRI) of semiconductors can be found from the Hervé–Vandamme relationship as: [ ] (n2 − 1)3∕2 dEg dB 1 dn =− + (1.22) TCRI = • n dT 13.6n2 dT dT where dB/dT ≈ 2.5 × 10−5 eV K−1 . TCRI is typically found to be positive (n increasing with temperature) and in the range of 10−6 to 10−4 K−1 . Although Eqs. (1.21) and (1.22) are popular, there are other very useful empirical and semiempirical relationships (see, e.g., [74]), some of which are summarized in Table 1.5 with appropriate references. 1.3.4

Refractive Index of Glasses

The Sellmeier equation with three terms have been found to represent the dispersion of n reasonably well for most glasses and ceramics. The coefficients in the Sellmeier equation have been listed at several websites [44] and handbooks.

11

12

1 Fundamental Optical Properties of Materials I

Table 1.4 FB coefficients for selected semiconductors [42] for four terms (i = 1 to 4). Bi (eV)

C i (eV2 )

n(∞)

E g (eV)

0.00405

6.885

11.864

1.950

1.06

0.01427

7.401

13.754

0.06830

8.634

18.812

0.17488

10.652

29.841

0.08556

4.589

5.382

2.046

0.60

0.21882

6.505

11.486

0.02563

8.712

19.126

0.07754

10.982

31.620

0.00652

7.469

13.958

2.070

2.17

0.14427

7.684

15.041

0.13969

10.237

26.567

0.00548

13.775

47.612

0.00041

5.871

8.619

2.156

1.35

0.20049

6.154

9.784

0.09688

9.679

23.803 44.119 1.914

0.65

1.766

1.27

1.691

0.30

1.803

0.12

Ai

Si

Ge

GaP

GaAs

GaSb

InP

InAs

InSb

0.01008

13.232

0.00268

4.127

4.267

0.34046

4.664

5.930

0.08611

8.162

17.031

0.02692

11.146

31.691

0.20242

6.311

10.357

0.02339

9.662

23.472

0.03073

10.726

29.360

0.04404

13.604

47.602

0.18463

5.277

7.504

0.00941

9.130

20.934

0.05242

9.865

25.172

0.03467

13.956

50.062

0.00296

3.741

3.510

0.22174

4.429

5.447

0.06076

7.881

15.887

0.04537

10.765

30.119

Note: First entry in the box is for i = 1, and the fourth is for i = 4.

Gladstone–Dale formula [82, 83] is an empirical equation that allows the average refractive index n of an oxide glass to be calculated from its density 𝜌 and its constituents as: ∑ n−1 pi ki = CGD , = p1 k1 + p2 k2 + · · · = 𝜌 i=1 N

(1.23)

1.3 Refractive Index and Dispersion

Table 1.5 Various selected simple relationships proposed between n and the bandgap E g . Relationship

Comment/Reference

n4 Eg = K; K = constant

Widely used, but has limitations. K ≈ 95 eV; 173 eV for Group IV elements [75]; K = 108 eV [76]. For a theoretical derivation and discussion of Moss’s rule, see [77].

)2

( 2

n =1+

A Eg + B

A ≈ 13.6 eV and B ≈ 3.4 eV. Hervé–Vandamme relationship. See text.

n = 4.084 + 𝛽Eg

𝛽 = −0.62 eV−1 . Proposed by Ravindra et al. [76], but has serious limitations for small and large n.

n = − ln(0.027Eg )

The Reddy Equation [78]. Based on the Duffy relationship Eg = 3.72Δ𝜒 op [79] and Δ𝜒 op = 9.8exp(−n) in [80], where Δ𝜒 op is optical electronegativity.

12.417 n2 = √ Eg − 0.365

See [81]. Equivalent to n4 (Eg − 0.365 eV) = 154. Similar to the Moss relation. Eg > 0.365 eV

Note: Eg is in eV.

where the summation is for various oxide components (each a simple oxide), pi is the weight fraction of the i-th oxide in the compound, and k i is the refraction coefficient that represents the polarizability of the i-th oxide. The right-hand side of Eq. (1.23) is called the Gladstone–Dale coefficient C GD . In more general terms, as a mixture rule for the overall refractive index, the Gladstone–Dale formula is frequently written as: n −1 n − 1 n1 − 1 w1 + 2 w2 + · · · , = 𝜌 𝜌1 𝜌2

(1.24)

where n and 𝜌 are the effective refractive index and effective density, respectively, of the whole mixture; n1 , n2 ,… are the refractive indices of the constituents; 𝜌1 , 𝜌2 ,… represent the density of each constituent; and w1 , w2 … are the weight fractions of the constituents. Gladstone–Dale equations for the polymorphs of SiO2 and TiO2 give the average n, respectively, as: n(SiO2 ) = 1 + 0.21𝜌 and n(TiO2 ) = 1 + 0.40𝜌

(1.25)

It is generally assumed that the refractive index can be related to the polarizability 𝛼 through the well-known Lorentz–Lorenz equation (equivalent to the Clausius–Mossotti equation in dielectrics), which involves the local field as n2 − 1 1 ∑ N𝛼 (1.26) = 2 n + 2 3𝜀o j j j in which 𝛼 j is the polarizability of a given type (species) (j) of atoms in the structure, and N j is the concentration of this species of atoms. Equation (1.26) includes both electronic and ionic polarizability and assumes that the local field is the Lorentz field, which depends on polarization through P/3𝜀o in the standard model for cubic crystals and noncrystalline solids. Ritland [84] assumed that the local field depends on the polarization as bP/𝜀o , where b is a numerical constant, and reformulated Eq. (1.26) in SI units

13

14

1 Fundamental Optical Properties of Materials I

as n2

n2 − 1 b ∑ N𝛼 = − 1 + (1∕b) 𝜀o j j j

(1.27)

and b is kept as a variable fitting parameter to the experimental data. Obviously, b = 1/3 is the usual Lorentz–Lorenz equation, but the best fits do not necessarily lead to 1/3 [84]. Most recent work on the refractive index of glasses has invariably used the Sellmeier equation, given its excellent fit to the dispersion data on glasses, as well as the adoption of the Sellmeier equation by some industrial glass manufacturers such as Schott [85]. Further, in some cases, the temperature dependence of the Sellmeier coefficients are also evaluated, so that dn/dT can be determined at different wavelengths [86–90]. While dn/dT is positive for many semiconductors, this is not generally true for glasses. 1.3.5

Wemple–DiDomenico Dispersion Relation

Based on the single oscillator model, the Wemple–DiDomenico (WD) model is a semi-empirical dispersion relation for determining the refractive index at photon energies below the interband absorption edge in a variety of materials [91, 92]. It is given by n2 = 1 +

Eo Ed Eo2 − (h𝜈)2

,

(1.28)

where 𝜈 is the frequency, h is the Planck constant, Eo is the single oscillator energy, and Ed is the dispersion energy, which is a measure of the average strength of interband optical transition. Ed can be written as Ed = 𝛽N c Za N e (eV), where N c is the effective coordination number of the cation nearest-neighbor to the anion (e.g. N c = 6 in NaCl, N c = 4 in Ge), Za is the formal chemical valency of the anion (Za = 1 in NaCl, 2 in Te, and 3 in GaP), N e is the effective number of valence electrons per anion excluding the cores (N e = 8 in NaCl, Ge; 10 in TlCl; 12 in Te; 91 /3 in As2 Se3 ), and 𝛽 is a constant that depends on whether the interatomic bond is ionic (𝛽 i ) or covalent (𝛽 c ): 𝛽 i = 0.26 ± 0.04 eV (e.g. halides NaCl, ThBr, etc., and most oxides, Al2 O3 , etc.), and 𝛽 c = 0.37 ± 0.05 eV (e.g. tetrahedrally bonded AN B8−N zinc blende and diamond type structures, GaP, ZnS, etc., and wurtzite crystals have a 𝛽 that is intermediate between 𝛽 i and 𝛽 c ). Further, empirically, Eo = CEg (D), where Eg (D) is the lowest direct bandgap and C is a constant, typically C ≈ 1.5. Eo has been associated with the main peak in the 𝜀′′r (h𝜈) versus h𝜈 spectrum. The parameters required for calculating n from Eq. (1.28) are listed in Table 1.6 [91]. It should be apparent that one can improve on the single oscillator model by adding a second oscillator term to Eq. (1.28) [93], and so on; although one loses the simplicity of the model, which the experimentalists like. Since its publication in 1971, the WD model has also been successfully applied to various amorphous semiconductors and glasses in addition to crystals, following the original arguments of Wemple [92]. In their original work, WD provided the single oscillator parameters for an extensive list of materials, some of which are summarized in Table 1.6, with further discussion in [94]. The ratio Eo /Eg has been observed by researchers to be very roughly in the range 1.5–2.5. The simple WD model is expected to hold in the small absorption region of the spectrum, that is, away from absorption bands due to interband transitions (h𝜈 < Eg ) or Reststrahlen absorption, etc. While it is apparent that the

1.3 Refractive Index and Dispersion

Table 1.6 Examples of parameters for Wemple–DiDomenico dispersion relationship in various materials. E o (eV) E d (eV) 𝜷 (eV) 𝜷

Material

Nc

Za

NaCl

6

1

8

10.3

13.6

0.28

𝛽i

Halides, LiF, NaF, etc.

CsCl

8

1

8

10.6

17.1

0.27

𝛽i

CsBr, CsI, etc.

TlCl

8

1

10

5.8

20.6

0.26

𝛽i

TlBr

CaF2

8

1

8

15.7

15.9

0.25

𝛽i

BaF2 , etc. Oxides, MgO, TeO2 , etc.

Ne

Comment

CaO

6

2

8

9.9

22.6

0.24

𝛽i

Al2 O3

6

2

8

13.4

27.5

0.29

𝛽i

LiNbO3

6

2

8

6.65

25.9

0.27

𝛽i

TiO2

6

2

8

5.24

25.7

0.27

𝛽i

ZnO

4

2

8

6.4

17.1

0.27

𝛽i

ZnSe

4

2

8

5.54

27

0.42

𝛽c

II-VI, Zinc blende, ZnS, ZnTe, CdTe

GaAs

4

3

8

3.55

33.5

0.35

𝛽c

III-V, Zinc blende, GaP, etc.

Si (Crystal)

4

4

8

4.0

44.4

0.35

𝛽c

Diamond, covalent bonding; C (diamond), Ge, 𝛽-SiC, etc.

SiO2 (Crystal)

4

2

8

13.33

18.10

0.28

𝛽i

Average crystalline form

SiO2 (Amorphous)

4

2

8

13.38

14.71

0.23

𝛽i

Fused silica

CdSe

4

2

8

4.0

20.6

0.32

𝛽 i –𝛽 c

Wurtzite

Note: Values extracted and combined from tables in [91].

WD relation can only be approximate, it has nonetheless found wide acceptance among experimentalists due to its straightforward simplicity. For example, in 2018 alone, it has been applied nearly 200 times to a wide variety of inorganic and organic material systems, particularly to semiconductor films. 1.3.6

Group Index

Group index is a factor by which the group velocity of a group of waves in a dielectric medium is reduced with respect to propagation in free space. It is denoted by N g and defined by N g = vg /c, where vg is the group velocity, defined by vg = d𝜔/dk, where k is the wave vector or the propagation constant. The group index can be determined from the ordinary refractive index n through [95] dn (1.29) d𝜆 where 𝜆 is the wavelength of light (in free space). Figure 1.4 illustrates the relation between N g and n in SiO2 . The group index N g is the quantity that is normally used in calculating dispersion in optical fibers, since it is N g that determines the group velocity of a propagating light pulse in a glass or transparent medium. It should be remarked that, although n vs. 𝜆 can decrease monotonically with 𝜆 over a range of wavelengths, N g can exhibit a minimum in the same range where the dispersion, dN g /d𝜆, becomes zero. The point dN g /d𝜆 = 0 is called the zero-material dispersion wavelength, which is around 1300 nm for silica, as apparent in Figure 1.4. Ng = n − 𝜆

15

16

1 Fundamental Optical Properties of Materials I

Figure 1.4 Refractive index n and the group index Ng of pure SiO2 (silica) glass as a function of wavelength. Source: Adapted from S.O. Kasap, 2017 [30].

1.49 1.48 Ng

1.47 1.46

n 1.45 1.44 500

700

900 1100 1300 1500 1700 1900 Wavelength (nm)

1.4 The Swanepoel Technique: Measurement of n and 𝜶 for Thin Films on Substrates 1.4.1

Uniform Thickness Films

In many instances, the optical constants are conveniently measured by examining the transmission through a thin film of the material deposited on a transparent glass or other (e.g. sapphire) substrate. The classic reference on the optical properties of thin films has been the book by Heavens [96]; the book is still useful in clearly describing what experiments can be carried out, and has a number of useful derivations such as the reflectance and transmittance through thin films in the presence of multiple reflections. Since then, numerous research articles and reviews have been published. Poelman and Smet [97] have critically reviewed how a single transmission spectrum measurement can be used to extract the optical constants of a thin film. In general, the amount of light that gets transmitted through a thin film material depends on the amount of reflection and absorption that takes place along the light path. If the material is a thin film with a moderate absorption coefficient 𝛼, then there will be multiple interferences at the transmitted side of the sample, as illustrated in Figure 1.5. In this case, some interference fringes will be evident in the transmission spectrum obtained from a spectrophotometer, as shown in Figure 1.6. One very useful method that makes use of these interference fringes to determine the optical properties of the material is called the Swanepoel method [98] which is based on earlier works of Manifacier et al. [99] and Hall and Ferguson [100]. Swanepoel has shown that the optical properties of a uniform thin film of thickness d, refractive index n, and absorption coefficient 𝛼, deposited on a thick substrate with a refractive index s, as shown in Figure 1.5, can be obtained from the transmittance T given by T=

Ax B − Cx cos 𝜑 + Dx2

(1.30)

where A = 16n2 s, B = (n + 1)3 (n + s2 ), C = 2(n2 − 1)(n2 − s2 ), D = (n − 1)3 (n − s2 ), 𝜑 = 4𝜋nd/𝜆, x = exp(−𝛼d) is an absorbance-type parameter, and n, s, and 𝛼 are all

1.4 The Swanepoel Technique: Measurement of n and α for Thin Films on Substrates

Incident monochromatic wave

Monochromatic light T=1

nair = 1 nfilm = n

d

Thin film

Thin film n* = n + iK

α>0

Glass substrate α=0

nsubstrate = s

Substrate T Detector

Figure 1.5 Schematic sketch of the typical behavior of light passing through a thin film on a substrate. On the left, oblique incidence is shown to demonstrate the multiple reflections. In most measurements, the incident beam is nearly normal to the film, as shown on the right. Full Transmission Spectrum from an a-Se Thin Film 1

Transmittance

0.7 0.6 0.5 0.4 0.3

TRANSPARENT REGION

0.8

STRONG ABSORPTION REGION

0.9

Interference Fringes

ABSORPTION REGION

0.2 0.1 0

Expt data Substrate 600

800

1000

1200 λ in nm

1400

1600

1800

2000

Figure 1.6 An example of a typical transmission spectrum of a 0.969-μm-thick amorphous Se thin film that has been vacuum coated on a glass substrate held at a substrate temperature of 50∘ C during the deposition.

17

1 Fundamental Optical Properties of Materials I

functions of wavelength 𝜆. Although x (with values of 1 under no absorption and approaching 0 under strong absorption) has been called “absorbance” in most studies using the Swanepoel technique as in the original papers, the usual definition of absorbance, however, is log10 (1/T), so that x, strictly, is not true absorbance. Eq. (1.30) assumes K 2 < < n2 , where n + iK is the complex refractive index of the film and normal incidence. What is very striking and useful is that all the important optical properties can be determined from the application of this equation; this will be introduced in the subsequent paragraphs. Before the optical properties of any thin film can be extracted, the refractive index of their substrate must first be calculated. For a glass substrate with very negligible absorption, that is, K ≤ 0.1 and 𝛼 ≤ 10−2 cm−1 , in the range of operating wavelengths, the refractive index s is, √( ) 1 1 s= + −1 (1.31) Ts Ts2 where T s is the transmittance value measured from the spectrophotometer. This expression can be derived from the transmittance equation for a bulk sample with little attenuation. With this refractive index s known, the next step is to construct two envelopes around the maxima and minima of the interference fringes in the transmission spectrum, as indicated in Figure 1.7. There will altogether be two envelopes that have to be constructed before any of the expressions derived from Eq. (1.30) can be used to extract the optical properties. This Full Transmission Spectrum 1 TS

TM

0.9

0.5 0.4 0.3

Tm Extreme Points

0.2 0.1 0

600

Minima

800

1000

TRANSPARENT REGION

0.6

T

ABSORPTION REGION

0.7

Maxima STRONG ABSORPTION REGION

0.8

Transmittance

18

Expt data Substrate Maxima Envelope Minima Envelope 1200 λ in nm

1400

1600

1800

2000

Figure 1.7 The construction of envelopes in the transmission spectrum of the thin a-Se film in Figure 1.6.

1.4 The Swanepoel Technique: Measurement of n and α for Thin Films on Substrates

can be done by locating all the extreme points of the interference fringes in the transmission spectrum and then making sure that the respective envelopes, T M (𝜆) for the maxima and T m (𝜆) for the minima, pass through these extremes, the maxima and minima, of T(𝜆) tangentially. From Eq. (1.30), it is not difficult to see that, at cos 𝜑 = ± 1, the expressions that describe the two envelopes are, Ax , B − Cx + Dx2 Ax Minima∶Tm = . B + Cx + Dx2

Maxima∶TM =

(1.32a) (1.32b)

Figure 1.7 shows two envelopes constructed for a transmission spectrum of an a-Se thin film. It can also be seen that the transmission spectrum is divided into three special regions according to their transmittance values: (i) the transparent region, where T(𝜆) ≥ 99.99% of the substrate’s transmittance value of T s (𝜆), (ii) the strong absorption region, where T(𝜆) is typical smaller than 20%, and (iii) the absorption region, in between the two latter regions as shown in Figure 1.7. The refractive index of the thin film can be calculated from the two envelopes, T M (𝜆) and T m (𝜆), and the refractive index of the substrate s through [ ] [ ] TM − Tm s2 + 1 2 2 1∕2 1∕2 n = N + (N − s ) ; N = 2s + , (1.33) TM Tm 2 where N is defined by the second equation on the right hand side of Eq. (1.33). T M and T m are assumed to be continuous functions of 𝜆 and x, so that values have to be at the same wavelength for use in Eq. (1.33). The derivation of Eq. (1.33) is based on considering 1/T m − 1/T M , which is 2C/A, and then substituting for C and A from earlier, and then solving for n. Since the equation is not valid in the strong absorption region, where there are no maxima and minima, the calculated refractive index has to be fitted to a well-established dispersion model for extrapolation to shorter wavelengths before it can be used to obtain other optical constants. Usually either the Sellmeier or the Cauchy dispersion equation is used to fit n vs 𝜆 experimental data in this range. Figure 1.8 shows the refractive indices extracted from the envelopes and fitted to the Sellmeier dispersion model with two terms. With the refractive index of the thin film corresponding to two adjacent maxima (or minima) at points 1 and 2, given as n1 at 𝜆1 and n2 at 𝜆2 , the thickness can be easily calculated from the basic interference equation of waves as follows: dcrude =

𝜆 1 𝜆2 2(𝜆1 n2 − 𝜆2 n1 )

(1.34)

where dcrude refers to the thickness obtained from the maxima (minima) at points 1, 2. As other adjacent pairs of maxima or minima points are used, more thickness values can be deduced, and hence an average value calculated. It is assumed the film has an ideal uniform thickness. The absorption coefficient 𝛼 can be obtained once x is extracted from the transmission spectrum. This can be done as follows: 𝛼=−

ln(x) dave

(1.35)

19

1 Fundamental Optical Properties of Materials I

Refractive index of the a-Se film in Figure 1.7 4 Fitted To Sellmeier Equation Crude Calculation From Envelopes Improved Calculation

3.8 3.6 3.4

2.8 2.6 2.4 2.2 2

TRANSPARENT REGION

3

STRONG ABSORPTION REGION

3.2 n

20

ABSORPTION REGION

800

600

1000

1200 λ in nm

1400

1600

1800

2000

Figure 1.8 Determination of the refractive index from the transmission spectrum maxima and minima shown in Figure 1.7. The solid black curve shows the fitted Sellmeier n vs. 𝜆 curve, which follows n2 = 3.096 + 2.943𝜆2 /[𝜆2 − (402.31)2 ], where 𝜆 is in nm.

EM −

√

2 EM −(n2 −1)3 (n2 −s4 )

2

s where x = ; EM = 8n + (n2 − 1)(n2 − s2 ); and dave is the average (n−1)3 (n−s2 ) TM thickness of dcrude . The accuracy of the thickness, the refractive index, and the absorption coefficient can all be further improved in the following manner. The first step is to determine a new set of interference orders, represented by m′ , for the interference fringes from the basic interference equation of waves; that is,

m′ =

2ne dave , 𝜆e

(1.36a)

where ne and 𝜆e are values taken at any extreme point, and m′ is an integer if the extremes taken are maxima or a half-integer if the extremes taken are minima. The second step is to get a new corresponding set of thickness values, d′ , from this new set of order numbers m′ , by rearranging Eq. (1.36a) as: d′ =

m′ 𝜆e 2ne

(1.36b)

From this new set of thickness values, d′ , a new average thickness, dnew , must be calculated before it can be applied to improve the refractive index. This can be done by

1.4 The Swanepoel Technique: Measurement of n and α for Thin Films on Substrates

ignoring those d′ that have values very different from the rest during averaging. With this new average thickness, a more accurate refractive index ne can be obtained from the same equation, ne =

m′ 𝜆 e 2dnew

(1.36c)

This new refractive index can then be fitted to the previous dispersion model again, so that an improved absorption coefficient 𝛼 can be calculated from Eq. (1.35). All these parameters can then be used in Eq. (1.30) to regenerate a calculated transmission spectrum T cal (𝜆), so that the root mean square error (RMSE) can be determined from the experimental spectrum T exp . The RMSE is calculated as follows: √ √ q √∑ √ (T − T )2 √ exp cal √ i=1 RMSE = , (1.37) q where T exp is the transmittance of the experimental or measured spectrum, T cal is the transmittance of the regenerated spectrum obtained through the Swanepoel calculation method, and q is the range of the measurement. Figure 1.9 shows the regenerated transmission spectrum of the a-Se thin film that appeared in Figure 1.6 using the optical constants calculated from the envelopes (as quoted in the caption of Figure 1.8). Full Spectrum of the a-Se Film in Figure 1.6 by the Swanepoel Method 1

Transmittance

0.7 0.6 0.5 0.4 0.3

TRANSPARENT REGION

0.8

STRONG ABSORPTION REGION

0.9

RMSE = 1.02%

ABSORPTION REGION 0.2

Expt data Regenerated Spectrum by Sellmeier Fitted n Substrate

0.1 0

600

800

1000

1200 1400 λ in nm

1600

Figure 1.9 Regenerated transmission spectrum of the sample in Figure 1.6.

1800

2000

21

22

1 Fundamental Optical Properties of Materials I

1.4.2

Thin Films with Non-uniform Thickness

The Swanepoel technique in the case of a film with a wedge-like cross-section, as shown in Figure 1.10, involves the integration of Eq. (1.30) over the thickness of the film in order for it to more accurately describe the transmission spectrum [101]. The transmittance then becomes, 𝜑

TΔd =

2 1 Ax dx 𝜑2 − 𝜑1 ∫𝜑1 B − Cx cos 𝜑 + Dx2

(1.38)

with 4𝜋n(d − Δd) 4𝜋n(d + Δd) , and 𝜑2 = , 𝜆 𝜆 where A = 16n2 s, B = (n + 1)3 (n + s2 ), C = 2(n2 − 1)(n2 − s2 ), D = (n − 1)3 (n − s2 ), 𝜑 has been defined after Eq. (1.30), x = exp(−𝛼d) corresponds x to the absorbance-type parameter calculated using the average film thickness d over the illumination region, n and s are the refractive index of the film and substrate, respectively, 𝛼 is the absorption coefficient, and Δd is the thickness variation throughout the illumination area, which has been called the roughness of the film (This nomenclature is actually confusing, since the film may not be truly “rough,” but may just have a continuously increasing thickness as in a wedge from one end to the other.) The first parameter to be extracted before the rest of the optical properties is Δd. Consider the transmission region where absorption is very small, that is, x is almost unity, and the spectrum exhibits clear maxima and minima. Eq. (1.38) can be modified by considering the maxima and minima, which are both continuous functions of 𝜆. In this way, we have, [ ( )] 2𝜋nΔd 𝜆 a 1+b −1 tan tan , (1.39a) Maxima∶TMd = √ √ 2𝜋nΔd 1 − b2 𝜆 2 1 − b [ ( )] 2𝜋nΔd 𝜆 a 1−b −1 , (1.39b) tan tan Minima∶Tmd = √ √ 2𝜋nΔd 1 − b2 𝜆 1 − b2 𝜑1 =

where a =

A , B+D

and b =

C . B+D

Monochromatic light T=1 Nonuniform film n* = n + iK

α>0

Glass substrate α=0 Refractive index = s T Detector

Δd d

Figure 1.10 System of an absorbing thin film with a variation in thickness on a thick finite transparent substrate.

1.4 The Swanepoel Technique: Measurement of n and α for Thin Films on Substrates

Notice that there is no x in these equations, since x = 1 was used. As long as 0 < Δd < 𝜆/4n, the refractive index, n and Δd, can both can be obtained by simultaneously solving Eqs. (1.39a) and (1.39b) numerically at various wavelengths. Since Eqs. (1.39a) and (1.39b) are only valid in the region of zero absorption, the refractive index, outside of the transparent region, must be obtained in another way. Theoretically, a direct integration of Eq. (1.38) over both Δd and x can be performed, although this would be analytically too difficult. Nevertheless, over the wavelength region where absorption is small (where one can still distinguish interference fringes), an approximation to the integration, according to Swanepoel, is as follows: [ ( )] ax 1 + bx 2𝜋nΔd 𝜆 −1 tan tan , (1.40a) Maxima∶TMx = √ √ 2 2𝜋nΔd 1 − b2 𝜆 1 − b x x [ ( )] 1 − bx ax 2𝜋nΔd 𝜆 −1 tan tan , (1.40b) Minima∶Tmx = √ √ 2𝜋nΔd 1 − b2 𝜆 1 − b2 x

Ax B+Dx2

x

Cx . B+Dx2

and bx = As long as 0 < x ≤ 1, numerically, there will only be where ax = one unique solution. Therefore, the two desired optical properties, refractive index, n and x (and hence 𝛼), can both be obtained when Eqs. (1.40a) and (1.40b) are solved simultaneously using the calculated average Δd. Eqs. (1.40a) and (1.40b) are valid for Δd ≪ d. As before, the calculated refractive index can be fitted to a well-established dispersion model, such as the Cauchy or Sellmeier equation, as shown in Figure 1.11, for Refractive Index of a Simulated Sample with Non-Uniform Thickness 4 Fitted To cauchy Equation Crude Calculation From Envelopes Improved Calculation

3.8 3.6

n

3.2 3 2.8 2.6 2.4 2.2 2

STRONG ABSORPTION REGION

3.4

ABSORPTION REGION n(λ) = 2.587 + (2.461 × 10–8 )λ–1 + (2.876 × 10–13)λ–2

600

800

1000

1200 1400 λ in nm

1600

1800

2000

Figure 1.11 The refractive index of a sample with d = 1 μm and Δd = 30 nm, and n fitted to a Cauchy equation in the figure. Eqs. (1.40a) and (1.40b) were used for the determination of n.

23

1 Fundamental Optical Properties of Materials I

extrapolation to shorter wavelengths. The thickness is calculated from any two adjacent maxima (or minima) using Eq. (1.34), and the absorption coefficient can be calculated from ln(xweak ) 𝛼weak = − (1.41) d where 𝛼 weak is the absorption coefficient in the weak and medium absorption region; xweak is the absorbance-like parameter (x) obtained from Eqs. (1.40a) and (1.40b); and d, as before, is the average thickness. According to Swanepoel, in the region of strong absorption, the interference fringes are smaller, and the spectrum approaches the interference-free transmission sooner. Since the transmission spectra in this region are the same for any film with the same average thickness, regardless of its uniformity, the absorption coefficient in the strong absorption region will thus be, 𝛼strong = −

ln(xstrong )

A−

√

(1.42)

d

(A2 −4Ti2 BD)

2T T

, Ti = T M+Tm , and T M and T m are the envelopes of maxima where xstrong = 2Ti D M m and minima, respectively, constructed from the measured spectrum. The accuracy of the thickness and refractive index can be further improved in exactly the same way as that for a film with uniform thickness for the computation of the new absorption coefficient, using Eqs. (1.40a) and (1.40b). Figure 1.12 shows the regenerated Transmission Spectrum of a Simulated Sample with Non-Uniform Thickness 1

0.8 0.7 0.6 0.5 0.4 0.3

STRONG ABSORPTION REGION

0.9

Transmittance

24

RMSE = 0.25% ABSORPTION REGION

0.2 Expt data Regenerated T Substrate

0.1 0

600

800

1000

1200 1400 λ in nm

1600

1800

2000

Figure 1.12 A regenerated transmission spectrum of a sample with an average thickness of 1 μm and average Δd of 30 nm, and a refractive index fitted to a Cauchy equation in Figure 1.11.

1.5 Transmittance and Reflectance of a Partially Transparent Plate

transmission spectrum of a simulated sample with nonuniform thickness using the optical constants calculated from the envelopes. Marquez et al. [102] have discussed the application of Swanepoel technique to wedge-shaped As2 S3 thin film and made use of the fact that a non-uniform wedge-shaped thin film has a compressed transmission spectrum. Figure 1.13 shows a flow chart that highlights the various steps involved in the extraction of the optical coefficients of thin film for both uniform and a nonuniform film thicknesses. Various computer algorithms based on the Swanepoel technique are available in the literature [103]. Further discussions and enhancements are also available [97, 104–106]. For example, one improvement in the strong absorption region is based on a so-called tangencypoint method as described in [106]. There are numerous useful applications of the Swanepoel technique for extracting the optical constants of thin films; some selected recent examples are given in [107–118].

1.5 Transmittance and Reflectance of a Partially Transparent Plate The transmittance of the thin film in Figure 1.6 is based on the interference of light waves within the thin film; it assumes that waves have a much longer coherence length than the thickness of the film, so that we may suitably add the optical fields of the waves. A film would be too thick if the coherence length of the waves is shorter than the thickness of the film. When we pass light through a transparent (or partially transparent) plate, we typically do not observe interference effects. Even if we have a thin film, we may not observe interference effects if the light source is incoherent. In these cases, we cannot add the electric fields to calculate the reflected and transmitted light irradiances; instead, we need to use reflectance and transmittance of the surfaces. Consider the multiple reflections shown in Figure 1.14. The first transmitted light intensity into the plate is (1 − R), and the first transmitted light out is (1 − R)(1 − R) = (1 − R)2 . However, there are internal reflections, so that the second transmitted light is R2 (1 − R)2 , so that the transmitted intensity through the plate is T plate = (1 − R)2 [1 + R2 + R4 + …], or Tplate =

2nno (1 − R)2 1−R = = 1 − R2 1 + R n2 + n2o

(1.43)

where R = (n – no )/(n + no ), and n is the refractive index of the plate and no that of the surrounding, as in Figure 1.14. One of the simplest ways to determine the refractive index of a medium as a plate is to measure the transmittance T plate , and then use √ n −1 −2 = Tplate + Tplate −1 (1.44) no The overall reflectance is 1 – T plate , so that Rplate =

(n − no )2 n2 + n2o

(1.45)

If the plate is partially transparent with some attenuation coefficient 𝛼, then, each time light traverses the thickness L of the plate, it experiences an attenuation factor exp(−𝛼L),

25

26

1 Fundamental Optical Properties of Materials I Uniform thickness film

Non uniform thickness film

Measure T(λ) for film on substrate

Measure T(λ) for substrate alone

Measure T(λ) for film on substrate

Measure T(λ) for substrate alone

Construct envelopes through extremes Find TM and Tm

Obtain s(λ)

Construct envelopes through extremes Find TM and Tm

Obtain s(λ)

Calculate N from TM, Tm, s and hence n at extremes at λextreme. Fit a dispersion relation, n = f(λ). Use Equation (1.33)

Obtain the refractive index n and Δd simultaneously from Equation (1.39a) and (1.39b) using only TM and Tm from the transparent region of the transmission.

Use adjacent pair of maxima and pair of minima to calculate a set of crude thickness values dcrude from Equation (1.34), then find the average dave

Obtain the refractive index n and x simultaneously from Equation (1.40a) and (1.40b) using only TM and Tm from the weak to medium absorption region (where one can still distinguish interference fringes) and the average Δdave calculated from previous step

Calculate x and hence α in the weak absorption region. From TM at maxima, n and s at λmaxima, find EM and then x. Use dave from above to find α = –ln(x)/dave.

Fit the refractive index, n obtained from the previous two steps (i .e, transparent to medium absorption region) to a dispersion relation, n = f(λ) to get the full spectrum range similar to the measured transmission spectrum

Find interference order m′ from extremes, m′ = 2nedave/λe where ne is n at λe = the extreme point λ. Make m′ integer for maxima and half integer for minima

Use adjacent pair of maxima and pair of minima in the zero to medium absorption region (where one can still distinguish interference fringes) to calculate a set of crude thickness values dcrude from Equation (1.34), then find the average dave

Using integer and half integer m′ find new thickness at each maximum and minimum, d′ = m′λe/2ne where ne is n at λe = λ at an extreme point. Find a new and better average thickness dave

Use x calculated in the weak to medium absorption region above to obtain α from Equation (1.41) in the same region, α in strong absorption region can be calculated from Equation (1.42)

Using this new dave and m′ calculate a new ne at each extreme λe. Obtain a better dispersion fit n = fnew(λ) to the new ne vs. λe data. Non-uniform thickness

Recalculate x using Equation (1.40a) and (1.40b) and use it to obtain a new absorption coefficient α in their respective absorption regions

Uniform thickness Recalculate x and hence α in the weak absorption region. From TM at maxima, new n and s at λmaxima, find EM and then x. Use new dave to find α = –ln(x)/dave

Use these parameters to regenerate the whole spectrum from Equation (1.30) to find the RMSE from Equation (1.37) with respect to the experimental transmission spectrum

All these parameters can then be used in Equation (1.38) to regenerate a transmission spectrum so that the root mean square error (RMSE) can be determined from the experimental spectrum using Equation (1.37)

Figure 1.13 A flow chart highlighting the steps involved in the Swanepoel technique.

1.6 Optical Properties and Diffuse Reflection: Schuster–Kubelka–Munk Theory

Incident beam

R4(1–R)

R3(1–R)

R(1–R)

(1–R) Thick transparent plate

R2(1–R)

no

(1–R)2

n

L

no

R2(1–R)2

Transmitted beam

R4(1–R)2

Detector

Figure 1.14 Transmitted and reflected light through a slab of material in which there is no interference.

resulting in a transmittance T plate = (1 − R)2 e−𝛼L + R2 (1 − R)2 e−3𝛼L + R4 (1 − R)2 e−5𝛼L + …, so that the series sums to (1 − R)2 e−𝛼L (1.46) Tplate = 1 − R2 e−2𝛼L

1.6 Optical Properties and Diffuse Reflection: Schuster–Kubelka–Munk Theory The treatise in Section 1.4 on the analysis of the transmission spectrum T(𝜆) assumes a smooth surface for the thin film and normal incidence. The propagating light wave through the film and its multiple reflections do not experience any scattering. The reflectance spectrum is simply 1 – T(𝜆). The situation is totally different if the film surface is rough or textured, and the bulk of the film exhibits scattering, which would be the case for particulate bulk material. The latter may be due to, for example, the polycrystallinity of the film, or the film may be a pressed powder layer or some composite with a mixture of two phases. Figure 1.15a shows a typical polycrystalline film sample with a rough surface whose optical properties are measured by diffused reflection. The nonspecular portion of light reflection is what constitutes the diffused reflection. There are many examples of diffused reflection from various forms of media, some of the most obvious being diffused reflection from rough surfaces, multiphase media, and multilayered translucent materials (e.g. paint coatings, paper, human skin, clothing, etc.). The instrumental setup is such that all light emitted over a wide solid angle is collected and the secularly reflected light is rejected—and hence the name. The experimental technique for implementing such measurements are well described in the literature (e.g. [119–127]), and there are various commercial instruments available from various vendors. The diffused reflectance spectrum is normally compared to a standard

27

28

1 Fundamental Optical Properties of Materials I

Incident light Diffused reflection cone

A particular incident ray

Specular reflection Diffused reflection

Rough surface Polycrystalline bulk

(a)

L

Inhomogeneous medium (b)

Figure 1.15 (a) Typical polycrystalline sample with a rough surface whose optical properties are measured by diffused reflection. (b) A simplified view of how diffused reflection builds up from scattering processes in the bulk.

by replacing the sample with a standard sample. Figure 1.15b shows a simplified view of how diffused reflection builds up from numerous refraction and reflection processes in the bulk so that the light is effectively scattered within the bulk. Although absorption is not shown, there would also be light extinction in the bulk. The treatise of diffused reflection is mathematically a complicated problem due to the inhomogeneity of the medium. The basic question is, “What are the suitable optical coefficients that can be used to describe the properties of the medium as related to diffused reflection?” This issue has been reviewed and discussed extensively in the literature. The Kubelka–Munk (KM) formalism [128, 129] is a theory that relates the diffuse reflectance R(𝜆) of a substance to multiple scattering and absorption within the substance. The treatise dates back to 1905 when Schuster quantitatively considered radiation traversing a foggy atmosphere [130], and hence the name Schuster–Kubelka–Munk (SKM) theory. The SKM formalism is widely used in various industrial [131–135] and biomedical applications [136, 137]. A recent practical review of diffuse reflectance with numerous references may be found in [126]. The KM theory addresses diffuse reflectance, and assigns two phenomenological coefficients labeled K and S to represent the absorption and scattering in the medium. K and S are used as coefficients that respectively indicate the amount of absorption and diffusion per unit thickness in the medium. K should not be confused with the extinction coefficient of the complex refractive index. It should be emphasized that K and S do not directly correspond to the intrinsic optical constants, that is, the absorption coefficient 𝛼, and the scattering coefficient s of the medium, although they are related. To describe the theory in simple terms, initially consider a medium such as a particulate layer (a layer of pressed powers), and assume that the sample is very thick (the sample thickness L is large, i.e. KL ≫ 1), so that there is no reflection from or transmission through the rear surface of the sample. A remission function (also known as the KM or SKM function) F(R) of diffuse reflectance R at a given wavelength is defined for the radiation absorption and scattering properties the medium of interest as F ≡ F(R) =

(1 − R)2 K = R S

(1.47)

1.6 Optical Properties and Diffuse Reflection: Schuster–Kubelka–Munk Theory

Figure 1.16 Medium with light absorption and scattering: K is the absorption and S is the scattering coefficient, that is, the fraction of light intensity absorbed or scattered, respectively, per unit distance. L is assumed to be large. I0 is the intensity of light entering the medium.

Diffused reflection RI0 measured z=0 z

I0 I

RI0 Forward light flux K, S

dz

J

L

Backward light flux

z=L

where the function F(R) represents a nearly steady-state intensity of diffuse reflectance R corresponding to an incident radiation at a wavelength 𝜆. Figure 1.16 presents a two-flux diffuse-reflectance model of a semi-infinite diffusely reflecting medium in which the incident light I 0 toward the +z direction is infinitesimally below the spectral refection surface medium [138]. The model considers the energy balance in a very thin layer (of thickness dz) in terms of diffuse intensities I and J, which are respectively along the forward (+z) and backward (−z) directions, as shown in Figure 1.16 [139], so that, over dz, dI = −(K + S)I + SJ (1.48) dz and dJ = (K + S)J − SI (1.49) dz Equations (1.48) and (1.49) can be applied as long as the particulate size is comparable to or smaller than 𝜆. K and S have been proposed to be proportional (but not equal) to the usual optical absorption coefficient 𝛼 and the scattering coefficient s, respectively, as discussed by Murphy [140, 141]; for example, K = 𝜀𝛼, where 𝜀 is defined in such a way that it is the average path length traveled by diffuse light in crossing dz. From Eqs. (1.48) and (1.49), it can be shown that (1 − R)2 K = (1.50) S 2R For an infinitely thick medium, R = J/I and Eq. (1.50) provides K/S. Equation (1.50) provides a useful way of using the diffusive reflectance spectrum to extract the absorption spectrum. The rearrangement of Eq. (1.50) gives an explicit expression for R as R = (1 + K∕S) − [(1 + K∕S)2 − 1]1∕2

(1.51)

The remission function F(R) and K for a weakly absorbing powdered medium of thickness L may be written in terms of fundamental optical parameters n and 𝛼 through [139], F=

2n2 𝛼L 3

(1.52)

K=

2n2 𝛼 3

(1.53)

and

29

1 Fundamental Optical Properties of Materials I

The particles are assumed to be spherical with rough surfaces, and the particle dimeter is larger than the wavelength. Notice that K is not the same as 𝛼. Although the relatively simple equations in the preceding text are attractive to experimentalists, a more rigorous treatise shows that the relationships between K and S and the material absorption and scattering coefficients 𝛼 and s involve the ratio 𝜇 of the total optical path length of a photon from its entry point (from one scattering process to the next) to its effective displacement from the entry point [142]. The problem immediately becomes analytically intractable without simplifying assumptions. In this model, K is proportional to 𝛼 through the ratio 𝜇. The Tauc law, which has been widely used for determining the bandgap of various semiconductors, has also been adapted in recent years (e.g. [143–148]) for the diffuse reflectance spectrum in terms of a transformed KM function h𝜈F(R), that is, [ ]p h𝜈F(R) = B(h𝜈 − Eg ) (1.54) where B is a constant (independent of photon energy) and p is an index, typically 1/3 to 2, that depends on the nature of the transition and the states that are involved. (The Tauc law is discussed in Chapter 3 in connection with amorphous semiconductors, but, in the form in Eq. (1.54), it represents a general description of absorption for photon energies just above the bandgap energy.) The rationale for Equation (1.54) is based F being proportional to 𝛼. Figure 1.17 shows a diffuse reflectance spectra within the UV and visible region (400 nm < 𝜆 < 860 nm) for a composite film of methylammonium iodide (CH3 NH3 I) and lead iodide (PbI2 ) in a 1 : 4 weight ratio. The Tauc plot is also shown and points to a bandgap Eg of 1.49 eV. Equation (1.54) has had success for estimating Eg for powder-form nanomaterials yielding experimentally supported values as reported in [150]. Alternative descriptions for characterizing the optical properties of coating materials have also been available for Eq. (1.51) in the literature. For example, the diffuse reflectance from a layer of finite thickness L on a substrate having a reflectance Rg is 80

×103 6 [h v F(R)]2

Diffuse reflectance (AU)

30

20kV X10,000

1µm

1

0000 13 41 SEI

0 0.4

0.6 0.8 Wavelength (µm)

4

1.0

0 1.15 1.20

1.30 1.40 1.50 Photon energy (hv) in eV

Figure 1.17 UV-visible diffuse reflectance spectra (left) and the Tauc-like plot (right) with p = 2 of a Perovskite film (550 nm thick) containing methylammonium iodide (CH3 NH3 I) and lead iodide (PbI2 ) in weight ratios of 1 : 4. The inset shows an SEM image of the microstructure, which consists of a random distribution of pores (voids) (AU is “arbitrary units”). The data were extracted, reanalyzed, and replotted from [149].

Acknowledgments

given as [140]: R=

1 − Rg [a − b coth(bSL)] a + b coth(bSL) − Rg

(1.55)

in which a = 1 + K/S and b = [a2 − 1]1/2 . The assumptions in Eq. (1.55) are that the light arrives from a medium that has the same refractive index as the layer, and that scattering is isotropic within the medium. In the case of a homogenous slab that has a thickness L, the diffuse reflectance R at a given wavelength can be written as R=

(1 − 𝛽 2 ) sinh(𝜅L) (1 + 𝛽 2 ) sinh(𝜅L) + 2𝛽 cosh(𝜅L)

(1.56)

where 𝛽 = [K/(K + 2S)]1/2 and 𝜅 = [K(K + 2S)]1/2 . Fitting this equation to the observed reflectance would yield 𝛽 and 𝜅, and hence K and S. However, Eq. (1.51), in many cases, has been found to be adequate for the interpretation of the spectra from high-emissive coatings with no significant effect of its covering power color building in visible and near-infrared spectral range [151]. The usefulness of the KM theory is due to the fact that the scattering and the absorption constants K and S tend to scale linearly with the concentration of colorants or pigments, and the overall contributions of colorants, inks, and pigments can be found by applying an additive rule, that is, adding the individual KL together for absorption and individual SL for scattering. The interpretation of diffuse reflectance spectra R(𝜆) and the extraction of meaningful optical coefficients continue to be an active research area. There have been several important reviews and useful criticisms of the simplified KM approach described above, which should be considered [141, 142, 152–154].

1.7 Conclusions This chapter has provided a semiquantitative explanation and discussion of the complex refractive index n* = n + iK; the relationship between the real n and imaginary part K through the Kramers–Kronig relationships; various common dispersion (n vs. 𝜆) relationships such as the Cauchy, Sellmeier, Wemple–DiDomenico dispersion relations; and the determination of the optical constants of a material in thin film form using the popular Swanepoel technique. The latter technique is based on the interference of waves in the thin film, and uses the maxima and minima in the transmittance spectrum to extract the dispersion relation for n and K. The transmittance of light through a thick semitransparent plate is also considered. The KM formulism is described in a simplified way to interpret diffuse reflection from inhomogeneous coating and/or rough surfaces. Examples are given to highlight the concepts and provide applications. The optical constants of various selected materials have been also provided in tables to illustrate typical values and enable comparisons.

Acknowledgments The authors are most grateful to Cyril Koughia for his helpful comments and assistance during the preparation of the first edition of this chapter.

31

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1 Fundamental Optical Properties of Materials I

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37

2 Fundamental Optical Properties of Materials II S.O. Kasap 1 , K. Koughia 1 , Jai Singh 2 , Harry E. Ruda 3 , and Asim K. Ray 4 1

Department of Electrical and Computer Engineering, University of Saskatchewan, 57 Campus Drive, Saskatoon, Canada College of Engineering, IT and Environment, B-Purple 12, Charles Darwin University, Ellengowan Drive, Darwin, Australia Department of Materials Science and Engineering, University of Toronto, 170 College Street, Toronto, Canada 4 Department of Electrical & Computer Engineering, Kingston Lane, Brunel University London, UK 2 3

CHAPTER MENU Introduction, 37 Lattice or Reststrahlen Absorption and Infrared Reflection, 40 Free Carrier Absorption (FCA), 42 Band-to-Band or Fundamental Absorption (Crystalline Solids), 45 Impurity Absorption and Rare-Earth Ions, 48 Effect of External Fields, 54 Effective Medium Approximations, 58 Conclusions, 61 References, 62

2.1 Introduction The optical properties of semiconductors typically consist of their refractive index n and extinction coefficient K or absorption coefficient 𝛼 (or, equivalently, the real and imaginary parts of the relative permittivity) and their dispersion relations, that is, their dependence on the wavelength 𝜆, of the electromagnetic radiation or photon energy h𝜈 and the changes in the dispersion relations with temperature, pressure, alloying, impurities, etc. This chapter is an extension of Chapter 1 on fundamental optical properties with emphasis on optical absorption. (Those general references quoted in Chapter 1 also apply to this chapter.) The elementary descriptions herein, as in Chapter 1, emphasize physical concepts and subsequent results rather than mathematical derivations. A typical relationship between the absorption coefficient and photon energy observed in a crystalline semiconductor is shown in Figure 2.1, where various possible absorption processes are illustrated. The important features in the 𝛼 versus h𝜈 behavior as the photon energy increases can be classified in the following types of absorptions: (a) reststrahlen or lattice absorption, in which the radiation is absorbed by vibrations of the crystal ions; (b) free carrier absorption (FCA) due to the presence of free electrons and holes, an effect that decreases with increasing photon energy; (c) an impurity absorption band Optical Properties of Materials and Their Applications, Second Edition. Edited by Jai Singh. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

2 Fundamental Optical Properties of Materials II

106 Absorption coefficient (1/cm)

38

Fundamental absorption Reststrahlen

105

Eg

104 Absorption edge

103 Impurity

102

Exciton (low T) Free carrier

10 1 0.001

Free carrier

0.01

0.1 1 Photon energy (eV)

10

100

Figure 2.1 Absorption coefficient is plotted as a function of the photon energy in a typical semiconductor to illustrate various possible absorption processes.

(usually narrow) due the various dopants; (d) exciton absorption peaks that are usually observed at low temperatures and are close to the fundamental absorption edge; and (e) band-to-band or fundamental absorption of photons, which excites an electron from the valence to the conduction band (CB). The type (e) absorption has a large absorption coefficient and occurs when the photon energy reaches the bandgap energy Eg . It is probably the most important absorption effect; its characteristics for h𝜈 > Eg can be predicted using the results of Section 2.4. The values of Eg , and its temperature shift, dEg /dT, are therefore important factors in semiconductor-based optoelectronic devices. In nearly all semiconductors, Eg decreases with temperature, hence shifting the fundamental absorption to longer wavelengths. The refractive index n also changes with temperature, as described in Chapter 1. There is a good correlation between the refractive index and the bandgap of semiconductors in which, typically, n decreases as Eg increases (see Chapter 1); wider bandgap semiconductors have lower refractive indices. The refractive index n and the extinction coefficient K (or 𝛼) are related by virtue of the Kramers–Kronig relations, described in Chapter 1. Thus, large increases in the absorption coefficient for h𝜈 near and above the bandgap energy Eg also result in increases in the refractive index n versus h𝜈 in this region. The optical and some structural properties of various semiconductors are listed in Table 2.1. The characteristics of some of these absorptions are described in the following sections. While these topics have been covered in extensive detail in various graduate-level textbooks in the past [1, 2], the approach here is to provide concise descriptions with more insight from an experimentalist’s point of view rather than a rigorous treatise. We have included simple electro-optic effects, since these have become particularly important in the last two decades with the advent of optical communications [3]. The last section in this chapter deals with the effective medium approximation to describe the optical properties of heterogeneous media, that is, mixtures of different phases. Such mixed phases occur frequently in various applications, for example a continuous medium that contains dispersed particulate matter, including air pores.

2.1 Introduction

Table 2.1 Crystal structure, lattice parameter a, bandgap energy E g at 300 K, type of bandgap (D = Direct and I = Indirect), change in E g per unit temperature change (dE g /dT) at 300 K, bandgap wavelength 𝜆g , and refractive index n close to 𝜆g .

Crystal a nm

Eg eV

dE g /dT Type meV K−1 𝝀g (𝛍m) n (𝝀g )

Diamond

D

0.3567

5.48

I

−0.05

0.226

2.74

1.1

Ge

D

0.5658

0.66

I

−0.37

1.87

4

27.6 42.4 (4 μm)

Si

D

0.5431

1.12

I

−0.25

1.11

3.45

13.8 16 (5 μm)

a-Si:H

A

Semiconductors

dn/dT ×10−5 K−1

Group IV

1.7–1.8

0.73

III–V Compounds AlAs

ZB

0.5661

2.16

I

−0.40

0.57

3.2

15

AlP

ZB

0.5451

2.45

I

−0.35

0.52

3

11

AlSb

ZB

0.6135

1.58

I

−0.3

0.75

3.7

GaAs

ZB

0.5653

1.42

D

−0.45

0.87

3.6

15

GaAs0.88 Sb0.12

ZB

1.15

D

GaN

W

0.3190 a 3.44 0.5190 c

D

6.8

1.08 −0.45

0.36

2.6

GaP

ZB

0.5451

2.24

I

−0.54

0.40

3.4

GaSb

ZB

0.6096

0.73

D

−0.35

1.7

4

In0.53 Ga0.47 As on InP ZB

0.5869

0.75

D

1.65

In0.58 Ga0.42 As0.9 P0.1 on InP

ZB

0.5870

0.80

D

1.55

In0.72 Ga0.28 As0.62 P0.38 ZB on InP

0.5870

0.95

D

1.3

32.8

InP

ZB

0.5869

1.35

D

−0.36

0.91

3.4–3.5 9.5

InAs

ZB

0.6058

0.35

D

−0.28

3.5

3.8

2.7

InSb

ZB

0.6479

0.18

D

−0.3

7

4.2

29

ZnSe

ZB

0.5668

2.7

D

−0.72

0.46

2.3

6.3 (4 μm) 6.30 (10 μm)

ZnTe

ZB

0.6101

2.25

D

0.55

2.7

II–VI Compounds

Note: A = Amorphous, D = Diamond, W = Wurtzite, ZB = Zinc blende. Typical values combined from different sources.

39

40

2 Fundamental Optical Properties of Materials II

2.2 Lattice or Reststrahlen Absorption and Infrared Reflection In the infrared wavelength region, ionic crystals reflect and absorb light strongly due to the resonance interaction of the electromagnetic (EM) wave field with the transverse optical (TO) phonons. The dipole oscillator model based on ions driven by an EM wave gives the complex relative permittivity as [4]: 𝜀ro − 𝜀r∞ (2.1) 𝜀r = 𝜀′r + i𝜀′′r = 𝜀r∞ + ( ), ( )2 1 − 𝜔𝜔 − i 𝜔𝛾 𝜔𝜔 T

T

T

where 𝜀ro and 𝜀r∞ are the relative permittivity at 𝜔 = 0 (very low frequencies or dc) and 𝜔 = ∞ (very high frequencies), respectively; 𝛾 is the loss coefficient representing the rate of energy transfer from the EM wave to optical phonons (per unit reduced mass of the crystal ions); and 𝜔T is a transverse optical phonon frequency related to the nature of bonding between the ions in the crystal, that is, 𝜔2T = 𝜔2o (𝜀r∞ + 2)∕𝜀ro + 2), in which 𝜔o 2 = 𝛽/Mr , 𝛽 is the force constant in “restoring force = −𝛽 × displacement,” and Mr is the reduced mass of the negative and positive ions in the crystal. It should be emphasized that the derivation of Eq. (2.1) based on a simple Lorentz oscillator model for ionic polarization, which is then used in the Clausius–Mossotti equation. The loss, 𝜀r ′′ and the absorption are maxima when 𝜔 = 𝜔T , and the wave is attenuated by the transfer of energy to the transverse optical phonons, thus the EM wave couples to the transverse optical phonons. At 𝜔 = 𝜔L , the wave couples to the longitudinal optical phonons. The refractive index n from ionic polarization vanishes, and the reflectance is minimum. Figure 2.2 shows the optical properties of AlSb in terms of n, K, and R vs wavelength [5]. The extinction coefficient K and reflectance R peaks occur over about the same wavelength region, corresponding to the coupling of the EM wave to the transverse optical phonons. At wavelengths close to 𝜆T = 2𝜋/𝜔T , n and K peak, and there is a strong absorption of light that corresponds to the EM wave resonating with the TO lattice vibrations, then R rises sharply. 𝜔T and 𝜔L are related through 𝜔2T = 𝜔2L (𝜀r∞ ∕𝜀ro ) which is called the Lyddane–Sachs–Teller relation. √ √ It relates the high- and lowfrequency refractive indices n∞ = 𝜀r∞ and no = 𝜀ro to 𝜔T and 𝜔L . The complex refractive index, n* = n + iK, then becomes [4] [ ] 2 2 𝜔 − 𝜔 L T n∗2 = 𝜀r = 𝜀r∞ 1 + 2 . (2.2) 𝜔T − 𝜔2 − i𝛾𝜔 (Note that n2 − K 2 = 𝜀′r and 2nK = 𝜀′′r .) Taking CdTe as an example, and substituting the values for 𝜀r∞ , 𝜔T , 𝜔L , and 𝛾 from Table 2.2 into the preceding expression, at 𝜆 = 70 μm or 𝜔 = 2.6909 × 1012 rad s−1 , one gets n = 3.20 and K = 0.00235. Although the preceding expression is usually sufficient to predict n* in the infrared for many compound semiconductors and ionic crystals, for low-bandgap semiconductors, one should also include the contribution from the free carriers.

λL

14

λT

14

12

12

10

10

λL

λT

90 80 70 60

8

8

6

6

4

K n

22

4

24

26 28 30 32 Wavelength (μm)

40 30

R (%)

20

K

2 0 20

n

50

2

34

36

0 38

10 0 16 18 20 22 24 26 28 30 32 34 36 38 40 Wavelength (μm)

Figure 2.2 Infrared refractive index n, extinction coefficient K (left), and reflectance R (right) of AlSb. Note: The wavelength axes are not identical, and wavelengths 𝜆T and 𝜆L corresponding to 𝜔T and 𝜔L , respectively, are shown as dashed vertical lines. Source: Data extracted from W. J. Turner and W. E. Reese, 1962 [5] with 𝜆L = 2𝜋(c/𝜔L ) and 𝜆T = 2𝜋(c/𝜔L ) from Table 2.2.

42

2 Fundamental Optical Properties of Materials II

Table 2.2 Values of the quantities required for calculating n and K from Eq. (2.2) for some selected crystals; values adapted and converted from the Handbook of Optical Constants of Solids, edited by E. D. Palik, 1985. Crystal

𝜺r∞

𝝎T x 1012

𝝎L x 1012

𝜸 x 1012

AlSb

9.6

60.09

63.97

CdTe

7.1

26.58

31.86

0.34 1.24

GaAs

11

50.65

55.06

0.45

InAs

11.7

41.09

45.24

0.75

InP

9.61

57.25

65.03

0.66

SiC

6.7

149.48

182.65

0.90

Note: 𝜔T , 𝜔L , and 𝛾 are in rad s−1 .

2.3 Free Carrier Absorption (FCA) An electromagnetic wave with sufficiently low frequency oscillations can interact with free carriers in a material and thereby drift the carriers. This interaction results in an energy loss from the EM wave to the lattice vibrations through the carrier scattering processes. Based on the Drude model, the complex relative permittivity 𝜀r (𝜔) due to N free electrons per unit volume is given by [6] 𝜀r = 𝜀′r + i𝜀′′r = 1 −

𝜔2p 𝜔2

+ i(𝜔∕𝜏)

; 𝜔2p =

Ne2 , 𝜀 o me

(2.3)

where 𝜔p is a plasma frequency, which depends on the electron concentration, and 𝜏 is the relaxation time of the carrier scattering process, that is, the mean scattering time. For metals where the electron concentration is very large, 𝜔p is of the order of ∼1016 rad s−1 , in the range of UV frequencies, and for 𝜔 > 𝜔p 𝜀r ≈ 1, the reflectance becomes very small. Metals lose their free-electron reflectance in the UV range, thus becoming UV transparent. The reflectance does not fall to zero because there are other absorption processes such as those due to interband electron excitations or excitations from core levels to energy bands. Plasma edge transparency, where the reflectance almost vanishes, can also be observed in doped semiconductors. For example, the reflectance of doped InSb has a plasma edge wavelength that decreases with increasing free carrier concentration [7]. Equation (2.3) can be written in terms of the conductivity 𝜎 o at low frequencies (DC) as: 𝜏𝜎o 𝜎o +i . (2.4) 𝜀r = 𝜀′r + i𝜀′′r = 1 − 2 𝜀o [(𝜔𝜏) + 1] 𝜀o 𝜔[(𝜔𝜏)2 + 1] ′′ In metals, 𝜎 o is high. At frequencies where √𝜔 1/𝜏, 𝛼 becomes proportional to N, the free carrier concentration, and 𝜆2 as: 𝛼 ∼ 𝜎o ∕𝜔2 ∼ N𝜆2 .

(2.8)

Experimental observations on FCA in doped semiconductors are generally in agreement with these predictions. For example, 𝛼 increases with N, whether N is increased by doping or by carrier injection [10, 11]. However, not all semiconductors show the simple 𝛼 ∝ 𝜆2 behavior. A proper account of the field-driven electron motion and scattering must consider the fact that 𝜏 will depend on the electron’s energy. The correct approach is to use the Boltzmann transport equation [12] with the appropriate scattering mechanism. FCA can be calculated by using a quantum mechanical approach based on second-order time-dependent perturbation theory with Fermi–Dirac statistics [13]. Absorption due to free carriers is commonly written as 𝛼 ∝ 𝜆p , where the index p depends primarily on the scattering mechanism, although it is also influenced by the compensation doping ratio, if the semiconductor has been doped by compensation, and the free carrier concentration. In the case of lattice scattering, one must consider scattering from acoustic and optical phonons. For acoustic phonon scattering, p ≈ 1.5; for optical phonon scattering, p ≈ 2.5; and for impurity scattering, p ≈ 3.5. Accordingly, the observed FCA coefficient will then have all three contributions as 𝛼 = Aacoustic 𝜆1.5 + Aoptical 𝜆2.5 + Aimpurity 𝜆3.5 .

(2.9)

Inasmuch as 𝛼 for FCA depends on the free carrier concentration N, it is possible to evaluate the latter from the experimentally measured 𝛼, given its wavelength dependence and p, as discussed by Ruda [14]. FCA coefficient 𝛼 (mm−1 ) for n-type GaP, n-type PbTe, and n-type ZnO are shown in Figure. 2.3. FCA in p-type Ge demonstrates how the FCA coefficient 𝛼 can be dramatically different from what is expected from Eq. (2.9). Figure 2.4a shows the wavelength dependence of the absorption coefficient for p-Ge over the wavelength range from about 2 to 30 μm [18]. The observed absorption is due to excitations of electrons from the spin-off band

43

44

2 Fundamental Optical Properties of Materials II

FCA coefficient, 1/mm 50

p-PbTe (p = 3.7 × 1018 cm–3) 77 K

ZnO (n = 5.04 × 1019 cm–3) ZnO (n = 2.33 × 1019 cm–3)

α ~ λ2

10 p-PbTe (p = 5 × 1017 cm–3) 77 K α ~ λ3 n-GaP (Nd = 3 × 1017 cm–3) 300 K

1

n-PbTe (n = 3.9 × 1017 cm–3) 77 K

ZnO (n = 1.02 × 1019 cm–3)

0.3 0.3

1

10

30

Wavelength (μm)

Figure 2.3 Free carrier absorption coefficient (1/mm) in n-GaP at 300 K [15], p and n type PbTe [16] at 77 K, and in doped n-type ZnO at room temperature [17] (Nd is the donor concentration.). 1 × 104 LH to HH

E k

SO to HH

α, 1/m

VB

SO to LH

HH LH Heavy-hole band Light-hole band

1 × 103 p-Ge at 300 K

SO Spin-off band

5 × 102 1

10

40

Wavelength (μm) (a)

(b)

Figure 2.4 (a) Free carrier absorption due to holes in p-Ge [18]. (b) The valence band of Ge has three bands; heavy hole, light hole, and spin-off bands.

to the heavy hole band, and from spin-off band to the light hole band, and from the light hole band to the heavy hole band, as marked in Figure 2.4b. FCA may also be combined with hot-carrier photodetection in silicon to enable sub-bandgap near infrared light to be detected at room temperature with very high responsivity and minimal noise-equivalent power [19].

2.4 Band-to-Band or Fundamental Absorption (Crystalline Solids)

2.4 Band-to-Band or Fundamental Absorption (Crystalline Solids) Band-to-band absorption or fundamental absorption of radiation occurs due to the photoexcitation of an electron from the valence band (VB) to the conduction band (CB). Thus, the absorption of a photon creates an electron in the CB and a hole in the VB, and requires the energy and momentum conservation of the excited electron, hole, and photon. In crystalline solids, as the band structures depend on the lattice wave vector k, there are two types of band-to-band absorptions corresponding to direct and indirect transitions. In contrast, in amorphous solids, where no long-range order exists, only direct transitions are meaningful. The band-to-band absorption in crystalline solids is described in the following text, and that in amorphous or disordered solids will be presented in Chapter 3. First the direct and then indirect transitions will be described here. A direct transition is a photoexcitation process in which no phonons are involved. As the photon momentum is negligible as compared to the electron momentum when the photon is absorbed to excite an electron from the VB to the CB, the electron’s k-vector does not change. A direct transition on a E-k diagram is a vertical transition from an initial energy E and wavevector k in the VB to a final energy E′ and wavevector k ′ in the CB where k ′ = k as shown in Figure 2.5a. The energy (E′ − Ec ) is the kinetic energy (ℏk)2 /(2me *) of the electron with an effective mass me *, and (Ev − E) is the kinetic energy (ℏk)2 /(2mh *) of the hole left behind in the VB. The ratio of the kinetic energies of the photogenerated electron and hole depends inversely on the ratio of their effective masses. The absorption coefficient 𝛼 is derived from the quantum mechanical transition probability from E to E′ , the occupied density of states at E in the VB from which electrons are excited, and the unoccupied density of states in the CB at E + h𝜈. Thus, 𝛼 depends on the joint density of states at E and E + h𝜈, and we have to suitably integrate this joint density of states. Near the band edges, the density of states can be approximated by a parabolic band, and the absorption coefficient 𝛼 is obtained as a function of the photon (αhν)2 (eV/cm)2

E

(αhν)2 (eV/cm)2

50

CB

× 108

× 109 4

E′ 25

Ec

GaAs

CdTe

2

Photon

Eg

3

1

Ev

E

VB k

k

0 hν 1.35 1.40 1.45 1.50 1.55 Photon energy (eV)

hν 1.4 1.8 2.2 Photon energy (eV)

0

Figure 2.5 (a) A direct transition from the valence band (VB) to the conduction band (CB) by the absorption of a photon. Absorption behavior represented as (𝛼h𝜈)2 vs photon energy h𝜈 near the band edge for single crystals of (b) p-type GaAs. Source: From I. Kudman and T. Seidel, J. Appl. Phys., 33, 771 (1962) (c) CdTe. Source: From A. E. Rakhshani, J. Appl. Phys., 81, 7988 (1997).

45

46

2 Fundamental Optical Properties of Materials II

energy as: 𝛼 h𝜈 = A(h𝜈 − Eg )1∕2 ,

(2.10)

where the constant A ≈ [(e2 /(nch2 me *)](2𝜇*)3/2 , in which 𝜇* is a reduced electron and hole effective mass, n is the refractive index, and Eg is the direct bandgap, minimum Ec – Ev at the same k value. Experiments indeed show this type of behavior for photon energies above Eg and close to Eg as shown Figure 2.5b for a GaAs crystal [20] and in Figure 2.5c for a CdTe crystal [21]. The extrapolation to zero photon energy gives the direct bandgap Eg , which is about 1.40 eV for GaAs and 1.46–1.49 eV for CdTe. For photon energies very close to the bandgap energy, the absorption is usually due to exciton absorption, especially at low temperatures, and it will be discussed later in this chapter. In indirect bandgap semiconductors such as crystalline Si and Ge, the photon absorption for photon energies near Eg requires the absorption or emission of phonons during the absorption process, as illustrated in Figure 2.6a. The absorption onset corresponds to a photon energy of (Eg − h𝜗), which represents the absorption of a phonon with energy h𝜗 (𝜗 is the phonon frequency). In this case, 𝛼 is proportional to [h𝜈 − (Eg − h𝜗)]2 . Once the photon energy reaches (Eg + h𝜗), then the photon absorption process can also occur by phonon emission, for which the absorption coefficient is larger than that for phonon absorption. The absorption coefficients for the phonon absorption and emission processes are given by [22]: 𝛼absorption = A[fBE (h𝜗)][h𝜈 − (Eg − h𝜗)]2 ; h𝜈 > (Eg − h𝜗),

(2.11)

[ ] 𝛼emission = A (1 − fBE (h𝜗) [h𝜈 − (Eg + h𝜗)]2 ; h𝜐 > (Eg + h𝜗),

(2.12)

and

α1/2

E 8 CB

(cm–1/2)

2 Phonon

Photon

kCB

Ec Eg

333 K

6

Phonon emission

4 Ev

hυ12

195 K

Phonon absorption

2

VB 1 k1

kVB

k

hν

0 0

1.0

1.1

1.2

1.3

Photon energy (eV)

Eg – hϑ

Eg + hϑ (a)

(b)

Figure 2.6 (a) Indirect transitions across the bandgap involve phonons. Direct transitions in which dE/dk in the CB is parallel to dE/dk in the VB lead to peaks in the absorption coefficient. (b) Fundamental absorption in Si at two temperatures. The overall behavior is well described by Eqs. (2.11) and (2.12).

2.4 Band-to-Band or Fundamental Absorption (Crystalline Solids)

where A is a constant, and f BE (h𝜗) is the Bose–Einstein distribution function at the phonon energy h𝜗, that is, f BE (h𝜗) = [(exp(h𝜗/k B T) − 1]−1 , where k B is the Boltzmann constant and T is the temperature. As we increase the photon energy in the range (Eg − h𝜗) < h𝜈 < (Eg + h𝜗), the absorption is controlled by 𝛼 absorption and the plot of 𝛼 1/2 vs h𝜈 has an intercept of (Eg − h𝜗). For photon energies h𝜈 > (Eg + h𝜗), the overall absorption coefficient is 𝛼 absorption + 𝛼 emission , but, at slightly higher photon energies than (Eg + h𝜗), 𝛼 emission quickly dominates over 𝛼 absorption , since [f BE (h𝜗)] > > [(1 − f BE (h𝜗)]. Figure 2.6b shows the behavior of 𝛼 1/2 versus photon energy for Si at two temperatures for h𝜐 near band edge absorption. At low temperatures, f BE (h𝜗) is small, and 𝛼 absorption decreases with decreasing temperature, as apparent in Figure 2.6b. Equations (2.11) and (2.12) intersect the photon energy axis at (Eg − h𝜗) and (Eg + h𝜗), which can be used to determine Eg . Examination of the extinction coefficient K or 𝜀r ′′ versus photon energy for Si in Figure 1.2 (Chapter 1) shows that absorption peaks at certain photon energies, h𝜈 ≈ 3.5 and 4.3 eV. These peaks are due to the fact that the joint density of states function peaks at these energies. The absorption coefficient peaks whenever there is a direct transition in which the E versus k curve in the VB is parallel to the E versus k curve in the CB, as schematically illustrated in Figure 2.6a, where a photon of energy h𝜈 12 excites an electron from state 1 in the VB to state 2 in the CB in a direct transition k1 = k2 . Such transitions where E versus k curves are parallel at a photon energy h𝜈 12 result in a peak in the absorption versus photon energy behavior and can be represented by the condition that (∇k E)CB − (∇k E)VB = 0.

(2.13)

This condition is normally interpreted as the joint density of states reaching a peak value at certain points in the Brillouin zone called van Hove singularities. Identification of peaks in K versus h𝜈 behavior involves the examination of all E versus k curves of a given crystal that can participate in a direct transition. In silicon, the 𝜀r ′′ peaks at h𝜈 ≈ 3.5 eV and 4.3 eV correspond to Eq. (2.13) being satisfied at points L, along in k-space, and X along in k-space, at the edges of the Brillouin zone. In degenerate semiconductors, the Fermi level EF lies in a band, for example, it lies in the CB for a degenerate n-type semiconductor. In these semiconductors, electrons in the VB can only be excited to states above EF in the CB rather than to the bottom of the CB. The absorption coefficient then depends on the free carrier concentration since the latter determines EF . Fundamental absorption is then said to depend on band filling, and there is an apparent shift in the absorption edge, called the Burstein–Moss shift. Furthermore, in degenerate indirect semiconductors, the indirect transition may involve a non-phonon scattering process, such as impurity or electron-electron scattering, which can change the electron’s wavevector k. Thus, in degenerate indirect bandgap semiconductors, absorption can occur without phonon assistance, and the absorption coefficient becomes: 𝛼 ∼ [h𝜈 − (Eg + ΔEF )]2 ,

(2.14)

where ΔEF is the energy depth of EF into the band measured from the band edge. Heavy doping of degenerate semiconductors normally leads to a phenomenon called bandgap narrowing and bandtailing. Bandtailing means that the band edges at Ev and Ec are no longer well-defined cut-off energies, and there are electronic states above Ev

47

48

2 Fundamental Optical Properties of Materials II

and below Ec whose density of states falls sharply with energy away from the band edges. Consider a degenerate direct band gap p-type semiconductor. One can excite electrons from states below EF in the VB, where the band is nearly parabolic, to tail states below Ec , where the density of states decreases exponentially with energy into the bandgap, away from Ec . Such excitations lead to 𝛼 depending exponentially on h𝜈, a dependence that is usually called the Urbach rule [23, 24], given by: 𝛼 = 𝛼0 exp[(h𝜈 − Eo )∕ΔE],

(2.15)

where 𝛼 0 and Eo are material-dependent constants, and ΔE, called the Urbach width, is also a material dependent constant. The Urbach rule was originally reported for alkali halides. It has been observed for many ionic crystals, degenerately doped crystalline semiconductors, and almost all amorphous semiconductors. While exponential bandtailing can explain the observed Urbach tail of the absorption coefficient versus photon energy, it is also possible to attribute the absorption tail behavior to strong internal fields arising, for example, from ionized dopants or defects.

2.5 Impurity Absorption and Rare-Earth Ions Impurity absorption can be registered as peaks of absorption coefficient lying below the fundamental (band-to-band) and excitonic absorption (Figure 2.1). They can mostly be related with the presence of ionized impurities or, simply, ions. The origin of these peaks lies in the electronic transitions either between the electronic states of an ion and CB/VB or intra-ionic transitions (e.g. within d or f shells, or between s and d shells, etc.). In the first case, the appearing features are intense and broad lines, while in the latter case their appearance strongly depends on whether these transitions are allowed or not by the parity selection rules. For allowed transitions, the appearing absorption peaks are quite intense and broad, while the forbidden transitions produce weak and narrow peaks. The general reviews of the topic may be found in studies by Blasse and Grabmaier [25], Henderson and Imbusch [26], and DiBartolo [27]. Optical spectra of rare-earth ions in various crystals and glasses have been well treated in a number of books, some of which are [26, 28–30]. In the following sections, we concentrate primarily on the absorption properties of rare earth (RE) ions, which are of prime importance for modern optoelectronics. Consider the absorption of radiation due to dopants or impurities in a material system, for example, Er3+ ions in a glass host such as a germanium–gallium–sulfide glass. Figure 2.7a–c show typical examples of the transmission spectra, absorption bands, and the absorption cross-section, respectively, for the 4 I15/2 –4 I13/2 band of Er3+ in (GeS2 )0.9 (Ga2 S3 )0.1 glass. Let N (m−3 ) be the number of dopants per unit volume, and 𝛼(𝜆) be the absorption coefficient for dopant excitations from a manifold centered at energy E1 to a manifold centered at E2 . The absorbed radiation power per unit area 𝛿I(𝜆) over a small distance 𝛿x is 𝛿I(𝜆) = 𝛿xI𝛼(𝜆). The optical absorption due to electronic transitions between manifolds is very often described by using an absorption cross-section 𝜎 a (𝜆) defined as 𝛿I(𝜆) = 𝛿xIN𝜎 a (𝜆), that is 𝜎a (𝜆) =

1 𝛼(𝜆) N

(2.16)

2.5 Impurity Absorption and Rare-Earth Ions

(c) 60

10

40 20

α, cm–1

8

(a)

0 36 34 6

12

6

2H 11/2

4F

(b)

4F 9/2

7/2

4

4I 13/2

Cross-section, ×10–21 cm2

Transmittance, %

80

4 4I

2

4S 3/2

4I 9/2

600

600

2

11/2

0 0

1000 1500 1600 1450 Wavelength, nm

1500

1550

1600

0

Figure 2.7 (a) Optical transmittance of (GeS2 )0.9 (Ga2 S3 )0.1 glass doped with 0.5 at.% of Er. (b) Optical absorption coefficient of Er3+ ions extracted from data in figure (a). Peaks correspond to optical transition from ground manifold 4 I15/2 to excited manifolds as marked in the figure. The meaning of symbols is explained in Figure 2.8. (c) Absorption cross-section for the 4 I15/2 –4 I13/2 transition.

which basically follows the wavelength dependence of the spectral absorption coefficient, as shown in Figure 2.7b,c. RE is the common name for the elements from lanthanum (La) to lutetium (Lu). They have atomic numbers from 57 to 71 and form a separate group in the periodic table. The specific feature of these elements is the incompletely filled 4f shell. The electronic configurations of REs are listed in Table 2.3. The RE may be embedded in different host materials in the form of divalent or trivalent ions. As divalent ions, REs generally exhibit broad absorption–emission lines related to the allowed 4f → 5d transitions. Table 2.3 Occupation of outer electronic shells for rare earth elements. 57 58

La

4s2

4p2

4d10

Ce

2

2

10

4s

2

4p

2

4d

10

– 1

4f

3

5s2

5p6

5d1

6s2

2

6

1

5d

6s2

6

5s

2

5p

59

Pr

4s

4p

4d

4f

5s

5p

–

6s2

60

Nd

4s2

4p2

4d10

4f4

5s2

5p6

–

6s2

Er

4s2

4p2

4d10

4f12

5s2

5p6

–

6s2

Yb

4s2

4p2

4d10

4f14

5s2

5p6

–

Lu

2

2

10

14

2

6

… 68 … 70 71

4s

4p

4d

4f

5s

5p

6s2 1

5d

6s2

49

2 Fundamental Optical Properties of Materials II

In the trivalent form, REs lose two 6s electrons and one of the 4f or 5d electrons. As a result of the Coulomb interaction of 4f electrons with the positively charged core and other electrons in this shell, and spin-orbit coupling, the 4f level becomes split into a complicated set of manifolds whose position, in first approximation, is relatively independent of the host matrix, because the 4f level is well screened by 5s and 5p shells from outer influence [31]. Figure 2.8 shows the energy level diagram of the low-lying 4fN states of trivalent ions embedded in LaCl3 . (This is the so-called Dieke diagram [32].) In the second approximation, the exact construction and precise energy position

40

2F

1S 0

6D 9/2

7/2 5/2

38

3P 2

4G 9/2

36

6I 7/2

2H 11/2

34

6P

32

5D 4

3M

3P 1 3P 0 1I 6

5/2 7/2

8

2K 13/2

10

2P 3/2

3L

3/2 5/2 7/2

2D

0.3

30

22

18

2G

4F 4S 3/2

9/2

2

6F 1/2 3/2 5/2 7/2 6H

9/2

9/2

2 3

4

3H 6 3H 2F 5 7/2

8

15/2

6 3/2 H15/2 1/2

7

13/2

6

11/2

13/2 11/2

11/2

5/2

9/2 5

2F 3 4 5/2 H4 I9/2 Ce Pr Nd

5I

4 Pm

7F

7F

2F 5/2 3H

6

13/2 11/2

5 4

4

13/2

7

0.8

4

11/2

9/2

0 1 2 3

6

3H

4I

5

7/2

7/2

5I

4I

3F

9/2

11/2

7/2

0

5I

4F 9/2

5/2

4 3

0.5

5S 2 5F 5

6F

1G 4

1G 4

2H 11/2

5

3/2

4F 5F 3/2 1 5/2 3K8 6 4F 2 7/2

4 4S3/2

3 2 1

5/2

0.4

3

5S 2

2

4

4F 9/2

5D 4

4

2H 7/2 9/2

3F

15/2

0

5F

1D 2

5

1

5/2

2H 11/2

12

6

3/2 7/2

2G 7/2

1D 2

14

8

2

7/2

5G 2H 4 9/2 4G 11/2 4I

4G 7/2 4F 3/2 4G 5/2

2G

2K15/6 5G′ 4 9/2 5 G11/2 3K 7

5D 3

5/2

3/2 1 0 4G9/2

16

10

5D 3

3P 2P 1/2 1I 2 4G11/2 6

20

3H6

5I–10

2D

24

2

2

2P 3/2

26

5G 3

5D

5/2 3/2

Wavelength (μm)

4D 1/2

28

Energy (103 cm–1)

50

3F

5

1.0 1.3 1.6

4

2.0 2.5 5.0

3 2 1

6H 7 5/2 F0 Sm Eu

8S 7F6 6H15/2 5I8 4I15/2 3H6 2F7/2 Dy Ho Er Tm Yb Gd Tb

Figure 2.8 Energy level diagram of the low-lying 4fN states of trivalent ions doped in LaCl3 . Source: From G. H. Dieke, Spectra and Energy Levels of Rare Earth Ions in Crystals, Wiley, New York, 1968.

2.5 Impurity Absorption and Rare-Earth Ions

of manifolds depend on the host material via the crystal field and via the covalent interaction with ligands surrounding RE ion. The ligand is an atom, molecule, radical, or ion with one or more unshared pairs of electrons that can attach to a central metallic ion (or atom) to form a coordination complex. Examples of ligands include ions (F− , Cl− , Br− , I− , S2− , CN− , NCS− , OH− , NH2 − ) or molecules (NH3 , H2 O, NO, CO). Some ligands that share electrons with metal ions or atoms form very stable complexes. The optical transitions between 4f manifold levels are forbidden by the parity selection rule which states that, for a permitted atomic (ionic) transition, the wave functions of initial and final states must have different parity, that is, parity must change in the transition. The parity is a property of a quantum mechanical state that describes the function after a mirror reflection. Even functions (states) are symmetric—identical after reflection (like a cosine function)—while functions odd (states) are antisymmetric (like a sine function). For an ion (or atom), which is embedded in host material, the parity selection rule may be partially removed due to the action of crystal field giving rise to “forbidden lines.” The crystal field is the electric field created by a host material at the position of an ion. The resulting absorption–emission lines are characteristics of individual RE ions and quite narrow because they are related to forbidden inner-shell 4f transitions. The intensities of absorption–emission lines are related to the so-called oscillator strengths. Judd–Ofelt (JO) analysis is based on calculating the oscillator strengths of an electric dipole (ED) transition between states of a trivalent RE ion embedded in different host lattices and fitting them to experimentally observed oscillator strengths by using so-called intensity or force parameters Ω2 , Ω4 , and Ω6 that depend on the host material [33, 34]. Experimental oscillator strength for a particular absorption band is given by fexp =

me c 2 𝛼(𝜈)d𝜈 𝜋e2 N ∫Band

(2.17)

where N is the concentration of RE ions in the host, 𝜈 is the frequency, and 𝛼(𝜈) is the experimentally observed absorption spectrum as given in Figure 2.7b. The possible states of RE ions are often referred to terms and written as 2S + 1 LJ , where L = 0, 1, 2, 3, 4, 5, 6... determines the electron’s total orbital angular momentum and is conventionally represented by letters S, P, D, F, G, I. The term (2S + 1) is called spin multiplicity and represents the number of spin configurations, while J is the total angular momentum, which is the vector sum of the overall (total) angular momentum and overall spin (J = L + S). This description in the literature is based on the assumption that Russell–Saunders coupling is approximately valid, even though RE ions are not expected to rigidly obey this rule (their atomic numbers are more than 40). The value (2 J + 1) is called the term’s multiplicity and is the number of possible combinations of overall orbital angular momentum and overall spin, which yield the same J. Thus, the notation 4 I15/2 for the ground state of Er3+ corresponds to term (J, L, S) = (15/2, 6, 3/2), which has a multiplicity 2 J + 1 = 16 and a spin multiplicity 2S + 1 = 4. The energy separation between different terms (different L- and S-values) is due to the Coulombic repulsion between the electrons in this 4f subshell. The latter energy separation depends on L and S, but not J. The energy split for a given L and S—for example, energy separation between 4 I15/2 , 4 I13/2 , and 4 I11/2 , etc., are due to the spin-orbit coupling, which depends on J and MJ . The energy levels that are finely separated within a given 2S+1 LJ are due to the Stark effect.

51

52

2 Fundamental Optical Properties of Materials II ′

If (2S +1 LJ′ ) represents the initial and (2S+1 LJ ) the final state of an RE ion, then the electric dipole transition has the oscillator strength that is in the form of fcal =

∑ 8𝜋me c 𝜒ed Ωk |⟨SLJ|U (k) |S′ L′ J ′ ⟩|2 , 2 3h𝜆(2J + 1)n k=2,4,6

(2.18)

where Ωk are the previously mentioned intensity parameters representing the influence of host material; 𝜒 ed = n(n2 + 2)2 /9 is the so-called local field correction factor for electric dipole moment transitions for a medium with a refractive index n; and the reduced ′ ′ ′ matrix operator components |⟨SLJ|U (k) |S L J ⟩|2 are calculated using the so-called intermediate coupling approximation, and are tabulated because they are practically independent of the host material. Some values of reduced matrix operator components for Er3+ ions are presented in Table 2.4, for which data were taken from [35, 36]. The key idea of JO analysis is to minimize the discrepancy between experimental and calculated values of oscillator strengths, f exp and f cal , respectively, by the appropriate choice of coefficients Ωk . As a possible way, oscillator strengths f exp and f cal are the measured and calculated strengths, respectively, for all possible optical absorption bands ∑ (shown in Figure 2.7b) and the integral error Bands |(f exp 2 – f cal 2 )/f exp 2 | is minimized by choosing Ω2 , Ω4 , and Ω6 . The values Ωk are then used to calculate the probabilities of radiative transitions and appropriate radiative lifetimes of excited states, which are very useful for numerous optical applications. For example, the radiative lifetime 𝜏 r of a transition may be calculated as 𝜏 r = 1 /A, where A is the probability of spontaneous emission per unit time, which can be expressed as A(J ′ ; J) =

( ) 64𝜋 4 e2 × [𝜒ed Sed (J; J ′ ) + 𝜒md Smd J; J ′ ] ′ 3 3h(2J + 1)𝜆

(2.19)

Table 2.4 The reduced matrix elements of U(k) (k = 2, 4, and 6) and typical wavelengths 𝜆 for the Er3+ transitions. (S,L)J

(S′ ,L′ )J′

(U[2] )2

(U[4] )2

(U[6] )2

4

4

I13/2

0.0188

0.11176

1.4617

4

F9/2

0

0.5655

0.4651

1

652

4

S3/2

0

0

0.2285

2

521

2

H11/2

0.7056

0.4109

0.0870

4

F7/2

0

0.1467

0.6273

3

487

4

F5/2

0

0

0.2237

4

450

4

F3/2

0

0

0.1204

2

H9/2

0

0.078

0.17

5

407

2

G11/2

0.9178

0.5271

0.1197

2

G9/1

0

0.2416

0.1235

6

2

K15/2

0.0219

0.0041

0.0758

2

G7/2

0

0.0174

0.1163

378 1520 652

I15/2

Band

𝝀 (nm)

1520

Source: Data extracted from M.J. Weber, Phys. Rev. 157, 1967, 262, 1967 and W.T. Carnall, P.R. Field and B.G. Wybourne, J. Chem. Phys., 49, 4424, 1968.

2.5 Impurity Absorption and Rare-Earth Ions

where Sed (J′ ; J) and Smd (J′ ; J) are the electric and magnetic line strengths of this transition, respectively. The factors 𝜒 ed = n(n2 + 2)2 /9 and 𝜒 md = n3 are local field correction factors for the electric and magnetic dipole moment transitions, respectively. Sed and Smd are given by ∑ Sed (J; J ′ ) = Ωk |⟨SLJ|U (k) |S′ L′ J ′ ⟩|2 (2.20) k=2,4,6

and Smd (J; J ′ ) =

(

ℏ 2mc

)2 ∑

Ωk |⟨SLJ||L + 2S||S′ L′ J ′ ⟩|2 .

(2.21)

k=2,4,6

The magnetic dipole contributions can be easily calculated according to [37] as {[ [ ]]}1∕2 2 2 2 2 (J + 1) − (L − S) ⟨SLJ||L + 2S||S L J ⟩ = (S + L + 1) − (J + 1) 4(J + 1) (2.22) ′ ′ ′

Further, since |⟨SLJ|U (k) |S L J ⟩|2 are given as in Table 2.4, Eq. (2.20), for example, for the 4 I13/2 to 4 I15/2 transition at around 1.55 μm becomes, ′

′

′

Sed = 0.0188Ω2 + 0.1176Ω4 + 1.4617Ω6

(2.23)

The intensity parameters Ω2 , Ω4 , and Ω6 are used to characterize and compare different materials. Ω2 is considered to be of a prime importance because it is the most sensitive to the local structure around the rare-earth ion and material composition, and it has been correlated with the degree of covalency [38]. Some values of Ωk (k = 2, 4, and 6) for Er3+ ions in different hosts are given in Table 2.5 as typical examples. Notice that Ω2 is lower for fluoride glasses that are more ionic. More Ωk values for different ions and host materials may be found, for example, in references [38, 46]. Reviews of Table 2.5 Selected examples of intensity parameters Ω2 (k = 2, 4 and 6) for Er3+ in various hosts from various sources. 𝛀2 ×10−20 cm2

𝛀4 ×10−20 cm2

𝛀6 ×10−20 cm2

RE ion

Host

Er3+

Fluorochlorozirconate

1.92 ± 0.3

0.88 ± 0.16

0.59 ± 0.08

[39]

Er3+

Fluoride glasses

2.91

1.27

1.11

[40]

Er3+

ZBLAN

2.91

1.78

1.0

[41]

Er

Silicate glasses

4.23

1.04

0.61

[40]

Er3+

Phosphate glasses

6.65

1.52

1.11

[40]

Er3+

Aluminate glass (66.7CaO)(33.3Al2 O3 )

7.33

1.78

0.47

[42]

Er3+

Silica glass (SiO2 > 96%)

8.15

1.43

1.22

[43]

Er3+

Ge-Ga-S chalcogenide glass Ge0.30 Ga0.04 S0.655 :Er0.005

11 ± 2

2.9 ± 0.2

1.6 ± 0.1

[44]

Er3+

Ge-Ga-Se chalcogenide glass Ge0.23 Ga0.08 Se0.67 S0.014 Er0.006

13 ± 2

3.4 ± 0.2

1.3 ± 0.1

[45]

3+

Ref.

53

54

2 Fundamental Optical Properties of Materials II

the Judd–Ofelt theory and its applications have been given by Walsh [47] (one of the most extensive reviews), Quimby and Miniscalo [48], Tanabe and coworkers [49–51], Goldner and Auzel [52] (includes the modified Judd–Ofelt), Hehlen et al. (comprehensive review) [53], Adam [54], as well as others—for example, [55, 56].

2.6 Effect of External Fields 2.6.1

Electro-Optic Effects

The application of an external field (E or B) can change the optical properties in a number of ways that depends not only on the material but also on the crystal structure. Electro-optic effects refer to changes in the refractive index of a material induced by the application of an external electric field, which therefore “modulates” the optical properties; the applied field is not the electric field of any light wave, but a separate external field. The application of such a field distorts electronic motions in atoms or molecules of a substance and/or the crystal structure, resulting in changes in the optical properties. For example, an applied external field can cause an optically isotropic crystal such as GaAs to become birefringent. Typically, changes in the refractive index are small due to the applied electric field. The frequency of the applied field has to be such that it appears to be static over the time a medium takes to change its properties, as well as the time the light requires to pass through the substance. The electro-optic effects are classified according to first- and second-order effects. If we consider the refractive index as a function of the applied electric field E, that is, n = n(E), we can, of course, expand it in a Taylor series. Denoting the electric field-dependent refractive index by n′ , one can write: n′ = n + a1 E + a2 E2 + … ,

(2.24)

where n represents the electric field-independent refractive index, and the coefficients a1 and a2 are called the linear and second-order electro-optic effect coefficients, respectively. Although one may consider including even higher terms in the expansion of Eq. (2.24), these are generally very small and have negligible effects within highest practical fields. The change in n due to the linear E-dependent term is called the Pockels effect, and that due to the E2 term is called the Kerr effect. The coefficient a2 is generally written as 𝜆K, where K is called the Kerr coefficient. Thus, the contributions to n′ from the two effects can be written as: (2.25a)

Δn = a1 E 2

Δn = a2 E = (𝜆K)E

2

(2.25b)

All materials exhibit the Kerr effect but not Pockels effect, because only a few crystals have non-zero a1 . For all noncrystalline materials (such as glasses and liquids), and those crystals that have a center of symmetry (centrosymmetric crystals such as NaCl), a1 = 0. Only crystals that are noncentrosymmetric exhibit the Pockels effect. For example, an NaCl crystal (centrosymmetric) exhibits no Pockels effect, but a GaAs crystal (noncentrosymmetric) does. The Pockels effect involves examining the effect of the field on the indicatrix, that is, the index ellipsoid, and requires the electro-optic tensor. For example, the change in the

2.6 Effect of External Fields

principal refractive index n1 along the principal axis x (where D and E are parallel) of the indicatrix is written as: 1 1 1 Δn1 = − n31 r11 E1 − n31 r12 E2 − n31 r13 E3 (2.26) 2 2 2 where Ej is the field along j; j = 1–3 corresponding to x, y, and z; and rij are the elements of the electro-optic tensor r, a 6 x 3 matrix. The Pockels coefficients for various crystals can be found in references [57–59]. The Pockels effect has found important applications in optical communications in Pockels cell modulators that typically use lithium niobate (LiNbO3 ). 2.6.2

Electro-Absorption and Franz–Keldysh Effect

Electro-absorption is a change of the absorption spectrum caused by an applied electric field. There are fundamentally three types of electro-absorption processes: the Franz–Keldysh effect (FKE), field-injected FCA, and the confined Stark effect. In FKE [60, 61], a strong constant applied electric field induces changes in the band structure, which change the photon-assisted probability of an electron tunneling from the maximum of the VB to the minimum of CB. FKE can be detected as a variation of absorption and/or reflection of light with photon energies slightly less than the corresponding bandgap. This effect was first observed in CdS, where the absorption edge was red-shifted with the applied electric field, causing an increase of absorption that could be detected even visually [62]. Later, FKE was observed and investigated in Ge [63], Si [64], and other semiconducting materials. In the dynamic Franz–Keldysh effect (DFKE), ultrafast band structure changes are induced in a semiconductor in the presence of a strong laser electric field. These changes include absorption below the band edge and oscillatory behavior in the absorption above. The steady-state effect is normally quite small, but the dynamic effect can be up to 40%, as found in thin films of GaAs [65]. In the presence of an electric field, the absorption coefficient 𝛼 can be written as [1]: [ √ 3∕2 ] 4 2m∗ (Eg − ℏ𝜔) , (2.27) 𝛼(𝜔) ∝ 𝛼0 (𝜔) exp − 3 ℏ2 eF where 𝛼 0 (𝜔) is the absorption coefficient in the absence of electric field, m* is the effective mass, Eg is the optical gap, e is the electron charge, and F is the applied electric field. It should be mentioned that the electric field–induced change in the absorption coefficient implies a change in the refractive index as discussed by Seraphin and Bottka [66]. In the FCA, the concentration of free carriers N in a given band is changed (modulated) by an applied voltage, to change the extent of photon absorption. In this case, the absorption coefficient is proportional to N and to the light wavelength 𝜆 raised to some power, typically 2–3. In the confined Stark effect, the applied electric field modifies the energy levels in a quantum well (QW). A QW is a thin crystalline semiconductor between two crystalline semiconductors as barriers, which can confine electrons or holes in the dimension perpendicular to the layer surface, while the movement in the other dimensions is not restricted. It is often realized with a thin crystalline layer of a semiconductor medium, embedded between other crystalline semiconductor layers of wider bandgap, for example, GaAs embedded in AlGaAs. The thickness of such a QW is typically about

55

56

2 Fundamental Optical Properties of Materials II

p

MQW i

Relative transmission n

100 %

0 E

0

Vr

Figure 2.9 A schematic illustration of an electroabsorption modulator using the quantum confined Stark effect in MQWs. The i-region has MQWs. The transmitted light intensity can be modulated by the applied reverse bias to the pin device, because the electric field modifies the exciton energy in the QWs.

