Diffusion processes in advanced technological materials

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DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

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DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

Edited by Devendra Gupta Emeritus Research Staff Member, IBM Research Division Thomas J. Watson Research Center Yorktown Heights, New York

NORWICH, NEW YORK, U.S.A.

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Copyright © 2005 by William Andrew, Inc. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system now known or to be invented, without permission in writing from the Publisher. Cover Art © 2005 by Brent Beckley / William Andrew, Inc. ISBN: 0-8155-1501-4 (William Andrew, Inc.) ISBN: 3-540-21938-2 (Springer-Verlag GmbH & Co. KG) Library of Congress Cataloging-in-Publication Data Diffusion processes in advanced technological materials / edited by Devendra Gupta. p. cm. Includes bibliographical references. ISBN 0-8155-1501-4 (William Andrew, Inc.) — ISBN 3-540-21938-2 (Springer-Verlag GmbH & Co. KG) 1. Diffusion. 2. Diffusion processes—Mathematical models. I. Gupta, Devendra. QD543.D492 2004 530.4’15—dc22 2004009918 Printed in the United States of America This book is printed on acid-free paper. 10 9 8 7 6 5 4 3 2 1 This book may be purchased in quantity at discounts for education, business, or sales promotional use by contacting the Publisher. Published in the United States of America by William Andrew, Inc. 13 Eaton Avenue Norwich, NY 13815 1-800-932-7045 www.williamandrew.com www.knovel.com (Orders from all locations in North and South America)

Springer-Verlag GmbH & Co. KG Tiergartenstrasse 17 D-69129 Heidelberg, Germany www.springeronline.com (Orders from all locations outside North and South America)

NOTICE To the best of our knowledge the information in this publication is accurate; however the Publisher does not assume any responsibility or liability for the accuracy or completeness of, or consequences arising from, such information. This book is intended for informational purposes only. Mention of trade names or commercial products does not constitute endorsement or recommendation for their use by the Publisher. Final determination of the suitability of any information or product for any use, and the manner of that use, is the sole responsibility of the user. Anyone intending to rely upon any recommendation of materials or procedures mentioned in this publication should be independently satisfied as to such suitability, and must meet all applicable safety and health standards. William Andrew, Inc., 13 Eaton Avenue, Norwich, NY 13815 Tel: 607/337/5080 Fax: 607/337/5090

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Dedication To Sudha Gupta, my wife for more than four decades; the late Harish Chandra Gupta, my brother, to whom I owe my early education; and David Lazarus, my valued advisor at the University of Illinois, to whom I owe my long career in diffusion.

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Contents Contributors .......................................................................................... xiii Preface ..................................................................................................xvii 1 Diffusion in Bulk Solids and Thin Films: Some Phenomenological Examples ............................................... 1.1 Introduction ................................................................................ 1.2 Diffusion in Single Crystals ...................................................... 1.2.1 Mathematical Basis .......................................................... 1.2.2 Atomistic Nature of Diffusion ......................................... 1.2.3 Pressure and Mass Dependence of Diffusion .................. 1.2.4 Linear Chemical Diffusion Regime: Finite Driving Force F on Individual Atoms ................... 1.2.5 Nonlinear Chemical Diffusion Regime ........................... 1.3 Structurally Inhomogeneous Samples ....................................... 1.4 Some Illustrative Experimental Data ......................................... 1.4.1 Diffusion Profiles in Au Having Variable Microstructure .................................................................. 1.4.2 Self-Diffusion Data in the Au Lattice .............................. 1.4.3 Self-Diffusion in the Au and Au-1.2 at.% Ta Alloy Grain Boundaries .............................................. 1.5 General Characteristics of Grain Boundary Diffusion .............. 1.5.1 Anisotropy of Diffusion in Grain Boundaries ................. 1.5.2 Diffusion Mechanisms in Grain Boundaries ................... 1.5.3 Interrelationship Among Grain Boundary, Lattice Diffusion, and Energy .......................................... 1.5.4 Grain Boundary Solute Segregation Effects .................... 1.6 Diffusion in Quasicrystalline and Amorphous Alloys ............... 1.6.1 Diffusion in Quasicrystalline Alloys ............................... 1.6.2 Diffusion in Amorphous Alloys: Metallic Glasses .......... 1.7 Summary .................................................................................... Acknowledgment .............................................................................. References .........................................................................................

1 1 2 2 8 15 20 23 27 33 34 38 41 44 44 46 48 52 56 57 60 63 64 64

2 Solid State Diffusion and Bulk Properties ................................... 69 2.1 Introduction ................................................................................ 2.2 Correlations with Bulk Properties ............................................. 2.2.1 The Melting Parameters ................................................... 2.2.2 Elastic Constants .............................................................. 2.2.3 Bulk Modulus .................................................................. 2.2.4 The Debye Temperature ...................................................

69 71 71 82 85 87

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2.2.5 Valence Bond Parameter ................................................ 2.2.6 Electron-to-Atom Ratio ................................................. 2.2.7 Summary of Empirical Correlations .............................. 2.3 Equilibrium Thermodynamic Calculation of Diffusion Parameters ............................................................... 2.3.1 The Activation Volume .................................................. 2.3.2 Activation Entropy and Diffusion Frequency ............... 2.3.3 Specific Heat of the Activated Complex ....................... 2.3.4 Magnitude of Estimated Values ..................................... 2.3.5 Reliability of Estimated Parameters .............................. 2.4 Summary .................................................................................. Acknowledgment ............................................................................ Appendix 2A. Taylor Series Expansion of ∆G* ............................ Appendix 2B. Evaluation of Errors in Estimated Parameters ....... 2B.1 ∆S* .................................................................................. 2B.2 u ...................................................................................... 2B.3 ∆C*V ................................................................................. 2B.4 ∆C*P ................................................................................. References .......................................................................................

88 91 92 93 94 97 99 99 104 105 105 105 107 108 108 108 109 109

3 Atomistic Computer Simulation of Diffusion ............................ 113 3.1 Introduction ............................................................................... 3.2 Atomic Interaction Models ....................................................... 3.2.1 Embedded-Atom Method ............................................... 3.2.2 Angular-Dependent Potentials ....................................... 3.2.3 More Accurate Methods ................................................. 3.3 Molecular Statics ..................................................................... 3.3.1 Simulation Block and Boundary Conditions ................. 3.3.2 Point-Defect Formation Energy ..................................... 3.4 Harmonic Approximation ........................................................ 3.4.1 Harmonic Entropy of Point Defects .............................. 3.4.2 Effect of Boundary Conditions on Point-Defect Entropy ..................................................... 3.4.3 Embedded Cluster Method ............................................ 3.4.4 Local Harmonic Approximation .................................... 3.4.5 Quasi-Harmonic Approximation .................................... 3.5 Equilibrium Defect Concentrations .......................................... 3.5.1 Elemental Solids ............................................................ 3.5.2 Non-Stoichiometric Compounds ................................... 3.5.3 Effect of Pressure ........................................................... 3.6 Transition Rate Calculations .................................................... 3.6.1 Transition State Theory .................................................. 3.6.2 Finding the Saddle Point ................................................ 3.7 Kinetic Monte Carlo Simulations .............................................

113 114 114 116 118 119 119 120 122 122 123 125 127 128 129 129 130 137 138 138 141 143

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3.7.1 Basic Idea of the Method ............................................... 3.7.2 Grain Boundary Diffusion ............................................. 3.7.3 Diffusion in Alloys and Compounds ............................. 3.7.4 On-the-Fly Monte Carlo Simulations ............................ 3.8 Molecular Dynamics ................................................................ 3.8.1 Calculation of Diffusion Coefficients ............................ 3.8.2 Diffusion Mechanisms in Grain Boundaries ................. 3.8.3 Diffusion Mechanisms in Intermetallic Compounds ..... 3.8.4 Accelerated Molecular Dynamics .................................. 3.9 Conclusions .............................................................................. Acknowledgment ............................................................................ References .......................................................................................

ix

143 145 149 151 153 153 155 157 162 165 166 166

4 Bulk and Grain Boundary Diffusion in Intermetallic Compounds ............................................................ 173 4.1 Introduction .............................................................................. 4.2 Crystal Structures and Point Defects in Ni, Ti, and Fe Aluminides ................................................................... 4.3 Diffusion Mechanisms in Intermetallics ................................. 4.4 Experimental Results on Bulk Diffusion in Ordered Aluminides ............................................................ 4.4.1 Ni3Al .............................................................................. 4.4.2 Ti3Al ............................................................................... 4.4.3 TiAl ................................................................................ 4.4.4 NiAl ............................................................................... 4.4.5 Fe-Al System ................................................................. 4.5 Discussion of Lattice Diffusion in Intermetallics ................... 4.6 Grain Boundary Diffusion ....................................................... 4.6.1 Ni3Al .............................................................................. 4.6.2 Ti3Al ............................................................................... 4.6.3 TiAl ................................................................................ 4.6.4 NiAl ............................................................................... 4.6.5 Fe3Al .............................................................................. 4.7 Summary .................................................................................. Acknowledgments ................................................................... References ................................................................................

173 174 182 188 189 196 199 205 213 221 224 225 227 229 229 232 232 234 234

5 Diffusion Barriers in Semiconductor Devices/Circuits ............. 239 5.1 Introduction .............................................................................. 5.2 Diffusion Barriers from the 1960s Through the 1990s ........... 5.3 Brief Review of Diffusion and the Influencing Material Factors ................................................... 5.3.1 Diffusion in the Lattice and Grain Boundaries .............

239 241 244 244

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5.3.2 Interdiffusion Between Two or More Materials in Contact ....................................................... 5.3.3 Role of Material Properties ............................................ 5.4 Diffusion Barrier Materials ..................................................... 5.4.1 Metal Nitrides, Carbides, and Borides as APDB Used with Al .................................................................. 5.4.2 Barriers Between the ILD and Cu ................................. 5.5 New Concepts in Affecting APDB Behavior at the Interfaces ............................................................................ 5.5.1 Alloying of Cu to Form an APDB at Interfaces/Surfaces ......................................................... 5.5.2 Self-Assembled Molecular Monolayers ........................ 5.5.3 Zero-Flux Diffusion Zones (Multicomponent Diffusion Effects) ........................................................... 5.6 Brief Discussion of an APDB for Low-k ILD Materials ........ 5.7 Summary .................................................................................. References .......................................................................................

246 248 260 260 262 269 269 270 272 275 277 278

6 Reactive Phase Formation: Some Theory and Applications ........................................................................... 283 6.1 Introduction .............................................................................. 283 6.2 Theoretical Considerations ....................................................... 284 6.2.1 One Phase Growing, Diffusion Controlled .................... 284 6.2.2 Two Phases Growing Simultaneously, Diffusion Controlled ...................................................... 287 6.2.3 Linear Parabolic Kinetics, One-Phase Growth, Oxides, Equilibrium, Plotting of Data ........................... 290 6.2.4 Linear-Parabolic Kinetics: Sequential Phase Growth, Grain Boundary Versus Lattice Diffusion ...... 292 6.2.5 First Phase Formed ........................................................ 294 6.2.6 Nucleation of the First or Second Phases ...................... 294 6.2.7 Nucleation-Controlled Reactions and Consequences: Sequence of Phase Formation, Bulk Samples, Stresses .... 301 6.2.8 Amorphous and Other Metastable Phases, Quasicrystals, Ternary Systems ..................................... 304 6.2.9 Other Effects: Grain Boundaries and Impurities, Diffusion Coefficients Varying with Composition ........ 307 6.3 Practical Problems in Electronic Technology .......................... 309 6.3.1 Titanium Disilicide, Activation Energy for the Motion of the Interface Between the C49 and C54 Phases, Nucleation of the C49 Structure ............... 310 6.3.2 Cobalt Disilicide, Entropy of Mixing, (Possible) Enthalpy and Density-of-State Effects in Ternary Reactions .......... 317 6.3.3 Nickel Monosilicide ....................................................... 323

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xi

6.4 Conclusions .............................................................................. 324 Acknowledgments .......................................................................... 326 References ....................................................................................... 327 7 Metal Diffusion in Polymers and on Polymer Surfaces ............ 333 7.1 Introduction .............................................................................. 7.2 Diffusion During Nucleation and Growth of Metal Films on Polymers ......................................................... 7.3 Metal-Polymer Interaction ....................................................... 7.4 Diffusion in the Polymer Bulk ................................................ 7.5 Summary and Conclusions ...................................................... Acknowledgments .......................................................................... References .......................................................................................

333 336 341 342 358 359 359

8 Measurement of Stresses in Thin Films and Their Relaxation ........................................................................... 365 8.1 Introduction .............................................................................. 8.2 Measurement Techniques ........................................................ 8.2.1 Substrate Curvature ........................................................ 8.2.2 X-Ray Diffraction .......................................................... 8.3 Stress Relaxation ..................................................................... 8.3.1 Experimental Observations ............................................ 8.3.2 Dislocation Plasticity ..................................................... 8.3.3 Diffusional Creep ........................................................... 8.4 Conclusion ............................................................................... References .......................................................................................

365 368 370 373 378 378 381 393 398 400

9 Electromigration in Cu Thin Films ............................................ 405 9.1 9.2 9.3 9.4 9.5

Introduction .............................................................................. Cu Interconnection Integration ................................................ Test Structure and Experiment ................................................ Microstructure .......................................................................... Theory ...................................................................................... 9.5.1 Drift Velocity ................................................................. 9.5.2 Diffusivity ...................................................................... 9.5.3 Effective Diffusivity and Microstructure ....................... 9.5.4 Electromigration-Induced Backflow .............................. 9.5.5 Partial Blocking Boundary ............................................ 9.6 Resistance and Void Growth .................................................... 9.7 Fast Diffusion Paths ................................................................. 9.7.1 Free Surface and Grain Boundary Diffusion ................. 9.7.2 Ambient Effect ............................................................... 9.7.3 Alloying Effect ..............................................................