Vr

5–20 nm. Such thin crystalline layers can be fabricated by the molecular beam epitaxy (MBE) or metal–organic chemical vapor deposition. The energy levels are reduced by an amount proportional to the square of the applied field. Without any applied bias, light with photon energy just less than the QW exciton excitation energy will not be significantly absorbed. When a field is applied, the energy levels are lowered and the incident photon energy is now sufficient to excite an electron and hole pair in the QWs. Therefore, the relative transmission decreases with the reverse bias, as shown in Figure 2.9 for a reverse biased pin device in which the i-region has multiple quantum wells (MQWs). The reverse bias increases the electric field in the i-region and which enhances photogeneration in this region. In practice, the effect of electroabsorption is used to construct an electroabsorption modulator (EAM), which is a semiconductor device to control the intensity of a laser beam via applied electric voltage. Most EAMs are made in the form of a waveguide with electrodes for applying an electric field in a direction perpendicular to the modulated light beam. For achieving a high extinction ratio, one usually exploits the quantum confined Stark effect in a QW structure. They can be operated at very high speeds; a modulation bandwidth of tens of gigahertz can be achieved, which makes these devices useful for optical fiber communications. With the recent interest in two-dimensional materials has come the realization that these monolayer systems can offer strong quantum confinement effects and, in particular, with regards to the quantum confined Stark effect in them. This is discussed by Sie et al. [67]. 2.6.3

Faraday Effect

The Faraday effect, originally observed by Michael Faraday in 1845, is the rotation of the plane of polarization of a light wave as it propagates through a medium subjected to a magnetic field parallel to the direction of propagation of light. When an optically inactive material such as glass is placed in a strong magnetic field and then a plane polarized light is sent along the direction of the magnetic field, it is found that the emerging light’s plane of polarization has been rotated, as shown in Figure 2.10. The magnetic field can be applied, for example, by inserting the material into the core of a magnetic coil. The induced specific rotatory power, given by 𝜃/L, where L is the length of the medium and 𝜃 is the angle by which the rotation occurs, is found to be proportional to the magnitude of the applied magnetic field, B, which gives the amount of rotation as, 𝜃 = VBL,

(2.28)

where V is the proportionality constant, the so-called Verdet constant, which depends on the material and wavelength of light.

2.6 Effect of External Fields

I θ

Input light E

Output light E′ θ

B Beam cross-section L

I Output light E″

Input light E′

2θ

θ

B

Figure 2.10 The Faraday effect. The sense of rotation of the optical field E depends only on the direction of the magnetic field for a given medium (given Verdet constant). If light is reflected back into the Faraday medium, the field rotates a further 𝜃 in the same sense to come out as E′′ with a 2𝜃 rotation with respect to E.

The Faraday effect is typically small. For example, a magnetic field of ∼0.1 T causes a rotation of roughly 1∘ through a glass rod of length 20 mm. It seems to appear that an “optical activity” has been induced by the application of a strong magnetic field to an otherwise optically inactive medium. There is, however, an important difference between the natural optical activity and Faraday effect. The sense of rotation 𝜃 in the Faraday effect, for a given material (Verdet constant), depends only on the direction of the magnetic field B. If V is positive, for light propagating parallel to B, the optical field E rotates in the same sense as an advancing right-handed screw pointing in the direction of B. The direction of light propagation does not change the absolute sense of rotation of 𝜃. If we reflect the wave to pass through the medium again, the rotation increases to 2𝜃. The Verdet constant depends not only on the wavelength 𝜆 but also on the charge-to-mass ratio of the electron and the refractive index, n(𝜆), of the medium through: (e∕me ) dn 𝜆 . (2.29) 2c d𝜆 The Verdet constant values for some materials are listed in Table 2.6. Notice that dn/d𝜆, and hence V is large over wavelengths near a polarization resonance peak but small far away from a resonance peak. The Faraday effect has found useful applications in photonics, for example, in optical isolators [68]. V =−

Table 2.6 Verdet constants of some selected materials from various sources, including, M. J. Weber, The Handbook of Optical Materials, CRC Press, 2003.

Material

Quartz 589 nm

Tb3 Ga5 O12 633 nm

ZnS 589 nm

ZnTe 633 nm

NaCl 589 nm

Crown glasses 633 nm

Dense flint glass (SF57) 633 nm

V (rad m−1 T−1 )

4.0

−134

65.8

188

10

4–6

20

57

58

2 Fundamental Optical Properties of Materials II

2.7 Effective Medium Approximations Effective medium approximations (EMAs) are commonly employed to describe the optical constants, such as the effective refractive index neff and effective extinction coefficient K eff of a composite medium containing a mixture of two or more phases. The mixture is an inhomogeneous medium. Quite often, the mixture has dispersed particles (second phase) in a host matrix material (the first phase), and one is interested in describing the composite medium in terms of neff and K eff . The dispersed phase may be of definite shape and size (such as dispersed spheres), or both the shape and size may follow some distribution. There are numerous examples of composite media such as porous dielectrics, dielectrics containing a mixture of two or more different ceramics, cermets, nanoparticles dispersed in a solution, mesocrystals, highly polycrystalline media (e.g. thin films), composite epoxies used in electronic packaging, polymer blends used in various engineering applications, polymer oxide composites in optics, etc. Polymer blends can be engineered just right to have the so-called “smart optical properties.” The optical properties of nanomaterials and nanoporous materials invariably involve a description that is based on treating the medium as a mixture of dielectrics, each of which has well-defined n and K, so the objective is to find neff and K eff for the whole medium, that is, “homogenize the heterogeneous medium.” What is the correct mixture rule? The physics of macroscopically inhomogeneous media has attracted much interest, and there are several treatises and reviews on the subject that include dielectric and optical properties (e.g. [69–81]). There are several mathematical descriptions of the optical properties of composite media, including the well-known Maxwell Garnett and Bruggeman models. These models have several implicit assumptions, which have been highlighted in a collection of works in reference [82] and others in [73, 74, 83]. Typically, for example, the size of an inclusion is assumed to be smaller than the light wavelength, so that we only need to consider the interaction of the electric dipole with the electromagnetic wave. There are at least three important EMAs: Lorentz–Lorenz, Maxwell Garnett, and Bruggeman formulations. The Lorentz–Lorenz refractive index mixture rule is based on summing the polarizabilities of the constituents of a mixture so that one can write the effective refractive index neff of a mixture to be determined from its constituents as [84, 85] n2eff − 1 n2eff + 2

= f1

n21 − 1 n21 + 2

+ f2

n22 − 1 n22 + 2

+···

(2.30)

where n1 and f 1 are the refractive index and the volume fraction of component 1, respectively, and n2 and f 2 are the refractive index and the volume fraction of component 2, respectively, and so on, and f 1 + f 2 + … = 1. The mixture rule in Eq. (2.30) is equivalent to using the Clausius–Mossotti equation in the field of dielectrics. The volume fraction can be easily obtained from the mole fraction. This model has been widely and successfully used by many researchers. For example, in a recent paper, Eq. (2.30) has been applied to the measured optical properties of a medium consisting of Pb(II) ions in an aqueous solution and then extracting the Pb(II) concentration [86]. The latter work involved developing a sensor, which consisted of a silicon-on-insulator (SOI) microring resonator coated with tetrasulfide-functionalized mesoporous silica (MPS) (S4-MPS) to detect Pb(II) ions in aqueous solutions from 10 ppb to 1 ppm.

2.7 Effective Medium Approximations

The Bruggeman effective medium approximation [81, 87] for a heterogeneous or composite material system having two components with refractive indices n1 and n2 is given by f1

n21 − n2eff n21 + 2n2eff

+ f2

n22 − n2eff n22 + 2n2eff

=1

(2.31)

where f 1 and f 2 are the volume fractions of phases 1 and 2, and f 1 + f 2 = 1. The composite medium need not have dispersed particles in a continuous matrix and, in general, can be considered to be an aggregate system in which any phase can touch any other phase, that is, there is no condition imposed on whether a phase should be dispersed with a spherical geometry, etc. Porous semiconductors exhibit physically interesting optical properties such as a bandgap shift, increased luminescence intensity, and improved photoresponsivity. Because of their potential applications in optoelectronics and chemical sensors, porous silicon layers have been widely investigated as a composite mixture of three components, namely silicon, silicon dioxide, and voids. Equation (2.31) with three components has been applied to calculate neff for nanoporous silicon over the wavelength range 2–5 μm where the extinction coefficient is small [88]. The results are shown in Figure 2.11 as neff vs porosity from experiments and the Bruggeman theory in Eq. (2.31) with three terms for Si, SiO2 , and air. It can be seen that the agreement is excellent over the entire 30–70% porosity range investigated. While the agreement is indeed very good in this particular case, the fits are not always this close as highlighted by the original authors for the case of mesoporous silicon and in [89]. Although the preceding description specifically considered the real part of the complex refractive index, it is possible to use the mixture rules for the complex refractive index and hence extract both the real and imaginary (extinction coefficient) parts, that is, neff and K eff . For example, in a recent study, gallium nitride films on sapphire substrates were thermally oxidized to nanoporous gallium oxide (Ga2 O3 )

Effective refractive index

2.4

Experiment

2.2 2.0 1.8 1.6 1.4

Bruggeman

1.2 30

40

50 60 Porosity (%)

70

80

Figure 2.11 Plot of the experimental effective refractive index vs porosity and as predicted from the Bruggeman theory for nanoporous silicon, which has been taken as a three-component medium with Si, SiO2 , and pores (air). Source: After M. Khardani, M. Bouaïcha, B. Bessaïs, phys. stat. sol. (c) 4, 1986–1990 (2007).

59

60

2 Fundamental Optical Properties of Materials II

under wet conditions. The ellipsometric spectra of Ga2 O3 films were then analyzed in terms of the Bruggeman effective medium approximation. Figure 2.12 shows the variation of effective refractive index neff and effective extinction coefficient K eff with the free space wavelength 𝜆. The value of neff for nanoporous Ga2 O3 film is 1.92 at a wavelength of ∼400 nm, and the effective extinction coefficient is negligibly small in the wavelength range of 300 nm < 𝜆 < 840 nm, supporting the use of this material as a transparent coating in optoelectronic devices [90]. Equation (2.31) can easily be extended to multicomponent composite material systems. The Maxwell Garnett equation [81, 91] applies to a composite medium where one can identify a host as a continuous medium in which there are dispersed spherical dielectric particles. Suppose that the dispersed spherical dielectric particles have an index nd and volume fraction f d in a continuous host matrix with an index nm ; then, this mixture rule is given by n2eff − n2m n2eff + 2n2m

= fd

n2d − n2m

(2.32)

n2d + 2n2m

in which f d is assumed to be small. Further, the dispersed particles do not touch each other to form a continuum. This equation has been modified by Polder and van Santen [92] to take into account that the dispersed particles may have an ellipsoid shape. The effect of the shape and anisotropy of the microscopic inclusion has been also discussed in [70, 93–96]. Further, the inclusion need not be dielectric—for example, metal particles dispersed in an insulating dielectric [97]. Mixture rules provide a convenient method of calculating the effective n and effective K for a mixture of two or more phases, and hence characterizing the composite medium. The equations can be used to search for the best composite medium for a particular application. Mirkhani et al. [98] have theoretically investigated the change in the refractive index of nanocomposites composed of Ga-doped ZnO nanoparticles (Ga-ZnO NPs) dispersed in a matrix that is composed of conductive polymer Poly(3,4-ethylenedioxythiophene) and Poly(styrene sulfonic acid) (PEDOT:PSS). The aim was to match the refractive index of GaZnO/PEDOT:PSS with that of an

Refractive index

Extinction coefficient

2.4

0.4 Bulk Ga2O3

neff

Keff

Bulk Ga2O3

0.2

2.0 Nanoporous Ga2O3

Nanoporous Ga2O3 1.6

0 300

400 500 600 700 Wavelength (nm)

800

300

400 500 600 700 Wavelength (nm)

800

Figure 2.12 Refractive index and the extinction coefficient vs wavelength for bulk and porous Ga2 O3 . Source: Data extracted and replotted from S. Kim, M. Kadam, J-H Kang, and S-W Ryu, Electron. Mater. Lett., 12, 596–602 (2016).

Acknowledgments

organic emissive layer (1.7–1.8), and the effective medium calculations showed that the volume fraction of the Ga-ZnO should be within the 45–71% range when the concentration of Ga in ZnO nanoparticles is between 0 and 4% (Chapter 15) [98]. A suitably chosen composite medium with a desired effective index of refraction, as determined by the mixture rules, can be used for the enhancement of light extraction and absorption efficiency in optical devices such as organic light emitting diodes (OLEDs) and organic solar cells (OSCs). In these applications, it is crucial to match the refractive indices of conductive anode electrode and the emissive layer. It is important to mention that the preceding description has neglected light scattering that would occur at interphase interfaces. The introduction of scattering losses need to also be considered in an accurate design of a composite medium. Various other formulae have been proposed for calculating the dielectric and optical properties of composite media, some of which are due to Looyenga [99], Monecke [100], and others, as reviewed in [74]. Nearly all EMA formulae have one common feature—that they are based on Maxwell’s equations in the static limit. Theoretical descriptions of composite media have always interested and intrigued researchers and will continue to do so, given their technological importance. The chapter on Ellipsometry (Chapter 15) has further discussion on the effective medium approach to the optical properties of inhomogeneous media.

2.8 Conclusions Selected optical properties of solids are briefly and semi-quantitatively reviewed in this chapter. The emphasis has been on physical insight rather than mathematical rigor. Where appropriate, examples have been given with typical values for various constants. A classical approach is used for describing lattice or reststrahlen absorption, infrared reflection, and FCA, whereas a quantum approach is used for band-to-band absorption of photons. Direct and indirect absorption and the corresponding absorption coefficients are discussed with typical examples, but without invoking the quantum mechanical transition matrices. Impurity absorption that occurs in RE doped glasses is also covered with a brief description of the Judd–Ofelt analysis and how this theory can be used to extract the lifetime of an upper level from absorption spectra measurements. Due to their increasing importance in photonics, the Kerr, Pockels, Franz–Keldysh, and Faraday effects are also described, but without delving into their difficult mathematical formalisms. Important mixture rules that describe the optical properties of composite media in terms of their constituents, such as the Lorentz–Lorenz, Bruggeman, and Maxwell Garnet equations, have been discussed with typical applications. Media composed of nanoparticles can be most easily described by mixture rules such as the Bruggeman rule.

Acknowledgments One of the authors (S.O. Kasap) thanks the University of Saskatchewan Centennial Enhancement Chair program for financial support.

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67

3 Optical Properties of Disordered Condensed Matter Koichi Shimakawa 1 , Jai Singh 2 , and S.K. O’Leary 3 1 2 3

Department of Electrical and Electronic Engineering, Gifu University, Gifu 501-1193, Japan College of Engineering, IT and Environment, Purple 12, Charles Darwin University, Darwin, NT 0909, Australia School of Engineering, The University of British Columbia, Kelowna, Canada

CHAPTER MENU Introduction, 67 Fundamental Optical Absorption (Experimental), 69 Absorption Coefficient (Theory), 74 Compositional Variation of the Optical Bandgap, 79 Conclusions, 80 References, 80

3.1 Introduction In a defect-free crystalline semiconductor, there exists a well-defined energy gap between the valence and conduction bands. In contrast, in an amorphous semiconductor, the distributions of conduction- and valence-band electronic states do not terminate abruptly at the band edges. Instead, some of the electronic states, referred to as tail states, encroach into the otherwise empty gap region [1]. In addition to tail states, there are other localized states deep within the gap region [2]. These localized tail states in amorphous semiconductors arise as a consequence of defects. The defects in amorphous semiconductors are considered to be all cases of departure from the normal nearest-neighbor coordination (or normal valence requirement). Examples of defects are broken and dangling bonds (typical for amorphous silicon), over- and under-coordinated atoms (such as “valence alternation pairs” in chalcogenide glasses), voids, pores, cracks, and other macroscopic defects. As these tail and deep defect states are localized, there exist mobility edges, which separate these localized states from their extended counterparts [3–5]. These localized tail and deep defect states are responsible for many of the unique properties exhibited by amorphous semiconductors. Despite years of intensive investigation, the exact form of the distribution of electronic states associated with amorphous semiconductors remains a matter for debate. While there are still some unresolved issues, there is general consensus that the tail states arise as a consequence of the disorder (weak and dangling bonds) present within the amorphous network, and that the breadth of these tails reflects the amount of disorder present Optical Properties of Materials and Their Applications, Second Edition. Edited by Jai Singh. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

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3 Optical Properties of Disordered Condensed Matter

[6]. Experimental results from Tiedje et al. [7] and Winer and Ley [8] suggest exponential distributions for the valence- and conduction-band tail states in hydrogenated amorphous silicon a-Si:H, although other possible functional forms [9] cannot be ruled out. Singh and Shimakawa [5, 10] have derived different effective masses for the charge carriers in their extended and tail states. This implies that the density-of-states (DOS) of the extended and tail states can be represented in two different parabolic forms. A representation of the current understanding of the distribution of electronic states, for the case of a-Si:H, is schematically presented in Figure 3.1 [5, 11–13]. The existence of tail states in amorphous solids has a profound impact upon the band-to-band optical absorption. Unlike the case of a crystalline solid, the absorption of photons in an intrinsic amorphous solid can also occur for photon energies below the optical gap, that is, ℏ𝜔 ≤ E0 , due to the presence of the tail states in the forbidden gap, E0 , denoting the optical gap, which is usually close to the mobility gap, that is, the energy difference between the conduction- and valence-band mobility edges. In a crystalline semiconductor, the energy and crystal momentum of an electron involved in an optical transition must be conserved. In an amorphous semiconductor, however, only the energy needs be conserved [4, 5]. As a result, for optical transitions caused by photons of energy ℏ𝜔 ≥ E0 in amorphous semiconductors, the joint density of states approach is not applicable [5, 14]. One must consider the product of densities of both the conduction and valence bands in calculating the corresponding absorption coefficient [5, 15]. Two approaches are presented here that are used for calculating the absorption coefficient in amorphous semiconductors. In the first approach, one assumes that the transition matrix element is independent of the photon energy. In the second approach, contrary to the first one, using the constant dipole approximation, the transition matrix element is found to be photon-energy dependent. Applying the first approach, one obtains the well-known Tauc’s relation for the absorption coefficient of amorphous semiconductors, that is, (𝛼ℏ𝜔)1/2 ∝ (ℏ𝜔 − E0 ), where 𝛼 is the absorption coefficient. However, applying the second approach, one obtains (𝛼/ℏ𝜔)1/2 ∝ (ℏ𝜔 − E0 ), that is, a slightly different functional dependence. The first approach has been used widely and successfully to interpret the experimental measurements corresponding to a wide variety of amorphous semiconductors of interest, some of which support Tauc’s relation, and others deviating from it. For the latter, as described in the following text, the concept of fractals and effective medium have been used to account for the deviations from the traditional Tauc relationship. In this chapter, we first review the application of the first approach and then consider those of the second. It will be shown that, through the first approach, both the fractal and

E2 Ec

Ect

Tail states Ev

Evt Ev2

Figure 3.1 A schematic illustration of the electronic energy states, E 2 , E c , E ct , E vt , E v , and E v2 , in amorphous semiconductors. The shaded region represents the extended states. Energies E 2 and E v2 correspond to the center of the conduction band and valence band extended states, E ct and E vt representing the end of conduction band and valence band tail states, respectively.

3.2 Fundamental Optical Absorption (Experimental)

effective medium theories are useful in explaining the optical properties of disordered forms of condensed matter.

3.2 Fundamental Optical Absorption (Experimental) Here we will consider the first approach, where it is assumed that the transition matrix element is independent of the photon energy. As the application of this approach to amorphous semiconductors, such as hydrogenated amorphous silicon (a-Si:H), is well known and can be found in many books [4, 5], its application to amorphous chalcogenides (a-Chs) and hydrogenated nanocrystalline silicon (a-ncSi:H) will be presented here. 3.2.1

Amorphous Chalcogenides

Analysis of the optical absorption spectra is one of the most useful tools for understanding the electronic structure of solids in any form, crystalline or amorphous. As found in a free electron gas, the DOS for both the conduction and valence bands are expected to be proportional to the square root of the energy in 3D materials [5]. Applying this to amorphous structures within the framework of the first approach leads to the well-known Tauc plot for the optical absorption coefficient 𝛼 as a function of photon energy ℏ𝜔, giving (𝛼ℏ𝜔)1/2 ∝ (ℏ𝜔 − E0 ) [16]. However, this quadratic energy dependence of the absorption coefficient on the photon energy is not always observed. For example, a linear energy dependence [(𝛼ℏ𝜔) ∝ (ℏ𝜔 − E0 )] for amorphous Se (a-Se) and a cubic energy dependence (𝛼ℏ𝜔)1/3 ∝ (ℏ𝜔 − E0 ) for multicomponent chalcogenide glasses have also been observed [4]. The deviations from the simple Tauc relation may be regarded as arising from the deviations of the DOS functions from a simple power law. The DOS in disordered matter, in general, may be described by taking into account the fractals that are known to dominate many physical properties in amorphous semiconductors [17, 18]. We revisit the classical problem for interpreting the optical properties of amorphous semiconductors on the basis of the form of DOS applicable to amorphous chalcogenides. We find that the fundamental optical absorption in amorphous chalcogenides can be written as ((𝛼ℏ𝜔)n ∝ (ℏ𝜔 − E0 )), where the value of n deviates from the Tauc value of 1/2 [19]. Typical examples are shown in Figures 3.2 and 3.3 for obliquely deposited amorphous As2 S3 and Se (hereafter a-As2 S3 and a-Se), respectively, which are given in the plot of (𝛼h𝜈)n vs h𝜈). The fitting to the experimental data for a-As2 S3 (Figures 3.2a,b) produces n = 0.70 before annealing (as deposited) and n = 0.59 after annealing near the glass transition temperature for the film. Note that similar values of n (0.73 for as deposited and 0.58 after annealing) are also obtained for a-As2 Se3 . The fitting to the data for a-Se (Figure 3.3) produces n = 1 before annealing (as deposited), and it remains unchanged after annealing at 30∘ C for 2 hours. Here, a brief derivation of the fundamental optical absorption coefficient is given in order to facilitate the interpretation of the experimental results given earlier. For inter-band electronic transitions, the optical absorption coefficient through the first approach can be written as [4]: 𝛼(𝜔) = B

∫

Nv (E − ℏ𝜔)Nc (E)dE ℏ𝜔

(3.1)

69

3 Optical Properties of Disordered Condensed Matter

6 × 103 Before annealing n = 0.70

(𝛼hν)n [eV cm–1]n

5 × 103 4 × 103 3 × 103 2 × 103 1 × 103 0 26

27

28

29

30

hν [ev] (a)

(𝛼hν)n [eV cm–1]n

1500

After annealing n = 0.59

900

300 0 2.6

2.7

2.8

2.9

3.0

hν [ev] (b)

Figure 3.2 The optical absorption spectra of obliquely deposited a-As2 S3 , where (𝛼h𝜈)n vs h𝜈 is plotted: (a) n = 0.70 before annealing, and (b) n = 0.59 after annealing at 170∘ C for 2 hours. Figure 3.3 The optical absorption spectra of obliquely deposited a-Se, where (𝛼h𝜈)n vs h𝜈 is plotted with n = 1, and it remains unchanged before and after annealing.

3 × 105 Before annealing n = 1 (𝛼hν)n [eV cm–1]n

70

2 × 105

1 × 105

0 2.0

2.1

2.2 hν [ev]

2.3

2.4

3.2 Fundamental Optical Absorption (Experimental)

where B is a constant which includes the square of the transition matrix element as a factor, and the integration is over all pairs of states in the valence N v (E) and conduction band states N c (E). If the density of states for the conduction and valence states is assumed to be Nc(E) = const (E − EC )s and Nv(E) = const (EV − E)p , respectively, then Eq. (3.1) produces [19]: 𝛼(𝜔)ℏ𝜔 = B′ (ℏ𝜔 − E0 )p+s+1

(3.2)

where B′ is another corresponding constant. This gives [𝛼(𝜔)ℏ𝜔]1∕n = B′

1∕n

(ℏ𝜔 − E0 )

(3.3)

where 1/n = p + s + 1. If the form of both N c (E) and N v (E) is parabolic, that is, p = s = 1/2 for 3D, then the photon-energy dependence of the absorption coefficient obtained from Eq. (3.3) becomes: [𝛼(𝜔)ℏ𝜔]1∕2 = B′

1∕2

(ℏ𝜔 − E0 )

(3.4)

which is the well-known Tauc’s relation for the absorption coefficient. Let us consider Eq. (3.3) and first discuss the simple case of a-Se, where n = 1 is obtained. In this case, the sum of ( p + s) should be zero for n = 1. This is only possible if the product of the DOS functions is independent of the energy. The origin of such DOS functions was argued a long time ago, but was unclear. A chain-like structure is basically expected in a-Se. The top of the valence states is known to be formed by p-lone pair (LP) orbitals (lone-pair interaction) of Se atoms. The interaction between lone-pair electrons should be 3D in nature, and, therefore, the parabolic DOS near the valence band edge can be expected, that is, p = 1/2. The bottom of the conduction-band states, on the other hand, is formed by the anti-bonding states of Se atoms. If the interaction between chains is ignored, the DOS near the conduction-band states may be 1D in nature, that is, s = −1/2. Hence, we obtain n = 1/( p + s + 1) = 1, producing a linear dependence of energy, that is, (𝛼ℏ𝜔) ∝(ℏ𝜔 − E0 ) [19]. Next, we discuss As2 S(Se)3 binary systems. These systems are suggested to have layered structures. The tops of valence-band states are formed from the LP band, and, hence, the parabolic DOS near the top of the valence band can also be expected in these systems, since LP–LP interactions occur in 3D space, as was already mentioned. Unlike a-Se, however, the bottom of the conduction band arises from a 2D structure in nature, if the layer–layer interactions can be ignored for the anti-bonding states. This means that the corresponding DOS function is independent of energy (s = 0). The value of n, in this case, should be given by 2/3, since p = 1/2 and s = 0 are predicted from the argument of space dimensions, and it is close to those observed for as-deposited oblique films of a-As2 S(Se)3 . It should be noted, however, that the layer–layer interactions are not ignored in the DOS of the conduction band [20]. The deviations from n = 2/3 or n = 1/2 may be attributed to the fractional nature of the DOS functions, that is, p or s cannot be given only values such as 1/2 (3D), 0 (2D), and − 1/2 (1D). In obliquely deposited As2 Se3 , for example, n = 0.70 (before annealing) produces p + s = 0.43. In order to interpret this result, we may need to discuss the DOS for fractal structures. The DOS for the extended states with energy E on d-space dimension in the usual Euclidean space perspective is given as: N(E)dE ∝ 𝜌d−1 d𝜌

(3.5)

71

72

3 Optical Properties of Disordered Condensed Matter

where 𝜌 is defined as (2m∗e E)1∕2 ∕ℏ, instead of the wave vector k, which is not a good quantum number in disordered materials, m∗e being the electronic effective mass. In a fractal space, on the other hand, a fractal dimension, D, is introduced, instead of d. The DOS for the extended states in the fractal space D can be given by: N(E)dE ∝ 𝜌D−1 d𝜌 ∝ E

D−2 2

dE

(3.6)

Note that D is introduced as M(r) ∝ rD, where M is the “mass” in a space, and hence D can take any fractional value (even larger than 3). A similar argument of fractional dimensionality on interband optical transitions has also been presented in anisotropic crystals applicable to low-dimensional structures [21]. As we have discussed already, the energy dependence of the DOS for the conduction band is expected to be different from that for the valence band in amorphous chalcogenides, because usually the space dimensionality for the valence band is larger than that for the conduction band. Therefore, we introduce Dv and Dc for the dimensionality of the valence band and the conduction band, respectively. Then p + s + 1 in Eq. (3.2) is replaced by Dv + Dc − 2 (3.7) 2 for fractional-dimension systems. From the value of n, that is, 2/(Dv + Dc − 2), Dv + Dc can then be deduced. In As2 S3 , for example, Dv + Dc is 4.86. After thermal annealing, the value of n tends to 0.5 for both the oblique and flat samples, that is, Dv + Dc ≈ 6, indicating that each Dv and Dc approach three dimensions. This is due to the fact that thermal annealing produces a more ordered and dense structural network. A similar argument has also been given for flatly deposited materials. A cubic energy dependence, n = 1/3, is often observed for multi-component materials, such as Ge–As–Te–Si [4], which gives Dv + Dc = 8 according to the preceding analysis. This higher fractal dimension may be related to “branching” or “cross-linking” between Te chains by introducing As, Ge, and Si atoms. The “branching” may be equivalent to a “Bethe lattice” (or “Cayley tree”), resulting in an increase in the spatial dimensions [18]. In summary, the fundamental form of the optical absorption spectrum, empirically presented by the relation(𝛼ℏ𝜔)n ∝ (ℏ𝜔 − E0 ), where n deviates from 1/2 in amorphous chalcogenides, is interpreted by introducing the DOS of fractals. The energy-dependent DOS form is not the same for the conduction band and the valence band. The presence of disorder can greatly influence the nature of the electronic DOS even for the extended states. Accordingly, the concept of DOS on fractal structures has successfully been applied to interpret the fundamental optical absorption spectra. Finally, we should briefly discuss the validity of Eq. (3.1) itself in which the transition matrix element is assumed to be independent of energy, that is, B is independent of energy. Dersch et al. [22], on the other hand, suggested that the transition matrix element is energy dependent and has a peaked nature near the bandgap energy. The absorption coefficient with and without the energy-dependent matrix element will be briefly discussed in Section 3.3. p+s+1=

3.2.2

Hydrogenated Nano-Crystalline Silicon (nc-Si:H)

In this section, we discuss the optical properties of nanocrystalline Si prepared by the plasma-enhanced chemical vapor deposition (PECVD) method. Since this material

3.2 Fundamental Optical Absorption (Experimental)

𝛼 (cm–1)

Figure 3.4 A comparison of the optical absorption coefficient of a-Si:H, nc-Si:H, and c-Si. The EMA results are given by solid circles.

106

a-Si:H nc-Si:H

104

c-Si

EMA (Xc = 0.8)

102

100 1.2

1.6

2.0

2.4

2.8

Energy [eV]

consists of both amorphous and crystalline phases, its structure is very complex and nonuniform. Therefore, it may be expected that the optical absorption cannot be described by Tauc’s relation (Eq. (3.4)) [16]. One of the interesting properties these materials have is the excess of optical absorption in the fundamental absorption region. The optical absorption coefficient for nc-Si:H is larger than that for crystalline silicon (c-Si), as has been reported in the infrared to blue region. This points to an advantage of using nc-Si:H to fabricate solar cells, because many more photons can be absorbed in the films. An example of this difference is shown in Figure 3.4. Three solid lines represent the experimental data for a-Si:H, nc-Si:H, and c-Si. According to Figure 3.4, a-Si:H has the highest optical absorption coefficient and c-Si has the lowest in the energy range E > 1.7 eV. Therefore, a sample of nc-Si:H (a mixture of both) should be expected to have an absorption in between the two. However, how to calculate the absorption in hydrogenated microcrystalline silicon (μc-Si:H) is still a matter of debate. Although the scattering of light is suggested to be an origin for the enhanced optical absorption, Shimakawa [23] took the effective medium approximation (EMA), in an alternative way, to explain the excess of absorption. Here, first we briefly introduce the EMA before proceeding with the discussion on enhanced absorption within nc-Si:H. The EMA predicts that the total network conductance 𝜎 m for composite materials in D dimensions follows the following condition: ⟨ ⟩ 𝜎 − 𝜎m =0 (3.8) 𝜎 + (D − 1)𝜎m where 𝜎 is a random variable of conductivity and ⟨. . .⟩ denotes spatial averaging. Assuming that a random mixture of particles of two different conductivities, for example, one volume fraction, C, has conductivity of 𝜎 0 , and the remainder has conductivity of 𝜎 1 , 𝜎 1 being substantially less than 𝜎 0 , simple analytical expressions for the dc conductivity and the Hall mobility as a function of C have been derived (see the pioneering works by Kirkpatrik [24] and Cohen and Jortner [25]). EMA has also been extended to calculate the ac conductivity, in which case 𝜎 in Eq. (3.1) becomes a complex admittance, that is, 𝜎* = 𝜎 1 + i𝜎 2 [26]. As the dielectric constant, 𝜀* = 𝜀1 − i𝜀2 = 𝜎 2 /𝜔 − i𝜎 1 /𝜔, is closely related to 𝜎*, the optical absorption coefficient 𝛼(𝜔) can be calculated using 𝛼(𝜔) = 4𝜋𝜎(𝜔)/cn, where c is the speed of light and n is the refractive index [23] of the material. The results obtained from the EMA for the crystalline volume fraction X c = 0.8 and D = 3 are shown by open circles in Figure 3.4. The frequency (energy)-independent

73

74

3 Optical Properties of Disordered Condensed Matter

refractive index, n0 = 3.9 for c-Si and n1 = 3.2 for a-Si:H, which gives the square root of the corresponding real part of their optical dielectric constants, are used in the calculation. The calculated results agree very well with the experimental data, except at an energy of around 1.7 eV (see Figure 3.4). This suggests that a mean field constructed through a mixture of amorphous and crystalline states dominates optical absorption within nc-Si:H. It may be noted that the multiple light scattering suggested earlier seems to be not so important in this energy range.

3.3 Absorption Coefficient (Theory) In the previous section, we discussed the absorption coefficient in relation to the band-to band absorption under the assumption that the transition matrix element was independent of the absorbed photon energy. The constant coefficient B′ in Eqs. (3.3) and (3.4) was not derived, and it could only be determined from fitting of the experimental data. In this section, the theoretical derivation of the absorption coefficient from both approaches will be given. An effort is made to clearly identify the differences between the two approaches and their applications. The absorption coefficient in Eq. (3.1), derived from the rate of absorption, can be written as in reference [5]: )2 E𝜐 +ℏ𝜔 ( 2𝜋e 1 |pcv |2 Nc (E)Nv (E − ℏ𝜔)dE (3.9) 𝛼= ∫Ec ncV𝜔 m∗e where n is the real part of the refractive index and V is the illuminated volume of the material, and pcv is the transition matrix element of the electron–photon interaction between the conduction and valence bands. Thus, the integrand consists of a product of three factors, all three of which depend on the photon energy and integration variable of energy E. Therefore, it becomes very difficult to evaluate the integral analytically. For crystalline materials, pcv is evaluated under the dipole approximation, but for amorphous solids, the first approximation introduced was to consider pcv as being independent of the photon energy, ℏ𝜔 [4], as described in Section 3.2. Then Eq. (3.9) reduces to the form of Eq. (3.1) and the whole analysis of the absorption properties presented in Section 3.2 is based on this approach. However, the resulting constant coefficient, B′ , which is an unknown in Eqs. (3.3) and (3.4), can now be determined. For the constant pcv , a popular form used is that shown in reference [4]: pcv = −iℏ𝜋(L∕V )1∕2

(3.10)

where L is the average interatomic spacing in the sample. Assuming that both the valence band and conduction band DOS functions have square-root dependencies on energy (as in a free-electron gas), one gets Tauc’s relation given in Eq. (3.4), with B′ as given in reference [5]: )] [ ( )2 ( ∗ ∗ 3∕2 m ) L(m e e 1 h B′ = (3.11) nc𝜀0 m∗e 2 2 ℏ3 where m∗h is the effective mass of a hole in the valence band. The advantage of Eq. (3.11) is that, if one knows the effective masses, then the so called Tauc’s coefficient, B′ , can be determined theoretically without having to fit Eq. (3.4) to any experimental data. As

3.3 Absorption Coefficient (Theory)

stated in Section 3.2, the absorption coefficient of many amorphous semiconductors, including chalcogenides, fit to Eq. (3.4) very well, but not all. Deviations from Tauc’s relation have been observed, and if one assumes a constant transition matrix element, pcv , in Eq. (3.9), then the only way these deviations can be explained is from the deviations in the squared-root form of the density of states, N c and N v , as discussed in Section 3.2 for various examples of chalcogenides. For instance, some experimental data for a-Si:H fit much better to a cubic root relation given by a formula in reference [4]: (𝛼ℏ𝜔)1∕3 = C(ℏ𝜔 − E0 )

(3.12)

and therefore the cubic root has been used to determine the optical gap E0 . Here, C is another constant. If one considers that the optical transition matrix element is photon-energy independent [5], one finds that the cubic root dependence on photon energy can be obtained only when the valence band and conduction band DOS depend linearly on energy. Using such DOS functions, the cubic root dependence has been explained by Mott and Davis [4]. Another approach to arrive at the cubic root dependence has been suggested by Sokolov et al. [27]. Using Eq. (3.12), they have modeled the cubic root dependence on photon energy by considering the fluctuations in the optical bandgap due to structural disorders. For arriving at the cubic root dependence, they finally assume that the fluctuations are constant over the range of integration, and then the integration of Eq. (3.12) over the optical gap energy produces a cubic root dependence on the photon energy. Although their approach shows a method of achieving the cubic root dependence, as the integration over the optical gap is carried out by assuming constant fluctuations, Sokolov et al.’s model is a little different from the linear DOS model suggested by Mott and Davis [4]. Let us now consider the second approach, where pcv is not assumed to be a constant. As stated earlier, for studying atomic and crystalline absorption, one uses the dipole approximation to evaluate the transition matrix element. This yields (see reference [5]): pcv = im∗e 𝜔reh

(3.13)

where reh is the average separation between the excited electron-and-hole pair. It may also be noted that, in the case where an electron-and-hole pair is excited by the absorption of a photon, m∗e should be replaced by their reduced mass, 𝜇, where 𝜇−1 = m∗(−1) + e mh∗(−1) . Let us use Eq. (3.13) for amorphous solids as well. Inserting Eq. (3.13) into Eq. (3.9), Cody [28] has derived the absorption coefficient in amorphous semiconductors. Accordingly, the absorption coefficient is obtained, as shown in reference [5]: [𝛼ℏ𝜔] = B′′ (ℏ𝜔)2 (ℏ𝜔 − E0 )2 where e2 B = nc𝜀0 ′′

[

(m∗e m∗h )3∕2 2𝜋 2 ℏ7 𝜈𝜌A

(3.14)

] 2 reh

(3.15)

Here, v denotes the number of valence electrons per atom, and 𝜌A represents the atomic density per unit volume. Eq. (3.14) suggests that [𝛼ℏ𝜔] depends on the photon energy in the form of a polynomial of order 4. Then, depending on which term of the polynomial may be more

75

76

3 Optical Properties of Disordered Condensed Matter

significant for which material, one can get square, cubic, fourth, or any other root of the dependence of [𝛼ℏ𝜔] on the photon energy. In this case, Eq. (3.14) may be expressed as: [𝛼ℏ𝜔]x ∝ (ℏ𝜔 − E0 )

(3.16)

where x ≤ 1/2. Thus, in a way, any deviation from the square root or Tauc’s plot may be attributed to the energy-dependent matrix element [5, 10]. However, this is a rather difficult issue to resolve unless one can determine the form of the DOS functions associated with the conduction and valence bands unambiguously. As a result, as the first approach of the constant transition matrix element has been successful for many samples over the last few decades, as has been demonstrated by many experimental groups, there appears to be a kind of prejudice in the literature in its favor. Let us now discuss how to determine the constants B′ in Eq. (3.11) and B′′ in Eq. (3.15), which involve the effective masses of electrons and holes, m∗e and m∗h , respectively. Recently, a simple approach [5, 29] has been developed to calculate the effective mass of the charge carriers in amorphous solids. Accordingly, different effective masses of the charge carriers are obtained in the extended and tail states. The approach applies the concepts of tunneling and effective medium, and one obtains the effective mass of an electron in the conduction band extended states, denoted by m∗ex , and in the tail states, denoted by m∗et , as in references [5, 29]: m∗ex ≈

EL m 2(E2 − Ec )a1∕3 e

(3.17)

m∗et ≈

EL m 2(Ec − Ect )b1∕3 e

(3.18)

and

where: EL = Here, a = N2 N

ℏ2 me L2

N1 N

(3.19)

< 1, N1 is the number of atoms contributing to the extended states,

< 1, N2 is the number of atoms contributing to the tail states, such that a + b = 1 b= (N = N1 + N2 ), and me is the free-electron mass. The energy E2 in Eq. (3.17) corresponds to the energy of the middle of the conduction extended states at which the imaginary part of the dielectric constant becomes a maximum and Ect is the energy corresponding to the end of the conduction tail states (see Figure 3.1). Likewise, the hole effective masses and m*hx in the valence extended and tail states are obtained, respectively, as: m∗hx ≈

EL m 2(Ev − Ev2 )a1∕3 e

(3.20)

m∗ht ≈

EL m 2(Evt − Ev )b1∕3 e

(3.21)

and

where Ev2 and Evt are the energies corresponding to the half width of valence extended states and the end of the valence tail states, respectively (see Figure 3.1).

3.3 Absorption Coefficient (Theory)

Table 3.1 The effective mass of electrons in the extended and tail states of a-Si:H and a-Ge:H, calculated using Eqs. (3.17) and (3.18) for a = 0.99, b = 0.01, and E ct = E vt = E c /2. E L is calculated from Eq. (3.18). All energies are given in eV. Note that, as the optical absorption coefficient is measured in cm−1 , the value for the speed of light is given in cm/s. L(nm)

E2

Ec

EL

a-Si:H

0.235a)

3.6b)

1.80c)

1.23

a-Ge:H

0.245a)

3.6

1.05d)

1.14

m∗ex

m∗et

0.9

0.34me

6.3me

0.53

0.22me

10.0me

E c –E ct

Note: a) Morigaki [30], b) Ley [31], c) Street [1], and d) Aoki et al. [32].

Using Eqs. (3.18) and (3.19) and the values of the parameters involved, different effective masses for an electron are obtained in the extended and tail states. Considering, for example, the density of weak bonds contributing to the tail states as 1 at.%, that is, b = 0.01 and a = 0.99, the effective mass and energy EL thereby calculated for hydrogenated amorphous silicon (a-Si:H) and germanium (a-Ge:H) are given in Table 3.1. For sp3 hybrid amorphous semiconductors, such as a-Si:H and a-Ge:H, the energies Ect and Evt can be approximated as: Ect = Evt = Ec /2. According to Eqs. (3.17), (3.18), (3.20), and (3.21), for sp3 hybrid amorphous semiconductors, such as a-Si:H and a-Ge:H, the electron and hole effective masses are expected to be the same. In these semiconductors, as the conduction and valence bands are two equal halves of the same electronic band, their widths are the same, and that gives equal effective masses for electron and hole [5, 29]. This is one of the reasons for using Ec t = Evt = Ec /2, which gives equal effective masses for electrons and holes in the tail states as well. This is different from crystalline solids, where m∗e and m∗h are usually not the same. This difference between amorphous and crystalline solids is similar to, for example, having direct and indirect crystalline semiconductors but only direct amorphous semiconductors. Using the effective masses from Table 3.1 and Eq. (3.15), B′ can be calculated for a-Si:H and a-Ge:H. The values thus obtained with the refractive index n = 4 for a-Si:H and a-Ge:H are B′ = 6.0 × 106 cm−1 eV−1 for a-Si:H and B′ = 4.1 × 106 cm−1 eV−1 for a-Ge:H, which are an order of magnitude higher than those estimated by fitting the experimental data [4]. However, considering the quantities involved in B′ (Eq. (3.11)), this can be regarded as a good agreement. In a recent paper, Malik and O’Leary [33] have studied the distributions of conduction and valence-band electronic states associated with a-Si:H. They have noted that the effective masses associated with a-Si:H are material parameters which are yet to be experimentally determined. In order to remedy this deficiency, they have fitted square-root DOS functions to experimental DOS data and found m∗h = 2.34 me and m∗e = 2.78 me . The value of the constant B′′ in Eq. (3.15) can now be calculated theoretically, provided reh is known. Using the atomic density of crystalline silicon and four valence electrons 2 2 = 0.9 Å , which gives reh ≈ 0.095 nm, less than half per atom, Cody [28] has estimated reh

77

78

3 Optical Properties of Disordered Condensed Matter

of the interatomic separation of 0.235 nm in a-Si:H, but of the same order of magnitude. 2 2 = 0.9Å , and extended-state effective masses, we get Using v = 4, 𝜌A = 5 × 1028 m−3 , reh ′′ 3 −1 −3 B = 4.6 × 10 cm eV for a-Si:H and 1.3 × 103 cm−1 eV−3 for a-Ge:H. Cody has estimated an optical gap, E0 = 1.64 eV, for a-Si:H, from which, using Eq. (3.14), we get 𝛼 = 1.2 × 103 cm−1 at a photon energy of ℏ𝜔= 2 eV. This agrees reasonably well with the value 𝛼 = 6.0 × 102 cm−1 used by Cody. If we use the interatomic spacing, L, in place of reh in Eq. (3.14), we get B′′ = 2.8 × 104 cm−1 eV−3 , and then the corresponding absorption coefficient becomes 3.3 × 103 cm−1 . This suggests that, for an estimate, one may use the interatomic spacing in place of reh , if the latter is unknown. Thus, both the constants B′ and B′′ can be determined theoretically, a task not possible earlier due to our lack of knowledge regarding the effective masses in amorphous semiconductors. The absorption of photons of energy less than the band gap energy, ℏ𝜔 < E0 , in amorphous solids involves the localized tail states, and hence follows neither Eq. (3.4) nor Eq. (3.12). Instead, the absorption coefficient depends on the photon energy exponentially, as shown in Chapter 2, giving rise to an Urbach tail. Abe and Toyozawa [34] have calculated the interband absorption spectra in crystalline solids, introducing the Gaussian site diagonal disorder and applying the coherent potential approximation. They have shown that an Urbach tail occurs due to static disorder (structural disorder). However, the current stage of understanding is that Urbach’s tail in amorphous solids occurs due to both thermal and structural disorders [28]. More recent issues in this area have been addressed by Orapunt and O’Leary [35]. Keeping this discussion in mind, the optical absorption in amorphous semiconductors near the absorption edge is usually characterized by three types of optical transitions corresponding to transitions between tail and tail states, tail and extended states, and extended and extended states. The first two types correspond to ℏ𝜔 ≤ E0 , while the third corresponds to ℏ𝜔 ≥ E0 . Thus, the absorption coefficient vs photon energy (𝛼 vs. ℏ𝜔) dependence has the corresponding three different regions A, B, and C that correspond to these three characteristic optical transitions shown in Figure 3.5. In the small optical absorption coefficient range A (also called the weak absorption tail [WAT]), where 𝛼 < 10−1 cm−1 , the optical absorption is controlled by optical transitions from tail-to-tail states. As stated earlier, the localized tail states in amorphous semiconductors arise from defects. To some extent, the absolute value of the absorption in region A may be used to estimate the density of defects in the material. In region B, where typically 10−1 cm−1 < 𝛼 < 104 cm−1 , the optical absorption is related to transitions from the localized tail states above the valence band edge to extended states in the conduction band, and/or from extended states in the valence band to localized tail states below the conduction band. Usually, the spectral dependence in this region follows the so-called Urbach dependence, given in Eq. (2.15) of Chapter 2. For many amorphous semiconductors, ∇E has been related to the breadth of the valence- or conduction-band tail states, and may be used to compare the “breadth” of such localized tail states in different materials; ∇E typically ranges from 50 to 100 meV for the case of a-Si:H. In region C, the optical absorption is controlled by optical transitions from extended states to the extended states. For many amorphous semiconductors, the 𝛼 vs. ℏ𝜔 behavior follows the traditional Tauc relation given in Eq. (3.4). The optical bandgap, E0 , determined for a given material from the 𝛼 vs. ℏ𝜔 relations obtained in Eqs. (2.15) (Chapter 2), (3.4) and (3.12), can vary, as shown in Table 3.2 for a-Si:H alloys.

3.4 Compositional Variation of the Optical Bandgap

𝛼 (cm–1) Tauc absorption

105

(ћω𝛼)1/2

104

C

103 102

Tauc absorption

Urbach edge

101 1 10–1 10–2 10–3

B WAT A

ћω

ћω Eo

Eg

03

Figure 3.5 Typical spectral dependence of the optical absorption coefficient in amorphous semiconductors. In the A and B regions, the optical absorption is controlled by the optical transitions between tail and tail, and tail and extended states, respectively; in the C region, it is dominated by transitions from extended to extended states. In domain B, the optical absorption coefficient follows Urbach rule (see Eq. (2.15) in Chapter 2). In region C, the optical absorption coefficient follows Tauc’s relation (Eq. (3.4)) in a-Si:H, as shown on the right-hand-side figure. Source: Reproduced by permission of Professor S. Kasap. Table 3.2 The optical bandgap of a-Si1–x Cx :H films obtained from Tauc’s (Eq. (3.4)), Sokolov et al.’s (Eq. (3.12)), and Cody’s (Eq. (3.14)) relations (3.27). E g at 𝜶 = 103 cm–1

E g at 𝜶 = 104 cm–1

Eg = Eo (Tauc)

Eg = Eo (Cody)

Eg = Eo (Sokolov)

𝚫E meV

a-Si:H

1.76

1.96

1.73

1.68

1.60

46

a-Si0.88 C0.18 :H

2.02

2.27

2.07

2.03

1.86

89

3.4 Compositional Variation of the Optical Bandgap 3.4.1

In Amorphous Chalcogenides

In Sections 3.2 and 3.3, the behavior of the optical absorption coefficient vs photon energy has been discussed. In this section, we discuss the effect of the compositional variation on the optical gap, E0 , of amorphous chalcogenide alloys. The bandgap varies with the composition and often exhibits extrema at certain stoichiometric compositions, for example, a minima in As2 S(Se)3 and a maximum in GeSe2 (see, e.g., ref. [36]). Applying the virtual crystal approach proposed by Phillips [37], Shimakawa has accounted for such compositional variations in the following form [36]: E0 = xE0 (A) + (1 − x)E0 (B) − 𝛾x(1 − x)

(3.22)

where A and B are composite elements in an Ax B1 − x alloy, and 𝛾 is referred to as the bowing parameter.

79

80

3 Optical Properties of Disordered Condensed Matter

Through the study of the optical gap in composite chalcogenides, amorphous chalcogenides can be classified into three types: (i) random bond network (RBN) type, (ii) chemically ordered bond network (CON) type, and (iii) indefinite network type. For the RBN type, A and B are taken as composite elements, for example, A = Sb and B = Se for Sbx Se1 − x . For the CON type, A and B are taken as the stoichiometric composition and the element is in excess, respectively—for example, A = As2 Se3 and B = Se in a Asx Se1 − x system. It is known that the optical gap of a-Gex Si1 − x : H can also be represented by Eq. (3.22) with 𝛾 = 0 [38]. Note also that the validity of Eq. (3.22), proposed by Shimakawa [36], has been confirmed in many a-Chs [39]. The main conclusions here are as follows: (i) the optical bandgap for amorphous semiconducting alloys is determined by the volume fraction and the optical gap of each element of the alloy, leading to the conjecture that a modified virtual crystal approach for mixed crystals is acceptable for amorphous systems; (ii) the classification into the three types presented earlier is supported by the effective medium approach (EMA), which has been applied to electronic transport in amorphous chalcogenides [40].

3.5 Conclusions The current understanding of the fundamental optical properties in some disordered semiconductors are briefly reviewed. Fundamental optical absorption in amorphous semiconductors, including the chalcogenides, cannot always be expressed by the well-known Tauc relationship. The deviation from the Tauc relationship has been discussed through two approaches—first with the energy-independent transition matrix element, and second with the energy-dependent matrix element. In the first approach, deviations from the Tauc relationship are overcome by introducing the density-of-states in fractal structures of amorphous chalcogenides. In the second approach, the deviations from Tauc’s relation are obtained through the use of the energy-dependent transition matrix element. This fractal nature may be attributed to the clustered layer structures in amorphous chalcogenides. The applicability of an EMA to the fundamental optical absorption spectra of nano-crystalline silicons is confirmed, and the effect of the compositional variation on the optical gap in amorphous semiconductors can also be connected to the validity of EMA.

References 1 Street, R.A. (1991). Hydrogenated Amorphous Silicon. Cambridge: Cambridge Uni-

versity Press. 2 Papaconstantopoulos, D.A. and Economou, E.N. (1981). Phys. Rev. B 24: 7233. 3 Cohen, M.H., Fritzsche, H., and Ovshinsky, S.R. (1969). Phys. Rev. Lett. 22: 1065. 4 Mott, N.F. and Davis, E.A. (1979). Electronic Processes in Non-crystalline Materials.

Oxford: Clarendon Press. 5 Singh, J. and Shimakawa, K. (2003). Advances in Amorphous Semiconductors. London

and New York: Taylor & Francis. 6 Sherman, S., Wagner, S., and Gottscho, R.A. (1996). Appl. Phys. Lett. 69: 3242.

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7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Tiedje, T., Cebulla, J.M., Morel, D.L., and Abeles, B. (1981). Phys. Rev. Lett. 46: 1425. Winer, K. and Ley, L. (1987). Phys. Rev. B 36: 6072. Webb, D.P., Zou, X.C., Chan, Y.C. et al. (1998). Solid State Commun. 105: 239. Singh, J. (2003). J. Mater. Sci. Mater. Electron. 14: 171. Jackson, W.B., Kelso, S.M., Tsai, C.C. et al. (1985). Phys. Rev. B 31: 5187. O’Leary, S.K., Johnson, S.R., and Lim, P.K. (1997). J. Appl. Phys. 82: 3334. Malik, S.M. and O’Leary, S.K. (2004). J. Non-Cryst. Solids 336: 64. Elliott, S.R. (1998). The Physics and Chemistry of Solids. Sussex: Wiley. Singh, J. (2002). Nonlinear Optics 29: 119. Tauc, J. (1979). The Optical Properties of Solids (ed. F. Abeles), 277. Amsterdam: North-Holland. Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. New York: Freeman. Zallen, R. (1983). The Physics of Amorphous Solids, 135. New York: Wiley. Nessa, M., Shimakawa, K., Ganjoo, A., and Singh, J. (2000). J. Optoelectron. Adv. Mater. 2: 133. Watanabe, Y., Kawazoe, H., and Yamane, M. (1988). Phys. Rev. B 38: 5677. He, X.-F. (1990). Phys. Rev. B 42: 11751. Dersch, U., Grunnewald, M., Overhof, H., and Thomas, P. (1987). J. Phys. C Solid State Phys. 20: 121. (a) Shimakawa, K. (2000). J. Non-Cryst. Solids 266–269: 223. (b) Shimakawa, K. (2004). Encyclopedia of Nanoscience and Nanotechnology, vol. 4, 35. Valencia: American Scientific Publishes. (c) Shimakawa, K. (2004). J. Mater. Sci. - Mater. Electron. 15: 63. Kirkpatrick, S. (1973). Rev. Mod. Phys. 45: 574. Cohen, M.H. and Jortner, J. (1973). Phys. Rev. Lett. 30: 699. Springett, B.E. (1973). Phys. Rev. Lett. 31: 1463. Sokolov, A.P., Shebanin, A.P., Golikova, O.A., and Mezdrogina, M.M. (1991). J. Phys. Condens. Matter 3: 9887. Cody, G.D. (1984). Semiconductors and Semimetals B 21: 11. Singh, J., Aoki, T., and Shimakawa, K. (2002). Philos. Mag. B 82: 855. Morigaki, K. (1999). Physics of Amorphous Semiconductors. London: World Scientific. Ley, L. (1984, 1984). The Physics of Hydrogenated Amorphous Silicon II (eds. J.D. Joannopoulos and G. Lukovsky), 61. Berlin: Springer. Aoki, T., Shimada, H., Hirao, N. et al. (1999). Phys. Rev. B 59: 1579. (a) Malik, S.M. and O’Leary, S.K. (2005). J. Mater. Sci. Mater. Electron. 16: 177; (b) O’Leary, S.K. (2004). J. Mater. Sci.: Mater. Electron. 15: 401. Abe, S. and Toyozawa, Y. (1981). J. Phys. Soc. Jpn. 50: 2185. Orapunt, F. and O’Leary, S.K. (2004). Appl. Phys. Lett. 84: 523. Shimakawa, K. (1981). J. Non-Cryst. Solids 43: 229. Phillips, J.C. (1973). Bond and Band in Semiconductors. New York: Academic Press. Bauer, G.H. (1995). Solid-State Phenomena, vol. 365, 44–46. Tichy, L., Triska, A., Barta, C. et al. (1982). Philos. Mag. B 46: 365. Shimakawa, K. and Nitta, S. (1978). Phys. Rev. B 17: 3950.

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4 Optical Properties of Glasses Andrew Edgar School of Chemical and Physical Sciences, Victoria University of Wellington, New Zealand

CHAPTER MENU Introduction, 83 The Refractive Index, 84 Glass Interfaces, 86 Dispersion, 88 Sensitivity of the Refractive Index, 90 Glass Color, 95 Fluorescence in Rare-Earth-Doped Glass, 102 Glasses for Fiber Optics, 104 Refractive Index Engineering, 106 Glass and Glass–Fiber Lasers and Amplifiers, 109 Valence Change Glasses, 111 Transparent Glass Ceramics, 114 Conclusions, 124 References, 124

4.1 Introduction Historically, glass was first valued for jewelry and decoration. As the last millennium developed, however, it became apparent that glass had two key attributes that made it an especially valuable material: (1) it could be worked into a variety of shapes by processes such as casting, drawing, molding, polishing, and blowing; and (2) it was transparent to visible radiation. This combination permitted the development of key technologies: manufacturing of containers with transparent walls, building and vehicle windows, and optical instruments and appliances such as spectacles, telescopes, cameras, and microscopes. It is difficult to appreciate from the context of the twenty-first century what a difference having a transparent, weather-proof, durable window material must have made to daily living in the Middle Ages, or to the ability to correct long- and short-sightedness with spectacles. In the twentieth century, the discovery of methods of making low-loss silica optical fiber revolutionized communications and computing technologies. In the twenty-first century, we can expect further developments based on these two key attributes of glass as a material: optical transparency (which can be

Optical Properties of Materials and Their Applications, Second Edition. Edited by Jai Singh. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

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extended from the visible to the ultraviolet [uv] and the infrared [IR]), and the ability to shape or engineer glass into an arbitrary geometry or structure. This chapter first outlines the scientific and technical background of optical glass and its applications before presenting a discussion of some topics of current interest. Sections 4.2–4.5 discuss the complex refractive index, its dispersion, and factors that affect it, such as temperature, stress, chemical composition, and magnetic field. In Sections 4.6 and 4.7, we describe the origins of color in glass, and the effects of doping with open-shell ions from the 3d and 4f series of the periodic table, before outlining the fluorescence properties of rare-earth-doped glasses. In Section 4.8, we outline the optical fiber application of glass. Finally, in Sections 4.9–4.12, we describe work on refractive index micro-engineering in glasses, glass fiber lasers and amplifiers, and on transparent glass ceramics.

4.2 The Refractive Index Well-annealed glass is generally an optically homogeneous (on the scale of the wavelength of light), non-magnetic, and isotropic material. The behavior of an electromagnetic wave propagating through glass can be described by a complex refractive index n-ik, or equivalently a complex dielectric constant 𝜀′ − i𝜀′′ , which are related through (n − ik)2 = 𝜀′ − i𝜀′′

(4.1)

or equivalently 𝜀′ = n2 − k 2

(4.2)

𝜀′′ = 2nk

(4.3)

and In Figure 4.1, we show how the refractive index for a representative glass, silica, varies with wavelength, with data taken from a handbook of optical constants [1]. The key features are a broad region where n is almost (but not quite) wavelength-independent (the transmittance “window” of the glass), bounded by a sharp rise in k at short wavelengths (the “uv edge”), and a sharp rise in k at longer wavelengths (the IR edge). It is shown in many texts [2] that, for a plane monochromatic electromagnetic wave of frequency 𝜔, polarized in the x direction and propagating through a non-conducting glass in the z direction, the electric field strength Ex is given by, Ex (z) = E(0)exp(−kz𝜔∕c)exp[i𝜔(t − nz∕c)]

(4.4)

The wavelength in the glass is given by 𝜆 = 𝜆0 /n, where 𝜆0 is the vacuum wavelength, and the amplitude of the wave decreases exponentially in the propagation direction, with an intensity decaying as exp(−𝛼z), where 𝛼 = 2𝜔k/c is called the absorption coefficient. In Figure 4.2, we show the absorption coefficient in silica glass as a function of wavelength. Clearly, the abrupt rise in k at the two edges gives rise through 𝛼 = 2𝜔k/c to strong absorption (note the logarithmic scale in Figure 4.2) marking the perimeter of the transmission window. We also note that the well-known absorption features at 1.39 and 2.7 μm due to hydroxyl impurities in silica glass are too small to appear in Figure 4.2, the 1.39 μm absorption peak being of the order of 10−5 cm−1 in high-purity SiO2 . Although these bands are very weak, over a long path length of the order of kilometers, they and other effects (Section 4.8) have a dominant influence on the choice of wavelength used for fiber-optic communications within the broad silica window. The sharp rise in k also

4.2 The Refractive Index

3

2.5

n k

n,k

2

1.5

1

0.5

0 0.1

1 Wavelength (µm)

10

Figure 4.1 Complex refractive index (n, k) versus wavelength for silica glass SiO2 . 106

Absorption Coefficient (cm–1)

105

104

1000

100

10

1 0.1

1 Wavelength (µm)

10

Figure 4.2 Absorption coefficient plotted as a function of wavelength for silica SiO2 .

contributes to strong reflections from glass surfaces at the window boundaries. For normal incidence, the intensity reflection coefficient from an air–glass interface is given by | (n − 1)2 + k 2 | | R = || | | (n + 1)2 + k 2 | and, in Figure 4.3, we show the effect of the edges on the reflection coefficient.

(4.5)

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4 Optical Properties of Glasses

1

0.1 Reflection Coefficient

86

0.01

0.001

0.0001 0.1

1 Wavelength (µm)

10

Figure 4.3 Reflection coefficient at normal incidence versus wavelength for an air–silica glass interface.

4.3 Glass Interfaces The complex refractive index of glasses is dominated in the window of transparency by the real part. In that case, the reflection and transmission coefficients for normal incidence at the interface between two glass media are readily calculated from Eq. (4.5) using the relative refractive index m = n2 /n1 for the refractive indices for light in the medium of incidence (n1 ) and transmission (n2 ); the result is shown in Figure 4.4. For light, which makes an angle of incidence 𝜃 𝜄 with the normal to the interface, the reflection and transmission coefficients depend upon the polarization of the incident light. Light with its electric field vector polarized in the plane defined by the incident, and reflected ray and the normal is referred to as p-polarized. Light with its electric field vector polarized perpendicular to that plane is called s-polarized. The reflection and transmission coefficients for intensity follow from Fresnel’s equations, and are given by [ ] −cos𝜃i + mcos𝜃t 2 , Ts = 1 − Rs (4.6) Rs = cos𝜃i + mcos𝜃t [ ] mcos𝜃i − cos𝜃t 2 , Tp = 1 − Rp (4.7) Rp = mcos𝜃i + cos𝜃t where m is the relative refractive index (assumed real) defined earlier, and 𝜃 t is the angle the transmitted ray makes with the normal, related to 𝜃 t through Snell’s law. Equations. (4.6) and (4.7) reduce to Eq. (4.5) for normal incidence, n1 = 1, and k = 0. The reflection and transmission coefficients are shown in Figure 4.5 for the case when the relative refractive index is 1.5. The reflection coefficient (Rp ) for the p-polarized wave

4.3 Glass Interfaces

Reflection (R) or Transmission (T) Coefficient

1

R T

0.8

0.6

0.4

0.2

0 10

1 Relative Refractive Index m = n2/n1

0.1

Figure 4.4 Reflection (R) and transmission (T) coefficients for intensity at normal incidence for light incident from medium 1 onto medium 2 plotted as a function of the relative refractive index.

Reflection (R) and Transmission (T) Coefficients

1.2

1

Rs Ts Rp Tp

0.8

0.6

0.4

0.2

0

0

20

40 60 Angle of Incidence (degrees)

80

Figure 4.5 Plot of reflection (R) and transmission (T) coefficients versus the angle of incidence in degrees for light incident from medium 1 onto medium 2 for a relative refractive index m = 1.5.

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4 Optical Properties of Glasses

100

Brewster Critical Angle (degrees)

88

80

Brewster Angle Critical Angle

60

40

20

0 0.2

0.4

0.6 0.8 1 Relative Refractive Index m

3

5

Figure 4.6 Brewster and critical angles as a function of relative refractive index m.

goes to zero at the Brewster angle 𝜃 B , which from Eq. (4.7) and Snell’s law can be calculated to be tan 𝜃B = m

(4.8)

If the refractive index on the incident side is greater than that on the “transmitted” side of the interface, then the incident beam will be fully reflected from the interface if the angle of incidence exceeds the critical angle 𝜃 c where sin 𝜃c = m

(4.9)

In Figure 4.6, we show the Brewster and critical angles as a function of the relative refractive index. In many applications of glasses involving interfaces, these two angles play a crucial role. For example, most gas lasers use windows cut at the Brewster angle to eliminate reflection losses for one polarization. The critical angle is the defining parameter for the propagation of guided waves down fiber optic cables (strictly for cables whose diameter is much greater than the wavelength of light).

4.4 Dispersion The dispersion is the derivative of the refractive index with respect to wavelength, dn/d𝜆, and plays a critical role in the design of many optical instruments. A change in refractive index results in a change of focal length in lenses or deviation in prisms, and so a non-zero dispersion gives rise to chromatic aberrations (e.g. colored fringes of images) in optical systems. To describe the dispersion quantitatively, a figure of merit known as the Abbe number is generally used, and is given by: n −1 𝜈d = d (4.10) nF − nc

4.4 Dispersion

where nd is the refractive index at the helium d-line (587.6 nm), nF at the blue line of hydrogen (486.13 nm), and nC (656.27 nm) at the red line of hydrogen. We note that sometimes the sodium D-line at 589.6 nm is substituted for the helium line, in which case the Abbe number is denoted by vD . An alternative figure of merit, 𝜈 e , is based on the e-line of mercury (546.07 nm), and the F′ and C′ lines of cadmium (479.99 and 643.85 nm, respectively). The fraction (Eq. (4.10)) expresses the refracting power of the glass relative to the dispersion. The Abbe number is a basic parameter for optical instrument designers who have to deal with problems of chromatic aberrations, and is commonly presented as an “Abbe diagram,” where the Abbe number is plotted as a function of refractive index for different glass families. An Abbe diagram for oxide glasses, based on data for the Schott range [3], is shown in Figure 4.7. Usually, glasses with a large refractive index have high dispersion; it is the latter which dominates in Eq. (4.10), and so such glasses have a low Abbe number and appear in Figure 4.7 to the upper right. Glasses with small refractive indexes usually have low dispersion and high Abbe numbers, and so appear in the lower left of Figure 4.7. For achromatic lens design, a second parameter called the relative partial dispersion, Pg,F , defined as (ng – nF )/(nF – nC ), is also used to achieve a more comprehensive match of glass dispersion [3] for different optical elements. The refractive index ng is that at the blue line of mercury (435.83 nm). The Abbe number is an indirect measure of the first derivative of the refractive index, and the partial dispersion is an indirect measure of the second derivative. The refractive index nd , Abbe number 𝜈 d , and partial dispersion Pg,F have been historically used as a three-parameter summary of the refractive index and its dispersion, and are still widely used for comparing different glasses, but for analytical work it is advantageous to return to the refractive index itself. Glass manufacturers generally measure the refractive index at a variety of wavelengths corresponding to atomic 2

1.9

nd

1.8

BAF/BASF/BALF/KZF SF/F/LF LAF/LASF/LAFN LAK SK/SSK K/BAK/ZK PK/FK

1.7

1.6

1.5

1.4 90

80

70

60

50

40

30

20

vd

Figure 4.7 Abbe diagram for common optical glasses. Note that the Abbe number increases from right to left. The boundary at 𝜈 d = 50 divides crown glasses (left) from flint glasses (right).

89

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4 Optical Properties of Glasses

emission lines and fit the data to a curve, which represents the sum of classical damped oscillator responses to an electromagnetic field. Assuming several such oscillators are responsible for the absorption process near the absorption edge of a material, the refractive index is often written as: ∑ B j 𝜆2 (4.11) n2 − 1 = 2 2 j (𝜆 − 𝜆j ) with the expectation that the formula (known as the Sellmeier formula) would only be a good representation for n for wavelengths well away from the resonance wavelengths 𝜆j . In that case, Eq. (4.11) can also be expressed as an (even) series in inverse powers of 𝜆 by expanding the functions ( )2 ( )4 𝜆j 𝜆j 𝜆2 + −… (4.12) = 1 + 𝜆 𝜆 (𝜆2 − 𝜆2j ) so that, n2 − 1 = A + A1 𝜆2 + A2 𝜆−2 + A3 𝜆−4 + A4 𝜆−6 + A5 𝜆−8 + …

(4.13)

Here, the notation follows that used by Schott [3], and the 𝜆 term (with a negative coefficient) is added to improve the fit toward the IR region of the spectrum. Manufacturers will generally quote the coefficients Ak , or Bj and C j = (𝜆j )2 , so that the refractive index can be computed for any wavelength, or they may simply quote the values of n at several wavelengths. The accuracy of Eq. (4.11) for interpolating the refractive index in the visible region is good to within about 1 part in 105 for a six-parameter fit. Clearly, the Abbe number, being defined in terms of visible wavelengths, is of less relevance in the IR range, and so equivalent alternatives are used for IR-transmitting glasses. The refractive index and Abbe number are used in a common commercial numeric designation of optical glasses, which summarizes their properties in a six- or nine-digit code. For example, a glass described as 805 254 would have a refractive index nd of 1.805 and an Abbe number of 25.4. Schott appends the density in g cm−3 to give a nine-digit code. The chemical composition is indicated by a letter code, which commonly contains “K” for crown glasses or “F” for flint glasses. Historically, crown glasses with a low dispersion were based on SiO2 –CaO–Na2 O, while flint glasses were based on SiO2 –PbO. The current spectrum of glasses contains many other combinations of elements, but they are still labeled as “low-dispersion crown glasses” if the Abbe number is greater than 50, and “high-dispersion flint glasses” if the Abbe number is lower than 50. There are few systematics to the rest of the letter labeling system, and individual manufacturers have their own systems. Nevertheless, as an example, the common Schott BK-7 borosilicate glass is a crown K glass; here, the “B” indicates the boron content, whereas the “7” is a manufacturer’s running number. 2

4.5 Sensitivity of the Refractive Index 4.5.1

Temperature Dependence

The refractive index is a function of variables other than wavelength, notably temperature. From the Sellmeier formula, assuming only a single resonance in the uv region, the

4.5 Sensitivity of the Refractive Index

refractive index variation with temperature may be parameterized as: ) ( )( E n(𝜆, T0 )2 − 1 + 2E ΔT dn 0 1 = D0 + 2D1 ΔT + 3D2 ΔT 2 + dt 2n(𝜆, T0 ) 𝜆2 − 𝜆20

(4.14)

where T 0 is a reference temperature, and the D, E, and 𝜆0 values are experimentally determined parameters. The significance of the temperature dependence of the refractive index lies in the implications for precision refractive optics. It is also important in high-powered laser applications where defocusing can occur in glass materials that have a significant absorption coefficient at the laser wavelength, and so show substantial localized heating in the focal zone. It is also important for applications such as glass etalons where temperature-independent conditions for interference are desirable. The refractive index in a sample of bulk glass must be homogeneous for many optical applications, but inhomogeneities known as Striae can arise during the casting process, arising from convective flow and differential cooling rates. Entrapped gas bubbles, undissolved crystalline inclusions, platinum particles from the melt crucible, and crystals arising from mild devitrification all give rise to light scattering, which is also detrimental to glass performance. The catalogs of glass manufacturers can be consulted for specifications describing Striae and scattering. 4.5.2

Stress Dependence

The refractive index is a function of the stress on a glass sample. The stress may be of internal origin, such as, for example, after a poorly executed annealing process, or external. Sometimes, applied external stress is an undesired consequence of an experimental situation, as in windows for cryogenic apparatus, which are subject to stress from the thermal contraction of the support structure. The key attribute of stress is that it makes the material optically birefringent—that is, the refractive index depends upon the polarization of the optical electromagnetic (EM) field. Since a stress field is described by a tensor, the birefringence is a tensorial quantity; however, in the simple case of uniaxial stress 𝜎, the refractive indices for light polarized parallel and perpendicular to the stress direction may be written as n∥ = n +

dn∥ d𝜎

𝜎

(4.15)

and dn⟂ 𝜎 (4.16) d𝜎 The derivatives in Eqs. (4.15) and (4.16) are often referred to as the photo-elastic coefficients. An alternative pair of birefringence parameters can be defined in terms of strain and related to those discussed earlier [3] through the theory of elasticity. Birefringence polarization modulators based on exciting a block of silica glass into longitudinal resonant oscillation at ultrasonic frequencies are available commercially. A three-component resonator system that uses a feedback loop to keep the oscillation amplitude constant, based on the design first used for internal friction measurements by Robinson and Edgar [4], is used commercially in ellipsometers. n⟂ = n +

91

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4 Optical Properties of Glasses

4.5.3

Magnetic Field Dependence—The Faraday Effect

In the presence of a magnetic field, normally optically isotropic materials such as glasses become birefringent. The refractive index depends on the relative orientation of the magnetic field, the propagation direction for the light, and the polarization. The simplest geometry to analyze and the most important in practical applications is that where the propagation direction of linearly polarized light is parallel with the magnetic field, and in this case the manifestation of the effect is that the plane of polarization is rotated through an angle 𝜃, proportional to the field strength H, after traversing a path length s through the glass. This effect is known as the Faraday effect, after its discoverer Michael Faraday. The proportionality constant is called the Verdet constant, V , and is defined through 𝜃 = VsH

(4.17)

Plane polarized light may be decomposed into two counter-rotating circularly polarized waves traveling in the same direction. The Faraday effect arises through a difference in the refractive indices n+ and n− for the right-hand and left-hand rotating components, respectively. The relationship in Eq. (4.17) follows from the more fundamental definition ) 2c VH (4.18) 𝜔 The Faraday effect has both diamagnetic and paramagnetic contributions; the former is present in all glasses, but it is small, almost temperature independent, and characterized by a positive value for V , while the latter only occurs when the glass contains paramagnetic ions, is relatively large, negative, and has a large temperature coefficient. Both effects are strongly wavelength dependent. The diamagnetic effect arises from the Zeeman splitting of excited states. For example, the splitting of an excited state which is twofold degenerate results in a relative shift of the two associated transitions from the ground state; each transition is allowed only for one or other beam polarization. From Eq. (4.11), the value of 𝜆j for each transition is different, and so therefore are the refractive indices n+ and n− . The general form of the wavelength and temperature dependence of the Verdet constant is generally written [5] in terms of the frequency of the transition 𝜈, the number of ions per unit volume N, and the absolute temperature T, as: [ ] ∑ B(n, g) C(n, g) A(n, g) 2 V = 4𝜋N𝜈 + (4.19) + 2 2 2 2 (𝜈 2 − 𝜈n,g ) (𝜈 2 − 𝜈n,g ) T(𝜈 2 − 𝜈n,g ) n,g (

n− − n+ =

where the sum runs over all the states g of the ground multiplet and excited state levels n between which transitions are possible. The T −1 dependence for the C term reflects the temperature dependence of the level populations within the ground multiplet, which mirrors that of the Curie law for magnetic susceptibility, and is a reasonable approximation near room temperature. However, deviations from Curie’s law, arising from crystal field splittings in the ground multiplets, are often observed for rare-earth ions [6], and the same is also true for Verdet constants. In Figure 4.8, we show the (reciprocal) Verdet constant for a phosphate glass [5] plotted as a function of temperature; the relationship may be fitted with a Curie–Weiss law which shows that V varies with temperature T as (T + 65 K)−1 , not as T −1 . Van Vleck and Hebb [7] proposed that Faraday rotation should

4.5 Sensitivity of the Refractive Index

3.5

–1/V (Oe cm/min)

3 2.5 2 1.5 1 0.5 0

0

50

100

150 200 Temperature (K)

250

300

350

Figure 4.8 Reciprocal of the Verdet constant for Ce3+ ions in a phosphate glass using data taken from Borelli [5]. The line is a least squares fit to the Curie Weiss law, which gives a variation as (T + 65 K)−1 .

be proportional to magnetic susceptibility, implying that their temperature dependencies should be the same even if they are not Curie-like, but a test of this hypothesis by Edgar et al. [8] in the case of Ce3+ ions in fluoride glasses found significant deviations, which were explained in terms of an inappropriate assumption in Van Vleck and Hebb’s analysis—that the excited state splitting of the Ce3+ ions was relatively small. Nonetheless, the temperature dependence of the magnetic susceptibility is a good indicator of what to expect for the Faraday rotation. The experimental wavelength dependence of the Faraday rotation for undoped and rare-earth-doped ZBLAN fluorozirconate glass [8] is shown in Figure 4.9 and illustrates several aspects of Eq. (4.19). (Members of the fluorozirconate glass family are commonly labelled by an acronym comprising the chemical symbols for their constituent cations, so that ZBLAN glass comprises zirconium, barium, lanthanum, aluminum, and sodium fluorides). First, the glass itself has a small positive Verdet constant arising from 10 Verdet Constant (Rad/T.m)

Undoped 0 –10

1I

6

3P 3P0 1

1D

8% Pr 2

10% Ce

–20 –30 –40 400

450

500

550 600 650 Wavelength (nm)

700

750

800

Figure 4.9 Verdet constants for undoped and rare-earth-doped fluorozirconate glass recorded at room temperature as a function of wavelength. Source: Reproduced with permission from A. Edgar et al., J. Non-Cryst Solids, 231, 257 (1998).

93

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4 Optical Properties of Glasses

optical transitions that make up the uv edge near 200 nm. For Ce3+ doping, the Verdet constant is large and negative, and shows the smooth wavelength dependence predicted by the C term in Eq. (4.19) that arises from the intense 4d-5f Ce3+ transitions in the uv around 270–350 nm. Finally, for Pr3+ , the main effect is again from the same 4d-5f transitions, which are further toward the uv region, but there are also weaker 4f-4f transitions in the visible region that cause deviations from Eq. (4.19). Both the deviation and the absorption at those wavelengths must be considered in any application. Yamane and Asahari [9] give a useful compilation of Verdet constants for diamagnetic and paramagnetic glasses. Diamagnetic glasses generally have smaller Verdet constants than paramagnetic ones, but they offer the advantage for practical applications that they have no absorption lines across the visible region of interest, and the Faraday rotation is nearly temperature independent. Paramagnetic glasses can have much larger Verdet constants, but they have a significant temperature dependence, and there can be absorption lines or bands in the area of interest. Glasses doped with 3d ions show strong broad absorption, and so are of little interest as Faraday rotation materials. For the rare earths, trivalent cerium and divalent europium do not generally have any absorption in the visible region, and have intense 4d-5f transitions in the near uv region, and so give a large Verdet constant. Terbium has a large ground state g-value, resulting in large ground state splittings, and so also has a large Verdet constant, but with the complication of weak 4f absorption lines in the visible region. Glasses that can accommodate a large fraction of terbium display a correspondingly large Verdet constant—for example, Suzuki et al. [10] report a borate glass containing 60 mol% Tb2 O3 with a Verdet constant of −234 rad/Tm at 633 nm. The very high fraction of terbium was achieved through containerless processing to inhibit crystallization. However, for fiber applications, a lower terbium fraction is necessary to avoid crystallization during drawing, and so smaller Verdet constants result: for example, Sun et al. [11] describe an all-fiber magnetic field sensor system based on terbium-doped silicate glass fiber with a Verdet constant of −24.5 rad/Tm at 1053 nm. The Faraday effect finds applications in optical isolators, modulators, and in electric current sensing. In the latter application [12], a fiber optic sensor is typically used to measure the large current flowing in overhead power lines. A few turns of the fiber are wrapped around the cable, and the current detected through the surface magnetic field it generates; the advantage over competing electrical means of current sensing is the optical isolation provided by the fiber optic cable from the high voltage. 4.5.4

Chemical Perturbations—Molar Refractivity

It has been known for centuries that the refractive index of a glass increases when heavy elements are added to it—heavy in the sense of those having a high atomic number, such as lead, lanthanum, or barium. These elements are commonly found in flint glasses. The underlying relationship between the bulk property and the atomic character can be discussed using the Clausius–Mossotti (or Lorentz–Lorenz) relationship, which is derived in most standard texts on electromagnetism (e.g. Griffiths [13]) 1 ∑ n2 − 1 N𝛼 (4.20) = 2 n + 2 3 𝜀0 i i i

4.6 Glass Color

where n is the refractive index, 𝜀0 is the permittivity of free space, and N i is the density of atoms or ions of atomic polarizability 𝛼 𝜄 . If the glass is composed of compounds whose refractive indices can be measured separately, then the Clausius–Mossotti relation implies that the glass refractive index can be computed from the “molar refractivities” defined as ( 2 ) nj − 1 MWj (4.21) MRj = 𝜌j n2j + 2 where MW j is the molecular weight of the compound and 𝜌j is the density, so that the glass refractive index can be computed as 𝜌glass ∑ n2 − 1 f MRj (4.22) = 2 n + 2 MWglass j j where f j is the mole fraction of compound j in the glass. In practice, the ions or atoms in a glass do not have a unique polarizability; for example, for oxygen ions, the polarizability depends on whether they are in bridging or non-bridging positions. Nonetheless, Eq. (4.22) does give a useful first-order guide as to the effect of different chemical perturbations on the refractive index.

4.6 Glass Color Glasses may appear colored in transmitted light due to a variety of mechanisms, including absorption by transition metal ions, band-edge cut-off, and colloidal precipitates. Band edge coloration simply implies that the fundamental edge for absorption has moved into the visible region of the spectrum, and many glasses based on anions other than oxygen, such as sulfur, selenium, and tellurium, qualify in this regard. Transition metal ion doping by elements from the 3d series with their open valence shell electronic structure results in absorption bands in the visible and infra-red regions arising from 3d-3d electronic transitions. In principle, rare-earth ion doping can also give rise to coloration, but the absorption per ion for the usual 4f-4f transitions is much weaker and narrower than for 3d-3d transitions. 4.6.1

Coloration by Colloidal Metals and Semiconductors

It has long been known that the addition of minute quantities of metals such as Cu, Ag, and Au to a glass can, under certain preparation conditions, give rise to a strong coloration. The classic example is “ruby glass,” which is a silicate glass containing colloidal gold. The effect arises from the strong perturbation that small particles of metal make to the propagation of EM waves through the glass, due to the huge difference in the dielectric constants of the glass and metal. The effect is to scatter and absorb the light, with the balance between these two effects, and the wavelength and angular dependence (for scattering) depending on the particle size and the wavelength. The theoretical basis for discussing the subject is Mie scattering theory, as described, for example, by Van der Hulst [14], Kerker [15], or Bohren and Hoffman [16]. The latter authors, for example,

95

96

4 Optical Properties of Glasses

show that the absorption and scattering cross-sections are, respectively, given for small spherical particles of radius a by ] [ kV 27 Cabs = (4.23) 𝜖 ′′ 3 (𝜖 ′ + 2)2 + 𝜖 ′′ 2 and [ ] k4V 2 27 2 (4.24) Cscat = |𝜀 − 1| 18𝜋 (𝜖 ′ + 2)2 + 𝜖 ′′ 2 where k is the wavenumber in the glass, 𝜀 = 𝜀′ − j𝜀′′ is the dielectric constant of the particle relative to the glass, and V (= 4 /3 𝜋a3 ) is the particle volume. The extinction coefficient is given by 𝛼 = N(Cscat + Cabs ),

(4.25)

where N is the particle concentration, and the light intensity I after traversing a thickness z of glass is I = I0 exp(−𝛼z).

(4.26)

The relationships in Eqs. (4.23) and (4.24) are only valid for particles which are small relative to the wavelength, in the sense that |𝜀ka| < 6 have matrix elements which are identically zero within the 4fn configuration, and so they are usually omitted from Eq. (4.27). For crystals, some of the remaining Bl m parameters are necessarily zero as a consequence of the crystal symmetry, but for glasses there is no such constraint, and furthermore the ligand field varies from site to site. The ligand field is a small perturbation as compared to the spin orbit or electrostatic energies, and this, together with the site-to-site distribution in ligand fields, means that resolved ligand field levels within an [S, L, J] multiplet are not generally observed in glasses, unlike in crystals. The effect is generally just to broaden the

97

4 Optical Properties of Glasses

Figure 4.10 Energy level diagram for Pr3+ ions.

4f2 levels

40

Energy (104cm–1)

98

20

4fn–1 5d1 and/or charge transfer levels

1I 6

1S

0

3P 2 3P 0 1D 2

3P 1

1G 3F 3

4 3F 4 3F 2 3H 6

3H 5

0

3H 4

transitions associated with each [S, L, J] free ion level, albeit with some unresolved structure. Thus, the absorption spectrum of any particular rare-earth ion does not vary substantially from one glass to another, because it is the unresolved ligand field splittings which are glass-specific. The absorption is also quite weak. This is also a direct result of the shielding effect of the outer 5s2 , 5p6 shells, which has two consequences with regard to transition intensities. Firstly, electric dipole transitions within a pure 4f configuration are strictly forbidden by Laporte’s parity rule [19, 20]. However, the odd parity (l odd) components in Eq. (4.27) have a special role in breaking this selection rule. They have no matrix elements within states drawn from a 4f configuration, and so do not give rise in the first order to any crystal field splitting, but they do admix states from excited configurations with the opposite parity into the 4f states. To the extent that the states now contain a small admixture of opposite parity states, electric dipole transitions between the “impure” 4f states are now permitted. Thus, electric dipole transitions are weakly allowed through configuration mixing. (We note that magnetic dipole transitions are allowed within a 4f configuration, but they are approximately six orders of magnitude weaker than allowed electric dipole transitions, and so the dominant effect is usually the electric dipole transitions induced by configuration mixing, as discussed here.) The second key effect is that the electron-vibrational interaction between the 4f electronic states and the dynamic glass environment is also weak due to the outer shell shielding. In the case of the 3d ions, the electron phonon coupling to odd parity

ABSORPTION COEFFICIENT (cm–1)

4.6 Glass Color

5

3P

ZBLA:Pr3+ (2 mol.%) T = 300K

3F 3

2

4 3P ,1I 1 6

3 3F 2

3P

3F 4

0

2 1

3H

5

3H

1D 2

6 1G

0

4

10 15 WAVENUMBER (103 cm–1)

5

20

25

Figure 4.11 Optical absorption spectrum of Pr3+ ions in ZBLAN glass Source: Reproduced from J.L. Adam and W.A. Sibley, J. Non-Cryst. Solids, 76, 267 (1985) by permission of Elsevier.

vibrations results in allowed “vibronic” transitions, as explained in the next section, giving rise to broad vibronic sidebands which are more intense than the purely “zero-phonon” electronic transitions. For the rare earths, this effect is much weaker, and consequently the optical absorption is much less than for the same concentration of a 3d transition metal ion. In Figure 4.11, we show the experimental optical absorption spectrum for Pr3+ ions in ZBLAN glass [21] which can be compared with the energy level scheme shown in Figure 4.10. The transitions from the 3 H4 ground state are labeled by the final state. It is evident that the ligand field splittings of the multiplets cannot be resolved. The combination of the transitions to the 1 D2 state in the yellow, and the 3 P states in the blue, results in the glass having a light green tinge. 4.6.3

Absorption by 3d Metal Ions

The 3d series of metal ions are characterized by progressive filling of the 3d shell as shown in Table 4.1. The unshielded outer shell electrons interact strongly with the ligands and the glass environment, unlike the case for the rare-earth ions. The effect of the crystal field is greater than that of spin orbit coupling for the 3d ions, and so we consider the effects of the crystal field first. In a crystal, the ligand field has a well-defined symmetry which markedly reduces the number of parameters in the Hamiltonian in Eq. (4.27) (in addition, the summation only contains terms up to and including Table 4.1 Electronic configurations for ions of the 3d series. d1

d2

d3

d4

V2+ 3+

Ti

V

3+

3+

Cr

Mn4+

d5

Cr2+ 3+

Mn

Mn2+ 3+

Fe

d6

d7

d8

d9

Fe2+

Co2+

Ni2+

Cu2+

99

4 Optical Properties of Glasses

l = 4 since the matrix elements are identically zero for l > 4 within the states of a 3d configuration), and the powerful formalism of point group theory may be brought to bear on the problem. For glasses, there is no such symmetry, but nonetheless we find that dopant ions are typically found in environments which are a distribution about an “average” environment. For the purpose of discussion, let us suppose that the average nearest-neighbor environment is octahedral with six identical ligands. The symmetry of a perfect octahedron then limits the ligand field to have the form ( ) √ 5 4 ) (4.28) (C 4 + C−4 H = B4 C04 + 14 4 so that only one parameter is required. Tanabe and Sugano [22] have calculated the eigenvalues of the Hamiltonian (Eq. (4.28)) for the electrostatic and ligand field terms for this case, and for the similar cases of tetrahedral and cubic coordination, assuming particular values for the intra-shell coulombic interaction specified by the Racah parameters B and C [20]. Their plots of energy versus cubic crystal field are a standard aid in understanding 3dn absorption spectra. A modern account is given by Figgis and Hitchman [20]. Ligand field theory abounds in different normalizations, and Tanabe and Sugano use an equivalent but proportional parameter Dq to parameterize the ligand field rather than B4 . In Figure 4.12, we show the Tanabe–Sugano diagram for d7 configuration in an octahedral field for C/B = 4.63 [23]. The predicted absorption spectra, ignoring the effects of non-octahedral ligand field perturbations, spin-orbit coupling, and vibrational interactions, may be read off the diagram with the known typical Dq and B values for different ligand coordinations. B(MgF2) = 975 cm–1 B(ZBLA) = 920 cm–1

Co2+ 60 ZBLA

MgF2

86K 2A

50

2g

40 2F

E/B

100

2A

20

1g

4A

30

2g 4T 1g

2T 1g 2T 2g

2G

4P

4T 2g

10 2E

g 4T 1g

4F

0

1.0

0.5 Dq/B

1.5

Figure 4.12 Energy levels for the d7 configuration. Source: Reprinted from Y. Suzuki et al. Phys Rev. B, 35, 4472. (1987) by permission of the American Physical Society.

4.6 Glass Color

The implicit assumption here is that the crystal field is dominated by the ligands, but this should be a better approximation for glasses than for crystals due to the lack of long-range order in glasses. Fuxi [24] gives a table of estimated values of Dq and B for different ions in different glasses. It should be noted that the observed spectra are not the sharp lines which might be expected by running a vertical line up Figure 4.12 at the expected value of Dq/B. At the very least, the distribution in crystal fields is about the average, and the spin–orbit coupling splits and shifts term sub-levels, resulting in substantial inhomogeneous broadening of the order of hundreds of cm−1 . An additional and substantial homogeneous broadening effect for the 3d ions arises from the effect of strong electron–phonon coupling. The broadening effect is best explained through the “configuration coordinate” model (e.g. see Figgis and Hitchman [20]), which accounts for transitions that involve both electronic transitions and phonon excitations. This means that substantial absorption occurs on the high-energy side of the purely electronic transition, and in fact the model shows that the most probable transition is typically not the “zero-phonon” transition but one which corresponds to the creation of several phonons. The overall result is that the absorption band is approximately a broad Gaussian envelope, comprising many individual electron–phonon excitations, whose center is displaced to the high-energy side of the purely electronic transition. (The reverse occurs for fluorescence; the emission band lies on the low-energy side of the zero-phonon line.) The strength of the optical absorption for 3d ions is greater than that of 4f ions since the configuration admixture effect is larger (because the 3d outer shell is unshielded); in addition, odd-parity vibrations enhance the transition probability also through configuration admixture effects. A selection rule which is important in determining the relative strengths of electronic transitions for the 3d series is the spin selection rule that ΔS = 0, which arises because the electric dipole operator in the transition probability expression does not involve spin. However, spin–orbit mixing of pure LS terms means that such spin-forbidden transitions can occur, albeit with a reduced intensity. In Figure 4.13, we WAVELENGTH (nm) 1000 600

ABSORPTION COEFFICIENT (cm–1)

2000 1.5

MgF2:Co2+ 2.0 × 1020cm–3

1.0

4T

2g

4A

0.5 0 8 6 4

0

2A 1g

4T 1g

86K

2g 4A 2g

2

86K

2g

ZBLA:Co2+ 3.3 × 1020cm–3 4T

400 4T 1g

2A

1g

× 1/4 5

10 15 20 WAVE NUMBER (103cm–1)

25

Figure 4.13 Optical absorption spectrum of Co2+ in ZBLA glass and MgF2 . Source: Reprinted from Y. Suzuki et al. Phys Rev. B, 35, 4472 (1987) by permission of the American Physical Society.

101

102

4 Optical Properties of Glasses

show the observed absorption spectrum for Co2+ ions in a ZBLAN glass and MgF2 . The major spin-allowed transitions can be directly interpreted with the aid of Figure 4.12. The strong absorption in the red–green region leaves most Co-doped glasses with a deep blue coloration. Figures 4.11 and 4.13 show the typical behavior that rare-earth ions show narrower sets of absorption lines than the 3d transition metal ions. Figure 4.13 also shows the typical behavior of glasses vs crystals: that ligand field fine structure is frequently resolved in crystals but rarely in glasses. In summary, transition metal ions give rise to strong broad absorption bands in glasses. The best guide to the interpretation and prediction of spectra is a knowledge of the average local coordination together with the energy level diagrams of Tanabe and Sugano [22].