405 407 409 411 414 414 415 416 418 419 419 422 422 427 428

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9.8

Lifetime Distribution ........................................................... 9.8.1 Single-Damascene Line on W Via ............................. 9.8.2 Dual-Damascene Line on W Line .............................. 9.9 Current Density Dependence ............................................... 9.10 Lifetime vs. Linewidth ........................................................ 9.11 Lifetime Scaling Rule .......................................................... 9.11.1 Single-Damascene Line ............................................ 9.11.2 Dual-Damascene Line .............................................. 9.12 Short-Length Effect ............................................................. 9.13 Reduced Cu Interface Diffusion .......................................... 9.14 Conclusion ........................................................................... References .....................................................................................

433 433 435 453 458 461 461 462 468 474 480 482

10 Diffusion in Some Perovskites: HTSC Cuprates and a Piezoelectric Ceramic .............................................................. 489 10.1 Introduction .......................................................................... 10.2 Cation Diffusion .................................................................. 10.2.1 Characteristics of YBCO Bulk and Thin-Film Specimens .............................................. 10.2.2 Diffusion and Interactions Between YBCO Thin Films and Substrates ................................................ 10.2.3 Self-Diffusion of the Constituent Cations (Y, Ba, and Cu) of YBCO ....................................... 10.2.4 Diffusion of Cation Impurities in YBCO ................ 10.3 Anion Diffusion in Several HTSC Cuprates ....................... 10.3.1 Oxygen Diffusion Data in HTSC Cuprates ............. 10.3.2 Comparison Between Cation and Anion Diffusion 10.4 Grain Boundary Diffusion and Solute Segregation Effects in Perovskites .......................................................... 10.4.1 Grain Boundary Self-Diffusion of Nonsegregating Cations in the YBa2Cu3O7
x Superconductor .......... 10.4.2 Grain Boundary Diffusion of a Segregating Cation in the YBa2Cu3O7
x Superconductor and a Piezoelectric Ceramic .............................................. 10.4.3 Grain Boundary Diffusion of Oxygen in the YBa2Cu3O7
x Superconductor ................................. 10.5 Summary .............................................................................. Acknowledgments ........................................................................ References .....................................................................................

489 491 491 493 498 503 509 510 513 514 515 520 522 524 524 525

Index .................................................................................................... 529

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Contributors Sergiy Divinski, professor at the Institut für Materialphysik, Universität Münster, received his Ph.D. from the Institute for Physics of Metals, Kiev, Ukraine in 1990. He received an Alexander-von-Humboldt Fellowship in 1998 and the Werner-Köster Award in 2002. His research interests are bulk and grain boundary diffusion in intermatallic compounds, metals and nanocrystalline materials. He has authored/co-authored numerous journal publications. François M. d’Heurle, an emeritus research staff member at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York, received his Ph.D. from the Illinois Institute of Technology, Chicago, in 1958. His research interests are thin film electromigration, solid-state reactions, and silicide formation. He has received awards from IEEE, AVS, AIP, TMS, and IBM, and Doctor honoris causa from the Royal Institute of Technology, Sweden. Franz Faupel, Professor of Multicomponent Materials at the University of Kiel, Germany, received his Ph.D. from the University of Göttingen in 1985. His research interests include diffusion and reactions in metalpolymer interfaces, organic thin films, and crystalline solids and amorphous metallic alloys. He is Chairman of the Metal Physics Division of the German Physical Society and Associate Editor of the Journal of Materials Research. He has published more than 140 research papers. Huajian Gao, a director at the Max Planck Institute for Metals Research in Stuttgart, Germany, received his Ph.D. from Harvard University in 1988. He is a member of the scientific advisory board of the Institutes for Mechanics and Metals Research of the Chinese Academy of Sciences. His research interests include the mechanics of thin films and of biological and bio-inspired materials. He has received ASME and Guggenheim fellowships and awards from the National Science Foundation, IBM, and ALCOA. He has published more than 100 papers. Patrick Gas, Vice Director of the Provence Laboratory for Materials and Microelectronics, Marseille, received his Ph.D. in materials science and his Doctorat ès Sciences from the University of Aix-Marseille in 1975 and 1982, respectively. His research interests are the thermodynamic and kinetic aspects of solid-state reactions and diffusion in microelectronic materials. He has authored or co-authored more than 100 publications.

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Lynne M. Gignac, a senior engineer at the IBM Watson Research Center, obtained her Ph.D. in materials science and engineering from the University of Arizona in 1988. Her research interests include analytical and microstructural characterization of microelectronic materials via electron and focused ion beam microscopy and general reliability of Si to substrate interconnections. Devendra Gupta, an emeritus research staff member at the IBM Watson Research Center, received his Ph.D. from the University of Illinois, Urbana-Champaign, in 1961, under Prof. D. Lazarus. His research interests include diffusion, mass transport, and defects in solids and thin films for microelectronic applications. A fellow of the American Physical Society, he has written more than 100 articles and edited six books in the field of diffusion. Christian Herzig, an emeritus professor at the Institut für Materialphysik, Universität Münster, received his Ph.D. from the University of Münster in 1968. His research interests include diffusion, lattice dynamics, and defects in intermetallic compounds and interfaces. He received the Gustav Tammann Memorial Award in 1982 and the Werner-Köster Award in 2002. Chao-Kun Hu, a research staff member in the Materials and Reliability Sciences Department at the IBM Watson Research Center, received his Ph.D. in physics from Brandeis University in 1979. He has written several book chapters and more than 130 papers. Yoshiaki Iijima, a professor at Tohoku University, Japan, received his Doctor of Engineering degree from Tohoku University in 1983. His research interests are hydrogen diffusion and storage in materials. Michael Kiene, an engineer at Advanced Micro Devices in Austin, Texas, and Dresden, Germany, received his Ph.D. in 1997 under Prof. Franz Faupel at the University of Kiel. His research interests are the microstructure and chemistry of metal-polyimide interfaces. Oliver Kraft, a director at the Institute for Materials Research at the Forschungszentrum Karlsruhe, Germany, and Professor of Reliability in Mechanical Engineering at the University of Karlsruhe, received his Ph.D. from the University of Stuttgart in 1995. His research interests range from advanced structural materials to thin film systems related to the reliability of microelectronic and MEMS devices.

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Christian Lavoie, a research staff member at the IBM Watson Research Center, received his Ph.D. from the University of British Columbia in 1995. He has written or co-authored more than 90 scientific publications and holds 20 patents. Radhey S. Mehrotra, Head of the Fast Reactor Fuels Section at the Bhabha Atomic Research Center, Trombay-Mumbai, India, received his Ph.D. from the Indian Institute of Technology, Mumbai, in 1990. His research and development activities are in the area of ceramic-based nuclear fuel systems. Yuri Mishin, Professor of Materials Science at George Mason University, Fairfax, Virginia, received his Ph.D. from the Moscow Institute of Steel and Alloys in 1985. His research interests are atomistic modeling and computer simulation of diffusion processes and interfaces. Shyam P. Murarka, an emeritus professor at Rensselaer Polytechnic Institute, received his Ph.D. in chemistry from Agra University, India, and his Ph.D. in metallurgy and materials science from the University of Minnesota in 1970. His research interests are diffusion and defects in metals, oxides/insulators, and semiconductors, and thin film metallization. Jean Philibert, an emeritus professor at the Université de Paris-Sud, Orsay, received his Docteur ès Sciences from the University of Paris in 1955. His research interests are structural transformations, electron probe microanalysis, diffusion, and plasticity. He has edited or written seven books and 190 publications. He has received the Grande Médalle Henry Le Chatelier (Société Francaise de Métallurgie), Presidential Award of the Microbeam Analysis Society, and Gold Medal from Acta Metallurgica; he is a member of the Academa Europea. Robert Rosenberg, Manager of the Materials and Reliability Sciences Department at the IBM Watson Research Center, received his Ph.D. from New York University in 1962. He is a charter member of the IBM Academy of Technology. His research interests are degradation mechanisms, reliability of metallic conductor films, and metal-dielectric integration issues. He has published more than 70 papers and edited seven treatises in related fields. Thomas Strunskus, a member of the research group of Prof. C. Wöll at Ruhr-University, Bochum, received his Ph.D. in physical chemistry from the University of Heidelberg in 1988, working with Prof. M. Grunzein.

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His research interests are the vapor deposition of polyimide and formation of metal/polymer interfaces. Axel Thran, a staff scientist at Philips Research in Hamburg, Germany, received his Ph.D. from the University of Göttingen. He has conducted research on diffusion of metals and gases in polymers, working in the group of Dr. Franz Faupel at the University of Kiel. Gyanendra P. Tiwari, a senior scientist in the Department of Atomic Energy, Board of Research in Nuclear Sciences at the Bhabha Atomic Research Center, Trombay-Mumbai, received his Ph.D. from Banaras Hindu University, Varanasi, India, in 1971. His research interests include diffusion in solids, inert gas behavior in solids, hydrogen embrittlement of steels, and characterization of powders. Vladimir Zaporojtchenko, currently working in the group of Prof. Franz Faupel at the University of Kiel, received his Ph.D. in physics under Prof. V.L. Ginzburg at the Academy of Science of the USSR in 1975. His research interests include growth of thin metal films on polymer, metalpolymer interfaces, and novel metal-polymer nanocomposites. He has published more than 100 research papers.

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Preface My 12-year-old granddaughter Nina Alesi once asked me, “Grandpa, you are a scientist at IBM, so what do you do?” I tried to reply, “Oh, I watch atoms move…” But before I could finish this sentence, my 7-year-old grandson Vinnie interjected, “Grandpa, do atoms play soccer?” This book is about the games atoms play in diffusion and various other properties of materials. While diffusion has been studied for more than 100 years in solids, its importance, excitement, and intellectual challenges remain undiminished with time. It is central to understanding the relationship between the structure and properties of naturally occurring and synthetic materials, which is at the root of current technological development and innovations. The diversity of material has led to spectacular progress in functional inorganics, polymers, granular materials, photonics, complex oxides, metallic glasses, quasi-crystals, and strongly correlated electronic materials. The integrity of complex materials packages is determined by diffusion, a highly interactive and synergic phenomenon that interrelates to the microstructure, the microchemistry, and the superimposed physical fields. While the various physico-chemical properties of the materials are affected by diffusion, they determine diffusion itself. This book, which is intended to document the diffusive processes operative in advanced technological materials, has been written by pioneers in industry and academia. Because the field is vast, it has only been possible to address some critical materials where systematic investigations have been conducted and reasonable understanding of the underlying processes has been reached. The book may be considered a sequel to Diffusion Phenomena in Thin Films and Microelectronic Materials, edited by Devendra Gupta and P. S. Ho, published in 1988 by Noyes Publications. Chapter 1 provides phenomenological examples of diffusion in bulk solids and thin films that have a variety of atomic arrangements, such as single crystals, poly-crystals, quasi-crystals, and noncrystalline amorphous solids. This is followed by discussions of relationships of solidstate diffusion with other bulk physical properties in Chapter 2. Chapter 3 discusses atomic computer simulations of diffusion processes in elemental solids, nonstoichiometric compounds, and grain boundaries. Chapter 4 discusses bulk and grain boundary diffusion in intermetallic compounds, which are important to super alloys for high-temperature applications. Principles of diffusion barriers used in semiconductor devices and circuits

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PREFACE

are elucidated in Chapter 5, with numerous examples. Chapter 6 is concerned with the theory and applications of reactive phase formation, with emphasis on silicides and oxide, which are so vital to the microelectronics industry. In Chapter 7, metal diffusion in polymers and on polymer surfaces is discussed, as polymers are also assuming importance in microelectronics applications. Chapter 8 covers the measurement of stresses in thin films and their relaxation behavior, since the latter is also a diffusive process. Chapter 9 delves into electromigration in Cu thin films. Copper is currently a metallization of choice on Si chips in high-end computers. Finally, Chapter 10 discusses diffusion of perovskites, which consist of the newly discovered high-temperature superconducting cuprates and some piezoelectric ceramics, used as actuators and sensors. The contributors and I hope that this book will be useful to people in both industry and universities who are interested in applications of diverse materials in various combinations. I edited and prepared the manuscripts electronically, possessing only rudimentary knowledge of word processors. Credit for the successful completion of the book goes to my family as a whole: my daughter Chitra and her husband Vincent Alesi, my older son Sudhir Gupta, and my younger son Devratna Gupta, who taught and helped me with various computer-related skills. It is a pleasure to thank Karen Ailor of Eugene, Oregon, a freelance writer and editor, for reading all the manuscripts carefully and bringing them into conformance with standard English. I am also grateful to James Leonard of the Strategic Knowledge Group at the IBM Thomas J. Watson Research Center in Yorktown, New York, for cheerfully providing the needed library support. Devendra Gupta Yorktown Heights, NY December 1, 2004

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1 Diffusion in Bulk Solids and Thin Films: Some Phenomenological Examples Devendra Gupta IBM T. J. Watson Research Center Yorktown Heights, New York

1.1

Introduction

The science of diffusion had its beginnings in the early nineteenth century, although the metal artisans of antiquity used the phenomenon to make such objects as the Damascus swords by the cementation process and gilded bronze and copper wares from gold amalgam. Mercury in the amalgam was later oxidized, leaving behind a gold gilding film. John Dalton in 1808 is usually credited as the earliest person to describe diffusion.[1] Thomas Graham studied the diffusion phenomenon in the 1828 to 1833 period with simple, elegant, and definitive experiments.[2] His major observations were that diffusion, or spontaneous intermixing of two gases in contact or separated by porous membranes, is effected by an interchange in position of indefinitely minute volumes. The mixing rate depends on the concentration difference and is inversely proportional to the square root of the density of the gas. Furthermore, he established that “diffusion followed a diminishing progression – at longer intervals of time the mixing process decreased,” and that the diffusion rates in liquids are much slower than in gases. These concepts later became the bases for mathematical treatment of the diffusion process. The next major landmark in the theory of diffusion came from Adolf Fick, a young pathologist at the University of Zürich. He was interested in movement of water confined by membranes, a basic process of organic life as we know it. His work titled “Über Diffusion” was published in the prestigious Poggerdorf’s Annalen der Physik.[3] Fick’s original formulation is still considered a basis for later modifications, notably by Albert Einstein,[4] on Brownian motion in liquids. Recognition of diffusion in solids had to wait until 1896, when W. C. Roberts-Austen reported the first measurement of Au in Pb metal.[5] Diffusion is now considered ubiquitous in all three states of matter. In materials of technological interest, diffusion has assumed

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DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

great importance in their design, fabrication, and performance. Diffusion research during the past hundred years is replete with numerous examples from industry such as sintering, power generation, lighting, metal forming, aviation, space, and information technology, to mention a few. The objective of this chapter is to provide some basic and useful phenomenological aspects of diffusion for engineering materials. It is illustrated by some model materials, in the context of enormously differing time, temperature, and length scales used from industry to industry. A comprehensive discussion of the driving forces for diffusion, both in linear and nonlinear regimes, is included, which should be relevant in the context of the current technological trends of using thinner and multilevel metallization schemes. Typical data on diffusion in bulk metals, semiconductors, and thin metallic films, given in tabular form, with discussions on the role of variable microstructure and chemistry, should be useful to the materials community in general. Because the field of diffusion is vast, it was impossible to provide a comprehensive survey of diffusion as a scientific discipline. The recent literature[6
11] contains discussions of topics that could not be covered here.

1.2

Diffusion in Single Crystals

1.2.1 Mathematical Basis This section discusses the general aspects of diffusion without considering the role of the microstructure of the samples, which is discussed in Sec. 1.3. The specimens only contain lattice point defects such as vacancies, divacancies, interstitials, and impurity atoms in thermal equilibrium. Textbooks by Shewmon,[10] Carslaw and Jaeger,[12] and Crank[13] present detailed mathematical solutions of specific diffusion experiments.

1.2.1.1 Fick’s First Law Fick’s first law defines the diffusion coefficient (D
) of the component 1 and flux in an inhomogeneous single-phase binary alloy due to concentration gradient as:




∂C J1 
D1 1  C1v, ∂x T

(1)

where J1 is the flux at time t of atoms of the component 1, C1 is its concentration, n is the velocity of mass that moves due to application of forces such

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as electromigration or a thermal or chemical potential gradient, and x is the distance into the sample taken parallel to the concentration gradient. If the concentration of the diffusant can be defined at the entrance and exit surfaces of the sample separated by distance x in steady-state condition, the diffusivity can be measured easily by monitoring the flux in the absence of a driving force. A number of such simple experiments can be found in the literature; for example, diffusion of Ni in Si has been studied by Thompson et al.[14] using such a scheme. Because the measurement of flux contains no information on the distribution of the diffusant inside the sample, an independent study of the microstructure of the specimen and its chemistry is usually conducted to describe the diffusion conditions and path in the physical sense. It is possible to measure the diffusion coefficient in the absence of external forces, including the chemical gradient, using isotope techniques. The diffusant is the isotope of the host species; it may be a radioactive tracer or a stable isotope. The latter is detected by mass spectroscopy techniques such as the one used in secondary ion mass spectroscopy. Such measurements are termed self-diffusion and are generally denoted by an asterisk or by the isotope used. Fick’s first law then reduces to:




∂C∗ J∗ 
D∗  . ∂x T

(2)

For diffusion in a chemical gradient, the measured diffusion coefficient is affected by the motion of all the atomic species and is consequently termed interdiffusion coefficient. As will be seen in Sec. 1.2.4, it contains thermodynamical contributions due to the changing chemistry of the alloy. The∼ interdiffusion or chemical diffusion coefficient is generally denoted by D; consequently, Fick’s first law is written as:




∼ ∂C J 
D  . ∂x T

(3)

In three-dimensional vector notations, the general statement of Fick’s first law is: ∼

J 
DC.

(4)

1.2.1.2 Fick’s Second Law Fick’s second law describes diffusion in a non-steady-state condition when the concentration changes with time but particles are neither created

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nor destroyed. Thus, we have the continuity equation: ∂C  
J. ∂t

(5)

Fick’s second law can be written by combining Eqs. (2) and (5), ∂C   D 2C, ∂t

(6)

for linear flow and constant D (implying that it is not a function of position). For variable D, Fick’s second law is written as:






∂C ∂ ∂C    D . ∂t ∂x ∂x

(7)

Fick’s second law is the basis of most diffusion measurements in solids in general. It has been widely used in samples in rod, plate, and thin-layer geometry. It has been used for measurements of diffusion in single-crystal specimens, and along grain boundaries and dislocations. These individual applications are discussed below.

1.2.1.3 Instantaneous Source Geometry Imagine an infinitesimally thin layer of diffusant of strength M deposited on the face of a cylindrical specimen so that its thickness, the Dirac d, is a lot less than the diffusion distance 2(Dt)1/2, corresponding to the initial condition. Assume that there is no flux on the surface or in the imperfections, such as grain boundaries and dislocations. When the boundary conditions: C(x, 0)  Md(x)

(8)

∂C  (0, t)  0 ∂x

(9)

and

are maintained, the Gaussian solution to Fick’s second law is given by:

)exp(
x24Dt), C(x, t)  (M2pDt

(10)

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where t is time of diffusion and C is concentration at x and t. This geometry has been widely used in the radioactive tracer work where a Fermi-Dirac d of diffusant distribution can be initially obtained and the mass and thickness involved are infinitesimally small. Much of the basic diffusion work in solids has been carried out in this way.

1.2.1.4 Thick-Layer Geometry Generally, it is not possible to obtain Dirac d function in nontracer diffusion experiments. Consequently, a thick geometry constitutes a situation where the initial thickness of the source of the diffusant (h) is of the order Dt). The initial source condition for the of the diffusion distance (2 thick-layer geometry for t  0 is given by: C(x, 0)  C0

for

hx0

(11)

xh

(12)

and C(x, 0)  0

for

For t  0, the boundary condition is: ∂C  (0, t)  0. ∂x

(13)

The solution is then given by:











C0 xh x
h C(x, t)   erf 
erf  . 2 2 Dt 2 Dt

(14)

Thick-layer geometries commonly arise during diffusion in thin-film couples as well, which is described in Sec. 1.4.1. (The error function, erf, and its complement, erfc, are defined by Shewmon[10] in the form: 2 erf (z)  1
erfc(z)  1
 p x where h  . 2Dt 

e z


h 2

0

dh,

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1.2.1.5 Infinite Couple with Constant Surface Composition When a diffusion couple is formed from two samples that have uniform concentrations of C0 and Cl, the initial condition near the interface at x  0 and t  0 is described as: C(x, 0)  Cl

for

x  0,

(15)

C(x, 0)  C0

for

x  0,

(16)

and the solution is given by:






1 x C(x, t)
C 0   erfc  . 2 2Dt  Cl
C0

(17)

1.2.1.6 Diffusion Under a Chemical Gradient: Boltzmann-Matano Analysis It has been known for a long time that diffusion under a chemical gradient produces a net flow of matter relative to the initial interface. In Fig. 1.1(a), such a diffusion couple consisting of an alloy AB and a pure metal A is shown schematically from the article by Lazarus.[15] To keep track of the transfer of matter, inert markers were placed at W, the initial interface. The two species diffused at unequal rates, and the mass balance was maintained by the compensating flow of vacancies. After diffusion, the markers were found displaced relative to the edges of the couple, and porosity was found at the original interface due to condensation of supersaturated vacancies. To calculate the diffusion coefficient, Eq. (7) was initially solved by by Matano[17] in 1933. Boltzmann in 1894[16] and derived empirically ∼ Their solution was based on a single D describing diffusion and bulk motion as the same process. Thus, for the condition that the difference , as is the case in real experibetween Cl and C0 is large and x  2Dt ments, the diffusion coefficient will vary with composition and along the distance x. In Fig. 1.1(b), a Matano plot (concentration versus the distance x) is shown schematically. The initial conditions are: C  C0

for x  0, at t  0,

(18)

C0

for x  0, at t  0.

(19)

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(a)

(b) Figure 1.1. (a) Diffusion in an inhomogeneous diffusion couple showing Kirkendall effect: formation of porosity, motion of the tungsten wire markers (W). The Matano interface position is shown at M. (After Lazarus[15]) (b) Procedure to compute inter∼ diffusion coefficient D. Matano interface position is first obtained by equating areas ∼ above and below the composition curve. D (C ) is then computed from the slope of the tangent at the chosen composition C and the hatched area under the curve, according to Eq. (20).

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Following the determination of the concentration profile, the location of the Matano interface along x is determined either graphically or by numerical techniques such that the ∼two areas under the curve are equal. The value of chemical diffusivity D is then computed at the desired composition C according to the equation:


  xdc.

1 dx D(C ) 
  2t dc ∼

C

C

(20)

0

The term (dxdc) is actually the slope of the curve at the concentration C . The integrated hatched area under the curve between CC0  0 and ∼ CC0  C is then computed. Reasonable values for D(C ) can be computed in the range 0.1  C  0.9. Outside this range, the errors in computing the slope and the area under the curve become rather large.[10] It is also possible to determine the diffusivities for the individual components in an alloy using Matano-type analysis. Smigelskas and Kirkendall[18] studied the motion of Mo markers in the CuZn/Cu diffusion couple and measured unequal diffusion coefficients for the Zn and Cu species. The marker velocity (vm) was related to the diffusivities as: ∂C(Zn) vm  (DZn
DCu) . ∂x

(21)

∼

The interdiffusivity D itself was later related by Darken[19] to the intrinsic diffusivities as: ∼

D  [DZn C(Cu)  DCu C(Zn)].

(22)

Because Eqs. (21) and (22) involve two unknowns, DZn and DCu can be determined individually. A complete description of the Darken’s analysis for interdiffusion in non-ideal solid solutions is given in Sec. 1.2.4.

1.2.2 Atomistic Nature of Diffusion 1.2.2.1 Diffusion Mechanisms A few possible mechanisms considered for atomic diffusion in single crystals are shown in Fig. 1.2. The thermodynamics and computer modeling of these diffusion mechanisms are discussed in Chapters 2 and 3, respectively. They are briefly discussed here.

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(a) Interchange and ring mechanism. The diffusive motion in this mechanism may take place by a correlated rotation of two or more atoms about a common center without involving a defect. This mechanism has been found energetically unfavorable in most solids. (b) Interstitial mechanism. Small interstitial atoms can readily diffuse by meandering in the interstices. Commonly,

Figure 1.2 Possible diffusion mechanisms in solids. (After Lazarus[15])

9

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gas atoms such as O, N, H, and C diffuse easily in the open lattices of BCC metals, for example, Fe, Ta, W, and Mo. There may be several variations of interstitial diffusion that involve displacing a neighboring atom to an interstitial position, termed interstitialcy or kickout mechanism, or a correlated motion of a line of displaced atoms, the crowdion mechanism. (c) Vacancy mechanism. Atomic diffusion into the missing atomic sites (vacancies) has been found to be the most favorable. Indeed, vacancies are present in pure metals and alloys at all temperatures; their concentration at the melting temperature (Tm) is about 0.01%. This process has been studied extensively in the past century by a variety of techniques, and the results support this mechanism overwhelmingly on a wide basis in metals, alloys, and oxides. The vacancies may also be in the form of dimmers (the divacancies), particularly near the melting temperature (Tm). (d) Sub-boundary mechanism. The diffusing atom moves along interconnecting dislocation pipes, which result from naturally occurring low-angle boundaries. This mechanism, discussed later, has been found to operate at low temperatures, typically 0.5 Tm, the absolute melting temperature, in metals such as Au, Ag, and Cu. (e) Relaxion mechanism. The diffusing atom moves more or less freely within a disordered group of atoms within the lattice. This mechanism has been ruled out in most crystalline solids but has been considered in recent years in the context of radiation damage in amorphous metallic alloys and some polymers. This section focuses on the diffusion mechanism by vacancies and to a lesser extent by interstitial atoms. Figure 1.3 shows the energy barrier as a function of atomic position for a jumping atom under zero driving force, as described by Manning.[20] Due to the absence of any driving force, the probability of the forward and backward jumps is the same. Hence, the velocity of the vacancy flow v in Eq. (1) is zero. A successful jump is executed when an atom gathers the free energy Gm, goes over the barrier, and occupies an equivalent equilibrium position by an exchange with a defect like a vacancy. The probability W of acquiring the energy Gm is given by the Boltzmann factor as: W  vo exp(
Gm kT ),

(23)

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Figure 1.3 Schematic diagram for diffusion showing displacement of an atom in the lattice when the driving force is zero. (After Manning[21])

where no is the atomic vibration frequency. The success of the jump, however, depends on the availability of a defect in the adjoining position, which is also given by a Boltzmann factor. Hence, the average probability Nd of finding the proper defect may be written as: Nd  Z exp(
Gf kT ),

(24)

where Gf is free energy necessary to form the defect and Z is a coordination factor that depends on the crystal type. As mentioned earlier, the vacancy concentration in most metals and alloys is generally ~0.01% at the melting temperature and even smaller at lower temperatures. The frequency of jumping Γ is obtained by multiplying Eqs. (23) and (24): Γ  Zvo exp
(Gm  Gf).