4.7 Fluorescence in Rare-Earth-Doped Glass Fluorescence in rare-earth-doped glass is the basis for many applications such as neodymium-doped glass lasers, up-converters, erbium-doped fiber amplifiers for telecommunications, and fiber lasers. Fluorescence can be excited by optical pumping via the parity-allowed transition into the 4fn-1 5d configuration or charge-transfer excited states lying in the ultraviolet region indicated in Figure 4.10. From there, relaxation to the excited sub-states of the 4fn configuration either via radiative or nonradiative (phonon-assisted) mechanisms can occur, followed by radiative (fluorescent) decay to the ground state either directly, or via a cascade through intermediate states. Alternatively, the fluorescence can be excited by pumping into the higher-lying states of the 4fn configuration directly, but this is a weaker process as it is parity forbidden. As with the absorption, the fine structure due to crystal field splittings of the free ion multiplets is not resolved; the glassy environment again gives rise to a distribution of crystal fields, and this, together with the small magnitude of the crystal field, gives rise to an inhomogeneous broadening of the lines rather than a splitting, although unresolved structure can sometimes be observed. The unresolved structure can be probed further using techniques such as fluorescent line narrowing (FLN), which rely on the power and narrow linewidth of modern lasers to isolate transitions due to a particular glass site from within a broad envelope. Since the ligand field splittings are rarely resolved for rare-earth ions in glasses, it is not usually possible to parameterize the effects with a small set of ligand field parameters as is typically done for rare-earth ions in crystalline solids. However, there is great interest in the relative magnitudes of the transition probabilities for the various possible fluorescent transitions, since these are essential inputs to any consideration of practical device performance. Judd [25] and Ofelt [26] have shown that it is possible to predict the intensity of any allowed transition from just three parameters, Ω2 , Ω4 , and Ω6 , which summarize the effect of admixtures of excited state configurations into the ground state, induced by odd-parity components of the ligand field. The electric dipole oscillator strength for absorption or emission is given by [22]: 8𝜋 2 m𝜐𝜒ed ∑ fed (a, b) = Ω |⟨a‖U t ‖b⟩|2 (4.29) 3h(2J + 1)n2 t=2.4.6 t

4.7 Fluorescence in Rare-Earth-Doped Glass

where J is the total angular momentum of the initial level, n is the refractive index of the glass, 𝜈 is the frequency of the transition, and 𝜒 ed = n(n2 + 2)2 /9 is a local field correction for electric dipole transitions. The quantities U 2 , U 4 , U 6 are double-reduced matrix elements of the unit tensor operator U t that are tabulated, for example, by Carnall et al. [27]. From the oscillator strengths, the radiative lifetime can be calculated as the probability per unit time for the decay as: 8𝜋 2 𝜐2 e2 n2 fed (a, b). (4.30) mc3 The Judd–Ofelt parameters Ωt thus serve as a three-parameter base from which all the electric dipole transition probabilities [L, S, J] ⇒ [L′ , S′ , J ′ ] within a given configuration may be calculated, albeit for the entire multiplet rather than individual ligand field levels; but, since these are usually unresolved, this is not an important limitation. The parameters Ωt are specific to a given configuration and given glass host, since they parameterize the electron–ligand interaction. They can themselves be estimated from the absorption spectrum. In principle, measurements of the absorption cross-section for just three transitions suffice; but, in practice, least squares fitting to measurements from several lines is usually performed. The measurements yield the absorption cross-section 𝜎 abs , which is related to the absorption oscillator strength by: A=

f =

mc 𝜎 (𝜈)d𝜈 𝜋e2 ∫ abs

(4.31)

For transitions that have a significant magnetic dipole character, the oscillator strength for magnetic dipole allowed transitions is given by: fmd =

h𝜈𝜒md |⟨a‖L + 2S‖b⟩|2 6(2J + 1)n2 mc2

(4.32)

where 𝜒 md = n3 is the local field correction for magnetic dipole transitions. The magnetic dipole oscillator strength should first be subtracted from the observed oscillator strength f abs to yield f ed = f abs − f md for use in determining the Judd–Ofelt parameters from Eq. (4.29). Tables of Judd–Ofelt parameters for rare-earth ions in heavy-metal fluoride glasses have been given by Quimby [28], and for Nd3+ and Er3+ ions in a variety of glasses by Fuxi [24]. While this discussion has focused on the radiative process of de-excitation, one should be aware that there are parallel non-radiative processes involving energy migration between the rare-earth ions (in sufficiently concentrated systems) and ubiquitous phonon–assisted decay, which act to reduce the lifetime estimated from purely radiative considerations. In Figure 4.14, we show the fluorescence from Pr3+ ions in ZBLA glass. This spectrum was generated by pumping from the ground state into the 3 P2 state. It is noticeable that the low-temperature spectrum shows slightly narrower bands, but the residual inhomogeneous ligand field broadening still prevents resolution of any ligand field splittings. Fluorescence from 3d ions is infrequently observed in glasses, with the notable exception of the 3d3 (eg Cr3+ ) and 3d5 (eg Mn2+ ) ions.

103

4 Optical Properties of Glasses

ZBLA:Pr3+ (2 mol.%) λexc = 441 nm

6

T = 297K

Figure 4.14 Luminescence spectrum for Pr3+ ions in ZBLA glass. Source: Reproduced from J.L. Adam and W.A. Sibley, J. Non-Cryst. Solids, 76, 267 (1985) by permission of Elsevier.

4 LUMINESCENCE INTENSITY (arb.units)

104

2

3F

4

3F

3

3H

6

3H

5

0 3H 6

30 3F

T = 13K 2

20

3H 3F

3

3F

4

5

3H

4

10

0 11

13 15 17 19 21 PHOTON ENERGY (103 cm–1)

4.8 Glasses for Fiber Optics The glasses used to make fiber optics for long-distance communications are necessarily characterized by a low extinction coefficient in the transmission window, so low that, in the case of silica, it is too small to appear in Figure 4.1. As a practical engineering matter, the attenuation is measured in units of dB/km rather than as an extinction coefficient in cm−1 , with the relation between the two being ( ) ) ( P(L) 1 dB = log10 = 43429𝛼 (4.33) attenuation per unit length km L P(0) where P(x) is the power at distance x from the origin of a fiber optic, and the numerical value is for the attenuation coefficient 𝛼 in units of cm−1 . Practical values of the minimum attenuation for silica are in the region of 0.2 dB/km or 5 × 10−6 cm−1 . This implies that the intensity is reduced by a factor of 10 after about 50 km of cable. The attenuation in the window region for fiber optic glasses contains contributions from several effects, which are shown schematically in Figure 4.15. The so-called “V-curve” [29] is defined by electronic edge absorption and Rayleigh scattering on the short wavelength side of the minimum, and multi-phonon absorption on the long wavelength side; these effects can be described semi-empirically by [30]: 𝛼 = A exp(−a∕𝜆) +

B + C exp(c∕𝜆) 𝜆4

(4.34)

4.8 Glasses for Fiber Optics

Extinction Coefficient (cm–1)

1

0.1

0.01

Extrinsic Absorption Multiphonon (eg OH–)

0.001

0.0001

10–5 0.1

Electronic absorption

Rayleigh Scattering 1 Wavelength (µm)

10

Figure 4.15 Schematic diagram for the spectral dependence of extinction coefficient of silica.

The first term of Eq. (4.34) describes the absorption associated with the band edge. In crystalline solids, the absorption associated with electronic band gap transition should, in principle, commence abruptly when the photon energy first exceeds the band gap energy. However, there is a weak tail (the “Urbach Tail”) extending into the band gap zone that is thought to arise principally from phonon-assisted electronic transitions, and from localized defect-related electronic states. The involvement of phonons relaxes the electron wavevector selection rules that otherwise confine transitions to be vertical on the usual energy band diagram. In glasses, the structural disorder means that the concept of an abrupt onset of electronic states must be abandoned, and there is a continuous transition from high-mobility bulk states, which lie well within the effective band, to localized low-mobility states, which lie in the nominal band gap. It is these latter states, together with phonon-assisted transitions, which give rise to the Urbach tail in glasses. However, this contribution is usually dominated, at least in wide-gap materials, closer to the V-curve minimum by the Rayleigh scattering arising from chemical and structural perturbations on a molecular scale in the glass. We can expect from Eq. (4.24) (specifically in the form of Eq. (4.40)) that this effect will scale as inverse of the fourth power of the wavelength, as the square of the refractive index deviation from the average, and as the volume of the micro-regions that show refractive index deviation. On the long-wavelength side of the V-curve, the attenuation is determined by multiple phonon absorption. In a crystal, such transitions are permitted through anharmonicity of the lattice vibrational potential energy, giving rise to a long-wavelength tail in the absorption associated with the infrared absorption bands. For different materials, the position of the infrared bands shifts, depending on the atomic masses involved. In a simple estimation, the frequency of the IR bands is given by 𝜔20 = K∕𝜇

(4.35)

105

106

4 Optical Properties of Glasses

where K is the force constant and 𝜇 is the reduced mass for the vibration. Thus, for glasses made from higher-mass atoms and/or with weaker bonds, we expect that the IR bands will move to longer wavelengths (lower frequencies), and that the multi-phonon edge will move with them. Thus, the V curve for heavy-atom glasses moves to longer wavelengths along with the IR bands, and so we observe minima at longer wavelengths for the IR-transmitting heavy-metal fluoride and chalcogenide glasses. Since the Rayleigh scattering contribution is not directly dependent on the atomic mass, the effect is that the V-curve minima are expected to be deeper for these glasses than for silica. So far, this potential for lower fiber-optic attenuation than for silica has remained just that, due to extrinsic scattering by impurities and inclusions. For the heavy-metal fluoride glasses, residual transition metal ion concentrations, particularly of Fe2+ , Co2+ , and Cu2+ , give rise to electronic transitions near the V-curve minima, and there are also wavelength-independent scattering contributions from metallic inclusions such as Pt particles originating from the crucible used for glass melting, and from inadvertent crystallization in the fiber drawing process. All of these dominate the contributions in Eq. (4.34) near the V-curve minimum, and so silica retains its position as the fiber optic glass of choice. Even for silica, there is the well-known hydroxyl impurity vibrational overtone near 1.39 μm (with weaker overtones at 1.25 and 0.95 μm), shown schematically in Figure 4.15, which lies very close to the V-curve minimum at 1.55 μm, so that, in practice, it is the two minima on either side of this impurity absorption which are of practical use, resulting in the two telecommunications windows 1.2–1.3 μm and 1.55–1.6 μm. While the operation of a fiber optic cable at the wavelength of minimum attenuation is clearly desirable for long-distance communication, it is also important that the dispersion be minimal, since otherwise distortion of a pulse shape occurs in the time domain, limiting the maximum bit rate. Since the physical factors that determine the position of the V-curve minimum and the wavelength for zero dispersion are quite different, there is no reason to expect the wavelengths of minimum extinction and zero dispersion to be the same. If we use the Sellmeier formula with just two terms to express the refractive index due to one electronic resonance in the uv region and a vibrational resonance in the infrared as B e 𝜆2 B v 𝜆2 − (4.36) n2 − 1 = (𝜆2 − 𝜆2e ) (𝜆2 − 𝜆2v ) then we find, by computing the dispersion dn/d𝜆, that there is a zero in the dispersion at approximately ( )1∕4 Be 𝜆2e 𝜆2v (4.37) 𝜆= Bv In the case of silica fiber, this zero is at 1.27 μm as compared to the extinction minimum at 1.55 μm.

4.9 Refractive Index Engineering By systematically manipulating the refractive index within a volume of glass, it is possible to fabricate devices that show useful optical properties. The classic example is the

4.9 Refractive Index Engineering

optical fiber with a cladding that has a lower refractive index than the core, giving a structure which guides light rays within the core by total internal reflection. In this case, the refractive index variation is achieved, in concept at least, by simple fusion of a cylinder of core material and a pipe of cladding material, followed by stretching. The most common fiber-optic cables for telecommunications are of the so-called “single mode” construction where the core is of such small diameter than only a single mode can propagate, which eliminates modal dispersion and thus optimizes bit rates. The success of fiber optics has generated interest in extending the manipulation of refractive indices to other situations. GRadient INdex (GRIN) fibers, where the refractive index is engineered to vary radially with a gradual rather than a step transition, have improved transmission properties as compared to large-core multi-mode step fibers with regard to modal dispersion. The gradation is generally on the scale of microns, but it is also possible to produce index gradients over a larger size scale of millimeters, giving rise to a family of GRIN optical elements [31]. For example, the simplest GRIN device is the rod lens, essentially a scaled-up version of the graded index fiber, in which a radial index gradient results in the periodic focusing of light along the axis of the rod. GRIN lenses are typically a few millimeters in diameter, with focal lengths that can be of the same scale, and are used as relay lenses or in endoscopes. One advantage of GRIN over conventional lenses is that they have planar faces, which assists in fabrication and systems integration. Other applications include fiber optic couplers and imaging, where arrays of GRIN micro-lenses can be used to image large objects at short object-detector distances [9]. The index profile can be produced in a variety of ways. In the ion exchange method, a borosilicate glass containing a highly polarizable ion such as thallium or cesium is immersed in a bath of a molten salt containing an alkali ion of lower polarizability—for example, sodium or potassium [9]. Ion exchange results in a change in the refractive index of the glass, which can be understood through the Clausius–Mosotti relation (Eq. (4.20)) as a consequence of the replacement of an ion of one polarizability with another. The radial variation in refractive index directly reflects the ionic diffusion profile. Thallium salts are toxic, and so thallium-free processes (e.g. Hornschuh et al. [32]) based on exchanges between the ions Li, K, Na, and Ag have been developed. The ion exchange process can also be used to generate thin planar waveguide structures on glass surfaces that are more suited to opto-electronic integration than conventional fibers. Photolithography can be used to define waveguide geometry. GRIN structures have also been produced in polymers, with the advantage that larger diameters are possible; for example, Koike et al. [33] have made GRIN contact lens structures from polymer GRIN material. Spatial variations of the refractive index on a much smaller size scale are generated in the inscription of Bragg gratings [34] into optical fibers. The motivation here is to produce a periodic variation in refractive index, that is, a grating, along the fiber length that can act as an interference filter at wavelengths related in the usual way to the grating period. In Figure 4.16, we show the reflectivity of a Bragg grating inscribed in silica [35]. Such filters find widespread use in wavelength-dependent multiplexing in fiber optic communications, and in fiber optic sensors [36]. In the limiting case, the grating acts as a mirror, and so can be used as an in-situ reflector for fiber lasers. The grating is usually generated by an interference pattern from a high-intensity uv laser (typically at 244 nm from a continuous wave (CW) frequency-doubled 488 nm argon ion laser), and

107

4 Optical Properties of Glasses

20 REFLECTIVITY (%)

108

16

BRAGG GRATING RESPONSE ANDREW D-TYPE FIBER L = 0.9 mm E = 200 mJ/cm2/pulse T = 20 min at 50 pps

16% REFLECTIVITY

12 8 4

FABRICATION BY CONTACT PHOTOLITHOGRAPHY Using 1.06 μm Pitch Zero-Order Nulled Phase Mask and KrF Excimer Laser (249 nm)

0 1525

1527

1529 1531 WAVELENGTH (nm)

1533

1535

Figure 4.16 Reflectivity of a Bragg grating inscribed in a silica optical fiber. Source: Reproduced from K.O. Hill et al., Appl. Phys.Lett., 62 1035 (1993), by permission of the American Institute of Physics.

the interference pattern produces a permanent or temporary refractive index change in the glass at the positions of the interference antinodes. The interference patterns can be generated using a phase mask [35], or by an arrangement such as a Lloyd’s mirror configuration. In the common silica-based fiber, the photosensitivity is established by germanium doping and enhanced by hydrogen loading [34]. Refractive index changes of the order of 10−4 –10−2 can be produced. The germanium doping introduces an absorption band at 242 nm, and it is clear that the basic interaction with the uv light is through the resulting breaking and reconstruction of Ge–O bonds. But the exact mechanism of refractive index change is still a matter of discussion, with two interpretations. One is that a periodic densification effect is responsible, with the refractive index being periodically modulated either directly due to the changed particle density (Eq. (4.20)), or indirectly, for example, by the resulting periodic core-cladding stress modulating the refractive index through the photo-elastic effect [37]. The other model involves microscopic defects with changed atomic polarizability such as color centers [38]. Bragg gratings have been observed in other glasses (e.g. Ce-doped ZBLAN), but the mechanisms can be expected to be quite different from those proposed for silica. However, the best prospects for refractive index engineering come with the micro-machining technique [39]. In this method, a high-energy and high-repetition-rate femto-second laser (e.g. Ti: Saphire) or oscillator is focused via a microscope objective onto a micrometer-scaled volume inside a material that is normally transparent at the laser wavelength. Within the focal volume, the intensity is so high that electron-hole pairs are generated through non-linear interactions, and the resulting bond disruptions can lead to permanent refractive index changes of the order of 10−3 through mechanisms that are probably similar to those applied to Bragg gratings. (Although we note that color centers have been discounted by Streltsov and Borelli [40] in the case of silica and borosilicate glass.) By scanning the focused spot parallel to the surface, and also normal to the surface (to an extent limited by the focal length of the microscope objective), three-dimensional structures can be written in the glass. If the change in refractive index is positive, these structures act as light waveguides.

4.10 Glass and Glass–Fiber Lasers and Amplifiers

1.518

n

1.516

1.514

1.512 –20

–10

0 x (μm)

10

20

Figure 4.17 Variation of refractive index across the width of light waveguide produced by laser micromachining. Source: Courtesy of Professor Mazur.

If the energy dumped into the focal spot is increased, the concentration of electronhole-pair multiplies in an avalanche process. Their energy is transferred to the lattice through the electron–phonon interaction, and rapid, highly localized heating occurs. The material within the focal volume is essentially rendered into a plasma that first expands and then cools, leaving behind a permanent structural modification of approximately spherical geometry in which the refractive index varies radially [41]. The refractive index is smaller at the core than at the outer perimeter, which has been radially compressed, and so is just the opposite to that described earlier. In front of and beyond the focal volume, the electric field strength is inadequate to trigger the effect. By tuning the energy, focusing, and repetition rate of the laser or oscillator, the diameter of the microsphere and the refractive index can be adjusted. Again, by scanning the material perpendicular to the beam with an XY stage, or along the axis of the beam, or all of these, two- or three-dimensional light-guiding structures can be machined into the glass. In Figure 4.17, we show how the refractive index varies over the width of a simple linear waveguide that has a tubular structure [42]. The light guiding is achieved within the walls of the tubular structure. The technique is of commercial interest for micro-machining in a range of transparent solids; multiplexers, Bragg gratings, and filters have all been fabricated using this technique. The technique can also be used for write-once mass storage [43]. Schaffer has recently reviewed femtosecond micromachining [44].

4.10 Glass and Glass–Fiber Lasers and Amplifiers Lasers and optical amplifiers based on crystalline materials find widespread applications in modern society. Many, though not all, are based on the spectroscopic properties of rare-earth ions in crystalline hosts—for example, the high-powered Nd:YAG laser.

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Since the energy level structure of rare-earth ions are relatively insensitive to the local atomic environment, it is not surprising that rare-earth-doped glass is also widely used as a lasing and optically amplifying medium, with the advantages of ease and versatility of fabrication, for example, into fibers. Long lengths of amplifying medium are also possible, which leads to high gains and efficiencies. Fiber lasers and optical amplifiers, in particular, have undergone extensive development in the past two decades, driven by the telecommunications and materials processing industries. Lasers and amplifiers based on glass rods or plates, rather than fibers, find more limited applications, such as the high-powered Nd:phosphate glass amplifiers operating at 1.053 μm used at the National Ignition Facility in the USA. A number of monographs [45–47] have appeared that describe the basis and operation of fiber lasers and amplifiers in particular, so only an overview will be given here. Optical amplifiers are required for long-distance fiber optic communication to restore signal levels after propagation over some tens or hundreds of kilometers. A typical system in silica-based fiber operates at the wavelength corresponding to the dip in attenuation near 1.55 μm (see Figure 4.15), and so any amplifying element should preferably show narrow-band amplification centered around this wavelength. Fortunately, trivalent erbium shows a fluorescent transition in the infrared between the excited 4 I13/2 multiplet and the ground 4 I15/2 multiplet that is very close to this wavelength, and the levels are very suited to three or four level amplification (depending on the temperature) by stimulated emission. At room temperature, three-level amplification is the basis for the so-called erbium-doped fiber amplifier (EDFA), where the amplifying medium is the erbium-doped core of a silica fiber. The amplifier is usually pumped by an InGaAs laser diode operating at 980 nm, which excites the Er3+ ions into the 4 I11/2 levels, from which the 4 I13/2 multiplet is populated by radiative or non-radiative decay. However, supplementary and direct pumping into the 4 I13/2 multiplet using 1480 nm light from a InGaSP diode laser is also frequently used. The emission spectrum for the resulting 4 I13/2 to 4 I15/2 fluorescence transition, which is the basis for optical amplification, is broadened to around 30 nm by inhomogeneous (crystal field) and homogeneous (phonon) effects, which permits wavelength-division multiplexing within the bandwidth, and hence optimal bit rates. EDFA amplifiers can have gains of over 50 dB and output powers of the order of hundreds of watts, and are essential components of any long-distance fiber optic communications network. Nonetheless, while silica is an excellent material for optical transmission, it is in many ways a sub-optimal host for rare-earth ions and optical amplification—the solubility of rare-earth ions is very low in this material and so only a small atomic fraction of dopant ions can be incorporated. Secondly, the phonon frequencies associated with the SiO2 network are very high, which means that few phonons are required to bridge any energy gap with correspondingly fast non-radiative relaxation processes that bypass radiative relaxation, reducing radiative efficiency. Consequently, other ions such as Pr3+ (which would otherwise be a suitable material for amplification in the 1.3 μm window [Figure 4.15]) are primarily used in other host glasses such as ZBLAN or tellurite glasses, where the low phonon frequencies corresponding to heavy host ions mean that radiative efficiencies are much higher. However, the difficulties of splicing different fibers together and the different pump wavelengths required mean that the silica-based EDFA amplifier remains dominant in telecommunications systems.

4.11 Valence Change Glasses

Turning to optical fiber lasers, these can also be constructed from rare-earth-doped glasses, but now with feedback typically provided by UV-inscribed Bragg gratings at either end of the fiber. A wide variety of wavelengths from the visible to the infrared are available by choosing different combinations of rare earths and transitions, and matching the Bragg grating period to the transition of interest. The long path lengths available with fibers results in high gains, and power levels of tens of watts in single-mode CW can be attained, and hundreds of watts in multimode operation. This leads to applications ranging from optical sources in communications and illumination, to welding and cutting, including surgical procedures. High-power infrared fiber lasers based on Yb3+ (∼1 μm) or Tm3+ (∼2 μm) find widespread use in industrial processing. The choice of host glass is governed by the same issues as for optical amplifiers: transparency in the visible or infrared, rare-earth solubility, non-radiative relaxation rates, and energy migration. Silicate and phosphate glasses have a higher solubility for rare-earth ions, while fluoride glasses such as ZBLAN offer, in addition, excellent transparency in the infrared. A second type of amplifier or fiber laser based on stimulated Raman scattering (SRS) rather than simulated emission has also been developed [48]. These devices operate on the Raman scattering sidebands of a pump laser and so depend upon the vibrational spectrum of the glass host. Although not as efficient, SRS fiber devices have the advantage that the operating wavelength is not limited by the energy level structure of the rare-earth ion in the simulated emission scheme. Rather, the emission wavelength is just that of the pump displaced by the Raman shift to the peak of the Raman sideband spectrum, or to a wavelength within the sideband determined by the Bragg grating reflectors. Thus, the emission or amplifying wavelength is determined primarily by the pump wavelength, and is not fixed by a rare-earth transition energy, a considerable advantage in some practical applications. In addition, SRS devices can be cascaded so that the output of one acts as the input to the next, so that multiple upshifts in wavelength can be achieved. Alternatively, multiple Bragg gratings within a single fiber construction can be used to produce the same effect. It is also possible to engineer the device so that multiple wavelength-shifted outputs are available with a comb-like output spectrum. Thus, SRS devices are particularly useful in wavelength division multiplexing technologies. The glass types that are used are typically silica with modified phonon spectra through high phosphorus or germanium doping [48]. Phosphorus doping is used with Bragg reflectors that select out the phonon peak at 13.2 THz, resulting, for example, in a pump laser wavelength at 1117 nm producing gain or an output at 1175 nm. In the case of Ge doping, a secondary peak in the phonon spectrum at 40 GHz is observed, so that the same pump wavelength would result in an output at 1313 nm. By fabricating the Bragg grating period appropriately, the SRS for this wavelength shift can be selected rather than on the main peak. This has the advantage that fewer cascaded units are necessary to achieve a given wavelength shift from the pump.

4.11 Valence Change Glasses Some rare-earth ions and transition metal ions can occur in different valence states. In the rare-earth series, europium and samarium can occur either as the trivalent or the divalent ion, depending on the host glass. For a given glass, it is possible to achieve

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partial or complete conversion from one form to another by oxidation or reduction processes, which are in general irreversible, or by irradiation with either uv or X-rays, which can in some cases be reversed by irradiation with a different wavelength, or thermally. These processes give rise to a useful method of imaging for X-ray or uv radiography. For example, samarium-doped fluorophosphate or fluoroaluminate glass as prepared in argon [49] shows exclusively the Sm3+ valence state in the fluorescence spectrum (Figure 4.18). Sm3+ 1.0 X-ray Intensity, a.u.

(a)

2 sec 0.8 0.6 0.4

1.0 0.8 0.6 0.4 0.2 0.0

0.2

30 40 50 60 70 80 hv, keV

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0.4 0.2 0.0 1.0

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550

600

650 700 750 Wavelength, nm

800

Absorbance, arb. un.

(b) Photoluminescence, arb. un.

112

0.0 850

Figure 4.18 The evolution of PL spectra of samarium-doped fluorophosphate glass under synchrotron X-ray irradiation. The irradiations times are (a) 2 s, (b) 50 s, and (c) 2000 s, respectively. Experimental data are shown by symbols. Emissions of Sm3+ and Sm2+ ions are shown by thin solid lines. The inset in (a) shows the spectrum of the X-ray irradiation. The broken line in (c) represents the X-ray-induced absorbance. Source: Reproduced from Okada et al., Applied Physics Letters 2011, 99, 121 105–8 (2011) by permission of the American Institute of Physics.

4.11 Valence Change Glasses

Irradiation with X-rays results in the partial conversion of Sm3+ to Sm2+ , as shown in the figure. By exciting the Sm2+ fluorescence selectively, and with appropriate filtration preceding the detection device, it is possible to record only the induced Sm2+ fluorescence. The intensity of the fluorescence directly reflects the incident X-ray dose. Thus, the material can be used for X-ray dosimetry, or for X-ray imaging. If an object is placed between an X-ray source and a flat plate made of Sm-doped fluorophosphate glass, the post-irradiation spatial variation of Sm2+ intensity displays the shadow or radiographic image of the object of interest. The stored image is stable for periods of weeks or more, but can be erased either with uv light or by heating. This radiation-induced valence change effect is potentially very useful for checking X-ray beam profiles in a new technique for the treatment of cancer, known as microbeam radiation therapy (MRT). In MRT, the X-ray beam passes through a venetian-blind structured collimator, so that the emerging X-rays are in the form of thin slices, some 50 μm thick, separated by typically 400 μm. It is critically important for the safety of the patient and the treatment efficacy that the resulting beam intensity cross-section closely follow the thickness profile of the collimator. In a beam quality control check, a sheet of the glass is placed behind the collimator, and, after exposure, the spatial variation of the Sm2+ fluorescence intensity is recorded using the a high-resolution scanning technique based on confocal microscopy, which can record the variation on a micron scale. Figure 4.19 shows the read-out from the glass after a 20 kGy exposure. The rare-earth-doped glass is thus a very viable 2-d dosimeter material for MRT beam quality assurance. Competing optical methods based on radiation-sensitive powders or polymers cannot achieve this because of light scattering in the detector, while single crystal methods are prohibitively expensive due to the cost of large area detectors. 1.6 X-rays

R(300 s), arb.un.

1.4 1.2

Air W

1.0

Glass

0.8 0.6 0.4 0.2 0.0 100

200

300

400 500 600 Distance, µm

700

800

Figure 4.19 The Sm2+ /Sm3+ conversion pattern induced by X-rays through a microslit collimator and read-out by photoluminescence confocal microscopy as the ratio R(300 s) = PL(Sm2+ )/PL(Sm3+ ). The line is a guide to eye. The inset shows the geometry of the experiment. The irradiation time is 5 min, which corresponds to ∼20 Gy. Source: Reproduced from Okada et al., Applied Physics Letters 99, 121 105 (2011) by permission of the American Institute of Physics.

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Figure 4.20 X-ray image of a semiconductor IC recorded in Sm-doped fluorophosphate glass. The image shows the square semiconductor IC, the heat sink adhesive, the surrounding track pads, and the fine bonding wires linking the pads to the IC. The white marks are scattering from scratches in the glass surface.

The X-ray doses required for a readily measurable effect with the Sm-doped fluorophosphate or fluoroaluminate glass are quite high, which is appropriate for MBT, but inadequate for routine medical radiography. However, for industrial radiography, the dose is not a problem, and very-high-resolution images may be obtained with quite simple equipment such as a professional-standard digital camera. Figure 4.20 shows an image of a semiconductor IC recorded with an Sm-doped fluorophosphate glass using a charge-coupled device (CCD)–based digital camera (intended for astronomy) equipped with suitable filters. The 25-μm-thick bonding wires leading to the IC are clearly visible. The active process in the fluorophosphate glass is deduced to be one involving ionization of the phosphate bonding network, with subsequent trapping of the electron at the Sm3+ ion: PO → POHC + e− Sm3+ + e− → Sm2+ where PO represents a phosphorus–oxygen bond in the un-irradiated glass, POHC (phosphorous-oxygen hole centre) represents a hole in that bond, and e− is the photo-ejected electron. The POHC gives rise to color center absorption, which partially overlaps the Sm3+ and Sm2+ emission bands, as seen in Figure 4.18, but this overlap can be compensated for in any measurement. In the fluoroaluminate glass, the degree of overlap with the equivalent color center bands is much less of a problem. Valence conversion after uv irradiation has been observed for a number of other polyvalent ions in the 3d, 4d, and 5d series [50].

4.12 Transparent Glass Ceramics 4.12.1

Introduction

Glass ceramics comprise glassy matrices containing crystallites, which may range in size from nanometers to microns, and in volume fractions up to several tens of percent. They

4.12 Transparent Glass Ceramics

are produced by thermally controlled nucleation and growth of crystals, often with the aid of a nucleating agent, from a pre-quenched glass. The crystals can impart improved mechanical properties to the host glass through their inhibition of crack propagation. In addition, the balance between the thermal expansion coefficients of the crystals (sometimes negative) and the host glass permits adjustment of the composite expansion coefficient, and it is possible to design materials such as the Schott product Zerodur , which has zero expansion coefficient at room temperature. The outstanding mechanical and thermal properties of these glass ceramics find applications in areas as widespread as artificial joints, machineable ceramics, magnetic disk substrates, and telescope mirror blanks. However, the major interest in the present context is in “nanophase” or “nanocrystalline” glass ceramics, where crystals of size ≤ 50 nm are uniformly dispersed in a transparent host glass, resulting (for favorable conditions, discussed in the following text) in a transparent glass ceramic. Beall and Pinckney [51] give a review of these materials. Generally, the crystals are themselves transparent, but it is worth pointing out that the first optical application for nanophase glass ceramics was based on metallic crystallites. Stookey et al. [18] pioneered the work on photochromic glasses where the reversible darkening and/or coloration in silver- and copper-doped borosilicate glasses is based on photochemical reactions involving silver and copper ions, and colloidal silver particles [3]. In this case, the optical effects are due to Mie scattering from the metallic particles described earlier (Section 4.6.1). The first major application of glass ceramics containing transparent crystals evolved from the earlier development of low-thermal-expansion coefficient glass ceramics containing “stuffed” 𝛽-quartz when it was found that these could be made in transparent form if the particle size was sufficiently small. The term “stuffed” here refers to 𝛽-quartz, where some of the Si4+ ions have been replaced by Al3+ ions and monovalent cations such as Li+ ; in the limit, this is the crystal 𝛽-eucryptite LiAlSiO4 . This immediately gives rise to the possibility of a transparent material able to survive high temperature without cracking, and this type of transparent glass ceramic now finds applications in fire-doors, stove tops, and cookware. However, in the past 20 years, there has been growing interest in transparent glass ceramics for photonic applications. These materials combine the ease of formation and fabrication of a glass with the desirable properties of a crystalline environment for photo-active dopant ions, such as those from the rare-earth series. Critical to most applications is the transparency of the glass, since in general each crystallite will act as a scattering center, and, in applications such a lasers and optical amplifiers, it is absolutely critical that any mechanism (scattering, absorption, etc.) that removes light intensity from the beam be minimal. In 1993, Wang and Ohwaki [52] triggered the current interest when they reported that a glass ceramic comprising Er3+ - and Yb3+ -doped fluoride crystals in an oxyfluoride glass was as transparent as the precursor glass, and showed 100 times the up-conversion efficiency when pumped at 0.97 μm, a result comparable with the best achieved in single crystals. The potential advantages of this new class of photonic material were appreciated by Tick et al. [53], who adapted the material for Pr3+ amplifier operation at 1.31 μm by replacing Yb with Y and Zn fluorides, and the Er3+ with Pr3+ . They showed that the glass ceramic had a greater quantum efficiency than the competing Pr3+ -doped ZBLAN fluoride glass, and remarkably low scattering, raising the potential of this material as a fiber amplifier. In a later paper, Tick [54]

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went further to suggest that, under appropriate conditions of particle size and volume fraction, glass ceramics could be ultra-transparent—that is, have a transparency similar to that of pure glass. These three papers acted as the catalyst for the substantial interest over the past decade in transparent glass ceramics for photonics applications. In the absence of an established theory, qualitative ideas guided the development of transparent glass ceramics. The ideas of effective media suggest that, if the size scale of the refractive index perturbations are much smaller than the wavelength of light in the medium, then the material should be described by a single effective dielectric constant (given, e.g., by the Maxwell–Garnett theory), and so there should be no scattering. Tick [54] has suggested the following four empirical criteria for a transparent glass ceramic: 1) 2) 3) 4)

The particle size must be less than 15 nm. The inter-particle spacing must be comparable with the crystal size. Particle size distribution must be narrow. There cannot be any clustering of the crystals.

One might also add that the refractive index difference must be small. If there is no refractive index mismatch, then the material is of course fully transparent. It is worth pointing out that this can only occur for cubic (i.e. non-birefringent) crystals, but can explain why some glass ceramics involving a single crystalline phase can be transparent for large crystallite sizes. Experimentally, particle sizes are often measured from the X-ray diffraction pattern line widths using the Scherrer formula [55]. This is necessarily a mean size estimate, gives no information on the size distribution, and is limited by the instrumental resolution to particle sizes less than about 100 nm. Transmission electron microscopy (TEM) measurements are better in this regard, and can give particle size distributions, mean spacings, and volume fractions, but are much more time-consuming. Estimates of volume fraction from the starting compositions are likely to be overestimates because of loss of volatile components during the melting process. For example, the oxyfluoride glass ceramics are susceptible to loss of fluoride through SiF4 volatilization. As an example of the insight which can be gained from TEM work, Dejneka [56] has found agreement with Rayleigh scattering for 7% volume fraction LaF3 nano-crystals of average size 15 nm, with a refractive index mismatch of 0.07 to an oxyfluoride host glass. However, a sample that contained 300-nm particles when quenched, and which was quite opaque, became more transparent on annealing. TEM studies showed that the effect of annealing was simply to fill in the spaces between the large crystals with small 20–30-nm crystals. This cannot be explained by Rayleigh scattering, and suggests that a cooperative effect is involved in the scattering process. 4.12.2

Theoretical Basis for Transparency

An understanding of light scattering processes in glass ceramics is clearly important to optimize the applications. We begin by examining the regime in which the particles act as independent scatterers. Mie first developed a theory of independent scattering from spherical particles, which reduces to the Rayleigh theory for particles much smaller than the wavelength of light. The Mie theory is exact in the sense that it is based on solutions of Maxwell’s equations resulting from the matching of electromagnetic waves at the surface of the sphere, but those solutions are in the form of approximations to the resulting

4.12 Transparent Glass Ceramics

infinite series. There is also the Rayleigh–Gans (or Rayleigh–Debye) theory, which can be applied to particles somewhat larger than those for which Rayleigh theory is appropriate, and for non-spherical geometries. It works by dividing a volume into Rayleigh scattering micro-volumes, and then adding the scattering from each micro-volume taking into account position-dependent phase shifts. The scattering cross-section for extinction is defined as the area which, if it were completely “black,” that is, completely absorbing, would remove an equivalent intensity from the beam per scattering center. For a spherical particle of refractive index n1 and radius a, immersed in a medium of refractive index nm , the scattering cross-section Csca for light of vacuum wavelength 𝜆 is ( )2 24𝜋 3 V 2 n2 − 1 Csca = (4.38) (𝜆∕nm )4 n2 + 2 where V is the particle volume and n is the relative refractive index n1 /nm . This is actually Eq. (4.24) expressed for a real dielectric constant and in terms of refractive indices. A light beam of initial intensity I 0 that passes through a thickness z of scattering particles with concentration N m−3 will emerge with intensity I where I = I0 exp(−NCsca z) = I0 exp(−az)

(4.39)

If the difference in refractive index |n1 -nm | is much less than the average n = (n1 + nm )∕2, then Eq. (4.38) can be approximated in the expression for the extinction coefficient 𝛼 = NC sca by: ) ( 𝜋 4 a3 128 (NV ) 4 (Δnn)2 NCsca = (4.40) 9 𝜆 which illustrates some of the key dependencies. The extinction coefficient depends linearly on the volume fraction, on the cube of the particle size, and on the square of the difference in refractive indices. Thus, the designer of transparent glass ceramics should seek to minimize all of these quantities, particularly the particle size and refractive index mismatch, to achieve transparency in the Rayleigh regime. Tick et al. [53] pointed out that, for the estimated volume fraction of particles (∼20–30%) present in the oxyfluoride glass ceramics, the Rayleigh or Mie theories for dilute and weak scatterers give results for the extinction coefficient that are orders of magnitude larger than those observed. Since a theory for nucleated glass ceramics was not available, Tick et al. [53] turned to the work of Hopper [57, 58] for light scattering in glasses that showed spinodal decomposition. Hopper showed that the extinction coefficient should be 𝛼 = (6.3x10−4 )k04 𝜃 3 (Δnn)2

(4.41)

where 𝜃 is the mean phase width and k 0 is the wavevector in free space. We note first that the Rayleigh-like wavelength dependence of the extinction coefficient as the inverse fourth power is preserved in this theory. Making the large conceptual step of applying the theory to nucleated crystals, we can equate 𝜃 to (a + W /2), where a is the particle radius and W the interparticle spacing. If we take a value of 50% for the volume fraction, Hopper’s theory predicts an extinction coefficient about two orders of magnitude lower than that obtained from the Rayleigh theory [53] . However, the validity of this theory applied to a glass showing nucleated rather than spinodal decomposition remains questionable.

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Hendy [59] has presented a more recent analysis of light scattering for the late-stage phase-separated materials and has deduced an extinction coefficient given by ) ( 14 Δn 2 8 7 𝛼= k a (4.42) ⟨𝜙⟩⟨1 − 𝜙⟩ 15 n where φ is the volume fraction of the phase, k = 2𝜋/𝜆, and the other terms have their usual meaning. This formula has a very different dependence on wavelength and particle size to that of the Rayleigh and Hopper models; in particular, note the very high orders for the wavelength and particle size dependence. As far as the author is aware, there have been no reports of a 1/𝜆8 wavelength dependence for the extinction coefficient observed experimentally. This would be expected to give rise to visible coloration effects, more pronounced than those for Rayleigh scattering. In fact, there have been very few attempts to investigate the wavelength dependence of the scattering and confront it with the theories, but this is difficult in the case of rare-earth or other doping in glass ceramics of practical interest, since electronic absorption can dominate the extinction coefficient in the uv region where the 𝜆−8 effect would be most pronounced. We have examined [60] the second and fourth of Tick’s criteria, namely that the interparticle spacing must be comparable with the particle size, and that there should be no clustering, by making use of the discrete dipole approximation (DDA), [61], which has been used for a variety of scattering simulations. The DDA models the response of a dielectric material by a set of dipoles located on the lattice points of a simple cubic lattice. The response of the dipoles to an incident electromagnetic field is calculated self-consistently, including dipole–dipole interactions. The polarizability of the individual dipoles can be chosen to model any dielectric structure. For a glass–ceramic simulation, we populated a 64 × 64 × 64 array with 2 × 2 × 2 nanocubes whose dipole polarizabilities were chosen to represent BaCl2 nanoparticles in a fluorozirconate glass matrix, and the entire supercube was taken to be embedded in a host glass whose dielectric constant was estimated by the Maxwell–Garnett formula for an effective medium [16]. The total nanocube volume fraction was varied, but the nanocube distribution was constrained to be either: (i) at random locations without overlap, but with at least one lattice spacing between particles; or (ii) also at random, but permitting clustering without overlap. The results of the simulation show that the scattering varies almost linearly with particle concentration up to 25% volume fraction. If transparency were to be associated with small inter-particle separation (equivalent to a high-volume fraction of particles), we would expect to see some sub-linearity as particle concentration increased. The effect of clustering was to increase the scattering in a supra-linear fashion, as would be expected from the high-power law dependence on particle size when some particles cluster together to form an effectively larger composite particle. As a general observation, these calculations do not support the notion that a high-volume fraction of scattering particles is a key factor in determining transparency in glass ceramics, but support the idea that agglomeration is detrimental. A new explanation of high transparency in glass ceramics was subsequently put forward by the present author [62] based on a core-shell model for the nanoparticles. The basic assumptions are that, in the nucleation and growth process, some ions move into the volume of the particle core from the surrounding shell, while others from the core move into the surrounding shell, such that the number and type of particles is conserved. It is also assumed for simplicity that the final core and shell are (separately)

4.12 Transparent Glass Ceramics

of homogeneous composition, and that the scattering is dominated by single-particle scattering events. The analysis is based on the Rayleigh–Debye approximation [15], and shows that the scattered intensity for unpolarized light can be expanded in a power series of k4 , where k = 2𝜋/𝜆 is the wavevector of the incident light. The first term in the expression for the scattered intensity I(𝜃) at distance r and angle 𝜃 to the incident direction is given by [ ] |2 | ∑ | (1 + cos2 𝜃)k 4 || S 2 I = I0 4𝜋s ΔNi (s)𝛼i ds|| (4.43) | 2 2r |∫0 | i | | where s is the radial distance from the center of a particle of shell radius S, ΔN i is the change in concentration of ion species i from the homogeneous glass, and 𝛼 i is the polarizability of that ion species. Under the assumption of conservation of ions within the core-shell particle, this term is identically zero, giving rise to the prediction of ultra-transparency for any glass ceramic that satisfies the conditions of the model. The basic reason for the predicted ultra-transparency is that the scattering is unaffected (to lowest order) by the specific distribution of ions within the core shell structure, so that a core shell has the same scattering as the uncrystallized glass—that is, no scattering. The second term in the series is ( ) [ ] |2 2(1 + cos2 𝜃)sin4 𝜃2 k 8 || S ∑ | 4 | I = I0 4𝜋s ΔNi (s)𝛼i ds|| (4.44) | 2 9r |∫0 | i | | and varies as k 8 in the same way as predicted by the Hendy model. The angular dependence shows strong backscatter, as shown in Figure 4.21. As far as the author is aware, this predicted wavelength dependence in a highly transparent glass ceramic has yet to be tested. The experimental difficulties in testing such a relation relate to the difficulty [60] of separating any weak scattering contribution to the Figure 4.21 Polar plot showing the angular dependence (dashed line) of the scattering predicted by the second term in the expansion series for the core-shell scattering analysis.

90

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4 Optical Properties of Glasses

extinction coefficient of the type predicted by Eq. (4.44), which would only be prominent at short wavelengths, from the large residual electronic absorption of rare-earth or transition metal impurities or dopants. At long wavelengths, the wavelength-independent scattering from a small fraction of large particles dominates. We note that there is some evidence, for other materials, for an inverse eighth-power dependence on wavelength, and for strong back scatter in early work on “anomalous scattering,” which is summarized by Goldstein [63]. The core-shell model analytical predictions have been checked in two ways: by a parallel analytical calculation based on Mie scattering from a coated sphere, and by a DDA calculation of the scattering from a single core-shell structure embedded in a binary composition glass [64]. These alternative analyses confirm the basic premise—that, if the crystallization process comprises nucleation and growth from the local environment resulting in a core-shell nano-crystallite, the resulting scattering is much less than would be expected from Rayleigh scattering from the core embedded in a uniform glass. 4.12.3

Rare-Earth-Doped Transparent Glass Ceramics for Active Photonics

Rare-earth-doped transparent glass ceramics have been an area of substantial activity, primarily driven by telecommunication applications. Dejneka [65] and Goncalves et al. [66] have presented reviews of photonic applications of rare-earth-doped transparent glass ceramics. These materials offer the following advantages for applications such as fiber amplifiers and up-conversion: 1. They can be almost as transparent as simple glasses. 2. The active rare-earth ions may selectively partition into the crystals. 3. The rare-earth ions are in a crystalline environment where their radiative decay characteristics are superior to those typically found in a glassy environment. This is especially the case for heavy-metal halides such as LaF3 and Pbx Cd1-x F2 . 4. The glass ceramics include silicate-based varieties that are compatible with bonding to silica fiber. A great deal of work has been done on the oxyfluorides containing Pbx Cd1-x F2 and LaF3 crystallites. In the latter case, it is perhaps not surprising that the nano-crystalline phase readily accommodates a variety of trivalent rare-earth ions, since there is excellent charge and ionic radius match. However, Dejneka [56] has made an interesting observation: that oxyfluorides doped with GdF3 , rather than LaF3 , show crystallization in a hexagonal tysonite phase that has not been previously reported; similarly, YF3 and TbF3 are also found in the hexagonal phase, although this is not the room temperature stable phase for these compounds. We have also found hitherto unreported phases, and phases unstable at room temperature and pressure in ZBLAN glasses containing chloride and bromide crystals. It seems likely that the increased role of surface as against volume energies for nanoscale crystallites stabilizes phases that would normally be unstable at STP, and that one can, in general, “expect the unexpected” in nanoscale glass ceramics. For the case of Pbx Cd1-x F2 , the nano-crystalline phase was identified from the X-ray diffraction (XRD) pattern as a solid solution of CdF2 and PbF2 . It is surprising that trivalent rare-earth ions selectively partition into these nano-crystals, given the mismatch of charge and ionic radius. Tihkomirov et al. [67] proposed that, in erbiumdoped oxy-fluorides, the Er3+ ions are involved in nucleating centers that comprise

4.12 Transparent Glass Ceramics

mixed-phase orthorhombic 𝛼-PbF2 : ErF3 nucleation centers for the subsequent growth of 𝛽-PbF2 nano-crystals. This mechanism can explain the selective uptake of rare-earth ions, and has considerable implications for the quantum efficiencies, since one can expect at least a region of high rare-earth concentration near the core of the nano-crystal that may be subject to concentration quenching. One would therefore expect a marked dependence of the optical performance of the material on the nucleation and growth conditions. Just this effect was observed by Mortier et al. [68] in Er3+ -doped germanate glasses, with clear evidence in the 4 I11/2 –4 I13/2 fluorescent transition for concentration-quenching effects. In contrast to these results, for sol-gel-derived oxyfluoride glasses, the rare-earth ion seems to limit nucleation and growth of the crystals [66], and so the role of the rare-earth ion is clearly process dependent. 4.12.4

Ferroelectric Transparent Glass Ceramics

Ferroelectric crystals from the ABO3 family such as LiNbO3 are widely used in non-linear electro-optics. But the expense of single crystal production has prompted a search for glass ceramic materials containing ferroelectric crystals that may be more cost-effective, and which also offer the usual glass advantages of fabrication versatility. Such materials can be used directly as an optically isotropic medium in Kerr birefringence modulators, or can be electrically poled at an elevated temperature, resulting in a linear electro-optic effect as required for a Pockels modulator and for second harmonic generation. A number of transparent glass ceramics containing ferroelectric crystals have been reported [69–71], and in a niobium–lithium–silicate glass containing sodium niobate crystals, a Kerr effect comparable with nitrobenzene, a standard Kerr cell liquid, has been observed [72]. Nonlinear properties of glasses are the subject of Chapter 10. 4.12.5

Transparent Glass Ceramics for X-ray Storage Phosphors

X-ray storage phosphor (XRSP) imaging plates [73, 74] are solid-state replacements for the photographic film used in medical, dental, and industrial radiography. The currently favored materials used in imaging plates are powdered crystalline BaFBr or CsBr doped with ∼1000 ppm divalent europium. The image is stored as a spatial variation of radiation-induced trapped electrons and holes, and is read out by stimulating their recombination with a raster-scanned red laser beam. The recombination energy is transferred to the europium ions and appears as blue photo-stimulated luminescence (PSL). Although the imaging plate technology offers many advantages over the traditional photographic film process such as re-usability, wider dynamic range, freedom from chemical developers, and direct digital image read-out, the spatial resolution is not as good as that of fine-grained photographic film. The reason is that the focused laser beam used in the read-out process in BaFBr is scattered by the powder grains, so that the PSL occurs not only from the region of the focusing spot, but also from the surrounding material, limiting the resolution to about 100 μm. While this is adequate for most applications, such as chest X-rays, it is inadequate for applications such as mammography and crack detection in materials testing. It may be thought that the problem could be overcome by embedding the material in a refractive index matching binder, but BaFBr is birefringent, so matching is not possible. For cesium bromide, a preparation process that results in

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an oriented needle-like structure in the image plate gives a somewhat improved spatial resolution through the light-guiding properties of the needles, but the improvement factor is no better than about two [75]. Consequently, the idea of a transparent XRSP such as a glass is very attractive for minimizing optical scattering and improving the spatial resolution in imaging plate radiography, but unfortunately it seems that the density of stable trapping centers in simple glasses examined so far is insufficient for a viable XRSP. However, it has been shown [76–79] that fluorozirconate glass ceramics containing europium-doped BaBr2 , Rb2 BaBr4 , RbBa2 Br5 , or BaCl2 nano-crystals show a significant XRSP effect, in the latter case up to 80% of that for BaFBr:Eu. The magnitude of the effect depends on the crystallite size, and it was suggested by Secu et al. [76] that there was a surface layer on the BaCl2 crystallites of about 7 nm in thickness which was not PSL active. Large crystallites of size ≅30 nm are therefore optimal for a large PSL effect. However, an increasing crystallite size also results in more light scattering [80], as shown in Figure 4.22, and so there is a trade-off between PSL efficiency and transparency, as shown in Figure 4.23. For samples that are still just transparent to the eye, the relative efficiency falls to about 10%. The spatial resolution of the glass ceramic, as specified by the modulation transfer function, is significantly better than for BaFBr [81, 82]. High-resolution images taken with a Eu-doped fluorochlorozirconate glass ceramic have been presented by Schweizer et al. [82], who also demonstrate that the same material can be used as a high-spatial-resolution phosphor/scintillator. The effect of light scattering for scanned laser beam image read-out from a glass ceramic X-ray storage phosphor has been simulated numerically [83]. The model used for the read-out process includes scattering of both read-out light and PSL, and absorption. Scattering was characterized by a macroscopic scattering length μ, such that the probability of a photon being scattered after traversing a length 𝜉 is given by ( ) −𝜉 (4.45) P(𝜉) = 1 − exp 𝜇 and the anisotropy in the scattering was approximated by a cone-like Henyey– Greenstein function [84]. The major conclusions were that the modulation transfer function is bimodal, as observed experimentally, with contributions from the effects of both the scattering and the finite laser beam diameter. The spatial resolution is optimized when the scattering length is much greater than the thickness, corresponding to a highly transparent glass ceramic, and is limited only by the beam diameter, but this case results in low read-out intensity, since most photons pass straight through the sample without interaction. Hence, a fully transparent X-ray storage phosphor material is not necessarily a practical solution. The other extreme, where the scattering

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Figure 4.22 Visual appearance of fluorochlorozirconate glass ceramic storage phosphor annealed at various temperatures (∘ C). Source: Reproduced from A. Edgar, G.V.M. Williams, M. Secu, S. Schweizer, J-M Spaeth. Radiat Meas. 2004;38(4–6):413–6 by permission of Elsevier.

4.12 Transparent Glass Ceramics

35 30

Transmittance –0.5

25 20

–1 15

Relative PSL (%)

Log (Transmittance at 633 nm)

0

10 –1.5 5 PSL –2

0 220

230 240 250 260 270 Annealing Temperature (°C)

280

Figure 4.23 PSL efficiencies relative to BaFBr: Eu(100 ppm) for annealed glass. The measured transmittance at 633 nm is also plotted for a 1.45-mm-thick sample. The lines are guides to the eye. Source: Reproduced from A. Edgar, G.V.M. Williams, M. Secu, S. Schweizer, J-M Spaeth. Radiat Meas.; 38(4–6):413–6 (2004) by permission of Elsevier.

length is much less than the sample thickness, also results in high spatial resolution, since the interactions are concentrated in a small volume near the surface, but again a low read-out intensity results, because only a thin layer of material near the surface is being activated. The optimal practical solution is a compromise between spatial resolution and read-out efficiency, corresponding to the case when the scattering length is approximately the same as the thickness. This results in a bimodal modulation transfer function as observed [83], which can be reproduced by an appropriate choice of parameters in the simulation. It was also shown that selective choice of additives (such as Co2+ ) to the glass matrix to absorb red light, but not blue, would result in an improved modulation transfer function by limiting the spatial spread of the red stimulating light. In an effort to understand the processes that lead to the nucleation and growth of barium chloride nano-crystals in this fluorozirconate glass host, Hendy et al. [85] and Edgar and Hendy [86] undertook molecular dynamics simulations of the quenching process for glass melts. In a ZBLN glass with up to 30% of the fluoride ions replaced by chloride ions, it was found that, for chlorine concentrations above about 20%, there was clustering into zirconium-rich/chlorine-poor and sodium- and barium-rich/chlorine-rich regions. Figure 4.24 shows the fluorine and chlorine ions for a glass containing 20% chlorine anions, showing the predicted phase separation for the anions. In a subsequent study [86] based on both chlorine-doped ZBLN and ZBLLi fluoride glasses, a similar propensity to phase separation was again found, but was most pronounced in the ZBLN glass. Although direct precipitation of a solid crystalline phase was not predicted and is, in any case, not expected with this sort of calculation, the existence of a phase separation is a clear portent of crystallization by a chloride phase. Given that ternary barium chlorides involving sodium or lithium are not known, this would most probably be barium chloride, as observed. The simulations do give a guide

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Figure 4.24 Anion distribution for a ZrBaLaNa glass showing fluorine distribution (left) and chlorine distribution (right). A chlorine-rich, fluorine-poor region can be seen at the lower left. Source: Reproduced from S.C. Hendy and A. Edgar, J Non-Cryst Solids;352(5):415–22 (2006) by permission of Elsevier.

to the optimal chlorine concentration, since the phase separation reached a maximum in the latter calculation at around 12–14% of the total anion content.

4.13 Conclusions The optical properties of glasses is a subject that has both mature and developing aspects; the historical use of glass in optical instruments means that a great deal of information has been gathered and analyzed regarding the bulk optical properties of traditional glass in the visible region. Yet, there are new and fascinating applications that have emerged in the past 30 years, dominated by fiber optics and the telecommunications industries. The revolution in human communications brought about by the Internet and the resources that it delivers are, of course, based on the high-speed, high-capacity optical links provided by optical pulse propagation on glassy silica fibers. But there have been other new developments: novel glass families optimized for optical performance in the infra-red and ultraviolet spectral regions, glass fiber lasers and amplifiers, a whole new class of glassy materials in the form of transparent glass ceramics, new physical phenomena such as PSL, and new technologies for refractive index engineering. The next few years promise to deliver equally intriguing developments in optical glass science.

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5 Concept of Excitons Jai Singh 1 , Harry E. Ruda 2 , M.R. Narayan 1 , and D. Ompong 1 1

College of Engineering, Information Technology and Environment, Charles Darwin University, Darwin, NT 0909, Australia Centre for Nanotechnology, and Electronic and Photonic Materials Group, Department of Materials Science, University of Toronto, Toronto, Ontario, Canada 2

CHAPTER MENU Introduction, 129 Excitons in Crystalline Solids, 130 Excitons in Amorphous Semiconductors, 135 Excitons in Organic Semiconductors, 139 Conclusions, 153 References, 154

5.1 Introduction An optical absorption in semiconductors and insulators can create an exciton, which is an electron–hole pair excited by a photon and bound together through their attractive Coulomb interaction. The absorbed optical energy remains held within the solid for the lifetime of an exciton. Because of the binding energy between the excited electron and hole, excitonic states lie within the band gap near the edge of the conduction band. There are two types of excitons that can be formed in nonmetallic solids: Wannier or Wannier–Mott excitons and Frenkel excitons. The concept of Wannier–Mott excitons is valid for inorganic semiconductors such as Si, Ge, and GaAs, because, in these materials, the large overlap of interatomic electronic wave functions enables electrons and holes to be far apart but bound in an excitonic state. For this reason, these excitons are also called large-radii orbital excitons. Excitons formed in organic crystals are called Frenkel excitons. In organic semiconductors/insulators or molecular crystals, the intermolecular separation is large and hence the overlap of intermolecular electronic wave functions is very small, and electrons remain tightly bound to individual molecules. Therefore, the electronic energy bands are very narrow and closely related to individual molecular electronic energy levels. In such solids, the absorption of photons occurs close to the individual molecular electronic states, and excitons are also formed within the molecular energy levels (see, e.g., reference [1]). Such excitons are therefore also called molecular excitons. For details of the theory of Wannier and Frenkel excitons, readers may like to refer to the book by Singh [1]. Another way of understanding excitons in Optical Properties of Materials and Their Applications, Second Edition. Edited by Jai Singh. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

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inorganic and organic semiconductors is on the basis of the dielectric constant of these two solids. For example, in silicon (Si), the static dielectric constant is 12, and that in organic solids and polymers is about 3–4. This gives the binding energy of excitons in organic solids to be nearly four times larger than that in Si, leading to the excitonic Bohr radius in Si to be four times larger than that in organic semiconductors. Thus, the excitonic Bohr radius is larger in inorganic semiconductors, and hence termed as large-radii orbital excitons. The excitonic concept from the theory point of view was initially developed only for crystalline solids, and it used to be believed that excitons cannot be formed in amorphous semiconductors. However, several experiments on photoluminescence measurements have revealed the existence of excitons in amorphous semiconductors as well, and the theory of excitons in such solids was subsequently developed. Here, the concept of excitons in crystalline solids is reviewed briefly first as it is well established, and then amorphous semiconductors will be considered.

5.2 Excitons in Crystalline Solids In Wannier–Mott excitons, the Coulombic interaction between the hole and electron can be viewed as an effective hydrogen atom with, for example, the hole establishing the coordinate reference frame about which the reduced-mass electron moves. If the effective masses of the isolated electron and hole are m∗e and m∗h , respectively, their reduced mass, 𝜇x , is given by: 𝜇x−1 = (m∗e )−1 + (m∗h )−1

(5.1)

Note that, in the case of so-called hydrogenic impurities in semiconductors (i.e. both shallow donor and acceptor impurities), the mass of the nucleus takes the place of one of the terms in Eq. (5.1), and hence the reduced mass is given to a good approximation by the effective mass of the appropriate carrier. In the case of an exciton as the carrier, effective masses are comparable, and hence the reduced mass is markedly lower—accordingly, the exciton binding energy is markedly lower than that for hydrogenic impurities. The energy Ex of a Wannier–Mott exciton is given by (e.g., see reference [1]): ℏ2 K 2 (5.2) − En 2M where Eg is the bandgap energy; ℏK the linear momentum; M (= m∗e + m∗e ) the effective mass associated with the center of mass of an exciton; and En is the exciton binding energy, given by: Rxy 𝜇 e4 𝜅 2 1 (5.3) En = x 2 2 2 = 2 2ℏ ε n n 1 , ε is the static dielectric constant of the solid, where e is the electronic charge, 𝜅 = 4𝜋ε 0 and n is the principal quantum number associated with the internal excitonic states n = 1 (s), 2 (p), etc., as shown in Figure 5.1. Rxy is the so-called effective Rydberg constant of an exciton, given by Ry (𝜇x /me )/ε2 , where Ry = 13.6 eV. According to Eq. (5.3), as stated earlier, the excitonic states are formed within the bandgap near the conduction band edge. However, as the exciton binding energy is very small—for example, a few meV in bulk Si and Ge crystals—exciton absorption peaks can be observed only at very low Ex = Eg +

5.2 Excitons in Crystalline Solids

E

Figure 5.1 Schematic illustration of excitonic bands for n = 1 and 2 in semiconductors. E g represents the energy gap.

Conduction Band n=2 n=1 Eg

Excitonic Band

K

Valence Band

temperatures. For bulk GaAs, the binding energy (n = 1) corresponds to about 5 meV. Following from the hydrogen atom model, the extension of the excitonic wave function can be found from an effective exciton Bohr radius ax , given in terms of the Bohr radius as a0 = h2 /𝜋e2 ; that is, ax = a0 (ε/𝜇x ). For GaAs, this corresponds to about 12 nm or about 21 lattice constants—that is, the spherical volume of the exciton radius contains ∼(ax /a)3 or ∼9000 unit cells, where a is the lattice constant of GaAs. As Rxy ≪ Eg and ax ≫ a, excitons in GaAs are large-radii orbital excitons, as stated in the preceding text. It should be noted that the binding energy of excitons in semiconductors tends to be a strong function of the bandgap. In Figure 5.2a,b are shown the dependencies of Rxy (exciton binding energy) and ax /a on the bandgap of semiconductors, respectively. In the case of excitons having large binding energy and correspondingly small radius (i.e. approaching the size of about one lattice parameter), the excitons become localized on a lattice site, as observed in most organic semiconductors. As stated earlier, such excitons are commonly referred to as Frenkel excitons or molecular excitons. Unlike Wannier–Mott excitons, which typically are dissociated at room temperature, Frenkel excitons are stable at room temperature. For the binding energy of Frenkel excitons, one may refer to Singh [1], for example. Excitons may recombine radiatively, emitting a series of hydrogen-like spectral lines as described by Eq. (5.3). In bulk crystalline (3D) semiconductors such as Si, Ge, and GaAs, exciton lines can be observed only at low temperatures. As the binding energy is small, typically ≤10 meV, excitons in bulk are easily dissociated by thermal fluctuations. The preceding discussion refers to the so-called free excitons formed between the conduction band electrons and valence band holes of a crystalline inorganic semiconductor or insulator. According to Eq. (5.2), such an exciton is able to move throughout a material with a center-of-mass kinetic energy (second term on the right-hand side). It should be noted, however, that free electrons and holes move with a velocity ℏ(dE/dk), where the derivative is taken for the appropriate band edge. To move through a crystal, both electron and hole must have identical translational velocity, thereby restricting the regions in k-space where these excitations can occur with (dE/dk)electron = (dE/dk)hole , commonly referred to as critical points.

131

5 Concept of Excitons

Exciton binding energy (meV)

1000

100

10

1

0.1 0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

3.0

3.5

4.0

Bandgap (eV) (a) Exciton radius/lattice constant

132

100.00

10.00

1.00 0

0.5

1.0

1.5

2.0

2.5

Bandgap (eV) (b)

Figure 5.2 Dependence of (a) exciton binding energy (R∗y ) (Eq. (5.3)) and (b) size in terms of the ratio of the excitonic Bohr radius to lattice constant (a∗B ∕a0 ) as a function of the semiconductor bandgap. Exciton binding energy increases concurrently with marked diminishment of exciton spreading as the bandgap increases. Above a bandgap of about 2 eV, the Wannier-based description is not appropriate.

In quantum wells and other structures of reduced dimensionality, the spatial confinement of both the electron and hole wave functions in the same layer ensures strong excitonic transitions of a few meV below the bandgap, even at room temperature. The binding energy of excitons, Ebx (α, n), in confined systems of dimension α, is given by references [2, 3]: Ebx (α, n) =

Rxy [n + (α − 3)∕2]2

(5.4)

Thus, the binding energy of excitons in quantum wells (α ≈ 2) and n = 1 increases to four times the value in 3D (α= 3). For quantum wires, α= 1, the binding energy becomes infinitely large, and for quantum dots (α= 0), it becomes the same as in 2D. It is this enhancement in the binding energy due to confinement that allows excitonic absorption and photoluminescence to be observed even at room temperature in quantum wells.

5.2 Excitons in Crystalline Solids

Furthermore, the observation of biexcitons [3, 4] and excitonic molecules also becomes possible due to the large binding energy. The ratio of the binding energy of biexcitons, Ebxx to that of excitons, Ebx for n = 1, is usually constant in quantum wells, and, for GaAs quantum wells, one gets Ebxx = 0.228Ebx [2, 3]. 5.2.1

Excitonic Absorption in Crystalline Solids

As exciton states lie below the conduction-band edges in crystalline solids, absorption to excitonic states is observed below the conduction-band edge. According to Eq. (5.2), the difference of energy between the bandgap and excitonic absorption gives the binding energy. As the exciton–photon interaction operator and pair of excited electron and hole and photon interaction operator depend only on their relative motion momentum, the form of these interactions is the same for band-to-band and excitonic absorption. Therefore, for calculating the excitonic absorption coefficient, one can use the same form of interaction as that for band-to-band absorption in Chapter 2, but one has to use the joint density of states for crystalline solids. Thus, the absorption coefficient associated with the excitonic states in crystalline semiconductors is obtained as (e.g. [5]): ( )1∕2 2 4e2 |pxv |2 (ℏ𝜔 + En − Eg )1∕2 , n = 1, 2, … (5.5) αℏ𝜔 = √ 2 𝜇 x εcℏ where ℏ𝜔 is the energy of the absorbed photon, pxv is the transition matrix element between the excitonic state and the valence band, and En is the exciton binding energy corresponding to a state with n = 1, 2, etc. (see Eq. (5.3)). Equation (5.5) is similar to the case of direct band-to-band transitions discussed in Chapter 2, and it is valid only for the photon energies, ℏ𝜔 ≥ Eg . There is no absorption below the excitonic ground state in pure crystalline solids. The absorption of photons to the excitonic energy levels is possible either by exciting electrons to higher energy levels in the conduction band and then by their non-radiative relaxation to the excitonic energy level, or by exciting an electron directly to the exciton energy level. Excitonic absorption occurs in both direct as well as indirect semiconductors. If the valence hole is a heavy hole, the exciton is called a heavy-hole exciton; conversely, if the valence hole is light, the exciton is a light-hole exciton. For large exciton binding energy, say corresponding to n = 1, the excitonic state will be well separated from the edge, Eg , of the conduction band, and then one can observe a sharp excitonic peak at the photon energy ℏ𝜔 = Eg − E1 . The absorption to higher excitonic states, corresponding to n > 1, may not be observed in materials with small binding energies, as these states will be located within the conduction band. The excitonic absorption and photoluminescence in GaAs quantum wells are shown in Figure 5.3a,b, respectively [3]. In Figure 5.3b are also shown the biexcitonic photoluminescence peaks (with superscript XX) observed in GaAs quantum wells. In the absorption spectra (Figure 5.3a), both light-hole (LHX ) and heavy-hole (HHX ) excitons are observed in quantum wells of different well widths indicated by the corresponding subscript. For example, HH100 X and HH100 XX mean heavy-hole exciton and biexciton peaks, respectively, in a quantum well of width 100 Å. It may be noted that the transition matrix element, pxv , in Eq. (5.5) is assumed to be photon-energy independent. It is the average of the linear relative momentum between

133

5 Concept of Excitons

(a) 2.5 HHx130

2.0

αd

1.5

HHx100 HHx80

HHx160

LHx130

LHx100

LHx160

1.0 0.5 0.0

(b) HHxx 160

HHx160

HHxx 130

PL intensity [Arb. units]

134

HHx130 HHx100

1.52

1.53

1.54

1.55

HHx80

1.56

1.57

1.58

Photon energy (eV)

Figure 5.3 (a) Low-temperature absorption, and (b) photoluminescence spectra in GaAs quantum wells of different well widths. (HHx100 and HHxx 100 denote heavy-hole exciton and biexcition, respectively, in a quantum well of width 100 Å.) The photoluminescence data are obtained using HeNe laser excitation [3].

electron and hole in an exciton. However, if one applies the dipolar approximation, the transition matrix element thus obtained depends on the photon energy [5, 6], which gives a different photon-energy dependence in the absorption coefficient. This aspect of the absorption coefficient is described in detail in Chapter 3. Excitonic absorption is spectrally well located and very sensitive to optical saturation. For this reason, it plays an important role in nonlinear semiconductor devices (nonlinear Fabry–Perot resonator, nonlinear mirror, saturable absorber, and so on). For practical purposes, the excitonic contribution to the overall susceptibility 𝜒 ex around the resonance frequency 𝜈 ex can be written as: 𝜒ex = −A0

(𝜐 − 𝜐0 ) + jΓex (𝜐 − 𝜐0 )2 + Γ2ex (1 + S)

(5.6)

with Γex being the line width and S = I/I S , the saturation parameter of the transition. For instance, in GaAs multiple quantum wells (MQWs), the saturation intensity I S is as low

5.3 Excitons in Amorphous Semiconductors

as 1 kW cm−2 , and Γex (≈3.55 meV at room temperature) varies with the temperature according to: Γ1 (5.7) Γex = Γ0 + ) ( ℏ𝜔LO exp 𝜅T − 1 where ℏΓ0 is the inhomogeneous broadening (≈2 meV), ℏΓ1 the homogeneous broadening (≈5 meV), and ℏ𝜔LO the longitudinal optical-phonon energy (≈36 meV). At high carrier concentrations (provided either by electrical pumping or by optical injection), an efficient mechanism of saturating the excitonic line is the screening of the Coulombic attractive potential by free electrons and holes. A number of more complex pairings of carriers can also occur, which may also include fixed charges or ions. For example, for the case of three charged entities, with one being an ionized donor impurity (D+ ), the following possibilities can occur: (D+)(+)(−), (D+)(−)(−), and (+)(+)(−) as excitonic ions, and (+)(+)(−)(−) and (D+)(+)(−)(−) as biexcitons or even bigger excitonic molecules (see, e.g., references [2, 3, 7]). Complexity abounds in these systems as each electronic level possesses a fine structure corresponding to allowed rotational and vibrational levels. Moreover, the effective mass is often anisotropic. Note that, when the exciton or exciton complex is bound to a fixed charge such as an ionized donor or acceptor center in the material, the exciton or exciton complex is referred to as a bound exciton. Indeed, bound excitons may also involve neutral fixed impurities. It is usual to relate the exciton in these cases to the center binding them—thus, if an exciton is bound to a donor impurity, it is usually termed a donor-bound exciton.

5.3 Excitons in Amorphous Semiconductors As stated earlier, the concept of excitons is traditionally valid only for crystalline solids. However, several observations in the photoluminescence spectra of amorphous semiconductors have revealed the occurrence of photoluminescence associated with the singlet and triplet excitons (see, e.g., reference [5]). Applying the effective-mass approach, a theory for the Wannier–Mott excitons in amorphous semiconductors has been developed in real coordinate space [7–10]. The energy W x of an exciton thus derived is obtained as: P2 (5.8) − En (S) Wx = Eg + 2M where Eg is the optical gap, P is the linear momentum associated with the exciton’s center of mass motion, and En (S) is the binding energy of excitons, given by: En (S) = where

𝜇x e 4 𝜅 2 2ℏ2 ε′ (S)2 n2

[ ]−1 (1 − S) ε (S) = ε 1 − A ′

(5.9)

(5.10)

with S being the spin of an exciton (S = 0 for singlet and S = 1 for triplet), and A is a material-dependent constant representing the ratio of the magnitude of the Coulomb

135

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5 Concept of Excitons

and exchange interactions between the electron and hole of an exciton. Equation (5.9) is analogous to Eq. (5.3) obtained for excitons in crystalline solids for S = 1. This is because Eq. (5.3) is derived within the large-radii orbital approximation, which neglects the exchange interaction and hence is valid only for triplet excitons [1, 11]. As amorphous solids lack long-range order, the exciton binding energy is found to be larger in amorphous solids than in their crystalline counterparts; for example, in hydrogenated amorphous silicon (a-Si:H), the binding energy is higher than in crystalline silicon (c-Si). This is the reason why it is possible to observe the photoluminescence of both singlet and triplet excitons in a-Si:H [12] but not in c-Si. According to Eq. (5.9), the singlet exciton binding energy corresponding to n = 1 becomes: (A − 1)2 𝜇x e4 𝜅 2 (5.11) E1 (S = 0) = 2A2 ℏ2 𝜖 2 and the triplet exciton binding energy becomes: 𝜇x e 4 𝜅 2 (5.12) 2ℏ2 ε2 From Eqs. (5.11) and (5.12), we get the relation between the singlet and triplet exciton binding energies as: E1 (S = 1) =

(A − 1)2 E1 (S = 1) (5.13) A2 For hydrogenated amorphous silicon (a-Si:H), A is estimated to be about 10 [5], which gives from Eq. (5.13) the result that the singlet exciton binding energy is about 81% of the triplet exciton binding energy. As 𝜇x is different in the extended and tail states, there are four possibilities through which an exciton can be formed in amorphous solids: (i) both the excited electron and hole are in their extended states; (ii) the excited electron is in the extended states and the hole is in the tail states; (iii) the excited electron is in the tail states and the hole is in the extended states; and (iv) both the excited electron and hole are in their tail states. Using the expressions of effective mass of charge carriers derived in the extended and tail states in Chapter 3, we get the effective mass of the electron and hole in the extended states in a-Si:H as m∗ex 0.34 me and in the tail states as m∗et = 7.1 me . This gives 𝜇x = 0.17me for possibility (i) when both the excited carriers are in their extended states; 𝜇x = 0.32me for possibilities (ii) and (iii) when one of the excited charge carriers is in the tail and the other in the extended states; and 𝜇x = 3.55 me for possibility (iv) when both electron and hole are in their tail states in a-Si:H. Using ε= 12, one obtains from Eq. (5.12) the triplet exciton binding energies in a-Si:H as 16 meV for possibility (i); 30 meV for possibilities (ii) and (iii); and 0.34 eV for possibility (iv). The corresponding singlet exciton binding energies in a-Si:H is obtained from Eq. (5.13) as E1 (S = 0) = 13 meV for possibility (i), 24 meV for possibilities (ii) and (iii), and 0.28 eV for possibility (iv). These energies are measured from the conduction band edge or the conduction band mobility edge, Ec , and are schematically shown in Figure 5.4. Accordingly, the triplet exciton states lie below the singlet exciton states. E1 (S = 0) =

5.3 Excitons in Amorphous Semiconductors

Conduction states Eb (n = 1; s = 0)

ΔEex

Eb (n = 1; s = 1)

ESinglet state ETriplet state

Eopt

Valence states

Figure 5.4 Schematic illustration of singlet and triplet excitonic states in amorphous solids. ΔE ex represents the energy difference between singlet and triplet excitonic states.

The excitonic Bohr radius is also found to be different for singlet and triplet excitons in amorphous semiconductors. Writing the exciton energy as: E1 (S) = −

𝜅e2 2ax (S)

(5.14)

we get from Eqs. (5.11)–(5.13): ax (S = 0) =

A2 a (S = 1) (A − 1)2 x

(5.15) m

which gives for a-Si:H with A = 10, ax (S = 0) ≈ 54 ax (S = 1), where ax (S = 1) = 𝜇 e a0 , x and a0 = 0.0529 nm is the Bohr radius. Accordingly, for triplet excitons in a-Si:H, we get ax (S = 1)= 3.73 nm for possibility (i), 1.98 nm for possibilities (ii) and (iii), and 0.18 nm for possibility (iv); and, for the corresponding excitonic Bohr radius for a singlet exciton, we get for possibility (i) ax (S = 0) = 4.66 nm, for possibilities (ii) and (iii) ax (S = 0) = 2.5 nm, and for possibility (iv) ax (S == 0) ≈ 0.233 nm. The excitonic Bohr radius plays a very significant role in the radiative recombination of excitons, because this is the average separation between the electron and hole in an exciton prior to their recombination. Therefore, the rates of spontaneous emission depend on the excitonic Bohr radius (see Chapter 7). 5.3.1

Excitonic Absorption in Amorphous Solids

In amorphous semiconductors, the excitonic absorption and photoluminescence can be quite complicated. According to Eq. (5.8), the excitonic energy level lies below the optical bandgap by an energy equal to the binding energy given in Eq. (5.9). However, as stated in the previous section, there are four possibilities of transitions for absorption in amorphous semiconductors: (i) valence-extended to conduction-extended states, (ii) valence-tail to conduction-extended states, (iii) valence-extended to conduction-tail states, and (iv) valence-tail to conduction-tail states. These possibilities will have different optical gap energies, Eg , and different binding energies. Possibility (i) will give rise

137

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5 Concept of Excitons

to absorption as in the free exciton states, possibilities (ii) and (iii) will give absorption in the bound exciton states because one of the charge carriers is localized in the tail states, and the absorption through possibility (iv) will create localized excitonic geminate pairs. This can be visualized as follows: if an electron – hole pair is excited by a high-energy photon through possibility (i) and forms an exciton, initially its excitonic energy level and the corresponding Bohr radius will have a reduced mass corresponding to both charge carriers being in the extended states. As such an exciton relaxes downward nonradiatively, the binding energy and excitonic Bohr radius will change because the effective mass changes in the tail states. When both charge carriers reach the tail states, possibility (iv), although the pair is localized, its excitonic Bohr radius will be maintained. In this situation, the excitonic nature breaks down as both charge carriers are localized, and the excitonic wave function cannot be used to calculate any physical property of these localized carriers. One has to use the individual localized wave functions of the electron and hole. For calculating the excitonic absorption coefficient in amorphous semiconductors, one can use the same approach as presented in Chapter 3, and similar expressions, such as Eqs. (3.4) and (3.14), will be obtained. This is because (i) the transition matrix element remains the same for excitonic absorption and for band-to-band free-carrier absorption, and (ii) the concept of the joint density of states applied to excitonic absorptions in crystalline solids is not applicable to excitonic absorptions in amorphous solids. Therefore, by replacing the effective masses of charge carriers by the excitonic reduced mass and the distance between the excited electron and hole by the excitonic Bohr radius, one can use Eqs. (3.4) and (3.14) for calculating the excitonic absorption coefficients for transitions corresponding to the preceding four possibilities in amorphous semiconductors. Thus, one obtains two types of excitonic absorption coefficients for amorphous solids. The first type is obtained by assuming that the transition matrix element is independent of the photon energy but depends on the excitonic Bohr radius as: 1∕2

[αℏ𝜔]1∕2 = Bx [ℏ𝜔 − Eg ]

(5.16)

where Bx =

𝜇x e 2 ax 4ncε0 ℏ3

(5.17)

The coefficient Bx in Eq. (5.17) is obtained by replacing m∗e and m∗h in Eq. (3.11) by 𝜇x , and L by ax . The absorption coefficient derived in Eq. (5.16) has the same photon-energy dependence as Tauc’s relation obtained for the band-to-band free carrier absorption (see Chapter 3). The second expression is obtained by applying the dipole approximation as: [αℏ𝜔] = B′x (ℏ𝜔)2 (ℏ𝜔 − Eg )2

(5.18)

where B′x =

3 𝜇xa 2 e2 x ncε0 2𝜋 2 ℏ7 𝜐𝜌A

(5.19)

Here, again, reh in Eq. (3.15) has been replaced by ax . Thus, the excitonic absorption coefficient depends on the photon energy in the same way as does the band-to-band absorption coefficient described in Chapter 3. This is probably the reason why distinct excitonic absorption peaks in amorphous semiconductors have not yet been observed

5.4 Excitons in Organic Semiconductors

to the best of our knowledge, but excitonic photoluminescence peaks have been observed [13].

5.4 Excitons in Organic Semiconductors The organic semiconductors consist of organic molecules, which are usually hydrocarbons. A hydrocarbon molecule usually consists of hydrogen and carbon atoms bonding sometimes with other atoms of oxygen and nitrogen, depending on the structures. The organic solids are formed by the weak Van der Waals forces, leading to weak bonding caused by the weak overlap of the electronic wave functions between neighboring molecules. The result of this weak bonding is that the inter-molecular separation in organic solids is usually much larger than those in inorganic solids, leading to much narrower electronic bands than in inorganic solids, and hence the energies of the valence and conduction bands of solids can be well approximated by those of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbitals (LUMOs) of individual molecules, respectively. In the earlier studies on excitons in organic solids, these used to be referred as molecular excitons, because both the excited electron and hole are excited on the same molecule to form Frenkel excitons [1, 14]. Although approximate, it is now fully accepted that the HOMO and LUMO of individual molecules represent the valence and conduction bands, respectively, and the corresponding energy gap represents the energy gap of an organic semiconductor. As described in the following text, if not all, most optical, electronic, and opto-electronic properties of organic semiconductors are studied using the HOMO and LUMO energies for the energy band energies. There are two classes of organic semiconductors: (i) low-molecular-weight compounds, and (ii) polymers. The common feature of both classes is that the constituent molecules are planar, and their carbon atoms are bonded through sp3 hybridization. In such a bonding, the 𝜎 bonds are formed between the planar s, px , and py orbitals, and these bonds are much stronger, contributing to the stability of the planar structure. The pz components of the p-bonds, known as the 𝜋-bonds, are loosely bonded and form delocalized conjugated bonds, which contribute to the electronic and optical properties of molecules. As a result, the low-energetic electron excitations of conjugated molecules occur through π-π*–type transitions from the HOMO to the LUMO energy levels. As described earlier, the energy gap between the LUMO and HOMO is identified as the energy band gap in organic semiconductors, which is usually between 1.5 and 3 eV. Such transitions cause the absorption or emission in the visible region of solar radiation. The energy gap may be controlled by the degree of conjugation of the individual systems, which opens various possibilities for the modification of optoelectronic properties of organic semiconductors [15, 16]. Another common aspect in the electronic and opto-electronic properties of organic semiconductors is that, as soon as electronic excitation occurs from HOMO to LUMO energy levels, either through optical excitations or charge carrier injections, the excited electron and hole pairs in organic semiconductors form Frenkel excitons instantaneously. This is because the dielectric constant of most organic solids is about 3–4, which is much lower than that of inorganic semiconductors—for example, it is 12 in Si. 2 This means that the binding energy EB = 𝜅eεr between the excited pair of electron (e)

139

140

5 Concept of Excitons

and hole (h) is at least four times larger than that in Si. (Here, k = (4𝜋εo )−1 = 9 × 109 , e is electronic charge, ε is the dielectric constant of the organic material, and r is the average separation between the excited e-h pair.) This is the reason why organic semiconductors are also known as excitonic semiconductors, and this is the key difference between inorganic semiconductors such as Si and Ge and organic semiconductors such as poly(thiophene) (P3HT) and poly(fluorene-alt-pyridine) (PFPy). In this section, a theoretical study of the optical properties of organic semiconductors is examined with a view to review its applications in devices such as organic solar cells (OSCs) and organic light-emitting devices (OLEDs). From the theory point of view, the operation of OSCs is opposite to that of OLEDs based on electroluminescence (EL): one shines light on OSCs to generate current, and one passes current in OLEDs to generate light. In practice, however, it is not a device with an identical structure on which if you shine light it will generate current and if you pass current it will generate light. The device structure of the organic layers in OSCs and OLEDs is very different from each other. For understanding the operation of OSCs and OLEDs, it is very important to understand the opto-electronic properties, such as photon absorption, photoluminescence, electroluminescence, and charge carrier generation in organic semiconductors. In Section 5.4.1, the theory of optical absorption is presented, and exciton generation, photoluminescence, and electroluminescence are presented in Section 5.4.2, followed by the exciton dissociation in Section 5.4.3. 5.4.1

Photoexcitation and Formation of Excitons

Absorption of light of energy larger than or equal to the bandgap of an organic solid results in the production of excited electron (e) and hole (h) pairs. An electron is excited to the LUMO, leaving a hole in the HOMO, which instantly forms an exciton due to the strong Coulomb interaction between e and h caused by the low dielectric constant (ε = 3–4) of organic solids. In organic solids, as stated earlier, the energy bandgap is equal to the energy difference between the LUMO and HOMO energy levels. As the electronic intermolecular interaction is weak in organics, the formation of such excitons is usually of Frenkel type [17–20]. There are two possible spin configurations for the formation of excitons: (i) singlet excitons, and (ii) triplet excitons, as shown in Figure 5.5. Absorption of a photon in any organic (inorganic) semiconductor can excite an electron from the HOMO (valence band) to the LUMO (conduction band), leaving a hole behind in the HOMO (valence band). Such an excited state will usually be a

(a)

+

,

,

–

(b)

Figure 5.5 Schematic diagrams of (a) one-spin configuration for the singlet; and (b) three-spin configurations for the triplet excitons.

5.4 Excitons in Organic Semiconductors

singlet because of the spin conservation. In organic semiconductors, usually a triplet state lies at a lower energy than a singlet state, and hence it can be excited from the singlet excited state through intersystem crossing (ISC) [21]. The mechanism of ISC will be described in the following text (Section 5.4.1.2). The triplet state can also be excited directly by photons of energy larger than or equal to the triplet mediated by the exciton–spin–orbit–photon interaction operator, as described in the following text. The mechanism of absorption of a photon and the formation of a singlet exciton due to exciton–photon interaction is presented in Section 5.4.1.1, and for a triplet exciton due to the exciton-spin-orbit-photon interaction in Section 5.4.1.2. 5.4.1.1

Photoexcitation of Singlet Excitons Due to Exciton–Photon Interaction

We consider an organic solid in which an electron is excited directly from HOMO to LUMO upon photon absorption without any change in its spin configuration. The inter̂Is , between radiation and a pair of electron and hole is given by [6]: action operator, H ) ( e e HIs = − (5.20) p − p .A m∗e e m∗h h where the subscript I and s denote interaction and singlet, respectively; m∗e and m∗h are the effective masses; and pe and ph are the linear momenta of the excited electron and hole, respectively; e is the electronic charge; and A is the vector potential of photons, given by: ∑ A0s (ε̂𝜆 e−i𝜔𝜆 t C𝜆+ + ε𝜆 C𝜆 ei𝜔𝜆 t ) (5.21) A= 𝝀

)1∕2 ( where A0s = 2ε nℏ2 V 𝜔 , n is the refractive index, V is the illuminated volume of the 0 𝜆 material, 𝜔𝜆 is the photon frequency, ε0 is the vacuum permittivity, ℏ is the reduced Planck’s constant, and ε̂𝜆 is the unit polarization vector of photons. C𝜆+ (C𝜆 ) is the creation (annihilation) operator of a photon in mode 𝜆. The first term of A represents the emission, while the second term relates to absorption of a photon. In this chapter, only the absorption term is considered from here onward, and emission will be considered in Chapter 6 on photoluminescence. Using the center of mass coordinate, Rx = (m∗e re + m∗h rh )∕M, where M = m∗e + m∗h and ̂as for singlet, can relative coordinate r = (re − rh ), the absorption interaction operator, H be written as [22, 23]: ( )1 2 e ∑ ℏ Has = − ei𝜔𝜆 t (ε𝜆 .p)C𝜆 (5.22) 2 𝜇x 𝜆 2ε0 n V 𝜔𝜆 where the H as denotes the absorption interaction operator, with only the second term of Eq. (5.21), and p = − iℏ∇r is the relative momentum between electron (e) and hole (h). The interaction operator in Eq. (5.22) can be written in the second quantized form as [22]: ( )1 ∑ 2 e ℏ ̂as = − ei𝜔𝜆 t Zlm𝜆 B+LHlm C𝜆 (5.23) H 2 𝜇x 𝜆 2ϵ0 n V 𝜔𝜆 [ ( ) ( ) ( ) ( )] + + − 12 + a+Ll − 12 dHm + 12 [5] is the creation operwhere B+LHlm = √1 a+Ll + 12 dHm 2 ator of a singlet exciton by exciting an electron in LUMO at site l and a hole in HOMO at

141

142

5 Concept of Excitons

site m, and Zlm𝜆 = ∫ 𝜙Ll (re )ε𝜆 .p𝜙H m (rh )dre drh . Applying Fermi’s golden rule and two-level ̂ approximation with Has in Eq. (5.23) as the perturbation operator, the rate of absorption of photons to generate singlet excitons in organic semiconductors is obtained as [22]: √ 4𝜅e2 εa2s (ELUMO − EHOMO )3 Ras = (5.24) 3ℏ4 c3 ε1.5 where 𝜅 = (4𝜋ε0 )−1 = 9 × 109 , c is the speed of light, n2 = ε is used to write Eq. (5.24) in terms of the static dielectric constant ε, and as is the excitonic Bohr radius of singlet excitons. 5.4.1.2

Excitation of Triplet Excitons

The triplet excitons can be excited in two ways in organic semiconductors—directly through strong exciton–spin–orbit–photon interactions, and indirectly through the ISC, as described in the following text. Direct Excitation to Triplet States Through Exciton–Spin–Orbit–Photon Interaction By absorp-

tion of a photon of adequate energy, an electron can be directly excited to a triplet spin configuration by flipping its spin through the electron–spin–orbit–photon interaction. Using the time-dependent exciton–spin–orbit–photon interaction operator derived in [24, 25] as a perturbation operator and Fermi’s golden rule, the rate Rat of triplet exciton absorption is obtained as [22, 24]: Rat =

32e6 Z2 𝜅 2 ε(ELUMO − EHOMO ) 𝜇x4 c7 𝜀0 a4t

(5.25)

where Z is the heaviest atomic number present in the organic molecule/solid, and at is the excitonic Bohr radius of triplet excitons. The rates of singlet and triplet absorption derived in Eqs. (5.24) and (5.25), respectively, are sensitive to the energy gap, (ELUMO − EHOMO ), of the organic material. For various organic materials used as donors in the fabrication of OSCs, the rates of formation of singlet and triplet excitons are calculated using their (ELUMO − EHOMO ) obtained from the experimental studies and presented in Table 5.1. The inverse of these rates gives , i = s or t). the time of formation of the corresponding exciton, (𝜏ai = R−1 ai According to Eqs. (5.24) and (5.25), Ras does not depend on Z, but Rat depends on Z2 . Thus, in the case of a triplet excitation, the higher the Z value, the faster will be Rat . Normally, organic materials consist of hydrocarbons, where carbon, C, has the highest atomic number. In experimental studies [5, 26, 27, 36, 37], it is found that incorporating the heavy metal atoms such as Ir in the donor organic materials enhances the photon–electron conversion efficiency of OSCs. Therefore, we have calculated the rates and corresponding formation times of singlet and triplet excitons with Z = 6 for C and Z = 77 for Ir, for several organic materials used as donors in OSCs, and the results are presented in Table 5.1. For all organic materials used as donors and listed in Table 5.1, the rate of excitation of singlet excitons is five orders of magnitude larger than that of triplet excitons. This is because Ras is highly sensitive to the absorption energy/energy gap Eg3 = (ELUMO − EHOMO )3 , while Rat is only linearly dependent on Eg . According to Table 5.1, TFB has the highest absorption energy of 3.59 eV and provides the fastest rate of absorption of 6.36 × 1010 s−1 for singlet excitons and 5.28 × 105 s−1 for triplet excitons

5.4 Excitons in Organic Semiconductors

Table 5.1 Rates of absorption of photons to form singlet excitons due to exciton–photon interaction and triplet excitons due to exciton–photon–spin–orbit interaction, calculated using Eqs. (5.24) and (5.25), respectively, at the listed energy gaps E g = E LUMO − E HOMO (eV) of various donor organic materials. Organic material

Eg

Ras 1010

𝝉 as 10−11

Rat (C) 103

𝝉 at (C) 10−4

Rat (Ir) 105

𝝉 at (Ir) 10−6

(eV)

(s−1 )

(s)

(s−1 )

(s)

(s−1 )

(s)

Ref.