(25)

For the three-dimensional case, the diffusion coefficient is expressed as: a2 D  Γ  f , 6

(26)

where a is the nearest neighbor atomic distance. For tracer diffusion measurements, which are discussed in Sec. 1.2.3, a correction term f 1 should be introduced for the non-random character of the jumps. The factor f is known as a correlation factor. Its full implications are discussed by Manning;[20, 21] it is briefly discussed in Sec. 1.2.3. Finally, the diffusion coefficient can be written as:




1 D   a2 Z f vo exp[(Sm  Sf)k]  exp
[( Hf  Hm)kT], 6

(27)

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where ∆Hf and ∆Hm are enthalpies for the formation and motion of the defect, and Sf and Sm are the corresponding entropy terms, respectively. The temperature dependence of the diffusion coefficient is generally of the Arrhenius type and is written as: D  Doe
(QkT),

(28)

1 Do   a2 Z f vo exp[(Sm  Sf)k]. 6

(29)

with




From the measurement of the diffusion coefficient, knowledge of the crystal geometry (the coordination number and the lattice parameter), vo (related to Debye qD), and an independent measurement of the correlation factor, f, it is possible to evaluate the total entropy factor. For the separation of formation and motion components of the entropy and enthalpy, separate experiments are needed, such as the resistance studies of the quenched-in defects, the positron annihilation, and the simultaneous length and lattice parameters.[10, 22] Note that the diffusion coefficient and the various factors involved therein are basic properties of the solids. Table 1.1 lists self-diffusion data for several important pure elements. While most metals diffuse by a vacancy mechanism, self-interstitial atoms are important for diffusion in covalently bonded solids like Ge and Si, at least at high temperatures.[23] When an interstitial atom is formed and a vacancy is left behind, it is known as a Frenkel defect. A notable example for the formation of Frenkel defects is radiation damage in solids. The energy involved in Frenkel defect formation equals the sum of the vacancy and interstitial formation energies less the binding energy. The Frenkel defect is usually very mobile and is accompanied by mass transport. The interstitial atoms may also be of the impurity type, which fit into the interstices in lattice easily by virtue of their small size. Consequently, their formation energy is negligible, and in Eq. (27), the terms Sf and ∆Hf may be omitted. Hence, diffusion of the foreign or impurity interstitials may be very fast. Gas atoms such as H, C, O, and N commonly occur as interstitials in BCC metals such as Fe, W, Nb, and Ta, and in HCP metals such as Ti, Zr, and Hf. In the Si lattice as well, most transition metal impurities enter as interstitial atoms, although their solubilities rarely exceed one part per million.[14, 23, 24]

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Table 1.1. Self-Diffusion in the Lattice in Some Pure Elementsa Element Aluminum Antimony ‘ c ⊥c Beryllium ‘ c ⊥c Cadmium ‘ c ⊥c Chromium Cobolt-Paramagnetic Copper Germanium Gold a-Hafnium Indium ‘ c ⊥c a-Iron-Paramagnetic g-Iron d-Iron Lead Lithium Magnesium ‘ c ⊥c Molybdenum Nickel Niobium Palladium Platinum Potassium Silicon Silver Sodium Tantalum a-Thallium ‘ c ⊥c b-Thallium b-Tin ‘ c ⊥c a-Thorium a-Titanium b-Titanium Tungsten g-Uranium Vanadium Zinc ‘ c ⊥c b-Zirconium

Activation Energy Q l (kJ/mol)

Frequency Factor Dol (10
4 m2/sec)

123 200.6 149.6 164.7 157.2 76.1 79.8 435 283.0 199 299 170 370 78.2 78.2 284 284 283.7 238.3 109 56.39 134.6 135.8 385.4 278 401.2 265.9 285.1 40.8 484 170 35 412.6 95.7 94.5 83.4 107.0 105.0 320.2 169 A  130.4; B  250.8 586 110.8 307.9 393.5 91.5 96.14 A  116; B  273

0.047 56.0 0.1 0.62 0.52 0.05 0.10 970 0.83 0.16 32 0.04 0.86 2.7 3.7 0.49 0.49 0.18 1.9 1.37 0.39 1.0 1.5 0.1 0.92 1.1 0.21 0.33 0.31 1460 0.04 0.004 0.124 0.4 0.4 0.7 7.7 10.7 1.2 6.6  10
5 (1) 3.58  10
4 ; (2) 1.09 1.88 1.12  10
3 0.036 214.0 0.13 0.18 (1) 8.5  10
5; (2) 1.34

13

Comments/ References

2 exponentials fitb

880 to 1360°C 1360 to 1830°C

2 exponentials fitb

a. Data are obtained from N. L. Peterson, Solid State Phys., 22:429–430 (1970), and from a more recent compilation in Gupta and Ho.[11] l
AkT l
BkT b. Dl  D01 e  D02 e .

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1.2.2.2 Effect of Substitutional Impurity Atoms The presence of substitutional impurity atoms in single crystals modifies diffusion of the host atoms in addition to their own diffusion, largely because of their interaction with the vacancies by virtue of their differing electronic structure. The former is known as solvent diffusion enhancement. In Fig. 1.4, a five-frequency model for vacancy–atom jumps in FCC crystal is shown in a dilute alloy after Manning[20, 21] and LeClaire.[25] The solute atoms by themselves diffuse using vacancies as first-nearest neighbors with the frequency w2, so that their diffusion coefficient D2 can be expressed as: D2  a2w2 f2 exp[
(Gvf  Gs
v)kT],

(30)

where Gs
v is the solute-vacancy-association free energy. The correlation factor f 2 is no longer constant and depends on all the jump frequencies in a complex way. Furthermore, the activation energy for D2 can also be perturbed to the extent of 25% of the value for the solvent self-diffusion. The general expression for solvent enhancement of diffusion, [(D1(c)
D1(0))D1], usually measured by the solvent tracer, may be written as: D1(c)  D1(0){1  b1c  b2c2  .....},

(31)

where c is the concentration of the solute and the subscript 1 is used to denote the solvent species. For a dilute alloy with 1% solute, the higher order terms are negligible and the diffusion enhancement is generally linear. The strength of enhancement, measured by the term b1, is related to the perturbation caused by the solute-vacancy binding to the first-nearest neighbor jump frequencies in an FCC lattice. Figure 1.4 shows that wo is the general atom frequency in the presence of an unbound vacancy. w1 and w2 are jumps in which vacancy remains bound to the solute atom. In the w3 jump, the vacancy dissociates from the solute atom. Finally, the w4 jump again brings the vacancy adjacent to the solute atom. Manning[21] and LeClaire[25] provided the following kinetic relationship among these jumps:






7 w w1 b1 
18  44    , 2 wo w3

(32)

where b1  (1/c)[(D1(c)
D1(0))D1(0)] in the linear regime. The vacancysolute binding energy is defined by the temperature dependence of VS w4w3  eEb kT. Furthermore, the frequencies w1 and wo are also temperature-

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Figure 1.4 Five-frequency exchange model for host atoms in a dilute alloy: The position of the solute atom is shown at S by a hexagon and of vacancy at V by a square. (After Manning[20] and LeClaire[25])

dependent, so that w1wo  eBkT. The solute-vacancy binding energy EbVS ≈ 10 kJ/mol (0.1 eV); perhaps B also has the same magnitude. The individual measurements EVS b and B are usually very complex and involve knowledge of correlation factors as well. In any event, while the enhancement factor is determined by exponential terms, it may not always be positive; at certain temperatures and compositions, it may even be negative. Figure 1.5 provides an example in the self-diffusion measurements in Pb and Pb-Sn alloys.[26] Initially, addition of the Sn solute results in de-enhancement; only at sufficiently higher concentrations and lower temperatures is real enhancement of the solvent species observed.

1.2.3 Pressure and Mass Dependence of Diffusion Pressure and mass dependence of diffusion are discussed briefly here. They are addressed in detail in Chapter 2.

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Figure 1.5 Relative enhancement of solvent (Pb) lattice diffusion in Pb-Sn alloys. (Gupta and Oberschmidt[26])

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1.2.3.1 Pressure Dependence The diffusion coefficient may be expected to be pressure-dependent in view of Maxwell’s thermodynamic relationship:


∂∂GP  V,

(33)

T

where volume V is to be associated with the diffusion process. From Eqs. (27) and (28), the dependence of D on the hydrostatic pressure may be written as: ∂ ln D ∆V ∂ ln a g f  
 
 ,
 ∂P kT ∂P 2

G

T

T

(34)

where gG is the Grüneisen compressibility coefficient. The activation volume may be further separated into a sum of the formation and motion terms, Vf  Vm, for processes involving point defects in solids, similar to Eq. (27). Furthermore, the activation volume, entropy, and enthalpy are interrelated by the volume thermal expansion coefficient a and the isothermal compressibility b given by Lawson[27] and Keyes,[28] respectively, as: S  a Vb

(35)

V  4b Q.

(36)

and

The values of ∆V are typically of the order of an atomic volume (Vo), even in closed packed lattices, which imply only a small relaxation around a vacancy. Table 1.2 lists activation volumes for some pure metals and elemental semiconductors. Rice and Nachtrieb[29] have shown that ∆V and Q can also be empirically related to the heat of melting Lmelt and the volume of melting Vmelt as: Q  Lmelt VVmelt .

(37)

Experimentally, measurable changes in the diffusivity (10%) occur at hydrostatic pressures of the order of a few kilobars in metals and at temperature 0.5Tmelt.

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Table 1.2. Isotope Effect for Various Diffusion Mechanisms in Some Elements Mechanism/ Host (Isotopes)

Correlation Factor 0f1

Kinetic Factor ( K) [Eq. (38)]

Ag/Ag*(110,105): Lattice

0.782

0.92

0.718(1000K)a 0.639(1150K)

0.9

Ag/Ag*(110,105): Grain boundary

...

...

0.46b

1.1c

Au/Au*(199,195) Lattice

0.782

0.9

0.704(1127K)d 0.653(1327K)d

0.9

Cu/Cu*(67,64) Lattice

0.782

0.87

0.669(1200K)a

0.727

0.5e

0.348(300–350K) f 0.33

0.5

0.86

0.26(1173K)g

Isotope Effect ( E)

Activation Volume ( VVo)

A. Vacancy FCC Metals

BCC Metals Na Diamond Lattice Ge/Ge*(77,71)

0.33

B. Interstitial

1.0

0.68

–

–

C. Divacancy

0.475

1.45

–

–

D. Interstitialcy

0.667

1.99

–

–

E. Interchange

0.969

2.66

–

–

a. N. L. Peterson, “Self diffusion in metals,” J. Nucl. Mater., 69–70:3–37 (1978) b. J. T. Robinson and N. L. Peterson, “Correlation effects in grain boundary diffusion,” Surf. Sci., 31:586 (1972) c. G. Martin, D. A. Blackburn, and Y. Adda, “Autodiffusion au joint de grains de bicristaux d’argen soumis a une press hydrstatique,” Phys. Status Solidi, 23:223 (1967) d. C. Herzig, H. Eckseller, W. Bussmann, and D. Cardis, “The temperature dependence of the isotope effect for self-diffusion and Co impurity-diffusion in Au,” J. Nucl. Mater., 69–70:61 (1978) e. M. D. Feit, “Dynamical theory of diffusion, II. Comparison with rate theory and impurity isotope effect,” Phys. Rev., B5:2145 (1972) f. J. N. Mundy, “Effect of pressure on the isotope effect in Na self-diffusion,” Phys. Rev., B3:2431 (1971) g. D. R. Campbell, “Isotope effect for self diffusion in Ge,” Phys. Rev., B12:2318 (1975)

As mentioned earlier, the total activation volume is actually composed of formation and motion components for the mobile defect and is analogous to Eq. (27). The individual components can be measured by monitoring resistance with pressure on and off. This technique is similar

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19

to quenching experiments.[22] The measurements of the activation volumes have been very useful in determining the nature of the diffusing defect and, in fact, supplement the measurement of the isotope effect and the underlying correlation factor, which are discussed in the following section. Thin metallic films held on substrate usually have a bi-axial tensile stress of the order 109 dynes/cm2 (108 Pa), as discussed in Chapter 8, which is of the magnitude mentioned above. Consequently, we may expect ∼10% enhancement of the diffusivity, which is minuscule and insignificant.

1.2.3.2 Mass Dependence In Eqs. (26) and (27), a correlation factor f was introduced to account for the nonrandom character of the atom-vacancy jumps in tracer diffusion measurements. Therein, a reverse jump has a greater than random probability since the vacancy is still located adjacent to the tracer atom. It measures the fraction of jumps that is effective in causing net displacement of the tracer atom. For self-diffusion in cubic lattices, f is a geometrical factor and depends on the crystal type and the mobile defect responsible for diffusion. It is also generally temperature-independent. The values of f for various diffusion mechanisms can be readily computed from the direction of jumps. Table 1.2 lists the values of the correlation factors for various diffusion mechanisms possible in FCC lattices. Since correlation factors differ markedly for different mechanisms, their measurement provides clues to the mobile defect responsible for diffusion. The correlation factor is measured by diffusing two radioisotopes simultaneously in a single crystalline sample. The resulting isotope effect ∆E is related to their diffusivities (Da and Db), masses (ma and mb), and a kinetic factor ∆K as:




m E [Da Db
1] b ma

12




1  f K.