PCBM

2.40

1.90

5.26

2.14

4.67

3.53

2.83

[26]

PFPy

2.94

3.49

2.86

2.62

3.81

4.32

2.31

P3HT

2.10

1.27

7.85

1.87

5.34

3.09

3.24

[27]

α-NPD

3.10

4.10

2.44

2.77

3.61

4.56

2.19

[28]

P3OT

1.83

0.84

11.9

1.63

6.12

2.69

3.72

[29]

Pt(OEP)

1.91

0.96

10.4

1.70

5.87

2.81

3.56

[30]

MEV-PPV

2.17

1.41

7.12

1.94

5.16

3.19

3.13

[31]

PPV

2.80

3.02

3.31

2.50

4.00

4.12

2.43

[32]

MDMO-PPV

2.20

1.46

6.83

1.96

5.09

3.23

3.09

[33]

PEOPT

1.75

0.74

13.6

1.56

6.40

2.57

3.89

[34]

PTPTB

1.70

0.68

14.8

1.52

6.59

2.50

4.00

BBL

1.90

0.94

10.6

1.70

5.90

2.79

3.58

F8BT

1.80

0.80

12.5

1.61

6.22

2.65

3.78

PFB

2.81

3.05

3.28

2.51

3.99

4.13

2.42

TFB

3.59

6.36

1.57

3.20

3.12

5.28

1.89

[35]

Abbreviations: PCBM, [6,6]-phenyl-C61 -butyric acid methyl ester; PFPy, poly(fluorene-alt-pyridine); P3HT, poly(thiophene); α-NPD, N,N′ -diphenyl-N,N′ -bis(1-naphthyl)-1-1′ biphenyl-4,4′′ diamine; P3OT, poly(3-octylthiophene-2,5-diyl); Pt(OEP), platinum octaethyl porphyrin; MEV-PPV, poly(2-methoxy-5-(2′ -ethylhexyloxy)-1,4-phenylenevinylene); MDMO-PPV, poly[2-methoxy-5-(3′ ,7′ -dimethyloctyloxy)-1,4-phenylenevinylene]; PEOPT, poly(3-(4′ -(1′′ ,4′′ ,7′′ -trioxaoctyl)phenyl)thiophene); PTPTB, poly-(N-dodecyl-2,5-bis(2′ -thienyl)pyrrole-(2,1,3-benzothiadiazole)); BBL, poly(benzimidazobenzophenanthroline ladder); F8BT, poly(9,9-dioctylfluorene-cobenzothiadiazole); PFB, poly(9,9-dioctylfluorene-co-bis-N,N-(4-butylphenyl)-bis-N,N-phenyl-1,4-phenylenediamine); TFB, poly(9,9-dioctylfluoreneco-N-(4-butylphenyl) diphenylamine).

as compared to other organic materials. Therefore, utilizing wide-band-gap materials can be expected to greatly enhance the singlet and triplet absorption rates in OSCs. According to Table 5.1, the rate of triplet excitation by incorporating Ir atom (Z = 77) increases by two orders of magnitude in all donors than without it. This clearly shows that the incorporation of heavy-metal atoms enhances the rate of triplet excitation in OSCs due to enhanced exciton–spin–orbit–photon interaction. Indirect Excitation of Triplet Excitons Through Intersystem Crossing and Exciton–Spin–Orbit– Phonon Interaction In this case, an exciton is already excited in the singlet exciton state

in an organic molecule of an organic solid, and the molecule also has a triplet state at a lower energy by ΔE such that the molecular vibrational energies of singlet and triplet overlap as shown in Figure 5.3. If the singlet exciton is excited at a higher molecular vibrational energy state of a molecule overlapping with a molecular vibrational energy

143

144

5 Concept of Excitons

state of the triplet state, then the singlet can get transferred to the triplet; however, the spin configuration will not flip to a triplet configuration. The flipping of the spin can only occur through the exciton–spin–orbit interaction. Accordingly, the mechanism of ISC can occur through the combination of excitons, molecular vibrations, and spin–orbit interactions. It is interesting to note that, although the ISC has been known for very long, no such interaction operator was known to exist in the literature until recently. We have derived a new exciton–spin–orbit–phonon interaction operator suitable for ISC in organic solids [21], as described in the following text. The stationary part of the spin–orbit interaction for an exciton in a molecule consisting of N atoms can be written as [21, 24]: ( ) N Zn e2 g𝜅 ∑ Zn (5.26) s ⋅ l + 3 sh ⋅ lhn , HSO = − 2 3 e en 2𝜇x c2 n ren rhn where se (sh ) is the electron (hole) spin, len = ren × pe is the electron angular momentum, and ren (pe ) is the position vector (orbital momentum) of the electron from the nth nucleus. Similarly, lhn = rhn × ph is the hole angular momentum, and rhn (ph ) is the position vector (orbital momentum) of the hole from the nth nucleus. For a non-rigid structure, Eq. (5.26) can be expanded in Taylor series about the equilibrium positions of molecules. Terminating the expansion at the first order, we get: 0 1 + HSOv HSO = HSO

(5.27)

0 where HSO is the zeroth-order term and represents the interaction in a rigid structure, 1 and HSOv is the first-order term that gives the interaction between exciton–spin– orbit–molecular vibration interactions, and it is obtained as: ( ) 3e2 g𝜅Z ∑ se ⋅ le sh ⋅ lh 1 Rnv , + 4 (5.28) HSOv = − 2𝜇x2 c2 n,v re4 rh

where Rnv is the molecular displacement from the equilibrium position due to the intramolecular vibrations. In Eq. (5.28), the subscript n on ren and rhn is dropped using −4 the approximation that the quantity within parentheses in Eq. (5.28) depends on ren −4 and rhn ; thus, the nearest and heaviest nucleus to the electron and hole is expected to play the dominant influence, and, as such, the presence of other nuclei may be neglected. This approximation helps in reducing the summation to only one nucleus for each electron and hole, and hence the subscript n will be dropped here onward. In carrying out the Taylor series expansion, it is further assumed that the distances ren and rhn of the electron and hole with reference to the individual nuclei of a molecule can be replaced by their distances re and rh , respectively, with reference to the equilibrium position of the individual molecules. This approximation may be regarded to be quite justified within the Born–Oppenheimer approximation regime. Applying the preceding approximation, Rnv = Rv can be expressed in the second quantization as [21, 38]: Rv = (qv0 − q00 )(b+v + bv ),

(5.29)

where b+v (bv ) is the vibrational creation (annihilation) operator in vibrational mode v.

5.4 Excitons in Organic Semiconductors

For expressing the operator in Eq. (5.28) in the second quantization, we can write the field operator for an electron in the LUMO and that of a hole in the HOMO, respectively, as: ∑ 𝜑LUMO aL (𝜎e ), (5.30a) 𝜓 ̂e = 𝜎

e ∑ 𝜑HOMO dH (𝜎h ), 𝜓 ̂h =

𝜎h

dH (𝜎h ) = a+H (−𝜎h ),

(5.30b)

where 𝜑LUMO and 𝜑HOMO are the wavefunctions of the electron in the LUMO and hole in the HOMO, respectively. Using Eqs. (5.30), the interaction operator in Eq. (5.28) can be expressed in the second quantization as: 24e2 g𝜅Z ∑ ̂1 ≈ − H (qv0 − q00 )aL (𝜎e )dH (𝜎h )𝛿𝜎e ,𝜎h (se .le + sh .lh )(b+v + bv ) (5.31) SOv 𝜇x2 c2 rx2 v,𝜎e ,𝜎h where rx is the average separation between the electron and hole in the exciton, and it is approximated as: ( r )−4 ⟨𝜑HOMO ∣ re−4 ∣ 𝜑LUMO ⟩ ≈ ⟨𝜑HOMO ∣ rh−4 ∣ 𝜑LUMO ⟩ ≈ x . (5.32) 2 Using the interaction operator in Eq. (5.28) as a perturbation operator and Fermi’s golden rule, the rate of intersystem K isc crossing is obtained as [21]: Kisc =

3072𝜋 2 𝜀6 𝜅 2 Z2 e4 ℏ3 (ΔE)2 𝜇x4 c4 a6x (ℏ𝜔v )3

s−1

(5.33)

where ΔE is the exchange energy difference between singlet and triplet energy states (see Figure 5.6), ax is the excitonic Bohr radius, and ℏ𝜔v is the molecular vibrational energy. It may be noted that it is the singlet excitonic Bohr radius ax that should be used in the calculation, which is the state prior to the transition. The derivation of the rate in Eq. (5.33) also clarifies how the phenomenon of ISC occurs. An exciton is first excited to the singlet exciton state, which is higher in energy than the triplet state. The vibrational Figure 5.6 Schematic illustration of a singlet excited state whose molecular vibrational energy states overlap with those of a triplet excited state that is at a lower electronic energy by ΔE.

ΔE Triplet excited state

145

146

5 Concept of Excitons exp Table 5.2 The calculated intersystem crossing rate K isc from Eq. (5.33) and experimental rates (Kisc ) for some OSC materials along with their highest atomic number (Z) and singlet–triplet energy difference (ΔE). Here we have used ε = 3, ax = 4.352nm, and 𝜔v = 8 × 1014 s−1 .

Organic material

Z

𝚫E (eV)

K isc (s−1 )

exp −1 Kisc (s )

NPD (Ir doped)

77

0.90

1.1 × 1011

[39]

CBP (Ir doped)

77

0.90

1.1 × 1011

[39]

P3HT

16

0.80

3.7 × 109

SubPc

9

0.71

9.2 × 108

9.1 × 108

[41]

F8BT

16

0.70

2.8 × 109

1.2 × 107

[42]

8

6

Ref.

[40]

Toluene

6

0.70

4.0 × 10

8.5 × 10

[43]

Naphthalene

6

1.47

1.8 × 109

5.0 × 106

[44]

1-Bromonaphthalene

35

1.30

4.7 × 1010

≈×109

[45]

Benzophenon

16

0.30

5.2 × 108

≈×1010

[44]

>×10

[46]

Platinum-acetylide

78 ′

10

0.80

8.8 × 10

11

′

Abbreviations: NPD, N,N -bis (naphthalen-1-yl)-N, N -bis(phenyl)-benzidine; CBP, 4,4′ -bis(9-carbazolyl)-1,1′ -biphenyl; P3HT, poly(3-hexylthiophene); SubPc, boron subphthalocyanine chloride; F8BT, poly(9,9-dioctylfluorene-cobenzothiadiazole).

states of both singlet and triplet overlap in energy as shown in Figure 5.6. Thus, the exciton–spin–orbit–molecular vibration interaction flips the spin to the triplet state, which then gets transferred to the triplet state. This is the reason why K isc in Eq. (5.33) vanishes if ΔE = 0. It is unlikely that the singlet and triplet states in any molecule can become isoenergetic, and, if that is the case, then the singlet absorption will dominate. The rate of intersystem crossing K isc in Eq. (5.33) is calculated for some organic materials used in the fabrication of OSCs and listed in Table 5.2. Also listed in Table 5.2 are the corresponding experimental values of K isc for comparison. According to Table 5.2, the calculated rates K isc are found to be in reasonable agreement with experimental results, and minor discrepancies may be attributed to the approximations used in deriving Eq. (5.33). The rate in Eq. (5.33) can be applied to calculate the ISC rate in any molecular solids. Comparing the rates derived in Eqs. (5.24), (5.25), and (5.33) as listed in Tables 5.1 and 5.2, we find that, in most organic solids, the rate of singlet absorption Ras is of the order of 1011 s−1 and formation time 𝜏 as ≥ 10 ps, and the rate of the direct triplet absorption is Rat ≈ 103 s−1 and 𝜏 at ≈ 10−4 s without the involvement of any heavy metal atoms, but with the incorporation of Ir as the heavy metal atom, we get Rat ≈ 105 s−1 and 𝜏 at ≈ 1 μs. In comparison, the rate of intersystem crossing K isc ≥ 108 s−1 (𝜏 isc ≈ 1 ns) without the metal atom and K isc ≥ 1011 s−1 (𝜏 isc = 1 ps) with Ir atoms involved. It is interesting to note that the rate of ISC to form triplet excitons is as fast as the rate of formation of singlet excitons. If the organic solids do not have any heavy metal atoms, then the rate of formation of triplet excitons via direct transitions is three orders of magnitude lower than that via the ISC. It may however be noted that there is some energy loss in exciting triplet excitons through the ISC, which is lost as thermal vibrational energy and may heat a device.

5.4 Excitons in Organic Semiconductors

5.4.2

Exciton Up-Conversion

In the preceding text, we have described the theory of ISC, where an exciton in singlet state converts into a triplet state through the exciton–spin–orbit interaction, which flips the spin and vibrational energy overlap that moves the exciton from singlet to triplet state. The reverse process of converting a triplet exciton into a singlet exciton is also possible and is applicable in the operation of OLEDs, where electrons and holes are injected from the opposite electrodes (see Chapter 7). As shown in Figure 5.5, there is statistically only one possible spin configuration for forming singlet excitons, but there are three spin configurations possible for forming triplet excitons. However, the radiative recombination of triplet excitons is not spin allowed unless one has strong spin–orbit interaction, which is usually not the case in organic solids. In such a situation, only the singlet excitons emit light, and that limits the internal quantum efficiency of OLEDs to 25%, because triplets cannot recombine and hence cannot emit radiation. If the triplets could easily be converted into singlets, then the internal quantum efficiency of 100% can be achieved. This is the principle of operation of OLEDs based on thermally activated delayed fluorescence (TADF) [47]. In organic solids, it is relatively easy to substitute another molecule (a donor) whose singlet energy may be close to the triplet state of the host solid (acceptor) and also to introduce heavy metal atoms to enhance the exciton–spin–orbit interaction. If the energy difference is very small, the up-conversion can occur even at room temperature. The thermal energy will be adequate to raise the vibrational energy of the triplet exciton to overlap with the singlet state of the donor to enable the transfer by flipping the spin through the exciton–spin–orbit interaction, as schematically shown in Figure 5.7. The rate of up-conversion also known as the reverse intersystem crossing (K RISC has recently been derived [48], and it should be the same as the rate of ISC (Eq. (5.33))), multiplied by the Boltzmann factor as: ( ) ΔEST , (5.34) KRISC = KISC exp − kb T where k b is the Boltzmann constant and T is temperature. For an efficient reverse ISC, the radiative lifetime of singlet state must be shorter than that of the triplet state, and the energy gap between singlet and triplet excited states must be smaller than 100 meV [47]. An example of thermally activated up-conversion material is 2-biphenyl-4,6-bis (12-phenylindolo [2,3-a]carbazole-11-yl)-1,3,5-triazine (PIC-TRZ), containing an indolocarbazole donor unit and a triazine acceptor unit [47]. Figure 5.7 Schematic illustration of thermally activated up-conversion or reverse intersystem crossing from a triplet exciton to singlet exciton state with energy difference ΔE ST . S0 , S1 , and T 1 are the ground, first excited singlet, and first excited triplet exciton states of an organic molecule.

∆EST S1 T1

S0

147

148

5 Concept of Excitons

Following the up-conversion to a singlet state, fluorescence occurs due to the radiative recombination of thus up-converted singlet excitons, known as the TADF. Using TADF, in an OLED, the generated triplet excitons, which cannot contribute to electroluminescence due to unfavorable spin configurations, can be made to recombine radiatively, and thus the internal quantum efficiency can be enhanced through the up-conversion. The synthesized molecule of PIC-TRZ exhibits a very small ΔEST , providing both efficient up-conversion from T1 to S1 levels and intense fluorescence that leads high electroluminescence (EL) efficiency. Both the processes of intersystem and reverse ISCs are nonradiative. The fluorescence- or radiative-recombination-related processes are described in more detail in Chapter 6. 5.4.3

Exciton Dissociation

As described in the preceding text, photo-excitations in organic semiconductors form excitons instantly, and excitons are the excited pairs of e and h bound in hydrogen-like electronic states. These are not free excited pairs of charge carriers, and, therefore, the formation of excitons in devices such as OSCs is not good for their operation, because one needs to dissociate excitons into free electron (e) and hole (h) excited pairs that can be transported to the opposite electrodes of OSCs to produce photocurrents. On the contrary, formation of excitons is good for OLEDs, because it enables radiative recombination and emission of light. In this section, the dissociation of excitons in the bulk heterojunction (BHJ) OSCs is described in detail. In the development of OSCs, first a single layer of organic semiconductor sandwiched between anode and cathode was prepared. In such a structure, there is no force other than the electrical force available due to the difference in work functions of the two electrodes to dissociate the excitons excited within the organic layer. The built-in electric field due to the difference in the work functions of the electrodes is given by: 𝜙c − 𝜙a , (5.35) r where 𝜙c and 𝜙a are the work functions of the cathode and anode, respectively, and r is the separation between them, which is about the total thickness of an OSC. However, as the exciton is an electrically neutral entity, the built-in electric field is expected to have little contribution in the exciton dissociation. This built-in electric field can only contribute in charge separation and collection after the excitons have dissociated. As a result, the single-layered OSCs have very poor power conversion efficiency (PCE). Then, in 1986, a bilayer concept was invented by Tang [49] with one layer of a donor material, which can easily give away its electrons, and the second layer of an acceptor material, which can easily accept electrons. The two layers have an interface between them and sandwiched between two electrodes. Many combinations of donor and acceptor materials have been invented and tried for OSCs. The important point in selecting the donor and acceptor materials is that the LUMO and HOMO of a donor molecule are at higher energies than those of an acceptor molecule, as shown in Figure 5.8. It is generally accepted that the generation of photo charge carriers in a bilayer OSC occurs through the following five processes in sequence [38, 50]: (i) photon absorption from the sun in the donor and/or acceptor excites electron–hole pairs that instantly form neutral Frenkel excitons; (ii) diffusion of the excited excitons to the donor–acceptor F=

5.4 Excitons in Organic Semiconductors

∆ELUMO

LUMO

LUMO

CT exciton (1) CT exciton (2) HOMO ∆EHOMO Donor

HOMO Acceptor

Figure 5.8 Schematic illustration of the formation of CT excitons at the D-A interface in a BHJ OSC. CT exciton (1) is formed when Frenkel exciton is excited in the donor, and CT exciton (2) is formed when Frenkel exciton is excited in the acceptor.

(D-A) interface; (iii) formation of charge transfer (CT) excitons at the D-A interface by transferring the electron to the acceptor, from excitons excited in the donor and/or by transferring the hole to the donor from excitons excited in the acceptor [51]; (iv) dissociation of the CT excitons at the D-A interface; and (v) transport and collection of the dissociated free charge carriers at their respective electrodes to generate photocurrent, which is the main purpose of any solar cell. In a bilayer structure, excitons are required to diffuse to the D-A interface to form CT excitons, leading to the subsequent dissociation. How exactly the dissociation of CT excitons takes place will be discussed later in text, but it may be noted that, an exciton being electrically neutral, it cannot be directed to move in any particular direction by any external or built-in electric field. Therefore, an exciton can only diffuse from one point to another in a random motion, and when it reaches the D-A interface, it will form a CT exciton. This requires that the exciton diffusion length LD be larger than the thickness of the donor or acceptor layers. In organic semiconductors, LD is short (∼10 nm), and hence the bilayer-structured OSCs also have very poor PCE. One way forward is to make the active organic layer from a blend of the donor and acceptor materials, and the structure thus obtained is called BHJ OSC. BHJ OSCs are one of the most promising alternative photovoltaic technologies due to the advantages of high absorption coefficient, light weight, flexibility, and the potential of low-cost solution process capability, etc. In a BHJ OSC also, the photogeneration of charge carriers occurs through the above five processes, but excitons do not have to diffuse to a D-A interface at a fixed distance. The BHJ OSCs have reached PCE more than 10% [43–46], with an expectation of achieving 15% in the very near future [45]. The processes 1–3 and 5, listed earlier for the operation of BHJ OSCs, have been quite well investigated and understood. The heterojunction structure is the only possible way of letting excitons diffuse and change into CT excitons at a nearby D-A interface. However, the process (iv) of the dissociation of CT excitons at the D-A interface assisted only by the built-in electric field can be very inefficient [52], as discussed in the following text, and hence the mechanism of dissociation has not been fully understood. Therefore, for an efficient dissociation of CT excitons into free electron and hole pairs, one must consider other possibilities. As both the donor and acceptor materials are organic semiconductors with similar dielectric constants, the formation of CT excitons from Frenkel excitons, excited either

149

150

5 Concept of Excitons

in the donor or acceptor material, neither makes the CT excitons loosely bound, nor does the electron and hole in a CT exciton become farther separated. This is because excitons excited in the donor molecules form CT excitons: (i) by electron transfer from the LUMO of the donor to the LUMO of the neighbor acceptor molecules, having a lower energy by ΔELUMO , across the D-A interface, and those excited in the acceptor molecules form CT excitons; and (ii) by hole transfer from the HOMO of the acceptor to the HOMO of the donor molecules, having a lower energy by ΔEHOMO , as clearly illustrated in Figure 5.8. For an efficient formation of CT excitons, it is required that both HOMO and LUMO levels of the donor be at higher energies than those of the acceptor. As the CT excitons are created by transferring the electrons and holes to lower energy states, it may be expected that the CT exciton states are even more stable than their predecessor Frenkel excitons, and hence cannot be dissociated in any way easier than the corresponding Frenkel excitons. The binding energy of singlet excitons is about 0.06 eV and that of a triplet is 0.7 eV in most organic solids. Therefore, the binding energy of the corresponding CT excitons is expected to be at least the same, if not larger. As the formation of a CT exciton involves two molecules—electron excited on an acceptor molecule and hole on a donor molecule—it is speculated that the electron and hole become farther apart in a CT exciton and hence can easily be dissociated due to the built-in electric field generated by the difference in the electrode work functions. A study on the dissociation of CT excitons by Devizis et al. [50] reveals that only charge pairs with an effective electron–hole separation distance of less than 4 nm are created during the dissociation of Frenkel excitons, which is about the same as the excitonic Bohr radius of singlet Frenkel excitons in organic solids [30]. Therefore, an exciton with a separation of 4 nm between their charge carriers is not dissociated yet, but it may dissociate after the formation of a CT exciton. As already described earlier, it is also to be noted that the built-in electric field due to the difference in work functions of electrodes cannot act efficiently on an electrically neutral particle, such as a CT exciton, and hence it cannot dissociate it. Therefore, the cause of dissociation of a CT exciton is puzzling and needs further investigation. In view of this, although the formation of a CT exciton is a prerequisite intermediate state for its dissociation, as it has been identified earlier [31, 53, 54], its dissociation requires some excess energy to overcome its binding, which is about the same as that of an exciton binding energy as explained earlier. It is usually assumed that, if the built-in electric field given in Eq. (5.35) is adequate to offer an energy larger than or equal to the binding energy of a CT exciton, it may be expected to dissociate. However, if CT excitons can be dissociated by this built-in electric field, then Frenkel excitons can also be dissociated, whether excited in the donor or acceptor, because, as discussed earlier, they have similar or even lower binding energies. However, that would mean that the formation of CT excitons at the D-A interface in BHJ OSCs plays no role, which is contrary to the observed higher efficiency in BHJ OSCs, and hence cannot be accepted. It has also been suggested that a CT exciton will subsequently dissociate following Onsager’s theory [55]. However, a CT exciton is a discrete quantum state holding a quantum of solar energy; it is not exactly a geminate pair of varying distance between the two charge carriers (e and h). In this situation, whether Onsager’s theory can be applied for the dissociation of an exciton becomes, on its own, a topic of debate and controversy. Although, there is a significant amount of work done in studying the morphology of the BHJ OSCs for efficient charge generation [52], not much has been discussed on the

5.4 Excitons in Organic Semiconductors

mechanism of dissociation of CT excitons. We have recently proposed [38, 56] that a CT exciton can be dissociated only if it is given an excess energy equivalent to or greater than its binding energy from an external source. The only possible source of such energy available in OSCs is due to the formation of CT excitons, which releases excess energy from the conversion of a Frenkel exciton (at higher energy) to a CT exciton (lower energy), as it has been modeled recently [28]. Thus, the dissociation of an exciton at the D-A interface can be regarded as a two-step process as described in the following text. 5.4.3.1

Conversion from Frenkel to CT Excitons

As an exciton is electrically neutral, it cannot be directed to move in any particular direction to reach the D-A interface. So while it is excited within a material sandwiched between an anode and cathode, it can diffuse in any direction randomly. To direct an exciton to move in any particular direction, one has to introduce an interface with an acceptor (another organic material with a lower LUMO energy) in that direction. In this way, a cascade kind of structure of LUMO and HOMO of donor and acceptor with their interfaces parallel to the electrodes can direct an exciton to move to the nearest D-A interface by forming a CT exciton, and then to the next D-A interface, and thus forming and remaining in a CT exciton state at each interface, as shown in Figure 5.9 for a cascade of three materials. In this case, in a blended structure of ternary materials, the excitons can be excited in each of the three materials. If the exciton is excited in the first donor, then the CT exciton can be formed at the first interface by transferring the electron to the LUMO of the first acceptor and thus releas1 . This CT exciton can then change to another CT exciton such ing an energy ΔELUMO that the electron is transferred to the second acceptor material by releasing an energy, 2 ΔELUMO , but the hole is still in the first donor material. In case the exciton was excited in the first acceptor material (middle donor/acceptor), then it has the possibility of forming

LUMO ∆E 1LUMO ∆E 2LUMO

LUMO

∆E 1HOMO HOMO

∆E 2HOMO HOMO

1 Figure 5.9 Schematic illustration of the formation of CT excitons in a ternary BHJ OSC, where ΔELUMO 2 1 2 and ΔELUMO represent the LUMO energy off-sets, and ΔEHOMO and ΔEHOMO represent the HOMO energy offsets at interfaces 1 and 2, respectively, counted from left to right.

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a CT exciton in two ways: (i) electron goes down to the third acceptor’s LUMO by releas2 ; and/or (ii) the hole goes up to the HOMO of the first donor by ing an energy ΔELUMO 1 releasing an energy ΔEHOMO . Finally, if the exciton is excited in the second acceptor (third material), it has only one way of forming a CT exciton at the second D-A interface by transferring its hole to the 2 , which may acceptor of HOMO of the middle material and releasing an energy ΔEHOMO then move to the first interface and form a CT exciton by transferring itself to the first 1 . donor and releasing an energy ΔEHOMO 5.4.3.2

Dissociation of CT Excitons

The dissociation of a CT exciton can only occur due to the mechanism of its formation, as discussed in our earlier work [28, 57]. Accordingly, as the formation of a CT exciton involves release of energy due to either transfer of an electron to a lower energy state of LUMO of the acceptor at a D-A interface or transfer of a hole to a higher HOMO level of the donor. If this released energy, usually in the form of molecular vibrational energy, impacts back on the CT exciton, it may dissociate, provided this energy is larger or at least equal to the binding energy of CT excitons. Thus, the formation of a CT exciton also involves its dissociation if the energy offsets are greater than or equal to the binding energy of the CT exciton. This may be the reason for assuming that the formation of a CT exciton leads to automatic dissociation of excitons, and hence the formation of CT state is a prerequisite state for the dissociation [50, 51]. Accordingly, the condition of dissociation of CT excitons can be given by [28]: ΔELUMO or ΔELUMO ≥ Eb , D ELUMO

(5.36)

A ELUMO

where ΔELUMO = − is the difference between the energy of donor LUMO, D A D A ELUMO , and that of acceptor LUMO, ELUMO ; ΔEHOMO = EHOMO − EHOMO is the differD A ence between the energy of donor HOMO, EHOMO , and that of acceptor HOMO, EHOMO ; and Eb is the binding energy of excitons (Figure 5.4). Based on the condition in Eq. (5.36) and using the newly derived exciton–molecular vibration interaction operator as a perturbation, the rate of dissociation of CT excitons is obtained as [28]: i,j

RD =

1 96𝜋 2 ℏ3 ε2 Ebi

[(EjD − EjA ) − Ebi ]2 (ℏ𝜔v )𝜇x a2xi

(5.37)

where i denotes parameters associated with S (singlet) and T (triplet) excitons, and j denotes LUMO when CT excitons are formed from excitons excited in the donor and HOMO when CT excitons are formed from excitons excited in the acceptor material. The rate given in Eq. (5.37) is applicable to the dissociation of both singlet and triplet excitons, but one has to use the corresponding parameters. In deriving the dissociation rate in Eq. (5.37), it is assumed that a singlet Frenkel exciton changes to a singlet CT exciton, and a triplet Frenkel exciton changes to a triplet CT exciton at the D-A interface. Accordingly, the exciton dissociation at a D-A interface may be regarded as a two-step process: (i) the formation of CT excitons at the interface; and (ii) if the condition in Eq. (5.36) is met, they may dissociate efficiently. Although the CT exciton formation at the D-A interface is a pre-requisite for its dissociation, it is not yet dissociated. As stated earlier, a CT exciton is no different in the binding energy than a Frenkel exciton, and,

5.5 Conclusions

similar to a Frenkel exciton, it is also an electrically neutral entity and hence cannot be influenced by the built-in electric field caused by the electrodes. Therefore, for an efficient dissociation of a CT exciton at the D-A interface, it is important that the condition in Eq. (5.36) be met. This is also supported by the experimental result by He et al. [53], who fabricated a BHJ OSC using PTB7 as donor and PC71 BM as acceptor, and reported a PCE of 9.2% for an inverted structure. We will not discuss here the effect of inverted structure on the PCE, but it is to be noted that the LUMO energy of PTB7 as donor is at −3.31 eV and that of PC71 BM as acceptor is at −4.3 eV, giving ΔELUMO = 0.99 eV, which is much larger than the binding energy of both singlet and triplet CT excitons, and hence satisfies the condition in Eq. (5.36) very well for dissociating excitons excited in the donor material. Likewise, for the dissociation of excitons excited in the acceptor PC71 BM with HOMO energy at −5.15 eV and HOMO of donor PTB7 at −6.1 eV, ΔEHOMO = 0.95 eV, which is also more than the binding energy of both singlet and triplet excitons, and satisfies the condition in Eq. (5.36). The dissociation rates calculated from Eq. (5.37) for excitons excited in the donor are 1.89 × 1014 s−1 for singlet and 4.82 × 109 s−1 for triplet excitons. The rates of dissociation for excitons excited in the acceptor are 1.64 × 1014 s−1 for singlet excitons and 3.16 × 109 s−1 for triplet excitons. These high rates imply faster dissociation of singlet than triplet excitons in PTB7: PC71 BM BHJ OSCs, leading to faster free charge carrier generation, resulting in enhanced photocurrent and hence higher power conversion efficiency. The concept of having ΔEji ≥ Ebi , I = S or T, and j = LUMO or HOMO for an efficient dissociation of CT excitons is very important, although it has not yet been fully realized.

5.5 Conclusions In this chapter, concepts of excitons in crystalline and amorphous solids, both inorganic and organic, are presented. Excitonic absorption in crystalline solids is reviewed. It is shown that, in amorphous solids, the excitonic absorption spectrum is similar to the band-to-band absorption spectrum. For example, the excitonic absorption in amorphous semiconductors also satisfies Tauc’s relation, as does band-to-band absorption. This is because: (i) the transition matrix element remains the same for excitonic absorption and for band-to-band free-carrier absorption, and (ii) the concept of the joint density of states applied to excitonic absorptions in crystalline solids is not applicable to excitonic absorptions in amorphous solids. In organic semiconductors, mainly Frenkel excitons are created, and the absorption of photons creating Frenkel excitons, both singlet and triplet, is described in detail. Two processes of exciting triplet excitons are described: (i) direct excitation to the triplet through the exciton–spin–orbit–photon interaction; and (ii) indirect excitation through the ISC caused by the exciton–spin–orbit–molecular vibration interaction. Likewise, two processes for the excitation of singlet excitons is described in organic solids: (i) direct excitation through the exciton–photon interaction; and (ii) indirect excitation through the up-conversion or reverse ISC. The process of dissociation of excitons in organic semiconductors at the donor– acceptor interface of BHJ OSCs is also described.

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References 1 Singh, J. (1994). Excitation Energy Transfer Processes in Condensed Matter. New

York: Plenum. 2 Singh, J., Birkedal, D., Lyssenko, V.G., and Hvam, J.M. (1996). Phys. Rev. B 53: 15909,

and references therein. 3 Birkedal, D., Singh, J., Lyssenko, V.G. et al. (1996). Phys. Rev. Lett. 76: 672. 4 Miller, R., Kleinman, D., Gossard, A., and Monteanu, O. (1982). Phys. Rev. B 25:

6545. 5 Singh, J. and Shimakawa, K. (2003). Advances in Amorphous Semiconductors. London 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

and New York: Taylor & Francis. Singh, J. and Oh, I.-K. (2005). J. Appl. Phys. 95: 063516. Singh, J. (1997). Nonlinear Optics, vol. 18, 171. Singh, J., Aoki, T., and Shimakawa, K. (2002). Philos. Mag. B 82: 855. Singh, J. (2002). J. Non-Cryst. Solids 444: 299–302. Singh, J. (2002). Nonlinear Opt. 29: 111. Singh, J. (2003). J. Mater. Sci. 14: 171. Elliot, R.J. (1962). Polarons and Excitons (eds. K.G. Kuper and G.D. Whitfield), 269. Edinburgh and London: Oliver & Boyd. Aoki, T., Koomedoori, S., Kobayashi, S. et al. (2002). Nonlinear Opt. 29: 273. Davydov, A.S. (1971). Theory of Molecular Excitons. New York: (Plenum. Brutting, ̀ W. (2005). Physics of Organic Semiconductors. Wiley-VCH. Swist, A. and Soloducho, J. (2012). Chemik 66: 289. Narayan, M.R. and Singh, J. (2012). Phys. Status Solidi C 9: 2386. Roncali, J. (2009). Acc. Chem. Res. 42: 1719. Kippelen, B. and Brédas, J.L. (2009). Energy Environ. Sci. 2: 251. Brédas, J.L., Norton, J.E., Cornil, J., and Coropceanu, V. (2009). Acc. Chem. Res. 42: 1691. Ompong, D. and Singh, J. (2016). Phys. Status Solidi C 13: 89. Narayan, M.R. and Singh, J. (2013). J. Appl. Phys. 114: 154515. Singh, J. and Williams, R.T. (eds.) (2015). Excitonic and Photonic Processes in Materials. Singapore: Springer, Ch. 8. Singh, J. (2007). Phys. Rev. B 76: 085205. Singh, J. (2011). Phys. Status Solidi A 208: 1809. Schulz, G.L. and Holdcroft, S. (2008). Chem. Mater. 20: 5351. Yang, C.-M., Wu, C.-H., Liao, H.-H. et al. (2007). Appl. Phys. Lett. 90: 133509. Singh, J. (2010). Phys Status Solidi C 7: 984. Shafiee, A., Salleh, M.M., and Yahaya, M. (2011). Sains Malays. 40: 173. Singh, J., Baessler, H., and Kugler, S. (2008). J. Chem. Phys. 129: 041103. Li, Y., Cao, Y., Gao, J. et al. (1999). Synth. Met. 99: 243. Da Costa, P.G. and Conwell, E. (1993). Phys. Rev. B 48: 1993. Thompson, B.C. and Fréchet, J.M. (2007). Angew. Chem. Int. Ed. 47: 58. Winder, C. and Sariciftci, N.S. (2004). J. Mater. Chem. 14: 1077. Bittner, E.R., Ramon, J.G.S., and Karabunarliev, S. (2005). J. Chem. Phys. 122: 214719. Xu, Z., Hu, B., and Howe, J. (2008). J. Appl. Phys. 103: 043909. Huang, J., Yu, J., Guan, Z., and Jiang, Y. (2010). Appl. Phys. Lett. 97: 143301. Narayan, M.R. and Singh, J. (2013). J. Appl. Phys. 114: 73510.

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The University of Utah. 41 Medina, A., Claessens, C.G., Rahman, G.M.A. et al. (2008). Chem. Commun.:

1759–1761. 42 Ford, T.A., Avilov, I., Beljonne, D., and Greenham, N.C. (2005). Phys. Rev. B 71:

125212. 43 Cogan, S., Haas, Y., and Zilberg, S. (2007). J. Photochem. Photobiol., A 190: 200. 44 Donald, L.P., Gary, M.L., George, S.K., and Engel, R.G. (2011). A Small Scale

Approach to Organic Laboratory Techniques, 415. Belmont, CA: Cengage Learning. 45 Klessinger, M. and Michl, J. (1995). Excited States and Photochemistry of Organic

Molecules, 255. New York: VCH. 46 Guo, F., Kim, Y.G., Reynolds, J.R., and Schanze, K.S. (2006). Chem. Commun.:

1887–1889. Endo, A., Sato, K., Yoshimura, K. et al. (2011). Appl. Phys. Lett. 98: 093302. Usman, S. and Singh, J. (2018). Org. Electron. 59: 121–124. Tang, C.W. (1986). Appl. Phys. Lett. 48: 183. Devizis, A., Jonghe-Risse, J.D., Hang, R. et al. (2015). J. Am. Chem. Soc. 137: 8192. Wright, M. and Uddin, A. (2012). Sol. Energy Mater. Sol. Cells 107: 87. Peumas, P., Yakimov, A., and Forest, S.R. (2003). J. Appl. Phys. 93: 3393. He, Z., Zhong, C., Su, S. et al. (2012). Nat. Photonics 6: 591. Lu, L., Zheng, T., Wu, Q. et al. (2015). Chem. Rev. 115: 12666. R.F. Service (2011). Science 332 (6027): 293. Singh, J., Narayan, M.R., Ompomg, D., and Zhu, F. (2017). J. Mater. Sci. - Mater. Electron. 28: 7095. 57 Narayan, M.R. and Singh, J. (2017). J. Mater. Sci. - Mater. Electron. 28: 7070–7076. 47 48 49 50 51 52 53 54 55 56

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6 Photoluminescence Takeshi Aoki Joint Research Center of High-technology, Department of Electronics and Information Technology, Tokyo Polytechnic University, Atsugi 243-0297, Japan

CHAPTER MENU Introduction, 157 Fundamental Aspects of Photoluminescence (PL) in Materials, 158 Experimental Aspects, 164 Photoluminescence Lifetime Spectroscopy of Amorphous Semiconductors by QFRS Technique, 175 QFRS on Up-Conversion Photoluminescence (UCPL) of RE-Doped Materials, 192 Conclusions, 197 References, 198

6.1 Introduction A radiative emission process in condensed matter is called luminescence, and such a luminescent material is sometimes called phosphors. Luminescence can occur through a variety of electronic processes unrelated to heat, among which photoluminescence (PL) and electroluminescence (EL) are most popularly used. The radiative process requires a non-equilibrium carrier concentration in the electronic band of solids or in the electronic state of an impurity or defect. If the non-equilibrium state (excited state) is created by photoexcitation (PE), the resulting luminescence is called PL, and if it is obtained by carrier injection through an electric field, it is called EL [1–6]. The study of luminescence from condensed matter is not only of scientific but also technological interest, because it forms the basis of solid-state lasers. Also, it is important for display panels in electronic equipment; lighting, such as in fluorescent lamps and phosphor-converted (white) light-emitting diodes (pcWLEDs); fluorescent paints; bioimaging; and photodynamic therapy (PDT). Undoubtedly, PL is frequently used as a non-destructive technique for material characterization or research in material science as well. PL spectroscopy is a sensitive tool for investigating both intrinsic electronic transitions between energy bands and extrinsic electronic transitions at impurities and defects of organic molecules, semiconductors, and insulators [6]. As a comprehensive review of PL spectroscopy used on all the condensed matter systems is beyond the scope of this chapter, it mainly focuses on a limited number of Optical Properties of Materials and Their Applications, Second Edition. Edited by Jai Singh. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

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topics related to PL recombination in disordered materials. In particular, the quadrature frequency resolved spectroscopy (QFRS) of PL is not popular, despite it being a powerful tool for PL lifetime analysis. Therefore, the QFRS of PL of amorphous semiconductors and up-conversion photoluminescence (UCPL) of rare-earth (RE) ion–doped glasses are addressed specifically in this chapter, and readers interested in other topics or further details should refer to the relevant references herein. Section 6.2 provides a brief outline of the fundamental aspects of PL in condensed matter. Section 6.3 focuses on the experimental aspects of PL, which ends with the fundamentals of the QFRS used for PL lifetime analyses. Section 6.4 deals with the QFRS of PL of amorphous semiconductors, in particular, hydrogenated amorphous Si (a-Si:H), including a-Ge:H. And, finally, Section 6.5 presents recent developments made by QFRS in studying the UCPL of RE-doped chalcogenides glass (ChGs).

6.2 Fundamental Aspects of Photoluminescence (PL) in Materials Conceptually, PL is the inverse of the absorption process of photons, where photons absorbed into a material transfer their energy to electrons in the ground state and energize them to excited states. Applying the old quantum theory and statistical mechanics to the two-energy-level model, Einstein predicted the occurrence of two radiative emission processes: stimulated photon emission and spontaneous photon emission after the photon absorption [5, 7, 8]. It is noteworthy that the former process is induced by an incoming photon and is the basis of lasing, whereas the latter occurs without any photon stimulation. In a strict sense, therefore, the photon absorption process is the stimulated photon absorption. If there are multiple number of excited states, then electrons excited to higher excited states rapidly relax non-radiatively to the lowest excited state S1 by thermalization, or by emitting phonons, and subsequently relax radiatively to the ground state S0 by emitting photons, as shown in Figure 6.1. Therefore, the absorbed photons are usually of higher energy than the emitted photons by the energy of thermalization or emitted phonons. The energy difference between the absorbed and emitted photons is called Stokes’ shift. PL includes fluorescence and phosphorescence, which are preferably used in the field of organic chemistry as well as biochemistry. Fluorescence occurs when a radiative transition from the excited state to the ground state is spin-allowed, that is, both the ground and excited states have the same spin multiplicity (singlet–singlet S1 –S0 or sometimes triplet–triplet states), and phosphorescence occurs when the transition is spin-forbidden, that is, the ground and excited states have different spin multiplicity (triplet–singlet T1 –S0 states). The spin-forbidden transitions can only occur through the spin–orbit coupling, which is usually weak, and hence such a radiative recombination has longer lifetime. As shown in Figure 6.1 for organic molecules, fluorescence and phosphorescence arise from radiative recombination of singlet and triplet Frenkel excitons, respectively [5, 9–11]. The fluorescence lifetime is usually much shorter (0.1 ∼ 10 ns) than the phosphorescence lifetime (1 ms ∼ 10 s) due to the spin selection rule. However, a delayed fluorescence possesses exceptionally longer lifetime similar to the phosphorescence lifetime, examples of which are semiconductor nanocrystals (NCs) such as CdSe, Cu+ :CdSe, and

6.2 Fundamental Aspects of Photoluminescence (PL) in Materials

S1

Intersystem Crossing Exchange Energy

EX

T1

Fluorescence ћω Phosphorescence ћω

S0

Figure 6.1 Illustration of radiative recombination processes in organic materials. S0 and S1 denote the ground and first excited singlet states, respectively. T1 denotes the first excited triplet state. E X is photoexcitation energy, and ℏ𝜔, fluorescence or phosphorescence emission energy. Here, competing nonradiative recombination processes are omitted.

CuInS2 [12]. However, it should be noted that the distinction between fluorescence and phosphorescence is not always clear [5, 13]. Sometimes, delayed fluorescence is also called phosphorescence. Metal–ligand complexes (MLCs), which contain a transition metal atom and one or more organic ligands, possess rather large intersystem crossing coefficients, owing to the large spin–orbit coupling caused by the heavy metal atoms, and therefore exhibit strong phosphorescence due to mixed singlet-triplet states (Figure 6.1). Accordingly, these MLCs display intermediate lifetimes of 10 ns ∼ 10 μs [13] and, in particular, fac-tris (2-phenylpyridine) iridium [Ir(ppy)3 ] demonstrates a very high internal quantum efficiency (QE) due to the emission from threefold degenerate triplet excitons [4, 14, 15], which work as organometallic triplet emitters in phosphorescent organic light-emitting diodes (OLEDs) with very high QE in green emission [16] (for theory, see Chapter 7). However, achieving the same level of QE in blue phosphorescent emitter still seems to be challenging [17]. 6.2.1

Intrinsic Photoluminescence

In inorganic semiconductors, PL is classified into two categories: intrinsic PL and extrinsic PL [1, 2, 6, 18, 19]. The intrinsic PL occurs mainly due to the band-to-band radiative transitions in a highly pure semiconductor even at a relatively high temperature, where, by absorbing a photon of energy higher than the band-gap energy EG , an electron is excited to the conduction band (CB), leaving a hole behind in the valence band (VB), and then they radiatively recombine to give rise to intrinsic PL, as shown in Figure 6.2a. The band-to-band transition that occurs in indirect-gap semiconductors such as Si and Ge is called an indirect transition, in contrast to the direct transition in direct-gap semiconductors such as GaAs. The latter transition does not need phonon-assistance, where the radiative recombination probability (rate) is much larger (high QE), and such materials are extensively applied to LEDs and semiconductor laser diodes. At low temperatures, an excited electron–hole (e–h) pair can form a free exciton (FE), or Wannier exciton through their Coulomb attractive force, as shown in Figure 6.2d, and then the PL occurs through the excitonic recombination in place of the band-to-band transition in the pure semiconductors. Beside the PL from FE, recombination of various exciton types—such as bound exciton (BE), excitonic polaron, self-trapped

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Conduction band (CB) D0 (or D+)

D0 EG

D0

A0 r

A0 (a)

(b)

(c)

Deep level

(d)

(e)

(f)

➀ ➁ (g)

Valence band (VB)

Figure 6.2 Schema of radiative recombination processes in semiconductors. (a) band-to-band e–h recombination; (b) neutral donor (D0 ) to valence band (VB) transition; (c) conduction band (CB) to neutral acceptor (A0 ) transition; (d) radiative recombination of free exciton (FE); (e) radiative recombination of bound exciton (BE), which is bound to D0 —recombination of BE bound to ionized donor D+ is also possible; (f ) donor–acceptor pair (DAP) recombination with separation r; (g) 1 and  2 is radiative. deep-level defect luminescence; either one of two transitions 

exciton (STE), and excitonic molecule—take place at low temperatures. Figure 6.2e shows an example of the radiative recombination of BE bound to neutral donor (D0 ) or ionized donor (D+ ). Excitons play even more important roles in semiconductor quantum wells, quantum wires, and quantum dots (QDs), where the excitonic PL can be observed even at much higher temperatures because the quantum confinement increases exciton-binding energy [4, 5, 20–22] (see Chapter 5). 6.2.2

Extrinsic Photoluminescence

An extrinsic PL is caused mostly by metallic impurities or defects (activators) intentionally incorporated into materials such as ionic crystals and semiconductors, and materials made luminescent in this way are called phosphors. The extrinsic PL occurs either between free carriers and impurity or defect states, or within the impurity or defect states (localized luminescence centers). The most important impurities are donors (Ds) and/or acceptors (As) in ionic crystals and semiconductors. In extrinsic semiconductors, where Ds or As are intentionally incorporated, PL occurs between an electron trapped by a neutral donor D0 and a hole at the top of VB (Figure 6.2b) or between an electron at the bottom of CB and a hole trapped by a neutral acceptor A0 (Figure 6.2c). In compensated semiconductors, where both Ds and As are ionized, or charged as D+ with the ionization energy ED and A− with the ionization energy EA , donor–acceptor pair (DAP) recombination occurs, as shown in Figure 6.2f. It is called distant DAP recombination, when the separation r between a D+ ion and an A− ion is much greater than the larger internal dimension, that is, the effective Bohr radius of D0 or A0 . In the distant DAP recombination, an excited free electron in CB gets bound on a D+ , accompanied by D+ → D0 , and a free excited hole in VB gets bound to an A− , accompanied by A− → A0 after band-to-band excitations. Thus, both the D and A become neutral D0 and A0 , respectively, and then the radiative recombination between the e–h pair bound to the DAP occurs, leaving the DAP charged as D+ and A− (Figure 6.2f ): D0 + A0 → D+ + A− + ℏ𝜔, where ℏ𝜔 is the emitted photon energy, but a potential energy

6.2 Fundamental Aspects of Photoluminescence (PL) in Materials

−e2 /𝜅r due to the Coulomb attractive force between the ionized DAP (D+ and A− ) remains, where 𝜅 is the static dielectric constant of host material [1, 4, 5, 18]. Therefore, ℏ𝜔 should be higher than the energy difference between D and A levels by e2 /𝜅r, and is given by: ℏ𝜔 = EG − ED − EA +

e2 kr

(6.1)

Since the D–A distance r can be multiples of the crystallographic lattice constant, the last term in Eq. (6.1) gives rise to a long series of sharp lines on the higher photon energy side of the PL spectrum. This was experimentally observed in the indirect gap semiconductor GaP doped with S (D) and Si (A) and first analyzed by Hopfield et al. [23]. The transition probability of a DAP recombination is proportional to the square of the overlap of hydrogen-like D and A wave functions. The hydrogen-like wave function is expressed as ∝ exp(−r/a), where a is the effective Bohr radius of a hydrogen-like impurity atom [1, 2, 4, 5]. If either of D0 or A0 has larger a, the square of the overlap proportional to the transition probability is approximately given by exp(−2r/a).Thus, the PL lifetime 𝜏 of DAP recombination depends on the DAP separation r as: ( ) 2r , (6.2) 𝜏 = 𝜏 0 exp a where 𝜏 0 is an electric–dipole transition time usually shorter than 10 ns. The distant DAP recombination shows the following features. First, according to Eq. (6.2), a decrease in D–A separation r increases the distant DAP recombination rate 1/𝜏, and thereby increases PL intensity. However, any decrease in r decreases the number of possible pairings of DAPs, leading to a reduction in PL intensity. Thus, the PL intensity should exhibit a maximum at a certain r. On the other hand, the PL emission energy ℏ𝜔 is a monotonically decreasing function of r (Eq. (6.1)), which is reflected in static PL spectroscopy (Section 6.3.1). It is observed in GaP crystal at 1.6 K that the PL spectrum is broad at r > 4 nm, with a maximum at r ≈ 5 nm and discrete at 1 < r < 4 nm, with a number of peaks corresponding to the D–A separations [24]. Since the recombination rate is slower for a DAP of larger r, PL intensity at lower ℏ𝜔 gets easily saturated and does not increase by increasing the excitation intensity similar to that observed at higher ℏ𝜔. Second, if we observe the temporal PL spectrum of the distant DAP recombination, the PL peak energy shifts to lower ℏ𝜔 as time elapses, since a DAP of smaller r which emits higher ℏ𝜔 will decay faster than that of a larger r which emits lower ℏ𝜔. This implies the importance of both PL lifetime spectroscopy and the static PL spectroscopy for a detailed study on the DAP recombination. The hydrogen-like nature of either D or A is essential to the distant DAP recombination; either D or A should be in a shallow impurity state: ED ≪ EG or EA ≪ EG . Therefore, the occurrence of this type of recombination is limited only to relatively low temperatures, which is not suitable for practical use except for the characterization of semiconductors [6]. In contrast, when both D and A are energetically deep, they are localized, and the PL of this type DAP recombination is strongly affected by phonons, and its mechanism is not entirely clear. Because of the deep impurity levels of D and A, however, the PL intensity

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is sometimes enough intense at room temperature to serve as practical phosphors, such as blue-emitting ZnS:Ag and green-emitting ZnS:Cu, Cl [4, 5]. Generally, defects and metallic impurities intentionally incorporated in ionic crystals and semiconductors often act as efficient phosphors (luminescence centers) [1, 2, 4, 5]. Color- or F-centers are optically active vacancies in ionic crystals such as alkali halides. Paramagnetic metal ions incorporated into host materials act as localized activators; actually, these are transition metal ions such as Mn2+ , Cr2+ , etc., and the rare earth (RE), or lanthanide ions of Nd3+ , Eu3+ , Tb3+ , Er3+ , etc. For the transition metal ions, optical transitions occur between the unfilled d-d states, and, for the RE ions, these occur between the unfilled f -f states (see Chapters 2 and 4; also [4, 5]). Though both d-d and f-f transitions are forbidden by the selection rules for electric dipole transitions, their forbidden character is altered by the crystalline electric field, and then these transitions become more or less allowed. Since 3d orbitals of the transition metal ions are the valence orbitals, any environmental perturbation is influential to the d-d transitions; the energy (color) of PL is rather dependent on the host materials. On the contrary, RE ions are formed when the outermost 6s electrons are removed, leaving the optically active 4f orbitals inside the filled 5s and 5p shells. This makes the unfilled 4f orbitals smaller in radius and less sensitive to the crystalline field, so that the optical spectra of doped rare RE ions are generally similar to those of free ions having narrow emission lines. A good example of this type of luminescence is from phosphors of pcLEDs, for which RE-doped materials such as (Y,Gd)3 (Al,Ga)5 O12 :Ce3+ (YAG:Ce), (Ba,Sr,Ga)2 SiO2 :Eu2+ , etc., have been developed [4, 5]. YAG laser is also an example, where Nd3+ ion is incorporated into the host of yttrium aluminum garnet (Y3 Al5 O12 , or YAG) as an activator for the stimulated emission at 1064 nm [8]. Instead of phosphors, the term fluorophores is used for organic dyes, biological fluorescent proteins such as green fluorescent protein (GPF), quantum dots (QDs) of CdS, InP, PbS, etc., and nanoparticles (NPs) activated by RE ions. The fluorophores serve as tracers in fluid measurement and as sensors for metal ions, gas molecules, pesticide residue, etc., in environmental measurements [13, 25, 26]. More importantly, they are used as stains, markers, and probes in bioimaging and medically as photosensitizers of PDT and photo-thermal therapy (PTT) [13, 27, 28]. 6.2.3

Up-Conversion Photoluminescence (UCPL)

Since 4f orbitals of the RE ions possess ladder-like energy levels and some of them have similar energy difference, an anti-Stokes process of UCPL occurs with two or more sequential photon absorption processes followed by a photon emission at an energy higher than that of the absorbed photons in RE-doped crystals, glasses, proteins, etc. [4, 29, 30]. The typical UCPL from RE ions can be categorized into three mechanisms: (i) GSA/ESA: the excitation of a single ion by ground state absorption (GSA) (Figure 6.3a), and successively followed by excited state absorption (ESA) (Figure 6.3b) to excite the uppermost state; (ii) GSA/ETU: one of two neighboring ions excited by GSA relaxes to the lower state (not necessarily the ground state) by transferring its energy: energy transfer (ET) to another ion to excite the uppermost state by energy transfer up-conversion (ETU) (Figure 6.3c); and (iii) PA: photon avalanche via cross-relaxation process (CRP) (Figure 6.3d), where an ion excited to its uppermost state relaxes to

6.2 Fundamental Aspects of Photoluminescence (PL) in Materials

(a)

(b)

(c)

(d)

Figure 6.3 Basic mechanisms important for up-conversion of photoluminescence (UCPL): (a) ground state absorption (GSA), (b) excited state absorption (ESA), (c) energy transfer up-conversion (ETU), and (d) cross-relaxation process (CRP).

the intermediate state by simultaneously exciting the lower state of a proximate ion to the intermediate state via ET and thereby the intermediate state doubles by a single transition from the uppermost to the intermediate state, giving rise to PA. However, the condition for the occurrence of PA is rather special [4, 30, 31]. Thus, the GSA/ESA, GSA/ETU, and composite GSA/(ESA + ETU) processes are commonly accepted as UCPL mechanisms of RE ions. Since the GSA/ESA occurs with two or more sequential photons’ absorption at a single RE ion, it is almost insensitive to RE ion concentration. In contrast, the GSA/ETU process arises from ET between two neighboring RE ions and thus depends on the RE ion concentration [4, 29, 30]. The RE-doped nanoparticles (NPs) that can emit the visible UCPL, excited through “optical transparent window of biological tissue: 700–1100 nm” by near infrared (NIR) light, serves as a promising fluorophore for biochemistry, molecular biology, PTT, and PTD [27, 28, 32]. Another notable potential of RE-doped NPs is modification of the solar-spectrum by up-conversion and/or down-conversion to improve the efficiency of solar cells [33, 34]. In addition, up-conversion can also be assisted thermally as applied in OLEDs, which is described in Chapter 7. 6.2.4

Other Related Optical Transitions

Intense light irradiation to some condensed matters induces two phenomena similar to UCPL; one is two photon absorption (TPA or 2PA), and the other is second-harmonic generation (SHG). TPA occurs in materials such as atomic vapor, chromatic molecules, and semiconductors, with simultaneous absorption of the two photons having their sum energy equal to the energy difference between a ground state and a real excited state, where the electronic excitation follows via a virtual intermediate state in the absence of a resonant intermediate state [35]. In contrast, the SHG occurs from simultaneous absorption of two photons in nonlinear optical crystal materials such as LiNbO3 and KH2 PO4 , followed by the electronic excitation from a ground state to a virtual excited state via a virtual intermediate state [8]. However, both phenomena are low in the conversion efficiencies due to the involvement of virtual intermediate states. Raman scattering is the scattering of a light by phonons in condensed matters. When a laser light of narrow linewidth with the excitation energy Ex enters the condensed matter having the phonon energy ℏ𝜔ph , a Stokes shifted emission appears at the energy ℏ𝜔 = Ex − ℏ𝜔ph in the spectrum of the scattered light, and an anti-Stokes light appears at the energy ℏ𝜔 = Ex + ℏ𝜔ph . When Ex is close to the energy of a real excited

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state, the Raman signals are enhanced, which is called resonant Raman scattering. Quantum-mechanically, in Raman scattering, a laser light excites electrons from the ground state to virtual excited states at Ex , which then relax radiatively to the ground states by emitting phonons (Stokes shift) or absorbing phonons (anti-Stokes shift). In resonant Raman scattering, the excited states are not virtual but real, like band-gap, d-d, or f-f electronic transition states. Since ℏ𝜔ph ≪ Ex , Raman spectroscopy needs narrow-linewidth excitation laser and high-resolution spectrometer. Occasionally in Raman spectroscopy, PL coexists with Raman signals, causing some confusion. This is solved by changing Ex in Raman spectroscopy and plotting the spectrum in an absolute wavelength [36]. Furthermore, mapping in conjunction with photoluminescence excitation (PLE) and PL spectroscopies can identify the difference between PL and Raman signals [6].

6.3 Experimental Aspects 6.3.1

Static PL Spectroscopy

PL contains information about emitted photon density (PL intensity), photon energy (wavelength), photon polarization, photon-emitting position, and photon lifetime. Except for emitted photon lifetime obtained from temporal PL measurements, the valuable information is available fundamentally from static PL measurements. The apparatus used for static PL spectroscopy [37, 38] is conventionally assembled as shown in Figure 6.4. The PL excitation source can be a laser beam, monochromatic light of a Xenon lamp, or Halogen lamp monochromatized by an additional (excitation) monochromator (MC) and LED light, and the PL excitation energy Ex is normally higher than the PL emission energy ℏ𝜔 due to Stokes shift. Here, it should be noted that the unit of photon energy is usually expressed in eV in physics and electronics, but frequently in wavenumber cm−1 in physics and chemistry; according to quantum mechanics, an NIR photon of wavelength 𝜆 = 1 μm in free space possesses the wavenumber 1/𝜆 = 104 cm−1 and the energy ℏ𝜔 = hc/𝜆 ≈ 1.24 eV with Planck’s constant h and light velocity c in free space. For a thin film on a substrate, the substrate surface should be roughened and the thickness d of the film should be as thick as possible, because the internal PL reflection causes interference effects on the PL spectrum. Interference at the film interfaces will occur at m𝜆 = 2nd with the refractive index n of the film and an interference multiple number m. If we plot PL spectrum in ℏ𝜔 = hc/𝜆, an interference fringe appears with a fringe distance hc/(2nd). Hence, the large d reduces the fringe distance. For example, in a-Si:H with n ≈ 3.6, hc/(2nd) amounts to ∼17 meV for d = 10 μm. The fringe is thereby obscured by the monochromator of resolution ∼30 meV; see Section 6.4.3 (details on the interference spectrum of thin films are given in Chapter 1). A PL-absorbing substrate should be avoided, since it will significantly affect the PL spectrum and reduce the intensity [39]. In some cases, the self-absorption of PL in the film also modifies the spectrum, which needs some corrections [1]. A grating monochromator is used to measure the PL spectrum: intensity vs. wavelength 𝜆, but a light of wavelength in higher orders such as 𝜆/2, 𝜆/3, etc., must be blocked by a long pass filter (LPF). However, care must be taken in the ultraviolet (UV) photoexcitation, where the blocking filter as well as other filters such as a neutral density filter

6.3 Experimental Aspects

Cryostat

Sample

Laser BPF

NDF

C1

C2

Lock-in Amp

Lens LPF Lens

Monochromator λ

PC

GPIB Bus

Detector

Figure 6.4 Experimental set-up of static PL measurement. C1, optical chopper at normal position; C2, optical chopper for residual PL decay measurement; BPF, band-pass filter; NDF, neutral density filter; LPF, long-pass filter; PC, personal computer.

(NDF) often fluoresces by a stray UV light. This is also the case even with UV-laser notch filters. The situation becomes more serious with a glass filter, which fluoresces even with a visible light excitation. Sometimes such an effect can be avoided or reduced by putting the filters at the output side of the monochromator. The PL sensitivity depends largely on the throughput of the monochromator as well as the sensitivity of the detector; a monochromator of a low f -number has high throughput at the expense of the resolution. Usually, focusing the light source at an input slit with the same f -number, we get the maximum throughput (optical matching). However, note that PL is not always a point source of light, but it has a finite size. If it is collected by a lens of a small f -number, for example, f/1.0, and incident on the slit of the monochromator with f/4.0 under the optical matching, the image of PL at the slit is magnified by 4 in this example. The PL image larger than the slit area loses some intensity, which is called a vignetting loss. Since we must trade between the throughput and the spectral resolution by adjusting the slit width as well, a grating monochromator between sample and detector often decreases the throughput by nearly three orders, even if it has low f -number. We can vary the 𝜆 of a monochromatized light by rotating the grating of the monochromator, manually or automatically with a stepping motor and a personal computer (PC)–controlled drive. However, a set of band-pass filters (BPFs) is an inexpensive and more efficient alternative for this, but sacrifices the arbitrary choice of 𝜆. Today, a charge-coupled device (CCD) is installed in a position of the output slit of the monochromator: a multi-channel spectrometer with electronically scanned 𝜆 by the CCD. Various handheld multi-channel spectrometers, equipped with USB interface and a coupler for optical fiber leading to low stray light, are commercially available from Ocean Optics, StellarNet, Hamamatsu, and so on.

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Another alternative in the range of NIR and far infrared (FIR) is the Fourier transform infrared (FTIR) spectrometer based on a Michelson interferometer, which is commercially available from Bruker Optics, Oriel Instruments (Newport). The method is called Fourier transform photoluminescence (FTPL) technique [5, 38, 40, 41]. Unlike the dispersive spectrometer, the FTIR spectrometer collects all wavelengths simultaneously (Felgett Advantage) and does not use the entrance and exit slits. Hence, the PL sensitivity is increased, and the measuring time is saved by the FTPL. Bignazzi et al. [42] have, however, given some warnings about the limitations of the FTPL. The PL can be detected by various detectors such as photomultiplier tube (PMT), photodiode (PD), and photoconductor. A PMT is the most sensitive detector with high-speed response, and is hence usable for time resolved spectroscopy (TRS), as described in the following text. Its spectral response is usually limited below a wavelength of ≈900 nm. However, NIR PMT: R5509-72 (Hamamatsu Photonics) and its series of InGaAsP/InP photocathode having sensitivity up to 𝜆 ≈ 1700 nm is superior to other NIR detectors such as Ge PD in sensitivity and time response. PMT output may be directly measured with an electronic dc current meter, but usually we modulate the excitation light at rather low frequency, typically from a few Hz to 10s of kHz by setting an optical chopper C1 in front of the sample (Figure 6.4), and we thereby detect the PL by a lock-in amplifier with a significantly high sensitivity and high signal-to-noise ratio (S/N). However, care must be taken in choosing the chopping frequency, so that the chopping period may last longer than the PL lifetime for the reason described in Section 6.3.5. Incidentally, by setting the chopper C2 between the sample and the detector, we can avoid the preceding problem, and also can observe the residual decay of PL with high sensitivity and S/N after turning off the PL excitation light (Figure 6.4) as described in Section 6.4.4, where we need a complete darkroom and must strictly avoid stray NIR light arising from an optical switch of the chopper and the temperature T-control heater of the cryostat when a sample is installed into a cryostat to measure the T-dependence of PL. At a very low PL intensity, one can operate PMT in the digital mode, called photon counting method [5, 13, 43]. At the very low light intensity, photons incident on the photocathode of PMT are separated in time, and thus the PMT output consists of separated electric pulses corresponding to the respective incident photons. Since the number of PMT output pulses is proportional to the amount of incident light, we can measure the light intensity by electronically counting the output pulses, in contrast to the analogue mode, where the PMT output pulses overlap each other and eventually can be regarded as electric current of superposed shot noises. Even in the dark, however, the PMT has dark current (noise pulses) arising from various sources. The amplitude of a noise pulse is generally lower than that of the incident photon, and thus the noise pulse can be removed by an electronic discriminator, which significantly enhances the S/N of the photon counting method. We may also apply an avalanche photodiode (APD), a sort of PDs [5, 8], to the photon counting instead of the PMT, but its gain is at most ∼100 in the linear mode operation below the bias voltage of avalanche breakdown, whereas that of the PMT amounts to ∼107 . However, the APD operated above the avalanche breakdown voltage works as a single-photon avalanche diode (SPAD) or Geiger-mode APD, and has a typical gain of 105 ∼ 106 . Once the SPAD is precipitated into the breakdown state by the absorption of a photon, that is, the ON state, it cannot recover to the OFF state by itself,

6.3 Experimental Aspects

because it is biased above the avalanche breakdown. Therefore, quenching by a resistor or more sophisticated active circuitry is necessary to restore it to the OFF state for a subsequent photon. Nowadays, a sort of solid-state photomultiplier composed of SPAD pixels, implemented with the number of 102 ∼ 104 in a matrix 106 and surpasses conventional PMT in several points: the internal QE (>0.8) of SPAD is higher than the external QE ( 1. Therefore, in the static PL spectroscopy with lock-in detection of the chopper frequency 𝜔, one should set 𝜔 such that it satisfies 𝜔𝜏 ≪ 1. However, by using the optical chopper in the position of C2 between the sample and the detector (Figure 6.4), we can avoid such a limitation, provided that the PL detecting system is completely shielded in a perfect darkroom from unwanted light—for example, NIR light from the optical switch to control the chopper frequency 𝜔 and provide reference signal, and that from a cryostat heater (Section 6.3.1). More precisely, 𝜏 is deduced by fitting R(𝜔) and 𝜃(𝜔) to data of amplitude and phase delay as functions of 𝜔. Materials having the multi-exponential decay as given by Eq. (6.12) are usually analyzed by nonlinear least squares method for fitting [63]. Such FRS with varying frequency 𝜔 is called variable-frequency fluorometry, and is used for organic and biological molecules and a variety of condensed matters [13]. If a luminescent material has a broad lifetime distribution, such as amorphous semiconductors, it will be more tedious to obtain the lifetime distribution by the FRS. In order to overcome this difficulty of FRS as well as disadvantages of the biased TRS, Depinna and Dunstan [64] devised the modified FRS, called quadrature frequency resolved spectroscopy (QFRS), as described in the following text. The theoretical principle of QFRS was fully established by Stachowitz et al. [56].

6.3.6

Quadrature Frequency Resolved Spectroscopy (QFRS)

As stated in the preceding text, QFRS [56, 64] is more suitable for analyzing the broad PL lifetime distributions of disordered (amorphous) materials and the UCPL of RE-ion-doped materials having multi-lifetimes. Here again, we employ the alternating-current circuit theory in Figure 6.5; by substituting s = i𝜔 and separating real and imaginary parts, the transfer function of Eq. (6.9) I(s) becomes: I(iω) =

1 𝜔𝜏 1 −i = 1 + iω𝜏 1 + (𝜔𝜏)2 1 + (ω𝜏)2

,

(6.16)

where the real part gives the in-phase output and the imaginary part gives the quadrature output iQ (𝜔) against the input of modulated PL excitation: G¯aei𝜔t .

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Then, as done in Eq. (6.13), here again we introduce the probability density P(𝜏) to write the quadrature part of the modulated PL as: ∞

iQ (𝜔) = Ga

P(𝜏)

∫0

𝜔𝜏 dτ, 1 + (𝜔𝜏)2

(6.17)

The function 𝜔𝜏/[1 + (𝜔𝜏)2 ] is single-peaked with maximum at 𝜔 = 𝜏 −1 , but it cannot be approximated as a delta function, because its integral from 𝜔 = 0 to ∞ diverges, and it is not a sharply peaked function. In order to solve this problem, we set 𝜔 = 10−x and 𝜏 = 10u (d𝜏 = ln 10 ⋅ 10u du) to get: ∞

ln10 ⋅ sech(ln10 ⋅ (x − u)) 𝜋 iQ (𝜔) = Ga 10u P(10u ) du ∫ 2 𝜋 −∞ ∞

=

𝜋 Ga 10u P(10u )𝜉(x − u)du , ∫ 2

(6.18)

−∞

where ξ(x) = ln10⋅sech(ln10⋅x) is the QFRS kernel for the logarithmic inverse angular fre𝜋 quency x = log10 𝜔−1 , and normalized for its integration from x = −∞ to ∞ to be unity, and peaks at x = 0 with FWHM of ≈1.14 (Figure 6.6). Thus, if the distribution function 10x P(10x ) is broad enough for 𝜉(x) to be regarded as a delta function, we may approximate the lifetime distribution in the logarithm of reciprocal 𝜔: x = log10 𝜔−1 as [56]: 𝜋 (6.19) iQ (𝜔) = iQ (10−x ) ≈ Ga10x P(10x ). 2 If this is not the case, we must recover the true lifetime distribution 10x P(10x ) by deconvoluting Eq. (6.18) to be reconciled with the QFRS data. However, the deconvolution is not always easy, because it often exaggerates the experimental noise. For this reason, two approximate methods of deconvolution are proposed in the following text. Method 1: Since the integration of sech x is tan−1 (sinh x), we can analytically convolute a simple rectangular function of x with the kernel. Assuming the lifetime distribution 10x P(10x ) to be a superposition of rectangular functions, similar to piled-up building blocks, we get an analytical expression of its convolution with𝜉(x). Then, fitting it to the QFRS data, we get approximate 10x P(10x ), again similar to piled-up building blocks [65]. Method 2: This is completely numerical. We assume the true lifetime distribution to be a linear combination of Gaussians, each having three unknown parameters, and then determine the parameters by a nonlinear regression, so that its numerical convolution with 𝜉(x) can reconcile with the data. Strictly, the true lifetime distribution is not always guaranteed to be the linear combination of Gaussians (Section 6.4.3). More specifically, we have derived the analytical form of the distant-pair (DP) recombination lifetime distribution with a single parameter of steady-state carrier concentration obtained from the theoretical impulse-response of the biased TRS given by Dunstan and Levin et al. [58, 59] separately and verified for the QFRS of a-Si:H (for details, see [66]). As mentioned in the preceding text, PL measurements for absolute QE of a film on opaque substrate is not established yet, but the deconvolution of QFRS spectrum gives the relative QE for a deconvoluted component by dividing its area by the total area of all the deconvoluted components. TRS and QFRS methods using the respective kernels 𝜆(x) and 𝜉(x) are mathematically equivalent [56], but QFRS has a decisive advantage in

6.4 Photoluminescence Lifetime Spectroscopy of Amorphous Semiconductors by QFRS Technique

the availability of a conventional lock-in amplifier with extreme sensitivity and S/N in the frequency range from 1 Hz to 100 kHz, corresponding to a five-decade lifetime from ∼1 μs to ∼0.1 s.

6.4 Photoluminescence Lifetime Spectroscopy of Amorphous Semiconductors by QFRS Technique 6.4.1

Overview

Using the QFRS technique, Depinna and Dunstan [64] first demonstrated the PL lifetime distribution of a-Si:H more precisely than that using TRS measurements, and also observed emissions from singlet and triplet bound-excitons in GaP crystals, well-separated in lifetime as well as emission energy. Although the static PL spectroscopy of a-Si:H was nearly established in the 1970s and early 1980s [39, 67], considerable debates over its detailed mechanisms still continue even today [68–70]. It has been generally agreed that, in an intrinsic a-Si:H at a low temperature T, photoexcited electrons and holes immediately thermalize in extended and band-tail states by the hopping process, and intrinsic PL arises from the radiative tunneling (RT) transition between electrons and holes localized in their respective tail states [39, 67, 70]. The PL lifetime is governed by the RT lifetime identical to Eq. (6.2) of the DAP recombination as: ( ) 2R , (6.20) 𝜏 = 𝜏0 exp a where R is the e–h separation. The pre-exponential factor 𝜏 0 is the electrical-dipole transition time usually expected to be ∼10−8 s, and a is the extent of the larger of the electron and hole wave functions (localization length): usually, the electron Bohr radius of a-Si:H ∼1 nm. However, as stated in the preceding text, whether the recombination is geminate (recombination of e–h pair created by a single photon) or nongeminate (recombination of e–h pair created by different photons) is still a controversial issue. This comes from the featureless distribution of the PL lifetime 𝜏 as well as the PL emission energy ℏ𝜔 due to the disorder in amorphous semiconductors. In fact, the optical absorption due to an exciton being a typical geminate e–h pair has not been observed in a-Si:H even at low temperatures, unlike in crystalline semiconductors. Thus, the precise measurement of lifetime is necessary to investigate the PL of a-Si:H at a sufficiently low generation rate G for photocarriers, which can be done using the QFRS technique [56, 64]. Using this technique, Bort et al. [71] have observed a transition in the recombination kinetics of PL in a-Si:H at low T at a generation rate G ≈ 1019 cm−3 s−1 ; the peak lifetime 𝜏 p , that is, the peak position of QFRS spectrum, is constant under the condition G < 1019 cm−3 s−1 , whereas 𝜏 p decreases as G increases for G > 1019 cm−3 s−1 . They have proposed a geminate recombination model at G < 1019 cm−3 s−1 and the distant-pair (DP), or nongeminate recombination model at G > 1019 cm−3 s−1 . By contrast, the light-induced electron spin resonance (LESR) intensity, known to be proportional to steady state density of photoexcited e–h pair n in the band-tail states, depends on G sublinearly for the whole range of 1015 < G < 1020 cm−3 s−1 [71–74]. If the PL is simply governed by the DP recombination based on Eq. (6.20), since n ≈ G𝜏 p holds,

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the contentious decrease of 𝜏 p with increasing G should give rise to the same sublinear G-dependence of n as the LESR intensity in the preceding whole range of G; however, this is not the case for G ≤ 1019 cm−3 s−1 [58, 71, 75]. Meanwhile, under the geminate condition (G < 1019 cm−3 s−1 ), Boulitrop and Dunstan [76] were the first to identify a double-peaked lifetime distribution of PL consisting of short-lived (∼μs) and long-lived (∼ms) components in a-Si:H by QFRS. The double-peak QFRS spectrum of a-Si:H was studied in further detail by Ambros et al. [77], expanding the high frequency limit to 2 MHz, corresponding to a lifetime of 𝜏 ≈ 0.1 μs. It is difficult to identify the two lifetime components on the basis of the RT model [76, 77]. Stachowitz et al. [78] have proposed the exciton involvement in the double-peak phenomenon, attributing the short- and long-lived components to singlet- and triplet-excitons, respectively. We have also observed a double-peak lifetime in the similar tetrahedral amorphous semiconductor of a-Ge:H, supporting the exciton model [69, 79, 80]. However, identifying the short-lived component of ∼𝜇s with singlet exciton recombination is a problem, in that the lifetime of a singlet exciton, or fluorescence lifetime, is normally less than 10 ns. In addition, a disadvantage of the QFRS is the minimum lifetime limited to the μs order due to the upper-limit frequency of the conventional lock-in amplifiers. Using the rf digital lock-in amplifier of the frequency range from 25 kHz to 200 MHz (SR844, Stanford Research System), we have developed a nanosecond QFRS system named dual-phase double lock-in (DPDL) QFRS, as described in the following text. 6.4.2

Dual-Phase Double Lock-in (DPDL) QFRS Technique

As illustrated in Figure 6.8a, we have developed the DPDL technique [65, 81, 82] to measure PL lifetime distribution from 2 ns to 5 μs, employing the SR844 rf digital lock-in amplifier and an electro-optic modulator (EOM, Conoptics) to modulate the laser beam from 0.5 Hz to 80 MHz. Since electromagnetic cross-talk between the EOM driver of high rf power (D) and the rf lock-in amplifier becomes serious at rf frequency 𝜔/(2𝜋) > 10 MHz, the laser beam was chopped at a low frequency of 𝜔m ≪ 𝜔 to discriminate the PL signal from the cross-talk by the double lock-in detection [65, 81]. Figure 6.8b represents the signal flow of DPDL. Instead of a complex exponential input ei (t) ∝ ei𝜔t in the real-coefficient Eq. (6.11) (Figure 6.5), employing its imaginary part, ei (t) ∝ sin(𝜔t), we obtain the output eo (t) ∝ R(𝜔) sin(𝜔t − 𝜃(𝜔)) from the imaginary part of Eq. (6.15), and then the doubly modulated PL signal becomes: S(t) = R(𝜔) sin(𝜔t − 𝜃(𝜔))• sin(𝜔m t),

(6.21)

where R(𝜔) and 𝜃(𝜔) are the PL amplitude and phase at 𝜔, respectively, in Eq. (6.15). When 𝜔m ≪ 𝜔, the quadrature part of Eq. (6.21) is given by R(𝜔)sin𝜃(𝜔) against the sinusoidal excitation, sin(𝜔t), which is proved to be proportional to iQ (𝜔) in Eq. (6.16). The time constant of LPF of the rf lock-in amplifier is set much greater than 𝜔−1 and much less than 𝜔m −1 . Thus, the rf lock-in amplifier outputs in-phase signal X(t) on the X-channel and quadrature signal Y (t) on the Y -channel, which are sinusoidal at the frequency of 𝜔m as: } X(t) = 1∕2R(𝜔) cos 𝜃(𝜔) sin(𝜔m t) . (6.22) Y (t) = 1∕2R(𝜔) sin 𝜃(𝜔) sin(𝜔m t)

Chopper BPF NDF 8Hz

sin(ωmt–ψ)

Cryostat

C

Sample Laser

Laser

EOM

FG DC-80MHz ~

D

Expander

XX Delay Line Lock-in Amp. X-out Ref. in XY RF Lock-in Amp. Ref. in YX Y-out Lock-in Amp. YY

sin ωt × sin ωmt

EOM

Sample

PL C

NIR-PMT

Lens NDF LPF

XY YX

DC-100MHz Amp

FG

XX

YY

Lock-in Amp.

PA RF Lock-in Amp.

Lock-in Amp.

Filter or Monochromator

sin ωt

X(t)

Y(t)

PMT S(t) = R(ω)sin(ωt – θ(ω))·sin(ωmt)

PC

PC GPIB bus (a)

(b)

Figure 6.8 (a) Experimental set-up of DPDL QFRS system. C, optical chopper; D, electronic driver for EOM (electro-optic modulator); FG, function generator; BPF, band-pass filter; LPF, long-pass filter; NDF, neutral density filter; NIR-PMT, near-infrared photomultiplier tube; PC, personal computer. Delay line: for compensation of instrumental phase-difference between signal and reference inputs of RF lock-in amp. (b) Block diagram of DPDL QFRS system. C, optical chopper with chopping frequency of 𝜔m and phase delay of 𝜓; FG, function generator; PA, preamplifier; PMT, photomultiplier tube; PC, personal computer. Solid line, electrical signal line; dotted line, optical pass. Source: Reproduced from T. Aoki, T. Shimizu, D. Saito, and K. Ikeda, J. Optoelectron. Adv. Mater., 7, 137 (2005) with permission from INOE.

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6 Photoluminescence

These signals are again synchronously detected at 𝜔m using the two digital lock-in amplifiers (SR830) with LPF time constants much greater than 𝜔m −1 . An instrumental phase shift 𝜓 is inserted between the chopped light and the synchronous signal of the chopper driver (Figure 6.8b). Hence, the X- and Y -channel outputs of the two lock-in amplifiers (X X , X Y , Y X , and Y Y ) are given by: XX XY YX YY

= = = =

1 R(𝜔) cos 𝜃(𝜔) cos 𝜓 ⎫ 4 ⎪ 1 R(𝜔) cos 𝜃(𝜔) sin 𝜓 ⎪ 4 ⎬. 1 R(𝜔) sin 𝜃(𝜔) cos 𝜓 ⎪ 4 1 R(𝜔) sin 𝜃(𝜔) sin 𝜓 ⎪ ⎭ 4

(6.23)

Eliminating 𝜓 in Eq. (6.23), we obtain R(𝜔) and 𝜃(𝜔) as follows: √ ⎫ R(𝜔) = 4 XX2 + XY2 + YX2 + YY2 ⎪ ( )⎬ . 1 −1 YX +XY −1 YX −XY 𝜃(𝜔) = 2 tan X −Y + tan X +Y ⎪ X Y X Y ⎭

(6.24)

However, as R(𝜔) and 𝜃(𝜔) include components of instrumental responses due to the EOM and its driver, the PL detecting system, and optical and electrical lengths, the quadrature signal should be calibrated. It should be noted that a delay-line (RF coaxial cable) is intentionally inserted between the function generator (FG) and the reference input (Ref. in) of the RF lock-in amplifier in order to reduce the instrumental phase-difference between the signal and reference inputs in Figure 6.8a. By measuring the instrumental values of the amplitude R′ (𝜔) and phase 𝜃 ′ (𝜔) of the modulated laser light reflected by a roughened Al plate instead of the sample as a reference, the intrinsic QFRS signal of PL is given by [82]: R(𝜔) sin(𝜃(𝜔) − 𝜃 ′ (𝜔))∕R′ (𝜔).

(6.25)

The nanosecond order of lifetime resolution of the DPDL QFRS has been confirmed by measuring the fluorescence lifetime ∼4 ns at room temperature of Rhodamine 6G in 1 × 10−6 mol l−1 aqueous solution in agreement with that obtained by TCSPC [81, 83]. 6.4.3

Exploring Broad PL Lifetime Distribution in a-Si:H by Wideband QFRS

In order to explore the wide lifetime-distribution of amorphous semiconductors, by using an acousto-optic modulator (AOM) and the lock-in amplifier (Stanford SR830), we have also expanded the lower-limit frequency of QFRS down to 1 mHz (𝜏 ≈ 160 s); it is operated in internal reference mode together with synchronous filtering in order to reduce the phase noise, where a single QFRS measurement of 𝜏 ≥ 30 s needs 6 h by setting its time constant at 3 ks [84]. Thereby, the wideband QFRS system combining the DPDL QFRS with the internal reference mode makes it possible to analyze lifetimes over almost 11 decades from 2 ns to 160 s [84]. In the following, we show that our wideband QFRS results reveal the exciton involvement as well as DP recombination in the triple-peaked lifetime structure of the intrinsic a-Si:H. Similar phenomena observed in chalcogenide amorphous semiconductors as well as a-Ge:H indicate that the triple-peak QFRS spectrum, or the coexistence of

6.4 Photoluminescence Lifetime Spectroscopy of Amorphous Semiconductors by QFRS Technique

geminate and nongeminate recombination, is universal among amorphous semiconductors [85]. In addition, the residual PL decay of a-Si:H persisting for more than 104 s is presented; the DP recombination kinetics as well as steady-state photocarrier concentration obtained from QFRS results agrees with those of LESR [86]. Films of intrinsic a-Si:H were deposited on roughened Al substrates, with thickness of 1 ∼ 9 μm and defect density ≤ 2.0 × 1016 cm−3 , and intrinsic a-Ge:H with thickness of ≈1 μm and defect density ∼1016 cm−3 . PL signals were detected by a Hamamatsu R5509-42 NIR PMT at photon energies ranging from 0.9 to 1.7 eV for a-Si:H of bandgap energy EG ≈ 1.8 eV and R5509-72 NIR PMT from 0.7 to 1.5 eV for a-Ge:H. An optical system of f /1.0 ∼ 2.0 optics was designed to collect PL emission into the PMTs. The QFRS spectra of dispersed PL were measured by a 10 cm and f/3.0 monochromator with a resolution ∼30 meV. 6.4.3.1

Effects of Excitation Intensity, Excitation, and Emission Energies

Figure 6.9a shows G-evolved QFRS spectra of PL of a-Si:H from 2 ns to 160 s excited at the above-bandgap excitation energy Ex = 2.33 eV for generation rate G from 2.5 × 1015 to 5.0 × 1022 cm−3 s−1 [87]. We can see that the peak-positions of the longand short-lived components are fixed at 𝜏 T ≈ 3 ms and at 𝜏 S ≈ 2 μs, respectively, even though G changes from 2.5 × 1015 to 4.1 × 1019 cm−3 s−1 ; this is the well-known τS G [cm–3s–1] 5.0 × 1022

τT

a-Si:H EX = 2.33[eV] T = 3.7[K]

2.0 × 1022

106 104

1.9 × 1021 4.1 × 1020 1.2 × 1020 τD

4.1 × 1019 1.3 × 1018 2.8 × 1017

EX = 1.81[eV] τD

T = 3.7[K] a = 10[A] τ0 = 10–8[s]

100 10–2 τT 10–4

6.5 × 1016

10–6

1.3 × 1016 2.5 × 1015 10–8

EX = 2.33[eV]

a-Si:H

102

PL Lifetime τ [s]

QFRS Signal [a.u.]

5.0 × 1021

10–6

10–4

10–2

100

102

10–8 1012

τS 1014

1016

1018

1020

1022

PL Lifetime τ [s]

Generation rate G [cm–3s–1]

(a)

(b)

1024

Figure 6.9 (a) G-evolved QFRS spectra from 2 ns to 160 s for a-Si:H at 3.7 K and E X = 2.33 eV with various G from 2.5 × 1015 to 5.0 × 1022 cm−3 s−1 . The two data at G of 2 × and 5 × 1022 cm−3 s−1 were taken by laser light condensed through a lens. Source: Reproduced from T. Aoki, J. Non-Cryst. Solids, 352, 1138 (2006) by permission of Elsevier. (b) Peak positions 𝜏 S , 𝜏 T , and 𝜏 D as functions of G at 3.7 K at E X = 2.33 eV (⚬) and 1.81 eV (•). Solid line is calculated from balance equation. Source: Reproduced from T. Aoki, T. Shimizu, D. Saito, and K. Ikeda, J. Optoelectron. Adv. Mater., 7, 137 (2005) with permission from INOE.

179

180

6 Photoluminescence

double-peak lifetime distribution observed under the so-called geminate condition G ≤ 1019 cm−3 s−1 [76, 77]. By extending the longer lifetime limit with the internal reference mode, a third peak higher than the other two peaks appears at 𝜏 D ≈ 20 s for G ≈ 2.5 × 1015 cm−3 s−1 (see Figure 6.9a). This peak might have been overlooked earlier due to the lack of the very-low-frequency QFRS. As G increases, 𝜏 D continuously shortens, and the peak merges with the 𝜏 T -component at G ≈ 1.3 × 1018 cm−3 s−1 . Figure 6.9b shows the peak-positions 𝜏 D , 𝜏 T , and 𝜏 S for 1013 < G < 1023 cm−3 s−1 at the above-bandgap excitation of Ex = 2.33 eV (denoted by ⚬) and the bandgap excitation of Ex = 1.81 eV (•); the continuous shortening of 𝜏 D with increasing G is a salient feature of DP recombination based on the RT model and the plot of 𝜏 D vs. G fits the curve calculated 1 from the balance equation based on Eq. (6.20) with n ≈ G𝜏D , R ≈ n− 3 , 𝜏 0 = 10−8 s, and a = 1 nm [82, 88–90]. Thus, the PL of intrinsic a-Si:H is triple-peaked in the lifetime distribution at a low temperature of 3.7 K and Ex = 2.33 eV under the geminate condition G ≤ 1019 cm−3 s−1 , with the third peak-position of lifetime 𝜏 D decreasing from ∼20 to ∼0.1 s as G increases from ∼1015 to ∼1018 cm−3 s−1 . The peak lifetimes 𝜏 D (•), shorter by nearly two orders of magnitude at Ex = 1.81 eV in Figure 6.9b, indicate the effect of reduced thermalization of electrons excited by Ex close to EG ≈ 1.8 eV [82]. At sufficiently low G, the three peaks have very distinct lifetimes, suggesting that the recombination events at 𝜏 S , 𝜏 T , and 𝜏 D occur via three independent channels as non-competing recombination. However, when the 𝜏 D -component begins to merge with the 𝜏 T -component as G approaches ∼1018 cm−3 s−1 (Figure 6.9a), the two recombination events at 𝜏 T and 𝜏 D are no longer independent. Indeed, further increasing G to ∼1.2 × 1020 cm−3 s−1 shifts the combined component of 𝜏 T and 𝜏 D to shorter lifetimes and merges it with the 𝜏 S -component at G ≈ 1022 cm−3 s−1 , leading to a single-peak structure. Similar triple-peaked lifetime structures are also observed in the G-evolved QFRS spectra of a-Ge:H excited by Ex = 1.81 eV under the geminate condition G < 1019 cm−3 s−1 at T = 3.7 K [84]. In Figure 6.10, plots (⚬) show the steady-state carrier concentration nD = 𝜂 D G𝜏 D of a-Si:H as a function of G, where 𝜏 D and the relative QE 𝜂 D of the 𝜏 D -component are obtained by deconvoluting the G-evolved QFRS spectra (Figure 6.9a). The metastable carrier density nmet denoted by × and + deduced by integrating the residual PL decay of Figure 6.20 described in Section 6.4.4 and LESR spin densities are denoted by ∇ [71], Δ [73], and • [74]. All results shown in Figure 6.10 agree with the sublinear G-dependence ∝ G0.2 [86, 87, 91]. The steady-state carrier concentrations nS = 𝜂 S G𝜏 S denoted by ◽ and nT = 𝜂 T G𝜏 T denoted by ▴ with the respective relative QEs 𝜂 S and 𝜂 T calculated from Figure 6.9a are plotted as functions of G in Figure 6.10. In this case, nT increases almost linearly with G under the geminate condition G ≤ 1019 cm−3 s−1 , while 𝜏 T and 𝜂 T remain nearly constant with G (Figure 6.9b). When G exceeds 1019 cm−3 s−1 , the plots of nT coalesce with the sublinear curve of nD ∝ G0.2 with nT ≈ nD ≈ 1017 cm−3 . At around G ≈ 1019 cm−3 s−1 , the two components are observed to merge, as shown in Figures 6.9a and 6.10. Similarly, an extrapolation of the plot of nS vs. G intersects the sublinear curve at around nS ≈ nD ≈ 1018 cm−3 (G ≈ 1023 ∼1024 cm−3 s−1 ); the coalescence of all the components is also seen at around G ≈ 5.0 × 1022 cm−3 s−1 in Figure 6.9a. Now, by attributing 𝜏 S - and 𝜏 T -components to singlet and triplet excitons, respectively—in particular, STEs [52, 68, 92] and the 𝜏 D component to the DP recombination, we shall explain all the results. The values of nS ≈ 1018 cm−3 and

6.4 Photoluminescence Lifetime Spectroscopy of Amorphous Semiconductors by QFRS Technique

1020

Metastable Carrier Density [cm–3]

Figure 6.10 The plots ⚬, ◽, and ▴ indicate steady-state carrier densities vs. generation rate G: nD, nS , and nT , respectively, obtained from the QFRS spectra of Figure 6.9. The plots × and + denote metastable carrier density nmet vs. G deduced by integrating residual PL decay for high-quality film A and defective film B of a-Si:H (Figure 6.20), respectively. The plots ∇, Δ, and • denote LESR densities obtained from References [71, 73, 74], respectively. The unit of plots Δ is arbitrary [73]. The dotted line indicates the sublinear G-dependence of nmet ∝ G0.2 . Source: Reproduced from T. Aoki, J. Optoelectron. Adv. Mater., 11, 1044 (2009) with permission from INOE.

1018

1016

PL Decay (high quality) PL Decay (defective) DP (QFRS)

a-Si:H

G0.2

1014

1012

1010

108 1013

Triplet Exciton Singlet Exciton 1015 1017 1019 1021 Generation Rate G [cm–3s–1]

1023

nT ≈ 1017 cm−3 at the coalescences give the average inter-pair distance between the singlet excitons as ∼0.5nS −1/3 , amounting to ∼5 nm, and that between triplet excitons ∼0.5nT −1/3 , amounting to ∼10 nm. The spatial wavefunction of a singlet exciton is symmetric due to the anti-parallel spins, while that of a triplet exciton having parallel spins is antisymmetric; the exciton radius is smaller on average for the symmetric spatial function than the antisymmetric function [10, 87]. Thus, the triplet exciton coalesces at a lower concentration nT as compared to the singlet exciton. If a singlet exciton radius aex is close to its inter-pair distance ∼5 nm, the exciton binding energy e2 /2𝜅aex amounts to be ∼5 meV with a dielectric constant of a-Si:H: 𝜅 ≈ 12, which explains the disappearance of the 𝜏 S component at the low T ≈ 25 K (see Section 6.4.3.2). Figure 6.11a shows the QFRS spectra of dispersed PL of a-Si:H at 3.7 K, Ex = 2.33 eV, and G ≈ 2.3 × 1017 cm−3 s−1 ; the two lifetime peaks 𝜏 S and 𝜏 T get shorter as PL emission energy ℏ𝜔 increases, whereas 𝜏 D remains constant. The plots of recombination rates 𝜏 S −1 and 𝜏 T −1 vs. ℏ𝜔 are approximately proportional to (ℏ𝜔)3 in contrast to 𝜏 D −1 (Figure 6.11b). The (ℏ𝜔)3 dependence of 𝜏 S −1 and 𝜏 T −1 suggests the involvement of STEs in the 𝜏 S - and 𝜏 T -recombination according to References [86, 93–96]; details are available in the following text. On the other hand, 𝜏 D −1 remains constant with ℏ𝜔, which is explained by the involvement of DP recombination governed by Eq. (6.20) being independent of ℏ𝜔 (Figure 6.11b) [86, 91]. The reason why the observed lifetime 𝜏 S of singlet exciton is of the μs order is explained by the difference in e–h orbital sizes of the STEs [93, 97]. Kivelson and Gelatt [93] gave the radiative transition rate 𝜏 S −1 of the singlet exciton with a Bohr radius aB *: [ ][ 3] 1 4 1 e2 𝜔 = 𝜅− 2 (6.26) a2h S, 𝜏s 3 ℏc c2 where e2 /(ℏc) ≈ 1/137 is the fine structure constant, c is the speed of light, and S is an overlap term ∼(2ah /aB *)3 . Hence, this equation corroborates the proportionality between 𝜏 S −1 and (ℏ𝜔)3 . Omitting S reduces the right side of Eq. (6.26) to the radiative

181

6 Photoluminescence

a-Si:H G = 2.3 × 1017[cm–3s–1] τT Ex = 2.33[eV] Eћω[eV] τ T = 3.7[K] S 1.60 τD 1.55 1.50 1.45 1.40 1.35 1.30 1.25 1.20 1.15 1.10 1.05 1.00 0.95 0.90 10–8 10–6 10–4 10–2 100 102 PL Lifetime τ [s] (a)

107 Ex = 2.33eV 1/τS Emission Rate 1 / τ [s–1]

QFRS Signal [a.u]

182

105

(ћω)3

103

1/τT

101

1/τD G = 2 × 1017cm–3s–1

10–1 0.9 1.0

1.5 ћω [eV] (b)

Figure 6.11 (a) QFRS spectra of the monochromatized PL for emission energy ℏ𝜔 from 0.90 to 1.60 eV under the geminate condition G = 2.3 × 1017 cm−3 s−1 for a-Si:H at 3.7 K and E x = 2.33 eV. Source: Reproduced from T. Aoki, J. Non-Cryst. Solids, 352, 1138 (2006) by permission of Elsevier. (b) Double-log plots of emission rates for three recombination processes (𝜏 S −1 , 𝜏 T −1 , 𝜏 D −1 ) vs. PL emission energy ℏ𝜔 obtained by deconvoluting (a). The straight line indicates the power law 1/𝜏 ∞ (ℏ𝜔)3 on double-logarithmic scale. Source: Reproduced from T. Aoki, K. Ikeda, S. Kobayashi, and K. Shimakawa, PML, 86, 137 (2006) by permission of Taylor & Francis.

electric dipole transition rate 𝜏 0 −1 of an electric dipole moment eah [94]. Since aB * ≈ 2.4 nm for a-Si:H by Kivelson and Gelatt [93] and aB * ≈ 5 nm by us as mentioned earlier, 𝜏 S ranges from 1.2 to 11 μs, with the photon energy ℏ𝜔 = 1.5 eV, 𝜅 = 12, and ah = 0.2 nm. Considering an order of magnitude uncertainty in Eq. (6.26), our experimental result of 𝜏 S ≈ 2 μs agrees quite satisfactorily with the estimated value. By fixing the frequency at 39 kHz (𝜏 S ≈ 4.1 μs), 58 Hz (𝜏 T ≈ 2.7 ms), and 1.1 Hz (𝜏 D ≈ 0.14 s), the PL spectra of QFRS signals of the three components were taken at G ≈ 2.8 × 1017 cm−3 s−1 and T = 3.7 K for a sample of thickness d ≈ 9.3 μm to avoid the interference effects (Figure 6.12). Though fixing 𝜏 S and 𝜏 T is not strictly correct due to their dependence on ℏ𝜔 (Figure 6.11b), the PL spectrum of the QFRS signal at 𝜏 S ≈ 4.1 μs is similar to that of 𝜏 T ≈ 2.7 ms, except for a shift to higher ℏ𝜔 by ∼40 meV. This has been confirmed by more correct plots of areas of the deconvoluted 𝜏 S - and 𝜏 T -components from Figure 6.11a vs. ℏ𝜔 [88]. The spectrum of the 𝜏 D -component is the lowest in ℏ𝜔, suggesting that this component arising from deeper states (Figure 6.12). An increase of Ex does not alter the QFRS spectra for 𝜏 S and 𝜏 T so much, but enhances the 𝜏 D -component as well as 𝜏 D at G ≈ 2.0 × 1015 cm−3 s−1 (Figure 6.13a). Figures 6.12 and 6.13a suggest that both the 𝜏 S - and 𝜏 T -components originate from similar recombination processes except the difference ∼40 meV in ℏ𝜔, which is attributed to the exchange energy between singlet- and triplet-excitons [22, 64, 80, 87]. The increase in 𝜏 D and 𝜂 D by increasing Ex is attributed to the thermalization (diffusion) of DPs, since

6.4 Photoluminescence Lifetime Spectroscopy of Amorphous Semiconductors by QFRS Technique

1.0

τD

τS

2.33 1.94 1.81 1.70 1.63 1.58 1.53 1.49

τD

0.6

τT

0.4

10–1

1

1.2

1.4 ћω [eV]

10–5 10–3 10–1 PL Lifetime τ [s] (a)

101

1.8

104

ηS

103 α

10–2

102

10–3

101 G = 2.0 × [cm–3s–1]

1015

10–4 10–7

1.6

ηD ηT

T = 3.7[K]

1.46 10–9

τS

0.2

1 Quantum Efficiency η [a.u.]

QFRS Signal [a.u]

Ex[eV]:

τT

Resolution

0.8

0.0

a-Si:H

Ex = 2.33[eV] G = 2.8 × 1017[cm–3s–1]

a-Si:H T = 3.7[K]

1.4

1.5 1.6 1.7 1.8 1.9 Excitation Energy Ex [eV]

Absorption Coefficient α [cm–1]

Normalized QFRS Signal [a.u.]