(38)

The kinetic factor ∆K is defined as the fraction of kinetic energy at the saddle point associated with atomic motion in the jump direction. To obtain reasonable precision in ∆E, a mass difference of at least 5% is sought between the two isotopes. The isotopes are generally co-diffused so that the errors in temperature, time, diffusion distance measurements, and so forth are the same and do not figure in the relative diffusivities. Under ideal conditions, the relative difference between the two diffusivities can be measured with a precision of ∼5%. This has been possible because a surprisingly large number of isotope combinations are found to

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fulfill the seemingly stringent mass criterion. In addition, imaginative schemes for nuclear counting have been used based on discrimination of decay time and energy of radioactive emissions, which are discussed in detail by Rothman.[30] The kinetic factor ∆K has the lower and upper bounds of zero and unity, respectively, for point defects. For FCC and HCP lattices, the measured values of ∆K are on the order of 0.9 for metals such as Cu, Ag, and Zn. For BCC lattices, smaller values of ∆K  0.68 are observed for metals like Na and Cr and are attributed to a relatively open saddle-point structure rather than a change of diffusion mechanism. Finally, note that the factor ∆K from the isotope effect and the activation volume measurements [Eq. (34)] show numerically similar values for the same crystal lattice and the diffusing defect. Thus the values of ∆K and ∆VVo are considered complementary in deciding the diffusion mechanism. For a vacancy mechanism in close-packed metals, both ∆K and ∆VVo 0.9. In open lattices like BCC, the corresponding values are of the order of 0.68 and 0.5, respectively, for the vacancy mechanism. For the interstitial diffusion mechanism, ∆K ∼ 1 and ∆VVo 0 may be expected. For extended defects, such as the divacancies, interstitialcy, and grain boundaries, as listed in Table 1.2, ∆K and ∆VVo  1 may be expected.

1.2.4 Linear Chemical Diffusion Regime: Finite Driving Force F on Individual Atoms In Fig. 1.6, the nonzero driving force F on individual atoms results in a net flux in the right-hand direction. Consequently, the mass velocity term v in Eq. (1) cannot be neglected. The driving force F makes the

Figure 1.6 Schematic diagram for diffusion showing displacement of an atom in the lattice when the driving force is finite, defined by the chemical potential gradient ∂m∂x. (After Manning[21])

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energy at successive lattice sites differ by: ∆Gm  aF.

(39)

Assuming that the energy barrier undergoes an equal change on either side, the net number of jumps can be written as: Γ
Γ
 Γ(ee
e
e),

(40)

where Γ has been defined in Eq. (25) and e is given by: 1 e   ∆Gm kT  aF2kT. 2

(41)

Combining Eqs. (40) and (41), the external force term may be written as:



  sinh aFkT. 2

(42)

Some atomic driving forces are listed in Table 1.3. These forces, with the exception of perhaps centrifugal force, are commonly encountered in Table 1.3. Driving Forces in Diffusion

Driving Force

Atomic Force (F)

Governing Parameter

Example

Electro-Migration

ZeE

Ze (effective charge)

Thin-film stripe failures under high currents

Thermo-Migration

Q* ∂T   T ∂x

Q* (heat of transport)

Soret effect in ULSI solder failures

g (activity coefficient)

Kirkendall void formation

m (effective molecular mass)

Creep in aero-engines, isotope enrichment

U (interaction energy)

Hydrostatic pressure, stresses in thin-film, diffusion creep

Chemical Inhomogeneity [non-ideal chemical potential gradient (∂m ∂x)]

∂
ng
kT  ∂x

Centrifugal Force

mw2r

Stress Field

∂U  ∂x

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the thin-film metallization used in the interconnection technology of modern computers. Each driving force deserves an exhaustive discussion in its own right, which may be found in earlier review articles by Shewmon,[10] Huntington,[31] and Tu.[32] Here, we discuss the case of the chemical composition variation that gives rise to chemical potential gradients in nonideal solid solutions. The phenomenon, though old, has assumed greater significance in the context of recent development of the artificially layered thin films known as manmade superlattices and multilayered thin films in general. The atomic force over N lattice planes due to the chemical potential gradient (∂m∂x) may be written as: kT F   [ln C2g 2
ln C1g1], aN

(43)

where γ1,2 are the chemical activity coefficients for the non-ideal solution and C1,2 are the concentration terms; the subscripts 1 and 2 refer to the specific component of the binary solution. Despite the large variation of concentrations and activity factors (for example, 10  C2g2C1g1  1000, over, say, 10 atomic planes), the force factor remains small in view of its logarithmic nature. Under these conditions, Eq. (42) can be simplified as: aF



    1, kT

(44)

which is known as the linear region. Hence, the drift term in Eq. (1) can be identified from its basic definition and random walk theory[21] as: C1v  C1 a2FkT

(45)

D1F v  . kT

(46)

or

The quantity D1 kT is denoted by B1. In general, it is a tensor. Equation (46) is commonly known as Nernst-Einstein relationship. For diffusion in concentrated alloys that display non-ideal behavior, Fick’s first law for the first component may then be expressed in a generalized form as: J1  B1 FC1.

(47)

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– For a multicomponent system where Fi is the partial molal free energy of the ith component, Eq. (47) assumes the form: 1 ∂
F Ji 
 i BiCi . N ∂x

(48)

From this generalized Fick’s first law and the non-ideal behavior of the alloy, Darken derived the following equation for chemical diffusion:[19]





d ln g1 ∼ D  (D1∗C2  D2∗C1) 1   , d ln C1

(49)

where D1* and D2* are the tracer diffusivities of the two components of the alloy, and g1 and g2 are the chemical activity coefficients. The terms in the square brackets are collectively known as the thermodynamic factor. It should now be obvious that without the knowledge of the thermodynamic factor, it is not possible to account fully for the observed chemical diffusion. Only Boltzmann-Matano-type diffusion experiments, coupled with marker velocity measurement alluded to in Sec. 1.2.1 [see Eqs. (21) and (22)], fully account for chemical diffusion. In the regular-solutiontype alloys, the factor g is independent of the composition, but the thermodynamic factor still remains finite. Only in ideal solutions do g  1 and the chemical diffusivities become identical with those obtained using the tracer techniques. Note that Darken’s Eq. (49) accounted for uphill diffusion against the composition gradients first observed in a Fe-0.4%C/Fe0.4%C-4%Si couple. The explanation of this effect is that addition of Si increases the chemical potential of the Fe-C system.[33] This observation underscores the fact that in the chemical diffusion, the basic driving force is the chemical potential gradient, and in some limiting cases it becomes the concentration gradient. The latter may be violated in the uphill diffusion, which should be a common occurrence in thin-film metallic alloy couples.

1.2.5 Nonlinear Chemical Diffusion Regime When sharp chemical potential gradients exist over small distances,

10 atomic spacing, Eq. (42) cannot be simplified linearly since the boundary condition in the inequality [Eq. (44)] is not satisfied. Such a situation arises in the spinodal decomposition of a single phase where the curvature of the free-energy composition plot ∂ 2 F∂C2 becomes negative.

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Decomposition by uphill diffusion occurs as it lowers the free energy. Other examples of the existence of sharp chemical potential are the artificially layered polycrystalline films investigation by Cook and Hilliard[34] and compositionally modulated amorphous thin films studied by Rosenblum et al.[35] and by Greer and Spaepen.[36] The spinodal solid-state transformation is known as continuous; that is, it does not involve any nucleation and is assisted by diffusion that occurs under a sharp chemical potential gradient. This subject is briefly discussed here. Cahn and Hilliard[37] and the recent review article by Doherty[38] provide background and in-depth discussion. For spinodal decomposition, Doherty[38] has shown that the Darken’s term is the controlling factor in determining the curvature of the freeenergy and composition given by:









1 1 ∂2F d ln g1 2  RT    1   , ∂C C1 C2 d ln C2

(50)

where the notations have the same meaning as previously defined. We only need to change the various chemical concentrations to keep them consistent with those used by Doherty[38] so that C1  1 ∼
C2  1
C. Combining Eqs. (49) and (50), the chemical diffusivity D in the spinodal decomposition may be written as: ∼ C C d 2F D  [D*1 C2  D2* C1] 12 , RT dC2

(51)

and the diffusion mobility MD as: CC MD  [D*1 C2  D*2 C1] 12 . RT

(52)

To describe the thermodynamics of the inhomogeneous solid solution, a correction term is introduced as:


 dF  dC




dF   inhomo dC

∂2C
2k . homo ∂x2

(53)

Herein, k is the gradient-energy coefficient  NkTcy 2, where Tc is the critical temperature below which the homogeneous alloy tends to unmix and y is the “interaction distance” of atoms.[39]

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From Eqs. (52) and (53), Fick’s two laws for the inhomogeneous solid solution may be rewritten as: ∼ ∂3C d 2fo  2M k   J 
MD C D ∂x3 dC 2

(54)

dC ∂2C ∂4C   MD f o 
2M k  , D dT ∂x2 ∂x4

(55)

and

∂fo where fo  (dFdC)homo and f o  . ∂C2 The nonlinear differential equation above has been solved by Cahn and Hilliard[37] and Cahn[39, 40] for spinodal decomposition. An important spinoff from this theory has been the understanding of diffusion at low temperatures in compositionally modulated thin films, as discussed by Cook and Hilliard.[34] Accordingly, a harmonic composition modulation C(t ) will grow or decay with time t as: ∼

C
C0  A0 exp [
Dlh2t]cosh x,

(56)

amplitude of the where C0 is the average composition, A0 is the initial ∼ modulation, h(2pl) is the wave number, and D∼ l is the effective interdiffusivity. The macroscopic interdiffusivity and Dl are related as: ∼

∼

Dl  D [1  2kh2f0].

(57)

∼

be assumed For small deviations from homogeneity D, f0, and k can ∼ as a function to be composition-independent and the measurement of D l ∼ of l gives D and kf0, thereby providing information on the thermodynamic nature of the solid solution as well. Cook, de Fontaine, and Hilliard[41] extended the model to cubic lattices and measured diffusion in Ag/Au and Ni/Cu modulated films.[41, 42] Earlier, Dumond and Youtz[43] had considered a multilayered structure as a periodic modulation of the electron density c(x), and described it by a Fourier series as: 






2p c(x)  Σ Am sin mx ,  m


(58)

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where Am is amplitude of the mth order Fourier component, Λ is the modulation length, and x is the growth direction. The intensity of the x-ray satellite peaks Im is proportional to the square of the Fourier coefficients: Im  A2m. The decrease of the satellite peak intensity is used to extract the diffusion coefficient via the linearized equation:[37]




2pm 2 d  (ln Im) 
2D  .  dt

(59)

The example of Ag/Au interdiffusion is provided from the investigation of Cook and Hilliard.[34] Figure 1.7 shows x-ray intensities vs. annealing

Figure 1.7 Logarithmic plots of the relative x-ray intensity of the satellite peaks (log IIo) vs. annealing time at 228.3°C in an Ag-Au modulated thin-film package of 3.35-nm wavelength with an average composition of 32 at.% Au. (From Cook and Hilliard[34])

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∼

Figure 1.8 Arrhenius plot of the interdiffusion coefficients (D ) in the Ag50Au50 modulated thin-film package (from Cook and Hilliard[34]) compared with the hightemperature data of Johnson[44] obtained in a bulk diffusion couple.

time plots at 228.3°C for 000, 111, 222
, and 222 satellite peaks in Ag-Au modulated films. The diffusion coefficients obtained are shown in ∼ Fig. 1.8; they agree extremely well with the interdiffusivities (D) measured by Johnson[44] in the Ag50Au50 alloy at high temperatures using chemical analytical techniques over 12 orders of magnitude. Even more remarkable is the fact that a diffusion coefficient as low as 10
24 m2 s
1 involving only one or two atomic jumps could be measured in a short period of a few hours!

1.3

Structurally Inhomogeneous Samples

The discussions in Sections 1.1 and 1.2 assumed that the diffusion samples were structurally homogeneous, that is, free from such extended

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defects as dislocations, grain boundaries (GBs), and interfaces. Consequently, diffusion occurred only in the lattice due to equilibrium point defects such as vacancies, interstitial atoms, and divacancies. Engineering materials, however, are polycrystalline in nature and contain non-equilibrium extended defects. Sutton and Balluffi provide a comprehensive treatment of the nature of grain boundaries and their kinetic behavior.[45] Diffusion along these defects is commonly six or more orders of magnitude more rapid than in monocrystalline samples. In thin-film packages used in the ultra-large-scale integration (ULSI) of microelectronic devices, the density of the structural defects, notably the GBs, is very high. Because the temperatures of device fabrication and performance are low, ≈100°C, the diffusive process in thin films may be largely controlled by the grain boundaries. In recent years, interfaces in the microelectronic back-end metallization/dielectric packages have also been considered paths for fast diffusion (see, for example, Chapter 9). Because an interface is sandwiched between two materials, it may be termed a heterogeneous boundary as opposed to the homogeneous grain boundary, which consists of the same material on either side. We use the interface and grain boundary terms interchangeably because their analytical treatment is almost identical, with only a small adjustment to account for lattice diffusion in differing materials in the case of the former. In polycrystalline materials, in general, the diffusion within the grains and along GBs occurs simultaneously. The coupling of these processes takes place at the GBs, which have an effective thickness d, of the order of atomic width ∼0.5 nm. The extent to which the lattice atoms make excursions into the GBs, or vice versa, determines the diffusion kinetic regime that prevails in the sample as a whole. Harrison[46] designated three types of kinetic regimes known as A, B, and C, which are shown in Fig. 1.9. The distinguishing feature of type-A kinetics is the extensive lattice diffusion, which causes the diffusion fields from the adjoining grains to overlap. The boundaries are shown here as parallel slabs with spacing 2L. In essence, the atoms in the lattice have made some excursions into the GBs so that their transport has been accelerated. The measurements in type-A kinetics would show enhanced apparent diffusivities (Dapp) with lower activation energies, which can be described by Hart’s equation:[47] Dapp  D1  gDb ,

(60)

where the l and b refer to the lattice and GB, respectively, and g is the site occupancy factor at the GBs so that g  4dL. The value of g in bulk materials is ∼10
5, but in thin films, values of 10
2 are not uncommon. Consequently, the second term becomes dominant in thin films.

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Figure 1.9 Schematic representation of Harrison’s A, B, and C kinetic regimes of grain boundary diffusion.[46] The diffusion source coincides with the top horizontal lines. The three regimes are temperature-sensitive.