Figure 6.12 PL spectra of QFRS signals at fixed lifetime 𝜏 S ≈ 4.1 μs, 𝜏 T ≈ 2.7 ms, and 𝜏 D ≈ 0.14 s for a-Si:H of thickness ∼9.3 μm at 3.7 K, E X = 2.33 eV, and G = 2.8 × 1017 cm−3 s−1 . All the peaks of spectra are normalized to unity: actual peaks for 𝜏 S and 𝜏 D are, respectively, ∼30% and ∼80% of that for 𝜏 T . Source: Reproduced from T. Aoki, J. Non-Cryst. Solids, 352, 1138 (2006) by permission of Elsevier.

100 2.0

(b)

Figure 6.13 (a) QFRS spectra of a-Si:H excited at various PL excitation energy E x with G ≈ 2.0 × 1015 cm−3 s−1 at 3.7 K. (b) PL excitation spectra (PLE) for three QFRS components: 𝜂 S (◾), 𝜂 T (▴), and 𝜂 D (•); each component is deconvoluted and normalized by PL excitation intensity; 𝛼 denotes absorption coefficient (cm−1 ) at 3.7 K; the sample is the same as in Figure 6.12. Source: Reproduced from T. Aoki, K. Ikeda, N. Ohrui, S. Kobayashi, and K. Shimakawa, J. Optoelectron. Adv. Mater., 9, 70 (2007) with permission from INOE.

an increase in Ex will extend the DP distance in band-tail states as noted for Figure 6.9b [82, 84]; also see Section 6.4.3.2. Figure 6.13b shows the PLE spectra for the three relative QEs: 𝜂 S , 𝜂 T , and 𝜂 D , together with the absorption coefficient 𝛼 (cm−1 ), where the relative QEs are the deconvoluted components from Figure 6.13a divided by the PL excitation intensity at each Ex . Fast decline in 𝜂 D compared to that in 𝜂 S and 𝜂 T by lowering Ex below the bandgap EG ≈ 1.83 eV (for 𝛼 = 103 cm−1 at 3.7 K), as well as the lowest PL spectrum of the 𝜏 D -component (Figure 6.12), indicates that the PL of the 𝜏 D -component has the largest Stokes shift due to DPs deeply trapped in tail states [87, 98]. The theory of such a large Stokes shift is given in Chapter 7.

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6 Photoluminescence

6.4.3.2

Temperature Dependence

Figure 6.14 shows the temperature (T)-evolved QFRS spectra of the a-Si:H film (S163) having a defect density of ∼1 × 1016 cm−3 excited at Ex = 2.33 eV and G ≈ 7.0 × 1016 cm−3 s−1 [91, 99, 100]. As T is raised from 3.7 K to 100 K, the 𝜏 T -component shifts to shorter lifetimes and disappears at T ≈ 75 K. The third peak at 𝜏 D , on the other hand, remains persistent up to ∼133 K, and 𝜏 D shortens continuously as T increases. The 𝜏 S -component disappears at T ≈ 25 K, but, concomitantly, another shoulder denoted by 𝜏 G ≈ 10−5 ∼ 10−4 s emerges between 𝜏 S and 𝜏 T , growing into a hump at higher T. Thus, a new double-peak structure with maxima at 𝜏 D and 𝜏 G appears in the QFRS spectra of a-Si:H at ∼80 K [91, 99, 100]. The reason for the disappearing 𝜏 S -component at T ≈ 25 K and 𝜏 T -component at T ≈ 75 K is explained by singlet- and triplet-excitons, respectively. This is because the binding energy of singlet excitons is smaller than that of the triplet-excitons by the exchange energy of ∼40 meV, as mentioned in Section 6.4.3.1. Moreover, T ≈ 75 K for the disappearance of the 𝜏 T -component corresponds to the onset of instability of

F = 0[kV/cm] a-Si:H(S163) Ex = 2.33[eV] G ≈ 7 × 1016[cm–3s–1]

Normalized QFRS Signal [a.u.]

184

τT τD

τD

10 14

20 25

τG

20

30 40 47 50 55 60 70 80 85 90 100 10–9

τT

7.5

τS

T [K] 3.7 15

G ≈ 4.3 × 1017 [cm–3s–1]

a-Ge:H Ex = 1.81eV τS T [K] 3.7

30 40 50 60 70 τG

10–7

10–5 10–3 PL Lifetime τ [s] (a)

85 10–1

10

10–9

10–7

10–5 10–3 10–1 PL Lifetime τ [s]

101

(b)

Figure 6.14 (a) Temperature T-evolved QFRS spectra of undoped a-Si:H film (S163) without application of electric field (F = 0 kV cm−1 ). PL was excited at E X = 2.33 eV with G ≈ 7 × 1016 cm−3 s−1 . Source: Reproduced from T. Aoki, N. Ohrui, C. Fujihashi, and K. Shimakawa, PML, 88, 9 (2007) by permission of Taylor & Francis. (b) a-Ge:H photoexcited at E x = 1.81 eV with G ≈ 4.3 × 1017 cm−3 s−1 for various temperature T. Source: Reproduced from T. Aoki, T. Shimizu, S. Komedoori, S. Kobayashi, and K. Shimakawa, J. Non-Cryst. Solids, 338–340, 456 (2004) by permission of Elsevier.

6.4 Photoluminescence Lifetime Spectroscopy of Amorphous Semiconductors by QFRS Technique

self-trapped holes forming the STE [92]. In contrast, the 𝜏 D -component gets enhanced, and 𝜏 D is shortened by elevating T, which is explained in detail in Section 6.4.3.3. A similar T-dependence of QFRS spectra is observed in a-Ge:H at Ex = 1.81 eV and G ≈ 4.3 × 1017 cm−3 s−1 , as shown in Figure 6.14b [84]; the 𝜏 S - and 𝜏 T -components disappear at ∼14 K, whereas the 𝜏 D -component survives up to ∼85 K. Another peak becomes noticeable at 𝜏 G ≈ 20 μs between 𝜏 S and 𝜏 T when T ≈ 20 K and the QFRS spectrum of a-Ge:H turns into a double-peak structure with maxima at 𝜏 G and 𝜏 D at T ≈ 20 ∼ 85 K. The lower thresholds of T for the disappearance of the 𝜏 S - and 𝜏 T -components in a-Ge:H may be attributed to the smaller exciton binding energy due to smaller bandgap of a-Ge:H as compared to a-Si:H [80]. Figure 6.15 shows the G-evolved QFRS spectra of a-Si:H at T ≈ 100 K and Ex = 2.33 eV, where 𝜏 G remains constant as G increases up to ≈1019 cm−3 s−1 , whereas 𝜏 D continues to decrease [82, 84, 100]. This suggests that a part of the photogenerated e–h pairs are not in the form of excitons, but remain as geminate pairs at 100 K; presumably, the spin effect on geminate pairs fades out at ∼100 K, but the Coulombic effect persists even above 100 K. The effects of electric field F, T, and Ex on the 𝜏 G component are explained by the classical Onsager model (see Section 6.4.3.3 and Reference [100]). 6.4.3.3

Effect of Electric and Magnetic Fields

Figure 6.16 shows the electric field F-evolved QFRS spectra (at T = 3.7 K) of the a-Si:H film (S163 used for Figure 6.14 in Section 6.4.3.2) deposited on Al2 O3 (sapphire) of high thermal conductivity and equipped with 0.5 mm spaced coplanar electrodes [99, 100]. a-Si:H T = 100[K] G [cm–3s–1] 6.7 × 1014

Ex = 2.33eV τG

τD

2.4 × 1015 Normalized QFRS Signal [a.u.]

Figure 6.15 G-evolved QFRS spectra of a-Si:H excited with various G = 6.7 × 1014 − 1.2 × 1021 cm−3 s−1 at T = 100 K and E x = 2.33 eV. Source: Reproduced from T. Aoki, T. Shimizu, S. Komedoori, S. Kobayashi, and K. Shimakawa, J. Non-Cryst. Solids, 338–340, 456 (2004) by permission of Elsevier.

9.8 × 1015 3.8 × 1016 1.5 × 1017 7.7 × 1017 4.0 × 1018 2.0 × 1019 1.0 × 1020 4.9 × 1020 1.2 × 1021 10–8

10–6

10–4 10–2 PL Lifetime τ [s]

100

102

185

6 Photoluminescence

T = 3.7[K]

Figure 6.16 Electric field F-evolved QFRS spectra of undoped a-Si:H film (S163) at T = 3.7 K. E X and G are the same as in Figure 6.14a. Source: Reproduced from T. Aoki, N. Ohrui, C. Fujihashi, and K. Shimakawa, PML, 88, 9 (2007) by permission of Taylor & Francis.

τT τD

F [kV/cm] Normalized QFRS Signal [a.u.]

186

τS

0 10 20 30 40 50 60 70 80 90 100 10–9

10–7

10–5

10–3

10–1

101

PL Lifetime τ [s]

Increasing the electric field F up to 100 kV cm−1 monotonically decreases the 𝜏 S and 𝜏 T components, but only slightly, and it enhances the DP component and shortens the lifetime 𝜏 D more significantly. In this case, no 𝜏 G component appears at all (Figure 6.16). On the other hand, raising T from 40 to 80 K (at F = 0 kV cm−1 ) increases the DP component and shortens 𝜏 D in Figure 6.14a [99]. Thus, there appears to be a strong resemblance between the T- and F-evolved DP lifetimes 𝜏 D as shown in Figure 6.17, where 𝜏 D is plotted simultaneously as a function of T at F = 0 kV cm−1 and F at T = 3.7 K in the logarithmic T and F scales for the samples S163 and S119 of a defect density ∼2 × 1016 cm−3 . The plots of 𝜏 D vs. T for T ≥ 35 K coincide with those of 𝜏 D vs. F for F ≥ 50 kV cm−1 by shifting along the bidirectional arrow (Figure 6.17), which has been interpreted on the basis of the theory of Shklovskii et al. [89, 101] and Monroe [102]. According to their theory, a photogenerated electron at low T makes energy loss hops (ELHs), and the electron either recombines with a localized hole separated by R in a recombination lifetime given by Eq. (6.20), or makes an ELH to another site at a distance r with a lower energy in a hopping time 𝜏 d , given by: ( ) 2r , (6.27) 𝜏d = 𝜈0−1 exp a where 𝜈 0 is the attempt-to-escape frequency of the order of phonon frequency ∼1012 s-1 . The hopping diffusion dominates the recombination at early steps of ELHs due to the significant difference between 𝜏 0 and 𝜈 0 −1 in respective Eqs. (6.20) and (6.27). After successive ELHs, R becomes of the same order as r, and also becomes larger than the

6.4 Photoluminescence Lifetime Spectroscopy of Amorphous Semiconductors by QFRS Technique

101

(S163)T = 3.7[K] (S119)T = 3.7[K] DP Lifetime τD [s]

Figure 6.17 DP lifetime 𝜏 D as a function of T for the samples (•) S119 and (⚬) S163 at F = 0 kVcm−1 and 𝜏 D as a function of F for the samples (×) S119 and (◽) S163 at T = 3.7 K. Bottom scale is logarithms of both T [K] and F [kVcm−1 ]. Bidirectional arrow indicates that the data for T ≥ 35 K and those for F ≥ 50 kVcm−1 coincide by shifting along it. Source: Reproduced from T. Aoki, N. Ohrui, C. Fujihashi, and K. Shimakawa, PML, 88, 9 (2007) by permission of Taylor & Francis.

100

10–1

F

(S163)F = 0[kV/cm]

T

(S119)F = 0[kV/cm]

10–2 10

20 50 T [K] & F [kV/cm]

100

characteristic length Rc = (a/2)ln(𝜈 0 𝜏 0 ). A computer simulation shows that the recombination competes with the hopping diffusion under a condition r and R > Rc , where it is diffusion-limited recombination with the recombination lifetime 𝜏 being indistinguishable from the hopping time 𝜏 d [89]. Moreover, as T is elevated above ∼20 K, the so-called the transport energy (TE) in the tail state starts to shift upwards to the mobility edge, and electrons localized in tail states deeper than TE hop up to the vicinity of the TE, which increases recombination paths as well as the hopping diffusion, eventually leading to the shortening of 𝜏 D [89, 102, 103]. Similar shortening of 𝜏 D and enhancement of 𝜂 D were observed by applying infrared of energy 0.3 eV and 75 mW cm−2 to a-Si:H at 3.7 K, which supports the hop-up of localized electrons [98]. The hopping length at TE is given by rt = 3𝜀0 a/(2kT), with the exponential decay parameter of the electron tail state 𝜀0 ≈ 25 meV [101]. When T is elevated to T d = 3𝜀0 /k ln(𝜈 0 𝜏 0 ) for rt = Rc , where the condition of the diffusion-limited recombination r and R > Rc no longer holds, one theoretically obtains T d ≈ 94 K for a-Si:H [99, 101]. The interplay between T- and F-dependent 𝜏 D is interpreted on the basis of the effective temperature theory by introducing the effective temperature T eff = 𝛾eFa/k with elementary charge e and shifting the data of either T- or F-dependent 𝜏 D along the bidirectional arrow to coincide with each other in Figure 6.17. Here, we obtain the proportional factor 𝛾 ≈ 0.6 with a = 1 nm, which is surprisingly close to 𝛾 ≈ 0.67 obtained from a hopping transport parameter: photoconductivity (PC) in a-Si:H [104, 105]. Strictly, F-dependent PC was not obtained at T = 0 K, but a finite T, and thus the influence of finite T and F on PC, was parametrized by a single quantity 𝛽: Teff (T, F) = [T 𝛽 + (γeFa∕k)𝛽 ]1∕𝛽

(6.28)

From T- and F-dependences of PC, 𝛽 = 2 was determined in a-Si:H [104, 105]. We have measured 𝜏 D for various values of F at the intermediate temperature T = 40 and 55 K for the samples S119 and S163, and optimized 𝛽 = 1.4, so that all the data of the two samples may conform on a line [99].

187

6 Photoluminescence

In contrast, the 𝜏 G -component decreases by increasing F up to 70 kV cm−1 , as well as by increasing T from 40 to 100 K, which can be expressed by the relevant thermalization length r0 from the classical Onsager theory for e–h pairs [100, 106]. The T- and F- dependences of the 𝜏 G component can be explained by the increase of r0 caused by raising T and/or Ex above EG ≈ 1.8 eV of a-Si:H from 1.95 to 2.33 eV [100]. A magnetic field is found to have little influence on the 𝜏 S -component up to 0.9 T, but it enhances the 𝜏 T -component and reduces the 𝜏 D -component in the QFRS spectra of a-Si:H at 3.7 K, Ex = 2.33 eV, and G ≈ 7.7 × 1016 cm−3 s−1 , as shown in Figure 6.18a,b [82]. Three QFRS signals at fixed lifetimes (𝜏 S ≈ 2 μs, 𝜏 T ≈ 3 ms, and 𝜏 D ≈ 80 ms) vs. the magnetic field intensity are plotted as a function of the magnetic field intensity in Figure 6.18b. The preceding effects of the magnetic field on the PL lifetimes are interpreted as follows. Robins and Kastner [107] observed an enhancement of triplet exciton recombination for both amorphous and crystalline As2 Se3 under a magnetic field by TRS. Although singlet exciton state is absent in these materials [81, 82, 107–109], we have confirmed the same magnetic field effect on a-As2 Se3 by QFRS as well [85]. According to Robins and Kastner’s explanation, the 𝜏 S component assigned to a singlet exciton of total spin S = 0 is unaffected by a magnetic field. In contrast, the transition of triplet exciton of total spin S = 1 to the singlet ground state (S = 0) is spin-forbidden, but only allowed by weak coupling with the singlet excited state, and hence its relaxation rate depends on the coupling strength, as mentioned about Figure 6.1 in Section 6.2. Application of a magnetic field induces Zeeman splitting in the triplet excited state, increasing the coupling strength to enhance the triplet exciton recombination. Thus, the enhancement of the 𝜏 T component under the magnetic field may be attributed to the triplet exciton recombination in a-Si:H as well as in a-As2 Se3 [82, 85]. By a distinct method of pulsed optically

1

1 G = 7.7 × 1016[cm–3s–1] a-Si:H 0[T] 0.9[T]

T = 3.7[K] τT

τS

Ex = 2.33[eV]

τD

0 10–6 10–5 10–4 10–3 10–2 10–1 100 101 102 PL Lifetime τ [s] (a)

τT = 3 × 10–3[s] QFRS Signal [a.u.]

QFRS Signal [a.u.]

188

τD = 8 × 10–2[s]

τS = 2 × 10–6[s]

0 10–4

10–3

10–2

10–1

100

Magnetic Field [T] (b)

Figure 6.18 (a) QFRS spectra of a-Si:H at 3.7 K, E x = 2.33 eV, and G = 7.7 × 1016 cm−3 s−1 with (•) and without (⚬) application of a 0.9 T magnetic field. (b) QFRS signals of a-Si:H at fixed lifetimes 𝜏 S = 2 μs, 𝜏 T = 3 ms, and 𝜏 D = 80 ms as functions of magnetic field intensity. Source: Reproduced from T. Aoki, T. Shimizu, D. Saito, and K. Ikeda, J. Optoelectron. Adv. Mater., 7, 137 (2005) with permission from INOE.

6.4 Photoluminescence Lifetime Spectroscopy of Amorphous Semiconductors by QFRS Technique

STE

DP

~5 meV ~40 meV

EC EG ≈ 1.8 eV

Figure 6.19 Schematic models of three types of radiative recombination in a-Si:H of bandgap E G ≈ 1.8 eV at low temperatures. STE, self-trapped exciton; S, singlet; T, triplet; STH, self-trapped hole; R, intra-distance of a DP; r, an ELH (energy loss hop) distance.

EV

T

S ћω

ћω

ћω STH ~0.4 eV R

r

detected magnetic resonance (pODMR), Lips et al. [110] have found a direct evidence of the triplet exciton in a-Si:H at a rather high G ≈ 2 × 1022 cm−3 s−1 . Meanwhile, the spin directions of the DPs are completely random in the absence of a magnetic field; the total spin S is not a good quantum number because of the lack of correlation in the pair. Application of a magnetic field aligns spins of carriers in average, and thus the number of spin-aligned pairs (spin-forbidden to recombine) exceeds that of antiparallel-spin pairs (spin-allowed to recombine), leading to the paramagnetism of DPs. Hence, the spin-aligned DPs will be prevented from recombination by the magnetic field, which decreases the 𝜏 D component (Figure 6.18a,b) [82]. A recombination model that takes into account all these in a-Si:H at a low T is presented in Figure 6.19. 6.4.4

Residual PL Decay of a-Si:H

The kinetics of spectrally integrated PL intensity I(t) was measured by turning off 2.33 eV excitation laser by an electro-mechanic shutter after sufficiently long time irradiation on a-Si:H films to allow I(t) to reach its steady state value. Subsequently, the residual PL decay I(t) was lock-in detected by chopping PL emission in the position of C2 of Figure 6.4, so that I(t) may be free from chopping frequency, where all experimental procedures were performed electronically [86]. Figure 6.20 shows the double-log plots log10 I(t) vs. log10 t after cessation of PL excitation at 3.7 K with various G from 2 × 1013 to 2 × 1019 cm−3 s−1 for a-Si:H films A and B. The film A has the defect density 2 × 1016 cm−3 , and the more defective film B of defect density 6 × 1016 cm−3 is excited at G = 2 × 1013 cm−3 s−1 for comparison. Except for the I(t) observed at the lowest G of 2 × 1013 cm−3 s−1 , all other PL decay curves of the film A converge asymptotically at ∼104 s, even though they are initially different by five orders of magnitude. As mentioned in Section 6.4.3.1, the metastable carrier density of the film A, nmet (×), was calculated for various G by the integration by substitution of I(t) with t = 10x and adjusting nmet to nD at G = 1016 cm−3 s−1 (inset of Figure 6.20) [86]; both nmet and nD show sub-linear G-dependences (∝ G0.2 ), agreeing with the LESR densities (∇[71], Δ[73], •[74]). These results support that LESR measures the photocarriers localized in band-tail states, which subsequently recombine to give the very-long-lived PL. However, as G decreases below ∼1016 cm−3 s−1 , nmet as well as the LESR densities deviates from the sub-linear dependence of G0.2 ; the deviation is larger for the more defective film B

189

6 Photoluminescence

Figure 6.20 Double-log plots of residual PL decay of a-Si:H at 3.7 K and E x = 2.33 eV with various values of G—(1A) G = 2.0 × 1019 cm−3 s−1 , (2A) 2.0 × 1018 , (3A) 2.0 × 1017 , (4A) 2.0 × 1016 , (5A) 2.0 × 1015 , (6A) 2.0 × 1014 , (7A) 2.0 × 1013 —for film A of defect density 2 × 1016 cm−3 ; (7B) G = 2.0 × 1013 cm−3 s−1 for film B of defect density: 6 × 1016 cm−3 . Plots are fitted to the derivative of a stretched exponential (SE) function (solid red curves). Inset is 10x I(10x ) vs. x = log10 t for G corresponding to (4A), (5A), and (6A). Source: Reproduced from T. Aoki, K. Ikeda, S. Kobayashi, and K. Shimakawa, PML, 86, 137 (2006) with permission from Taylor & Francis.

103 4A

101

10xI(10x) [a.u.]

1A. 2A. 3A. PL Intensity I(t) [a.u.]

190

10–1

2.0

6A 0 –2

4A.

5A

1.0

0

2 X = log10 t

4

5A.

10–3

6A. 7A. 7B.

10–5

10–7 10–2

10–1

100

101 102 Time t [s]

103

104

105

(+) shown in Figure 6.10. Actually, the PL decay curve of the defective film (curve 7B in Figure 6.20) deviates more pronouncedly from the asymptotically converged curves at t ≈ 103 s. Thus, non-radiative recombination at the defect sites probably participates in shortening the fate of metastable carriers, when nmet becomes comparable with the defect density [86, 101, 111]. In order to avoid the contribution of excitonic or geminate recombination to the early stage of the PL decay at times t < 1 s in Figure 6.20, we have fitted the derivative of a stretched exponential (SE) function Kt 𝛽-1 exp[−(t/𝜏 0 )𝛽 ] to each PL decay I(t) for t > 10 s after the cessation of PL excitation. Here, 𝜏 0 is the effective recombination time, and 𝛽 is a dispersion parameter (0 < 𝛽 < 1). It is seen that I(t) after ∼10 s obeys the derivative of SE function for various G (≤2 × 1019 cm−3 s−1 ), and various T (described below in Figure 6.22), which manifests the DP recombination governed by time-dispersive monomolecular reaction. Figure 6.21a,b show, respectively, 𝛽 and 𝜏 0 (I-shape bars) vs. log10 G obtained by the non-linear regress for Figure 6.20 [91]. The lengths of the I-shape bars indicate the extents of 𝛽 and 𝜏 0 , where the nonlinear least square fit keeps the 𝜒 2 value within 1.5 times its minimum value. The squares (◽) in Figure 6.21a,b represent, respectively, the 𝛽 and 𝜏 0 values estimated by Morigaki [112] from the data of LESR kinetics of Yan et al. [74] for various G. Here, we notice a very close similarity in the two kinetic parameters 𝜏 0 and 𝛽 between PL and LESR. Since 𝜏 0 is relevant to the derivative of SE function at t around 𝜏 0 , a measuring time shorter than 𝜏 0 may cause a noticeable error. We have measured the PL decay over t > 104 s, while the LESR decay was measured only until t ≦ 2 × 103 s. Thus, one order difference of 𝜏 0 in magnitude between the PL and

6.4 Photoluminescence Lifetime Spectroscopy of Amorphous Semiconductors by QFRS Technique

0.6 PL 0.5

LESR

β

0.4 0.3 0.2 0.1 0.0 15

16

17

18

19

20

19

20

Log(G) [cm–3s–1]

Log(τ0) [s]

(a) 5 4 3 2 1 0 –1 –2 –3 –4 –5 –6 –7 –8 15

PL LESR

16

17

18

Log(G) [cm–3s–1] (b)

Figure 6.21 Parameters of SE function (I-shape bars) obtained by curve-fitting of Figure 6.20. Squares (◽) obtained by Morigaki from LESR decay [112]. (a) Dispersion parameter of SE function 𝛽 vs. G; (b) effective recombination lifetime 𝜏 0 vs. G. Source: Reproduced from T. Aoki, J. Optoelectron. Adv. Mater., 11, 1044 (2009) with permission from INOE.

LESR kinetics arises from the differences in the measuring time t, Ex , and T, and its exponential dependence on 1/𝛽. For details, see Reference [91]. Figure 6.22 demonstrates the PL decay I(t) at various T under the geminate condition of G ≈ 2.0 × 1016 cm−3 s−1 , where the PL decay can still be represented by the derivative of SE function. At an elevated T, photogenerated electrons diffuse apart by thermally activated hop-ups as well as hop-downs in the tail states and recombine with holes liberated from the self-trapping, which is the so-called dispersive bimolecular recombination. In fact, both 𝛽 and 𝜏 0 decrease with raising T, and I(t) approaches a power law (curve T4). The overall features of the PL decay can be given by the combined effects of radiative and nonradiative recombinations. Nevertheless, Figure 6.23 shows that the plots of integrated I(t), nmet (Δ) vs. T, agree with the LESR data (+) from Reference [113], and (◾) from Reference [114] as well as nD (⚬) obtained from the T-evolved QFRS spectra [87].

191

6 Photoluminescence

PL Intensity I (t) [ a.u.]

Figure 6.22 Double-log plots of residual PL decay of a-Si:H film A with G = 2.0 × 1016 cm−3 s−1 and E x = 2.33 eV at various temperatures T; (T1) 3.7 K, (T2) 100 K, (T3) 130 K, (T4) 150 K. Plots are fitted to the derivative of an SE function (solid red curves). Source: Reproduced from T. Aoki, K. Ikeda, S. Kobayashi, and K. Shimakawa, PML, 86, 137 (2006) by permission of Taylor & Francis.

G = 2.0 × 1016[cm–3s–1]

T1. T2.

10–1

T3. T4. 10–3 3.7[K]

10–5

150[K] 130[K] 100[K]

10–7 10–2 10–1

100

101 102 Time t [s]

103

104

105

1017 a-Si:H Carrier Con. n [cm–3]

192

1016

QFRS PL Decay

1015

LESR [a.u.] 1 LESR [a.u.] 2 1014

0

20

40

60 80 100 120 Temperature T [K]

140

160

Figure 6.23 Plots of steady-state carrier concentrations n vs. T; the plots nmet (Δ) estimated from residual PL decay, (⚬) from QFRS [84, 87] and the LESR intensities in arbitrary units as functions of T for a-Si:H; (+) from Ref. [113], (◾) from [114]. Source: Reproduced from T. Aoki, J. Non-Cryst. Solids, 352, 1138 (2006) by permission of Elsevier.

6.5 QFRS on Up-Conversion Photoluminescence (UCPL) of RE-Doped Materials Unlike amorphous semiconductors with broad PL spectra and lifetime distributions, RE ions in condensed matters possess ladder-like energy levels and emit narrow PL spectra similar to line spectra of isolated atoms due to the f-f transitions. The relaxation time between the two levels of an RE ion ranges from μs to ms.

6.5 QFRS on Up-Conversion Photoluminescence (UCPL) of RE-Doped Materials

We assume an M-level model of the RE ion energy, and the population density N i of i-th (i = 0, 1, 2,…M−1) level of RE ions is obtained from a set of M simultaneous rate equations, which are intrinsically nonlinear, since UCPL involves multi-photoexcitation steps. Hence, a set of nonlinear differential equations for N i cannot be solved in a closed form, and are therefore analyzed either numerically or approximately [29, 30, 115, 116]. As mentioned in Section 6.3.3, a biased TRS is technically difficult to realize, because a small impulse response of PL cannot be obtained by a phase-sensitive (lock-in) technique [58, 59]. However, the QFRS sinusoidally modulates the CW excitation flux density 𝛷 with a perturbation 𝜙, which induces sinusoidal perturbation ni in the time-invariant population density N i at each i-th level, and detects it by the lock-in technique with very high S/N [56, 64]. Here, N i is normalized by the total density of RE ions N ion (cm−3 ) and numerically calculated from the M-1 nonlinear rate equations ∑M−1 with the time-derivative Ṅ i = 0 and i=0 Ni = 1, where i = 0 denotes the ground state. On the assumption |𝜙| ≪ 𝛷, the second-order perturbation terms such as ni 𝜙 and ni nj can be neglected. Accordingly, the equation of the vector n = [n0 , n1 , n2 ,…nM−1 ]T is linearized in the set of M linear differential equations, with T denoting “transpose.” Furthermore, we subject the linearized equation of n to Laplace transform by setting ̃(s) and 𝜙̃ ∝ exp[ist]with i2 = −1 to Laplace transform of n and 𝜙, respectively, and n obtain a set of algebraic equations expressed in the M × M matrix form: ̃ , (sI − A − W − 𝛷P)̃ n(s) = 𝜙PN

(6.29)

where I is an M × M identity matrix; N = [N 0 , N 1 , N 2 , …N M − 1 ]T ; and A, W , and P correspond, respectively, to the radiative and nonradiative relaxation terms, the energy-transfer (ET) term involved with ETU and/or CRP, and the photoexcitation term (PE) consisting of GSA and ESA parameters (for details, see Reference [117]). Hence, we by multiplying Eq. (6.29) by the inverse matrix (sI − A − W − 𝛷P)−1 : can figure out ñ𝜙(s) ̃ ̃(s) n = (sI − A − W − 𝛷P)−1 PN 𝜙̃ Thereby, we have the transfer function

̃ ni (s) 𝜙̃

(6.30) for the modulated UCPL at the i-th level, and ̃ n (s)

the QFRS signal is obtained from its imaginary part –Im[ i𝜙̃ ] by setting s = i𝜔. Now, we shall limit ourselves to the typical three-level model of the UCPL GSA/(ESA + ETU) of Er3+ ions, as shown in Figure 6.24, where level 0 corresponds to the ground state Er3+ : 4 I 15/2 ; level 1, to the intermediate excited state 4 I 11/2 ; and level 2, to the uppermost excited states of three-bundled 4 F 7/2 , 2 H 11/2 /4 S3/2 manifolds [117, 118], and the matrices A, W , and P are given by: ⎡0 k1 ⎤ ⎡−𝜎0 0 0⎤ ⎡0 2wN 1 0⎤ bk 2 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ A = ⎢0 −k1 (1 − b)k2 ⎥ , W = ⎢0 −4wN 1 0⎥ , P = ⎢ 𝜎0 −𝜎1 0⎥ ⎢0 0 ⎥ ⎢ 0 𝜎 0⎥ ⎢0 2wN 0⎥ −k2 ⎦ ⎦ ⎣ ⎦ ⎣ ⎣ 1 1

(6.31)

with the relaxation rate k 1 (s−1 ) from level 1 to 0; the relaxation rate at level 2, k 2 (s−1 ), with the branching ratio b (e.g. bk 2 is the relaxation rate from level 2 to 0, as shown

193

6 Photoluminescence

0

4I

ETU

15/2

bk2

(1–b)k2

w

Figure 6.24 Schematic of three-level model for green UCPL Er3+ (2 H11/2 / 4 S3/2 → 4 I15/2 ) of mixed GSA/(ESA + ETU) at 975 nm excitation. 0: ground state (4 I15/2 ). 1: intermediate state (4 I11/2 ). 2: uppermost state (three bundled 4 F 7/2 , 2 H11/2 / 4 S3/2 ). The kinetic parameters k1 , k2 , b, w, 𝜎 0 , and 𝜎 1 are described in text (color online).

N1 + n1

k1

4I 11/2

ESA

1

N2 + n2

GSA

4F 7/2 2H 11/2 4S 3/2

σ1 (Φ + ϕ)

2

σ0(Φ + ϕ)

194

N0 + n0

in Figure 6.24); the ETU parameter, w (s−1 ); and the GSA and ESA cross-sections, 𝜎 0 ̃ n (s) and 𝜎 1 (cm2 ), respectively. Thereby, the transfer function of UCPL 2𝜙̃ is expressed as a linear combination of two fractional functions Ri (𝛷)/[s + Ri (𝛷)] and coefficients ai (𝛷) (i = 1, 2) [117, 118]: ̃ (s + k1 + 𝜎0 𝛷)𝜎1 N1 + 𝜎0 𝜎1 N0 𝛷 + 2wN 1 (𝜎0 N0 + 𝜎1 N1 ) n2 (s) = (s + k + 𝜎0 𝛷 + 𝜎1 𝛷 + 4wN 1 )(s + k2 ) − [(1 − b)k2 − 𝜎0 𝛷](𝜎1 𝛷 + 2wN 1 ) ̃ 𝜙 1 R1 (𝛷) R2 (𝛷) = (6.32) a (𝛷) + a (𝛷), s + R1 (𝛷) 1 s + R2 (𝛷) 2 The fractional function Ri (𝛷)/[s + Ri (𝛷)] is the transfer function of the system having the relaxation rate Ri (𝛷), or the relaxation (lifetime) time 𝜏 i = Ri (𝛷)−1 in Section 6.3.3 (Eq. (6.9)); Ri (𝛷) is obtained by setting the denominator of Eq. (6.32) to 0. Thus, the QFRS on the three-level model produces two relaxation rates R1 (𝛷) at level 1 and R2 (𝛷) at level 2, which become k 1 and k 2 , respectively, as 𝛷 → 0. In contrast to the ordinary TRS [120], 𝜏 i of QFRS depends on 𝛷, and R1 (𝛷) is approximated in the first order of 𝛷 for 𝜎 0 𝛷/k 1 ≪ 1 to be: [ }] { 2w𝜎0 (1 + b) 𝛷 = k1 + ̂ S𝛷, (6.33) R1 (𝛷) ≈ k1 + 𝜎0 + 𝜎1 b + k1 𝜎1 with the slope Sˆ and R2 (𝛷) ≈ k 2 due to k 2 ≫ k 1 [117, 118]. Moreover, the ratio of the QFRS peak 𝛾(𝛷) = a2 (𝛷)/a1 (𝛷) gives ETU contribution against ESA expressed by the 2w𝜎 ratio k 𝜎 0 [117–119]. As 𝛾(𝛷) is a complicated function of 𝛷, only 𝛾(0) is given here as: 1 1

( )( ) 2w𝜎0 −1 k1 k1 γ(0) = 1 − − . 1+ k2 k1 σ1 k2

(6.34)

We have applied QFRS on the green UCPL of Er-doped Ge28.1 Ga6.3 S65.6 chalcogenides glass (ChGs) while systematically varying Er-doping from 0.01 to 0.5 at% as well as 975 nm excitation power P∝𝛷 (for experiment details, see References [117, 118]).

6.5 QFRS on Up-Conversion Photoluminescence (UCPL) of RE-Doped Materials

(a)

Normalized QFRS Signal (a.u.)

P (mW)

(b)

P (mW)

42.5

42.5

42.5

15.3

15.3

15.3

11.3

11.3

11.3

9.8

9.8

9.8

7.8

7.8

7.8

5.3

5.3

5.3

2.3

τ2

τ1

10–6 10–5 10–4 10–3 10–2

2.3

τ2

τ1

10–6 10–5 10–4 10–3 10–2 Lifetime τ (s)

(c)

P (mW)

2.3

τ2

τ1

10–6 10–5 10–4 10–3 10–2

Figure 6.25 Pump power (975 nm laser) P-evolved QFRS spectra of green (2 H11/2 /4 S3/2 → 4 I15/2 , ∼550 nm) UCPL in (a) 0.02, (b) 0.1, and (c) 0.5 at% Er-doped GeGaS ChGs. Red curves are best-fit curves of spectra for P = 2.3 mW by nonlinear regression of Eq. (6.32), with red bars indicating the corresponding magnitudes a1 (𝛷) and a2 (𝛷). The dotted lines are guides for eyes of P-evolved lifetimes 𝜏 1 and 𝜏 2 (color online). Source: Reproduced from L. Strizik, V. Prokop, J. Hrabovsky, T. Wagner, and T. Aoki, J. Mater. Sci: Mater. Electron. 28, 7053 (2017) by permission of Springer.

Figure 6.25 demonstrates the P-evolved QFRS spectra of UCPL of the samples for the excitation power P = 2.3–42.5 mW (roughly corresponding to the photon-flux density: 𝛷 ≈ 2.3 × 1021 –4.25 × 1022 cm−2 s−1 ), exhibiting a salient feature of two lifetime components of 𝜏 1 and 𝜏 2 . An increase of P shortens the long lifetime 𝜏 1 from 1.1 to 0.6 ms, in contrast to the short lifetime 𝜏 2 —remaining fairly constant at several tens of μs. The red curves are obtained by the nonlinear regression for the imaginary part of Eq. (6.32) with s = i𝜔 as a model function with unknown parameters k 1 , k 2 , and w to fit the QFRS spectra at the lowest P (2.3 mW) [117]. Thereby, the red bars indicate the amplitudes a1 (𝛷) and a2 (𝛷) corresponding to the 𝜏 1 and 𝜏 2 components, respectively; clearly, the ratio 𝛾(0) decreases as Er concentration N Er increases, predicting an increase of w according to Eq. (6.34) [117, 118]. As P increases, however, the model function [Eq. (6.32)] does not give a good fit to the QFRS spectra. Hence, we have introduced the lifetime distributions of two Gaussians centered around 𝜏 1 and 𝜏 2 at the higher P by the convolution techniques mentioned in Section 6.3.6 [118]. Figure 6.26a shows R1 (𝛷) vs. 𝛷 giving k 1 ≈ 970 s−1 in average. However, similarly obtained k 2 varies almost linearly from ∼1.5 × 104 to ∼2.3 × 104 s−1 with increasing N Er , which is attributed to the ET from the uppermost three-bundled states 4 F 7/2 , 2 H 11/2 /4 S3/2 to the absorption edge of the ChGs host [117, 119]. The GSA cross-section 𝜎 0 was estimated in two ways: 𝜎 0 = 7.2 × 10−21 cm2 by Judd-Ofelt (JO) analysis (see Chapters 2 and 4), and 𝜎 0 = 5.9 × 10−21 cm2 by the absorption spectra (GSA). The values of 𝜎 1 /𝜎 0 = 1.4 and b = 0.7 are determined from the JO analysis [117].

195

Relaxation rate R1(Φ ) = τ1–1(×103 s–1)

6 Photoluminescence

1.7 1.6 1.5 1.4 1.3 1.2 1.1

0.5 % Er 0.1 % Er 0.02 % Er

1.0 0.9 0

1

2

3

4

5

Photon-flux density Φ (×1022cm–2 s–1) (a) 14 Macroscopic ETU parameter WEr (×10–18cm3 s–1)

196

12

Slope R1 (Φ)JO

Ratio γ (0)JO

Slope R1 (Φ)Exp.

Coleman et al.

10 8 6 4 2 1019

1020

Er concentration NEr

(cm–3)

(b)

Figure 6.26 (a) Relaxation rate R1 (𝛷) at level 1 (4 I11/2 ) vs. excitation photon-flux density 𝛷 of 975 nm laser. (⧫) for 0.02, (◽) 0.1, and (⚬) 0.5 at% Er-doped GeGaS ChGs. The lines were obtained by linear fits of Eq. (6.33) to data at low 𝛷. (b) Macroscopic ETU parameter W Er = w/NEr vs. Er concentration NEr (cm−3 ) of the green (∼550 nm) UCPL. (⚬) obtained from slope Sˆ of the R1 (𝛷) vs. 𝛷 at low 𝛷 with 𝜎 0 = 7.2 × 10−21 cm2 and 𝜎 1 /𝜎 0 = 1.4 (JO analysis), (◊) with 𝜎 0 = 5.9 × 10−21 cm2 (GSA) and 𝜎 1 /𝜎 0 = 1.4, (•) from peak ratio 𝛾(0) at P = 2.3 mW and (◽) from Reference [116]. Source: Reproduced from L. Strizik, V. Prokop, J. Hrabovsky, T. Wagner, and T. Aoki, J. Mater. Sci: Mater. Electron. 28, 7053 (2017) by permission of Springer.

Figure 6.26b shows the macroscopic ETU parameter W Er = w/N Er (cm3 s−1 ) plotted as a function of N Er . Here, the symbols ⚬ and ◊ denote W Er obtained from Sˆ in Eq. (6.33) with 𝜎 0 = 7.2 × 10−21 cm2 (JO analysis) and with 𝜎 0 = 5.9 × 10−21 cm2 (GSA), respectively, and the symbol • denotes W Er derived from the ratio 𝛾(0) of Eq. (6.34). The plots ⚬ and • are similar to those of Coleman et al (◽) by TRS in GeGaS:Er ChGs [116], whereas those of ◊ deviate considerably from others. Probably, the error in w∝(Sˆ − 𝜎 0 − b𝜎 1 ) is exaggerated when Sˆ approaches 𝜎 0 + b𝜎 1 , requiring more accurate determination of 𝜎 1

6.6 Conclusions

and b. Furthermore, the differences among the data of W Er are partly addressed to the determination of N Er , which was estimated from volumetric mass density of the glass under the assumption of the homogeneous distribution and full ionization of Er3+ in the GeGaS matrix. Therefore, N Er is not always identical to N ion (cm−3 ), and, moreover, inaccurate N Er causes noticeable error in determining 𝜎 0 = 5.9 × 10−21 cm2 from the GSA spectra. Nevertheless, the obtained w predicts that ETU dominates ESA only for the 2w𝜎 0.5% Er-doped sample by the criterion ETU ≷ ESA according to the ratio k 𝜎 0 ≷ 1 [117]. 1 1 The UCPL of Er3+ ions excited at 975 nm emits red light (4 F 9/2 → 4 I 15/2 , ∼660 nm) as well as green light. We have made QFRS on the red UCPL of the preceding samples and analyzed the data by a 5 × 5 matrix equation on the five-level model (M = 5) by adding two 4 I 13/2 and 4 I 9/2 manifolds, which demonstrates the versatility of Eq. (6.30). For details, please see Reference [119].

6.6 Conclusions The PL of condensed matter is reviewed, and its state of the art is described as far as possible. Though QFRS is a counterpart in the frequency-domain against TRS in the time-domain and a powerful tool to investigate widespread PL lifetime distribution, it is not so prevalent. Thus, this chapter mainly focuses on the QFRS on PL lifetime distribution: the broad lifetime distribution of amorphous semiconductors, in particular, a-Si:H; and the narrow but multi-lifetime distribution of RE-doped ChGs. The wideband QFRS technique has revealed the triple-peak lifetime distribution in PL of a-Si:H as well as a-Ge:H at low temperatures T (∼3.7 K) and low generation rates G ( Eh , but gets localized in the tail states for Ev < Eh . ̂ xp (Eq. (7.3)) can be written Using Eqs. (7.2), (7.4), and (7.6), the interaction operator H in the second quantized form as: ( )1∕2 e ∑ ℏ ̂ Hxp = − Qcv c+𝜆 (7.8) 𝜇x 𝜆 2𝜀0 n2 V 𝜔𝜆

207

208

7 Photoluminescence, Photoinduced Changes, and Electroluminescence in Noncrystalline Semiconductors

where Qcv = N −1

∑

exp[−ite ⋅ Rel ] exp[−i th ⋅ Rhm ]Zlm𝜆 Bcvlm (S = 0)

(7.9)

l,m

where Bcvlm (S = 0) =

∑ 𝜎e ,𝜎h

acl (𝜎e )dvm (𝜎h )𝛿𝜎e ,−𝜎h is the annihilation operator of a singlet

exciton of spin S = 0 by annihilating an electron in the conduction c states at site l and a hole in the valence v states at site m, and the summations over 𝜎 e and 𝜎 h represent combinations of spins to form singlet and triplet excitons [5]. Zlm𝜆 is given by: Zlm𝜆 =

∫

𝜙∗l (re )𝜀̂𝜆 ⋅ p𝜙m dre drh

(7.10)

It may be noted that it is not necessary to use the exciton operator Bcvlm (S = 0) in ∑ Eq. (7.9); one can also use the product of fermion operators 𝜎e ,𝜎h acl (𝜎e )dvm (𝜎h )𝛿𝜎e ,−𝜎h instead, provided the condition of singlet spin state 𝛿𝜎e ,−𝜎h is maintained. This is because the integral in Eq. (7.10) is non-zero only if the spin configuration of the photoexcited electron and hole is in singlet state; in other words, it is non-zero only for singlet excitons. We now consider a transition from an initial state with one singlet exciton created by exciting an electron at a site, say, l, and hole at site m, and where any other site is assumed to have zero excitons to a final state with one created photons and no excitons. We also assume that there are no photons in the initial state. The transition matrix element is then obtained as [19]: ( )1∕2 ∑ ℏ ̂ xp ∣ i⟩ = − e ⟨f ∣ H pcv (7.11) 𝜇x 𝜆 2𝜀0 n2 V 𝜔𝜆 where pcv = N −1

∑

exp[−ite ⋅ Rel ] exp[−ith ⋅ Rhm ]Zlm𝜆 𝛿l,m

(7.12)

l,m

Following Eq. (7.10), the transition matrix element in Eq. (7.11) is also non-zero only for singlet excitons and vanishes for triplet excitons. Also, the same transition matrix element is obtained even if one considers the electron and hole field operators in Eqs. (7.4) and (7.6), respectively, to be spin independent. That means the transition matrix element of the radiative recombination of a singlet exciton or an excited free electron–hole pair remains the same as given in Eq. (7.11), but that it is zero for a triplet spin configuration. Therefore, the formalism developed here onward for calculating the rates of spontaneous emission is applicable to singlet excitons, type I and singlet type II geminate pairs, and nongeminate pairs. Let us first derive Zlm𝜆 . As stated earlier in Chapter 3, there are two approaches, which are used in amorphous solids, to evaluate this integral. The integral actually determines the average value of the relative momentum between the excited electron–hole pair. In the first approach, it is assumed to be a constant and independent of the photon energy as [1, 5]: ( )1∕2 L Zlm𝜆 = 𝜙∗l (re )𝜀̂𝜆 ⋅ p𝜙m dre drh = Z1 = 𝜋h (7.13) ∫ V

7.2 Photoluminescence

where L is the average bond length in a sample and Z1 denotes the matrix element obtained from the first approach. This form was first introduced by Mott and Davis [1] and has been used widely since then. In the second approach, the integral is evaluated using the dipole approximation as [5, 33–35]: Zlm𝜆 =

∫

𝜙∗l (re )𝜀̂𝜆 ⋅ p𝜙m dre drh = Z2 = i𝜔𝜇x |re−h |

(7.14)

where Z2 denotes the transition matrix element obtained from approach 2; ℏ𝜔 = E′ c − E′ v is the emitted photon energy; and |re−h | = ⟨l|𝜀̂𝜆 ⋅ r||m⟩ is the average separation between the excited electron–hole pair, which also can be assumed to be site independent. Thus, in both approaches, the integral becomes site independent and can be taken out of the summation in Eq. (7.11). In the case of singlet excitons and type II geminate pairs, it can be easily assumed that |re − h | = aex , the excitonic Bohr radius, but for the case of type I geminate and nongeminate pairs, |re − h | is the average separation between e and h. It may be noted here that the value of the integral obtained from the first approach, Z1 , leads to the well-known Tauc’s relation for the absorption coefficient observed in amorphous semiconductors, and that obtained from the second approach is used to explain the deviations from Tauc’s relation in the absorption coefficient observed in some amorphous semiconductors [1, 33, 34] (also see Chapter 3 of this volume). Using Eqs. (7.13) and (7.14), pcv in Eq. (7.12) can be written as: ∑ pcv = Zi N −1 exp[−ite ⋅ Rel ] exp[−ith ⋅ Rhm ]𝛿l,m , i = 1, 2 (7.15) l,m

Now the derivation of pcv in Eq. (7.15) for non-crystalline solids depends on four possibilities, which are considered separately in the following text. (i) Extended-to-extended states transitions Rearranging the exponents in Eq. (7.15) as: exp[−ite ⋅ Rel − ith ⋅ Rhm ] = exp[−ite ⋅ (Rel − Rhm ) − i(te ⋅ th ) ⋅ Rhm ]

(7.16)

and identifying the fact that Rel − Rhl = aex , the excitonic Bohr radius (the separation between e and h in an exciton prior to their recombination, which is assumed to be site independent), the first exponential becomes site independent. It can be taken out of the summation signs, and then the matrix element in Eq. (7.15) becomes: pcv = Zi N −1 exp[−ite ⋅ aex ]𝛿te ,−th , i = 1, 2

(7.17)

where 𝛿te ,−th represents the momentum conservation in the transition. The square of the transition matrix element then becomes: |pcv |2 = |Zi∗ Zi | = |Zi |2 , i = 1, 2

(7.18)

(ii) Transitions from extended to tail states Here, we consider the electron e excited in the extended state and the hole h in the tail state. For this case, Eq. (7.15) can be written as: ∑ exp[−ite ⋅ Rel ] exp[−th′ ⋅ Rhm ]𝛿l,m , i = 1, 2 (7.19) pcv = Zi N −1 l,m

209

210

7 Photoluminescence, Photoinduced Changes, and Electroluminescence in Noncrystalline Semiconductors

where ∣ th′ ∣= th′ =

√ 2m∗h (Eh − Ev )∕ℏ

(7.20)

Rewriting Eq. (7.19) as: pcv = Zi N −1

∑

exp[−ite ⋅ (Rel − Rhm )] exp[−i(te − ith′ ) ⋅ Rhm ]𝛿l,m , i = 1, 2

l,m

(7.21) and simplifying it in the same way as Eq. (7.16), we get: pcv = Zi exp[−ite ⋅ aex ]𝛿te ,ith , i = 1, 2

(7.22)

Equation (7.16) gives the same expression for |pcv |2 as in Eq. (7.18) for the possibility (i), and in this case the momenta also remain conserved. (iii) Transitions from tail to extended states We consider an electron e excited in the tail and the hole h in the extended states. For this case, the transition matrix element can be derived in a way analogous to that for the extended-to-tail states. pcv is obtained as: pcv = Zi exp[−ith ⋅ aex ]𝛿ite′ ,th , i = 1, 2 where ∣ te′ ∣= te′ =

√ 2m∗e (Ec − Ee )∕ℏ

(7.23)

(7.24)

Here, again, |pcv |2 remains the same as in Eq. (7.18). (iv) Transitions from tail-to-tail states Here, we consider that both electron e and hole h are excited in the tail states. In this case, an exciton loses its usual excitonic character and behaves like a type II geminate pair, as explained earlier. This is because the localized form of wave functions of e and h does not give rise to an exciton. For this case, using the localized form of the electron and hole wave functions, pcv is obtained as: ∑ pcv = Zi N −1 exp[−te′ ⋅ Rel ] exp[−th′ ⋅ Rhm ]𝛿l,m , i = 1, 2 (7.25) l,m

Here again, one can rearrange the exponential as: ∑ pcv = Zi N −1 exp[−te′ ⋅ (Rel − Rhm )] exp[−(te′ + th′ ) ⋅ Rhm ]𝛿l,m , i = 1, 2

(7.26)

l,m

which becomes: pcv = Zi exp[−te′ ⋅ aex ]𝛿te′ ,−t′ , i = 1, 2 h

(7.27)

Assuming that the relative momentum of the electron, ℏt′ e , will be along the direction of aex at the time of recombination, |pcv |2 from Eq. (7.27) becomes: |pcv |2 = |Zi |2 exp[−2te′ aex ] 𝛿te′ ,−t′

h

(7.28)

7.2 Photoluminescence

7.2.2

Rates of Spontaneous Emission

Using Eq. (7.11) and applying Fermi’s golden rule, the rate Rsp (s−1 ) of spontaneous emission can be written as [19, 36–40]: ( ) ∑ 1 2𝜋e2 |p |2 f f 𝛿(E′ − Ev′ − ℏ𝜔𝜆 ) (7.29) Rsp = 2 2 2𝜀0 n V 𝜔𝜆 E′ ,E′ cv c v c 𝜇x c

v

where f c and f v are the probabilities of occupation of an electron in the conduction state and a hole in the valence state, respectively. The rate of spontaneous emission in Eq. (7.29) can be evaluated under several conditions, which will be described in the following text. (i) Rate of spontaneous emission under two-level approximation Here, only two energy levels are considered as in atomic systems. An electron takes a downward transition from an excited state to the ground state; no energy bands are involved. In this case, f c = f v = 1, and then, denoting the corresponding rate of spontaneous emission by Rsp12 , we get: Rsp12 =

𝜋e2 |p12 |2 𝛿(ℏ𝜔 − ℏ𝜔𝜆 ) 𝜀0 n2 𝜔𝜇x2

(7.30)

where ℏ𝜔 = E2 − E1 , which is the energy difference between the excited (E2 ) and ground (E1 ) states. In this case, p12 is derived using the dipole approximation (Eq. (7.14)) as: p12 = i𝜔𝜇x ⟨𝜀̂𝜆 .r⟩

(7.31)

where r is the dipole length, and ⟨…⟩ denotes integration over all photon modes l. Substituting Eq. (7.31) into Eq. (7.30), and then integrating over the photon wave vector k using 𝜔 = kc/n, we get [8, 37]: √ 4𝜅e2 𝜀𝜔3 |re−h |2 (7.32) Rsp12 = 3ℏc3 where 𝜀 = n2 is the static dielectric constant, ∣re − h ∣ is the mean separation between the electron and hole, and 𝜅 = 1/(4𝜋𝜀0 ). It may be noted that, for deriving the rate of spontaneous emission within the two-level approximation, pcv with Zlm𝜆 in the form of Eq. (7.13) has not been used to the author’s knowledge. The expression of the rate of spontaneous emission obtained in Eq. (7.32) is well known [8, 11, 36], and it is independent of the electron and hole masses and temperature. As only two discrete energy levels are considered, the density of states is not used in the derivation. Therefore, although by using |re − h | = aex , the excitonic Bohr radius, the rate Rsp12 has been applied to a-Si:H [8, 11], it should only be used for calculating the radiative recombination in isolated atoms, not in condensed matter. (ii) Rates of spontaneous emission in amorphous solids In applying Eq. (7.29) for any condensed-matter system, it is necessary to determine f c and f v and the density of states. On the one hand, it may be argued that short-time PL can occur before the system reaches thermal equilibrium, and that therefore no equilibrium distribution functions can be used for the excited charge carriers. In this case, one should use f c = f v = 1 in Eq. (7.29) for condensed matter as well. On the other hand, as the carriers are excited by the same energy photons, even in a

211

212

7 Photoluminescence, Photoinduced Changes, and Electroluminescence in Noncrystalline Semiconductors

short time delay, they may be expected to reach thermal equilibrium among themselves, but not necessarily with the lattice. Therefore, they will relax according to an equilibrium distribution. As the electronic states of amorphous solids include the localized tail states, it is more appropriate to use the Maxwell–Boltzmann distribution for this situation. We will consider here radiative recombination under both nonequilibrium and equilibrium conditions. 7.2.2.1

At Nonthermal Equilibrium

Considering f c = f v = 1 and substituting |pcv |2 (Eq. (7.12)) derived above from the two approaches in Eq. (7.29), we get the rates, Rspni , for the possibilities (i)–(iii) as: ( ) ∑ 2𝜋e2 1 Rspni = 2 𝛿(Ec′ − Ev′ − ℏ𝜔𝜆 ), i = 1, 2 (7.33) |Zi |2 2 2𝜀0 n V 𝜔𝜆 𝜇x E′ ,E′ c

v

where the subscript spn in Rspni denotes rates of spontaneous emission derived at the non-equilibrium. For evaluating the summation over Ec′ and Ev′ in Eq. (7.33), the usual approach is to convert it into an integral by using the excitonic density of states, which can be obtained as follows. Using the effective mass approximation, the electron energy in the conduction band and hole energy in the valence band can be, respectively, written as: p2 (7.34) Ec′ = Ec + e ∗ 2me and Ev′

= Ev −

p2h

(7.35)

2m∗h

Subtracting Eq. (7.35) from Eq. (7.34) and then applying the center of mass and relative coordinate transformations (see Eq. (7.3)), we get the excitonic energy Ex in the parabolic form as: p2 P2 (7.36) + Ex = Eo + 2M 2𝜇x where E0 = Ec − Ev is the optical gap; P and M = m*e + m*h are linear momentum and mass of an exciton associated with its center of mass motion, respectively; and the last term is the kinetic energy of the relative motion between e and h with linear momentum p, which contributes to the exciton binding energy through the attractive Coulomb interaction potential between them [41]. The exciton density of states g x then comes from the parabolic form of the second term associated with the center of mass motion as: ( ) V 2M 3∕2 (Ex − E0 )1∕2 (7.37) gx = 2𝜋 2 ℏ2 Using Eq. (7.37), the summation in Eq. (7.33) can be converted into an integral as: ∑ 𝛿(Ex′ − ℏ𝜔𝜆 ) = Ij = gx 𝛿(Ex′ − ℏ𝜔𝜆 )dE′x (7.38) ∫ ′ E E′ x

x

which gives Ij =

( ) V 2M 3∕2 (ℏ𝜔𝜆 − E0 )1∕2 2𝜋 2 ℏ2

(7.39)

7.2 Photoluminescence

Substituting Eq. (7.39) into Eq. (7.33) we get: ( ) 2𝜋e2 1 j |Zi |2 Ij Θ(ℏ𝜔𝜆 − E0 ), Rspni = 2 2𝜀0 n2 V 𝜔𝜆 𝜇x

i = 1, 2

(7.40)

where Θ(ℏ𝜔𝜆 − E0 ) is a step function used to indicate that there is no radiative recombination for ℏ𝜔𝜆 < E0 , and j denotes that these rates are derived through the joint density of states. Having derived the rates of spontaneous emission using the excitonic density of states, it is important to remember that the use of such joint densities of states for amorphous semiconductors does not give the well-known Tauc’s relation [1, 5] in the absorption coefficient [34]. Therefore, by using it in calculating the rate of spontaneous emission, one would violate the Van Roosbroeck and Shockley relation [42] between the absorption and emission. For this reason, the product of individual electron and hole densities of states is used in evaluating the summation in Eq. (7.27) for amorphous semiconductors. This approach has proven to be very useful in amorphous solids as it gives the correct Tauc’s relation [1, 5, 33, 43]. Denoting the summations over E′ c and E′ v in Eq. (7.33) by I and using the product of individual densities of states, I can be evaluated by converting the summations into integrals as: Ev +ℏ𝜔𝜆

I=

∫Ec

Ev

∫E′ −Ev

gc (Ec′ )gv (Ev′ )𝛿(Ec′ − Ev′ − ℏ𝜔)dE′c dE′v

(7.41)

c

where g c (E′ c ) and g v (E′ v ) are the densities of states of the conduction and valence states, respectively, and these can be written within the effective-mass approximation as: ( ) V 2m∗ 3∕2 1∕2 Eq , q = c, v (7.42) gq (E′ ) = 2𝜋 2 ℏ2 where m* is the effective mass of the corresponding charge carrier, and q = c (conduction) and q = v (valence) states. Substituting Eq. (7.42) into Eq. (7.41), the integral can be evaluated analytically to give [5]: V 2 (m∗e m∗h )3∕2

(ℏ𝜔 − E0 )2 (7.43) 4𝜋 3 ℏ6 Using Eqs. (7.39) and (7.43) in Eq. (7.33) and substituting the corresponding Zi, we get the two rates in nonthermal equilibrium for possibilities (i)–(iii) as: I=

Rspn1 =

e2 L(m∗e m∗h )3∕2 4𝜀0 ℏ3 n2 𝜇x2 (ℏ𝜔)

(ℏ𝜔 − E0 )2 Θ(ℏ𝜔𝜆 − E0 )

(7.44)

and Rspn2 =

(m∗e m∗h )3∕2 e2 |re−h |2 2𝜋 2 𝜀0 n2 ℏ7

V ℏ𝜔(ℏ𝜔 − E0 )2 Θ(ℏ𝜔𝜆 − E0 )

(7.45)

For excitonic transitions, it is more appropriate to replace m∗e and m∗h in Eqs. (7.44) and (7.45) by the excitonic reduced mass 𝜇x and use ∣re − h ∣ = aex . It may also be noted that the volume V is appearing in the formula for Rspn2 (Eq. (7.45)). This is inevitable through the second approach, and it has been tackled earlier by Cody [35] by defining 2N 0 /V = 𝜈𝜚A , where N 0 is the number of single spin states in the valence band, and thus 2N 0 becomes the total number of valence electrons occupying N 0 states, 𝜈 is the

213

214

7 Photoluminescence, Photoinduced Changes, and Electroluminescence in Noncrystalline Semiconductors

number of coordinating valence electrons per atom, and 𝜌A is the atomic density per unit volume. Thus, replacing V by V/N 0 in Eq. (7.45), one can get around this problem. We thus obtain: e2 L𝜇x Rspn1 = (7.46) (ℏ𝜔 − E0 )2 Θ(ℏ𝜔𝜆 − E0 ) 4𝜀0 ℏ3 n2 𝜇x2 (ℏ𝜔) and Rspn2 =

𝜇x3 e2 a2ex ℏ𝜔(ℏ𝜔 − E0 )2 Θ(ℏ𝜔𝜆 − E0 ) 2𝜋 2 𝜀0 n2 ℏ7 𝜈𝜌A

(7.47)

It may be emphasized here that E0 , defined as the energy of the optical gap, is not always the same in amorphous solids. It depends on the lowest-energy state within the conduction band from where the radiative recombination occurs. As the rates derived in Eqs. (7.46) and (7.47) do not have any peak value, it is not possible to determine E0 from these rates. Likewise, using |pcv |2 (Eq. (7.28)) in Eq. (7.33) for possibility (iv), we get the rates of spontaneous emission from the two approaches for the tail-to-tail states transitions as: Rspnti = Rspni exp(−2te′ ∣ re−h ∣), i = 1, 2

(7.48)

where the subscript spnt of Rspnt stands for the spontaneous emission at nonequilibrium from tail-to-tail states. re−h is the average separation between e and h; for excitons, re−h = aex , where aex is the excitonic Bohr radius in the tail states, and for a singlet exciton it is given by [6, 10]: aex =

5𝜇𝜺 a 4𝜇x 0

(7.49)

where m is the reduced mass of an electron in the hydrogen atom, and a0 = 0.529 Å is the Bohr radius. Results for the rates of spontaneous emission obtained in the preceding text are valid in the nonthermal equilibrium condition, as stated earlier. However, unless one knows the relevant effective masses of charge carriers and the value of E0 , these rates cannot be applied to determine the lifetime of PL. The effective mass of charge carriers in their extended and tail states can be determined [5], as also described in Chapter 3 of this volume, but not E0 . For this reason, it is useful first to derive these rates under thermal equilibrium as well, as shown in the following text. 7.2.2.2

At Thermal Equilibrium

Assuming that the excited charge carriers are in thermal equilibrium among themselves, the distribution functions, f c and f v , can be given by the Maxwell–Boltzmann distribution function, as given in reference [44]: [ ] (E − EFn ) fc = exp − e (7.50) 𝜅B T and

[ ] (EFp − Eh ) fv = exp − 𝜅B T

(7.51)

where Ee and Eh are the energies of an electron in the conduction and a hole in the valence state, respectively, and EFn and EFp are the corresponding Fermi energies. 𝜅 B is

7.2 Photoluminescence

the Boltzmann constant and T temperature of the excited charge carriers. The product f c f v is then obtained as: [ ] (ℏ𝜔 − E0 ) (7.52) fc fv ≈ exp − 𝜅B T where Ee − Eh = ℏ𝜔 and EFn − EFp = E0 are used. It may be noted that Eq. (7.52) is also an approximate form of the Fermi–Dirac distribution obtained for Ee > EFn and Eh < EFp , and it has been used widely [36, 38–43] for calculating the rate of spontaneous emission in semiconductors. Substituting Eq. (7.52) in the rate in Eq. (7.29), the integral in Eq. (7.41) is obtained as: [ ] V (m∗e m∗h )3∕2 (ℏ𝜔 − E0 ) 2 I= (ℏ𝜔 − E ) exp − (7.53) 0 𝜅B T 4𝜋 3 ℏ6 Using Eq. (7.53), the rates of spontaneous emission at thermal equilibrium for possibilities (i)–(iii) are obtained as: [ ] (ℏ𝜔 − E0 ) , i = 1, 2 (7.54) Rspi = Rspni exp − 𝜅B T And for the possibility (iv), we get: [ ] (ℏ𝜔 − E0 ) , Rspnti = Rspni exp − 𝜅B T

i = 1, 2

(7.55)

Rates in Eqs. (7.54) and (7.55) have a maximum value, which can be used to determine E0 , as described in the following text. 7.2.2.3

Determining E0

For determining E0 , we assume that the peak of the observed PL intensity occurs at the same energy as that of the rate of spontaneous emission obtained in Eqs. (7.54) and (7.55). The PL intensity as a function of ℏ𝜔 has been measured in a-Si:H. From these measurements, the photon energy corresponding to the PL peak maximum can be determined. By comparing the experimental energy thus obtained with the energy corresponding to the maximum of the rate of spontaneous emission, we can determine E0 . For this purpose, we need to determine the energy at which the rates in Eqs. (7.54) and (7.55) become maximum. Defining x = ℏ𝜔 − E0 , (x > 0) and 𝛽 = 1/𝜅 B T and then setting dRspi /dx = 0 (i = 1, 2), we get x01 and x02 from Eq. (7.54) and Eq. (7.55), respectively, at which the rates are maximum, as [19]: √ 1 − 𝛽E0 (1 − 𝛽E0 )2 2E0 + ± (7.56) x01 = 2𝛽 4𝛽 2 𝛽 and x01

3 − 𝛽E0 = ± 2𝛽

√

(3 − 𝛽E0 )2 2E0 + 4𝛽 2 𝛽

(7.57)

where only the + sign produces x > 0. It may be noted that both Eqs. (7.54) and (7.55) give the same expression for x01 and x02 as derived in Eqs. (7.56) and (7.57). Using

215

216

7 Photoluminescence, Photoinduced Changes, and Electroluminescence in Noncrystalline Semiconductors

x0 = Emx − E0 , where Emx = ℏ𝜔mx is the emission energy at which the PL peak intensity is observed, the corresponding E0 value is obtained from Eqs. (7.56) and (7.57) as: E01 =

Emx (−1 + 𝛽Emx ) 1 + 𝛽Emx

(7.58)

E01 =

Emx (−3 + 𝛽Emx ) 1 + 𝛽Emx

(7.59)

and

Using this in Eqs. (7.54) and (7.55), one can calculate the rates of spontaneous emission from the two approaches for all four possibilities. The time of radiative recombination or radiative lifetime, 𝜏 ri , is then obtained from the inverse of the maximum rate (𝜏 ri = 1/Rspi , I = 1, 2) obtained from Eqs. (7.54) and (7.55), calculated at ℏ𝜔mx = Emx . 7.2.3

Results of Spontaneous Emission and Radiative Lifetime

Applying the preceding theory, rates of spontaneous emission are calculated in this section for a-Si:H, (i) within two-level approximation, (ii) under nonequilibrium with the joint and product density of states, and (iii) under equilibrium with the product density of states. Although the rate derived within the two-level approximation is strictly valid only for isolated atomic systems, for applying it to a condensed-matter system, one requires 𝜔 and |re − h |. For applying it to the excitonic radiative recombination, one may be able to assume |re − h | = aex , but it is not clear at what value of 𝜔 one should calculate the rate to determine the radiative lifetime. For this reason, the rate Rsp12 in Eq. (7.32) derived within the two-level approximation will be calculated later at the end of this section. Rates in Eqs. (7.46)–(7.48), derived under nonthermal equilibrium with the product density of states, cannot be used to calculate the radiative lifetime unless one knows the value of E0 , which cannot be determined from these expressions. Therefore, here we can only present the results for the rates of spontaneous emission derived under the thermal equilibrium in Eqs. (7.54) and (7.55). We will use Eq. (7.54) to calculate the rates, Rspi , I = 1, 2, for possibilities (i)–(iii), and Eq. (7.55) to calculate Rspt1 and Rspt2 for possibility (iv) in a-Si:H. As the non-radiative relaxation is very fast (in ps), only excitonic recombinations can be expected to contribute to PL from possibilities (i)–(iii). All free excited carriers (type I and II geminate pairs and nongeminate pairs) are expected to have relaxed to their tail states before recombining radiatively, and hence they can contribute to PL only through possibility (iv). For calculating the effective masses of charge carriers, we consider a sample of a-Si:H with 1 at% weak bonds contributing to the tail states (i.e. a = 0.99 and b = 0.01), using L = 0.235 nm [4], E2 = 3.6 eV [45, 46], Ec = 1.80 eV [2], and Ec − Ect = 0.8 eV [47]. Using these in Eqs. (3.17)–(3.21), we arrive at m∗ex = 0.34 me and m∗et = 7.1 me , respectively, for a-Si:H. For determining E0 from Eqs. (7.58) and (7.59), we need to know the values of Emx and the carrier (exciton) temperature T. Wilson et al. [8] have measured PL intensity as a function of the emission energy for three different samples of a-Si:H at 15 K and at two different decay times of 500 ps and 2.5 ns. Stearns [9] and Aoki [12, 49, 50] have also measured it at 20 K and 3.7 K, respectively. Emx estimated from these spectra is given in Table 7.1.

7.2 Photoluminescence

Table 7.1 Values of E mx estimated from the maximum of the observed PL intensity from three different experiments (Figure 7.2) and the corresponding values of E 0 calculated from Eqs. (7.58) and (7.59) (note that both expressions produce the same value). E mx (eV)

E 0 (eV)

Experiment

T (K)

500 ps

2.5 ns

500 ps

2.5 ns

Sample 1 [11]

15

1.428

1.401

1.425

1.398

Sample 2 [11]

15

1.444

1.400

1.441

1.397

Sample 3 [11]

15

1.448

1.405

1.445

1.402

Sample 4 [9]

20

1.450

1.447

Sample 5 [50]

3.7

1.360

1.359

The next problem is to find the temperature of the excited carriers before they recombine radiatively. This can also be done on the basis of the three experimental results on PL [9, 11, 48–50]. The measured energy, Emx, of maximum PL intensity is below the mobility edge Ec by 0.4 [8, 11] to 0.44 eV [50]. This means that most excited charge carriers have relaxed down below their mobility edges and are not hot carriers anymore before the radiative recombination. It is therefore only logical to assume that the excited charge carriers in these experiments are in thermal equilibrium with the lattice, and E0 should be calculated at the lattice temperature. The assumption is very consistent with the established fact that the carrier–lattice interaction is much stronger in a-Si:H [5] than in crystalline Si. Using the experimental values of Emx and the corresponding lattice temperature, the values of E01 calculated from Eq. (7.58) and those of E02 calculated from Eq. (7.59) are also listed in Table 7.1. Both the approaches, Eqs. (7.58) and (7.59), produce identical results for E0 . It should be noted, however, that E0 increases slightly with the lattice temperature. Having determined the effective mass and E0 , we can now calculate the rates of spontaneous emission for transitions through possibilities (i)–(iv). (i) Possibility (i)—Both e and h excited in extended states As explained in the preceding text, possibilities (i)–(iii) are only applicable to the singlet excitonic recombinations. In possibility (i), the excitonic reduced mass is obtained as 𝜇x = 0.17 me and the excitonic Bohr radius as 4.67 nm. Using n = 4, the calculated Rsp1 is plotted as a function of the emission energy, and at a temperature T = 15 K for one of the samples of a-Si:H from Wilson et al. [11] in Figure 7.1a and Rsp2 for the same sample in Figure 7.1b. Similar curves are obtained for all the samples at other temperatures as well, but the magnitudes of the rates vary. The radiative lifetime can be obtained from the inverse of the maximum value of the rate at any temperature and E0 . The maximum value of the rates is obtained at the emission energy, Emx , at a given temperature. The rates and the corresponding radiative lifetimes thus calculated are listed in Table 7.2. As can be seen from Table 7.2, the radiative lifetime is found to be in the ns time range at 15 and 20 K and in the ms range at 3.7 K. (ii) Possibilities (ii)–(iii)—Extended-to-tail states recombinations For possibilities (ii) and (iii), where one of the charge carriers of a singlet exciton is in its extended state and the other in its tail state, we get 𝜇x = 0.32me and

217

7 Photoluminescence, Photoinduced Changes, and Electroluminescence in Noncrystalline Semiconductors

Figure 7.1 Rates of spontaneous emission plotted as a function of the emission energy for E 0 = 1.445 eV at a temperature of 15 K and calculated from (a) method 1 and (b) method 2 for calculating the matrix elements are listed in Table 7.2. As can be seen from Table 7.2, the radiative lifetime is found to be in the ns time range at 15 and 20 K, and in the ms range at 3.7 K.

Rate (108 s–1)

1.50 1.25 1.00 0.75 0.50 0.25 0 1.445

1..45 1.455 1.46 Emission energy (eV)

1.465

(a) 2.00 Rate (108 s–1)

218

1.50 1.00 0.50 0 1.445

1.455 1.46 1..45 Emission energy (eV)

1.465

(b)

aex = 2.5 nm. Using these and the other quantities as in possibility (i), we get from Eq. (7.54) (i = 1), Rsp1 = 2.94 × 108 s−1 , which gives 𝜏 r1 = 3.41 ns at 15 K, 5.22 × 108 s−1 giving 𝜏 r1 = 1.92 ns at 20 K, and at 3.7 K we get Rsp1 = 1.47 × 107 s−1 , which gives 𝜏 r1 = 0.1 μs. Likewise, from Eq. (7.54) (i = 2), we get Rsp2 = 3.45 × 108 s−1 and 𝜏 r2 = 2.9 ns at 15 K, and Rsp2 = 1.08 × 109 s−1 giving 𝜏 r2 = 0.92 ns at 20 K. At 3.7 K, we obtain Rsp2 = 1.62 × 107 s−1 and 𝜏 r2 = 0.1 μs. Thus, the radiative lifetimes for all the three possibilities (i)–(iii) are in the ns time range at 15 and 20 K, but in the μs time range at 3.7 K. It is not possible from these results to determine where exactly the PL is originating from in these experiments—that is, whether it is from the extended or tail states. In order to determine the origin of PL, one also needs to know the Stokes shift in the PL spectra. For example, if the excitonic PL is occurring through radiative transitions from extended-to-extended states, then the observed Stokes shift observed in the PL spectra should only be equal to the exciton binding energy. For this reason, it is important to determine the exciton binding energy corresponding to all the four transition possibilities. The ground-state singlet exciton binding energy Es in a-Si:H is obtained as [5, 10]: Es =

9𝜇x e4 20(4𝜋𝜀0 )2 𝜀2 ℏ2

(7.60)

This gives Es ∼ 16 meV for possibility (i), 47 meV for possibilities (ii) and (iii), and 0.33 eV for possibility (iv). The known optical gap for a-Si:H is 1.8 eV [2], and the

7.2 Photoluminescence

E0 estimated from experiments is about 1.44 eV at 15 K [11], 1.45 eV at 20 K [9], and 1.36 eV at 3.7 K [49] (see Table 7.1). Considering that the PL in a-Si:H originates from excitonic states (possibility (i)), we find a Stokes shift of 0.36 eV, 0.35 eV, and 0.44 eV at temperatures 15 K, 20 K, and 3.7 K, respectively. Such a large Stokes shift is not possible due only to the exciton binding energy, which at most is in the meV range for possibilities (i)–(iii), as given earlier. As the nonradiative relaxation of excitons is very fast, not much PL may occur through possibility (i) even at a time delay of 500 ps measured by Wilson et al. [11] and Stearns [9]. It may possibly occur from transitions only through possibilities (ii)–(iv), which means at least one of the charge carriers has relaxed to the tail states before recombining radiatively. This can be easily explained from the following text: as stated earlier, the charge carrier–lattice interaction is much stronger in a-Si:H than in crystalline Si [5]. As a result, it is well established [4] that an excited hole gets self-trapped very fast in the tail states in a-Si:H. Thus, a Stokes shift of about 0.4 eV observed experimentally at 3.7, 15, and 20 K is due to the relaxation of holes in excitons to the tail states plus the excitonic binding energy, which is in the meV range and hence plays a very insignificant role. The radiative lifetimes of such transitions calculated from the present theory and listed in Table 7.2 fall in the ns time range at temperatures 15–20 K, which agrees very well with those observed by Wilson et al. [8, 11] and Table 7.2 The maximum rates, Rsp1 and Rsp2 , of spontaneous emission and the corresponding radiative lifetime calculated using the values of E mx and E 0 for the five samples in Table 7.1. Sample

T (K)

Rsp1 (s−1 )

1a)

15

1.58 × 108

6.34 × 10−9

2.05 × 108

4.88 × 10−9

15

8

−9

8

4.82 × 10−9

8

2a)

Rsp2 (s−1 )

𝝉 r1 (s)

1.56 × 10

6.42 × 10

𝝉 r2 (s)

2.07 × 10

3a)

15

1.55 × 10

6.43 × 10

2.00 × 10

5.00 × 10−9

4a)

20

2.77 × 108

3.62 × 10−9

3.72 × 108

2.69 × 10−9

6

−6

6

0.12 × 10−6

8

2.93 × 10−9

8

5a) 1b)

3.7 15

8

7.77 × 10

8

2.97 × 10

−9

0.13 × 10

−9

3.37 × 10

8.38 × 10 3.43 × 10

2b)

15

2.93 × 10

3.41 × 10

3.46 × 10

2.89 × 10−9

3b)

15

2.92 × 108

3.42 × 10−9

3.47 × 108

2.89 × 10−9

20

8

−9

9

0.92 × 10−9

7

0.62 × 10−7

7

4b) 5b)

3.7

8

5.22 × 10

7

1.47 × 10

−9

1.92 × 10

−7

0.68 × 10

1.08 × 10 1.62 × 10

1c)

15

1.04 × 10

0.96 × 10

1.18 × 10

0.85 × 10−7

2c)

15

1.03 × 107

0.97 × 10−7

1.19 × 107

0.84 × 10−7

15

7

−7

7

0.84 × 10−7

7

0.47 × 10−7

6

1.82 × 10−6

3c) 4c) 5c)

20 3.7

7

1.03 × 10

7

1.84 × 10

6

0.51 × 10

−7

0.97 × 10

−7

0.55 × 10

−6

1.96 × 10

1.19 × 10 2.14 × 10 0.55 × 10

Note: Results are given for transitions involving extended–extended states (possibility (i)), extended–tail states (possibilities (ii) and (iii)), and tail–tail states (possibility (iv)). a) Extended–extended states transitions (possibility (i)). b) Extended–tail or tail–extended states transitions (possibilities (ii)–(iii)). c) Tail–tail states transitions (possibility (iv)).

219

220

7 Photoluminescence, Photoinduced Changes, and Electroluminescence in Noncrystalline Semiconductors

Stearns [9]. At 3.7 K, the calculated radiative lifetime is found to be in the μs range, which also agrees very well with Aoki et al.’s experimental results of the singlet exciton’s radiative lifetime [49] (see Figures 6.11, 6.13, and 6.14). (iii) Possibility (iv)—Tail-to-tail states recombinations For possibility (iv), as a type II singlet geminate pair, when both e and h are localized in their tail states, we get 𝜇x = 3.55 me , aex = 0.223 nm, and t ′ e = 1.29 × 1010 m−1 . Using these in Eq. (7.55), with i = 1, one gets Rspt1 = 1.03 × 107 s−1 , which gives 𝜏 r1 = 0.97 × 10−7 s at 15 K, Rspt1 = 1.84 × 107 s−1 , which gives 𝜏 r1 = 0.55 × 10−7 s at 20 K, and Rspt1 = 0.51 × 106 s−1 with the corresponding 𝜏 r1 = 2.0 μs at 3.7 K. Using Eq. (7.55) with i = 2, we get Rspt2 = 1.18 × 107 s−1 , which gives 𝜏 r2 = 0.85 × 10−7 s at 15 K, and Rspt2 = 2.14 × 107 s−1 , which gives 𝜏 r2 = 0.47 × 10−7 s at 20 K. The corresponding results at 3.7 K are obtained as Rspt2 = 0.55 × 106 s−1 and 𝜏 r2 = 2.0 μs. Wilson et al. [11] have also observed the lifetime in the μs time range, which may be attributed to transitions through possibility (iv). Aoki et al. [49] have observed another PL peak at 3.7 K (𝜏 G , Figure 6.14) with a radiative lifetime in the μs range but slightly longer than the singlet exciton radiative lifetime (𝜏 s ) at 3.7 K. According to the radiative lifetime calculated here (Table 7.2), the PL peak at 𝜏 G may be attributed to transitions from the tail-to-tail states from a type II singlet pair (possibility (iv)), because the corresponding radiative time is about 2 μs, longer than the radiative lifetime for possibilities (i)–(iii). A geminate pair of type I may be expected to have a larger separation and hence longer radiative lifetime. Indeed, in Figure 6.13, a peak, denoted by 𝜏 D , appears at much larger lifetime of ms, and Aoki et al. have attributed this peak to the distant or nongeminate pairs. According to the present theory, this peak may be attributed to the type I geminate pairs or distant pairs, but it is difficult to distinguish between the two (see the following text). For possibility (iv), we have calculated the rates using the same value of E0 as for possibilities (ii) and (iii), which may not be correct. However, as E0 cannot be measured experimentally, it is difficult to find a correct value for it for possibility (iv). One way to find a value of E0 for possibility (iv) is from the expected corresponding Stokes shift, when both the charge carriers have relaxed to their tail states. This can be done as follows [31]. The measured Stokes shift in a-Si:H is found to be 0.440 at 3.7 K [49] and 0.350 eV at 20 K [9] for possibilities (ii)–(iii). Here, we consider only these two different experimental values as examples. For possibility (iv), when both e and h are in their tail states, a double Stokes shift can be assumed. Accordingly, one gets the Stokes shift of 0.880 eV at 3.7 K and 0.700 eV at 20 K. Then, considering EC = 1.8 eV in a-Si:H, the PL peak is expected to occur at Emx = 1.8 − 0.880 = 0.920 eV at 3.7 K and 1.1 eV at 20 K, which give, from Eqs. (7.58) and (7.59), E0 = 0.918 eV at 3.7 K and 1.094 eV at 20 K. Using these values of E0 , the maximum value of rates at ℏ𝜔 = Emx are calculated, and the inverse of these reveals the corresponding radiative lifetimes (𝜏 r = 1/Rsp ). For type I geminate pairs, the lifetime is calculated for e–h separation, reh = 2L and 3L, where L = 0.235 nm, the average interatomic separation in a-Si:H; and, for the type II geminate pairs, reh = aex is used. The radiative lifetimes thus calculated in a-Si:H using the experimental value of Emx at 3.7 K from Aoki et al.’s experiment [49] and at 20 K from Stearns’ experiment [9] are listed in Table 7.3. At 3.7 K, for the type II geminate pairs, the radiative lifetime

7.2 Photoluminescence

Table 7.3 Rates of spontaneous emission and corresponding radiative lifetimes calculated from Eq. (7.55) (with i = 2) for geminate pairs of types I and II in a-Si:H. Radiative lifetime (𝝉 r ) Geminate I E 0 (eV) E mx (eV)

T (K)

Geminate II

reh = 2L

reh = 3L

reh = aex

0.918

0.920

3.7

1.10 ms

0.15 s

15.00 μs

1.094

1.100

20

6.20 μs

0.87 ms

88 ns

Note: Rates for type I are calculated for the average e–h separation, re–h = 2L and 3L (L = 0.235 nm) at the same E0 and Emx values.

is obtained in the ms time range, which agrees with the lifetime 𝜏 G observed by Aoki et al. At 20 K, the same lifetime is in the ns time range, which agrees with the lifetime observed by Stearns. Here, it is important to note that the input values used to calculate the rate of spontaneous emission at 3.7 and 20 K are obtained from two different experiments, namely references [9, 49], and therefore are different. Aoki et al. have not observed any radiative lifetime in the ns time range at any temperature, including 20 K as observed by Stearns. The value of Emx is not available to the author at 3.7 and 20 K from the same experiment. The results for the type I geminate pairs calculated at reh = 2L and 3L are in the ms to s time range, which agree with the radiative lifetimes, 𝜏 D , measured by Aoki et al. [48] but attributed to nongeminate pairs (see Figure 6.13). It is not possible to distinguish between the lifetime of type I geminate pairs and nongeminate pairs unless one can distinguish the corresponding separation between e and h in each. Therefore, it is possible that the measured lifetime is actually due to type I geminate pairs, because the type II geminate pairs were not known then [50]. In this view, the lifetime 𝜏 G measured in the ms time range can be associated to the type II geminate pairs, 𝜏 D in the ms to s range to type I geminate pairs, and any other longer lifetimes to nongeminate pairs. We have discussed here only the type II singlet geminate pairs, because, without consideration of the spin–orbit coupling, the transition matrix element (Eq. (7.11)) vanishes for a triplet spin configuration. As the magnitude of the spin–orbit coû xp in Eq. (7.3), the radiative lifetime of triplets pling is usually smaller than that of H is expected to be longer. The excitonic Bohr radius of triplet excitons and triplet type II geminate pair may not be very different, and therefore the lifetime 𝜏 T (see Figure 6.13) may be attributed to both. As the rates of recombination of nongeminate pairs or distant pairs have the same expression as in Eq. (7.55), it is not possible to distinguish between nongeminate and geminate pairs of the type I. The only difference can be the average separation between carriers, as it is well established that nongeminate pairs are usually formed from those geminate pairs excited by higher-energy photons, the separation between e and h in nongeminate pairs being larger than the critical separation given by rc ≤ n−1/3 ≤ 2rc for small generation rate [29, 30]. According to the present theory, a larger separation will yield an exponentially smaller recombination rate and hence larger radiative lifetime in the time range of seconds [50] (see Figure 6.13).

221

222

7 Photoluminescence, Photoinduced Changes, and Electroluminescence in Noncrystalline Semiconductors

The radiative lifetime for reh ≥ 3L indeed produces such longer lifetimes at 3.7 K. One may therefore conclude that the excited pairs with a separation of 3L or more may be classified as nongeminate pairs. The longer lifetime obtained for possibility (iv) is because of the exponential factor exp(−2te′ |re−h |) appearing in the rate of recombination in Eq. (7.55) through Eq. (7.48), which is proportional to the probability of quantum tunneling a barrier of height Ec for a distance |re−h |. Therefore, the expression also implies that the rate of spontaneous emission reduces, and hence the radiative lifetime is prolonged in possibility (iv) due to the quantum tunneling through a distance of |re−h | before the radiative recombination. In evaluating the transition matrix element of the exciton–photon interaction operator (Eq. (7.3)), which depends only on the relative momentum between e and h in an exciton, the wave functions of e and h are used instead of the( wave) function of an exciton [10, 41]—for example, in the form of Ψ(R, r) = exp i Pℏ. R 𝜙(r). From this point of view, the theory presented here appears to address the recombination between an excited pair of an electron and a hole, not that between an electron and a hole in an exciton. The fact that the interaction operator is independent of the center of mass momentum P means that the integral Zlm𝜆 (Eq. (7.10)) always gives the average value of the separation between e and h. In an exciton, the separation is the excitonic Bohr radius, and in an e-and-h pair, it would be their average separation. As far as the recombination in amorphous solids is concerned, therefore, this is the only difference between an excitonic recombination and free e and h recombination. (iv) Results of two-level approximation As we have determined the energy, Emx , at which maxima of PL peaks have been observed in the five samples of a-Si:H, we can calculate Rsp12 in Eq. (7.32) and the corresponding radiative lifetime at these photon energies for a comparison. Here, we will use |re−h | = aex and 𝜔 = Emx /ℏ in Eq. (7.32). For extended-to-extended state recombination, aex = 4.67 nm, and, using Emx = 1.4 eV for Wilson et al.’s sample, we get w = 2.1 × 1015 Hz, Rsp12 = 7.4 × 1010 s−1 , and 𝜏 r = 1/Rsp12 = 13 ps. For Stearn’s sample, with Emx = 1.45 eV leading to 𝜔 = 2.2 × 1015 Hz, we get Rsp12 = 8.5 × 1010 s−1 and the corresponding 𝜏 r = 12 ps. For Aoki’s sample, with Emx = 1.36 eV and the corresponding 𝜔 = 2.0 × 1015 Hz, we get Rsp12 = 6.2 × 1010 s−1 and 𝜏 r = 16 ps. Thus, within the two-level approximation, one cannot get any results for the radiative lifetime longer than ps, which does not agree with the experimental results. This is a further clear indication of the fact that the two-level approximation is not valid for condensed-matter systems. 7.2.4

Temperature Dependence of PL

The maximum rates calculated from Eqs. (7.54) and (7.55) at the emission energy Emx increase exponentially as the temperature increases. This agrees very well with the model used by Wilson et al. [11] to interpret the observed temperature dependence in their PL intensity. Such a temperature dependence is independent of the origin of the PL, whether the recombination originates from the extended or tail states. However, the maximum rates obtained in Eqs. (7.54) and (7.55) vanish at T = 0, because x01 (Eq. (7.56)) and x02 (Eq. (7.57)) become zero as T → 0, which is in contradiction with the observed

7.2 Photoluminescence

rates [9, 11]. The reason for this is that one cannot derive the rate at 0 K from Eqs. (7.54) and (7.55). At 0 K, the electron-and-hole-probability distribution functions become unity; f c → 1 for Ee < EFn and f v → 1 for Eh > EFp as T → 0, and using this the rates at 0 K become equal to the pre-exponential factors as obtained in Eqs. (7.46)–(7.48). Thus, the temperature dependence of the rates obtained here can be expressed as: Rspi (T) = R0i {[1 − Θ1 (T)] + exp[−(ℏ𝜔 − E0 )∕𝜅B T]Θ2 (ℏ𝜔 − E0 )}, i = 1, 2

(7.61)

where R0i are the pre-exponential factors of Eqs. (7.54) and (7.55) corresponding to the four possibilities. The first step function, Θ1 (T), is used to indicate that the first term of Eq. (7.61) vanishes for T > 0, and the second step function Θ2 (ℏ𝜔 − E0 ) shows that there is no spontaneous emission for ℏ𝜔 − E0 < 0. For an estimate of R0i , we have used the values of Emx and E0 obtained at T = 3.7 K, which gives R01 = 7.77 × 106 s−1 , 1.47 × 107 s−1 , and 0.51 × 106 s−1 for extended–extended, extended–tail, and tail–tail states, respectively, and the corresponding values for R02 are obtained as 8.38 × 106 s−1 , 1.62 × 107 s−1 , and 0.55 × 106 s−1 . Wilson et al. [11] have fitted their observed rates to the following model: 1 = 𝜈1 + 𝜈0 exp(T∕T0 ) 𝜏r

(7.62)

and the best fit has been obtained with v1 = 108 s−1 , v0 = 0.27 × 106 s−1 , and T 0 = 95 K. Stearns [9] has obtained a best fit to his data with v1 = 2.7 × 108 s−1 , v0 = 6.0 × 106 s−1 , and T 0 = 24.5 K. According to Eq. (7.62), 𝜏r−1 = 𝜈1 at T = 0. Comparing these results with Eq. (7.61), one should have R0i = v1 , but R0i is obtained an order of magnitude smaller than v1 . Also, in Eq. (7.62), v1 > v0 , whereas, according to Eq. (7.61), v1 = v0 = R0i . These discrepancies may be attributed to the fact that Eq. (7.62) is obtained by fitting to the experimental data and not by any rigorous theory. As a result, both Eqs. (7.61) and (7.62) produce similar results for the radiative lifetime, but individual terms on the right-hand sides contribute differently. This is also apparent from the fact that, according to Eq. (7.62), v1 contributes to the rates also at nonzero temperatures, but, according to Eq. (7.61), R0i contributes only at T = 0 K. Stearns [9] has fitted the observed rate of emission at the emission energy of 1.43 eV to Eq. (7.62). We have plotted the calculated rates in Eq. (7.54) as a function of the temperature for possibility (ii) at the same emission energy (1.43 eV), and our results are shown in Figure 7.2a,b, respectively. Aoki et al. [49] have measured the QFRS spectra of a-Si:H and a-Ge:H at various temperatures (see Figure 6.14), where all PL peaks appearing at 𝜏 S (μs), 𝜏 G (μs), 𝜏 T (ms), and 𝜏 D (0.1 s) show a shift toward a shorter time scale as the temperature increases. This behavior is quite consistent with the temperature dependence obtained from the present theory as well. 7.2.5

Excitonic Concept

Another question may arise here. Why is the concept of excitons necessary to explain the radiative lifetime when the excited charge carriers can recombine without forming excitons? This can be answered as follows: Instead of using the reduced mass of excitons and the excitonic Bohr radius, if one uses the effective masses of e and h and the distance between them, one can calculate the radiative lifetime of a free excited electron–hole pair (not an exciton) from Eqs. (7.54) and (7.55). The radiative lifetime thus obtained

223

Rate (1010 s–1)

7 Photoluminescence, Photoinduced Changes, and Electroluminescence in Noncrystalline Semiconductors

Figure 7.2 Rates plotted as a function of temperature at an emission energy of 1.43 eV and calculated for possibility (ii) from Eq. (7.54) in (a) i = 1 and (b) i = 2.