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In type-B kinetics, the grain boundaries are assumed to be isolated, and the flux at large distance in the x direction approaches zero. Effectively, the GB and lattice fields are confined individually to themselves. The leakage of mass from the grain boundaries is only in the x direction, and the contribution of the GBs to the diffusion within the grains [at the top of Fig. 1.9(b)] is generally neglected in the analysis. Type-B kinetics is commonly exploited in systematic experimental work, which is discussed in Sec. 1.4. In type-C kinetics, lattice diffusion is considered negligible, and significant atomic transport occurs only within the boundaries. Atkinson and Taylor,[48] Hwang and Balluffi,[49] and, more recently, Surholt et al.[50] have carried out type-C diffusion experiments. Type-C kinetics should be of great significance for the mass transport in thin films at low temperatures. Therefore, the mathematical analyses for GB diffusion will differ from situation to situation, depending on the thickness of the sample, its grain size, and the relative contribution of the lattice diffusion as dictated by the temperature of annealing. The mathematical solutions for GB diffusion have been previously reviewed in detail.[51] Here, we only discuss some salient solutions that have commonly been used to evaluate experimental data and that have yielded reliable diffusion coefficients. Fisher,[52] for the first time, analyzed the coupled lattice and GB diffusion problem for a simple slab configuration in the type-B kinetic regime shown in Fig. 1.10. The crucial approximation in the Fisher solution was that the concentration Cb within the boundary changed so slowly with time t that the term ∂Cb∂t could be set equal to zero. Subsequently, Whipple[53] and Suzuoka[54] have solved the problem exactly, removing this limitation. In discussing the solutions, it is more convenient to use reduced variables defined as follows: y x
(d 2) d Db h   , x  , a   , ∆  , and b  (∆
1)a. t D t D 2 D t Dl  l l  l (61) The Fisher solution for the slab geometry may be written as:






2

∂ ln C dDb  0.56  ∂y


 4D l t

12

,

(62)


with the condition that ∂Cb∂t  0 in the slab, where C is the concentration of the diffusant in the sectioning method of determining the diffusivities.

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Figure 1.10 Grain boundary diffusion measurements by sectioning methods: Is concentration contours are averaged in a section of thickness ∆Y. The profile is analyzed according to Eqs. (63) and (64) where the angle φ is well-defined.

The Fisher solution is written in this form for easy comparison with the Whipple and Suzuoka solutions. The solution for the slab geometry of the GB, according to Whipple[53] and Suzuoka,[54] is given by:







∂ ln C dDb   ∂y65





 
53

4D l t

12

∂ ln C  [∂(hb)
12]65



53

.

(63)

For large values of b  30, the third term converges to (0.78)53 and (0.72b.008)53 for infinite and instantaneous source conditions, respectively. Consequently, the asymptotic Whipple and Suzuoka solutions can be approximated as:







∂ ln C dDb  0.661  ∂y65



5 3

4D l t

12

.

(64)

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The larger finite values of b imply better resolution between the GB and lattice diffusion processes. Physically, it ensures the type-B kinetic regime, as shown in Fig. 1.9. LeClaire[55] and Cannon and Stark[56] have pointed out the discrepancies between the Fisher and Whipple-Suzuoka analyses. Even for large b, the Fisher solution underestimates the GB diffusivity by a factor of 1.5 to 2.7. For b  30, the ratio Db (Fisher)/Db (true) depends on the value of b, which results in erroneous measurement of the activation energy as well. Consequently, the Fisher solution has rarely been used in recent years. The Fisher solution, however, has the advantage that it involves linear power of the penetration distance; hence the origin of the penetration is not needed. It is therefore very useful in thinfilm couples where the interface may not be well defined. Furthermore, we have never experienced a problem keeping b large in thin films since enough diffusant or radioactive tracer is trapped in the GBs due to the fine grain structure even at low temperatures. The penetration distance in thin-film diffusion is also small (1 mm) and measurements are conducted at low temperatures over a limited range. Under these conditions, the quality of linear or 65 power fit to the experimental data is usually comparable, and the difference between the use of Fisher or Whipple-Suzuoka solutions is only about 10 to 50%, with higher values obtained from the Fisher solutions. For the GB diffusion measurements in polycrystalline bulk specimens, however, a large discrepancy is found between the use of the two solutions. Use of Whipple-Suzuoka is recommended only when the penetration distance does not exceed a few percent of the average grain diameter. In diffusion work in thin films, it is not always possible to maintain a semi-infinite specimen thickness condition compared to the diffusant penetration distance. Gilmer and Farrell[57, 58] have shown that a correction must be made to account for the finite thickness (
) of the thin-film specimen to obtain true diffusivities. Figure 1.11 shows ratios of the effective . Dbeff to Db for various values of bho
2 where ho 
D l t True value of Db occurs when bho
2 ≈ 0.1. Physically, this implies that very large values of b result in flat diffusion profiles, so that diffusant has reached the thinfilm substrate interface and may even have been reflected. Consequently, both b and ho need to be optimized so that the profiles have sufficiently large slope in the type-B kinetic regime. The Arrhenius dependence of the grain boundary diffusion is similar to that in the lattice [see Eq. (27)] and may be written as: 1 dDb ∼  a2 Z f vo exp[(Sbm  Sbf)k]  exp
[(Hbf  Hbm)kT]. (65) 4

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[57, Figure 1.11 Ratio of Dbeff to Db for various values of bh
2 o . (After Gilmer and Farrell )

58]

The entropy and enthalpy terms in grain boundaries are much smaller than in the lattice so that the grain boundary diffusion coefficient is many orders of magnitude larger. Section 1.4 discusses this difference.

1.4

Some Illustrative Experimental Data

This section uses some critical experimental results accumulated during 50 years of research to illustrate the topics discussed in the preceding sections. It is not possible to review this vast and complex field. The intention here is to provide insight into the diffusion processes of the engineering materials. We will show examples of diffusion in single crystals, epitaxial thin films, polycrystalline bulk and thin-film metals, and dilute alloys. We will discuss the interplay of the various diffusion processes in specimens with variable microstructures and the influence of impurity atoms.

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1.4.1 Diffusion Profiles in Au Having Variable Microstructure 1.4.1.1 Self-Diffusion Profiles in Au Single Crystals We consider the case of self-diffusion in an Au single crystal measured by Makin et al. using an 198Au radioactive tracer.[59] This has been the basis of comparison for a variety of subsequent studies of the kinetics of point defects, notably, quenched-in resistance,[10, 15] positron annihilation,[22] and nuclear magnetic resonance.[9] In Fig. 1.12, a tracer distribution curve in an Au single crystal annealed at 772°C for 1.033  106 sec is shown yielding a diffusion coefficient of 1.80  10
14 m2/sec using the Gaussian solution [Eq. (10)]. The serial sectioning was carried out on a lathe, which required long annealing times and limited the lowest temperature of the measurements to 600°C. The parameters for self-diffusion are listed in Table 1.1.

1.4.1.2 Self-Diffusion Profiles in Au Polycrystalline Bulk Specimens A sputtering technique was introduced to measure small diffusion coefficients in bulk as well as thin-film specimens.[60, 61] In Fig. 1.13(a) and (b), typical 195Au tracer profiles are shown in polycrystalline bulk Au specimens. Figure 1.13(a) shows a 195Au tracer profile in the first 3 mm depth in an Au specimen annealed at 500°C for 4 hours, yielding a diffusion coefficient of 2.61  10
17 m2/sec. This distribution is within the grains and free from contributions of grain boundaries and even dislocations because of the small diffusion length in the lattice. It will be seen in Sec. 1.4.2 that the diffusion coefficients extracted from such tracer profiles agree well with the data obtained in a single Au crystal. Note that diffusion could be measured at very low temperatures (250°C) down to 10
22 m2/sec, which was considered to be impossible before the introduction of the sputtering technique. Because bulk Au specimens in general contain large-angle grain boundaries and subgrains within the grains, in addition to the lattice defects, the penetration profiles typified by Fig. 1.13(b) show three segments relating to diffusion along these paths. As the profile is traversed from right to left, the diffusivities along these paths are progressively smaller because of the larger activation energies involved. In thin films, however, one or more diffusion paths may be absent, depending on their microstructure. The diffusion coefficients extracted from the three regions are discussed in Sec. 1.4.2.

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Figure 1.12 198Au tracer profile in an Au single crystal at 772°C annealed for 48 hours over a depth of 400 mm. (After Makin et al.[59])

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Figure 1.13(a) 195Au tracer diffusion profile in a polycrystalline Au specimen annealed at 500°C for 4 hours; obtained by sputtering. The total profile is obtained over a depth of 3 mm. (Gupta and Tsui[60])

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Figure 1.13(b) 195Au tracer profile in a polycrystalline Au specimen showing three regions of diffusion in the lattice, sub-boundaries (dissociated dislocations), and grain boundaries. (Gupta[61])

1.4.1.3 Self-Diffusion in Au Epitaxial Films Figure 1.14 shows 195Au tracer penetration profiles for epitaxial Au films grown on (001) MgO.[62] Despite very long annealing times, the profiles maintain steep slopes because the diffusion process is very slow. Furthermore, the epitaxial films display two regions of diffusion: At small penetration distance, the diffusion is related to the lattice diffusion; at deeper depths, to dissociated dislocations.

1.4.1.4 Self-Diffusion in Polycrystalline Au Films As shown in Fig. 1.15, the 195Au profiles in polycrystalline Au films grown on fused quartz substrates[63] exhibit extremely fast diffusion at much lower temperatures and shorter annealing times compared to the single-crystalline bulk or epitaxial films of Au. The penetration profiles in epitaxial and polycrystalline Au films were analyzed according to the Whipple-Suzuoka asymptotic solution [Eq. (64)] to extract the combined diffusivity dDb for dissociated dislocations and

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Figure 1.14 195Au tracer profile in epitaxial Au films held on MgO substrates over a depth of 0.7 mm. Note the two regions of diffusion in the lattice and dissociated dislocations. (Gupta[62])

high-angle GBs, respectively. The activation energies for diffusion along these two paths were determined to be 111 kJ/mol (1.16 eV) and 85 kJ/mol (0.88 eV), respectively. The former is noticeably larger, which has ramifications on the core structure of dissociated dislocations and grain boundaries.

1.4.2 Self-Diffusion Data in the Au Lattice In Fig. 1.16, self-diffusion coefficients in the Au lattice measured by a variety of techniques[59, 61, 62, 64–66] are shown over 10 orders of magnitude. Within the experimental errors, all the data agree with the monovacancy model analyzed by Seeger and Mehrer.[67] However, the data in

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Figure 1.15 195Au tracer profiles in polycrystalline Au films held on fused SiO2 substrates over a depth of 1 mm. Note short periods of annealing and diffusion in grain boundaries only. The blank run on the extreme right shows a shallow and steep profile to which no diffusion process can be assigned. (Gupta and Asai[63])

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Figure 1.16 Self-diffusion in Au from various investigations. The monovacancy analysis of Seeger and Mehrer[67] is also shown.

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the Au epitaxial films[62] are consistently higher because of the high density of dissociated dislocations, ∼1014 lines/m2, so that diffusion of the lattice atoms also gets accelerated, in accordance with Eq. (60). Similarly, diffusion in the lattice of the polycrystalline bulk specimens remains unperturbed because of the large grain size (∼100 µm) and spacing of  dissociated dislocations in relation to the diffusion length 2D l t of ∼100 nm. (Bulk metals commonly have a dislocation density on the order of 1010/m2 at the most.)

1.4.3 Self-Diffusion in the Au and Au-1.2 at.% Ta Alloy Grain Boundaries In Fig. 1.17(a) and (b), self-diffusion in polycrystalline bulk Au and thin films is shown in the lattice and grain boundaries. The effect of 1.2 at.% Ta addition on the two self-diffusion processes is also shown.[68] Ta enhances solvent lattice diffusion according to the expectation of the five-frequency model discussed in Sec. 1.2.2 [Eq. (32)]. The change in the activation energy by about 27 kJ/mol (0.28 eV) may be attributed to the sum of vacancy-solute binding (EbVS) and the vacancy motion energy difference (B). The former may be ∼10 kJ/mol. The effect of adding Ta to Au grain boundary diffusion is even more complex because it shows enhancement at high temperature and suppresses diffusion at lower temperatures. This is due to the solute segregation effect common to most polycrystalline metallic alloys and results in lowering of the grain boundary energy. The details of this process are discussed in Sec. 1.5. In Fig. 1.17(b), self-diffusion along dissociated dislocations, the subgrain boundaries, is also displayed on a comparative basis in pure Au and the Au-1.2 at.% Ta alloy. The two data are similar, within experimental errors, with respect to the activation energies. This is because dissociated dislocations constituting the sub-boundaries are in the lowest energy

configuration with a shorter Burger’s vector of (a/6)[21 1 ], and any further relaxation on addition of solute is unlikely. Furthermore, as seen in Fig. 1.17(b), the pre-exponential factor for diffusion in the sub-boundaries of the Au-1.2 at.% Ta alloy is about 8 times larger than that in the epitaxial Au films. The difference may be attributed to Ta segregation at the subboundaries, similar to the high-angle grain boundaries discussed in Sec. 1.5.4. Indeed, in grain boundary diffusion measurements, a triple-product sdDb instead of dDb in Eq. (64) is measured in alloys, where s is the grain boundary segregation factor.