6 4 2 0 0

50

100

150 T (K)

200

250

300

200

250

300

(a)

10 Rate (1010 s–1)

224

8 6 4 2 0 0

50

100

150 T (K) (b)

for such a recombination is also in the ns range for possibilities (i)–(iii). For possibility (iv), of course, as the separation between e and h is expected to be larger, one may get smaller rates and hence a longer radiative lifetime. However, there are two important observations that support the formation of excitons: (i) A free e–h pair will relax down to the tail states in a ps time scale, resulting in PL only from possibility (iv), which is not supported by the observed PL (see Figures 6.11–6.13); and (ii) PL has a double-peak structure appearing in the time-resolved spectra corresponding to the singlet and triplet excitons. No such spin-correlated peaks can be obtained with the recombination of free electron–hole pairs. The theory presented here for the radiative recombination from the tail states is different from the radiative tunneling model [2], where the rate of transition is given by Rspt ∼ R0 exp(−2d/d0 )—d being the separation between e and h; d0 being the larger of the extents of the electron and hole wave functions, usually considered to be 10–12 Å; and R0 being the limiting radiative rate, expected to be about 108 –109 s−1 . Here, d, being the distance between a pair of excited charge carriers (not an exciton), is not a fixed distance, such as aex; it can be of any value. As a result, the radiative tunneling model cannot explain the appearance of the double-peak structure in PL originating from the tail states. Although the rate of recombination from the tail states derived in Eq. (7.55) also has a similar exponential dependence as in the radiative tunneling model, it depends on the excitonic Bohr radius, which is a constant, and hence the exponential factor becomes a constant. This explains very well the peak structure in PL.

7.3 Photoinduced Changes in Amorphous Chalcogenides

Kivelson and Gelatt [7] have developed a comprehensive theory for PL in amorphous semiconductors based on a trapped-exciton model. A trapped exciton considered by them is one in which the hole is trapped in a localized gap state and the electron is bound to the hole by their mutual Coulomb interaction. The model is thus similar to possibility (ii) considered here, but possibilities (i) and (iii) have not been considered by them. The square of the dipole transition matrix element is estimated by them to be ∼(2ah )5 /(aB )3 . This is probably one of the main differences between the present theory and that of Kivelson and Gelatt. The other important difference between the present theory and that of Kivelson and Gelatt is in possibility (iv), involving recombinations in the tail states. For this possibility, Kivelson and Gelatt have followed the radiative tunneling model and used the exponential factor as exp(−2d/a∗B ). As discussed in the preceding text, such an exponential dependence on varying d cannot explain the peak structures observed in PL spectra. It is quite clear from the expressions derived in Eqs. (7.54) and (7.55) that rates do not depend on the excitation density. This agrees very well with the measured radiative lifetime by Aoki et al. [48], who have found that the radiative lifetime of singlet and triplet excitons is independent of the generation rate (see Chapter 5). This provides additional support for the PL observed in a-Si:H to be due to excitons. One of the speculations, as described earlier, has been that the shorter PL time in the ns range observed by Wilson et al. may be due to the high excitation density used in the TRS measurements. The present theory clarifies this point very successfully—that it is due not to the excitation density but to the combined effect of temperature and fast nonradiative relaxation of charge carriers to lower energy states that one does not observe any peak in the ns time range at 3.7 K. Considering distant-pair recombinations, Levin et al. [30] have studied PL in amorphous silicon at low temperatures by computer simulation. They have also used the radiative tunneling model for recombinations in the tail states, but they have not considered radiative recombination of excitons. Therefore, the results of their work also cannot explain the occurrence of the double-peak structure in PL.

7.3 Photoinduced Changes in Amorphous Chalcogenides One of the photoinduced changes observed in amorphous semiconductors, particularly in amorphous chalcogenides (a-Chs), is the reduction in the bandgap by illumination, and it is commonly known as PD [5, 24, 51, 52]. In some cases, PD is also accompanied by photoinduced volume expansion (PVE) in the material (see Chapter 7). There are two kinds of PD observed in a-Chs, one that disappears by annealing but remains even if the illumination is stopped [26], and the other that disappears once the illumination is stopped [27]. The former is called metastable PD and the latter transient PD, which has been observed only recently. Much effort has been devoted to understanding the phenomena of PD and PVE in the last two decades [5, 53–56]; however, no model has been successful in resolving all issues observed. One of the recent models is repulsive electronic interaction [57]. As chalcogenides, similar to As2 Se(S)3 , have layered structures, the charge carriers excited by illumination in these materials move in these layers. The hole mobility is larger than the electron mobility in these materials, and hence photo-generated electrons reside mostly in the conduction-tail states while

225

226

7 Photoluminescence, Photoinduced Changes, and Electroluminescence in Noncrystalline Semiconductors

holes diffuse away faster in the un-illuminated region through valence-extended and -tail states. Therefore, the layers that absorb photons during the illumination become negatively charged, giving rise to a repulsive inter-layer Coulomb interaction, which increases the inter-layer separation and causes PVE. The same force is assumed to induce an in-plane slip motion that increases the LP–LP interaction and causes PD. However, the repulsive Coulomb force model is qualitative, and more recent calculations [58] indicate that the force is too weak to cause PVE. More recently [59] a unified description of the photo-induced volume changes in chalcogenides based on tight-binding (TB) molecular dynamics (MD) simulations of amorphous selenium has been put forward (see Chapter 7). For causing movements in an atomic network, one requires the involvement of lattice vibrations, of which (surprisingly) none of the models of PD and VE has given any account. However, the involvement of lattice vibrations has recently been considered [60] in inducing photo-structural changes in glassy semiconductors. In 1988, Singh [61] discovered a drastic reduction in the bandgap due to the pairing of charge carriers in excitons and exciton–lattice interaction in nonmetallic crystalline solids. The magnitude of reduction in the bandgap varies with the magnitude of the exciton–phonon interaction, which is different in different materials; the softer the structure is, the stronger the carrier–phonon interaction. It has been established that, due to their planar structure [5], the carrier–phonon interaction in a-Chs is stronger than in a-Si:H, which satisfies the condition for the occurrence of Anderson’s negative-U [62] in a-Chs but not in a-Si:H. This is the basis of the hole-pairing model for the creation of light-induced defects in a-Chs [5]. In this section, based on Holstein’s approach [63], the energy eigenvalues of positively and negatively charged polarons and paired charge carriers, created by excitations due to illumination, are calculated. It is found that the energies of the excited electron (negative charge) and hole (positive charge) polarons become lowered due to the carrier–phonon interaction [64]. Also, the like-excited charge carriers can become paired because of the negative-U effect [62] caused by the strong carrier–lattice interaction, and the energy of such paired states is also lowered. Thus, the hole polaronic state and paired-hole states overlap with the lone-pair and tail states in a-Chs, which expands the valence band and reduces the bandgap energy and hence causes PD. The formation of polarons, as well as the pairing of holes, increases the bond length on which such localizations occur, which causes VE. The energy eigenvalues of such polarons and pairing of charge carriers have been calculated. 7.3.1

Effect of Photo-Excitation and Phonon Interaction

Based on Holstein’s approach [63], we consider here a model amorphous solid in the form of a linear chain of atoms. The electronic Hamiltonian in such a chain can be ̂ ph ), and charge carrier–phonon inter̂ el ), phonon (H written as a sum of charge carrier (H ̂ I ) energy operators in the real coordinate space as [5, 65]: action (H ̂ =H ̂ el + H ̂ ph + H ̂I H

(7.63)

In a-Chs, as the lone-pair orbitals overlap with the valence band, the combined valence band becomes much wider than the conduction band. As the effective mass of charge carriers is inversely proportional to the corresponding band width [5] (see Eqs. (3.17)

7.3 Photoinduced Changes in Amorphous Chalcogenides

and (3.20)), the hole’s effective mass becomes smaller than the electron’s effective mass in a-Chs. As a result, holes move faster than electrons in these materials. However, in the tail states, the charge carriers become localized. In such a system, we consider that a photon is absorbed to excite an electron (e) in the conduction state and hole (h) in the valence state. Considering the effect of strong carrier–phonon interaction, the excited hole may become a positive-charge polaron and an electron may become a negative-charge polaron. Furthermore, two excited electrons and two excited holes can become paired and localized on a bond due to the effect of negative-U, a fact which has already been established [5, 62]. Pairing of holes on a bond breaks the bond and creates a pair of dangling bonds, which is used to explain the creation of light-induced defects in a-Ch [5]. Let us first consider the case of an excited hole in the valence band. The eigenvector of such an excited hole can be written as: | | ∑ | +| Cl d0l (7.64) |0 > |h, 0 >= | | l | | where h denotes the hole and 0 denotes valence band; and the conduction band is + (=a0l ) is the denoted by 1. C l represents the probability amplitude coefficient, d0l creation operator of a hole, a0l is the annihilation operator of an electron in the valence states on site l, and |0> represents the vacuum state with all valence states completely filled and all conduction states completely empty. Using Eqs. (7.63) and (7.64), one can ̂ 0 ≥ Wh |h, 0 >, to get a secular equation such as: solve the Schrödinger equation, H|h, ) ( ∑ ℏ2 ∇2n 1 ∑ h 0 M𝜔m 𝜔n xm xn − Al xl − Eh Cl + Th (Cl−1 + Cl+1 ) Wh Cl = − + 2M 2 m,n n (7.65) where W h is the energy eigenvalue of the hole; the first two terms within the parentheses correspond to the kinetic and potential energies of nuclear vibrations; xn is the nth bond length of a diatomic molecule in the chain vibrating with a frequency 𝜔n ; Ahl is the hole–phonon coupling coefficient; Eh0 = Ehl , a constant of energy of a hole localized at site l and hence site independent; and T h is the hole-transfer energy between nearest neighbors from l to {l ±1} in the chain. Although the vibration frequency 𝜔m has a subscript m, being the intramolecular vibrational frequency of identical molecules, it is site independent. Considering only the diagonal terms in the vibrating potential, multiplying Eq. (7.65) by Cl∗ and then summing over all l, we get: ∑ ℏ2 ∇2n 1 ∑ ∑ M𝜔2m x2m − Al xl |Cl2 | − Eh0 + Th (Cl−1 + Cl+1 )Cl∗ (7.66) Wh = − + 2M 2 n m l ∑ ∗ ∑ 𝜕W 2 where l Cl Cl = l |Cl | = 1 is used. Setting 𝜕x h = 0, the energy eigenvalue, W h , can q

be minimized with respect to the bond length, xq . This gives the bond length at the minimum energy as: j

x0q =

Aq |Cq(0) |2

(7.67) M𝜔2 where the superscript (0) denotes the value of quantities at the minimum energy, and the subscript q denotes the qth bond at which the hole is localized. The subscript q has

227

228

7 Photoluminescence, Photoinduced Changes, and Electroluminescence in Noncrystalline Semiconductors

been dropped from the frequency 𝜔 as it is independent of bond sites. The bond length increases by x0q due to the localization of the hole on the bond. Substituting Eq. (7.67) into Eq. (7.66), we calculate the minimum energy of the hole, denoted by Wh0 , as: Wh0 = −Eh0 − 2Th + Ehq

(7.68)

where Ehq represents the hole–polaron binding energy obtained as: 2

Ehq =

(Ahq ∕M𝜔2 )2 48Th

(7.69)

This is the energy by which the energy of an excited hole is lowered from the free-hole state energy, which is at = −Eh0 − 2Th . This means that the hole-energy state moves upward in the energy gap by releasing an energy Ehq to phonons. 7.3.2

Excitation of a Single Electron–Hole Pair

The generation of excited electrons and holes occurs in pairs through photoexcitations. The excited electrons may also form polarons in the conduction states, and their energy will also be lowered. For a single pair of excited charge carriers on our linear chain, the secular equation can be written as: ] [ 1∑ l+X l e h 2 2 Weh Cl = (Ee − Eh ) − (Al+X − Al ) + M𝜔 xm − Ueh Cl 2 m − Te (Cl+X+1 + Cl+X−1 ) + Th (Cl+1 + Cl−1 )

(7.70)

where W eh is the energy eigenvalue of the excited pair; Eel = Ee0 represents the energy of an electron at site l and is also site independent; U eh is the Coulomb interaction energy between the excited pair; T e is the energy of electron transfer between nearest neighbors; M is the atomic mass and 𝜔 the frequency of vibration between nearest-neighbor atoms; X is the distance between the excited pair; and xl is the bond length between nearest neighbors of site l. Minimizing the energy with respect to xq , we obtain from Eq. (7.70) as [5]: Weh = Ee0 − Eh0 − Ueh − 2(Te − Th ) − Eeq − Ehq

(7.71)

where Ee0 and Eh0 are the site-independent energies of the excited electron and hole, respectively, without the lattice and Coulomb interactions between the excited charge carriers, and Eeq is the electronic polaron-binding energy, which can be obtained from Eq. (7.69) by replacing the subscript h by e. The energy of an excited pair of charge carriers without the lattice interaction is given by: Weh = Ee0 − Eh0 − Ueh − 2(Te − Th )

(7.72)

In the excitation of a free pair of charge carriers, one usually neglects U eh , and then the 0 energy Weh is close to the optical energy gap in most materials. Subtracting Eq. (7.72) from Eq. (7.71), we get the reduction in the optical gap due to the formation of an excited pair of polarons as: 0 ΔW = Weh − Weh = −Eeq − Ehq

(7.73)

7.3 Photoinduced Changes in Amorphous Chalcogenides

The lowering of an excited pair’s energy also means that the bond on which such polaronic formation occurs will be stretched by x(0) q , and the bond may break. A bond breaking due to such single excitation has been recently demonstrated by numerical simulation ([66]; see also Chapter 7) as well. This can also be seen from Eq. (7.67)—that the bond becomes stretched due to an excitation in which a hole is localized on the bond, and hence the bonding becomes weaker. 7.3.3

Pairing of Like Excited Charge Carriers

Here, we consider that two pairs of electrons and holes are excited in the chain. Assuming that the separation between electron and hole in one excited pair is x and that in the other is x′ , and that the separation between two holes is X, the secular equation analogous to Eq. (7.70) can be written as [5, 64]: W (x, x′ , X)Cl = [(Eel+x − Ehl ) + (Eel−X+x − Ehl−X ) − (Ael+x xl+x − Ahl xl ) 1∑ ′ − (Ael−X+x′ xl−X+x′ − Ahl−X xl−X ) + M𝜔2m x2m + U12 )]Cl 2 m ′

− Te (Cl+x+1 + Cl+x−1 + Cl−X+x′ +1 + Cl−X+x′ −1 ) + Th (Cl+1 + Cl−1 + Cl−X+1 + Cl−X−1 )

(7.74)

where W (x, x′ , X) is the energy eigenvalue of the two excited pairs of electrons and holes, and U ′ 12 is the total Coulomb interaction between the two pairs of excited charge carriers. It is well established that Se-based chalcogenides have linear chain-like structures and hence are flexible, and in such structures the carrier–phonon interaction is considered to be very strong. Such a strong carrier–phonon interaction can induce pairing of like-excited charge carriers on a bond due to Anderson’s U effect [62], known for valence-band electrons. In most solids, usually one of the two interactions, electron–phonon or hole–phonon, is stronger than the other. Therefore, here we first consider the case that the hole–phonon interaction is stronger, and therefore that two holes can be paired on a bond and the two excited electrons will form two polarons on the chain. The pairing of electrons can also be studied in an analogous way, as described in the following text. In solving Eq. (7.74), it is assumed that the energy eigenvalues of two electronic polarons are already derived in an analogous way as for the hole polaron (Eq. (7.68)). Our interest here is to calculate the energy eigenvalue of the excited state with two holes localized on the same bond, that is, X = 0. In this case, the secular equation (Eq. (7.74)) reduces to: ] [ 1∑ 0 l h 2 2 ′ W2h(x,x′ ,0) Cl = 2We − 2Eh + 2Al xl + M𝜔m xm + U12 Cl + 2Th (Cl+1 + Cl−1 ) 2 m (7.75) where the subscript 2h on W 2h(x, x′ , 0) denotes the localization of the two holes, and We0 is the energy of an electronic polaron derived in an analogous way as in Eq. (7.68) for a hole polaron, and is obtained as: We0 = Ee0 − 2Te − Eeq

(7.76)

229

230

7 Photoluminescence, Photoinduced Changes, and Electroluminescence in Noncrystalline Semiconductors

Following the steps used to derive Eqs. (7.65) and (7.66), here again we can minimize the energy with respect to xq to get: 0 = 2We0 − 2Eh0 + 4Th + Uh − Ehh W2h ′ = 2(Ee0 − Eh0 ) − 4(Te − Th ) + U12 − 2Eeq − Ehh

where

( Ehh

1 = 6Th

(Ahq )2

(7.77)

)2 (7.78)

M𝜔2

The bond length of a bond on which such a pairing occurs becomes twice as large as when only a single-hole polaron is formed, and is given by: 2Ahq Cq∗ Cq

= 2x0q (7.79) M𝜔2 The energy of two excitations without the charge carrier–phonon interaction can be written as: xhh q =

00 ′ W2h = 2(Ee0 − Eh0 ) − 4(Te − Th ) + U12

(7.80)

Thus, the energy of a pair of excitations with two holes localized on a bond is lowered by ΔE, given as: 00 0 ΔE = W2h − W2h = 2Eeq + Ehh

(7.81) 0 W2e ,

In an analogous way, one can derive the energy eigenvalue, of a pair of excited electrons localized on an antibonding orbital of a bond and two-hole polarons localized separately elsewhere as: 0 ′ W2e = 2(Ee0 − Eh0 ) − 4(Te − Th ) + U12 − 2Ehq − Eee

(7.82)

where Eee can be obtained from Eq. (7.78) by replacing the subscript h by e. It may be noted here that, unlike the case of pairing of holes on a bond, pairing of electrons on a bond does not break the bond. In this case, the two-hole polaron’s energy states move upward in the bandgap, and the paired-electron energy states in the conduction band move downward, resulting in a narrowing of the bandgap. We have shown earlier that the localization of an excited hole on a bond increases its bond length, and the bond can break. Such a localization occurs by the formation of a hole polaron due to strong interaction between the hole and lattice vibrations. The hole-polaron state has an energy lower than the free-hole state and moves upward, mixing with the lone-pair orbitals in chalcogenides that widens the valence band and narrows the bandgap. Such a strong charge carrier–phonon interaction is possible in a-Chs because of their linear flexible structure and weak coordination, which can also induce pairing of excited holes on a bond. In this case, the bond length of a bond increases twice as much as in the case of polaron formation, and their binding energy is eight times larger than the polaron binding energy. Such paired-hole states contribute significantly to both PD and PVE. PD is caused by the lowering of the paired-hole state energy, and such a state widens the valence band even further and hence reduces the bandgap. As the bond length expands twice as much as in the case of a single-hole polaron, this contributes significantly to VE as well. Moreover, pairing of holes on a bond breaks the bond because

7.3 Photoinduced Changes in Amorphous Chalcogenides

of the removal of covalent electrons and causes photo-induced bond breaking in a-Chs, as has already been established. Let us arrive at some estimates of the possible reduction in the bandgap due to formation of polarons and bipolarons (paired holes) on a bond. Depending on the material, ΔE (Eq. (7.81)) can be in the region of a fraction of an eV. We have estimated ΔE in As2 S3 as follows. The energy of lattice vibration of a bond can be written as [5]: 1 E(q) = E0 + M𝜔2 (q − q0 )2 (7.83) 2 where E0 and q0 are the energy and the interaction coordinate, respectively, at the minimum of the vibrational energy. The vibrational force along the interaction coordinate can be obtained as: ( ) 𝜕E A= = −M𝜔2 q0 (7.84) 𝜕q q=0 Using this in Eqs. (7.69) and (7.78), we obtain: ( )2 2 2 Ehp Ehh 1 M𝜔 q0 =8 = Th Th 6 Th

(7.85)

For As2 S3 , the phonon energy of the symmetric mode is 344 cm−1 [4]. Using this and applying Toyozawa’s criterion [66] of strong carrier–phonon interaction as Ehp ≥ T h , we get T h = 12.33 meV from Eq. (7.85), which gives Ehp = 12.33 meV, Ehh = 98.40 meV, and ΔE = 0.123 eV. This agrees very well with the observed reduction in the bandgap in As2 S3 of about 0.16 eV [5, 67]. A similar narrowing in the bandgap is expected when two excited electrons become paired on an anti-bonding orbital and two-hole polarons are formed elsewhere. In this case, however, as no bond breaking may occur, the substance will go back to the original state after the exciting energy source (illumination) is stopped. In materials where the excited pairs of charge carriers form only a pair of polarons, without any pairing of like-charge carriers, the reduction in bandgap will be equal to Eep + Ehp ∼ 2Ehp = 25 meV. Reductions in the bandgap in various materials, estimated from experimental data [67], are listed in Table 7.4; accordingly, most observed reductions are found to be in the range of 0.02–0.17 eV. Table 7.4 Bandgap reduction, ΔE estimated from the observed data [67] in various amorphous materials. Amorphous material

𝚫E (eV)

As2 S3

0.16

GeS2

0.17

S

0.12

As2 Se3

0.07

Se

0.06

GeSe2

0.03

As2 Te3

0.06

Sb2 S3

0.02

231

232

7 Photoluminescence, Photoinduced Changes, and Electroluminescence in Noncrystalline Semiconductors

Thus, it is shown in the preceding text that, in materials with strong carrier–lattice interaction, the excited charge carriers can form polarons; also, like-excited-charge carriers can become paired as self-trapped bipolarons on a bond because a paired like-charge carrier excited state is more energetically stable. The planar structure of chalcogenides with weak coordination enables these materials to be more flexible and hence possess stronger carrier–phonon interaction. The energy states of both polarons and bipolarons are lower than that of excited free-charge carriers. Thus, the energy of a hole polaron and bipolaron (paired holes) moves up further in the lone-pair orbitals and -tail states, which expands the valence band. Likewise, the energy states of electron polaron and bipolaron (paired electrons) in the anti-bonding orbitals lower the conduction mobility edge down. These effects together are responsible for a reduction in the bandgap, and hence PD. It should be noted that all the preceding possibilities might not occur together in the same material. In materials where the electron–lattice interaction is larger than the hole–lattice interaction, the formation of electron polaron and bipolaron will be more efficient, and materials with stronger hole–lattice interaction will have hole polaron and bipolaron formation more efficient. Accordingly, a varying degree of PD is expected to occur in different materials, and this is quite in agreement with the results listed in Table 7.4. It has been established [5] that the pairing of holes on a bond breaks the bond as soon as two excited holes become localized on it. It is also possible for a bond to be broken due to localization of a single hole on a bond. The bond breaks due to the removal of covalent electrons, and a pair of dangling bonds is created. This is the essence of the pairing-hole theory of creating light-induced defects in a-Chs. Such light-induced defects are reversible by annealing. However, the pairing of excited electrons does not break a bond; it only reduces the bandgap, and such an excited state will revert to the original material after the illumination has been switched off. This concept can be applied to explain both metastable and transient PD. The former occurs due to either formation of hole polarons or pairing of holes, or both, such that bonds are broken, and which cannot be recovered by stopping the illumination. It remains metastable, and the material reverses back to its original form by thermal annealing. The latter occurs due to pairing of electrons and/or formation of polarons without any bond breaking, and then the material reverses back to its original form after the illumination is stopped. Usually, transient PD is observed more than is metastable PD. This is because there are three processes contributing to transient PD—pairing of electrons, formation of positive-charge polarons (without bond breaking), and negative-charge polarons—as compared to only two possible channels of formation of positive-charge polarons (with bond breaking) and pairing holes contributing to metastable PD.

7.4 Radiative Recombination of Excitons in Organic Semiconductors The photoexcitation and formation of excitons have been discussed in Section 4.4 in detail. The non-radiative processes of the formation of charge transfer (CT) excitons and dissociation of excitons have also been discussed in Section 4.4. In this section, mechanisms of radiative recombination of singlet and triplet excitons are described with a view of their applications in OLEDs. In organic semiconductors and polymers, when the

7.4 Radiative Recombination of Excitons in Organic Semiconductors

electron and hole pairs are created either by photoexcitation or carrier injection, they usually form Frenkel excitons instantaneously due to the high binding energy between them caused by the low dielectric constant, in the range of 3–4. As stated earlier in Section 4.4, this high binding energy enables excitons to recombine radiatively by emitting light. However, only the radiative recombination of singlet excitons is spin-allowed and that of triplet excitons is spin-forbidden. Therefore, the radiative recombination from triplet excitons can only occur through spin–orbit interaction, which is known to be weak in organic materials. The emission of light due to the radiative recombination of singlet excitons is called fluorescence and due that due to triplet excitons is called phosphorescence. 7.4.1

Rate of Fluorescence

The rate of spontaneous emission of singlet excitons or fluorescence derived within the two-level approximation as given in Eq. (7.32) can also be applied to organic semiconductors. The rate of fluorescence, also known as the prompt fluorescence, denoted by RPF , because this is much prompter than the rate of phosphorescence, can be written as: (7.86)

RPF = Rsp12

where Rsp12 is the rate of spontaneous emission from a singlet exciton derived in Eq. (7.32), and |re−h | = axs ∕𝜀

(7.87) 5axt , 4

with axt as where axs is the Bohr radius of singlet excitons, which is given by axs = the triplet excitonic Bohr radius given by axt = 𝜇𝜀 a , where 𝜇 is the reduced mass of 𝜇x o electron in hydrogen atom, ao = 0.0529 nm is the Bohr radius [68, 69], 𝜀 is the static dielectric constant, and 𝜇x = 0.5 me is the reduced mass of excitons in organic solids Thus, we get the singlet exciton Bohr radius as: axs =

5𝜇𝜀 a 4𝜇x o

(7.88)

In organic semiconductors, the rate of prompt fluorescence is usually of the order of 108 s−1 [70], which gives the fluorescence time, 𝜏 PF = 1/RPF , in the ns range. 7.4.2

Rate of Phosphorescence

The rate of radiative recombination of a triplet exciton can be calculated using the newly invented time-dependent exciton–photon–spin–orbit interaction operator [71]. Within the two-level approximation, applicable to most organic solids, the rate of spontaneous emission from the radiative recombination of triplet excitons is obtained as [68, 72]: Rspt =

32e6 Z2 𝜅 2 𝜀ℏ𝜔12 𝜇x4 c7 𝜀0 a4xt

s−1

(7.89)

where Z is the highest atomic number in the organic molecules, 𝜅 = (4𝜋𝜀0 )−1 , ℏ𝜔12 = ELUMO − EHOMO is the emitted energy equal to the energy difference between the excited triplet (LUMO) and ground state (HOMO) in an organic solid, 𝜀o is the 𝜇 vacuum permittivity, and ℏ is the reduced Planck’s constant. Using 𝜇x = 0.5, 𝜀 = 3, we

233

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7 Photoluminescence, Photoinduced Changes, and Electroluminescence in Noncrystalline Semiconductors

get axt = 6a0 (see Eq. (7.88)) and other standard parameter values in Eq. (7.89), and we can write the rate of phosphorescence, RPhos , as: RPhos = Rspt = 25.3Z2 ℏ𝜔12 s−1 ( ℏ𝜔12 in eV)

(7.90)

For emission in the range of 2.00–2.5 eV, in an organic compound doped with iridium (Ir) with Z = 77, we get the rate of phosphorescence RPhos ∼ 105 s−1 , which is two orders of magnitude higher than in any undoped organic solid and three orders of magnitude lower than the rate of prompt fluorescence in Eq. (7.86). The time of phosphorescence 𝜏 Phos = 1/RPhos is in the μs range in organic solids incorporated with Ir, and in a fraction of ms range in undoped ones. 7.4.3

Organic Light Emitting Diodes (OLEDs)

In a basic OLED, a layer of organic material is sandwiched between two electrodes, as shown in Figure 7.3. In organic solids, the valence and conduction bands are presented by the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO), respectively. When an external voltage is applied in the positive bias, electrons are injected from the cathode in the LUMO and holes from anode to the HOMO of the organic layer, and they instantly form Frenkel excitons due to the lower dielectric constant leading to a higher binding energy (see Figure 7.3). Although the formation of excitons due to higher binding energy is good for their radiative recombination and hence the emission of light, also called the electroluminescence, excitons are formed in both singlet and triplet states. Due to the multiplicity of the triplet configuration, the statistical probability of forming singlet and triplet excitons is in the ratio of 1 : 3, and, as triplet excitons cannot recombine radiatively, the emission efficiency, also called internal quantum efficiency (IQE), of such a simple OLED can at most be only 25%. The rate of spontaneous emission of singlet excitons or fluorescence is given in Eq. (7.86), which has the radiative lifetime in the ns range. Therefore, it is very important to capture the emission of triplet excitons in OLEDs. For this, there are two ways in which the emission from triplet excitons can be achieved: (i) by incorporating heavy metal atoms, such as iridium (Ir), palladium (Pd), or platinum (Pt), in the organic compound, which increases the spin–orbit interaction by increasing the atomic number Z in the rate of phosphorescence described in Eqs. (7.89) and (7.90); and (ii) by converting triplet excitons into singlet excitons through either the reverse intersystem crossing (RISC) described in Chapter 5 (see Eq. (5.34)) or triplet–triplet annihilation (TTA) [41]. Using the first method, the radiative lifetime of triplet excitons LUMO CATHODE e Exciton Light emission h ANODE HOMO Organic layer

Figure 7.3 A schematic structure of a simple OLED and its operation mechanism.

7.4 Radiative Recombination of Excitons in Organic Semiconductors

gets shortened to the μs time range, which is still longer than that of the prompt fluorescence. Thus, the IQE can be increased from 25% to 100% by using one of these two methods; however, through TTA, as two triplets interact and upconvert into one singlet, the maximum contribution can only be 37.5% out of 75% triplet excitons. Through the other two processes, one can achieve 100% IQE. However, it may be noted that both the processes—that is, direct conversion to singlet through spin–orbit interaction (see rates in Eqs. (7.89) and (7.90)) and RISC (Eq. (5.34))—require the incorporation of heavy metal atoms for efficient transfer, but the cost of heavy metal atoms, for example, Ir or Pt, is quite high, and hence the use of such OLEDs becomes expensive [73, 74]. In this regard, the method of conversion of triplet excitons into singlets, known as up-conversion, leading to TADF has been explored extensively in the recent past, and has emerged as one of the most attractive technologies for lighting [73, 75]. As described in Chapter 5, the conversion from triplet to singlet or vice-versa involves two mechanisms: (i) spin flip; and (ii) least possible energy difference between singlet and triplet, denoted by ΔEST , also known as exchange energy. The process may not be very efficient if ΔEST is large and the spin cannot be flipped. TADF OLEDs based on heavy metal atoms are commonly called second-generation OLEDs. 7.4.3.1

Second- and Third-Generation OLEDs: TADF

The mechanism of TADF is schematically presented in Figure 7.4, where first a triplet exciton at T1 gets thermally elevated with energy ΔEST and then converted into a singlet exciton at S1 , through the RISC. The radiative lifetime of the singlet exciton (prompt fluorescence) is assumed to be much faster than that of the nonradiative intersystem crossing to go back to the triplet state. The singlet exciton radiatively recombines with the ground state S0 and emits fluorescence. According to Figure 7.4, the radiative lifetime of TADF can be obtained from the rate of TADF as [70]: −1 −1 τTADF = R−1 TADF = RPF + kRISC

(7.91)

As RPF is much faster than k RISC , the time 𝜏 TADF ≈k RISC , which is usually much slower than the radiative life time of prompt fluorescence (𝜏 PF = k PF −1 ), and hence TADF occurs with a delay in time—this is the reason for calling it delayed fluorescence. However, in 2013, Adachi et al. [76] have proposed the third generation of OLEDs, where two molecules are synthesized, such that ΔEST is very small between them. By blending such organic molecules as donor (with singlet state S1 ) with the acceptor (with triplet state T1 ) in the active layer, Adachi’s group has achieved very efficient TADF OLEDs, as shown in Figure 7.4. Although it is not fully understood yet how the spin −1

Figure 7.4 Schematic presentation of the mechanism of TADF in OLEDs. A triplet exciton gets thermally elevated to singlet S1 , which then recombines radiatively to the ground S0 state.

S1 ΔEST

kRISC T1

RPF RPhos

S0

235

236

7 Photoluminescence, Photoinduced Changes, and Electroluminescence in Noncrystalline Semiconductors

gets flipped without the involvement of any heavy metal atoms, there are two possible speculations [77]: (i) as ΔEST is very small, the smaller spin–orbit interaction achieved in such organic solids may be adequate to flip the spin; and (ii) the molecular structure of such donor acceptor molecular blending may be such that it is able to enhance the spin–orbit interaction adequately enough to make the spin flipping very efficient.

7.5 Conclusions It is demonstrated that the effective-mass approach can be applied to amorphous structures to understand many electronic and optical properties that are based on the free-carrier concept. Using the effective-mass approximation, it is shown that the excitation-density-independent PL observed in a-semiconductors arises from the radiative recombination of excitons, types I and II geminate pairs, and nongeminate pairs. Both the Stokes shift and radiative lifetimes should be taken into account in determining the PL electronic states. Although the radiative lifetime for possibilities (i)–(iii) are of the same order of magnitude, the Stokes shift observed in PL suggests that these recombinations occur from extended-to-tail states (possibility (ii)) in a-Si:H. The singlet radiative lifetime is found to be in the ns time range at temperatures >15 K and in ms range at 3.7 K, and triplet lifetime in the ms range. In possibility (iv), carriers have to tunnel to a distance equal to the excitonic Bohr radius, and hence the radiative lifetime gets prolonged. The PL from possibility (iv) can arise from two types of geminate pairs, excitonic (type II) and nonexcitonic (type I), and nongeminate or distant pairs. It is also shown that the effective mass of a charge carrier changes in amorphous semiconductors as it crosses its mobility edge. This also influences the radiative lifetime of an exciton as it crosses the mobility edges. A large Stokes shift implies a strong carrier–lattice interaction in a-Si:H, and therefore PL occurs in thermal equilibrium. The results of two-band approximation and nonequilibrium are therefore not applicable for a-Si:H. Finally, using the effective mass-approximation, it is shown quantum mechanically that it is the strong carrier–lattice interaction in a-Chs that causes all three phenomena: creation of the light-induced defects, PD, and VE. The results of the present theory agree qualitatively with those obtained from the molecular dynamics simulation (see Chapter 7). Finally, a theory of electroluminescence from organic semiconductors is presented in Section 7.4, including the calculation of the rates fluorescence and phosphorescence. A theory of light emission in OLEDs is also presented along with the operation of the second- and third-generation OLEDs based on the thermally activated and time-delayed fluorescence (TADF). The theory of TADF is also briefly presented.

Acknowledgments The author has benefited very much from discussions with Professors T. Aoki, K. Shimakawa, Keiji Tanaka, and Sandor Kugler during the course of this work. The work was originally supported by the Australian Research Council’s large grants (2000–2003) and IREX (2001–2003) schemes, and a bilateral exchange grant (2005/06) by the Australian Academy of Science and the Japan Society for the Promotion of Science (JSPS).

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8 Photoinduced Bond Breaking and Volume Change in Chalcogenide Glasses Sandor Kugler 1 , Rozália Lukács 2 , and Koichi Shimakawa 3 1 2 3

Department of Theoretical Physics, Budapest University of Technology and Economics, H-1521, Budapest, Hungary Norwegian University of Life Sciences, Ås, Norway Department of Electrical and Electronic Engineering, Gifu University, Gifu, 501-1193, Japan

CHAPTER MENU Introduction, 241 Atomic-Scale Computer Simulations of Photoinduced Volume Changes, 243 Effect of Illumination, 244 Kinetics of Volume Change, 245 Additional Remarks, 248 Conclusions, 249 References, 249

8.1 Introduction Chalcogenide glasses (ChGs), which contain group VI elements S, Se, and Te, but no O, exhibit various reversible and irreversible changes in structural and electronic properties during illumination. These properties include photoinduced volume change, photodarkening (PD), photoinduced crystallization and amorphization, photoinduced fluidity, and anisotropic optomechanical effects, etc. These photoinduced changes are unique to glass phases and are not observed in any crystalline structures of chalcogenide glasses. Films of ChGs can either expand (a-As2S3, a-As2Se3, etc.) or shrink (a-GeSe2, a-GeSe2, etc.). The time scale of photoinduced volume changes is in minutes. These processes are several orders of magnitude slower than those in data storage devices (DVDs) that utilize the properties of phase change materials. Data storage is affected by fast, reversible phase changes between crystalline and amorphous states of chalcogenide alloy systems, such as In–Sb–Te, Ge–Sb–Te, Ag–In–Sb–Te, Ge–Sb–Te, etc. In the recent years, interest in amorphous Se, the model material of chalcogenide glasses, has increased due to its applications as, for example, a good flat-panel X-ray image detector. A selenium atom contains six valence electrons (4s2 4p4 ). From s, px , py , and pz , only four independent orbitals can be constructed, that is, two out of four must be doubly occupied. A hydrogen selenide molecule has a H–Se–H molecular structure, with a bond angle of 90∘ . As a first approximation, we can explain the bonding process in Optical Properties of Materials and Their Applications, Second Edition. Edited by Jai Singh. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

8 Photoinduced Bond Breaking and Volume Change in Chalcogenide Glasses

the following way: two near-perpendicular sigma bonds can be constructed by putting an electron in px and another in py atomic orbitals, which are overlapped by 1s orbital of hydrogen atoms. Other two unshared electrons occupy the pz atomic orbital, forming a non-bonding lone pair (LP) state. The remaining two electrons remain untouched by the s atomic orbital. In the condensed phase, selenium atoms can form a chainlike structure. It can be concluded that only two electrons in px and py orbitals form sigma bonds overlapping with similar orbitals of adjacent atoms. Bond angles are slightly larger than 90∘ , because the s atomic orbital is not untouched; a little contribution of s orbital is needed to form near-perpendicular sigma bonds and an LP. The energy of these non-bonding LP states lies somewhere between electron bonding and antibonding energy levels. It was believed for a long time that photoinduced volume changes (usually called photoinduced volume expansion; PVE) and PD (PD: bandgap decreases during illumination) were two different sides of the same phenomenon in amorphous chalcogenides. It was expected therefore that one-to-one correlation should exist between PVE and PD. Systematic studies of PVE has been first performed by Hamanaka, et al. [1], showing that PD always accompanies PVE. Tanaka [2], however, has experimentally shown, from the time evolution of these two phenomena, that the time constants of PD and PVE in a-As2 S3 are different. It has also been observed that PVE saturates earlier than PD, suggesting that these two phenomena are not directly related to each other. A theoretical description of light-induced metastable defects creation, PD and PVE, in amorphous chalcogenides has been reviewed by Singh [3]. To understand the dynamics of PVE and PD during illumination, in-situ measurements have been performed [4–6]. A real-time in-situ surface height measuring system, called the physical vapor deposition (PVD), has been developed using the Twyman–Green interferometer with image analysis technologies [7, 8]. As shown by the dotted line in Figure 8.1, a small amount of PVE is found in (flatly deposited) a-Se films, and the height increases rapidly by 2.5 nm (Δd/d = 0.5%) with illumination (532 nm in wavelength and 90 mW cm−2 of power density). As soon as the light is switched off after 800 seconds of illumination, the height decreases by 2 nm, and then gradually decreases to the original height in 400 seconds (shown by the dotted line in Figure 8.2). 3.5 Change of thickness [nm]

242

3 a

2.5 2 1.5 1 0.5 0

–0.5

0

50

100

150 Time [s]

200

250

300

Figure 8.1 Time development of measured volume expansion in amorphous Selenium (dotted line) [7] and its fit using our model (solid line).

8.2 Atomic-Scale Computer Simulations of Photoinduced Volume Changes

Change of thickness [nm]

3

a

2.5 2 1.5 1 0.5 0 –0.5 800

850

900

950

1000 1050 1100 1150 1200 Time [s]

Figure 8.2 The measured shrinkage of a-Se (dotted line) [7] and the fitted curve (solid line) after switching off the illumination.

8.2 Atomic-Scale Computer Simulations of Photoinduced Volume Changes Some years ago, we have proposed a description of the photoinduced volume changes based on tight-binding (TB) molecular dynamics (MD) simulations on amorphous selenium. A typical configuration can be seen in Figure 8.3. For the description of atomic interactions between selenium atoms, a TB model developed by Molina et al. [9] has been used. The velocity Verlet algorithm has been applied in order to follow the motion of atoms with a time step equal to t = 2 fs. The temperature was controlled via the velocity-rescaling method. Samples containing 162 atoms were prepared by “cook and quench” sample preparation procedure. They had an initial density of 4.33 g cm−3 . We X

Z

Y

Figure 8.3 Snapshot of a glassy selenium network. The sample can move in z-direction. Arrow shows the thickness of the sample.

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8 Photoinduced Bond Breaking and Volume Change in Chalcogenide Glasses

prepared them from liquid phase by rapid quenching. At the beginning, we set the temperature of the system to 5000 K for the first 300 MD steps. During the following 2200 MD steps, we linearly decreased the temperature from 700 to 250 K, driving the sample through the glass transition temperature and reaching the condensed phase. Then we set the final temperature to 20 K and relaxed the sample for 15 ps in 7500 MD steps. The closed box shown in Figure 8.3 was opened in the z-direction at the 3000th MD step. In order to model the photoinduced volume changes, the periodic boundary conditions were lifted along the z direction at this point. This procedure provided us the slab geometry with periodic boundary conditions in only two dimensions. After that, the system was relaxed for another 40 000 MD steps (80 ps) to obtain a stable configuration. Samples prepared at 20 K had final densities in the range of 3.95–4.19 g cm−3 . The number of coordination defects ranged from 3% up to 12%. Most of these defects were located on the surfaces. The structure mainly consisted of branching chains, but some rings could also be found. The sample could expand/shrink in the z-direction. The measure of sample thickness is defined as the distance between centers of masses of 15–15 surface atoms on both sides, as shown by the horizontal arrow in Figure 8.3.

8.3 Effect of Illumination Photoexcitation generates an electron–hole pair when a photon is absorbed. This process can be modeled by transferring an electron from the highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO). In computer simulations, it was assumed that, immediately after photon absorption, the electron and the hole became separated in space on a femtosecond time scale [10]. The Coulomb interaction between the electron and hole was neglected. The influences of excited electrons and of holes were treated independently. First, an extra electron was added into the LUMO (excited electron creation), and next, an electron in the HOMO (hole creation) was annihilated. When an additional electron was added in the LUMO, a bond-breaking event occurred. In the majority of cases, a covalent bond between a twofold-coordinated atom and a threefold-coordinated atom was broken. However, sometimes, bond breaking between two twofold-coordinated neighbor atoms was also observed. Our localization analysis reveals that the LUMOs are usually localized at such sites (at a two- or three-fold coordinated atom) before bond breaking. The change of bond lengths alternates between shrinkage and elongation in the vicinity of the broken bond due to bond breaking. If the electron on the LUMO is deexcited, then all the bond lengths are restored to their original values. Reversible thickness change in a-Se sample is observed during several photoexcitations. A very different behavior is observed during hole creations [10]. Interchain weak bonds are formed after creating a hole, thereby causing contraction in the sample. This always happens near the atoms where the HOMO is localized. Since the HOMO is usually localized in the vicinity of a onefold-coordinated atom, the interchain bond formation often takes place between a onefold-coordinated atom and a twofold coordinated atom. Sometimes the formation of interchain bonds between two twofold-coordinated atoms has also been observed. Hartree–Fock ab initio Raman spectra calculations and Raman spectroscopic measurements have been carried out on amorphous selenium in order to identify the

8.4 Kinetics of Volume Change

characteristic vibrational mode due to sigma bonds [11]. In the Raman spectra, the peak occurring around 250 cm−1 corresponds to 0.234 nm covalent bonds vibrational modes in the amorphous Se [12]. As the Raman intensity varies in time due to illumination, a large number of covalent bonds break, and they get re-formed again after stopping the illumination. The difference between the Raman spectra of a 10-minute illuminated sample and after-40-minute relaxed sample was significant [11]. The Raman spectrum measurement provides strong experimental evidence for the photoinduced bond breaking process.

8.4 Kinetics of Volume Change 8.4.1

a-Se

Volume expansion and shrinkage are additive quantities; namely, the expansion by a thickness d+ is proportional to the number of excited electrons ne , so that d+ = 𝛽 + ne ; likewise, the shrinkage d− is proportional to the number of created holes nh , so that d− = 𝛽 − nh . The parameter 𝛽 + (𝛽 − ) is the average thickness change caused by an excited electron (hole). The time-dependent thickness change is given by d(t) = d+ (t) − d− (t) = 𝛽 + ne (t) − 𝛽 − nh (t). Assuming that the number of electrons is equal to number of holes, ne (t) = nh (t) = n(t), we get d(t) = (𝛽+ − 𝛽− )n(t) = 𝛽n(t)d(t) = (𝛽+ − 𝛽− )n(t) = 𝛽n(t)

(8.1)

where 𝛽 = 𝛽 + − 𝛽 − is a characteristic constant of chalcogenide glasses related to the photoinduced volume change, and it is a unique parameter for each sample. The sign of this parameter determines whether the material expands or shrinks. The number of photoexcited electrons and holes is proportional to the time duration of a steady-state illumination. Their generation rate G depends on the photon absorption coefficient and the number of incoming photons. After photon absorption, the excited electrons and holes randomly diffuse and eventually recombine. A phenomenological non-linear rate equation for this process is given as: dn(t) (8.2) = G − Cn(t)2 dn(t)∕dt = G − C n(t)2 dt where C is a constant. Using Eq. (8.1), the solution for the time-dependent thickness change in Eq. (8.2) is given by: √ √ G (8.3) tanh( GC t). d(t) = 𝛽 C The prefactor of tanh(x) (marked by “a” in Figure 8.1) is measurable after long-time illumination (tanh(∞) = 1), and, in this limiting situation, only one fitting parameter √ ( GC) remains to be determined. Figure 8.1 shows the measured time evolution of the surface height (dotted line), and the best fit (solid line) is obtained in the interval of 0–300 seconds illumination. The photoinduced expansion of amorphous selenium films was measured in situ using an optoelectronic interference, enhanced by the image processing [7]. After the light is switched off, the rate Eq. (8.2) reduces to: dn(t) = −Cn(t)2 dn(t)∕dt = −C n(t)2 dt

(8.4)

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8 Photoinduced Bond Breaking and Volume Change in Chalcogenide Glasses

with the solution d(t) = a(aCt𝛽 −1 + 1)−1

(8.5)

where a is equal to d(t) when the illumination is switched off. Figure 8.2 shows the measured time evolution of the surface height (dotted line) [7], and the best fit (solid line) is obtained in the interval of 0–400 seconds after switching off the light. 8.4.2

a-As2 Se3

In flatly deposited films of a-As2 Se3 , the surface height increases by 10 nm (Δd/d = 2%) in 200 seconds of illumination by a laser (532 nm in wavelength and 90 mW cm−2 of power density). After 600 seconds, the illumination was switched off. The surface height started decreasing and settled in 200 seconds at 2 nm less than its height before the light was switched-off. In the case of obliquely deposited films, the surface height decreased by 12 nm (Δd/d = 2.4%) in 3 × 104 seconds [7]. Similar results have also been observed in other amorphous chalcogenides. In the ideal case (a-Se), we assumed that each original local structure was reconstructed after the electron–hole recombination. However, the result of a measured volume change on flatly deposited a-As2 Se3 film is quite different from that in the case of ideal Selenium (see Figures 8.4 and 8.5). To explain the difference, we must take into account a large number of irreversible (metastable) changes in the local atomic arrangement—that is, after turning off the light, a part of local configuration remains the same [13]. The total volume change includes both the reversible (drev (t)) and irreversible (dirr (t)) changes, and it can be written as: (8.6)

dtotal (t) = drev (t) + dirr (t).

The reversible part follows Eqs. (8.2) and (8.4) during and after the illumination, with the corresponding solutions given in Eqs. (8.3) and (8.5), respectively. Let us consider the irreversible component. During the illumination, the generation rate of irreversible microscopic change is time dependent. An upper limit exists for the 10 Change of thickness [nm]

246

a+b b

8

Metastable part

6 4 2 0

Transient part

0

100

200

300 Time [s]

400

a

500

Figure 8.4 Time development of volume expansion of a-As2 Se3 . Thin solid line is the measured curve, and thick solid line is the fitted line. Lower dashed curve is the best fit of reversible part, while upper one is that of the irreversible part.

8.4 Kinetics of Volume Change

Change of thickness [nm]

10 b

8 6 4 2 0

1000

1500

2000

Time [s]

Figure 8.5 The measured decay (thin solid line) and the fitted theoretical curves (thick solid line) for the shrinkage as a function of time after the illumination has stopped.

maximum number of electrons and holes causing irreversible changes, and are denoted by ne,irr,max and nh,irr,max . To simplify the derivation, let us assume that ne,irr,max = nh,irr,max ; then, we can write the electron generation rate for the irreversible component as: Ge (t) = Ge,irr (ne,irr,max − ne (t)).

(8.7)

Note that there is no recombination term in Eq. (8.7). In this case, the irreversible expansion is governed by: dirr (t)∕dt = Girr − Cirr dirr (t).

(8.8)

Girr is the generation rate at the beginning of illumination. The solution of Eq. (8.8) is obtained as: dirr (t) = (Girr ∕Cirr ) ( 1 − exp {−Cirr t }).

(8.9)

Here, at the steady state (t = ∞), we get: d(t = ∞) = (Girr ∕Cirr ) = b.

(8.10)

Here, d(t = ∞) = b is a measurable quantity. Now, we have got only two fitting parameters to determine—a in Eq. (8.5) and b in Eq. (8.10). The parameter a in Eq. (8.5) is associated with when the light is switched off, so it is not obtained at t → ∞ as b is obtained. Using Eqs. (8.3) and (8.9) in Eq. (8.6), the best fit for the total volume expansion (dtotal (t)) from Eq. (8.6), reversible (transient) (drev (t)) in Eq. (8.3), and irreversible (metastable) (dirr (t)) in Eq. (8.9) parts are displayed as a function of the illumination time in Figure 8.4. After illumination, there is no volume change caused by the irreversible microscopic effects. Figure 8.5 shows the initial shrinkage after switching off the illumination, which is similar to what is obtained in Figure 8.2 for the reversible case. This model is a good explanation of a measured volume change on flatly deposited a-As2 Se3 , but the result of obliquely deposited film seems to be quite different, as shown in Figure 8.6, where a permanent volume shrinkage was observed. In order to understand this discrepancy, we have made an additional computer simulation for structures of different preparation methods [14]. The final conclusion of the analysis is that oblique

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8 Photoinduced Bond Breaking and Volume Change in Chalcogenide Glasses

0

Experimental data –A*tanh(B*t)–C*(1–exp(–D*t))

Height [nm]

–2 –4 –6 –8 –10 –12 –14 0

0.5

1

1.5

2 2.5 Time [s]

3

3.5

4 × 104

Figure 8.6 Volume changes in function of time of obliquely deposited a-AsSe thin film. Symbols are the measured volume changes during the illumination, and solid line is the best fit of a d(t) = (−6.4)*tanh(3*10−4 *t) + (−15.2)*(1 − exp(−1.6*10−5 *t)) function. The prefactors of tanh and (1 − exp) have negative values!

samples of about 1000 atoms contain large voids and have a more porous structure than a flatly deposited film. Large voids can collapse due to bond breakings, which can cause shrinkage during light illumination. If we consider that the signs of prefactors in Eqs. (8.2) and (8.9) are negative, we again get an excellent fit with the measured data [15] as it is displayed in Figure 8.6.

8.5 Additional Remarks Instead of Debye-type normal exponential relaxations, several times stretched exponential functions are fitted to the measured data in disordered systems. These curves usually fit quite well, but sometimes the physical understandings of these approaches are absent. In our case, we also tried to fit a stretched exponential function, and we obtained relatively good correlation, as shown in Figure 8.7. This example demonstrates that two different processes can sometimes be fitted by one stretched exponential function. Experimental Experimentaldata data M*(1–exp(–t/N)^P)) M*(1–exp(–t/N)^P))

0 –2 Height [nm]

248

–4 –6 –8 –10 –12 –14 0

0.5

1

1.5

2 2.5 Time [s]

3

3.5

4 × 104

Figure 8.7 Volume changes of the same obliquely deposited a-AsSe thin film fitted by the d(t) = (−20)*(1 − exp(−(t/3.17*104 )0.5 )) stretched exponential function.

References

8.6 Conclusions The photoinduced covalent bond breaking can be observed in chalcogenide glasses, caused by excited electrons, whereas holes contribute to the formation of inter-chain bonds in the vicinity where these excited electrons and holes are localized. The interplay between photoinduced bond breaking and inter-chain bond formation leads to either volume expansion or shrinkage. We have developed a universal macroscopic model of temporal development that is able to describe the photoinduced expansion as well as the shrinkage in chalcogenide glasses prepared by different methods. It is convenient to follow the time development of photoinduced volume change during and after the illumination.

References 1 Hamanaka, H., Tanaka, K., Matsuda, A., and Iijima, S. (1976). Solid State Commun. 2 3 4 5 6 7 8 9 10 11 12 13 14 15

19: 499. Tanaka, K. (1998). Phys. Rev. B 57: 5163. Singh, J. (2007). J. Optoelectron. Adv. Mater. 9: 50. Ganjoo, A., Ikeda, Y., and Shimakawa, K. (1999). Appl. Phys. Lett. 74: 2119–2122. Ganjoo, A., Shimakawa, K., Kamiya, H. et al. (2000). Phys. Rev. B 62: R14601. Ganjoo, A. and Shimakawa, K. (2002). J. Optoelectron. Adv. Mater. 4: 595–604. Ikeda, Y. and Shimakawa, K. (2004). J. Non-Cryst. Solids 539: 338–340. Kugler, S. and Shimakawa, K. (2015). Amorphous Semiconductors. Cambridge University Press. Molina, D., Lomba, E., and Kahl, G. (1999). Phys. Rev. B 60: 6372. Hegedus, J., Kohary, K., Pettifor, D.G. et al. (2005). Phys. Rev. Lett. 95: 206803-1-4. Lukacs, M.V., Shimakawa, K., and Kugler, S. (2010). J. Appl. Phys. 107: 073517–1-5. Dash, S., Chen, P., and Boolchand, P. (2017). J. Chem. Phys. 146: 224506. Hegedus, J., Kohary, K., and Kugler, S. (2006). J. Non-Cryst. Solids 352: 1587. Lukacs, R., Hegedus, J., and Kugler, S. (2008). J. Appl. Phys. 104: 103512. Lukacs, R. and Kugler, S. (2011). Jpn. J. Appl. Phys. 50: 091401.

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9 Properties and Applications of Photonic Crystals Harry E. Ruda and Naomi Matsuura Centre for Nanotechnology, University of Toronto, 170 College Street, Toronto, Canada

CHAPTER MENU Introduction, 251 PC Overview, 252 Tunable PCs, 255 Selected Applications of PC, 260 Conclusions, 265 References, 265

9.1 Introduction Photonic crystals (PCs) are periodic, dielectric, composite structures in which the interfaces between the dielectric media behave as light scattering centers. PCs consist of at least two component materials having different refractive indices, which scatter light due to their refractive index contrast. The one-, two-, or three- dimensional (1D, 2D, or 3D) periodic arrangement of the scattering interfaces may, under certain conditions, prevent light with wavelengths comparable to the periodicity dimension of the PC from propagating through the structure. The band of forbidden wavelengths is commonly referred to as a “photonic band gap” (PBG). Thus, PCs are also commonly referred to as PBG structures. PCs have great potential for providing new types of photonic devices. The continuing demand for photonic devices in the areas of communications, computing, and signal processing, using photons as information carriers, has made research into PCs an emerging field with considerable resources allocated to their technological development. PCs have been proposed to offer a means for controlling light propagation in small, sub-micron-scale volumes – the photon-based equivalent of a semiconductor chip – comprising optical devices integrated together onto a single compact circuit. Proposed applications of PCs for the telecommunication sector include optical cavities, high-Q filters, mirrors, channel add/drop filters, superprisms, and compact waveguides for use in so-called planar lightwave circuits (PLCs). Practical applications of PCs generally require human-made structures, as photonic devices are designed primarily for light frequencies ranging from the ultraviolet to the Optical Properties of Materials and Their Applications, Second Edition. Edited by Jai Singh. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

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near-infrared (IR) regime (i.e. ∼100 nm to ∼2 μm, respectively), and PCs having these corresponding periodicities are not readily available in nature. 1D PCs in this wavelength range may be easily fabricated using standard thin film deposition processes. However, 2D and 3D PC structures are significantly more difficult to fabricate, and remain among the more challenging nanometer-scale architectures to realize with cost-effective and flexible patterning using traditional fabrication methodologies. Recently, there has been considerable interest in PC-based devices that has driven advanced fabrication technologies to the point where techniques are now available to fabricate such complex structures reliably on the laboratory scale. In addition to traditional semiconductor nanostructure patterning methods based on advanced patterning/etching techniques developed by the semiconductor industry, novel synthesis methods have been identified for 2D and 3D periodic nanostructured PC arrays. There are several excellent reports reviewing these fabrication techniques in the literature [1–4], and this growing field has already been the subject of numerous recent reviews, special issues, and books in the area of theoretical calculations (both band structure and application simulations), 2D PC structures, 3D PC structures, and opal-based structures [5–7]. Recently, there has been great interest in exploring the use of PCs for applications in the active field of telecommunications, such as in the area of PLCs (e.g., for optical switching). In such applications, the PC properties should be adjustable to create “tunable” PBGs. This development increases the functionality of all present applications of PCs by allowing the devices in such applications to be adjustable, or tunable. We review here recent developments in the engineering of tunable nanometer-scale architectures in 2D and 3D. This review aims to organize this ever-changing volume of information such that interested theorists can design structures that may be easily fabricated with certain materials, and such that technologists can try to meet existing fabrication “gaps” and problems with current systems.

9.2 PC Overview 9.2.1

Introduction to PCs

The simplest PC structure is a multilayer film, periodic in 1D, consisting of alternating layers of material with different refractive indices. Theoretically, this 1D PC can act as a perfect mirror for reflecting light with wavelengths within its PBG, and for light incident normal to the multilayer surface. 1D PCs are found in nature, as seen, for example, in the iridescent colors of abalone shells, butterfly wings, and some crystalline minerals [4], and in human-made 1D PCs (i.e. also known as Bragg gratings). The latter are widely used in a variety of optical devices, including dielectric mirrors, optical filters, and in optical fiber technology. The center frequency and size (i.e. frequency band) or so-called stop band of the PBG depends on the refractive index contrast (i.e. n1 /n2 , where n1 and n2 represent the refractive indices of the first and second materials, respectively) of the component materials in the system. This multilayer film is periodic in the z-direction and extends to infinity in the x- and y- directions. In 1D, a PBG occurs between every set of bands, at either the edge or at the center of the Brillouin zone – PBGs will appear whenever n1 /n2 is not equal to unity [7]. For such multilayer structures, the corresponding PBG diagrams show that

9.2 PC Overview

the smaller the contrast, the smaller the band gaps [7]. In 1D PCs, if light is not incident normal to the film surface, no PBGs will exist. It is also important to note that at long wavelengths (i.e. at wavelengths much larger than the periodicity of the PC), the electromagnetic wave does not probe the fine structure of the crystal lattice and effectively sees the structure as a homogeneous dielectric medium. The phenomenon of light waves traveling in 1D periodic media was generalized for light propagating in any direction in a crystal periodic in all three dimensions, in 1987, when two independent researchers suggested that light propagation in 3D could be controlled using 3D PCs [8, 9]. By extending the periodicity of the 1D PC to 2D and 3D, light within a defined frequency range may be reflected from any angle in a plane in 2D PBG structures, or at any angle in 3D PBG structures. Since the periodicity of the PCs prevents light of specific wavelengths (i.e. those within the PBG) from propagating through them in a given direction, the intentional introduction of “defects” in these structures allows PCs to control and confine light. Propagation of light with wavelengths that were previously forbidden can now occur through such “defect states” located within the PBG. Defects in such PCs are defined as regions having a different geometry (i.e. spacing and/or symmetry) and/or refractive index contrast from that of the periodic structure. For example, in a 2D PC comprising a periodic array of dielectric columns separated by air spaces, a possible defect would include the removal of a series of columns in a line. Specific wavelengths of light forbidden from propagating through defect-free regions would then be able to propagate through the line defect, but not elsewhere. Indeed, by appropriately eliminating further columns, light may be directed to form optical devices, including, for example, a low-loss 90∘ bend in a 2D waveguide, as shown in the theoretical simulation [7]. Clearly, photons controlled and confined in small structures, of size on the order of the wavelength of light, using the extremely tight bend radii offered by PCs would facilitate miniaturization and the fabrication of PLCs [7]. In addition, since the periodicity of the PC gives rise to the existence of band gaps, which change the dispersion characteristics of light at given frequencies, defect-free PCs give rise to other interesting phenomena, including highly dispersive elements, through the so-called superprism effect [10]. Possible designs of PC-based optical devices have been extensively explored using such properties [1, 11], generating much excitement in the optical telecommunications field [6].

9.2.2

Nanoengineering of PC Architectures

Most of the promising applications of 2D and 3D PCs depend on the center frequency and frequency range for the PBGs. A so-called “complete,” “full,” or “true” PBG is defined as one that extends throughout the entire Brillouin zone in the photonic band structure – that is, for all directions of light propagation for photons of appropriate frequency. An incomplete band gap is commonly termed a “pseudo-gap” or a “stop band,” because it only occurs in the reflection/transmission along a particular propagation direction. A complete gap occurs when stop band frequencies overlap in all directions in 3D. The center frequencies and stop band locations of the PBGs critically depend on the unit cell structure [7–9, 12, 13]. In particular, the PC properties depend on the symmetry of the structure (i.e. the unit cell arrangements), the scattering element shape within the unit

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b) Scattering center shape

a) Structure symmetry

a

z r y x

d) Topology

c) Fill factor e) Refractive index contrast

Figure 9.1 PBG parameters that affect the frequency and associated stop band. In particular, key PBG parameters for the 2D periodic arrangement of one material (i.e. the rods, in gray) embedded in a second material (i.e. air), periodic in the x-y plane (shown in the central figure) include the structure symmetry, scattering center shape, fill factor, refractive index contrast, and topology.

cell, the fill factor (i.e. the relative volume occupied by each material), the topology, and the refractive index contrast (Figure 9.1). It has been shown that a triangular lattice with circular-cross-section scattering elements in 2D, or a face-centered cubic/diamond lattice with spherical scattering elements in 3D, tend to produce larger PBGs [7]. Also, as discussed above, the dielectric contrast is an important determining factor with 2D and 3D PC structures. The lower the structure dimension, the more readily are PBGs manifested since an overlap of the PBGs in different directions is more likely (i.e. something which is a certainty in 1D structures). In 3D structures, calculations have determined that the minimum dielectric contrast (n1 /n2 ) required to obtain a full PBG is about two [2, 12, 14]. For full PBGs, the ideal structure typically consists of a dielectric–air combination, to obtain both the greatest dielectric contrast as well as reduced losses associated with light propagation in optical materials other than air [7]. The impact of the fill factor, structure symmetry, and scattering element shape on the size and location of the PBG is complex [7, 15–18] and will not be discussed in

9.3 Tunable PCs

detail here. However, it is clear that the ability to adjust, or tune, one or more of these parameters and thus tune the PC properties is a very exciting development for future PC device applications. 9.2.3

Materials Selection for PCs

The physical architecture of a given PC is just one of the design considerations – other important ones are the optical properties of the PC materials. In particular, the refractive index of a given material and its electronic band gap determine the performance and appropriate range of frequencies of PC devices fabricated from such a material. For large PBGs, component materials need to satisfy two criteria; first, a high-refractive-index contrast, and second, a high transparency in the frequency range of interest. It may be a challenge to satisfy both the criteria at optical wavelengths. Suitable classes of PC materials include conventional semiconductors and ceramics, since at wavelengths longer than their absorption edge (or electronic band gap), they can have both high refractive indices and low absorption coefficients. In addition, these materials have other very useful electronic and optical properties that may complement the functions served by the presence of the PBG. Another, often overlooked, consideration for PC material selection is the ability to translate the desired “bulk,” single-crystal properties of well-known materials into nanometer-scale PC properties. Such structures typically have extremely large surface areas, and the microstructure of the constituent PC elements must be controlled during fabrication. Consequently, the final properties of the PC elements often vary from those of the bulk properties. This is extremely relevant when considering the functionality of final PC devices, such as those relying on electronic properties (e.g. lasing). Finally, it is interesting to note that there is a strong correlation between the fabrication methods selected and the component materials. For example, “top-down” dry-etching of semiconductors (e.g. Si, GaAs, InP) into nanostructured 2D arrays is well characterized and relatively commonplace, whereas the same cannot be said for ceramic materials. The opposite is generally true for chemical or “bottom-up” synthesis using sol-gel technology, in which ceramic PCs are relatively easy to fabricate when sol-gel based infiltration techniques are used (e.g. in 3D inverse opal fabrication). It is also clear that the fabrication technique primarily determines the PC structure that can be fabricated, which in turn determines the PC properties.

9.3 Tunable PCs Although conventional PCs offer the ability to control light propagation or confinement through the introduction of defects, once such defects are introduced, the propagation or confinement of light in these structures is not controllable. Thus, discretionary switching of light, for example, or re-routing of optical signals, is not available with fixed defects in PCs. There are two approaches that have been pursued to tune the properties of PCs: (1) tuning the refractive index of the constituent materials or (2) altering the physical structure of the PC. In the latter case, the emphasis has been principally on changing the lattice constant, although other approaches are also relevant (i.e. including, for example, the fill factor, structure symmetry, and scattering element shape). In the text below, we discuss the state of the art in both of these areas.

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9.3.1 Tuning PC Response by Changing the Refractive Index of Constituent Materials We discuss recent progress in using four approaches within this category – namely, tuning the PC response by using (1) light, (2) applied electric fields, (3) temperature or electrical field, and (4) changing the concentration of free carriers (using the electric field or temperature) in semiconductor-based PCs. 9.3.1.1

PC Refractive Index Tuning Using Light

One approach to modifying the behavior of PCs is to use intense illumination of the PC by one beam of light to change the optical properties of the crystal in a nonlinear fashion; this in turn can thus control the properties of the PC for another beam of light. An example of a nonlinear effect includes the flattening of the photon dispersion relation near a PBG, which relaxes the constraints on phase matching for second- and third-harmonic generation [19]. In addition, light location near defects can enhance a variety of third-order nonlinear processes in 1D PCs [20]. A new approach for PBG tuning has been proposed based on photo-reversible control over molecular aggregation, using the photochromic effect in dyes [21]. This can cause a reversible change over the photonic stop band. Structures that were studied included opal films formed from 275-nm-diameter silica spheres, infiltrated with photochromic dye: two dyes were considered, namely, 1,3-dihydro-1,3,3-trimethylspiro-[2H-indol-2,3′ -[3H]-naphth[2,1-b][1,4]oxazine] (SP) and cis-1,2-dicyano-1,2-bis(2, 4, 5-trimethyl l-3-thienyl)ethene (CMTE). For the SP dye–opal, a reversible 15 nm shift in the reflectance spectrum was observed following UV irradiation, which was ascribed to changes in the refractive index due to resonant absorption near the stop band. Smaller shifts (of about 3 nm) were observed for the CMTE dye–opal. The recovery process on cessation of UV illumination was quite slow, taking about 38 s for the SP dye. Nonlinear changes in the refractive index have also been studied in PCs comprising 220-nm-diameter SA polystyrene spheres infiltrated with water [22]. In these studies, the optical Kerr effect was used to shift the PBG. 40 GW cm−2 of peak pump power at 1.06 μm (35 ps pulses at 10 Hz repetition rate) was used to shift the PBG 13 nm. The large optical nonlinearity originates from the delocalization of conjugated π-electrons along the polymer chains, leading to a large third-order nonlinear optical susceptibility. The time response was measured as a function of the delay time between the pump and the probe, and confirmed a response time of several picoseconds. 9.3.1.2

PC Refractive Index Tuning Using an Applied Electric Field

There have been a number of studies focusing on using ferroelectric materials to form PCs [23–26]. The application of an electric field to such materials can be used to change the refractive index and tune the PC optical response. For example, lead lanthanum zirconate titanate (PLZT) inverse opal structures have been fabricated by infiltration using 350-nm-diameter polystyrene sphere templates and annealing at 750 ∘ C [25]. The films were formed on indium tin oxide (ITO)-coated glass to enable electric-field-induced changes in reflectivity to be measured due to the electro-optical effect. Applying voltages of up to about 700 V across films of thickness of about 50 mm only achieved a few nanometers of peak shift, attributed to the very modest changes in

9.3 Tunable PCs

refractive index from the applied field (i.e. from 2.405 to 2.435, as a result of the bias). It should be noted, however, that the intrinsic response of the electro-optical effect in these materials is known to be in the gigahertz range and hence highly suitable for rapid tuning. Other reports include the formation of inverse opal barium strontium titanate (BST) PCs using infiltration of polystyrene opals [23, 24]. BST is most interesting in that it provides a high-refractive-index material and a factor-of-at-least-two-times-higher breakdown field strength than PLZT, and hence offers a much wider range of the applied field for tuning. Reports of the use of other ferroelectric materials include a high-temperature infiltration process for the ferroelectric copolymer, poly(vinylidene fluoride-trifluoroethane), infiltrated into 3D silica opals with sphere diameters of 180, 225, and 300 nm [26]. 9.3.1.3

Refractive Index Tuning of Infiltrated PCs

In this case, the approach has been to consider modulation of the refractive index of a PC infiltrated with a tunable medium – in particular, the most popular approach has been to use liquid crystals to infiltrate porous 2D and 3D PC structures. Such liquid crystals can behave as ferroelectrics whose refractive index may be tuned using either an applied electric field, or by thermal tuning. Reported results for this approach, using 3D inverse opal structures, have been restricted to infiltration of a ferroelectric liquid crystal material into a silicon/air PC [27]. Reported changes in the refractive index of the liquid crystal are from 1.4 to 1.6 under an applied field [27]. However, since the ferroelectric liquid has a higher refractive index than air, the infiltration results in a significant decrease in the refractive index contrast. This means that the original full PBG of the inverse opal silicon PC no longer exists, and the practical utility of the silicon structure as a PC is effectively lost. Theoretical simulations have shown that partial surface wetting of the internal inverse opal surface can retain the full photonic band in silicon [27], but it is questionable whether such a complex structure can ever be practically fabricated. Finally, the presence of ferroelectric liquid crystal surrenders one of the main advantages of the original concept for PC structures: permitting light propagation in air [28]. The temperature tuning of the liquid crystal material was shown to result in very small changes in refractive index (change in n < 0.01 over a 70 ∘ C change) and thus only provide minimal shifts in the transmittance through the PC over a large temperature range [29]. 9.3.1.4 PC Refractive Index Tuning by Altering the Concentration of Free Carriers (Using Electric Field or Temperature) in Semiconductor-Based PCs

An elegant way to rapidly tune the PBG of semiconductor-based PCs is to adjust the refractive index by modulating the free carrier concentration using an ultra-fast optical pulse [30]. Using this approach, the reflectivity of a two-dimensional silicon-based honeycomb PCs with 412 nm air holes (100 μm in length) in a 500 nm periodic array was studied with a pump-probe approach [30]. By varying the delay between the pump and probe beams, the speed of PBG tuning was measured to be about 0.5 ps. The reflectance relaxation (corresponding to the return of the PBG to its original position) occurred on a timescale of 10–100 ns, characteristic of recombination of excess electrons and holes. Although these results are very encouraging, this approach cannot suppress, or correct for, light scattering losses caused by structural imperfections, which remain important considerations for currently fabricated PCs.

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9.3.2

Tuning PC Response by Altering the Physical Structure of the PC

The second approach that we discuss for tuning the response of a PC is based on changes to its physical structure. We discuss the following approaches for tuning with this approach using (1) temperature, (2) an applied magnetic field, (3) strain/deformation, (4) piezoelectric effects, and (5) micro-electro-mechanical systems (MEMS) [i.e. actuation]. 9.3.2.1

Tuning PC Response Using Temperature

An example of this approach is a study of the temperature tuning of PCs fabricated from self-assembled polystyrene beads [31]. The PBG in these structures was fine-tuned by annealing samples at temperatures from 20 to 100 ∘ C, resulting in a continuous blue shift in the stop band wavelength from 576 to 548 nm. New stop bands appeared in the UV transmission spectra when the sample was annealed above about 93 ∘ C, the glass transition temperature of the polystyrene beads. 9.3.2.2

Tuning PC Response Using Magnetism

An example of this approach is the use of an external applied magnetic field to adjust the spatial orientation of a PC [32]. This can find application in fabricating photonic devices such as tunable mirrors and diffractive display devices. These authors fabricated magnetic PCs by using monodispersed polystyrene beads self-assembled into a ferromagnetic fluid composed of magnetite particles, with particle sizes smaller than 15 nm [32]. On evaporation of the solvent, a cubic PC lattice was formed with the nanoparticles precipitating out into the interstices between the spherical polystyrene colloids. These authors then showed how the template could be selectively removed by calcination or wet etching, to reveal an inverse opal of magnetite – such structures are being proposed for developing magnetically tunable PCs. 9.3.2.3

Tuning PC Response Using Strain

The concept in this case is quite straightforward – deforming or straining the PC changes the lattice constant or arrangement of dielectric elements in the PC with a concomitant change in the photonic band structure. Polymeric materials would appear to be best suited to this methodology, owing to their ability to sustain considerable strains. However, concerns of reversibility and speed of tuning clearly would need to be addressed. Theoretical predictions for the influence of deformation on such systems include a report on a new class of PCs based on self-assembling cholesteric elastomers [33]. These elastomers are highly deformable when subjected to external stress. The high sensitivity of the photonic band structure to strain and the opening of new PBGs are discussed in this paper [34]. Charged colloidal crystals were also fixed in a poly(acrylamide) hydrogel matrix to fabricate PCs whose diffraction peaks were tuned by applying mechanical stress [35]. The PBG shifted linearly and reversibly over almost the entire visible spectral region (from 460 to 810 nm). Modeling of the photonic band structure of 2D silicon-based triangular PCs under mechanical deformation was also reported [35]. The structures considered comprised a silicon matrix with air columns. The authors showed that while a 3% applied shear strain provides only minor modifications to the PBG, uniaxial tension can produce a considerable shift. Other modeling includes a study of how strain can be used to tune

9.3 Tunable PCs

the anisotropic optical response of 2D PCs in the long wavelength limit [36]. Their calculations showed that the decrease in dielectric constant per unit strain is larger in the direction of the strain than normal to it. Indeed, the calculated birefringence is larger than that of quartz. They suggest that strain tuning of this birefringence has attractive application in polarization-based optical devices. To appreciate the sensitivity of such structures to mechanical tuning, it is instructive to refer to some recent work on PMMA inverse opal PC structures that were fabricated using silica opal templates [37]. Under the application of uniaxial deformation of these PCs, the authors found a blue shift of the stop band in the transmission spectrum – the peak wavelength of the stop band shifted from about 545 nm in the undeformed material to about 470 nm under a stretch ratio of about 1–6. Another practical approach that has been applied to physically tuning PC structures is that of thermal annealing. One such study showed how the optical properties of colloidal PCs composed of silica spheres can be tuned through thermal treatment [38]. This was attributed to both the structural and physiochemical modifications of the material on annealing. A shift in the minimum transmission from about 1000 nm (un-annealed) to about 850 nm (annealed at about 1000 ∘ C) was demonstrated, or a maximum shift in Bragg wavelength of about ∼11%. A quite novel application of strain tuning was recently reported [39]. The authors studied 2D PCs composed of arrays of coupled optical microcavities fabricated from vertical-cavity-surface-emitting laser structures. The influence of strain, as manifested by shifts in the positions of neighboring rows of microcavities with respect to each other, corresponded to alternating square or quasi-hexagonal shear strain patterns. For strains below a critical threshold value, the lasing photon mode locked to the corresponding mode in the unstrained PC. At the critical strain, switching occurred between square and hexagonal lattice modes. Finally, there has been a proposal for using strains in a PC to tune the splitting of a degenerate photon state within the PBG, suitable for implementing tunable PC circuits [40]. The principle they applied is analogous to the static Jahn–Teller effect in solids. These authors showed that this effect is tunable by using the symmetry and magnitude of the lattice distortion. Using this effect, they discussed the design of an optical valve that controls the resonant coupling of photon modes at the corner of a T-junction waveguide structure. 9.3.2.4

Tuning PC Response Using Piezoelectric Effects

In this section, we discuss using piezoelectric effects to physically change PC structures and hence tune them. A proposal was made for using the piezoelectric effect to distort the original symmetry of a two-dimensional PC from a regular hexagonal lattice to a quasi-hexagonal lattice under an applied electric field [41]. The original bands decomposed into several strained bands, dependent on the magnitude and direction of the applied field. In the proposed structures, the application of ∼3% shear strain is shown to be suitable for shifting 73% of the original PBG, which they refer to as the tunable bandgap regime. An advantage of such an approach is that such structures are suggested to be capable of operation at speeds approaching the megahertz level. Another report [42] discusses the design and implementation of a tunable silicon-based PBG microcavity in an optical waveguide, where tuning is accomplished using the piezoelectric effect to strain the PC – this was carried out using integrated piezoelectric

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microactuators. These authors report a 1.54 nm shift in the cavity resonance at 1.56 μm for an applied strain of 0.04%. There have also been reports of coupling piezoelectric-based actuators to PCs. One such report [43] discusses a poly(2-methoxyethyl acrylate)-based PC composite directly coupled to a piezoelectric actuator to study static and dynamic stop band tuning characteristics – the stop band of this device could be tuned through a 172 nm tuning range, and could be modulated at up to 200 Hz. 9.3.2.5

Tuning PC Response Using MEMS Actuation

There have been a number of interesting developments in this field, including PC-based devices comprising suspended 1D PC mirrors separated by a Fabry Perot cavity (gap) [44]. When such structures are mechanically perturbed, there can be a substantial shift in the PBG due to strain. The authors discuss how a suite of spectrally tunable devices can be envisioned based on such structures – these include modulators, optical filters, optical switches, wavelength-division multiplexing (WDM), optical logical circuits, variable attenuators, power splitters, and isolators. Indeed, the generalization of these concepts beyond 1D was discussed in a recent patent [45] and covers tunable PC structures. This report [45], as well as others [46], considered an extension of these ideas to form families of micromachined devices. The latter authors [46] modeled and implemented a set of micromachined vertical resonator structures for 1.55 μm filters comprising two PC (distributed Bragg reflector [DBR]) mirrors separated by either an air gap or semiconductor heterostructures. Electromechanical tuning was used to adjust the separation between the mirrors and hence fine-tune the transmission spectrum. The mirror structures were implemented using strong-index-contrast InP/air DBRs, giving an index contrast of 2.17, and weak-contrast (i.e. an index contrast of 0.5) silicon nitride/silicon dioxide DBRs. In the former case, a tuning range of over 8% of the absolute wavelength was achieved – varying the inter-membrane voltage up to 5 V gave a tuning range of over 110 nm. Similar planar structures are discussed in other papers [47], except where the mirrors are formed from two slabs of PC separated by an adjustable air gap (also see [45]). These structures are shown to be able to perform as either flat-top reflection or all-pass transmission filters, by varying the distance between the slabs, for normally incident light. Unlike all previously reported all-pass reflection filters, based on Gires–Tournois interferometers using multiple dielectric stacks, their structure generates an all-pass transmission spectrum, significantly simplifying signal extraction and optical alignment. Also, the spectral response is polarization independent owing to the 90∘ rotational symmetry of their structure.

9.4 Selected Applications of PC In this section, a selection of some interesting application areas for PCs are discussed focusing on integrated optics or PLCs, with the notable omission of other important areas such as microwave-based PCs and sensing applications (some of which importantly cover the field of biotechnology). Regretfully, these omissions are necessary, as covering any one of the many other fields would inevitably entail another such special review to do the subject justice.

9.4 Selected Applications of PC

To appreciate the application possibilities within this limited scope, recall that some of the key attractive properties that PCs possess for PLCs include, in particular, an ability to strongly confine light [1], as well as unique dispersive properties [10]. The property of confinement may be exploited to realize compact channel waveguides and sharp bends and also achieve high isolation between adjacent channels (i.e. so-called low cross-talk) [7]. Dispersive properties, on the other hand, readily lend themselves to optical functions including wavelength separation (e.g. the so-called superprism effect) as well as pulse shape modification (e.g. pulse compression) [1]. It should be noted that as far as the maturity of technology is concerned, current PLCs are very simple systems. Typically, they are composed of components having sizes in the millimeter to centimeter range, permitting a very limited number of different functions to be integrated together on a chip, which is very much reminiscent of the state of electronic ICs in the early 1960s. 9.4.1

Waveguide Devices

As discussed above, there are a number of different types of waveguide devices that have been developed in PCs. These include both channel waveguides and coupled resonant optical cavity (CROW) structures [48]. The practical implementation of these structures relies on achieving low losses. Planar PCs are able to achieve ∼4–10 dB/unit cell, and so ∼30 dB can be achieved within a few cells. There are three key loss mechanisms to consider in these structures: (1) scattering losses at structural imperfections in the PCs, (2) out-of-plane losses, and (3) losses caused by TE/TM coupling. When considering scattering losses at imperfections, it should be recognized that intrinsically PC waveguides should not suffer any in-plane losses as such structures are designed to prohibit all propagating modes at the operating wavelength. One therefore expects that scattering from such imperfections should have a much lower impact in PC structures than for ridge waveguides having similar dimensions. On the other hand, in multimode waveguides, this type of scattering can excite higher-order modes and therefore become more significant. PLCs are ideally made as two-dimensional devices, with the third dimension assumed theoretically to be infinite. However, in real finite PLC structures, out-of-plane losses from diffraction and scattering provide the dominant loss mechanism in PC waveguides. In the case of structures formed by etching hole arrays into a solid semiconductor structure, losses originate from poor guiding by holes and their depth being insufficiently long. This causes part of the waveguide mode to be scattered away. Such losses can be minimized by increasing the aspect ratio of the etched holes. In TE/TM mode coupling, such losses can occur in cases such as when the PC surrounding a channel waveguide provides a bandgap for only the TE polarization. In this case, once the confined TE mode is converted to TM, it no longer experiences the photonic bandgap and is therefore free to leak out of the channel waveguide and propagate through the crystal. CROWs are quite distinct from conventional waveguides and have no real analogy with traditional guided wave devices. The waveguide is formed through an array of coupled defects in the PC [48]. In some sense, these structures are analogous to quantum electronic structures such as resonant tunneling structures in which extended states are coupled to standing-wave-type cavity states to control the transmission flux through the structure. In the case of CROWs, by varying the type of defects and their spacing,

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one can control the propagation of light from defect to defect, and this allows for the tailoring of the wavelength response of the structure as well as its group velocity [48]. Some impressive experimental results have been demonstrated for light propagation in such structures. Importantly, such structures overcome some of the problems of the more conventional PC-based waveguiding devices, showing very little propagation loss as well as providing a means to guide light around sharp bends for ultra-compact optical device and systems applications. Such structures have been successfully demonstrated in the microwave regime, where CROWs are real competitors for conventional PC channel waveguides [49]. Although somewhat unglamorous, the problem of coupling fiber to PCs presents one of the biggest engineering challenges for implementing PC-based PLCs in practice. The problem originates from the fact that fibers possess a circular cross section of ∼5–7 μm diameter as distinct from the PC waveguides, which have cross sections of only several hundreds of nanometers. This leads to a severe mode mismatch between the fiber and PC, and is responsible for insertion losses, which can be ∼20–30 dB. Various approaches to solving this problem have been discussed in the literature. 9.4.2

Dispersive Devices

Light guiding in PC-based structures can be supplemented by unique dispersive devices using the same systems. This again provides a distinct advantage for PCs over traditional alternatives for realizing compact, highly functional optical systems for a variety of applications. One of the most important aspects of this is waveguide dispersion using the superprism effect. That is for performing beam steering, multiplexing, and de-multiplexing (MUX/DEMUX) for dense wavelength-division multiplexing (DWDM), for example. Such dispersion effects in periodic structures have been known for some time [50] and were rediscovered in PCs for application in WDM systems. The superprism effect, as described by Kawakami and coworkers [10], uses the band structural asymmetry near the Γ-point, where the wavevector changes much more dramatically with change in frequency in the Γ-K direction as compared with the Γ-M direction. These workers show that an angular dispersion of 50∘ can be observed for a change in input wavelength from 900 to 1000 nm. As discussed above, PC systems are in principle scalable to any wavelength range. Thus, when such a system is scaled to the 1.55 μm regime for telecommunications applications, this dispersive behavior would correspond to ∼2∘ for the 50 GHz channel spacing in a typical DWDM system. If one assumes a series of output waveguides that are laterally separated by 5 μm, a distance of ∼150 μm would be required to separate the channels. This is an impressively small distance compared with current phased array waveguides, where dimensions are typically on the millimeter to centimeter scale. 9.4.3

Add/Drop Multiplexing Devices

Some of the key components in DWDM systems are the so-called add/drop devices, which allow for the selective removal or addition of a particular wavelength channel to an optical data stream, thereby allowing all the channels to be fully utilized. The first PC-based add/drop device was proposed in 1997 [51] and was based on the resonance created by two defects in intersecting PC waveguide devices, enabling coupling

9.4 Selected Applications of PC

between two channel waveguides at the resonance frequency. It should be noted that such devices are closely related to the CROW structures discussed above. By analogy with quantum electronics, whereas CROWs involve a series of coupled cavities, these devices appear as the equivalent of a single isolated quantum well/dot structure coupled to their waveguides. Unfortunately, experimental realization of this approach has proved to be particularly challenging. The main reason for this is that the resonance condition is strongly dependent on the size of the defect, making it impractical to control the dimensions of the defect to the required tolerance. 9.4.4

Applications of PCs for Light-Emitting Diodes (LEDs) and Lasers

The concept of using microstructured mirrors for developing semiconductor laser structures is fairly mature. Periodic microstructures have been commonly used in distributed feedback (DFB) and DBR lasers, as well as for vertical cavity structures (e.g. vertical-cavity-surface-emitting lasers [VCSELs] using Bragg stacks). However, the dielectric contrast in such structures is typically less than 1% as compared with almost 4:1 in PC-based structures. This is particularly significant as it can lead to much shorter interaction lengths of ∼1 μm as compared with hundreds of micrometers for conventional DFB/DBR structures. Importantly, this reveals the possibility of creating edge emitting laser elements with a very small optical volume, as is discussed below. Our discussions begin by considering how PCs can be fruitfully applied to improve the performance of LEDs. The principal active material for LEDs remains semiconductors with reported internal quantum efficiencies as high as 99.7% [52]. The utility of these materials then depends on how efficiently this generated light can be extracted. As a consequence of Snell’s law, only the radiation falling within a small cone (i.e. ∼16∘ for GaAs) can escape – everything else is totally internally reflected. A simple calculation shows that the extraction efficiency scales as 1/4n2 , where n is the semiconductor refractive index. Thus, the small cone of light allowed by Snell’s law represents only a small percentage (e.g. about ∼2%) of the total available solid angle for GaAs, explaining the low observed external quantum efficiency of ∼2–4% in standard commercial LEDs. High-brightness LEDs have recently come to market and are finding applications in display and lighting applications owing to their superior brightness and lifetime compared with incandescent sources. These devices make use of light extracted from more than one facet. Another approach is to place the active layer on a low-index substrate and then roughen it. The resulting surface scatters a large fraction of the light out of the material, and measured efficiencies of as high as ∼30% have been reported [52]. However, one of the most promising approaches to developing LEDs is based on PCs, as these structures offer the potential to improve both the extraction efficiency of the light as well as control over the direction of light emission from the structures. As regards the latter, PCs enable one to suppress emission in unwanted directions, and enhance it in the desired directions. Such structures have been predicted to be suitable for producing light emitters with external quantum efficiencies exceeding 50% [1]. Two main factors for controlling the emission are (1) increased extraction and photon recycling and (2) altering the fundamental emission process. One of the key motivations for studying PCs is their promise for strong spontaneous emission enhancement – i.e. owing to the fact that the spontaneous emission rate of

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an excited atom can be increased by placing it in a microcavity. In particular, such a microcavity can be formed by creating a defect within a PC structure, as discussed above. The maximum enhancement factor is achieved when the radiating dipole is so oriented as to experience the maximum interaction with the cavity mode. The cavity enhancement factor f P or Purcell factor is given by (6/𝜋 2 )(Q/V ), where Q is the quality factor of the cavity, and V = v/(l/2𝜋)3 is the effective volume of the cavity taking into account the fact that the emitter is usually embedded in a material of index n and that the number of cavity modes is approximately quantized in integral multiples of half wavelengths, 𝜆/2, in a cavity of volume v. PC microcavities offer high-Q microcavities with an exceptionally low mode volume that is suitable for very high spontaneous emission enhancement factors. Recently, it was shown that one could indeed control the timing of light emission from CdSe nanocrystals (i.e. quantum dots) embedded in a titania inverse opal PC. Londahl et al [53] found that by changing the PC lattice constant, one could accelerate/decelerate the rate of spontaneous emission from the quantum dots by as much as a factor of two. Speeding up of the spontaneous emission could lead to more efficient light sources such as LEDs, while slowing it down could help create more efficient solar cells. The application of PCs for developing lasers also appears to be very promising. Typical semiconductor-based lasers usually have between 104 and 105 modes, of which only one is the desired lasing mode. Clearly therefore, a lot of light is wasted before stimulated emission occurs. In addition, since all of these modes are still there after the onset of lasing, they contribute to noise. PC structures offer a means for reducing the number of available modes and thus reducing both the lasing threshold condition and also dramatically reducing the laser noise characteristics. This may be described by reference to the so-called 𝛽-factor, where 𝛽 = Γ𝜆4 /8𝜋vn3 Δ𝜆. 𝛽 quantifies the mode ratio of the desired mode to all the other modes and determines the laser threshold, and Γ is the confinement factor (the ratio of the gain volume to cavity volume). Defects can be used in the PCs to place modes in an otherwise forbidden spectral regime – for example, one can artificially create a mode within the forbidden bandgap by using a defect, and the 𝛽-factor will then assume a value of unity because the single allowed mode becomes the lasing mode. The principle of realizing a laser using a PC structure is quite straightforward then – one needs to design the PC such that the defect frequency coincides sufficiently closely with the gain peak of the respective emitter material. One other interesting development in PCs is related to harnessing surface plasmons. This may be applied to develop sensors, light emitters, or light detecting structures. In particular, we focus on the application of plasmonic structures for light emitters. Plasmonics rely on plasmon-related effects in metallic structures and can produce exciting phenomena when applied to light-emitting devices. Surface plasmons, quanta of electron oscillations at metal–dielectric interfaces, have been used to explain the super-transmission effect where a transmission of ∼4% was measured for a thin metal film perforated with holes comprising ∼2% area fill factor and size well below the cutoff wavelength (150 nm for 1.55 μm light emission) [54]. The best explanation to date for this is that the incident light excites the top surface plasmons, which then couple to the surface plasmon on the other side of the metal film. This latter surface plasmon mode couples to radiation modes, making the whole process appear as if the incident light was directly transmitted through the metal film. Surface plasmon modes also have been observed in small metal particles with the possibility of enhancing interactions between

References

light and other structures, such as for photodetectors found using such an approach. The concept of developing PCs from such plasmonic systems has obvious attractiveness as a means of using such resonances to transfer light from an active semiconductor structure to the outside world, improving light extraction efficiency – one of the key problems for light source design. The design of such systems would rely on matching the plasma resonance frequency to that of the emission, and by using control over the PC structure, control the coupling.

9.5 Conclusions This chapter discussed the principles and properties of PCs, both passive and tunable, with an emphasis on the application of such structures to PLCs. New phenomena in these systems are offering a glimpse of the far-reaching prospects of developing photonic devices that can, in a discretionary fashion, control the propagation of modes of light in an analogous fashion to how nanostructures have been harnessed to control electron-based phenomena. Analogous to the evolution of electronic systems, one can anticipate a path toward development of compact, active, integrated photonic systems based on the technology outlined in this review. We discuss proposals and indeed demonstrations of a wide variety of PC-based photonic devices with applications in areas including communications, computing, and sensing, for example. In such applications, PCs can offer both a unique performance advantage, as well as potential for substantial miniaturization of photonic systems.

Acknowledgments The authors gratefully acknowledge support from NSERC, OCE, CSA, and AFOSR.

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269

10 Nonlinear Optical Properties of Photonic Glasses Keiji Tanaka∗ Department of Applied Physics, Graduate School of Engineering, Hokkaido University, Sapporo, Kita-ku, N13W8, Japan

CHAPTER MENU Introduction, 269 Photonic Glass, 271 Nonlinear Absorption and Refractivity, 272 Nonlinear Excitation-Induced Structural Changes, 280 Conclusions, 285 Addendum: Perspectives on Optical Devices, 286 References, 288

10.1 Introduction In this article, we define photonic glass as a high-purity inorganic glass that can be applied to photonics. There are many types of applications, the most innovative undoubtedly being optical fibers, which now densely surrounds the earth for communication purposes. Optical fiber is also used in functional devices such as Bragg reflectors, optical amplifiers, and lasers. In addition, integrated glass waveguides are extensively developed for optical signal processing such as wavelength divisions. In these applications, not only linear but also nonlinear optical properties play important roles [1–3]. On the one hand, large nonlinearities are needed for all-optical switches, power stabilizers, soliton fibers, supercontinuum generators, etc. [4, 5]. Besides, nonlinear optical excitations are promising for fabricating three-dimensional memories and waveguide devices [4]. On the other hand, we may need a small nonlinearity in high-intensity glass lasers and non-dispersive optical fibers [1, 2]. Despite the wide range of applications, glass science is still less mature than crystal science [6–11]. This is because of three inherent characteristics of the glass . First, due to the lack of long-range structural periodicity, x-ray diffraction patterns are less informative, and explicit atomic structures cannot be determined. Second, also due to the lack of periodicity, electron wave functions and atomic vibrations tend to localize, with the result that the wave numbers are no longer good quantum parameters. Bloch wave functions cannot be postulated, in principle. The conventional one-electron *Emeritus Professor (Email address: [email protected]). Optical Properties of Materials and Their Applications, Second Edition. Edited by Jai Singh. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

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approximation is largely limited, and only the energy density of states could have physical meanings. Under these circumstances, we may analyze types of many-body problems for electrons and atoms using computer simulations. However, simulated results cannot necessarily provide universal insights because of restricted calculating conditions, such as the enormously fast quenching speeds employed in digitized molecular dynamics. Third, the glass is thermodynamically quasi-stable, having glass-transition temperatures that vary with preparation procedures and post-preparation treatments, as exemplified in the As2 S3 case [12]. The quasi-stability also causes variations in macroscopic properties [7–11]. It is difficult to envisage an ideal glass structure, even in a statistical sense, which is in marked contrast to the situations in ideal crystals and gases, which are perfectly periodic and completely random, respectively [11]. These three features still pose many unresolved problems. As for linear optical properties, for instance, we cannot yet give definite interpretations of the origins of the Tauc optical gap [6, 11], minimal Urbach-edge energy of ∼50 meV [11, 13], midgap absorption [7, 14], etc. For optical nonlinearity in photonic glasses, although we have obtained substantial amount of data for bulk samples, only rough theoretical frameworks have been outlined [4, 5]. Here, we will review three subjects pertaining to optical nonlinearity in photonic glasses. After a brief summary of the photonic glass in Sections 10.2, Section 10.3 focuses on a unified understanding of the third-order nonlinearity. Optical nonlinearity in glasses is analyzed using semiconductor terminology, which may be complementary to the dielectric approach [4] widely employed among glass scientists. In the present scheme, as illustrated in Figure 10.1, the relationship among atomic structures (atom and bonding), electronic structures, and optical absorption (𝛼, Eg , and 𝛽) and refractivity (n0 and n2 ) spectra becomes clearer. Spectral dependence is obtained in a straightforward way. For other nonlinear properties such as the second-order nonlinearity in poled glasses [15] and the large nonlinearity in nano-particle dispersed glasses [16], the reader may refer a recent article by the author [17]. In Section 10.4, we will consider the role of nonlinear excitations in photo-induced phenomena. It is often asserted that when a photo-induced phenomenon is induced by light of photon energy less than the bandgap (ℏ𝜔 < Eg ), nonlinear excitation takes place. However, this may be a hasty conclusion, n0 (3) α, Eg

Boling (6)

Moss

n2 (5)

(7) Sheik – Bahae

(2)

β (4)

electronic structure

bonding

atoms

Figure 10.1 Relations between atomic structure (atoms and bonding), electronic structure, absorption spectra (𝛼 and 𝛽), and refractive index spectra (n0 and n2 ). n0 , 𝛼 and E g are linear properties, and n2 and 𝛽 are nonlinear. Solid arrows represent Eqs. (10.2) and (10.4); double-solid arrows represent linear and nonlinear Kramers–Krönig relations, Eqs. (10.3) and (10.5); and dashed arrows show the Moss rule (E g ∼ n0 ), Boling’s relation, Eq. (10.6), and Sheik-Bahae’s relations including Eq. (10.7).

10.2 Photonic Glass

because it neglects midgap states, whose existence is inherent to the glasses. We will see that photoexcitation mechanisms in glasses are not as simple as those in crystals. Section 10.5 presents conclusions. Finally, the Addendum sketches the personal views of the author on photonic devices from the standpoint of future perspectives.

10.2 Photonic Glass Several types of photonic glasses are available at present, which include oxides [7, 9], chalcogenides [3, 8–11], halides [18], and their mixtures. As is well known, studies on nonlinear effects in halide glasses have been limited due to their smallness [5, 17–20], despite the wider transparency in wavelength, and accordingly, these will not be included in this article. On the other hand, both oxide and chalcogenide glasses consist of the group VIb (16) atoms (O, S, Se, and Te), which are characterized by s2 p4 valence-electron configurations, so that their atomic and electronic structures can be considered in a unified way [3, 6, 11]. Figure 10.2 shows schematic atomic networks in SiO2 and As2 S3 . The structure of such simple glasses can be grasped at four atomic scales [6, 11]. First, it is demonstrated that the so-called short-range structure, which includes the atomic coordination number, bond length, and bond angle, is similar to that in the corresponding crystal. Second, the medium-range order, which encompasses atomic structures of 0.5–3 nm, is believed to exist, but the actual structures have not been elucidated, because of limited experimental methods. Small rings, such as 3-membered Si—O rings (Figure 10.2a), probably exist in SiO2 [6, 7, 21], and distorted layer structures (Figure 10.2b) are proposed for As2 S(Se)3 and GeS(Se)2 glasses [8, 10, 11]. Third, there are point-like defects such as E’ centers (Figure 10.2a) in SiO2 and D0 in GeS2 , which are neutral dangling bonds with typical densities of ppm levels so that their characteristics (a)

(b)

Figure 10.2 Atomic structures of (a) SiO2 and (b) As2 S3 glasses. In (a), Si and O are shown as a solid and an open circle with four- and twofold coordination. In (b), As and S are shown as a solid and as open circles with three- and twofold coordination. The bond lengths are 0.16 nm in (a) SiO2 and 0.23 nm in (b) As2 S3 , so that the side lengths of these illustrations are 2–3 nm. Note that near the centers, (a) includes a small ring and an E’ center and (b) contains wrong bonds (As—As and S—S). In the PbO-SiO2 glass, Si atoms in SiO2 (a) are replaced by Pb with some changes in their atomic coordination.

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have conventionally been analyzed through electron-spin signals [6–11]. In covalent stoichiometric glasses such as As2 S3 , Raman-scattering spectra show unambiguously wrong bonds (Figure 10.2b), i. e. homopolar bonds in stoichiometric glasses, which exist with typical density of a few at.% [6, 8, 11, 12]. Several other defects such as charged defects (D+ and D− ) have been proposed [6–11], which still remain only good working hypotheses, due to difficulties in experimentally confirming their existence. Lastly, it is plausible that the glass structure is inhomogeneous or fluctuating at scales of ∼10 nm or wider [11, 13, 22], specifically in multi-component systems, although such structures have been ruled out in many cases, which may cause intricate problems. Constituent atoms with noncrystalline connectivity determine the electronic structure at three levels [6, 11]. First, short-range bonding governs the optical bandgap energy Eg , which is in the range 4–10 eV for oxides and 1–3 eV for chalcogenides (see Figure 10.6). Accordingly, the latter are regarded as a type of amorphous semiconductors, although the sulfide is electrically insulating. For simple stoichiometric glasses such as SiO2 and As2 S3 , the origins of the valence and conduction bands are known, as will be described in Section 10.3.2. The bandgap energy is a major factor in determining the refractive index n0 of a material, which is ∼1.7 in oxides and ∼2.5 in chalcogenides, which is consistent with the empirical Moss rule, n0 4 Eg ≈ 77 [23]. Second, the medium-range and inhomogeneous structures are assumed to affect the density of states at the band edges, which govern the steepness of the exponential Urbach edges [13]. Third, defects such as dangling bonds and wrong bonds are likely to produce gap states, which are responsible for residual optical absorption [12, 24]. We here emphasize that defect-induced absorption is inherent to glasses. For an ideal dielectric crystal, which is conceptually obtained through infinitesimally slow cooling of a melt to 0 K, we can envisage complete transparency arising ultimately from zero-gap states. By contrast, the glass is prepared through freezing at around the glass-transition temperature, so that it necessarily contains structural disorders such as strained normal bonds and defects. The defective bonds are likely to produce gap states, which give rise to midgap absorption peaks in oxides [6, 7] and weak absorption tails in chalcogenides [11, 14].

10.3 Nonlinear Absorption and Refractivity 10.3.1

Fundamentals

The polarization P induced by an electric field E in a material with electric susceptibility 𝜒 can be written in cgs units as [4, 5]: P = 𝜒 (1) ∶ E + 𝜒 (2) ∶ E ⋅ E + 𝜒 (3) ∶ E ⋅ E ⋅ E,

(10.1)

where the first term 𝜒 :E depicts the conventional linear response, in which the complex parameter 𝜒 (1) represents the linear refractive index n0 and the absorption coefficient 𝛼, and other terms give nonlinear responses. The overall features are illustrated on a frequency axis in Figure 10.3. Here, we should note that the glass is macroscopically isotropic (centrosymmetric) and accordingly cannot provide even-order nonlinear effects arising from 𝜒 (2) :E⋅E, with notable exceptions in poled glasses [4, 5, 15–17]. (Such added nonlinearities are in general smaller than those in selected crystals, while long (1)

10.3 Nonlinear Absorption and Refractivity

χ(1)

χ(2)

ω

0

2ω

3ω

χ(3)

Figure 10.3 Roles of electric susceptibilities, 𝜒 (1) , 𝜒 (2) , and 𝜒 (3) in the frequency scale upon excitation with light of frequency 𝜔.

and wide samples, including fibers and film forms, may compensate for the smallness.) Then, in the ordinary case, the third-order nonlinearity 𝜒 (3) :E⋅E⋅E becomes the most important nonlinear term, and in some cases the fifth-order effects may appear as well [25–27]. The third-order nonlinearity provides at least three effects [4, 5]. The first effect is third-harmonic generation (see Figure 10.3). Suppose the frequency of a light field E is 𝜔, then E3 contains the term 3𝜔, and this component may be detected if a phase-matching condition for the fundamental and the overtone light, n0 (𝜔) = n0 (3𝜔), could be fulfilled. However, in isotropic materials such as glass, the condition in general cannot be satisfied, resulting in low generation efficiency. In addition, if 𝜔 is at near infrared, 3𝜔 is likely to be located at ultraviolet, at which frequency a glass may not be transparent. Accordingly, such overtone generation has limited applications in practice [28, 29]. The second effect is the so-called optical Kerr effect, including self-(de) focusing, which arises from the intensity-dependent refractive index change n2 I, where I is the light intensity. Such a change appears because the third term of Eq. (10.1), 𝜒 (3) :E⋅E⋅E, can be rewritten as n2 I E, in which n2 is proportional to 𝜒 (3) ; n2 (cm2 /W) ≈ 0.04𝜒 (3) (esu)/n0 2 [5]. The third effect is two-photon absorption, which is schematically illustrated in Figures 10.4b and b’. As is well known, the absorbed light intensity is proportional to the time average of E⋅dP/dt, which contains the term of 𝛽I 2 (∝ E4 ), i.e. two-photon absorption, where 𝛽 is related to the imaginary part of 𝜒 (3) . These two quantities, n2 I and 𝛽I 2 with the proportionality factors of n2 and 𝛽, appear to be very useful in controlling light at optical communication frequencies. Quantitatively, however, 𝜒 (3) in bulk glasses is very small [1–5]. For instance, as described later, n2 ≤ 10−15 m2 W−1 , so the condition n0 = n2 I can be fulfilled only CB

CB

CB

CB

CB

VB

VB

VB

VB

VB

(a)

(a′)

(b)

(b′)

(c)

Figure 10.4 Schematic illustrations of (a) one-photon, (b) two-photon, (b’) resonant two-photon, and (c) two-step absorptions from the valence band (VB) to the conduction band (CB) in a semiconductor. (a’) shows a midgap absorption. Note that the resonant two-photon absorption (b’) is a one-step process, which occurs resonantly with midgap states, while the two-step absorption (c) consists of two successive one-photon processes.

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when incident light is as intense as ∼1015 W m−2 ≈ 102 GW cm−2 , which may damage the glass (see Figure 10.10). To suppress such a breakdown, we could focus ultrashort light pulses onto a small spot (∼μm) and propagate them in long (or thick) samples, such as optical fibers and waveguides. Besides, promising tactics for obtaining efficient nonlinearity are to employ resonant enhancements of the electric field using excitons, plasmons, and/or advanced device structures [1–3, 17]. Alternatively, we may adopt less stringent modulation criteria, an example of which will be described later. Optical responses are formulated in terms of quantum mechanics as follows (see Figure 10.1): The one-photon absorption (Figures 10.4a and a’) coefficient 𝛼(ℏ𝜔) for amorphous materials is conventionally written as [6, 11] 𝛼(ℏ𝜔) ∝ ∣< 𝜑f ∣ H|𝜑i >|2

∫

Df (E + ℏ𝜔)Di (E) dE,

(10.2)

where H is the photon–electron interaction Hamiltonian, the simplest form being proportional to r; 𝜑(r) is the electron wave function, D(E) is the density of states, and the subscripts i and f represent an initial and a final state. Here, the momentum conservation has been neglected taking localized wave functions into account, and the transition matrix amplitude | < 𝜑f |H|𝜑i > | is assumed to be independent of ℏ𝜔 for simplicity. The refractive index spectrum n0 (ℏ𝜔) can then be calculated from 𝛼(ℏ𝜔) using a Kramers–Krönig relation as [30] n0 (𝜔) − 1 = (c∕π)℘

∫

{𝛼(Ω)∕(Ω2 − 2 )} dΩ,

(10.3)

where ℘ denotes the principal value of the integral. Since the absorption spectrum 𝛼(𝜔) is governed by Eg , the Moss rule (n0 4 Eg ≈ 77) [23] works as the simplest approximation of Eq. (10.3) (see Figure 10.8). When photons with ℏ𝜔 ≈ Eg /2 are absorbed, we may envisage two-photon absorption, as illustrated in Figure 10.4b. Under similar assumptions as those employed in deriving Eq. (10.2), the two-photon absorption coefficient 𝛽(ℏ𝜔) can be written as [30, 31] 𝛽(ℏ𝜔) ∝ ∣

∑

< 𝜑f ∣ H ∣ 𝜑n >< 𝜑n ∣ H|𝜑i >∕(Eni − ℏ𝜔) |2

n

∫

Df (E + 2ℏ𝜔)Di (E) dE, (10.4)

for degenerate cases, i.e. when the two photons have the same energy. Here, Eni = En – Ei , where the subscript n represents an intermediate state with energy En . The two-photon absorption governs the intensity-dependent refractive index n2 , which can be calculated using a nonlinear Kramers–Krönig relation for a nondegenerate case. In the present context of degenerate cases, we may express it as a rough approximation in the form [4] n2 (𝜔) ≈ (c∕π)℘

∫

{𝛽(Ω)∕(Ω2 − 𝜔2 )}dΩ,

(10.5)

which appears to be practically useful if 𝛽(Ω) is located in a narrow region at sufficiently higher frequencies than a calculated n2 (𝜔) spectrum, an example of which is shown later. Figure 10.5 shows typical linear and nonlinear spectra of absorption and refractivity for a direct-gap semiconductor with a bandgap energy of Eg [4, 31, 32]. The absorption edge of multi-photon absorption processes is located at Eg /m, where m is the number of photons simultaneously absorbed. Accordingly, two-photon absorption rises at Eg /2.

10.3 Nonlinear Absorption and Refractivity

Figure 10.5 Schematic illustrations of absorption spectra, 𝛼 and 𝛽, and refractivity spectra, n0 and n2 , governed by one- (𝛼 and n0 ) and two-photon (𝛽 and n2 ) processes in an ideal crystal. Excitonic effects are neglected.

n0

α β

n2 0

0

Eg/2

Eg ћω

Note that, in ideal crystalline semiconductors, all the electronic states in the vicinity of band edges are extended, so that a simple band theory can be applied. This situation also applies for indirect-gap semiconductors, although the spectral shape markedly changes [33, 34]. However, the situation becomes intricate in a disordered material. That is, midgap states with localized wave functions influence the absorption in at least three ways. First, the state may work as a site for linear absorption, as illustrated in Figure 10.4a’. The second is through resonant two-photon absorption. If a gap state that satisfies Eni − ℏ𝜔 = 0 in Eq. (10.4) exists, the two-photon absorption at around the gap state can resonantly be enhanced, as in Figure 10.4b’. Third, as illustrated in Figure 10.4c, a midgap state may cause two-step absorption, a successive one-photon absorption process, which can occur if the lifetime of the intermediate states is long enough [35]. The two-step absorption is assumed to occur at the Urbach-edge region in As2 S3 [36] and to govern a defect formation process in SiO2 [24]. It is needless to say that not only two-, but multi-photon absorptions can be envisaged in similar ways. However, theoretical considerations regarding these processes in glasses, including midgap states, still remain to be fully explored. 10.3.2

Two-Photon Absorption

Figure 10.6 shows one- and two-photon absorption (attenuation) spectra of SiO2 [34, 37], As2 S3 [36], and Se [38]. We see that these three glasses possess the same qualitative features. In the one-photon spectra, the Urbach edge appears at ℏ𝜔 ≥ 8.0, 1.9, and 1.4 eV in SiO2 , As2 S3 , and Se, which are consistent with the Tauc gaps at Eg ≈ 9, 2.4, and 2.0 eV [6, 11], respectively. Below the edge, residual absorption (attenuation) seems to exist. However, as exemplified for SiO2 , the spectral shapes have not been reproduced among several studies, probably reflecting varying absorption due to defects and impurities and also light scattering due to structural fluctuations. On the other hand, two-photon absorption appears at ℏ𝜔 ≥ Eg /2, which may have peaks at ∼5.8, ∼2.0, and ∼1.5 eV, respectively. It is interesting to note that, consistent with the absorption spectrum in amorphous Se, Enck [39] has detected two-photon photocurrents using 1.2 eV light pulses.

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103 x = 68 x = 38

α/cm–1, β/cm·(GW)–1

102 101

SiO2

100

Se

10–1 PbO–SiO2

10–2 10–3

As2S3 0

1

2

3

4 5 6 Photon energy/eV

7

8

9

Figure 10.6 Spectral dependence of the one- and two-photon absorption coefficients, 𝛼 (dashed lines) and 𝛽 (solid lines), respectively, in SiO2 , As2 S3 , Se, and two xPbO-(100-x)SiO2 glasses with x = 38 and 68. 𝛼 in SiO2 at ℏ𝜔 ≤ 8 eV may be influenced by light scattering, where some different spectra have been reported, so that 𝛼 can be read as attenuation.

The spectral features described above can be understood as follows [40]: In SiO2 , as illustrated in Figure 10.7, the top of the valence band is composed of the lone-pair electrons of O 2p states, and the bottom of the conduction band is governed by anti-bonding orbitals consisting of Si 3p and 3d states [6, 7]. The symmetry of the initial and final wave functions is different, i.e. ≠ 0, and accordingly, one-photon absorption can occur from the valence band to the conduction band. The fact that the two-photon spectrum appears at an energy around half of the optical gap (∼9 eV) suggests that the two-photon transition also occurs from the valence band to the conduction band. Similar interpretations are applicable to As2 S3 and Se as well. The spectral shape of the two-photon absorption shown in Figure 10.6 may resemble that of the theoretical curve in Figure 10.5, although the correspondence is not accurate. Specifically, the peak at ∼2.0 eV in As2 S3 may be deceptive, since the decrease in 𝛽 at ℏ𝜔 ≥ 2.0 eV occurs with distortion of light pulses, suggesting two-step absorption [36]. In addition, the observed peaks in SiO2 and Se are substantially sharper than the corresponding theoretical ones shown in Figure 10.5, where the vertical axis is plotted on a linear scale. The reasons for this discrepancy remain to be studied.

Cu 3d

DOS

276

Pb 6s

O 2p

Pb 6p

Na 3s Si 3p 3d

10 eV Energy

Figure 10.7 Density of states (DOS) of the valence (O 2p) and conduction (Si 3p and 3d) bands in SiO2 . Also added are Cu, Pb, and Na states in SiO2 . Occupied states are shaded.

10.3 Nonlinear Absorption and Refractivity

We also see in Figure 10.6 that the 𝛽 spectra are exponential, ∼exp(ℏ𝜔/E𝛽 ), below their peaks. Besides, in As2 S3 at ℏ𝜔 ≈ 1–2 eV, we notice an interesting correlation: the exponential 𝛽 spectrum appears nearly parallel to the weak absorption tail, which has the form 𝛼 ∝ exp(ℏ𝜔/EW ) [11, 13]; E𝛽 (≈ 150 meV) ≈ EW (≈ 200–300 meV) [36]. Note that this comparison is meaningful, since all the data of As2 S3 in the figure have been obtained using a single, high-purity ingot. Although such correlations may also exist in Se and SiO2 , the limited experimental spectra cannot confirm their existence. The nearly parallel exponential spectra in As2 S3 , E𝛽 ≈ EW , suggest the important role of gap states. That is, the two-photon absorption process is likely to occur resonantly with the gap states, which give rise to the weak absorption tail in the linear absorption. We can derive the rough equality as follows: In Eq. (10.4), approximating 1/(Eni − ℏ𝜔) as ∑ 𝛿(Eni − ℏ𝜔), we see that n 1/(Eni − ℏ𝜔) behaves as a density of states, which could imply 𝛽 ∝ Dn (E), where Dn (E) is the density of state of the intermediate (gap) states. On the other hand, the gap state can also govern the final state Df (E) in the linear absorption, given by Eq. (1.2), which may lead to 𝛼 ∝ Df (E). These two relations are consistent with the observation E𝛽 ≈ EW . Figure 10.6 also includes the spectra for PbO-SiO2 glasses, which present three notable features [41, 42]. First, 𝛼(ℏ𝜔) has an exponential Urbach edge at around ℏ𝜔 ≈ 3 eV (𝛼 ≈ 1 ∼ 100 cm−1 ), which shifts nearly in parallel to lower energies with the PbO content. Second, on the other hand, 𝛽(ℏ𝜔) appears to be non-exponential. There exist fairly sharp increases at ∼2.0 eV, which seem to be common to all the compositions. This threshold suggests that two-photon transitions in these glasses occur between the states that are separated in energy by more than ∼4 eV, which is substantially higher than the Urbach edge located at ∼3 eV. Finally, the maximal 𝛽 in PbO-SiO2 increases by ten times from 1 to 10 cm/GW for nearly twice an increase in the PbO content from 38 to 68 at.%. The first and second features of 𝛼(ℏ𝜔) and 𝛽(ℏ𝜔) spectra can be understood as follows [40, 42]: The electronic structure of PbO-SiO2 is illustrated in Figure 10.7, which suggests that the one-photon absorption edge ∼3 eV is governed by intra-atomic 6s → 6p transitions in Pb. This interpretation can explain the red-shifting one-photon absorption edge with an increase in the PbO content, since the 6s and 6p bands arising from the Pb atoms probably become broader-reflecting enhanced interatomic interactions. However, such a transition cannot be assumed for two-photon absorption, because it cannot occur between s and p states in a single atom (see Eq. (10.4); note that H ∝ r). Then, a possible two-photon transition with a small photon energy is assumed to be O (2p) → Pb (6p), which can account for the different threshold in 𝛽(ℏ𝜔) at ∼2 eV from that in 𝛼(ℏ𝜔). This interpretation is also consistent with the dramatic 𝛽 increase with an increase in the PbO content, which is ascribable to resonant two-photon absorption. Two-photon absorption can resonantly be enhanced by the Pb (6s) states, as the term / (EPb6s,O2p − ℏ𝜔) possibly governs the transition probability in Eq. (10.4). Note that since Pb (6s) is an occupied state, it cannot commit to two-step absorption (Figure 10.4c), as is experimentally confirmed [42]. In short, in contrast to the situation in simple glasses such as SiO2 , the three atomic levels (O 2p, Pb 6s, and Pb 6p) seem to play dominant roles in the optical properties of PbO-SiO2 glasses. Comparatively, defective structures are less important in this system. Similar interpretations are probably applicable to other heavy-metal oxide glasses containing Bi2 O3 , etc. [43–46].

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10 Nonlinear Optical Properties of Photonic Glasses

10.3.3

Nonlinear Refractivity

Since measurements of n2 are relatively difficult, several relations for estimations of n2 using linear optical properties have been proposed [4, 17, 47]. These relations can be grouped into two types. One type is specifically applicable to transparent materials, and the most famous may be Boling’s relation [48], given by n2 (10−13 esu) ≈ 391 (nd − 1)∕𝜈d 5∕4 ,

(10.6)

where nd is the refractive index at the He d-line (𝜆 = 588 nm), and 𝜈 d is the Abbe number. This relation is known to provide satisfactory agreements in materials with a small nd (≤ 1.7). In the other type of relation, which applies to semiconductors, the nonlinear properties are connected with the bandgap energy Eg . For instance, Sheik-Bahae and Van Stryland [31, 32] have shown for many (∼30) crystals with Eg ≈ 1–10 eV that the relation given by n2 n0 = K G(ℏ𝜔∕Eg )∕Eg 4

(10.7)

provides a good approximation, where K is a material-independent constant (∼6 × 10−11 (cm2 W−1 )(eV)7/2 ), and G(ℏ𝜔/Eg ) represents a universal spectral curve; the functional form varies with nonlinear mechanisms. (A corresponding expression for 𝛽 is also given by them.) As shown by the solid line in Figure 10.8, this relation also gives satisfactory fits to many data of oxide and chalcogenide glasses [34, 49, 50], where for Eg we take the 101

15Na2O·85GeO2

PbO–SiO2

GeS2

As2S3

10–2

As2Se3

Ag20As32Se48

10–1

Bi2O3–SiO2

100

35La2S3·65Ga2S3

10

n0

10–3

SiO2

n2n0·(KG)–1

278

10–4

1

5

1 10

Eg/eV

Figure 10.8 Linear and nonlinear refractivity, n0 (open symbols) and n2 (solid symbols), in some oxide (circles), sulfide (triangles), and selenide (squares) glasses as a function of the optical gap E g . The solid and dashed lines depict Sheik-Bahae’s relation (Eq. (10.7)) and the Moss relation n0 4 E g ≈ 77.

10.3 Nonlinear Absorption and Refractivity

Tauc gap or the photon energy at 𝛼 = 103 cm−1 . This result suggests that irrespective of crystalline or noncrystalline solids, n2 is mostly determined by Eg , which is governed by the short-range atomic structure. We can assume that with a scale of optical wavelengths, there is no marked difference between crystals and glasses, provided that anisotropic effects in the former could be neglected. The disordered structure in glasses, including the gap states, tends to cause only secondary effects on n2 , which may explain a large deviation, e.g. in Ag20 As32 Se48 having n2 ≈ 8 × 10−16 m2 W−1 [26], from the line in Figure 10.8. Moreover, as illustrated in Figure 10.8, the Moss rule (n0 4 Eg ≈ 77) [23] also applies to these glasses, which further reinforces the importance of Eg . It may be interesting to examine if we can relate n2 (ℏ𝜔) to 𝛽(ℏ𝜔) in real materials. Since 𝛽(ℏ𝜔) data are limited to sub-gap photon-energy regions, we here approximate the spectra in As2 S3 and SiO2 by Gaussian curves, which are shown by dashed lines in Figure 10.9. Then, n2 (ℏ𝜔) can be calculated using the nonlinear Kramers–Krönig relation, Eq. (10.5), as shown by the solid lines in the figure. We see that the calculated n2 (ℏ𝜔) shapes resemble the theoretical curve for the crystalline semiconductor shown in Figure 10.5. We also see in Figure 10.9 that the agreements between the calculated n2 (ℏ𝜔) and the experimental n2 [5], plotted by open symbols, are surprisingly good. This agreement reinforces the fact that Eq. (10.5) can be practically employed for estimating n2 (ℏ𝜔), and that n2 in the near-infrared region is governed by two-photon absorption, which has also been confirmed in crystalline semiconductors [5]. Similar insights have been obtained also for PbO-SiO2 glasses [42]. In conclusion, irrespective of crystals and non-crystals, the bandgap energy Eg is the primary factor governing the nonlinear absorption and refractivity. The electronic density of states, including gap states, and the form of wave functions play secondary roles, and the details remain to be studied. Besides, since the transition amplitudes of oneand two-photon absorptions have different forms, further studies of these processes will provide valuable insights into related states. The idea described here provides an answer to a frequently asked question: “Why does As2 S3 possess higher n2 than SiO2 at optical communication wavelengths, ℏ𝜔 ≈ 1 eV? ” In the present context, this is because a higher n2 arises from the greater

n2/cm2·(GW)–1, β/cm·(GW)–1

102 100 10–2

As2S3

SiO2

10–4 10–6 10–8

0

1

2

3

4

5

6

7

ћω/eV

Figure 10.9 Two-photon absorption spectra 𝛽(ℏ𝜔) in As2 S3 (solid circles) and SiO2 (solid triangles), fitted Gaussian profiles (dashed lines), calculated n2 (ℏ𝜔) (solid lines), and experimental n2 (open circle and triangle). 𝛽(ℏ𝜔) data are the same as those shown in Figure 10.6 .

279

280

10 Nonlinear Optical Properties of Photonic Glasses

𝛽 (∼10 cm GW−1 ) at a lower photon-energy region in As2 S3 [34]. The greater 𝛽 can then be connected with the smaller Eg , because Eq. (10.4) gives 𝛽 ∝ Eg −2 , provided that the denominator (Eni − ℏ𝜔)2 governs the gross features. (Precise analyses give 𝛽 ∝ Eg −3 [49, 50].) A higher n2 at a smaller value of Eg can also be derived from Eq. (10.7), which leads to n2 ∝ Eg −4 . We can also evaluate the feasibility of n2 in bulk materials at the communication wavelength 𝜆 ≈ 1.5 μm [50]. As Eg corresponds to the electronic cutoff wavelength 𝜆g through a simple equation, Eg [eV]⋅𝜆g [μm] ≈ 1.24, the Sheik-Bahae’s relation (Eq. (10.7)) predicts for materials with Eg ≥ 1.2 eV (𝜆g ≤ 1 μm) that n2 ≤ 10−15 m2 W−1 (= 10 cm2 TW−1 ), which is consistent with reported values [26, 27, 51–55]. This result suggests that for obtaining practical nonlinearity of Ln2 I/𝜆 ≈ 𝜋 in optical waveguides with a light propagation distance L of 1 cm, we need light sources with an intensity I of ∼0.1 GW cm−2 , or 1 W μm−2 , which can be attained using presently available laser diodes.

10.4 Nonlinear Excitation-Induced Structural Changes 10.4.1

Fundamentals

Oxide and chalcogenide glasses are known to exhibit a variety of photo-induced phenomena [7–11]. These phenomena tend to yield metastable structures, whereas it is challenging to identify the atomic structures and transformations occurring in disordered glass networks. Some of the photo-induced changes can be recovered by thermal annealing at the glass-transition temperature T g . It should be noted that in comparison with photo-induced phenomena in crystals, such as F center formation in alkali halides, two features add complexities to such phenomena in glasses. One is that the photo-induced change in a glass depends upon many factors such as the photon energy and (peak) the intensity of illuminating light, continuous wave (cw) or pulse (pulse duration and repetition), light spot size (bulk and interfacial stress effects), light polarization, illuminating temperature T i , and illumination atmosphere (air, vacuum, etc.). For instance, many photo-induced transformations become smaller at higher T i , ultimately disappearing at T i ≈ T g [10, 11, 21]. The other feature that adds complexity is that a glass property varies from sample to sample, reflecting the quasi-stability, which substantially affects photo-induced phenomena. In fact, radiation effects in SiO2 , one of the simplest oxide glasses, cannot universally be grasped due to minute, but crucial, deviations in stoichiometry and impurities such as H, O2 , OH, and Cl [7, 21, 24, 56–58]. In a chalcogenide, As2 S3 , it is known that evaporated and annealed films undergo different photo-induced transformations of polymerization (stabilization) and disorder increases (destabilization), respectively [8–11]. In short, we are confronted with a variety of exposure conditions and samples for a single material, and as a result, it is rather difficult to obtain universal insights. Notable photo-induced effects are produced through linear excitations by bandgap photons with ℏ𝜔/Eg ≥ 1, and also through nonlinear (or linear) optical excitations by photons with ℏ𝜔/Eg < 1. Such low-energy photons are able to penetrate more than ∼1 cm into bulk samples, due to small linear absorption, which is favorable for producing volume-related effects. For instance, in Ge-doped SiO2 fibers with a diameter of ∼100 μm, Bragg reflectors are inscribed using ultraviolet excimer lasers [1]. Also, in

10.4 Nonlinear Excitation-Induced Structural Changes

oxide [58–60] and chalcogenide [3, 53, 61–64] glasses, waveguides can be fabricated by scanning near-infrared laser light. At these photon energies, since linear absorption is relatively small, nonlinear excitations could be responsible for photo-electronic excitations. However, the fundamental mechanisms of the photo-induced phenomena remain to be explored. In general, a photo-induced phenomenon occurs through energy transfer from photons to electron-lattice systems, and to atomic structures. Accordingly, the process may be divided into two parts, photo-electronic excitation and electro-structural change, and the latter can be regarded as either athermal (polaronic) or thermal. In an athermal process, the temperature rise in a material, which inevitably occurs to some degrees under illumination, can be neglected, but it is decisive in the thermal process including heat-spike generation. Otherwise, we may envisage instantaneous photo-electro-structural evolutions in electronic melting under ultrahigh excitations, the concept was originally proposed by Van Vechten et al. [65]. Permanent damage may occur under such excitations. In this section, we will make a comparative study of the athermal photo-induced phenomena induced by one- and multi-photon processes in the oxide SiO2 and the chalcogenide As2 S3 . Nevertheless, at the outset, it may be valuable to estimate the temperature rise induced by light illumination. The temperature rise ΔT can be evaluated very roughly for two extreme cases: Upon single pulse irradiation with a peak intensity I and a pulse duration 𝜏 (≥ 10 ps, which is a typical electron-lattice relaxation time), the temperature rise in a sample can be estimated as ΔT ≈ Q/cV , where Q (≈ 𝛼I𝜏) is the absorbed light energy, c the specific heat, and V (≥ Sd) a related volume, which can be evaluated from the light spot area S, a characteristic distance d (the sample thickness or light penetration length such as 𝛼 −1 or (𝛽I)−1 ), and the thermal diffusion length (k𝜏/c)1/2 , where k is the thermal conductivity. For instance, when an SiO2 bulk sample (k ≈ 10−2 W cm−1 ⋅K and c ≈ 2 J cm−3 ⋅K) is exposed to a light pulse with a width of 1 ns, peak intensity of 1 kW, spot diameter of 10 μm (≈ 1 GW cm−2 ), and ℏ𝜔 = 8 eV (𝛼 −1 ≈ 1 mm), one obtains ΔT ≈ 5 K. For light pulses of duration shorter than 1 ns, the thermal diffusion length (k𝜏/c)1/2 becomes shorter than ∼25 nm, which may be neglected for all practical purposes [66]. On the other hand, a steady-state temperature rise in an absorbing sample exposed to cw or high-repetition pulses with a time-averaged intensity I can be estimated from ΔT ≈ Q/(2𝜋kr) [67], where Q (≈ 𝛼IV ) is the absorbed light power, and r is the radius of the light spot. If Q = 1 mW and r = 10 μm, this gives ΔT ≈ 10 K for SiO2 . Experimentally, the temperature rise can be probed using thin-film thermocouples or infrared thermometers, or it can be estimated from the intensity ratio of the (anti-)Stokes Raman-scattering spectra. 10.4.2

Oxides

Figure 10.10a summarizes the structural responses of SiO2 to light excitations with varied photon energies and intensities. The results are obtained at room temperature, i.e. T i /T g ≈ 300/1500 = 0.2, using three types of light sources: 1) Laser pulses of fs-ps duration and ℏ𝜔 ≈ 1–2 eV (ℏ𝜔/Eg ≈ 0.1–0.2) [57–60, 68–79]. 2) Excimer laser pulses of ∼10 ns and ℏ𝜔 ≈ 5–8 eV (ℏ𝜔/Eg ≈ 0.5) [24, 58, 80–82]. 3) (Super-) bandgap excitations (ℏ𝜔/Eg ≥ 1) including x- and γ-rays [83–85].

281

10 Nonlinear Optical Properties of Photonic Glasses

d

G

d E′& –ΔV

M

+ΔV

K

ablation –ΔV –Δn WB

103

d c–Si

E′

100

β

1 Ti:sapphire

KrF ArF

Eg

F2

m 0

5

(b) T

ablation Δn

G

Δn

Δn

PD

m 0

1

100

+ΔV

Ti:sapphire Nd:YAG

1

103

PD

Δn

β

K

α

Δn Δn WB WB

M

10–3 15

10

ћω/eV

ћω/eV

PD Δn +ΔV

2

α/cm–1, β/cm·(GW)–1

Intensity/W·cm–2

α

d

T

Eg

As–As

α/cm–1, β/cm·(GW)–1

(a)

Intensity/W·cm–2

282

10–3

3

Figure 10.10 Photo-induced phenomena scaled by the excitation photon energy ℏ𝜔 and the (peak) light intensity in (a) SiO2 and (b) As2 S3 . T, G, M, etc. on the left-hand side vertical axis represent TW, GW, MW, etc. Also shown are the one- and two-photon absorption spectra, 𝛼 (solid lines), and 𝛽 (dashed lines), respectively. The scales are shown on the right vertical axis. Photon energies of related laser light are indicated on the horizontal axis as Nd:YAG, Ti:sapphire, KrF, ArF, and F2 . E g represents the bandgap energy. In the abbreviations, E’ indicates an E’-center formation, +ΔV a volume expansion, -ΔV a volume contraction, Δn a refractive index change, WB a wrong-bond formation, PD a photo-darkening, d damage. In the shaded region in (b), photo-darkening, refractive index increase, and volume expansion occur.

Experiments using (super-)bandgap illumination in (3) are limited, probably because of the large bandgap energy of Eg ≈ 9 eV. Takigawa et al. [83] report for synthetic SiO2 windows that a single shot of 9.8 eV laser light (Ar excimer laser, 5 ns) with ∼300 mJ cm−2 (≈ 60 MW cm−2 ) power causes topological surface damage and crystalline Si formation. They also mention that lower-energy shots with ℏ𝜔 ≈ 5 ∼ 8.5 eV do not produce crystalline Si, despite the marked surface damage. Awazu and Kawazoe [21] have demonstrated that super-bandgap excitations with energies 14 and 18 eV, which are obtained from an undulator-equipped synchrotron, produce small Si—O rings in thermally grown amorphous SiO2 films. Akazawa [84] has discovered that under super-bandgap excitations (synchrotron radiation of 100–300 eV), amorphous SiO2 films evaporate while accumulating Si—Si wrong bonds. In short, a (super-)bandgap excitation tends to produce Si wrong bonds and evaporation. These phenomena in SiO2 resemble photo-enhanced vaporization of As2 O3 , which appears efficiently in heated As2 S3 under super-bandgap illumination [86]. By contrast, it seems that a photo-darkening phenomenon (see Section 10.4.3), which is induced in covalent chalcogenide glasses by bandgap illumination, does not exist in SiO2 [10, 24] and GeO2 [87].

10.4 Nonlinear Excitation-Induced Structural Changes

On the other hand, many studies in the above experiments (1) and (2) have demonstrated some changes in small photon-energy excitations with ℏ𝜔/Eg ≈ 0.1–0.8. Specifically, three-dimensional optical fabrications have been explored using intense fs-ps lasers with ℏ𝜔 ≈ 1 eV under various conditions [58, 60, 66, 68, 69, 72–78].When the incident pulse is more intense than the order of J cm−2 levels, or GW/cm2 for 1 ns (or TW/cm2 for 1 ps) pulses, both surface and/or internal damage occur with ablation [21, 58, 59, 68, 72, 73, 76, 78]. However, when the peak intensity is around 1 MW/cm2 , three types of phenomena may appear: (1) creation of defective structures such as Si—Si bonds [79] with small rings [21], (2) volume changes [7, 74, 75, 80], and (3) refractive index changes [7, 58–60, 74–77]. The volume and the refractive index changes have been correlated through the Lorentz-Lorenz relation [7, 74, 75]. Nevertheless, to the best of the author’s knowledge, the polarization effects induced by pulse light have scarcely been taken into account in these changes [76, 77]. Is the multi-photon excitation really responsible in these photo-structural phenomena? It is plausible that in Ge-doped SiO2 , the dopants play the role of color centers, which cause linear absorption [58, 80] as illustrated in Figure 10.10a’. However, in pure SiO2 , it seems still unclear whether the excitation is triggered by one- or by multi-photon processes. For a two-photon excitation to be dominant, at least the condition 𝛽I ≫ 𝛼 must be satisfied. For instance, at ℏ𝜔 ≈ 5 eV, since 𝛼 = 10−2 cm−1 and 𝛽 = 10−1 cm GW−1 (see Figure 10.6), the light intensity must be greater than ∼100 MW cm−2 . This intensity level is located in Figure 10.10a near the boundary between the damage and the nondestructive structural modifications. Here, we should note that 𝛼 is evaluated using a weak probe light, but the condition 𝛽I ≫ 𝛼 should be satisfied under intense illumination that may produce transitory midgap absorption at defective sites [70] accompanying an increase in 𝛼. The intense illumination also produces a temperature rise, which is likely to enhance band-edge absorption. Accordingly, it is not straightforward to examine the 𝛽I-𝛼 criterion under practical conditions. In addition, we should distinguish the two-photon from the two-step absorption (see Figure 10.4), which may exhibit different response times. In fact, Kajihara [24] has proposed that the exposure with 7.9 eV photons to SiO2 causes the formation of E’ centers through a two-step absorption process rather than through two-photon absorption. Finally, how is the excitation converted to structural transformations? To answer this question experimentally, time-resolving photo-structural studies will be valuable for understanding the conversion process from photo-electronic excitation, transitory defect formation, and to the modifications in macroscopic properties [77]. 10.4.3

Chalcogenides

The chalcogenide glasses have an optical gap in the range 1–3 eV, so that visible light can induce several types of prominent structural changes [3, 8–11]. At the outset, however, it is important to distinguish whether the phenomena arise from photo-induced physical (bond alternation) or chemical (oxidation) processes in as-deposited films and annealed samples including bulk glasses and fibers. In the annealed samples, the phenomena of photo-darkening (evaluated as red shifts ΔE in the optical absorption edge, which can be recovered by annealing), related refractive index increase, and volume expansion have attracted continuous interest [8–11, 88]. This is because these changes are simple, athermal, inherent to covalent chalcogenide glasses such as Se and As2 S3 ,

283

284

10 Nonlinear Optical Properties of Photonic Glasses

and promising for optical applications including holographic memories and functional devices [10, 11, 53]. For understanding the fundamental mechanism of photo-darkening and related phenomena, spectral studies are valuable [10, 11]. For instance, in As2 S3 with Eg ≈ 2.4 eV, it is known that bandgap illumination at room temperature (T i /T g ≈ 300/450 ≈ 0.7) produces an edge shift ΔE of ∼50 meV and a refractive index increase of ∼0.02 (see Table 10.1), which are quantitatively related through the linear Kramers–Krönig relation (Eq. (10.3)); and in addition, a volume expansion of ∼0.4%. It has also been demonstrated that even an intense sub-bandgap illumination can produce notable changes. In As2 S3 , continuous illumination of light with a photon energy of 2.0 eV (ℏ𝜔/Eg ≈ 0.8) and an intensity greater than 100 W cm−2 can produce photo-darkening that is comparable to that induced by bandgap illumination, and a seemingly more prominent volume expansion. Pulsed light of energy 1.65 eV (ℏ𝜔/Eg ≈ 0.7) and a peak intensity of 100 MW cm−2 can also induce the optical and volume changes [89, 90]. Tanaka has proposed, taking a marked red shift of the photo-conduction spectrum into account, that these intense sub-gap illuminations excite enough carriers to completely fill band-tail states, which probably exerts similar effects to those by bandgap illumination [89]. However, the half-gap illumination (ℏ𝜔 ≈ 1.2 eV ≈ Eg /2) exhibits different features in As2 S3 [89, 91, 92] as explained in what follows: If the illumination is weak, no detectable changes appear within experimental time scales. But if a sample is illuminated by intense pulses that satisfy the condition 𝛽I » 𝛼, the excitation gives rise to a refractive index increase, but no discernible photo-darkening, 0 ± 5 meV, as shown in Table 10.1 and Figure 10.10b. Besides, no changes in sample thicknesses (≈10–30 μm) are detected with an accuracy of ∼100 nm. On the other hand, Raman-scattering spectroscopy demonstrates that the two-photon excitation causes a density increase (a few percent) in wrong bonds, As—As and S—S. In these experiments performed at ∼300 K, the temperature rise induced by the excitation is estimated to be ∼10 K, which can practically be neglected in comparison with the glass-transition temperature of ∼450 K; the process appears to be athermal. It has also been reported that near-infrared light pulses induce similar refractive index increases in glasses such as As2 Se3 (Eg ≈ 1.8 eV) [93], Ga-La-S (Eg ≈ 2.6 eV) [62, 64, 94], Ge-Sb-Se (Eg ≈ 2 eV) [95], etc. [96]. We may envisage that these observations resemble the fs-ps pulse effects in SiO2 described in Section 10.4.2. Ablation-related phenomena also emerge upon exposure to more intense near-infrared pulses [52, 97]. Table 10.1 Photo-darkening (red shift) ΔE and refractive index increase Δn induced in As2 S3 glass by pulsed light with energies of 1.17 and 2.33 eV and also by continuous-wave (cw) bandgap light of ∼2.4 eV for comparison [89]. ℏ𝝎(eV)

𝚫E(meV)

𝚫n

𝜶 (cm−1 )

𝜷 (cm W−1 )

Intensity (W cm−2 )

1.17

0±5

0.005

10−3

10−10

109

2.33

20

0.003

300

?

107

cw

50

0.02

∼500

∼0.05

Related parameters (𝛼, 𝛽, and excitation light intensity) are also listed. 𝛽 at 2.33 eV cannot be evaluated due to two-step absorption [36]. In the pulse experiments, ΔE and Δn are evaluated at a fixed absorbed photon number of 1023 –1024 cm3 , which may not provide saturated changes.

10.5 Conclusions

Conduction Band

Valence Band

Figure 10.11 Schematic illustrations of the bandgap excitations with one-photon (left) and resonant two-photon (right) processes. One-photon excitations may cause modifications in the medium-range structures, which broaden the valence band, and as a result, photo-darkening occurs. Two-photon excitations may produce localized defect centers such as small As clusters.

To understand the above two-photon excitation effect, we must answer at least two questions: (1) Why does the excitation produce the wrong bonds? And (2) why does the refractive index increase? To answer the first question, we may follow the idea that two-photon absorption can be resonantly enhanced by midgap states if the energies satisfy Eni − ℏ𝜔 ≈ 0 in Eq. (10.4). Otherwise, we envisage two-step absorption. Here, the midgap states are ascribable to the anti-bonding states of As—As bonds [36]. In such cases, resonant two-photon (step) absorption tends to occur spatially selectively around the defective sites, and as a result, As clusters may grow through some presently unspecified mechanisms. On the other hand, bandgap illumination probably causes defect-unrelated excitations resulting in different structural changes, such as bond twisting [8, 10, 11], which probably broadens the valence band. In short, as illustrated in Figure 10.11, reflecting the different transition probabilities between the one- and two-photon excitations, different structural changes may occur. Question (2) can be addressed as follows. Taking the linear Kramers–Krönig relation (Eq. (10.3)) into account, we may assume that the refractive index increase should be accompanied by some absorption increase due to photo-darkening (reduction in Eg ), but experimentally no edge shift has been detected. No volume changes have been detected as well. These facts may suggest the idea that inhomogeneous structures containing As and S clusters produce some stresses, which cause the increase in refractive index through photo-elastic effects [92]. Otherwise, we envisage that it occurs due to the emergence of some midgap absorption processes. In order to examine these notions, further studies such as composition dependence of the photo-induced refractive index change in the As—S system will be valuable.

10.5 Conclusions After briefly reviewing photonic glasses, we have highlighted two topics on nonlinear optical properties of oxide and chalcogenide glasses. One is a unified understanding of the absorptive and dispersive third-order nonlinearity, and the other is the role of nonlinear excitations in photo-induced phenomena.

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Optical nonlinearity in glasses is governed by the bandgap energy and is modified by detailed electronic structures. The role of the bandgap energy is similar to that in crystals, since the short-range structure in a glass is nearly the same as that in its crystalline counterpart. However, a difficult, but case-dependent problem is that the nonlinearity in glasses is affected by the band-tail and midgap states, having spatially localized wave functions, which could cause resonant multi-photon and/or multi-step absorption. Different photo-structural changes appear in a glass depending upon the photon energy and light intensity. When the light intensity is weak and ℏ𝜔 ≈ Eg /2, one-photon absorption by midgap states can be held responsible for polaronic changes. If the intensity is high, we should also take (resonant) two-photon and two-step processes into account. If ℏ𝜔 ≈ Eg , band-to-band excitations become dominant. These photoelectronic excitations seem to cause unique photo-structural changes. In SiO2 , the midgap excitation creates defective structures, which may produce volume contraction and refractive index increase, whereas (super-)bandgap illumination tends to produce Si—Si homopolar bonds. In As2 S3 , pulsed infrared laser light (∼Eg /2) produces wrong bonds, As—As and S—S, probably through localized resonant two-photon excitations, which is in contrast to the fact that bandgap illumination gives rise to photo-darkening, refractive index increase, and volume expansion. These photo-induced structural modifications are inherent to the metastable material, and further studies including polarized-light nonlinear excitations will be interesting for the creation of artificially tuned atomic structures.

10.A Addendum: Perspectives on Optical Devices So far, several types of photonic glass devices, passive and active, have been commercialized, and it would be valuable to shed light on some fundamental functions. At present, most of the devices have been developed for optical communications [1–3, 98], and the rest are for other purposes such as laser components and scientific experiments. For instance, a common passive element is the optically inscribed fiber Bragg filter, which is utilized for wavelength selection, etc. In addition, with regard to active optical devices, some of which are listed in Table 10.A.1, rare-earth-doped fiber amplifiers have widely been employed in wavelength-division multiplexing (WDM) systems. How can we characterize such optical devices? A comparison of the properties of an electron in semiconductors and a photon in photonics glasses reveals contrasting features in three facets; charge, speed, and scale. The electron has the charge, which is the origin of strong interaction with electric and Table 10.A.1 Active optical devices. Device

Nonlinear (bulky)

Others

Source/amplifier

Supercontinuum generator Overtone generator Stimulated Raman-scattering amplifier

Rare-earth doped

Controller/modulator

n2 (Kerr) switch 𝛽 stabilizer (intensity limiter)

(Electro-optical)

10.A Addendum: Perspectives on Optical Devices

magnetic fields. By contrast, the photon (electromagnetic wave) interacts only weakly with a material (atoms and bonding) through the permittivity. Naturally, nonlinear interactions tend to become weaker. Moreover, we can manipulate electrons and holes in semiconductor bipolar devices (or a majority carrier in unipolar devices), while the photonics has no counterpart of holes; i.e. photonic devices are nonpolar. Hence, we cannot produce optical elements such as the pn junction, a basic electronic component. As regards speed, the electron is able to move between zero (trapped) and ∼108 cm s−1 (under an electric field of 105 V cm−1 and a mobility of 103 cm2 (V s)−1 ), while the photon has a nearly fixed speed of ∼1010 cm s−1 . These two characteristics explain a principal outcome: the electron is suitable for signal processing at frequencies below the gigahertz range, and the photon is convenient for teleportation. Otherwise, we may utilize standing light waves in photonic structures, and so forth, although such devices are likely to possess retarded resonant responses. For the scale, an electron can be confined into nanometer wells, while a photon has a wavelength of a micrometer. These scales govern the related device dimensions. For instance, it will be impossible to devise optical circuits with integration scales of nanometers. As regards the macroscopic shape of optical devices, most of them have fiber forms, with functional two- and three-dimensional waveguides remaining to be developed. One of the reasons is that the nonlinearity presently obtained needs light–matter interaction lengths longer than ∼1 m, which are insufficient for compact devices of ∼1 cm or less [49, 50, 54, 55, 99]. As a consequence, the practical usage of integrated nonlinear optical circuits awaits innovative materials and device architectures. For the material, we have understood that the optical nonlinearity in bulk solids is governed by the bandgap energy. Hence, further enhancements of the nonlinearity need more sophisticated materials in combination with novel structures, examples being particle-doped glasses [3, 17, 100], micro-structured fibers [19, 101, 102], tapered fibers [102, 103], nanowires [104], resonant structures including spheres [105], rings [29, 106], photonic crystals [107–109], etc. For instance, it has been demonstrated that a Se-doped zeolite yields three-orders-of-magnitude-higher nonlinearity than that in bulk Se glass [50], with a proposed origin of exciton confinement [38]. Besides, plasmonic enhancements of electric fields offer tempting ideas [17, 110]. We also point out a common feature and a difference in electronic and optical devices: The common feature can be found in many of the component structures, such as diode or triode types. For instance, optical isolators, supercontinuum and overtone generators, and intensity stabilizers using nonlinear absorption can be regarded as a diode type consisting of only an input and an output. On the other hand, as illustrated in Figure 10.A.1, Nonlinear fiber

Figure 10.A.1 A triode-type optical device. INPUT

Isolator

OUTPUT

Filter PUMP

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Figure 10.A.2 A Mach-Zhender–type logic waveguide with nonlinear optical gates G1 and G2.

G1 input G2 output

stimulated Raman-scattering amplifiers, optical switches utilizing n2 , etc. are of the triode type: in the same way as field-effect transistors having gate, source, and drain. Such analogies will be valuable for improving and developing optical devices. For instance, the optical isolator and a pn-junction diode serve a similar function, although the former is still much larger. The intensity stabilizer may be analogous to a Zener diode, although the former function remains comparatively inefficient. We can also directly see that a Mach-Zhender interferometer with nonlinear couplers, illustrated in Figure 10.A.2, could operate as a dual-gate logic element. By contrast, a marked difference, or an advantage of the optical system, is that it can possess spectral (and polarization) multiplicity. Many signals having different wavelengths can be processed simultaneously in a single line, a property that has already been utilized in the WDM system equipped with fiber amplifiers of rare-earth doped and/or stimulated Raman-scattering types. In addition, supercontinuum generators are becoming unique light sources [1–4, 20, 101–104]. Spectral hole-burning memories have been continuously studied [111]. Nevertheless, it should be emphasized that the resonant structure including photonic-crystal devices [107–109] is in principle incompatible with multi-spectral operation. Under these circumstances, we have been exploring ultrafast-response (∼ps), compact, wide-band (Δ𝜆 ≥ 10 nm), nonlinear optical devices with sufficient stability [3, 101, 112]. In future, the optical components may be integrated with electronic (semiconductor) and electro-optical (ferroelectric) circuits on monolithic and/or hybrid chips, the concept originally proposed by Miller nearly a half century ago [113]. Or, the ultimate target will be totally optical systems, in which all the signal processing and transmission are accomplished via multi-spectral photons, and the electricity is fed just for power supply to tunable and/or supercontinuum light sources.

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in sub-wavelength diameter As2 Se3 chalcogenide fiber tapers. Opt. Express 15: 10324–10329. Foster, M.A., Turner, A.C., Lipson, M., and Gaeta, A.L. (2008). Nonlinear optics in photonic nanowires. Opt. Express 16: 1300–1320. Elliott, G.R., Hewak, D.W., Murugan, G.S., and Wilkinson, J.S. (2007). Chalcogenide glass microspheres; their production, characterization and potential. Opt. Express 15: 17542–17553. Ferrera, M., Razzari, L., Duchesne, D. et al. (2008). Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures. Nat. Photonics 2: 737–740. Suzuki, K. and Baba, T. (2010). Nonlinear light propagation in chalcogenide photonic crystal slow light waveguides. Opt. Express 18: 26675–26685. Kuznetsov, A.I., Miroshnichenko, A.E., Brongersma, M.L. et al. (2016). Optically resonant dielectric nanostructures. Science 354 (2472). Hussein, H.M.E., Ali, T.A., and Rafa, N.H. (2018). A review on the techniques for building all-optical photonic crystal logic gates. Opt. Laser Technol. 106: 385–397. Ung, B. and Skorobogatiy, M. (2011). Extreme optical nonlinearities in chalcogenide glass fibers embedded with metallic and semiconductor nanowires. Appl. Phys. Lett. 99: 121102. Khan, A.A., Jabar, M.S.A., Jalaluddin, M. et al. (2015). Spectral hole burning via Kerr non linearity. Commun. Theor. Phys. 64: 473–478. Curry, R.J., Birtwell, S.W., Mairaj, A.K. et al. (2005). A study of environmental effects on the attenuation of chalcogenide optical fibre. J. Non-Cryst. Solids 351: 477–481. Miller, S.E. (1969). Integrated optics: an introduction. Bell System Tech. J. 48: 2059–2069.

295

11 Optical Properties of Organic Semiconductors Takashi Kobayashi 1,2 and Hiroyoshi Naito 1,2 1

Department of Physics and Electronics, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, Osaka, Japan The Research Institute for Molecular Electronic Devices, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, Osaka, Japan 2

CHAPTER MENU Introduction, 295 Molecular Structure of π-Conjugated Polymers, 296 Theoretical Models, 298 Absorption Spectrum, 300 Photoluminescence, 304 Non-Emissive Excited States, 306 Electron–Electron Interaction, 309 Interchain Interaction, 314 Conclusions, 320 References, 321

11.1 Introduction Organic materials have fascinating optical properties and have been extensively investigated in many research fields associated with light. Compared to inorganic materials, a great advantage of organic materials is their variety. In inorganic materials, repeating units consist of either the same atoms or a few different atoms at the most, so there is not much room to control one property of the material while keeping the rest unchanged. In organic molecules, there are an infinite number of combinations of atoms, and therefore it is possible to design organic materials to have desirable properties. In addition, molecular arrangement and its dimensionality can also be controlled. Therefore, organic materials have received considerable research attention as a model system for understanding light–material interaction. From the application viewpoint, for fabricating light-emitting devices and nonlinear optical devices, π-conjugated polymers are the most promising materials because of their good processability, high fluorescence yield, large optical nonlinearity, and ultrafast class relaxation time. In this chapter, we will present the fundamental optical properties of π-conjugated polymers, including the underlying physics, and some related optical spectroscopic techniques.

Optical Properties of Materials and Their Applications, Second Edition. Edited by Jai Singh. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

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11.2 Molecular Structure of 𝛑-Conjugated Polymers Although π-conjugated polymers are mainly composed of only carbon and hydrogen atoms, there are many kinds of polymer backbones. A few representative polymer backbones are shown in Figure 11.1. The most important feature of π-conjugated polymers is that unsaturated carbons lie on a chain, over which π electrons are delocalized one-dimensionally. The side chains provide thermal stability, solubility, hydrophilicity, and many other properties to the polymer backbones, but the fundamental optical (and electrical) properties are determined by the π electrons. In this section, we will first review the chemical structure of π-conjugated polymers, and then introduce some theoretical approaches in the next section. Carbon consists of an atomic nucleus and six electrons, and their electronic configuration is 1s2 2s2 2p2 in the ground state. To form four bonding orbitals, the carbon is first promoted to the configuration of 1s2 2s 2p3 , and then one electron in the 2s orbital and some in the 2p orbitals are hybridized. For instance, in diamond, all of the four electrons in the L-shell form four identical sp3 hybridized orbitals to bond with four carbon atoms. In π-conjugated polymers, instead of forming the sp3 hybridized orbitals, one electron in the 2s orbital and two electrons in the 2p orbitals are hybridized to three identical sp2 orbitals, which are in the same plane and are arranged at an angle of about 120∘ (see Figure 11.2a). Since they do not have angular momentum in the bonding direction, they are called σ orbitals. Each carbon atom is combined with other carbon atoms and hydrogen atoms by the σ orbitals to form the backbone of π-conjugated polymers (see Figure 11.2b). These σ orbitals have maximum amplitude in the bonding direction, so that overlap of the σ orbitals of adjacent carbon atoms is expected to be large. In fact, the σ bonds are strong and cannot be excited by photons in the visible range. On the other hand, the remaining pz orbital in the carbon atom is called the π orbital, and does not have its maximum amplitudes in the bonding direction, so that the π orbital overlaps only slightly with other π orbitals of the neighboring carbon atoms above and below the plane containing the σ bonds to form π bond(s). The π bonds are thus relatively weak. The semiconducting properties of π-conjugated polymers are (a) Polyacetylene

(b) Polydiacetylene H C

H C

(c) Polythiophene S C

C

C

C

H

C H

n

H

H C C

C

C C

C C

H

(e) Polyfluorene H

H

H C C

C

C

H H n

H

n

(d) Poly(p-phenylene vinylene) H

C

C C

C C

C C

H

H

H

C

C C

C C H

H

C

H

n

Figure 11.1 Some backbone structures of π-conjugated polymers.

n

11.2 Molecular Structure of π-Conjugated Polymers

120°

π bond

pz orbital

sp2orbitals

C

C

C

C

C

σ bond (a)

(b)

Figure 11.2 (a) sp2 hybridized orbitals and pz orbital. (b) σ and π bonds in a single chain of polyacetylene. (In this figure, hydrogen atoms and σ bonds connecting carbon and hydrogen atoms are omitted.)

determined by the electrons in the π bonds. In fact, it is possible to extract an electron from and add one into the π bond without breaking up the molecule. Usually, electrons in the σ bonds do not contribute to the semiconducting properties of the polymers. An electron in the linked π orbitals (the π bond) and one in the σ orbitals (the σ bond) are sometimes called the π electron and σ electron, respectively. Let us consider a π-conjugated polymer with an infinite polymer chain, where the linked π orbitals form the π band. Each orbital has two states, i.e., the spin-up and -down states, but supplies only one electron. Thus, one may expect that the π band is a half-filled band, and therefore π-conjugated polymers should show efficient conductivity like metals. However, in the actual polymers, the intervals between carbon atoms are alternately modified to form bond alternation in order to reduce the total energy (see Figure 11.3). This alternation splits the π band into two equal half bands separated by a band gap; the lower half is fully filled, and the upper one is completely empty. Although this bond alternation increases the elastic energy, the energy of the electrons is lowered sufficiently to compensate for the increase. Consequently, π-conjugated polymers H

H

H

H

H

H

C

C

C

C

C

C

C

C

C

C

H

H

H

H

Empty band

Band gap

Half-filled band (a)

Filled band (b)

Figure 11.3 (a) In the case where carbon atoms are equally spaced, the linked π orbitals form a half-filled band. In the above chemical structure, the solid lines are the σ bonds, and the dashed lines indicate the linked π orbitals. (The dashed lines are not π bonds, because each carbon atom has only four bonding orbitals.) (b) In the case where the intervals between carbon atoms are alternately modulated, the half-filled band splits into a filled band and an empty band separated by a band gap.

297

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11 Optical Properties of Organic Semiconductors

show the electronic properties of semiconductors or insulators. The gap energy usually corresponds to the energy of a photon in the visible light range, and thus the response of π-conjugated polymers to visible light is exclusively due to π electrons. It may be noted that the bond alternation in the polymer backbone is depicted using a single line as well as a double line, but the bond alternation does not include purely single bonds. The interatomic distances between carbon atoms are modified only slightly by the alternation. Several electron diffraction measurements have revealed that the bond lengths are modified by less than 5% from the mean length [1]. Therefore, π electrons are considered to be delocalized on the whole polymer backbone. Although an infinite polymer chain is considered here, the same scenario can be applied to finite chains and small π molecules as well. In such cases, the electric bands are represented by the molecular discrete energy levels: the bottom level of the empty (conduction) band and the top level of the filled (valence) band are represented by the lowest unoccupied molecular orbital (LUMO) and the highest occupied molecular orbital (HOMO), respectively.

11.3 Theoretical Models Although a single chain of π-conjugated polymers can be considered as a onedimensional semiconductor, its theoretical treatment is not simple because of some interactions. In inorganic materials, it is usually assumed that nuclei do not move even after excited states are created. However, in π-conjugated polymers, it is essential to take into consideration coupling between electronic states and bond orders. To do this, the tight binding Hamiltonian including the electron–phonon (or electron–lattice) interaction is used as [2] ∑ 1∑ + K(un+1 − un )2 − tn+1,n (Cn+1 Cn + Cn+ Cn+1 ), (11.1) H= 2 n n where K is the elastic constant, un is the displacement of the carbon atom at site n, and C + n and C n are electron creation and annihilation operators, respectively. The second term in Eq. (11.1) has the same form as in the tight binding model, but the transfer integral, t n+1,n , is defined to be proportional to the interval between the carbon atoms at n + 1 and n using the following relation: tn+1,n = t0 − 𝛼(un+1 − un ),

(11.2)

where 𝛼 is a parameter indicating coupling strength between an electron and nucleus, and t 0 is a constant. This model can be applied to polymers with degenerate ground state, like polyacetylene: in polyacetylene, cis, and trans forms have the same ground state energy (see Figure 11.4a). However, in π-conjugated polymers with ring structures such as polyfluorene and polythiophene, the ground state of geometrical isomers is not degenerate (see Figure 11.4b). In order to include this non-degeneracy effect in such polymers, the following transfer integral is alternatively used: tn+1,n = [1 + (−1)n 𝛿0 ] t0 − 𝛼(un+1 − un ),

(11.3)

where 𝛿 0 is a dimensionless bond alternation parameter and is introduced here to solve the non-degenerate ground states. Calculations using these Hamiltonians have

11.3 Theoretical Models

E

E

Q

Q

H

H

H

H

H

H

H

H

H

H

H

S

S

H H

(a)

H

H

H

(b)

Figure 11.4 Examples of π-conjugated polymers with (a) degenerate and (b) non-degenerate ground states.

succeeded in explaining topological charged excited states, such as solitons, polarons, bipolarons, and molecular deformations due to photoexcitations, which are discussed in later sections. It is known that the electron–electron interaction is also essential in understanding the electronic and optical properties of π-conjugated polymers, and it is taken into account using the following Hubbard-Peierls Hamiltonian [3]: ∑ + + tn+1,n (Cn+1,σ Cn,σ + Cn,σ Cn+1,σ ) H=− n,σ

+

∑ n

U𝜌n↓ 𝜌n↑ +

1 ∑∑ V 𝜌 𝜌 ′, 2 n≠m σ,σ′ m,n n,σ m,σ

(11.4)

where U is the on-site Hubbard repulsion (the nearest-neighbor hopping integral), V is the nearest-neighbor charge density–charge density interaction, 𝜌n,σ = C+ n,σ Cn,σ , and σ indicates spin (up or down). We do not solve this Hamiltonian in this chapter because it is still difficult to solve it for a realistic π-conjugated polymer consisting of more than a few hundred sites and containing significant structural disorders. Here, we would only like to stress that electron–phonon and electron–electron interactions play an essential role in the electronic structure of π-conjugated polymers and that all of their optical and electronic properties cannot be described within the framework of band theory or the effective mass approximation, which are very efficient theoretical approaches to understand the basic optical and electronic properties of inorganic semiconductors. However, it is also true that many similarities exist between inorganic and organic materials. For instance, according to the one-dimensional exciton theory developed for inorganic semiconductors, discrete exciton energy levels appear below the continuum state, and most of the oscillator strength concentrates on the lowest exciton level. These features are observed in π-conjugated polymers as well (see Figure 11.5). Therefore, more familiar and intuitive theories for inorganic materials can be approximately used for organic materials as long as their applicable range is taken into account. The electron–phonon interaction is taken into account throughout this chapter, but the effect of electron–electron interaction is only briefly reviewed in Section 11.7.

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11 Optical Properties of Organic Semiconductors

Continuum band

Figure 11.5 Schematic energy band structure in π-conjugated polymers.

Exciton levels

Ground state

11.4 Absorption Spectrum

1.2

0.25

1.0

0.20

0.8

0.15

0.6 0.10 0.4 0.05

0.2 0.0 1.5

2.0

2.5

3.0

3.5

4.0

Photon Energy (eV)

4.5

0.00 5.0

PC Yield (arb. units)

Absorption measurement is the most fundamental spectroscopy and is very helpful in understanding the characteristic features of π-conjugated polymers, including large electron–phonon interactions. Figure 11.6 shows the absorption spectrum of spin-coated film of polyfluorene, where a broad and featureless band is observed at 3.2 eV. This band corresponds to the transition from the ground state to the lowest excited state, and the broad width of the band results from significant inhomogeneous broadening. In π-conjugated polymers, ideally, π electrons are delocalized over the whole polymer backbone, but structural disorders, for example, chemical defects and twist of the polymer backbone, limit the delocalization of π electrons and raise the resonance energy of the polymer. Since actual polymers have a large distribution of the delocalization length of π electrons, especially in disordered films and in solution, such a broad and featureless absorption spectrum is often observed. For analyzing the absorption spectrum in π-conjugated polymers, the simple band picture without Coulomb interaction (electron–electron interaction) is not valid; this, in fact, can be confirmed from a comparison between absorption and photoconductivity (PC) yield spectra (see Figure 11.6). The PC yield indicates the probability that an absorbed photon generates a pair of charged carriers. If the Coulomb interaction is negligible in a system, a photoexcitation always produces a pair of oppositely charged carriers, which will contribute to a photocurrent in the system. On the other hand, in a system with electron–electron interaction, the excited pairs of charge carriers can form excitons due to their Coulomb interaction, and then an excess energy will be necessary to separate such a pair from each other. This required excess energy can be estimated

Absorption (arb. units)

300

Figure 11.6 Absorption and photoconductivity (PC) yield spectra of spin-coated film of a polyfluorene derivative. The exciton binding energy can be estimated from the difference between the onsets of absorption and PC yield spectra.

11.4 Absorption Spectrum

from the photon energy difference between the onsets of absorption and PC yield spectra and is called the “exciton binding energy.” The exciton binding energy of the spin-coated film of a polyfluorene derivative is estimated to be about 0.1 eV from the comparison between absorption and PC yield spectra shown in Figure 11.6. In many other π-conjugated polymers, similar exciton binding energies have been reported [4, 5]. Although the band in the PC yield spectrum in Figure 11.6 corresponds to the continuum state shown in Figure 11.5, it is difficult to recognize the corresponding band in the absorption spectrum. This is because most of the oscillator strength concentrates on the lowest exciton level, which is a major feature of one-dimensional systems. Some π-conjugated polymers have a tendency to form an ordered packing structure. While polymer chains usually adopt a randomly twisted or random-coil conformation in disordered films and solutions, they adopt a more planar or sometimes a fully planar conformation(s) in ordered films. An improvement in the chain planarity increases the delocalization length of the π electrons, which reduces the resonance energy and suppresses inhomogeneous broadening. Although ordered films of a polyfluorene derivative can be prepared [6, 7], we here show the results of a polythiophene derivative and its isomer; in these, identical side chains are attached to the polythiophene backbones in two different ways [8]. In the solid state, the side chains attached in one way promote crystallization to form an ordered packing structure, whereas the chains attached in the other way prevent the crystallization. Figure 11.7 shows the absorption spectra of ordered and disordered films prepared by spin-coating from their chloroform solutions. The absorption spectrum of the disordered film is almost the same as that of its chloroform solution. The difference between the two absorption spectra is basically attributed to the difference in the chain planarity. The influence of the interchain interaction is not significant and is discussed in Section 11.8. The absorption spectrum of the ordered film is red-shifted with respect to that in solution and show a well-resolved vibronic structure, which consists of several bands with the same energy interval and is due to the electron–phonon interaction. In order to explain this vibronic structure, we have Disordered

Absorbance

Ordered

H2 C

S

2.0

CH3 O C H2

C H2

*

C H2

CH3

n

2.5 3.0 Photon Energy (eV)

3.5

Figure 11.7 Absorption spectra of ordered and disordered films of a polythiophene derivative. The inset is the chemical structure of the derivative. Source: Reprinted Figure 1 with permission from T. Kobayashi et al., Phys. Rev. B 62, 8580. Copyright (2000) The American Physical Society.

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11 Optical Properties of Organic Semiconductors

E

E

E

e

e

ћω0

Abs.

g

g Q

Q ∆Q (a)

PL E

302

(b)

Figure 11.8 Schematic illustration of vibronic potential energy curves for (a) absorption and (b) photoluminescence processes. In this figure, e and g indicate electronic excited and ground states. ℏ𝜔0 is the phonon energy of the associated mode, and ΔQ is the difference between the potential minima of the g and e curves. If ΔQ is zero, i.e. the case of no electron–phonon interaction, the transitions to the higher vibrational levels are forbidden.

schematically illustrated the energy potential curves in Figure 11.8, where a few vibrational energy levels of a phonon mode are indicated by the horizontal lines, and the potential minimum of the excited state is slightly shifted from that of the ground state. The magnitude of this shift, ΔQ, represents the strength of electron–phonon interaction in the system. In the system with nonzero ΔQ, transitions from the zero vibrational level in the ground state to the excited vibrational levels in the excited state are allowed, and discrete transition bands, i.e. a vibronic structure, appear in the absorption spectrum. The transition to the zero vibrational level in the excited state is called a “0-0 transition” and corresponds to a purely electronic transition. On the other hand, the other transitions are called “0-1 transition,” “0-2 transition,” etc. After a photoexcitation corresponding to a 0-n transition, an electronic excited state is created, and n phonons are emitted. In many π-conjugated polymers, the associated phonon mode is a C=C stretching mode with a phonon energy of around 0.18 eV. Such a vibronic structure is always observed in their ordered films. Although π-conjugated polymers have many phonon modes, most of their phonon energies are much less than 0.18 eV, and hence their contributions can be included in inhomogeneous broadening. The vibronic structure can be simulated by taking into account the electron–phonon interaction and the associated phonon mode [9]. In a system without electron–phonon interactions, the transition matrix element can be calculated simply from m=

∫

Ψ0e (r)∗ (−er)Ψ0g (r)dr,

(11.5)

where Ψ0 g and Ψ0 e are electronic wave functions for the ground and excited states, respectively, and r is the electronic coordinate. However, in a system with significant electron–phonon interactions, the electronic state is influenced by the displacement of atoms, which can be taken into account by adding the following term as a perturbation in the Hamiltonian: Hint (r, Q) = −u(r) Q,

(11.6)

11.4 Absorption Spectrum

where Q is the generalized coordinate for the associated phonon mode. In Eq. (11.6), we neglect the higher term of Q for simplicity. The perturbed wave function is then written as Φin (r, Q) = Ψi (r, Q) ⋅ 𝜉in (Q),

(11.7)

where i and n indicate the electronic state and vibrational level, respectively, and 𝜉 in describes the vibrational wave function of atoms. This perturbed wave function is referred to as the “vibronic wave function.” Using this vibronic wave function, the transition matrix element of transitions from the ground vibrational level of the ground state to the nth vibrational level of the excited state results in M0n =

∫

=

∫

Φg0 (r, Q)∗ (−er)Φen (r, Q − ΔQ)drdQ Ψg (r)∗ (−er)Ψe (r)dr ×

=m×

∫

∫

𝜉g0 (Q)∗ 𝜉gn (Q − ΔQ)dQ

𝜉g0 (Q)∗ 𝜉gn (Q − ΔQ)dQ,

(11.8)

where we focus on the transition only from the zero vibrational level in the ground state. In Eq. (11.8), the first factor is the same as that given in Eq. (11.5), which involves only transitions between electronic states. The second factor as an integral in Eq. (11.8), which depends on the vibrational levels, determines the vibronic structure. Since the absorption (and photoluminescence (PL)) intensity is proportional to the square of the transition matrix element, the following formula is more practical: | |2 F0n = || 𝜉g0 (Q)∗ 𝜉gn (Q − ΔQ)dQ|| . |∫ |

(11.9)

Equation (11.9) is referred to as the Franck–Condon factor and is denoted by F 0n . Using the harmonic oscillator approximation, Eq. (11.9) can be simplified into the following form: e−S Sn , (11.10) n! where the Huang–Rhys parameter, S, represents the strength of the electron–phonon interaction for an associated phonon mode, and is given by F0n (S) =

ΔQ 2 . (11.11) 2 In Figure 11.9, we show some vibronic structures calculated using Eq. (11.10) for several Huang–Rhys parameters. Equation (11.10) is the same as the Poisson distribution, and always gives the maximum intensity for the 0-S transition. For instance, in a case where S = 2, the 0-2 transition has the maximum intensity. However, it is still difficult to reproduce perfectly the observed absorption spectrum of π-conjugated polymers using Eq. (11.10). This is because the spectrum has a large inhomogeneous broadening even in ordered films and contributions from higher excited states. S=

303

304

11 Optical Properties of Organic Semiconductors

Figure 11.9 Examples of vibronic structure calculated using Eq. (11.10) for several Huang–Rhys parameters, S. S = 0.5

S =1

S =2

S =5

0 1 2 3 4 5 6

11.5 Photoluminescence Many π-conjugated polymers show PL in the visible spectral range. For instance, polyfluorene, poly(p-phenylene vinylene) (PPV), and polythiophene are known to be blue, green, and red emitters, respectively. Some of their derivatives have good fluorescence yield and are expected to be used in fabricating light-emitting devices. In these materials, after excited states are created by photoexcitation, they immediately relax into the lowest excited state (Kasha’s rule) and then emit light. Therefore, their PL straightforwardly reflects the nature of the lowest excited state. In the PL spectrum, a vibronic structure also appears due to transitions from the lowest vibrational level in the excited state to vibrational levels in the ground state (see Figure 11.8b). This process can also be described using Eq. (11.10), and then only the Huang–Rhys parameter determines the vibronic structure under the harmonic oscillator approximation. Therefore, symmetrical absorption and PL spectra are expected to be observed. This symmetry is called “mirror image.” Two examples calculated using Eq. (11.10) are shown in Figure 11.10. The difference between the absorption and the PL maxima is called the Stokes shift, which results from many relaxation mechanisms of the polymer chain occurring after the photoexcitation. However, as shown in the lower figure of Figure 11.10, a vibronic structure with a large Huang–Rhys parameter could be the main reason for the larger Stokes shift. In this case, the Stokes shift is roughly estimated to be 2ℏ𝜔0 S, where ℏ𝜔0 is the associated phonon energy. After the photoexcitation, the bond alternation is slightly modified to reduce the total energy of the excited state by forming a self-trapped excited state. The structure of the self-trapped excited state in polythiophene is shown in Figure 11.11, where

11.5 Photoluminescence

Stokes shift ABS.

S = 0.5

PL

Figure 11.10 Absorption and PL spectra calculated using Eq. (11.10). The difference between their maxima is called the Stokes shift. These are ideal cases where only one excited state appears in the observed spectral range. In actual π conjugated films, the observed absorption spectrum is not entirely in agreement with one expected from the PL spectrum, because the observed absorption spectrum contains inhomogeneous broadening and contributions from higher excited states.

Stokes shift

PL

ABS.

S = 2.5

Photon Energy H

H

H

H

H

H

S S

S H

H (a)

S

H

H e– S e+

S

H

H (b)

Figure 11.11 Bond arrangements (a) in the ground state and (b) in the self-trapped excited state for polythiophene.

an excited state is depicted as localized on one thiophene unit for clarity. However, an actual excited state is still delocalized within several thiophene units, although the delocalization length is shorter than that of the π electrons in the ground state. This self-trapping starts immediately after the photoexcitation and completes within 10 ps [8]. Therefore, the observed PL is emitted from the self-trapped state, and this reduction in energy also contributes to the observed Stokes shift. In disordered films of π-conjugated polymers, the π electrons are not fully delocalized: the twist of the polymer backbone limits delocalization of the π electrons. In such cases, films should be considered as an ensemble of segments with various π-conjugation lengths. The resonance energy of a segment is lowered as its delocalization increases. Therefore, after the photoexcitation, excited states migrate from shorter segments to longer ones prior to emission of PL. This energy migration can be an additional reason for the large Stokes shift observed in disordered films. However, PL does not always originate from the longest segment in disordered films. As an excited state migrates to longer segments, it becomes more difficult to find further longer segments nearby. Within its lifetime, an excited state can only migrate to segments with a certain length of the π conjugation. When a disordered film is excited by photons of high enough energy, the PL spectrum is independent of the excitation photon energy. On the other hand, when the

305

11 Optical Properties of Organic Semiconductors

photoluminescence (arb.units)

306

13

14

15 16 17 18 wavenumber (103 cm–1)

19

20

Figure 11.12 Site-selective fluorescence measurements on a disordered PPV film. The different PL spectra were obtained by varying the excitation energy (indicated by the spikes at the high-energy side), starting at the top far from resonance and moving into resonance going down the figure. Source: Reprinted Figure 4 with permission from H. Bässler and B. Schweitzer, Acc. Chem. Res. 32, 173. Copyright (1999) The American Chemical Society.

photon energy is less than the threshold, PL shows dependence on the photon energy. In this condition, those segments to which an excited state cannot migrate from shorter segments can be directly excited, and PL from these segments can be observed. Such a measurement is called “site-selective fluorescence measurement,” and an example of such measurements is shown in Figure 11.12 [10, 11]. Another factor that increases the Stokes shift is the interchain interaction, which is discussed in Section 11.8.

11.6 Non-Emissive Excited States As in inorganic semiconductors, it is possible to generate charge carriers in π-conjugated polymers also by doping. In π-conjugated polymers, these carriers are called solitons, polarons, or bipolarons, where bond alternation is modified, and charge is delocalized within 10–30 carbon atoms [2] as illustrated in Figure 11.13. Solitons are formed in the π-conjugated polymers that have a degenerated ground state (Figure 11.8a) such as polyacetylene and polydiacetylene, whereas polarons and bipolarons are formed in those π-conjugated polymers with non-degenerated ground states (Figure 11.8b),

11.6 Non-Emissive Excited States

e+ Neutral state

Soliton

e+ Polaron Neutral state

e+ e+ Bipolaron

Figure 11.13 Charged excited states in π-conjugated polymers. In these polymers, charge induces modification of the bond alternation to form stable excited states.

such as polyfluorene and polythiophene. These carriers govern conductivity in doped π-conjugated polymers. In undoped π-conjugated polymers, solitons or polarons can be created by photoexcitations, but they decay nonradiatively within a lifetime of microseconds to milliseconds. Thus, the formation of such charged excitations serves as one of the nonradiative decay channels in some π-conjugated polymers. Generation of triplet excited states via intersystem crossing is also one of the nonradiative decay channels because phosphorescence is extremely weak in conventional π-conjugated polymers [12], which do not contain heavy metals. The lifetime of triplet excited states at a very low temperature is comparable to that of the charged states and is several orders of magnitude longer than that of singlet excitons (around nanoseconds). Such long-lived photoexcitations in π-conjugated polymers can be experimentally observed by continuous-wave (cw) photo-induced absorption (PA) measurements. In PA measurements, a pump beam creates photoexcitations, and a probe beam detects the transitions of photoexcitations to higher excited states. Cw PA measurements are usually conducted using a cw laser and a lock-in-amplifier with a mechanical chopper, so that ultrafast photoexcitations contribute little to cw PA signals, and only long-lived photoexcitations appear in the observed spectrum. In Figure 11.14, we show cw PA spectra of disordered and ordered films of poly(3-hexylthiophene) (P3HT) [13, 14]. In this work, two isomers were used to prepare the ordered and disordered films. The lower figures in Figure 11.14 are photo-induced absorption-detected magnetic resonance (PADMR) spectra [13, 14], from which it is possible to know the spin number of each PA band. In a disordered P3HT film, only a broad PA band is observed, which can be assigned to a triplet–triplet transition because the spin number is 1. In contrast, in an ordered P3HT film, the PA band with spin 1 disappears, and several PA bands with spin 1/2 appear instead. As mentioned above (see Figure 11.3), the band gap in π-conjugated polymers is opened because of bond alternation, which is locally modified after polaron formation. As a result, one localized state is separated from the bottom of the conduction band and another from the top of the valence band. This situation is illustrated in Figure 11.15. In the case of polarons with a positive charge, the lower localized state is occupied by an electron whereas the higher one is empty, and thus new two transitions become possible. These transitions are in fact observed below the band

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11 Optical Properties of Organic Semiconductors

(a)

3 3 T

2

2

1

1

105(δT/T)

103(–ΔT/T)

PA

0

0

–1

–1 Spin 1

–2

–2 PADMR

–3

–3 4

2 PA

2 DP1 0

(×1/4)

IEX

P2

1 DP2

P1

0 Spin 1/2

–2

–1

–4

104(δT/T)

(b)

103(–ΔT/T)

308

–2 PADMR

–6

0

0.5

1 1.5 Photon Energy(eV)

2

–3

Figure 11.14 (Upper) PA and (lower) PADMR spectra of (a) disordered and (b) ordered P3HT films. In the upper figures, the PA bands correspond to transitions of the long-lived excited states. The lower figures show the spin number of each PA band. Source: Reprinted Figure 5 with permission from O. J. Korovyanko et al., Phys. Rev. B 64, 235122. Copyright (2001) The American Physical Society.

P2

P1

Neutral State

Polaron

Figure 11.15 Energy structure of polarons. Solid lines in the left side correspond to the bottom of the conduction band and the top of the valence band. Polarons have two localized states. Solid arrows represent electrons (spin direction), and broken arrows indicate transitions appearing within the band gap after the polaron formation.

11.7 Electron–Electron Interaction

gap, as shown in Figure 11.14b; the corresponding PA bands are labeled P1 and P2. From the fact that these P1 and P2 bands are also observed in isolated polythiophene chains in an inert polystyrene matrix, these PA bands are attributed to one-dimensional polarons. Ordered P3HT films contain the disordered and crystallized portions of the chains. One-dimensional polarons are considered to form in the disordered portions. In the crystallized portions, on the other hand, polarons delocalizing over several neighboring chains are formed. Such delocalized polarons have two localized states that are slightly shifted due to the interchain interaction, and thus exhibit one PA band lower than the P1 band and the other higher than the P2 band [14]; these bands due to the delocalized polarons are denoted by DP1 and DP2 in Figure 11.14b. Note that in addition to the PA bands with spin 1/2, a PA band with spin zero of the interchain excitons (IEX) is also observed at 1.7 eV, and is attributed to interchain excitations different from conventional singlet excited states. After all, many kinds of long-lived, non-emissive excited states may be created by photoexcitation in films of π-conjugated polymers.

11.7 Electron–Electron Interaction To properly understand the optical properties of π-conjugated polymers, it is also essential to consider the electron–electron interaction. One example showing the importance of the electron–electron interaction is the large exciton binding energy, as discussed in Section 11.4. In this section, we present another example related to the fluorescence yield, along with some experimental techniques used for investigation of the excited-state structure of π-conjugated polymers, which is influenced by the electron–electron interaction. We begin with a brief review of a basic concept of group theory that is applicable to π-conjugated polymers. As a simple model of π-conjugated polymers, let us consider a polyene, C2n H2n+2 (see Figure 11.16a). This molecular structure possesses C2h group symmetry and is invariant under the operation of space inversion at the symmetry center or rotation about the symmetry axis though 180∘ . In group theory, wave functions whose sign changes and remains unchanged after the rotation are labeled “b” and “a,” respectively. Furthermore, wave functions whose sign changes and remains unchanged after the space inversion are labeled “u” and “g,” respectively. The pz orbital has a shape represented by two touching spheres with opposite signs (see Figure 11.16a), and thus the sign of the pz orbital reverses after the space inversion. Considering this fact, the sign of π electrons consisting of 2n pz orbitals can be easily determined. Figures 11.16b and c are two examples of combinations of 2n pz orbitals having different symmetry. In Figure 11.16b, all the “+” spheres are located in the +z direction. In this case, the sign of this π electron level does not change after the rotation but changes after the space inversion. Thus, this π electron level has au symmetry. Similarly, the π electron level shown in Figure 11.16c has bg symmetry. When all combinations of six pz orbitals are examined, you will find that all π electron levels in the molecule can be classified into “au ” or “bg ” states. As illustrated in Figure 11.17, this molecule has 2n energy levels of π electrons, and therefore au and bg states appear alternately from the bottom to the top. Since each level has spin-up and -down states, this molecule has 4n states for 2n π electrons. Thus, the electronic configuration of this molecule is determined by the way that 2n π electrons are arranged to

309

310

11 Optical Properties of Organic Semiconductors

H

–

H

H

–

C

+

C

C

+

– –

C

+

+ H

H

–

–

+

C

x

C

z

H

+ H

y H

pz orbital

(a) An example of bg state

An example of au state

+ +

+ +

+

+ +

+

– +

– –

The symmetry center (b)

(c)

Figure 11.16 Symmetry of electronic states in C2n H2n+2 (in the case of n = 3). (a) The molecular structure of C2n H2n+2 with the emphasized pz orbitals. The shape of pz orbital is expressed by two touching spheres. In this figure, the touching points are in the plane containing the carbon atoms, and since the two spheres of each pz orbital above and below the plane have opposite signs, they are distinguished by “+” and “–.” (b), (c) Two examples of combinations of 2n pz orbitals. bg au LUMO

bg

HOMO

au bg au Ground Ag state Excited Bu state

Excited Ag state

Figure 11.17 Symmetry of configuration of C2n H2n+2 (in the case of n = 3). The molecule has 2n energy levels of π electrons, and in the ground state, 2n energy levels are occupied by 2n π electrons from the bottom (the left). The middle and left configurations are examples of the excited state of the molecule.

occupy 4n states. In Figure 11.17, three examples of electronic configurations are illustrated. The symmetry of a configuration can be calculated by multiplying the symmetry of 2n π electrons using the following relations: a × a = b × b = a,

g × g = u × u = g,

a × b = a × b = b,

g × u = u × g = u.

(11.12)

From this simple calculation, we find that a system with C2h symmetry has only Bu and Ag states (here, we use capital letters for the symmetry of configurations). In the ground state, 2n π electrons fill only half of the 4n states starting from the bottom, as shown in the left side of Figure 11.17. When one of the π electrons from HOMO is excited into LUMO, the symmetry of the configuration becomes Bu , as shown in the middle of Figure 11.17. Furthermore, when the excited π electron is further excited into the next lowest unoccupied level, the symmetry becomes Ag again. Therefore, the system

11.7 Electron–Electron Interaction

has an electronic energy structure as 1Ag (ground state), 1Bu , 2Ag , 2Bu ,…, nAg , and nBu appear from the lowest to the highest levels. This is always true whenever one electron theory, such as the Hartree–Fock approximation, is valid for the system. However, if electron–electron interaction is not negligible, the electronic configuration in the right side may become a more stable state than that in the middle in Figure 11.17, and then the 2Ag state will appear as the lowest excited state. Since the ground state is the 1Ag state, transitions from the 1Ag state into Bu states are one-photon allowed, and transitions into Ag states are one-photon forbidden. Thus, Ag states are absent in the conventional absorption spectrum. According to Kasha’s rule, photoexcitations into higher excited states immediately relax nonradiatively to the lowest excited state. If the lowest excited state is the 1Bu state, a PL with a radiative lifetime in the picosecond to nanosecond range would be observed. If the lowest excited state has Ag symmetry, the radiative lifetime increases dramatically, and most of the photoexcitations preferably decay nonradiatively. Therefore, the symmetry of the lowest excited state is one of the most important factors in determining the fluorescence yield. One of the experimental techniques that is suitable for a study on Ag states is the two-photon absorption (TPA) technique. When an intense laser pulse is irradiated into a sample, an excited state at an Ag state is created with a very low probability via simultaneous TPA, which is one of the nonlinear optical responses (see Figure 11.18). From the energy conservation law, the separation between the Ag state and the ground state is equal to twice the incident photon energy. In the TPA process, Bu states are forbidden. Thus, with this TPA technique, Ag states are selectively investigated. It is, however, difficult to measure the TPA cross section with a satisfactory signal-to-noise (S/N) ratio. To improve the S/N ratio, the linear dependence of the TPA cross section on the laser intensity is usually used. If a sample has a good fluorescence yield, the two-photon excitation (TPE) coefficient is alternatively measured; the TPE measurements are technically much easier than the TPA ones. In the TPE measurements, excited states are created via TPA, and then fluorescence from the lowest Bu state is measured. In this case, the square dependence of the fluorescence intensity on the laser intensity is utilized to determine the TPE coefficient with a good S/N ratio. Figure 11.19a shows the absorption and TPE spectra of thin films of a polyfluorene derivative [15]. In this figure, the 2Ag state at around 3.9 eV lies above the 1Bu (I) band in the absorption spectrum. A similar excited-state structure with the 1Bu state as the lowest excited state is observed in other luminescent π-conjugated polymers such as PPV [16]. On the other hand, Figure 11.19b shows the absorption and TPA spectra of a single crystal of a polydiacetylene derivative [17]. Polydiacetylene is known to be a non-luminescent π-conjugated polymer and has a fluorescence yield less than 10−5 [18]. In this case (Figure 11.19b), the Ag state is of lower energy than the lowest 1Bu state. Other non-luminescent π-conjugated polymers Figure 11.18 Two-photon absorption process. In the one-photon process, transition between states with the same symmetry is forbidden, and transition between states with opposite symmetries is allowed. In the two-photon process, this selection rule reverses: Transition from the ground (1Ag ) state to a higher Ag state is allowed, but in this process it is required that the energy interval between the two Ag states is equal to the sum of the photon energies of the two incident photons.

2Ag ћω

ћω 1Ag

311

300 1.0 250 0.8

I α2 (cm/GW)

200

0.6

7 Sample 1 Sample 2 6 α (cm–1) 5 4

150

0.4 0.2 0.0 2

×5

3 100

2

50 II 3 4 Photon Energy (eV) (a)

5

0 1.5

α (cm–1 × 105)

11 Optical Properties of Organic Semiconductors

Absorbance

312

1 2 Energy (2hv eV) (b)

0 2.5

Figure 11.19 (a) Squares represent the TPE spectrum of a polyfluorene derivative, whereas the solid line indicates the absorption spectrum. In this figure, “I” and “II” are major absorption bands, i.e. Bu states. Source: Reproduced with permission from S. Ikame et al., Phys. Rev. B 75, 035209. Copyright (2007) The American Physical Society. (b) The solid line is the absorption spectrum of a single crystal of a polydiacetylene derivative (right scale). The empty and filled circles are TPA coefficients obtained from two different samples, and the dashed line serves as a guide for the eye. Below 1.9 eV, the vertical scale is expanded to make clearer the two resonances below the lowest Bu state at 2.0 eV. Source: Reprinted with permission Figure 2 from B. Lawrence et al., Phys. Rev. Lett. 73, 597. Copyright (1994) The American Physical Society.

such as polyacetylene also have such an excited-state structure [19]. Thus, the absence of PL in these polymers is attributed to Ag states being below the lowest Bu state. It should be noted that although the electron–electron interaction is still important in the luminescence of π-conjugated polymers, ring structures in the polymer backbone such as phenyl and five-membered rings are known to stabilize the 1Bu state [20]. The electroabsorption (EA) measurement has also been used as an experimental method to investigate Ag states in π-conjugated polymers. In this measurement, an AC electric field with frequency f is applied to the film, and any change in the absorption coefficient at frequency 2f is detected with a lock-in amplifier. In this process, the applied electric field couples the Ag and Bu states and modifies the energy separation between the ground and excited states. This modification results in a very small absorption spectral change (the absorption coefficient changes by about 0.1% at most). As an example, we show in Figure 11.20 the EA spectrum of thin films of poly(p-phenylene ethynylene) (PPE) [21]. In the EA spectra of many π-conjugated polymers, a signal due to the red shift of the lowest absorption band appears (at around 2.5 eV in the case of Figure 11.20). This red shift is called the Stark shift and results from the strong repulsion between the 1Bu state and higher Ag states; generally, the repulsion between the 1Ag and 1Bu states does not contribute efficiently because of their large energy interval. Theoretically, the EA spectrum is described by the third-order nonlinear susceptibility, which is calculated using the third-order perturbation theory and is proportional to the product of four transition dipole moments [22]. In the two-level model, the dipole transition moment between the two states has to be used four times to calculate the third-order nonlinear susceptibility (see Figure 11.21a) and consequently

11.7 Electron–Electron Interaction

0.4

3

0.2 2 0

10–4 (ΔT/T)

Figure 11.20 Example of EA spectrum of π-conjugated polymers. The solid and dashed lines are the experimental and theoretical ones, respectively. The inset shows in more detail the EA feature at high photon energy. Source: Reproduction with permission from M. Liess et al., Phys. Rev. B 56, 15712. Copyright (1997) The American Physical Society.