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Figure 1.17(a) Enhancement of 195Au diffusion in the Au-1.2 at.% Ta alloy compared with that in pure Au. The activation energies are 143 and 170 kJ/mol, respectively. See also Fig. 1.4 for the mechanism.[68]

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Figure 1.17(b) Grain boundary and sub-boundary diffusion of 195Au tracer in Au-1.2 at.% Ta alloy compared with that in pure Au in the boundaries of both kinds. The activation energies are 121 and 92 kJ/mol for grain boundaries in Au-Ta alloy and pure Au, respectively. The activation energy for sub-boundaries in both the cases is 121 kJ/mol. (After Gupta and Rosenberg[68, 89])

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1.5

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General Characteristics of Grain Boundary Diffusion

This section discusses general characteristics of grain boundary diffusion and relates these characteristics to the lattice diffusion. It also touches on some important properties of grain boundaries, as well as their anisotropy and energy. Because this subject is vast, it cannot be covered here in depth. A recent special issue of Interface Science[69] discusses the latest developments in the field. Sutton and Balluffi’s book[45] deals in depth with the microstructure, defects, and kinetic properties in the various interfaces in crystalline solids. The handbooks by Kaur et al.[70] and Volume 26 in the Landholt-Börnstein series[71] contain discussions and data on various materials.

1.5.1 Anisotropy of Diffusion in Grain Boundaries A grain boundary is essentially a two-dimensional transition region between two crystals that have different orientations but the same periodicity. The transition region leads to a very complex crystallographic description of the grain boundary since 9 degrees of freedom or independent variables are involved, which may be specified as 3 for the misorientations, 3 for the choice of the GB plane, and 3 for the rigid-body translation. Figure 1.18(a) through (c) shows grain boundary diffusion of 119Sn radioactive tracer in polycrystalline Pb foils.[72] Contact auto-radiograms were made on the entrance and exit sides of the foil after annealing at 90°C for 100 hours, at which time the 119Sn tracer emerged at the back surface. To prevent cross talk between the two surfaces, the Pb foil thickness (64 mm) was chosen to exceed 10 times the half-absorption thickness for 10 keV x-rays from 119Sn with an absorption coefficient r  1000 cm
1. The auto-radiograms are compared with the grain boundary microstructure. Only a few grain boundaries, the high-angle ones, are seen in the auto-radiograms through which the 119Sn tracer diffused. The microstructure is in fact much finer, showing a variety of grain boundaries. Notably, the twin boundaries are totally absent in the auto-radiograms. This is obviously due to variable diffusion kinetics in grain boundaries of differing orientations. Grain boundary diffusion in polycrystalline Pb was subsequently measured by serial sectioning technique using 203Pb radioactive tracer.[73] In Fig. 1.19, GB self-diffusion coefficients (dDb cm3/sec) using the 203 Pb radioactive tracer from Gupta and Kim[73] are shown in an Arrhenius plot and are compared with earlier data of Stark and Upthegrove[74] as a

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Figure 1.18 Contact autoradiograms of the 119mSn tracer after diffusion in a Pb foil at 90°C. (A) The plated or entrant surface, (B) the exit surface, and (C) the microstructure. The field of view is 1.25 cm in all cases. Note the similarity in the grain structure. (After Gupta and Campbell[72])

Figure 1.19 Grain boundary diffusion in polycrystalline Pb and Pb bicrystals.[73, 74]

function of orientation. The data in polycrystalline Pb are seen to go through mostly the high-angle grain boundaries. These observations are in accordance with the studies of Turnbull and Hoffman,[75] and Couling and Smoluchowski,[76] on anisotropy of GB diffusion and its dependence on the orientation angle q. In Ag tilt boundaries,[100] diffusion of 110Ag showed anisotropy of ∼15 for parallel and perpendicular directions at

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450°C. The anisotropy progressively declined for orientations near Σ  13 (q  23 degrees), Σ  17 (q  28 degrees), Σ  5 (q  36 degrees), and so forth. The anisotropy of diffusion within the GB plane can be understood on the basis of the distribution and nature of GB dislocations, which predict channels of free volume parallel to the tilt axis in largeangle boundaries. The diffusion coefficients and penetration distance along GBs are also expected to show discontinuities and cusps as a function of misorientation angle similar to the GB energies because the contributing factor in both cases should be the coincident lattice sites and the GB dislocations. The activation energy for GB diffusion, however, appears to remain independent of the orientation and QbQl ≈ 0.4 (±20%), as in the case of GB diffusion in Pb, where Qb  42 kJ/mol (0.44 eV) and Ql  106 kJ/mol (1.1 eV). Thus, the anisotropy and orientation dependence of GB diffusion may be largely due to changes in the preexponential term Dbo.

1.5.2 Diffusion Mechanisms in Grain Boundaries This section briefly examines the structure of grain boundaries and the associated defects to explain possible diffusion mechanisms in grain boundaries. This subject is exhaustive and complex, and only a qualitative description of the gross features of grain boundaries is possible here. Sutton and Balluffi[45] and Balluffi[77] discuss this subject in depth. Grain boundaries have been shown to be a world in themselves, with a potpourri of defects such as the vacancies, interstitials, and dislocations. Because a grain boundary is a transition structure between two crystals, perfectly two-dimensional periodic arrays would only develop at some discrete choices of GB variables. To account for any arbitrary choice of orientations and translations, incorporation of line defects is required. Extensive grain boundary structure studies have been conducted over the past several decades[78, 79] using techniques such as high-resolution transmission microscopy and computer stimulation, from which the following broad features of grain boundaries have emerged (see Chapter 3). First, the grain boundary is found to be relatively narrow, on the order of a few atomic distances only. Second, the atomic packing in the GB core is only slightly less dense than in the perfect crystal. Although the core structure has finite excess volume, much of it may be considered to be relaxed by the development of a variety of atomic groupings and localized configurations. Some boundaries have short wavelength periodicity running parallel to the coincident site

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lattice (CSL) and are generally of low energy compared to the boundaries with long periodicity; the former may be termed special boundaries. In the case of arbitrary misorientation between two crystals, the grain boundary minimizes its energy by preserving patches of the special low-energy-ordered boundary and introducing therein arrays of intrinsic grain boundary dislocations. It is also possible to have partial grain boundary dislocations and extrinsic grain boundary dislocations in the grain boundaries. There is indirect experimental evidence from which a localized structure of vacancies may be inferred with little relaxation of the neighboring atoms. First, there was the measurement of pressure dependence of the GB self-diffusivities in Ag by Martin et al.,[80] which resulted in a value of ∆VbVo  1.1, where Vo is the activation volume for GB self-diffusion defined analogous to Eq. (34) and Vo is the atomic volume. The value of ∆Vb is of the order of atomic volume in grain boundaries and compares very favorably to its counterpart in the lattice (Table 1.2). Second, the isotope effect measurement for self-diffusion in Ag grain boundaries by Robinson and Peterson[81] has also resulted in a relatively large value of ∆Eb ∼ 0.46, which is temperature-insensitive. The corresponding value of the isotope effect for self-diffusion in lattice is 0.65  ∆El  0.71; the variation is due to some temperature dependence (Table 1.2). If the mobile defects responsible for GB diffusion were partially dissociated or highly relaxed vacancies, a correlated movement of a number of atoms would be required for diffusion to occur. Therefore, only a small isotope effect value would have been observed. The experimental data, however, neither support nor rule out contribution of interstitial atoms to grain boundary diffusion. Atomistic computer modeling of point defects in grain boundaries, using various techniques such as kinetic Monte Carlo and molecular dynamics, have shown that vacancies as well as interstitial atoms are stable, depending on the misorientation of the boundary and the host crystal lattice.[82] Both defects have similar attractive binding energies in the grain boundaries, are mobile, and account for diffusion under certain conditions. Suzuki and Mishin[82] have studied the role of interstitial atoms and vacancies in Cu grain boundaries in detail. They noted that diffusion in the Σ  5(310) [001], commonly considered a typical grain boundary, was dominated by the interstitial atoms, whereas other – – 2)[011], Σ  11(31 1)[011], boundaries, the Σ  5(210)[001], Σ  9(12 – – Σ  7(231)[111], and Σ  13(341)[111], were dominated by vacancies. In any event, grain boundary diffusion is many orders of magnitude faster than in the lattice. In Sec. 1.5.3, a thermodynamic model is discussed to explain this difference.

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1.5.3 Interrelationship Among Grain Boundary, Lattice Diffusion, and Energy As mentioned in Sec. 1.5.2,[45] grain boundaries have lower density of the interfacial material, which implies additional energy associated with the microstructure and consequently an easier atomic diffusion. In the past four decades, several investigators, notably Borisov et al.,[83] have attempted to correlate the energy of the interfaces with diffusion kinetics. The terms grain boundary and interface are used interchangeably because this discussion will also include interfaces in several multiphase alloys. The basic postulate in obtaining interface energy (gi ) from selfdiffusion data is that it is the difference between the Gibbs free energies (∆Gl and ∆Gi) for vacancy diffusion in the lattice and the interface, as shown schematically in Fig. 1.20. The difference is positive because diffusion in interfaces is many orders of magnitude faster than that in the adjoining lattice. Hence the interface energy can be written as: 1 gi   (∆Gl
∆Gi) 2

(66)

The factor of 1/2 enters because the interface energy is shared between the two adjoining crystals. The Arrhenius nature of the coefficient, D, is given by Eqs. (27) through (29). The subscripts l and i denote the lattice and grain boundary diffusion processes, respectively. The difference between the free energies, ∆Gl and ∆Gi , is then expressed as: 1 1 gi (Jmol)   (∆G

∆Gi)   RT {
n (DiD
)} 2 2

(67)

or gi (Jm2)  r{(Q

Qi)  RT
n(Doi Do
)},

(68)

where Dol and Doi (m2/sec) are the pre-exponential terms for lattice and interface diffusion, respectively; the conversion parameter r  12a2No, with a the mean distance between atoms at the interface, to the first approximation, equal to one lattice parameter; and No is Avogadro’s number. Ql and Qi, and R, the gas constant within the brackets, are to be taken in J/mol. To test that grain boundary energies computed from diffusion data are indeed meaningful, diffusion parameters and the resulting interface energies obtained from Eq. (68) for several pure metals are compared in Table 1.4 with the direct measurements of interface energies using nondiffusion

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Figure 1.20 Schematic representation of grain boundary energy (g i) and free energies for diffusion in lattice and grain boundaries. Note the decrease in g i by ∆G upon addition of a solute. (After Borisov et al.[83])

techniques such as the zero creep and transmission electron microscopy (TEM). Figure 1.21 shows a plot of the two kinds of interfacial energies. Despite the semi-empirical nature of the basic postulate in Eqs. (66) through (68), the two kinds of interfacial energies agree well, within a few percent. Because the diffusion processes, in general, are related to the absolute melting temperature of the host,[84] the grain boundary energies should also be thus related. Physically, such a dependence stems from the cohesive energy of the solid that controls the atomic motions as well as the energy of the grain boundary itself. Chapter 2 discusses this topic in detail. The

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Table 1.4. Diffusion in Lattice (Ql , Dol), GB (Qi, dDoi), and GB Energies (g i) in Pure Metals

Lattice Host (Tracer) Au(195Au) Au* Ag(110Ag) Ag(110Ag) Ag* Cu(110Ag) Cu(64Cu) Cu(67Cu) Cu* Ni(63Ni) Ni* Pb(203Pb) Pb(119Sn) Pb*

Grain Boundary

D°l Qi DD°i Ql (kJ/mol) (10
4 m2/s) (kJ/mol) (10
16 m3/s) 169.6 ... 169.6 169.6 ... 199.4 196.8 199.4 ... 277.5 277.5 100 99.4 ...

0.04 ... 0.04 0.04 ... 0.16 0.1 0.16 ... 0.92 0.92 0.16 0.41 ...

84.8 ... 77.1 74.4 ... 75.2 84.75 91.5 ... 170.5 ... 44.3 39.5 ...

3.1 ... 0.13 0.31 ... 23 11.6 29 ... 222 ... 61 73 ...

Energy(gi) (mJ/m2) 396-0.016T 400(1173K) 382(1173K) 392(1173K) 400(1173K) 776-0.123T 783-0.134T 692-0.0054T 590(1123K) 1126-0.214T 1373-0.4T 193-0.004T 206-0.03T 200(588)

Reference a b(*) c d e(*) f g h b i(*) j(*) k l m(*)

* Actual measurements a. D. Gupta, Metall. Trans., 8A:1432 (1977) b. J. E. Hilliard, M. Cohen, and B. L. Averbach, Acta Metall., 8:26 (1959) c. D. Turnbull and R. E. Hoffman, Acta Metall., 2:419 (1954) d. J. T. Robinson and N. L. Peterson, Acta Metall., 21:1181 (1973) e. A. P. Greenough and R. King, J. Inst. Metall., 79:415 (1951 f. G. Barreau, G. Brunel, G. Cizeron, and P. Lacombe, Mem. Sci. Rev. Metall., 68:357 (1971) g. T. Surholt and C. Herzig, Acta Mater., 45:3817 (1997) h. D. Gupta, Defect Diffusion Forum, 156:43 (1998) i. W. Lange, A. Hassner, and G. Mischer, Phys. Status Solidi, 5:63 (1964) j. L. E. Murr, R. J. Horilev, and W. N. Lin, Philos. Mag., 22:52 (1970) k. D. Gupta and K. K. Kim, J. Appl. Phys., 51:2066 (1980) l. K. K. Kim, D. Gupta, and P. S. Ho, J. Appl. Phys., 53:3620 (1982) m. K. T. Aust and B. Chalmers, Proc. Royal Soc., A204:359 (1950)

activation energies for diffusion in the lattice and grain boundaries, respectively, are related to the absolute melting temperature (Tm) of the host[51] as: Ql  142 Tm Jmol (±20%) and

Qb  71 Tm Jmole (±20).

(69)

Consequently, Eq. (68) can be rewritten as: 71Tm (mJm 2), gb  2 2 a No

(70)

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Figure 1.21 Comparison of diffusion-related grain boundary energies in pure metals with those measured directly.

neglecting contributions of the log of pre-exponential terms in the diffusion coefficients. As seen in Table 1.4, the temperature coefficient of gi in pure metals is small and negative, implying a positive entropy ≈ 0.25 mJ/m2 °C for grain boundaries. The value of gb ≈ 0.24Tm

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estimated from Eq. (70) and its small temperature coefficient are in agreement with the estimates that McLean[85] arrived at on the basis of grain boundary melting.