1

–0.2 3

3.5

4

0 PPE

–1 2

3 Energy [eV]

4

3.5

(a) 2

1

1Bu 1Ag

0

–1

(b) mAg 1Bu

5 10–4 (ΔT/T)

Figure 11.21 EA spectra simulated using (a) two-level model, (b) three-level model, and (c) three-level model taking into consideration asymmetric inhomogeneous broadening and the vibronic structure. In the insets, energy levels and the ways to choose four transition dipole moments to simulate the EA spectra are depicted. In (c), an example of asymmetric inhomogeneous broadening is also illustrated. Source: Reprinted Figure 6 with permission from M. Liess et al., Phys. Rev. B 56, 15712. Copyright (1997) The American Physical Society.

2.5

1Ag 0

–5

(c)

Energy shift distribution

2

mAg 1Bu

1

1Ag 0

–1 2.0

2.5

3.0 3.5 Energy [eV]

4.0

313

314

11 Optical Properties of Organic Semiconductors

a blue shift in the absorption band is always anticipated. In a three-level model, two kinds of transition dipole moments can be used (see Figure 11.21b), and a term using both transition dipole moments two times each also contributes to the third-order nonlinear susceptibility. This term, which explains the observed red shift, becomes dominant in actual π-conjugated polymers where the energy difference between the 1Ag and 1Bu states is much larger than that between the 1Bu and mAg states. As shown in Figure 11.21b, the highest level (the mAg state) can be found as a small positive peak at 3.3 eV. When inhomogeneous broadening and vibronic replicas are taken into consideration in the simulation using the asymmetric Gaussian function and the Franck–Condon factor, respectively, the experimental EA spectrum of π-conjugated polymers can be reproduced well (see the dashed lines in Figure 11.20) [21]. By fitting the simulated spectrum to experimental one, it is possible to determine the energy levels essential to describe the nonlinear property and dipole moments between these levels.

11.8 Interchain Interaction Some π-conjugated polymers form a well-ordered packing structure in the solid state as mentioned above (see Figure 11.7). In such films, inhomogeneous broadening is suppressed, and their optical properties are expected to be changed due to the interchain interaction. Evidence of the interchain interaction presented so far in this chapter is the transitions due to delocalized polarons (indicated by DP1 and DP2 in Figure 11.14), which are charged states delocalizing among several polymer chains [13]. The influence of the interchain interaction can also be seen in more fundamental optical properties, such as absorption and PL properties. Before discussing the interchain interaction in π-conjugated polymers, we briefly review a theory developed for aggregates or single crystals of small molecules to get a basic picture of the intermolecular interaction, which will be helpful in understanding the interchain interaction in π-conjugated polymers. Each organic molecule has many electronic energy levels that are determined from the chemical structure. Most of these original characters are preserved even in the presence of the intermolecular interactions. Therefore, we can describe the effect of intermolecular interaction using perturbation theory. At first, we consider a simple system consisting of two identical molecules (see Figure 11.22). When the molecules are close enough, each energy level of the isolated molecules split into two levels due to intermolecular interaction. The energy of the interacting system can be written as Esystem = Emolecule + ΔD ± ΔE,

(11.13)

where Emolecule is the energy of the isolated molecule, ΔE is the splitting energy due to the intermolecular interaction, and ΔD indicates an energy shift due to other effects, for example, the effects described by the dielectric constant of the surrounding medium. Generally, ΔD cannot be estimated quantitatively and is treated as a negative constant. If the intermolecular interaction is described by the dipole-dipole interaction, we obtain (M ⋅ r)(M2 ⋅ r) M ⋅M , (11.14) ΔE = 1 3 2 − 3 1 r r5

11.8 Interchain Interaction

2∆E Emolecular

Emolecular+∆D

Ground state Interacting molecules

Isolated molecule (a)

(b)

+

= 0

+

=

+

=

+

=

+

= 0

+

=

(c)

(d)

Figure 11.22 (a) Intermolecular interaction and the resultant energy splitting. On the left are shown the energy levels of an isolated molecule, and on the right are the energy levels of the interacting two molecules. 2ΔE is the splitting energy due to the intermolecular interaction, and ΔD is the energy shift due to other effects. (b)–(d) Molecular arrangements and dipole moments. The ellipses and the inside arrows indicate the interactive molecules and their transition dipole moments. Below the ellipses, their energy levels are depicted. In (b) and (c), the upper and lower energy levels of the split branches, respectively, do not have any transition dipole moment because the two transition dipole moments of individual molecules cancel each other, and only one band is observed in their absorption spectra. On the other hand, in (d), both branches have some transition dipole moments, and two absorption bands would be observed.

where M1 and M2 are transition dipole moments of the molecules, and |r| = r is the distance from one molecule to the other. From this equation, we find that ΔE depends not only on the distance between the two molecules but also on the relative angle between them. Three arrangements of a pair of molecules and their energy diagrams are illustrated in Figures 11.22b–d. In Figure 11.22b, the molecules are in a line, in (c) they are aligned parallel to each other, and in (d) they are in a non-parallel arrangement. In part (b), the lower branch corresponds to a parallel arrangement of two dipole moments, and the higher branch corresponds to an antiparallel arrangement. Since the transition dipole moments are vectors, they cancel each other in the antiparallel arrangement, and then the parallel arrangement has twice the oscillator strength of the isolated molecule. Consequently, in part (b), only the lower branch is observed in the absorption spectrum. In part (c), the antiparallel arrangement appears in the lower branch, and the parallel arrangement appears in the higher branch. Therefore, in this case, the lower branch has no transition dipole moment. In a non-parallel arrangement as shown in Figure 11.22d, the splitting energy may become smaller than the former two cases, and both branches have nonzero transition dipole moments. This means that two branches would be found

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11 Optical Properties of Organic Semiconductors

in the absorption spectrum. The magnitude and direction of the total transition dipole moment of two interacting molecules are calculated by the linear combination of each transition dipole moment. This simple model can also be applied to molecular aggregates where more than two molecules are interacting. When the molecules align in a head-to-tail manner as in Figure 11.22b, an energy band is formed, and the oscillator strength of the system concentrates on the lowest energy level. In such a case, excited states (Frenkel excitons) are delocalized over several molecules and become less sensitive to a slight misalignment of the molecules and local vibrations. This effect is called the motional narrowing effect [23] and reduces inhomogeneous broadening and suppresses the vibronic structure. As a result of the aggregation, a narrow absorption band red-shifted with respect to that of the isolated molecules is observed. This type of aggregate is called J-aggregates. On the other hand, aggregates with a side-by-side molecular arrangement as in Figure 11.22c are called H-aggregates and exhibit an absorption band that is relatively blue-shifted. Frequently, ΔE > ΔD, so that the observed absorption band is blue-shifted with respect to that of the isolated molecule. Another feature of H-aggregate is a suppressed fluorescence yield. In H-aggregates, most of the excited states preferably decay through nonradiative channels, such as by phonon emission, because of the zero transition dipole moment of the lowest excited state. In actual H-aggregates, misalignment of the molecules weakly induces PL from the lowest excited state. In such a case, the observed Stokes shift is equal to 2ΔE. In contrast, in J-aggregates, an enhanced transition dipole moment of the lowest excited state increases the fluorescence yield, and the Stokes shift ideally becomes zero. For a more detailed study on J-aggregates, readers may refer to Kobayashi’s book [24] and a review by Mobius [25]. In this chapter, we present two recent works as examples. The first one is reported by Liess et al. [26], who have synthesized three merocyanine dyes that have the same chromophore backbone but different substituents. Since these merocyanine dyes in dilute solution exhibit very similar absorption spectra, it is confirmed that the substituents do not directly alter their optical properties. However, the substituents significantly influence the packing structures of the molecules, and consequently they exhibit entirely different absorption spectra in the solid state. Interestingly, two of these absorption spectra are typical to J- and H-aggregates. In fact, the red-shifted one is due to the J-aggregate formation, and the blue-shifted one is due to the H-aggregate formation (see Figure 11.23). It should be noted that in the J-aggregates, the molecules are slightly tilted alternately to form a zigzag conformation so that the spectral shift, i.e. ΔE, is relatively small compared to that in the H-aggregates counterpart. The second example is of J- and H-aggregate formations of a nitroazo dye (see Figure 11.24) [27]. When the dye is thermally evaporated onto a glass substrate, an absorption spectrum that can be explained in terms of H-aggregation is observed. Tanaka et al. [27] have also deposited the dye onto a one-dimensionally aligned polytetrafluoroethylene (PTFE) thin layer, which can be prepared by thermal evaporation of PTFE and then rubbing with a cloth or by a friction-transfer method. On the aligned PTFE thin layer, the dye not only aligns macroscopically but also forms J-aggregates. The dye probably has the potential to form two different packing structures in the solid state, and one may be more stable than the other depending on the surface energy and/or the surface structure. These two examples clearly show that intermolecular interaction has a strong impact on the absorption spectrum.

11.8 Interchain Interaction

H

M

500

600

J

λ (nm)

700

Figure 11.23 Absorption spectra of substituted merocyanine dyes in solution (M; monomer) and in solid state (H- and J-aggregates). Double wall arrows represent the dipole moment of the molecule and the molecular arrangements. Source: Reprinted the table-of-contents graphic with permission from A. Liess et al., Nano Lett. 17, 1719. Copyright (2017) The American Chemical Society.

(a)

(b)

(c)

Absorbance (arb. unit)

Figure 11.24 Absorption spectrum of (a) H-aggregates of a nitroazo dye deposited on a glass substrate, (b) the isolated dye in solution, and (c) J-aggregates of the dye formed on an aligned PTFE thin layer. Since the J-aggregates are highly oriented, the spectrum was taken with polarized light. Source: Reprinted Figure 4 with permission from T. Tanaka et al., Langmuir 32, 4710. Copyright (2016) The American Chemical Society.

0 300

400

500 600 Wavelength (nm)

700

800

It should be noted that the concept shown in Figure 11.22 can also be applied to single crystals of small molecules. According to the theory of Frenkel excitons, all unit cells have the same transition dipole moment in exciton levels with k = 0, which is the essential condition for optical absorption. First, we consider the case where a unit cell contains only a single molecule. If the intermolecular interaction between two neighboring molecules aligning in a head-to-tail manner is dominant, the level with k = 0 appears at the bottom of the exciton band. This situation is similar to that illustrated in Figure 11.22b. On the other hand, if the interaction in the face-to-face direction is dominant, the level with k = 0 appears at the top of the exciton band, and the transition dipole moment at the bottom is completely canceled out, as expected from Figure 11.22c. In the case where two equivalent molecules exist in a unit cell, the two energy levels with k = 0 would have nonzero transition dipole moments, as illustrated in Figure 11.22d. In fact, two absorption bands separated by the intermolecular interaction energy ΔE are observed in some molecular crystals. This separation in molecular crystals is called Davydov splitting [28]. In the case of anthracene, for example, the splitting energies of

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11 Optical Properties of Organic Semiconductors

the lowest excited state with a moderate transition dipole moment and the third lowest excited state with a huge transition dipole moment are about 0.03 and 2 eV, respectively [29]. Since the two branches usually have the transition dipole moments with different directions, the relative intensity of the two absorption bands changes depending on the crystal plane. Can this concept be straightforwardly applied to ordered films of π-conjugated polymers? Some conjugated polymers form well-ordered packing structures in the solid state and exhibit clear X-ray diffraction patterns, which indicate that the polymer chains are aligned in a side-by-side manner as shown in Figure 11.22c. π-conjugated polymers usually have a large transition dipole moment between the ground state and the lowest excited state. Therefore, if the interchain interaction is large, it is expected that ordered samples will exhibit optical properties similar to those of the H-aggregates of small molecules. However, the actual ordered films of a polyfluorene derivative, for example, exhibit good fluorescence yield and a very small Stokes shift [6], which must be equal to 2ΔE if H-aggregates were formed. In the case of polythiophene derivatives, their ordered films have suppressed fluorescence yield in the solid state, but show red-shifted absorption spectra and small Stokes shifts with respect to those in solution (see Figure 11.25a). These features are apparently inconsistent with the expectations based on the analogy of the H-aggregates of small molecules. Because of these inconsistencies, the interchain interaction in π-conjugated polymers has remained controversial for many years. In the case of π-conjugated polymers that can form ordered packing structures in the solid state, the chain planarity is significantly improved during the solidification. The improvement in the chain planarity induces a large spectral red shift because of elongation of the delocalization length of the π-electrons (excitons). Therefore, the effect of the interchain interaction cannot be identified from a simple comparison between solution and solid-state samples. This is in striking contrast to aggregates of small molecules, in which the differences in optical properties between solid-state and solution samples can be ascribed to intermolecular interaction. For small molecules, ΔE is given in Eq. (11.14), which is derived from the point dipole approximation. This approximation is valid only when the length of a molecule (or its wave function) is sufficiently short compared to the distance between the two molecules, which is obviously invalid for π-conjugated polymers. From recent studies by Spano et al. [30, 31], it has been revealed that the interchain interaction in π-conjugated polymers is either weak or negligible depending on the interchain distance and the delocalization length of π electrons. Along with the interchain interaction, a topic that has been discussed for a long time is the PL spectral shape (a vibronic structure) of ordered polythiophene films (Figure 11.25a), whose 0-0 transition is much smaller than that expected from the simplified Franck–Condon factor, i.e. Eq. (11.10). In contrast, the PL spectral shape of ordered polyfluorene films can be perfectly reproduced using Eq. (11.10) [7]. Spano et al. [30, 31] have developed a theoretical model that considers interchain (excitonic) and electron–phonon interactions to understand the interchain interaction and the PL spectral shape of ordered films of π-conjugated polymers. Using the theoretical model, it is found that the vibronic structure gets slightly modified when the interchain interaction is in the weak regime. On the other hand, when the interchain interaction is strong enough, vibronic replicas should disappear because of the motional narrowing effect, as observed in the J- and H-aggregates of small molecules (see Figure 11.23, for example). The theoretical model also shows that the change in the vibronic structure

PL (a.u.)

Figure 11.25 (a) PL and absorption spectra of an ordered P3HT film measured at 6 K. The dashed line represents an absorption spectrum of its chloroform solution at room temperature. (b) PL and excitation spectra of ordered thin films of low-MW P3HT at 6 K. The dashed and solid lines indicate the results of Form I and II, respectively. Note that MWs of P3HT used were (a) 13 300 and (b) 3000. Source: Reproduced with permission from T. Kobayashi et al., Nanoscale Res. Lett. 12, 368. Copyright (2017) Springer.

Absorbance (a.u.)

11.8 Interchain Interaction

1.5

2.0 2.5 Photon Energy (eV)

3.0

PL (normalized)

Excitation Intensity (a.u.)

(a)

1.5

2.0 2.5 Photon Energy (eV) (b)

3.0

due to electron–phonon interaction is more noticeable in PL than in absorption. So we here discuss only the PL spectral shape. In the H-aggregates, the bottom of the exciton band appears at k = π so that an optical transition from the bottom to the electronic ground state with zero vibrational level (k = 0) is forbidden. However, transitions to the nonzero vibrational levels are allowed because the momentum is conserved through the involvement of an optical phonon mode (the Einstein model) [30, 31]. As a result, in the absence of any disorder, it is expected that H-aggregates of π-conjugated polymers will show PL without a 0-0 transition. In actual samples, misalignment of the polymer chains and residual slight twists of the polymer backbones increase the 0-0 transition in the PL spectrum. The observed lower fluorescence yield and small Stokes shift are also explained in terms of H-aggregates with the weak interchain interaction (ΔE). Recently, Kobayashi et al. [32] have reported an experimental study on the interchain interaction using polymorphic modifications of P3HT. Low-molecular-weight (MW) P3HT is known to form two different packing structures in the solid state. In one structure (Form I), which is commonly formed by high-MW P3HT, planar polymer chains stack in a face-to-face manner with a stacking (interchain) distance of 3.8 Å, and each stack is separated by around 16 Å by the inert hexyl side chains. The X-ray study has revealed that in the other structure (Form II), the stacking distance increases to 4.4 Å, whereas the separation between the stacks slightly decreases [33]. Since the chain planarity is similarly improved in both packing structures, a comparison between the two polymorphic modifications makes it possible to focus on the effects of the stacking distance in P3HT ordered samples. Kobayashi et al. [32] have succeeded in fabricating

319

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thin films that mainly consist of either Form I or II and compared their optical properties (see Figure 11.25b). The Form I sample shows a PL spectrum that is very similar to that known as the PL spectra of ordered film of conventional (i.e. high-MW) polythiophene derivatives. This means that the difference in MW is minor in the PL properties. On the other hand, the Form II sample exhibits a different PL spectrum that is blue-shifted by more than 0.1 eV with respect to that of the Form I sample and has a larger and narrower 0-0 transition. The narrow width indicates that the observed PL is not emitted from a disordered portion contained in the sample. Thus, the blue shift and the enhanced 0-0 transition can be attributed to the further weaker interchain interaction due to the larger stacking distance. The Form II sample has larger fluorescence yield and a smaller Stokes shift compared to those of the Form I counterpart. All of these differences can be consistently explained in terms of the weaker interchain interaction. In other words, the differences in optical properties of the Form I and II samples can be regarded as an experimental evidence of the interchain interaction in π-conjugated polymers. As mentioned above, ordered films of a polyfluorene derivative exhibit very efficient PL with a vibronic structure that can be reproduced with the simplified Franck–Condon factor [7]. In those films, the interchain interaction can be completely ignored. This is because the interchain distance is even larger than that in the P3HT Form II sample [34, 35], and the delocalization length of π electrons is also larger; the latter can be confirmed by the very narrow PL band observed at a very low temperature [6]. A theoretical treatment of the competition between inter- and intrachain excitations in ordered films of π-conjugated polymers can be found in Ref. [31].

11.9 Conclusions π electrons delocalizing on a one-dimensional network of carbon atoms govern the optical properties of π-conjugated polymers in the visible range. To treat the π electrons theoretically, electron–phonon and electron–electron interactions should be included in the Hamiltonian, and the interchain interaction may have to be considered for the case where the polymer chains form a well-ordered packing structure. However, it is still difficult to solve such Hamiltonians because of the huge size of the molecule and the structural disorders (twists of the polymer backbones and misalignment in the packing structure), which also have a large impact on the optical properties. The electron–phonon interaction determines the vibronic structure in the absorption and PL spectra, and the importance of the electron–electron interaction is confirmed by the large exciton binding energy as well as the appearance of a one-photon forbidden excited (Ag ) state below the lowest one-photon allowed (1Bu ) state in certain π-conjugated polymers. In ordered films, π-conjugated polymers may form H-aggregates due to the weak interchain interaction. Low fluorescence yield and a PL spectrum with a suppressed 0-0 transition are signatures of the interchain interaction. In disordered films and solutions, the polymer backbones are randomly twisted. Such backbones should be treated as ensemble of segments with various delocalization lengths of π electrons and thus exhibit different optical properties. Consequently, an improvement in the chain planarity suppresses the inhomogeneous broadening observed in absorption and PL spectra. Along with these fundamental optical properties of π-conjugated polymers, some associated experimental techniques are described in this chapter.

References

References 1 Drenth, W. and Wiebenga, E.H. (1955). Acta Crystallogr. 8: 755. 2 Heeger, A.J., Kivelson, S., Schrieffer, J.R., and Su, W.-P. (1988). Rev. Mod. Phys. 60:

781. 3 Baeriswyl, D., Campbell, D.K., and Mazumdar, S. (1992). An overview of the theory

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

of π-conjugated polymers. In: Conjugated Conducting Polymers (ed. G. Kiess), 7–133. Berlin: Springer-Verlag. Pakbaz, K., Lee, C.H., Heeger, A.J. et al. (1994). Synth. Met. 64: 295. Köhler, A., dos Santos, D.A., Beljonne, D. et al. (1998). Nature 392: 903. Ariu, M., Lidzey, D.G., Sims, M. et al. (2002). J. Phys. Condens. Matter 14: 9975. Asada, K., Kobayashi, T., and Naito, H. (2006). Jpn. J. Appl. Phys. 45: L247. Kobayashi, T., Hamazaki, J., Arakawa, M. et al. (2000). Phys. Rev. B 62: 8580. Henderson, B. and Imbusch, G.F. (1989). Optical Spectroscopy of Inorganic Solids. Oxford: Clarendon Press. Heun, S., Mahrt, R.F., Greiner, A. et al. (1993). J. Phys. Condens. Matter 5: 247. Bässler, H. and Schweitzer, B. (1999). Acc. Chem. Res. 32: 173. Monkman, A.P., Burrows, H.D., Hartwell, L.J. et al. (2001). Phys. Rev. Lett. 86: 13583. Korovyanko, O.J., Österbacka, R., Jiang, X.M. et al. (2001). Phys. Rev. B 64: 235112. Österbacka, R., An, C.P., Jiang, X.M., and Vardeny, Z.V. (2000). Science 287: 839. Ikame, S., Kobayashi, T., Murakami, S., and Naito, H. (2007). Phys. Rev. B 75: 035209. Frolov, S.V., Bao, Z., Wohlgenannt, M., and Vardeny, Z.V. (2000). Phys. Rev. Lett. 85: 2196. Lawrence, B., Torruellas, W.E., Cha, M. et al. (1994). Phys. Rev. Lett. 73: 597. Soos, Z.G., Etemad, S., Galvao, D.S., and Ramasesha, S. (1992). Chem. Phys. Lett. 194: 341. Orlandi, G., Zerbetto, F., and Zgierski, M.Z. (1991). Chem. Rev. 91: 867. Barford, W. (2005). Electronic and Optical Properties of Conjugated Polymers. Oxford: Clarendon Press. Liess, M., Jeglinski, S., Vardeny, Z.V. et al. (1997). Phys. Rev. B 56: 15712. Boyd, R.W. (1992). Nonlinear Optics. San Diego: Academic Press. Knapp, E.W. (1984). Chem. Phys. 85: 73. Kobayashi, T. (ed.) (1996). J-aggregates. Singapore: World Scientific. Mobius, D. (1995). Adv. Mater. 7: 437. Liess, A., Lv, A., Arjona-Esteban, A. et al. (2017). Nano Lett. 17: 1719. Tanaka, T., Ishitobi, M., Aoyama, T., and Matsumoto, S. (2016). Langmuir 32: 4710. Davydov, A.S. (1971). Theory of Molecular Excitons. New York: McGraw-Hill. Schwoerer, M. and Wolf, H.C. (2007). Organic Molecular Solids. Weinheim: Wiley-VCH. Spano, F.C. and Silva, C. (2014). Annu. Rev. Phys. Chem. 65: 477. Spano, F.C. (2006). Annu. Rev. Phys. Chem. 57: 217. Kobayashi, T., Kinoshita, K., Niwa, A. et al. (2017). Nanoscale Res. Lett. 12: 368. Prosa, T.J., Winokur, M.J., and McCullough, R.D. (1996). Macromolecules 29: 3654. Chen, S.H., Chou, H.L., Su, A.C., and Chen, S.A. (2004). Macromolecules 37: 6833. Surin, M., Hennebicq, E., Ego, C. et al. (2004). Chem. Mater. 16: 994.

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12 Organic Semiconductors and Applications Furong Zhu Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China

CHAPTER MENU Introduction, 323 Anode Modification for Enhanced OLED Performance, 327 Flexible OLEDs, 345 Solution-Processable High-Performing OLEDs, 353 Conclusions, 368 References, 369

12.1 Introduction Silicon-based transistors and integrated circuits are of central importance in the current microelectronics industry, which serves as an engine to drive progress in today’s electronics technology. However, there is a great need for significant new advances in the rapidly expanding field of electronics. New materials and innovative technologies are predicted to lead to developments beyond anything we can imagine today. The demand for more user-friendly electronics is propelling efforts to produce head-worn and hand-held devices that are flexible, lighter, more cost-effective, and more environmentally benign than those presently available. Electronic systems that use organic semiconductor materials offer an enabling technology base. This technology has significant advantages over the current silicon-based technology because it allows for an astonishing amount of electronic complexity to be integrated onto lightweight, flexible substrates for the production of a wide range of entertainment, wireless, wearable-computing, and network-edge devices. For example, organic light-emitting devices (OLEDs) that use a stack of organic semiconductor layers have the potential to replace liquid crystal displays (LCDs) as the dominant flat panel display device. This is because OLEDs have high visibility by self-luminescence, do not require backlighting, and can be fabricated into lightweight, thin, and flexible displays. A combination of organic electronics and existing microelectronics technologies also opens a new world of potential for electronics. The possible uses include a wide variety of industrial, medical, military, and other consumer-oriented applications.

Optical Properties of Materials and Their Applications, Second Edition. Edited by Jai Singh. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

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Cathode Electron Transport Layer Emissive Layer Hole Transport Layer

Transparent Cathode Electron Transport Layer Emissive Layer

ITO Hole Transport Layer Glass Substrate Anode Si Substrate (a)

(b)

Figure 12.1 (a) A conventional bottom-emitting OLED is made on a transparent substrate, e.g. a glass or clear plastic substrate, and (b) a top-emitting OLED (TOLED) requires a semitransparent top cathode. A TOLED can be made on both transparent and opaque substrates.

12.1.1

Device Architecture and Operation Principle

A typical OLED is constructed by placing a stack of organic electroluminescent and/or phosphorescent materials between a cathode layer that can inject electrons, and an anode layer that can inject “holes.” Polymeric electroluminescent materials have been used in OLEDs, and are referred to as polymer light-emitting devices. A conventional OLED has a bottom-emitting structure, which includes a metal or metal alloy cathode, and a transparent anode on a transparent substrate, enabling light to be emitted from the bottom of the structure (Figure 12.1a). An OLED may also have a top-emitting structure, which is formed on either an opaque substrate or a transparent substrate. A top-emitting OLED (TOLED) has a relatively transparent top cathode so that light can be emitted from the side of the top electrode (Figure 12.1b). Figure 12.1 shows a cross-sectional view of (a) a typical bottom-emitting OLED and (b) a top-emitting OLED. In a multilayered OLED, the organic medium consists of a hole-transporting layer (HTL), a light emissive layer (EML) and an electron-transporting layer (ETL). Indium tin oxide (ITO) is often used as the transparent anode due to its high optical transparency and electric conductivity. The cathode in the OLEDs is made of low-work-function metals or their alloys, e.g. MgAg, Ca, LiAl, etc. A schematic energy level diagram for an OLED under bias is shown in Figure 12.2. When a voltage of proper polarity is applied between the cathode and anode, “holes” injected from the anode and electrons injected from the cathode combine radiatively to release energy as light, thereby producing electroluminescence. The light usually escapes through the transparent substrate. Different organic semiconductor functional layers in an OLED can be optimized separately for carrier transport and luminescence. A large number of conducting small molecular materials and conjugated polymers have been used as the charge-transporting or emissive layer in OLEDs. The molecular

12.1 Introduction

Figure 12.2 Schematic energy level diagram of an OLED, showing the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of the hole-transporting layer (HTL) and electron-transporting layer (ETL), which are also referred to as affinitive energy level (EA) and ionization potential (IP). ΦITO and Φm represent the work function of the ITO anode and metallic cathode. ΔEh and ΔEe are the barrier heights at the ITO/organic and organic/cathode interfaces. The hole electron current balance in an OLED is set by the size of the barriers at the two electrodes.

m

vacuu

EA

Φm

O

LUM

ΦITO

∆Ee

IP

O HOM ∆Eh ITO

Org

Metal

structures of some commonly used organic semiconductor materials are illustrated in Figure 12.3. A small-molecule-based OLED device has a typical configuration of glass/ITO/HTL/organic emissive layer/ETL/Mg:Ag mixture/Ag. The multilayer thin film device is fabricated using the thermal evaporation method in a vacuum system. The thermal evaporations usually start at the base pressure of 10−6 mbar or lower. The fabrication of polymeric OLEDs involves solution processes, and the devices can be made using spin-coating or inkjet printing methods. After the deposition of organic semiconductor layers, a thin electron injector is evaporated following a 150–300 nm thick Al or Ag layer to inhibit oxidation of the cathode contact. The device operation principle and the fabrication process of an organic-semi conductor-based light-emitting device are different from those known for a conventional light-emitting device made with an inorganic semiconductor. It is very difficult to form a stable organic semiconductor p-n junction, as the organic materials cannot be doped reproducibly to form p-type and n-type semiconductors. The interfaces of organic semiconductor p-n junctions are easily degraded or even destroyed by chemical reaction and/or inter-diffusion. For this reason, OLEDs are usually designed having a p-i-n configuration, where the emissive layer is nominally intrinsic, although in practice it gets automatically doped. The second major difference between an OLED and an inorganic LED is the nature of the charge carrier transport, recombination, and luminescence processes. In an inorganic semiconductor, charge transport is delocalized and described in terms of Bloch states within the single-electron band approximation. However, charge transport in amorphous organic semiconductors is characterized by the localization of electronic states to individual molecules and occurs via thermally activated hopping processes [1]. 12.1.2

Technical Challenges and Process Integration

Organic semiconductors are finding increasing use in organic electronics, including flat panel displays, organic transistors, photodetectors, and solar cells. However, processing

325

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12 Organic Semiconductors and Applications

N O O N

Al

N

N

N

O

Alq3

NPB

n n

R

PPV

R PFO

O

n

O

S

n SO3H

PEDOT

PSS

Figure 12.3 Molecular structures of some small molecular and polymeric semiconductors commonly used in OLEDs. Tris(8-hydroxyquinoline) aluminum (Alq3 ) is used as an electron-transporting and emissive layer. 𝛼-napthylphenylbiphenyl (NPB) is used as a hole transport layer (HTL). Poly(p-phenylene vinylene) (PPV) and poly(fluorine) (PFO) are typical fluorescent polymers. Poly(styrene sulfonate)-doped poly(3,4-ethylene dioxythiophene) (PEDOT) is often used as a HTL in polymeric OLEDs.

these materials with the desired uniform thin film patterns or multilayer structure is challenging. A spin-coating process is probably the simplest method of producing thin films of a few hundred angstroms, but these films are amorphous with low carrier mobility and may have pinholes. Films produced using the Langmuir–Blodgett method are typically highly ordered, but this method is best suited for thin film formation. Inkjet printing can also be used to produce patterned polymeric thin films, but such films are amorphous. Although the printable transparent conducting and functional organic semiconductor thin film technologies are still in the research stage and have not yet been used in products, the related technologies thus developed are promising for next-generation products. Each currently available thin film deposition method has advantages and disadvantages, depending on the requirements. In addition, for many

12.2 Anode Modification for Enhanced OLED Performance

applications it would be desirable to produce patterned films, but photolithographic techniques cannot be used to pattern these polymeric films. Thus, the challenge is to obtain highly ordered patterned polymeric thin films. New techniques need to be developed for depositing patterned, high-quality pinhole-free ultrathin organic films for electronics.

12.2 Anode Modification for Enhanced OLED Performance Transparent conducting oxide (TCO) thin films have widespread applications due to their unique properties of high electrical conductivity and optical transparency in the visible spectrum range. The distinctive characteristics of the TCO films have been applied in anti-static coatings, heat mirrors, solar cells [2, 3], flat panel displays [4], sensors [5], and OLEDs [6–8]. A number of materials such as ITO, tin oxide, zinc oxide and cadmium stannate are used as TCO in many optoelectronic devices. It is well known that the optical, electrical, structural, and morphological properties of TCO films have direct implications for determining and improving the device performance. The properties of TCO films are usually optimized accordingly to meet the requirements in various applications involving TCO. The light scattering effect due to the usage of textured TCO substrates shows an enhanced absorbance in thin film amorphous silicon solar cells [9, 10]. The ITO contact used in LCDs comprises a relatively rough surface in order to promote the good adhesion of subsequently coated polymeric layer on its surface. However, a rough ITO surface is detrimental for OLED applications. The high electric fields created by the rough anode can cause shorts in thin functional organic layers. 12.2.1

Low-Temperature High-Performance ITO

ITO is one of the widely used materials for TCO. Thin films of ITO can be prepared by various techniques, including thermal evaporation deposition [11, 12], direct current (dc) and radio frequency (rf ) magnetron sputtering [13, 14], electron beam evaporation [15], spray pyrolysis [16], chemical vapor deposition [17], dip-coating techniques [18, 19], and the recently developed pulsed laser deposition method [20, 21]. Among these techniques, magnetron sputtering is one of the more versatile techniques for ITO film preparation. This technique has the advantage of fabricating uniform ITO films reproducibly. Both reactive and non-reactive forms of dc/rf magnetron sputtering can be used for film preparation. ITO films prepared by the dc/rf magnetron sputtering method often require heating the substrate at an elevated temperature during the film deposition or an additional post-annealing treatment at a temperature of over 200 ∘ C. High-temperature processes for ITO preparation is unsuitable in some applications. For instance, the organic-color-filter-coated substrates for flat panel displays, flexible OLEDs made with polyester, polyethylene terephthalate (PET), and other plastic foils are not compatible with a high-temperature plasma process. Therefore, the development of high-quality ITO films with smooth surfaces, low resistivity, and high transmission over the whole visible spectrum range at low processing temperatures for flat panel displays and flexible OLEDs is quite a challenge indeed. A number of techniques have been used to

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prepare ITO films at low processing temperatures. Ma et al. have deposited ITO films on polyester thin films over a substrate temperature of 80–240 ∘ C by reactive thermal evaporation [22, 23]. Laux et al. have prepared ITO films on glass substrates at room temperature by plasma ion-assisted evaporation [24]. Wu et al. [25] have used pulsed laser ablation to fabricate ITO films on glass substrates at room temperature. ITO films prepared by the radio frequency (rf ) and direct current (dc) magnetron sputtering methods on polycarbonate [26] and glass substrates [27] at low processing temperatures are also reported. ITO films fabricated by the rf and dc magnetron sputtering methods usually require a low oxygen partial pressure in the sputtering gas mixture when both alloy and oxidized targets are used [26, 27]. In the following discussion, a comprehensive study on the morphological, electrical, and optical properties of ITO films fabricated in our laboratory by rf magnetron sputtering using hydrogen–argon mixtures at a low processing temperature will be described. The addition of hydrogen in the sputtering gas mixture affects the overall optical and electric properties of ITO films considerably. Atomic force microscopy (AFM), X-ray photoelectron spectroscopy (XPS), secondary ion mass spectroscopy (SIMS), the four-point probe technique, the Hall effect, and optical measurements were used to characterize the morphological, electrical, and optical properties of the ITO films thus made. The mechanism of ITO film quality improvement due to addition of hydrogen in the sputtering gas mixture is discussed. 12.2.1.1

Experimental Methods

Thin films of ITO were prepared by rf magnetron sputtering on microscopic glass slides using an oxidized ITO target with In2 O3 and SnO2 in a weight ratio of 9:1. The background pressure in the sputter chamber was lower than 1.0 × 10−7 Torr. The deposition rate of the films prepared by rf magnetron sputtering can be controlled by the sputtering power and the substrate temperature [14]. In this study, a fixed power density of about 1.2 W cm−2 for ITO film preparation was used. The deposition process was carried out in a hydrogen–argon gas mixture at low temperature; i.e. the substrate was not heated during and after the film deposition. The total pressure of the sputtering gas was kept constant at 3.0 × 10−3 Torr during the film preparation. The hydrogen partial pressure was varied over the range 1 × 10−5 – 2.0 × 10−5 Torr. The thickness of ITO films deposited on glass substrates for morphological, electrical, optical, and spectroscopic characterizations was maintained at the same value of about 250 nm so that the measured properties of the ITO films are comparable. The thickness of the ITO films was measured by a KLA-Tencor Alpha-Step 500 profilometer. The sheet resistance of the films was determined using a four-point probe method. The charged carrier concentration and mobility of the films were characterized by Hall effect measurements using the van der Pauw technique. Wavelength-dependent absorption and transmission of ITO films were measured by a PerkinElmer spectrophotometer over the wavelength range 0.3–2.0 μm. The surface morphology of the ITO films was investigated in a DI Dimension 3000 Atomic Force microscope. The SIMS depth profiles are acquired using a CAMECA IMS 6f ion microprobe. Cs+ primary ions with an energy of 15 keV and negatively charged secondary ions were used in the SIMS analyses. XPS measurements were performed using a VG ESCALAB 220-i electron spectrometer. The Mg-K𝛼line at 1253.6 eV was chosen as the X-ray source in the

12.2 Anode Modification for Enhanced OLED Performance

XPS measurements. The position of all XPS peaks was calibrated using C 1 s with a binding energy Eb = 284.6 eV. 12.2.1.2

Morphological Properties

The rf magnetron sputtering method to deposit ITO films on glass substrates over the hydrogen partial pressure range 0–2.0 × 10−5 Torr was used. The substrate holder was not heated, and the substrate temperature during the film preparation was observed to be less than 50 ∘ C. The influence of the hydrogen partial pressure on the surface morphological properties of the ITO films was investigated by AFM. Figure 12.4 shows a typical AFM image of ITO films prepared with (a) argon, (b) hydrogen partial pressures of 7.0 × 10−6 Torr, and (c) 2.0 × 10−5 Torr. Figure 12.4 shows the similar grainy surface of the ITO films prepared at different hydrogen pressures. The average granule size of ITO films prepared at the hydrogen pressures of 7.0 × 10−6 Torr (Figure 12.4b) and 2.0 × 10−5 Torr (Figure 12.4c) had smaller dimensions than those of ITO film sputtered with argon shown in Figure 12.4a. The decrease in granule size corresponds to an increase in film smoothness. As the thickness of ITO films prepared at different hydrogen partial pressures was maintained at the same value of about 250 nm, the possible morphological difference due to thickness variations is negligible. The results, shown in Figure 12.4, actually reflect the influence of the hydrogen partial pressure on the morphological property of ITO films. The root mean square (rms) roughness of ITO films was also estimated from AFM images measured over an area of 300 nm × 300 nm as illustrated in Figure 12.4. The corresponding rms values of ITO films prepared with (a) argon, (b) hydrogen partial pressures of 7.0 × 10−6 Torr and (c) 2.0 × 10−5 Torr are 1.44, 1.13, and 0.92 nm, respectively. It is clear that the ITO film prepared with argon gas had a higher rms roughness value than that of ITO films prepared with hydrogen–argon mixtures using the same deposition parameters. This effect of hydrogen partial pressure on film morphology could be used to produce smooth ITO film with suitable opto-electrical properties in applications that are not compatible with the high-temperature process. In the application of flexible panel displays, for example, ITO is often required to be coated on the transparent plastic substrates at low processing temperatures to avoid deformation of the plastic substrates. A smooth ITO anode is also desired in a flexible OLED with a multilayered thin film configuration. An ITO anode with a smooth surface can minimize electrical shorts in the thin functional organic layers in OLEDs that are very often in the range 100–200 nm. ITO films formed by rf magnetron sputtering at low temperatures is usually amorphous. There is no lattice matching for ITO growth on glass substrates. Under this circumstance, three-dimensional (3D) nucleation is the dominant film formation mechanism. The film grown on a glass substrate is most likely formed by the coalescence of islands from nucleation sites [28]. Sun et al. [29] have investigated the initial growth mode of ITO on glass over the substrate temperature range 20–400 ∘ C. They suggested that ITO films with an amorphous structure formed at a substrate temperature below 150 ∘ C are due to a 3D nucleated growth mode similar to the Volmer–Weber mechanism. The layer-by-layer growth mode only occurred at substrate temperatures over 200 ∘ C. In this study, the ITO films were fabricated at a low processing temperature of about 50 ∘ C. The 3D-growth mode from nucleation sites is therefore expected, although the nucleation density may be varied due to the different hydrogen partial pressures

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20 nm

(a)

300 μm 200 μm 100 μm

20 nm

(b)

300 μm 200 μm 100 μm

(c)

20 nm

330

300 μm 200 μm 100 μm

Figure 12.4 AFM images obtained over an area of 300 μm × 300 μm for ITO films grown on glass substrates prepared at a low processing temperature with (a) argon gas and hydrogen partial pressures of (b) 7.0 × 10−6 Torr and (c) 2.0 × 10−5 Torr.

12.2 Anode Modification for Enhanced OLED Performance

used. The initial growth mode may affect the ultimate properties of thin films. However, the surface morphological property of the ITO films thus prepared is also related to the energetic ion particles and reactive sputter species in the plasma atmosphere created by the rf magnetron sputtering. In such a plasma environment, the growing film is subjected to various forms of bombardment involving ions, neutral atoms, molecules, and electrons [30]. The morphological changes that occurred in ITO films prepared with hydrogen–argon mixtures, as shown in Figure 12.4, were probably due to the presence of the additional reactive hydrogen species in the sputtering atmosphere. As hydrogen was introduced to the sputtering gas mixture during the film preparation, the growing flux during the magnetron sputtering created a significant amount of energetic hydrogen species with energies over the range 10–250 eV [31]. These reactive hydrogen species could remove weakly bound oxygen from the depositing film [13]. The bombardment of the sputtering particles and hydrogen species on the depositing film could cause the reduction of the indium atoms in the ITO film [32]. The energetic hydrogen species present could also react with the growing clusters containing intermediates such as Inx Oy , adsorbed O, reduced indium atoms, and sub-oxides like In2 O. The weakly absorbed oxygen and possible reduced interstitial metal atoms in the ITO film may be removed or re-sputtered by the reactive hydrogen species in the plasma process. Therefore, the nucleation growth kinetics and surface reaction rates on ITO islands formed via the nucleation sites are altered by reactive hydrogen species. The effect of reactive hydrogen species on the depositing ITO film can reduce the size of clusters in comparison with that of the ITO film prepared with only argon gas under the same conditions. As a consequence, the addition of the hydrogen in the sputtering mixture shows a reduced effect on oxide, and ITO films deposited with hydrogen–argon mixtures exhibited a smoother morphology. 12.2.1.3

Electrical Properties

Figure 12.5 shows the dependence of the electrical property of ITO films on the hydrogen partial pressure in the gas mixture. It can be seen that the resistivity of the ITO films changed considerably over the hydrogen partial pressure range 0–2.0 × 10−5 Torr used in the film preparation. The resistivity of the ITO films decreased initially with the hydrogen partial pressure and reached its minimum value of 4.66 × 10−4 Ωcm at an optimal 6.0 Film resistivity (×10–4 Ωcm)

Figure 12.5 Resistivity of ITO films as a function of hydrogen partial pressure.

5.5

5.0

4.5

4.0 0

5

10

15

20

Hydrogen partial pressure (×10–6 Torr)

331

35

N μ

34

4.5

33 4.0

32 31

3.5

30 29

3.0 0

Carrier concentration, N (1020/cm3)

12 Organic Semiconductors and Applications

Hall mobility, μ (cm2/V sec)

332

5 10 15 20 Hydrogen partial pressure (×10–6 Torr)

Figure 12.6 Carrier mobility and concentration of ITO films as functions of hydrogen partial pressure.

hydrogen pressure of about 7.0 × 10−6 Torr. A further increase in the hydrogen partial pressure above the optimal value was shown to increase the film resistivity. The existence of a minimum resistivity was also observed in ITO films prepared at an elevated temperature of 300 ∘ C [13]. Figure 12.5 shows that the relative minimum resistivity of ITO film prepared under the optimal conditions was about 11% lower than that of the film deposited with argon gas under the same conditions. This shows that the usage of the hydrogen–argon mixture had a direct effect in improving the electrical property of ITO films fabricated by the rf magnetron sputtering method at a low processing temperature. The charge carrier mobility and concentration in ITO films were measured by the Hall effect using the van der Pauw technique. The measured carrier mobility, 𝜇, and concentration, N, in ITO films as functions of the hydrogen partial pressure are plotted in Figure 12.6. It can be seen that both 𝜇 and N are very sensitive to the hydrogen partial pressures used in the film preparation. The results in Figure 12.6 show that ITO films prepared at the optimal hydrogen partial pressure of 7.0 × 10−6 Torr, which produced ITO films with the lowest resistivity as shown in Figure 12.5, exhibited the maximum carrier concentration and minimum mobility values. As the electrical conductivity is proportional to the product of 𝜇 and N, this implies that the low resistivity of ITO films prepared at the optimal hydrogen partial pressure was due to the higher carrier concentration in the conduction mechanism. Figure 12.6 shows that ITO films fabricated at a hydrogen partial pressure of 7.0 × 10−6 Torr have a high carrier concentration of 4.59 × 1020 cm−3 . In ITO films, both tin dopants and ionized oxygen vacancy donors provide the charge carriers for conduction. The number of oxygen vacancies that provide a maximum of two electrons per oxygen vacancy plays a dominant role in determining the charge carrier density in an ITO film with high oxygen deficiency. It is also affected by the deposition conditions such as the sputtering power, substrate temperature, Sn/In composition in the target, and sputtering species in the plasma during the film preparation. Banerjee et al. [33] investigated the effect of oxygen partial pressure on conductivity of the ITO films prepared by electron beam evaporation using a hot pressed powder target with a weight ratio of In2 O3 to SnO2 of 9:1. They found that the increased film conductivity was due to an enhancement in the Hall mobility, but the carrier concentration

12.2 Anode Modification for Enhanced OLED Performance

decreased with the oxygen partial pressure. A similar correlation between oxygen partial pressure and carrier concentration in ITO films prepared by pulsed laser deposition was also observed by Kim et al. [34]. Experimental results reveal that the improved electrical properties of ITO films fabricated at the optimal oxygen partial pressure were due to the increased carrier mobility in the film. The decrease in carrier concentration was attributed to the dissipation of oxygen vacancies when oxygen was used in the gas mixture during the preparation. When the ITO films were prepared in the presence of reactive hydrogen species, however, the carrier mobility of the ITO film had a relative low value. This implies that the mechanism of improvement in the conductivity of ITO films is dependent on the deposition process. The enhancement in the conductivity of ITO films prepared at a presence of hydrogen can be attributed to a high carrier concentration in comparison with ITO films made without hydrogen in the gas mixture. The above analyses are consistent with the previous results obtained from ITO films prepared at high substrate temperatures [13]. The above analyses based on the electrical measurements and morphological results suggest that the film quality improvement was due to the presence of reactive energetic hydrogen species in the sputtering plasma when the hydrogen–argon mixture was used. The relative low film resistivity was attributed to the higher oxygen deficiency and possible good contacts between different domains, as the film was denser when it was prepared with hydrogen. Thin films of ITO with high charge carrier concentration may have some advantages for OLED applications. ITO is an n-type wide bandgap semiconductor. The Fermi level, Ef , of ITO films is located at about 0.03 eV below the conduction band minimum [35]. It has an upward surface band bending with respect to its Ef [36] due to a surface Fermi-level-pinning mechanism. Therefore, the effective barrier for hole injection at the interface of ITO HTL should include both the upward surface band bending of ITO and the band offset between the Ef of ITO and the ionization potential of HTL. It has been reported that the surface band bending of ITO decreases with the increase in the carrier concentration in ITO films [37]. Lesser surface band bending lowers the effective energy barrier for carrier injection when it is used as an anode in an OLED [38]. It was demonstrated that the electroluminescence (EL) efficiency of OLEDs fabricated with ITO having a higher carrier concentration was always higher than that of those fabricated with ITO having a lower carrier concentration [39]. The increase in EL efficiency reflects an enhanced hole injection in the device. It can be considered that the ITO anode with a high carrier concentration has a smaller surface band bending, which lowers the effective energy barrier for hole injection in OLEDs. Therefore, ITO films prepared under an optimal hydrogen partial pressure of 7.0 × 10−6 Torr by the rf magnetron sputtering method at a low processing temperature are preferable in practical applications. 12.2.1.4

Optical Properties

In parallel with the morphological and electrical analyses, the transmission spectra of the ITO films deposited at different hydrogen partial pressures were examined over the wavelength range 0.3–2.0 𝜇m. Figure 12.7 shows the wavelength-dependent transmittance, T(𝜆), of the ITO films prepared with argon gas (solid curve) and hydrogen partial pressures of 7.0 × 10−6 Torr (dashed curve) and 2.0 × 10−5 Torr (dotted curve). Except for obvious deviations in the infrared wavelength region, the T (𝜆) of the films prepared at different hydrogen partial pressures also shows a slight difference over the short wavelength range.

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12 Organic Semiconductors and Applications

100 80

T(λ) (%)

334

60 40

With argon H2 partial pressure 7.0 × 10–6 Torr

20

H2 partial pressure 2.0 × 10–5 Torr

0 500

1000 1500 Wavelength (nm)

2000

Figure 12.7 Wavelength-dependent transmittance, T(𝜆) of ITO films prepared at a low processing temperature with argon (solid curve) and hydrogen partial pressures of 7.0 × 10−6 Torr (dashed curve) and 2.0 × 10−5 Torr (dotted curve).

Figure 12.7 reveals that the short wavelength cutoff in the T(𝜆) of ITO films prepared at the optimal hydrogen partial pressure of 7.0 × 10−6 Torr shifts toward shorter wavelengths in comparison with that in the T(𝜆) of ITO films prepared with argon gas. The shift of the short wavelength cutoff in T(𝜆) is related directly to the variation in the bandgap in the ITO films. To better understand the shift of the short wavelength cutoff in T(𝜆), the wavelength-dependent absorbance, A(𝜆), of ITO films was measured to estimate their optical bandgaps. Using the absorption coefficient, 𝛼, derived from the measured A(𝜆) of the ITO films, the optical bandgap Eg can be estimated. ITO is an ionic-bound degenerated oxide semiconductor. Usually, the following relation is used to derive Eg for heavily doped oxide semiconductors [3, 13]: 𝛼 2 ≈ (h𝜈 − Eg )

(12.1)

where h𝜈 is the photon energy. Figure 12.8 shows the photon energy dependence of 𝛼 2 for ITO films prepared at different hydrogen partial pressures. Extrapolation of the linear region of the plot to 𝛼 2 at zero gives the value of Eg . It can be seen from Figure 12.8 that Eg values for ITO films prepared with argon at hydrogen pressures of 7.0 × 10−6 Torr and 2.0 × 10−5 Torr are 3.75, 3.88, and 3.73 eV, respectively. Figure 12.8 shows that ITO films prepared with argon and a hydrogen partial pressure of 2.0 × 10−6 Torr have similar optical bandgaps, as both films also have similar carrier concentrations as shown in Figure 12.6. A higher Eg value of 3.88 eV is obtained for the film prepared at the optimal hydrogen pressure of 7.0 × 10−6 Torr. The widening of the bandgap can be attributed to the increase in the carrier concentrations in ITO film prepared at the optimal hydrogen pressure. ITO is a degenerate semiconductor. The conduction band is partially filled with electrons. The Fermi level, Ef , of ITO is very close to the conduction band minimum. The Fermi level shifts upward and overlaps or even is located within the conduction band when the charge carrier concentration increases. As Ef moves to a higher energy in the conduction band, the electronic states near the conduction band minimum are fully occupied. Thus, the energy

12.2 Anode Modification for Enhanced OLED Performance

300

With argon

α2 (1012 cm–2)

H2 partial pressure 7.0 × 10–6 Torr 200

H2 partial pressure 2.0 × 10–5 Torr

100

0 3.00

3.25

3.50

3.75

4.00

4.25

4.50

Photon energy (eV)

Figure 12.8 Square of absorption coefficient, 𝛼 2 , plotted as a function of the photon energy for ITO films prepared under different conditions. Extrapolation of the straight region of plot to 𝛼 2 at zero gives the bandgap E g = 3.88 eV for ITO films prepared at the optimal hydrogen partial pressure of 7.0 × 10−6 Torr.

level of the lowest empty states in the conduction band moves to the higher energy positions as well. Therefore, electrons excited from the valence band to those available electronic states in the conduction band require higher energy. This implies that the effective increase in the energy gap of ITO films prepared at the optimal hydrogen partial pressure of 7.0 × 10−6 Torr was due to an increase in the carrier concentration. The bandgap broadening due to an increased carrier concentration in the ITO film is also known as the Moss–Burstein effect, which means that the lowest states in the conduction band are filled by excess charge carriers [3]. In this case, the increase in charge carriers was due to the increase in the oxygen vacancies in the ITO films. This analysis is in a good agreement with the results obtained from electrical measurements shown in Figure 12.6. The transmission spectrum of ITO films prepared at a hydrogen partial pressure of 7.0 × 10−6 Torr (Figure 12.7) shows a considerable decrease in the near-infrared region in comparison with that measured for ITO films prepared with argon. In this region, the free carrier absorption becomes important for the transmittance and reflectance of ITO films. The optical behavior of ITO films in the infrared region can be explained by the Drude theory for free charge carriers [3, 40]. The appreciable reduction in transmission over the infrared range for ITO films prepared at the optimal hydrogen partial pressure was due to an increased carrier concentration. Since the thickness of ITO films was maintained at the same value, the difference in T(𝜆) observed in Figure 12.5 is an indication of the variation in the refractive index of the ITO films. Bender et al. [41] have calculated the refractive index of ITO films prepared by a dc magnetron sputtering method at different oxygen partial pressures. They found that the refractive index of ITO films increases with increasing oxygen flow used in the film preparation. A similar result was also obtained by Wu et al. [25], who showed that the refractive index of ITO films decreases with increasing carrier concentration in ITO films. Generally, the index

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12 Organic Semiconductors and Applications

of refraction of an ITO film can be represented by [41] n2 = 𝜀opt −

4𝜋Ne2 m∗ 𝜔20

(12.2)

where n is the index of refraction, 𝜀opt is the high frequency permittivity, m* is the effective mass of the electron, and 𝜔0 is the frequency of the electromagnetic oscillations. ITO films prepared with hydrogen–argon mixtures in this work had high carrier concentrations in their conduction mechanism. According to Eq. (12.2), a reduction in the refractive index of ITO films would be expected due to an increase in the carrier concentration, N, in films prepared in the presence of hydrogen. As such, the refractive index of an ITO film prepared at the optimal hydrogen partial pressure of 7.0 × 10−6 Torr would have the lowest n value as it has the maximum carrier concentration, as shown in Figure 12.6. This implies that a film becomes denser as its refractive index decreases. On the basis of the information obtained from the above morphological and electrical studies on ITO films, it can be considered that the average density of the film prepared at the optimal hydrogen partial pressure of 7.0 × 10−6 Torr is higher than that of the film sputtered with argon gas. Then, on the basis of the information obtained from AFM measurements, it may be inferred that a denser ITO film will have fewer internal voids in the bulk and fewer irregularities on its surface when it is fabricated under optimal conditions. 12.2.1.5

Compositional Analysis

The chemical binding energies of In3d5/2 and Sn3d5/2 for different ITO films were examined using XPS measurements. The energy positions of In3d5/2 and Sn3d5/2 peaks measured for the films deposited at different hydrogen partial pressures were all constant at 445.2 and 487.2 eV, respectively. There were no evident shoulders observed at the high-binding-energy side of the In3d5/2 peaks, which may relate to the formation of In-OH–like bonds in the ITO films prepared in the presence of the hydrogen [42]. The same binding energy positions and almost identical symmetric XPS peak shapes of In3d5/2 and Sn3d5/2 observed from different ITO films suggest that the chemical states of indium and tin atoms in the films remained in an ITO form. The atomic concentration of ITO films prepared at different hydrogen partial pressures was also estimated. The result shows that the stoichiometry of ITO films prepared at different hydrogen pressures was very similar. The variation in the hydrogen partial pressure used in this work did not seem to affect the chemical structure of ITO films significantly. The bulk composition of ITO films was also examined by SIMS measurements. Figure 12.9 shows a typical depth profile of ITO films prepared at the optimal hydrogen partial pressure of 7.0 × 10−6 Torr. The x-axis in Figure 12.9 shows the sputter depths of films that were calculated using the product of the sputtering time and an average sputtering rate of ∼0.36 nm s−1 . The depth profile steps occurring at the ITO surface and the boundary between the ITO and the glass substrate were due to the influence of the interfacial effects. It is obvious that the profiles of the ITO elements O, In, and Sn had stable counts throughout the entire depth profile region measured by SIMS. The steady distribution of ITO elements shown in Figure 12.9 confirms that the ITO films thus prepared were very uniform.

12.2 Anode Modification for Enhanced OLED Performance

Counts/sec

106

oxygen

105 ITO deposited under optimal conditions 104 Sn

103 102

Si

In

101 0

50

100

150 200 250 Sputter depth (nm)

300

350

Figure 12.9 SIMS depth profile of typical ITO film prepared at an optimal hydrogen partial pressure of 7.0 × 10 −6 Torr.

It is well known that the tin dopants and ionized oxygen vacancy donors govern the charge carrier density in an ITO film. In order to understand the mechanism of the carrier concentration variations in different ITO films, SIMS is also used to measure the relative concentration of oxygen and tin in films prepared at different hydrogen partial pressures. In order to compare the relative oxygen and tin content in different films, the intensities of oxygen and tin ions measured by SIMS are normalized to the corresponding intensities of indium acquired in the same measurements. Figure 12.10 plots the normalized depth profiles of oxygen in ITO films prepared with (a) argon gas and the hydrogen partial pressures of (b) 7.0 × 10−6 Torr and (c) 2.0 × 10−5 Torr. It can be seen that there is a slight difference in the normalized SIMS oxygen counts measured for the films fabricated at different hydrogen partial pressures. Figure 12.10 shows clearly that the ITO film deposited with only argon gas had a relatively higher oxygen content than those measured from films prepared with hydrogen–argon mixtures. This indicates that the addition of hydrogen to the sputtering gas mixture has the effect of removing the weakly bound oxygen atoms from depositing ITO films and increases the number of oxygen vacancies. This result is similar to the effect of hydrogen on ITO films prepared at a higher processing temperature of 350 ∘ C, as reported in a previous study [13]. SIMS was also used to examine the relative concentration of tin element for ITO films prepared at different hydrogen partial pressures. The corresponding normalized tin concentration in films prepared with argon and at hydrogen partial pressures of 7.0 × 10−6 Torr and 2.0 × 10−5 Torr are plotted as curves (a), (b), and (c) in Figure 12.11, respectively. Figure 12.11 reveals that ITO films fabricated with hydrogen–argon mixtures had a slightly higher relative tin content than that those obtained from an ITO film sputtered with argon gas. In particular, the ITO film prepared at the optimal hydrogen partial pressure of 7.0 × 10−6 Torr – curve (b) in Figure 12.11 – had the highest tin content. A tiny deviation in the normalized tin concentration in different ITO films may not change the chemical structure in ITO films considerably, but it impacts the efficient doping level in the material. During the film formation, tin atoms will substitute for indium atoms in the lattice, leaving the tin atoms with one electron more than the requirement for bonding.

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12 Organic Semiconductors and Applications

5 × 106

Oxygen

Counts/sec

4 × 106

(a)

3 × 106 (b)

(c)

2 × 106 (a) With argon

(b) H2 partial pressure 7.0 × 10–6 Torr

1 × 106

(c) H2 partial pressure 2.0 × 10–5 Torr 0 0

50

100

150

200

250

300

Sputter depth (nm)

Figure 12.10 Comparison of SIMS depth profiles of normalized relative oxygen concentration in ITO films prepared at a low processing temperature with (a) pure argon gas and hydrogen partial pressures of (b) 7.0 × 10−6 Torr and (c) 2.0 × 10−5 Torr.

Sn

2500

(b)

2000 Counts/sec

338

(c)

(a) 1500

(a) With argon

1000

(b) H2 partial pressure 7.0 × 10–6 Torr (c) H2 partial pressure 2.0 × 10–5 Torr

500 0

50

100

150 200 250 Sputter depth (nm)

300

350

Figure 12.11 Comparison of SIMS depth profiles of normalized relative tin concentration in ITO films prepared at a low processing temperature with (a) pure argon gas and hydrogen partial pressures of (b) 7.0 × 10−6 Torr and (c) 2.0 × 10−5 Torr.

This electron will be free in the lattice and act as a charge carrier. Apart from an increase in the number of oxygen vacancies and hence an increase in the number of charge carriers, the substitution of indium by tin atoms also improves the film conductivity. On the basis of the measured carrier concentration results, as given in Figure 12.6, it can be inferred that an enhanced doping level was achieved in an ITO film prepared at the optimal hydrogen pressure of 7.0 × 10−6 Torr, as shown in curve (b) in Figure 12.11. ITO films fabricated by rf or the dc magnetron sputtering method usually require an annealed substrate at temperatures in the range 200–300 ∘ C when both an alloy and oxidized targets are used. A high processing temperature enhances the crystallinity in the ITO film and hence increases the carrier mobility and film conductivity. Low-processing-temperature ITO is a prerequisite for fabricating flexible OLEDs or

12.2 Anode Modification for Enhanced OLED Performance

top-emitting OLEDs that precludes the use of a high-temperature process. An ITO anode with a smooth surface can minimize electrical shorts in the thin functional organic layers in OLEDs that are very often in the range 100–200 nm. Our results demonstrated that the process thus developed offers an enabling approach to fabricate ITO films with smooth morphology and low sheet resistance on a flexible plastic substrate at a low processing temperature. As such, a process that produces smooth ITO films with high optical transparence and high electric conductivity at a low processing temperature will be of practical and technical interest. 12.2.2

Anode Modification

Much progress has been made in OLEDs since the discovery of light emission from electroluminescent polymers. It is appreciated that the physical, chemical, and electrical properties at both anode and cathode interfaces in OLEDs play important roles, though still not well understood, in determining the device operating characteristics and the stability of these devices. Different substrate treatment techniques, which may involve ultrasonic cleaning of the ITO anode in organic solutions, the exposure of the pre-cleaned ITO to either ultraviolet (UV) irradiation or oxygen plasma treatment, are used for device fabrication [43–45]. It has been reported that oxidative treatments, such as oxygen plasma or UV ozone, can effectively increase the work function of ITO [6, 46] and also form a negative surface dipole facing toward the ITO. This may then lead to enhance hole injection and increase device reliability [8, 47, 48]. Ultraviolet photoelectron spectroscopy and Kelvin probes are often employed to investigate the changes in work function or surface dipole of ITO due to different surface treatments [49, 50]. It shows that an increase in the ITO work function is closely related to the increase in its surface oxygen content due to oxidative treatment [6, 51, 52]. The effect of ITO surface treatments on improvement of OLEDs, although widely investigated, is both important and not fully understood. A better understanding of the mechanism of oxygen plasma treatment on an ITO anode in OLED performance is of practical interest. In addition to the appropriate ITO anode cleaning and treatment, it is also shown that charge injection properties in OLEDs can be modified in different ways. At the cathode side, the use of low-work-function metals [7], and the introduction of electron-transporting materials with high electron affinity [53] or a-few-hundred-angstroms-thick conducting polymer layers have been investigated [50, 54–65]. At the same time, the charge injection can also be improved at the anode side by adding an HTL [57] by various treatments of ITO [6, 8, 58, 59]. An ITO anode with self-assembled monolayers can bring significant enhancements in luminous efficiency [50, 60–62]. Modifying the electrode/organic interface using various insulating layers can also substantially enhance the electroluminescence performance of the devices [63–67]. The following discussion will focus on an understanding of anode modification for enhanced carrier injection properties. In situ four-point probe methods in conjunction with XPS and time-of-flight secondary ion spectrometry (TOF-SIMS) measurements were used to explore the relation between a bilayer ITO/insulating interlayer anode and improvement of OLED performance. It is found that oxidative treatment induces a nanometer-thick oxygen-rich layer on the ITO surface. The thickness of this ultrathin oxygen-rich layer deduced from a dual layer model is consistent with the XPS and

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12 Organic Semiconductors and Applications

TOF-SIMS measurements. The enhancement in OLED performance correlates directly with such an interlayer of low conductivity. It serves as an efficient hole-injecting anode. ITO-coated glass with a thickness of 120 nm and a sheet resistance, Rs, of about 20 Ω/square is used in the device fabrication. The ITO used for OLEDs underwent wet cleaning processes including ultrasonication in the organic solvents, followed by oxygen plasma or UV ozone treatment. In this work, wet-cleaned specimens without undergoing any further oxidative treatment are marked as “non-treated” ITO. Wet-cleaned specimens further treated by oxygen plasma under different conditions prior to OLED fabrication are marked as “treated” ITO. The experiments were carried out in a multi-chamber vacuum system equipped with an ITO sputter chamber, an oxygen plasma treatment chamber, and an evaporation chamber. The system is also connected to a glove box purged by high-purity nitrogen gas to keep oxygen and moisture levels below 1 ppm. The system allows the substrate to be transferred among different chambers without breaking the vacuum. The samples can also be moved from an evaporation chamber to the glove box without exposure to air. The four-point probe is placed inside the glove box for an in situ measurement of any changes in ITO sheet resistance, Rs , due to oxygen plasma treatment. A 15-nm-thin ITO film was also deposited on glass for treatment study. This is to boost the effect of oxygen plasma treatment on the variations in Rs for a more accurate measurement. A phenyl-substituted poly (p-phenylenevinylene) (Ph-PPV) [68] was used as an emissive polymer layer. The single-layer testing device has a configuration of ITO (100 nm)/Ph-PPV (80 nm)/Ca (5 nm)/Ag (300 nm). The change in Rs of ITO films due to different oxygen plasma treatments was monitored by the in situ four-point probe measurement. All plasma-treated ITO films were found to have a higher sheet resistance than non-treated films. The increase in sheet resistance observed in a 15-nm-thick ITO was more apparent than that in a 120-nm-thick ITO treated under the same conditions. The process for oxygen plasma treatment was optimized on the basis of the EL performance of OLEDs, and it was chosen accordingly by varying the oxygen flow rates at a constant plasma power of 100 W and a fixed exposure time of 10 minutes. 12.2.3

Electroluminescence Performance of OLEDs

Figure 12.12 shows the current density–voltage (J–V ) characteristics of a set of identical devices built on ITOs treated with wet cleaning, oxygen plasma, and UV ozone. It shows the effect of different treatments on the device performance. For ITOs treated with oxygen plasma and UV ozone, the devices generally have a similar maximum luminance of ∼50 000 cd m−2 . For a wet-cleaned device, the luminance is ∼20 000 cd m−2 . In terms of the current density, the device treated with oxygen plasma and UV ozone treatments has lower values compared to the wet-cleaned ITO. In this case, the oxygen plasma treatment was not optimized. The J–V characteristics measured for a set of identical OLEDs made on ITOs treated by oxygen plasma at different oxygen flow rates of 0, 40, 60, and 100 standard cubic centimeters per minute (sccm) are illustrated in Figure 12.13. There are obvious differences in the EL performance of OLEDs with regard to the anode treatments. For instance, at a given constant current density of 20 mA cm−2 , the luminance and efficiency of identical devices made with oxygen plasma treatments falls within the range 600–1000 cd m−2

12.2 Anode Modification for Enhanced OLED Performance

Current Density (mA/cm2)

1800 UV ozone Wet-cleaned Oxygen plasma

1500 1200 900 600 300 0 0

2

4

6

8 10 Voltage (V)

12

14

16

18

Current Density (mA/cm2)

Figure 12.12 J–V characteristics of identical devices made on wet-cleaned, oxygen plasma, and UV-ozone-treated ITO substrates.

0 nm (non-treated) 0.6 nm (40 sccm) 0.9 nm (60 sccm) 3.1 nm (100 sccm)

600

400

200

0

0

2

4

6

8

10

12

14

16

18

Operating Voltage (V)

Figure 12.13 J–V characteristics of identical devices made on ITO anodes treated under different oxygen plasma conditions.

and 5.0–11.0 cd A−1 , respectively. These values are 560 cd m−2 and 5.0 cd A−1 , respectively, for the same device fabricated on a non-treated ITO anode. The enhancement in hole injection in the device made on a treated ITO anode is clearly demonstrated.The J–V characteristics, shown in Figure 12.13, indicate that there is an optimal oxygen plasma treatment process. The best EL performance was found in the OLED made on an ITO anode treated with an oxygen flow rate of 60 sccm used in this study. It is considered that the enhancement in EL performance of OLEDs due to oxygen plasma treatment is attributed to the improvement in the ITO surface properties. The improvement in device performance is explained by a low barrier height at the ITO/polymer interface and better ITO/polymer adhesion. This includes an increased work function, an improved surface morphology [6, 69], and a less affected bulk ITO property [8]. In order to explore the correlation between the variations in Rs and the possible compositional changes on the treated ITO surfaces, the surface contents of non-treated and treated ITO films were examined using XPS and TOF-SIMS.

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12 Organic Semiconductors and Applications

For surface compositional analyses, both non-treated and treated ITO samples were covered with a 5-nm-thick lithium-fluoride (LiF) capping layer before the samples were taken out for XPS and TOF-SIMS measurements. This protective LiF layer was to prevent any possible contamination on the ITO surfaces in air and was removed by argon ion sputtering in the XPS and TOF-SIMS measurements. Therefore, the changes in Rs observed by in situ four-point probe measurements and variations in the surface contents obtained by ex situ spectroscopic analyses on ITO surfaces due to oxygen plasma are being compared. In the XPS measurements, the In3d5/2 , Sn3d5/2 , and O1s peaks of non-treated and treated ITO films were examined. The binding energies of these peaks were found to be 445.2, 487.2, and 530.3 eV, respectively, for all ITO surfaces. This implies that no major chemical changes occur on an ITO surface during the oxygen plasma treatment. However, there was a considerable increase in the ratio of O/(In+Sn) for an ITO as the oxygen flow rate was increased. For example, the ratio of O/(In+Sn) obtained from an ITO surface treated by oxygen plasma with a flow rate of 60 sccm was almost 10% higher than the value obtained from a non-treated ITO surface. This implies that the change in Rs corresponds closely with the increase in the oxygen content on an ITO surface. The increase of the oxygen concentration on an ITO surface was found to correlate strongly with the increase in its work function [43, 44]. The comparison of TOF-SIMS depth profiles of the normalized relative oxygen concentration from the surfaces of non-treated ITO and ITO-treated films with a flow rate of 60 sccm is shown in Figure 12.14. It has been reported that oxygen-plasma-treated ITO has a smoother surface than the surface of the non-treated ITO [6]. In this work, the depth profile for 18 O ion was used for analyses because the intensity of 16 O ion counts was so high that it saturated the machine. As shown in Figure 12.14, from the 18 O ion, the relative oxygen concentration on the plasma-treated ITO surface is obviously higher. On the basis of the sputter rate used in the depth profile measurements, it appears that oxygen plasma treatment can induce a few-nanometers-thick oxygen-rich ITO region

18O

TOF-SIMS depth profile (Arb. Units)

342

Non-treated ITO Oxygen-plasma-treated ITO 0

50

100 150 200 Sputtering time (s)

250

300

Figure 12.14 Comparison of TOF-SIMS depth profiles of relative oxygen concentration on the non-treated and oxygen-plasma-treated ITO surfaces.

12.2 Anode Modification for Enhanced OLED Performance

in the vicinity of the ITO surface. However, the precise thickness of this region is difficult to determine directly by XPS or TOF-SIMS due to the influence of the interfacial effects occurred during the argon ion sputtering. ITO is a ternary ionic-bound degenerate semiconducting oxide. The conductivity of oxygen-deficient ITO is governed by tin dopants and ionized oxygen vacancy donors. In an ideal situation, free electrons can result either from the oxygen vacancies acting as doubly charged donors, providing two electrons each, or from the electrically active tin ionized donor on an indium site [11, 70]. The additional oxidation on an ITO surface by oxygen plasma may cause the dissipation of oxygen vacancies. Therefore, such an oxygen plasma treatment results in a decrease in electrically active ionized donors in a region near the ITO surface, leading to an overall increase in Rs as manifested by the in situ four-point probe measurement. By comparing the highly conducting bulk ITO, it can be considered that the treated ITO anode can be portrayed using a dual layer model. In order to gain an insight into the relation between the variation in sheet resistance and an optimal oxygen plasma process for enhanced hole injection, a set of 15-nm-thick ITO films was coated on glass substrates using the same deposition conditions in the sputter chamber. Each of the thin ITO films was transferred to a connected chamber for oxygen plasma treatments and was then passed to the glove box for an in situ sheet resistance measurement. The changes in the sheet resistance, ΔR, between the treated and non-treated ITO were 19, 30, and 89 Ω/square at the oxygen flow rate of 40, 60, and 100 sccm, respectively. The thickness of the oxygen-plasma-induced low-conductivity layer, x, is estimated to be 0.6, 0.9, and 3.1 nm at oxygen flow rates of 40, 60, and 100 sccm, respectively. Table 12.1 summarizes the changes in ΔR and the corresponding estimated thicknesses of the oxygen-plasma-induced low-conductivity layer, x, obtained for ITO films treated under different conditions. The measured ΔR correlates directly with the thickness of this low-conductivity layer, and both increase with the oxygen flow rate. The oxygen plasma treatment that was optimized for the OLED performance, in this case 60 sccm, induces a nanometer-thick resistive layer on an ITO surface. The above analysis based on the XPS, TOF-SIMS, and the electrical results suggest that the presence of an oxygen-plasma-induced nanometer-thick resistive layer on an ITO surface can also account for the enhanced OLED performance. Identical OLEDs made on ITOs, modified with LiF layers of thicknesses in the range 0–5.0 nm, were fabricated. Figure 12.15 shows the J–V characteristics of devices made with different LiF layer thicknesses. A comparison with the devices made without this Table 12.1 Oxygen-plasma-induced low-conductivity layer thickness and ΔR for the ITO films treated under different conditions.

𝚫R (𝛀/sq)

Oxygen-plasma-induced low-conductivity layer thickness, x (nm)

0

—

0

40

19

0.6

60

30

0.9

100

85

3.1

Oxygen flow rate (sccm)

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12 Organic Semiconductors and Applications

1000 Current Density (mA/cm2)

344

Bare ITO 0.5 nm LiF 1.5 nm LiF 5.0 nm LiF

800 600 400 200 0 0

4

8

12

16

Voltage (V)

Figure 12.15 J–V characteristics of devices made with different LiF interlayer thicknesses.

layer demonstrates that the former has a higher EL brightness operated at the same current density. At a given constant current density of 20 mA cm−2 , the luminance and efficiency for devices with 1.5-nm-LiF-coated ITO were 1600 cd m−2 and 7 cd A−1 , respectively. The corresponding values were 1170 cd m−2 and 5.7 cd A−1 , respectively, for the same devices made with only an ITO anode. The results demonstrate that the presence of an ultrathin LiF layer between the ITO and polymer favors efficient operation of LEDs. These improvements are attributed to an improved ITO/polymer interface quality and a more balanced carrier injection that improves the device efficiency. One possible mechanism for the above enhancement in device performance can be obtained from tunneling theory. LiF is an excellent insulator with a large bandgap of about 12 eV. However, the presence of an ultrathin insulating interlayer between ITO and polymer enhances hole injections. This indicates that the potential barrier for hole injection that was present in the device with a 1.5-nm-thick LiF interlayer was thus decreased via tunneling. Recently Zhao et al. [71] reported that insertion of an insulation LiF interlayer between the ITO anode and the HTL induces energy level realignment at the anode/HTL interface. There exists a triangle barrier at the ITO/HTL interface, and when a forward bias is applied on an OLED in the presence of an LiF interlayer, a voltage drop across the LiF lowers the ITO work function and hence also reduces the triangle barrier at the ITO/HTL interface. Although LiF also introduces an additional barrier, the overall effective interfacial barrier height can be reduced under an optimal interfacial modification condition, leading to enhanced carrier injection. The possible chemical reactions at the interface should also be taken into account for a more consistent explanation. They reveal that enhanced hole injection with the bilayer ITO/LiF anode altered the internal electric field distribution in the device. The enhanced injection for holes due to the tunneling effect could induce the change in the potential difference across both electrodes for carrier injection, leading to an improvement in the balance of the hole and electron injections. As the thickness of the insulating interlayer increases, the probability of carrier tunneling decreases, leading to a weaker carrier injection process. To achieve a given luminance, therefore, the applied voltage needs to be increased with increasing interlayer

12.3 Flexible OLEDs

thickness. As it becomes thicker, both the current density and EL brightness decreased because of reduced tunneling, but more balanced hole and electron injections were achieved so that the EL efficiency could reach its maximum value. For instance, an increased EL efficiency of 7.9 cd A−1 for a device with a 1.5-nm-thick LiF indicated a more balanced injection of both types of carriers, which was less optimal in the case of a device with a bare ITO, as shown in Figure 12.15. The devices presented in this study clearly portray this behavior. An ITO anode modified with an insulating interlayer for enhanced carrier injection in OLEDs has been demonstrated [67, 71, 72]. Also, attempts are made to explore the use of ultrathin inorganic insulating layers to modify ITO in OLEDs. Ho et al. [61] have reported that a 1–2-nm-thick insulating self-assembled-monolayer on ITO significantly alters the injection behavior and enhances EL efficiency, which is consistent with our results presented here. These improvements are attributed to an improved ITO/polymer interface quality and a more balanced carrier injection that improves device efficiency. Apart from the current understanding of the formation of a negative dipole facing the ITO due to oxygen plasma or UV ozone treatment, it seems that oxidative treatment modifies an ITO surface effectively by reducing the oxygen deficiency to produce a low-conductivity region. As such, in this case, oxygen-plasma-treated ITO behaves somewhat similar to the specimens where there is an ultrathin insulating interlayer serving as an efficient hole injection anode in OLEDs. The improvement in the EL performance of OLEDs is correlated directly to the thickness of the low-conductivity interlayer between the anode and the polymer layer. The results show that the best EL performance comes from the device with a nanometer-thick low-conductivity layer induced by an optimal oxygen plasma treatment. This is consistent with the use of an ultrathin parylene or LiF layer modified bilayer ITO/interlayer anode for enhanced performance of OLEDs [67, 73].

12.3 Flexible OLEDs Current OLED technologies employ rigid substrates, such as glass, which limits the “moldability” of the device, restricting the design and spaces where OLEDs can be used. The demand for more user-friendly displays is propelling efforts to produce head-worn and hand-held devices that are flexible, lighter, more cost-effective, and more environmentally benign than those presently available. Flexible thin film displays enable the production of a wide range of entertainment-related, wireless, wearable-computing, and network-enabled devices. The display of the future requires that it should be thin in physical dimensions, large-sized, flexible, and full color at a low cost. These demands are sorely lacking in today’s display products and in technologies such as the plasma display and LCD technologies. OLEDs [74–77] have recently attracted attention as display devices that can replace LCDs because OLEDs can produce high visibility by self-luminescence. The OLED stands out as a promising technology that can deliver the above challenging requirements. Next-generation flexible displays are going to be commercially competitive due to their low power consumption, high contrast, light weight, and flexibility. The use of thin flexible substrates in OLEDs will significantly reduce the weight of flat panel displays and provide the ability to bend or roll a display into any desired shape. To date, much

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effort has been focused on fabricating OLEDs on various flexible substrates [78–82]. The plastic substrates usually do not have negligible oxygen and moisture permeability. The barrier properties of these substrates are not sufficient to protect the electroluminescent polymeric or organic layers in OLEDs due to the penetration of the chemically reactive oxygen and water molecules into the active layers of devices. Therefore, the plastic substrates with an effective barrier against oxygen and moisture penetration have to be fabricated before this simple vision of flexible display can become a reality [83]. Polymer-reinforced ultrathin glass sheet is one of the alternative substrates for flexible OLEDs. In this section, we will discuss the results of OLEDs fabricated on flexible ultrathin glass sheet with polymer reinforcement coating and flexible plastic substrates. 12.3.1

Flexible OLEDs on Ultrathin Glass Substrate

A polymer-reinforced ultrathin glass sheet is one of the alternative substrates for flexible OLEDs. It has been found that the flexibility and handling ability of an ultrathin glass sheet can be improved significantly when it is reinforced. The reinforcement polymer layer has the same shrinkage direction as that of an ITO layer deposited on the opposite side of the substrate. ITO-coated ultrathin glass with a reinforcement polymer layer has high optical transmittance and is suitable for device fabrication. The work presented in Section 12.2.1 in this chapter indicates that the ITO film developed at a low processing temperature is suitable for OLED applications. The improvement in OLED performance correlates directly with the ITO properties. Plastic foils and ultrathin glass sheets with reinforced polymer layers are not compatible with a high-temperature plasma process. Therefore, the successful development of high-quality ITO films with a smooth surface, high optical transparency and electric conductivity at a low temperature provides the possibility for developing flexible displays. We have used flexible ultrathin glass sheets with a 250-nm-thick smooth ITO coating and a sheet resistance of about 20 Ω/square to fabricate phenyl alkoxyphenyl PPV copolymer-based OLEDs. In order to compare the EL performance of the OLEDs thus fabricated, we also fabricated an OLED with an identical structure using a commercial ITO-coated rigid glass substrate and used it as a reference. Both OLEDs were fabricated using the same conditions and had an active emitting area of 4 mm2 on glass substrates with dimensions of 5 cm × 5 cm. Figure 12.16 shows the current and luminance of both types of OLEDs as functions of the operating voltage, i.e. J–V and luminance–voltage (L–V ) characteristics. The solid curves in Figures 12.16a,b represent the device characteristics measured from the reference OLED, and the corresponding dashed curves represent those of the OLED made with an ITO-coated flexible thin glass sheet. As can be seen from Figure 12.16, there are some differences in the J–V curves of the two OLEDs at operating voltages higher than 8 V. This can probably be attributed to the less uniform organic films on ITO-coated ultrathin glass sheets. There was a warp in ITO-coated thin glass sheets due to a mismatch in the coefficients of thermal expansion between the ITO film and the ultrathin glass sheet. The polymer films spun on warped substrates over an area of 5 cm × 5 cm might be less uniform in comparison with those coated on the rigid substrate under the same conditions. This problem can be overcome by attaching the flexible glass sheet on a rigid substrate. In general, the forward current

12.3 Flexible OLEDs

Current density (mA/cm2)

(a)

1800 1600

OLEDs on 1.1 mm glass

1400

OLEDs on UTG

1200 1000 800 600 400 200 0 0

2

4

6 8 10 12 Operating voltage (V)

14

16

14

16

(b) 60000 OLEDs on 1.1 mm glass OLEDs on UTG

Luminance (cd/m2)

50000 40000 30000 20000 10000 0 0

2

4

6 8 10 12 Operating voltage (V)

Figure 12.16 (a) J–V and (b) L–V characteristics measured for the OLEDs fabricated with 50 μm flexible ultrathin glass (UTG) sheets and ITO-coated rigid glass substrate.

density measured from both OLEDs, shown in Figure 12.16a, exhibit similar behavior in the operating voltage range 2.5–7.5 V. A maximum luminance of 4.8 × 104 cd m−2 and an efficiency of 5.8 cd A−1 , measured for OLEDs fabricated on a reinforced flexible glass sheet, at an operating voltage of 7.5 V was obtained. The electroluminescent performance of OLEDs made with 50-μm-thick flexible borosilicate glass sheets is comparable to that of identical devices made with commercial ITO coated on rigid glass substrates. It is envisaged that further improvements in device performance can be achieved by optimizing the fabrication processes and encapsulation techniques. 12.3.2

Flexible Top-Emitting OLEDs on Plastic Foils

In the past decade, the display industries have experienced an extremely high growth rate. In the recent market report by Stanford Research, the compounded annual growth rate of OLEDs from 2003 to 2009 is 56%, which means that it will grow from US$300 million in 2003 to US$3.1 billion in 2009. OLEDs have recently attracted attention as display devices that can replace LCDs because OLEDs can produce high visibility by

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self-luminescence. OLEDs do not require back lighting, which is necessary for LCDs, and can be fabricated into lightweight, thin, and flexible display panels. A typical OLED is constructed by placing a stack of organic electroluminescent and/or phosphorescent materials between a cathode layer that can inject electrons and an anode layer that can inject holes. When a voltage of proper polarity is applied between the cathode and anode, holes injected from the anode and electrons injected from the cathode combine radiatively and emit energy as light, thereby producing electroluminescence. 12.3.2.1

Top-Emitting OLEDs

A conventional OLED has a bottom-emitting structure, which includes a reflective metal or metal alloy cathode, and a transparent anode on a transparent substrate, enabling light to be emitted from the bottom of the structure. An OLED may also have a top-emitting structure, which is formed on either an opaque substrate or a transparent substrate. Unlike the conventional OLED structure, top-emitting OLEDs can be made on both transparent and opaque substrates. One important application of the top device structure is monolithic integration of a top-emitting OLED on a polycrystalline or amorphous silicon thin film transistors (TFTs) used in active-matrix displays, as illustrated in Figure 12.17. The top-emitting OLED structures therefore increase the flexibility of device integration and engineering. Efficient and durable TOLEDs are required in high-resolution display applications. TOLEDs can be incorporated into an active-matrix display integrated with amorphous or poly-silicon TFTs, which have the important advantage of having on-chip data and scan drivers, allowing for ultrahigh pixel resolution ( 20% has been reported for an OLED using MoO3 HIL, which was achieved due to efficient hole injection at the interface between the HIL and a deep HOMO material, e.g. 4,4′ -Bis(carbazol-9-yl)biphenyl (CBP) [102]. Many studies have revealed that the TMO HIL possesses a high work function and a strong molecular electronegativity, allowing the extraction of electrons from the HOMO level of the neighboring organic layer to its deep conduction band [103–106]. In comparison with the solution-processed PEDOT:PSS HIL, the TMO-based HIL is usually prepared by sputtering or the thermal evaporation method. However, in all-solution fabrication process technologies, e.g. roll-to-roll or inkjet printing, the vacuum-involved fabrication process has a limitation in practical applications. Therefore, a lot of effort has been devoted to developing a solution-processed TMO HIL. Different approaches, e.g. the sol–gel method [107–110], direct dissolution of TMO powder in solvent [111, 112], and synthesis of TMO nanoparticles (NPs) solution [113, 114], have been reported to realize the solution-processed TMO HIL. However, the TMO HIL prepared by the sol–gel method requires a high sintering temperature, which is not suitable for devices made on flexible plastic substrates. For example, the MoO3 sol–gel process requires a sintering temperature of 275 ∘ C [109]. The solubility of the TMO powders in the solvent can be a problem, because most TMO powders do not dissolve in organic solvents easily. The synthesis of TMO NPs solution can be an adequate approach for solution-processed TMO HIL. But the films prepared by TMO NPs have a large surface roughness and a high density of pinholes, causing high leakage current [113, 114]. Recently, the use of hybrid HIL in organic optoelectronic devices has been reported, e.g. blending the PEDOT:PSS and TMOs [111, 115, 116]. High-performance organic solar cells with a hybrid HIL have been demonstrated to show promising results as compared to the device using pure TMO HIL [115–117]. In processing, the hybrid HIL solution can improve the wetting ability and the film formation property on the polymer surface [117]. In addition, a blending film with a pinhole-free morphology can improve interfacial contact and thus enhance the device efficiency and stability [115]. The use of solution-processed hybrid HIL has potential for application in large-area OLEDs. Thus, the solution-processed hybrid HIL plays an important role in the performance of OLEDs. This section discusses the performance of OLEDs with a solution-processed hybrid HIL. The hybrid PEDOT:PSS and MoO3 NPs hybrid HIL show a superior hole injection characteristic at the interface between the HIL and the low-lying HOMO material of CBP, leading to superior device efficiency comparable to the performance of OLED using

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–2.5

–2.6

TmPyPB TmPyPB

CBP

Ir(ppy)2acac

LiF/AI MoO3 NPs

MoO3-PEDOT:PSS

–4.9

PEDOT:PSS

ITO

–3.0 Energy (eV)

356

–5.24 –5.6

–5.74 –5.98

Ir(ppy)2acac CBP MoO3-PEDOT:PSS ITO Glass

–6.0 –6.7 (a)

(b)

Figure 12.23 The schematic diagrams of (a) the energy levels of all functional materials used in OLEDs, and (b) OLED’s structural configuration.

the vacuum-evaporated MoO3 HIL. The surface property of the hybrid HIL is determined, and the underlying physics is discussed. MoO3 thin film or the blending film of PEDOT:PSS and MoO3 is used as the HIL to modify the ITO in OLEDs. An HTL of CBP and an ETL of TmPyPB are used as the organic stacks. A simplified trilayer device, having a general configuration of anode/interlayer/HTL/ETL/cathode was used in this experiment. A phosphorescent dopant Bis-(2-phenylpyridine)-(acetylacetonate)-iridium(III) (Ir(ppy)2 acac) with a peak of EL emission at 520 nm was doped in the host material of CBP [102]. The schematic diagrams of the energy levels of the functional materials and the structural configuration of the OLED are presented in Figure 12.23. A bare ITO glass substrate, with a sheet resistance of 15 Ω/sq., was cleaned by ultrasonication sequentially with diluted detergent, deionized water, acetone, and isopropanol each for 20 minutes. The MoO3 NPs solution was synthesized according to the reported procedure [118]. The MoO3 NPs solution (0.1 mol ml−1 , dissolved in ethanol) was mixed with the PEDOT:PSS (CleviosTM P VP Al 4083) solution, with different volume ratios of PEDOT:PSS to MoO3 NPs pf 1:1, 2:1, 3:1, and 4:1. A 30-nm-thick hybrid HIL was formed on the ITO/glass substrate. A 10-nm-thick MoO3 HIL on ITO/glass was also formed using pure MoO3 NP solution for comparison studies. After the annealing at 120 ∘ C for 10 minutes, the samples were loaded into a N2 purged glove box, which is connected to a 10-source evaporator for device fabrication. All functional layers, as shown in Figure 12.23, were deposited by thermal evaporation in a high vacuum system with a base pressure of less than 5.0 × 10−4 Pa. OLEDs with an evaporated MoO3 HIL and a solution-processed MoO3 HIL have the same device configuration of ITO (80 nm)/MoO3 (10 nm)/CBP (50 nm)/CBP:Ir(ppy)2 acac (7%, 15 nm)/TmPyPB (55 nm)/LiF(1 nm)/Al(100 nm). For OLEDs with a hybrid PEDOT:PSS-MoO3 HIL, the devices have a configuration of ITO (80 nm)/hybrid HIL with different volume ratios of PEDOT:PSS to MoO3 (1 : 1, 2 : 1, 3 : 1 and 4 : 1, 30 nm)/CBP (30 nm)/CBP:Ir(ppy)2 acac (7%, 15 nm)/TmPyPB (55 nm)/LiF(1 nm)/Al(100 nm). All OLEDs were encapsulated in the glove box and then characterized in ambient conditions. The current density–voltage–luminance (J–V –L) characteristics of OLEDs were measured by a Keithley source measurement unit (Keithley Instruments Inc.,

12.4 Solution-Processable High-Performing OLEDs

Model 236 SMU), which was calibrated using a silicon photodiode. The EL spectra were measured by a spectra colorimeter (Photo Research Inc., Model 650) spectrophotometer. The surface morphology of HILs was characterized by AFM (Multimode V), which was performed in the tapping mode. The surface electronic properties of different HILs were analyzed using XPS, and UPS measurements were performed in the system equipped with an electron spectrometer (Sengyang SKL-12) and an electron energy analyzer (VG CLAM 4 MCD) operated at the base pressures of 2 × 10−9 mbar. The XPS spectra were measured by an achromatic Mg K𝛼 excitation (1253.6 eV) at 10 kV, with an emission current of 15 mA. The UPS spectra were measured by a He discharge lamp (He I radiation of 21.22 eV), and the samples were biased at −5.0 eV during the measurement in order to observe the low-energy secondary cutoff. The performance of a set of structurally identical OLEDs with different HILs was investigated. Figure 12.24a shows the J–V characteristics of OLEDs with different HILs inserted between ITO and CBP. Without the interlayer at the anode/organic interface, the OLED obviously shows an extremely low current density without emission over the entire operating voltage range from 0 to 9 V. This indicates that the hole injection is not ideal due to a high injection barrier at the ITO/CBP interface [119]. An obvious increase in the injection current in the OLED is obtained after the insertion of a MoO3 HIL between ITO and CBP, as shown in Figure 12.24a. This indicates that the MoO3 HIL makes good contact with CBP [102]. OLEDs made in the presence of a MoO3 layer, prepared by either vacuum evaporation or the solution process, have a comparable current density and a low turn-on voltage of ∼3.0 V. At a low driving voltage, the luminance of OLEDs with a solution-processed MoO3 HIL is slightly lower than that of OLEDs with a vacuum-evaporated MoO3 HIL, as shown in Figure 12.24b, due to a high leakage current. The inset in Figure 12.24a depicts the J–V characteristics of both devices in a semi-log plot. OLEDs with a solution-processed MoO3 NPs HIL have a high leakage current, e.g. about two orders of magnitude higher than that of OLEDs with a vacuum-evaporated MoO3 HIL at an operating voltage 3.2 V). In addition, the efficiency of OLEDs fabricated with different MoO3 -based HILs was also studied. Figure 12.25 shows the power efficiency as a function of the luminance for a set of structurally identical OLEDs made with different HILs. The power efficiency of the OLEDs having a hybrid MoO3 -PEDOT:PSS HIL is the highest compared to the efficiency of OLEDs with either an evaporated or a solution-processed MoO3 HIL, e.g. 89.2 lm W−1 @ 100 cd m−2 and 73.5 lm W−1 @ 1000 cd m−2 . The overall power efficiency of the hybrid HIL-contained OLEDs is almost two times higher than that of OLEDs with a solution-processed MoO3 HIL. Figure 12.26 depicts the EQE as a function of the luminance for a set of structurally identical OLEDs with different HILs. A high EQE

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12 Organic Semiconductors and Applications

(a) 80

102

Current density (mA/cm2)

101 100 60

10–1 10–2

–3 40 10 10–4

0

2

4

6

8

10

20

0 (b) 105

104 Luminance (cd/m2)

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103

102 Bare ITO Vacuum-evaporated MoO3

101

Solution-processed MoO3 Hybrid MoO3-PEDOT:PSS

100

2

4

6 Voltage (V)

8

10

Figure 12.24 (a) J–V and (b) L–V characteristics of a set of OLEDs fabricated using different HILs. Inset in Figure 12.24a is the semi-log plot of J–V characteristics of OLEDs with different MoO3 anode interlayers, vacuum-evaporated (green triangle), solution-processed (red circle), and hybrid MoO3 -PEDOT:PSS (inverted blue triangle).

for OLEDs with a hybrid HIL, e.g. >20%, was obtained over a wide luminance range from 10 to 10 000 cd m−2 . The EQE of OLEDs with a hybrid