1.5.4 Grain Boundary Solute Segregation Effects Equations (67) and (68) are also applicable to the interface diffusion in alloys. G. B. Gibbs pointed out[86] that in diffusion measurements in alloys in the Harrison’s regime B, a triple product sdDi is measured where s(T ) is the solute segregation factor. The segregation factor has its own Arrhenius temperature dependence. The alloy addition results in reduction of the interface energy denoted by g io and increase in the free energy for diffusion, as shown in Fig. 1.20. This is the well-known Gibbs absorption effect.[87] To extract grain boundary segregation parameters, self-diffusion measurements of the solvent species are required in both the lattice and the interface. From the difference of the interface energies between the pure host and the alloy, (g i
g io ), it has been possible to compute the enthalpy and entropy for the solute segregation using Gibb’s adsorption isotherm[87] combined with the McLean statistics.[85] The free energy for solute grain boundary segregation, ∆G , is given by: (gi
g io)  ∆g i  RT ln [1  Co exp(∆G RT)].

(71)

From the interfacial free-energy decrement, ∆G , and the solute concentration, Co, the segregation coefficient in the interface, s  CbCo, can be readily obtained as a function of temperature according to McLean’s statistics: Cb  Co exp(∆G RT)[1  Co exp(∆G RT)],

(72)

where Cb is solute concentration in the interface. Furthermore, the Arrhenius parameters for solute segregation can also be written as: ∆G  (∆H
T∆S ).

(73)

Note that a monolayer coverage (Cb) is implicit in Eq. (72), beyond which it is not valid. The limiting value of s depends on the input of Co. It also determines the magnitude of the entropy DS in Eq. (73), but the enthalpy DH remains unaffected. It is also possible to obtain s in Eq. (72) directly by measuring selfdiffusion in the Harrison’s B and C regimes separately in which sdDi and

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Figure 1.22 Temperature dependence of grain boundary energies in pure Au bulk and thin films and Au-1.2 at.% Ta alloy determined from the self-diffusion (195Au) measurements. Note the signs of the slopes.

Di are measured. Comparing the two quantities and assuming a value for d leads to the values of s as a function of temperature. Such an approach has been used by Surholt et al.[50] for Ag and Au segregation in Cu grain boundaries. We have discussed the interrelationships among diffusion, solute segregations, and interfacial energies in a number of diverse materials such as dilute alloys, concentrated alloys, a lamellar eutectic system, and intermetallic compounds in a recent article.[88] In Sec. 1.4.3, the effect of Ta solute addition in Au on the lattice and grain boundary diffusion were discussed qualitatively.[89] The data shown are examined further according to Eq. (71) (see Fig. 1.22). The decrements of the grain boundary energies, ∆g i, from their values in pure Au upon addition of 1.2 at.% Ta, were first

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computed according to Eq. (67). McLean’s statistical analysis, namely, Eqs. (71) and (72), was then applied to the decrements to obtain s as a function of temperature. Finally, the segregation parameters were computed according to Eq. (73). The resulting parameters are listed in Table 1.5, where data from a large number of alloys studied in a similar way are also listed. The solute segregation parameters in grain boundaries depend on the elastic strain energy associated with the size difference between the solute and the host atoms and on the electrochemical difference between the two. The former has been examined by McLean[85] and by Wynblatt and Ku.[90] Accordingly, the solute elastic strain contribution, ∆He, is given by: ∆He  [24p k Gr1ro (r1
r0)2(4 Go ro  3k r1)],

(74)

where k is the bulk modulus of the solute, G is the shear modulus of the solvent, and ro and ri are the effective radii of the solvent and the solute atoms, respectively. These values were obtained from Smithells and Brandes.[91] The elastic strain enthalpies ∆He for various alloys are listed in Table 1.5 and displayed by a solid curve in Fig. 1.23 as a function of the relative atomic size difference. The data points are for the ∆H values obtained from diffusion data. It is clear that most binary species, such as Pb-Sn, Pb-In, Cu-Ag, Cu-Au, Cu-Bi, and Cu-Sb, follow the computed elastic strain energy curve shown by the solid curve in Fig. 1.23. In the Cu-Bi and Cu-Sb alloys, although the atomic size difference is large, their electronic characteristics are similar. Consequently, their binding energies deviate little from the computed elastic strain energies where the actual atomic size difference is implicit. Briefly, the factors that promote solid solution in an alloy also prevent grain boundary segregation such as the close proximity of the solute and solvent species in the Periodic Table, similarity of the atomic size, the electronic structure in various atomic shells, and the transition state of the elements and the electro-negativity. On the other hand, the binary species with large valence differences show large deviations from the elastic strain energy curve. Hondros et al.[92, 93] have provided a relatively complete treatment of this difference; its discussion is beyond the scope of this chapter. Contributions arising from variable electronic configurations can be clearly seen in the cases of the Au-Ta and Cu-Cr alloys. In both alloys, the difference in atomic size is negligible but the binding energy is extremely large, owing to the addition of the Ta and Cr transition metals, which are also almost totally immiscible in Au and Cu solvents, respectively.

0.0013 0.021 0.63 3.9 ... ... 0.16 0.08 0.61 3.59 3.59

142.5 160 195 213.2 ... ... 199.4 191 194.4 303 303

121.4 115.4 121.2 121.2 ... ... 133.9 81.24 69.1 168 187

~40 5000 9500 13000 13000 ... ... 180 21.1 14 3270 12400

~50

1.7 51 23 1.6 7  106

1.3 4 1.3 1.6 1–3 ~1 1.6 0.9 4.1 ... ...

...

0.9 3.4 0.7 0.7 ...

135  0.094T 130  0.125T 125  0.106 144  0.06T 150 (T  400K) 233 (T  400K) 106  0.39T 314  0.364T 353  0.315T 353  0.315T ... ... 418  0.043T ... ... 888  0.028T 754  0.107T 32 83.6 62.7 65.3 66 107 42.3 9.7 39.5 ... ...

...

3.5 16 9.0 9.0 ...

H (kJ/mol)

15.4 0.3 22.3 55 55 88 58 22.3 15.4 ... ...

...

1.0 2.3 1.0–2.3 1.0–2.3 ...

He (kJ/mol)

d e e e f f g h i j j

c

a a, b a a c

Ref.

D. Gupta and J. Oberschmidt, in Diffusion in Solids: Recent Developments (M. A. Dayananda and G. E. Murch, eds.), Metall. Soc AIME, Warrendale, PA (1985), p. 121 K. K. Kim, D. Gupta, and P. S. Ho, J. Appl. Phys., 53:3620 (1982) D. Gupta, K. Vieregge, and W. Gust, Acta Mater., 47:5 (1999) D. Gupta, Philos. Mag., 33:189 (1976) G. Barreau, G. Brunel, G. Cizeron, and P. Lacombe, Mem. Sci. Rev. Metall., 68:357 (1971) E. D. Hondros, in Proc. Interfaces Conf., Australian Inst. Metals, Butterworths, London (1969), p. 77 D. Gupta, C.-K. Hu, and K. L. Lee, Defect Diffusion Forum, 143–147:1397 (1997) T. Surholt, Y. M. Mishin, and C. Herzig, Phys. Rev., B50:3577 (1994) S. Divinski, M. Lohmann, and C. Herzig, Acta Mater., 49:249 (2001) S. Frank, J. Rusing, and C. Herzig, Intermetall, 4:601 (1996)

0.16

100

31.8 41.4 43.4 31.8 80

S (R)

gi (mJ/m2)

8:36 AM

a. b. c. d. e. f. g. h. i. j.

0.15 0.0027 0.0071 0.00015 0.16

85.8 80.9 84.8 69.4 100

sdDi (10
16 m3/sec) Qi (kJ/mol) sdDlo

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Pb-5In Pb-5Sn Pb-5In-1Au Pb-2.7In-3.7Sn Pb-62Sn (Lamellae) Pb-62Sn (equiaxed) Au-1.2Ta Cu-1Cr Cu-0.13Ti Cu-0.08Zr Cu-Sb (AES) Cu-Bi (AES) Cu-Sn Cu-Au Cu-Ag Ni3Al Ni3Al(B)

Alloy

Dl (10
4 m2/sec) Ql (kJ/mol) Dl o

Table 1.5. Arrhenius Parameters for Diffusion and Solute Segregation in Some Alloys

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Figure 1.23 Grain boundary binding energy parameters from diffusion measurements and elastic strain energies computed from Eq. (74) in some alloys. (See also Table 1.5.)

1.6

Diffusion in Quasicrystalline and Amorphous Alloys

This section discusses diffusion in some noncrystalline solids to show that long-range diffusion is possible and much of the formalism discussed in the context of crystalline solids is applicable to these meta-stable systems as well. Discussion about diffusion in these alloys is an introduction to the subject rather than a comprehensive review. Quasicrystalline and amorphous phases are relatively new materials discovered in the middle to late twentieth century by Schechtman et al.[94] and Duwez et al.,[95] respectively. Discovery of alloys with five-fold crystallographic symmetry was very tantalizing to the physics and material community because it contradicted the classical laws of crystallography. The quasicrystals are actually quite stable and display many interesting properties. In 1960, Duwez and co-workers successfully made the first metallic glass consisting of metal and metalloid atomic species, notably the Pd80Si20 glass, by melt-spinning accompanied by rapid quenching. Metallic glasses have x-ray diffraction patterns similar to their silicate counterparts, showing only two halos of short-range order and absence of peaks for long-range

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order. They show high tensile strengths exceeding those of crystalline materials, and they are magnetically soft. They can be produced continuously in ribbons and tapes so that many and varied applications are possible. Another innovation of great importance was the production of bulk metallic glasses[96] so that the alloy can be cast in any shape and size. The state of defects and diffusion in these meta-stable phases has been intriguing because of their aperiodic atomic arrangement or absence of translational symmetry. Obviously, their recrystallization behavior, phase separation, and ultimate stability would be diffusion controlled. This section discusses diffusion processes in these materials and compares them with the self-diffusion processes in crystalline solids, particularly the metals.

1.6.1 Diffusion in Quasicrystalline Alloys Quasicrystalline alloys display five-fold symmetry, contrary to the accepted laws of classical crystallography, where it is forbidden. Quasicrystals are classified according to the dimensionality of their quasiperiodic order: three-dimensional, icosahedral; two-dimensional, decagonal; and one-dimensional, packing. The AlPdMn and AlNiCo ternary alloys are two common examples of the icosahedral and decagonal quasicrystalline systems. The two systems are denoted by the prefixes I and D, respectively. Another classification of the quasicrystal is based on the constituting elements: the Al-transition-metal group and the FrankKaspar group.[97] The quasicrystalline phase in the former is represented by a group of Zn-Mg-RE (Ho Y) alloys. Diffusion of several radioactive tracers (such as 63Ni, 57Co, 54Mn, 103Pd, and 65Zn) has been studied in the I-AlPdMn, I-ZnMgHo, and D-AlNiCo quasicrystals by investigators in Germany and Japan. We will discuss their important results and conclusions for diffusion processes operative in these novel phases. Nakajima and Zumkley[98] have studied diffusion of 68Ge, 63Ni, 60Co, 51 Cr, and 54Mn in I-Al70Pd21Mn9 single quasicrystals. Diffusion of 65Zn, 114m In, 59Fe, and 54Mn in I-Al70Pd21Mn9, D-Al72.6Ni10.5Co16.9, and I-ZnMgHo quasicrystals has been reviewed recently.[98
100] For our discussion, the data for 65Zn, 57Co, and 63Ni tracers, representing fast and slow diffusion of the majority and minority atomic species in I and D types of quasicrystals, have been selected. These data are shown in Fig. 1.24, and their diffusion parameters are listed in Table 1.6. It is seen that diffusion coefficients of all diffusing tracers follow linear Arrhenius relationships over wide temperature ranges. The diffusion parameters are within the permissible limits typical for metals and alloys; that is, the activation energy Q ∼ 142Tm (kJ/mol) ± 20% and the pre-exponentials factors Do ∼ 10
4 m2/sec within a factor of 100 (see Table 1.1). The signs and the

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Figure 1.24 Diffusion of radioactive tracers in several icosahedral (I) and decagonal (D) quasicrystals. (See Table 1.6 for compositions and diffusion parameters.)

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Table 1.6. Diffusion in Some Noncrystalline Alloys Alloy

Diffusant

Do (10
4 m2/s) Q (kJ/mol) References/Remarks

65

0.27 2.86 53.0 2.2 3900.0 28.0

Quasicrystals (a) I-AlPdMn (b) I-ZnMgHo (c) I-AlPdMn (d) D- AlNiCo (e) I-AlPdMn (f) D-AlNiCo

Zn Zn 63 Ni 63 Ni 60 Co 57 Co 65

Amorphous 110m (a) Pd19Si81 Ag 2  10
6 57 60 (b) Co76.7Fe2Nb14.3B7 Co, Co 1.0  10
3 57 (c) Co86Zr14 Co, 60Co 1.1  10
3 95 (d) Co92Zr8 Zr ... (e) Ni46Zr54 Hf (SIMS) 7.4  10–13 (f) Ni54Zr46 Cu (SIMS) 3.3  10
3 (g) ZrTiCuNiBe B, Fe ...

121.3 150.0 209.0 266.5 266.5 251.7

a; fast diffuser b; fast diffuser c; slow diffuser d; slow diffuser e; slow diffuser f; slow diffuser

125.3 221.6 159.0 ... 79.6 149.3 ...

f g; ∆VVo ∼ 0, ∆E ∼ 0 h; ∆E ∼ 0 i; ∆VVo ∼ 0.9 j; ∆VVo ∼ 0.5 k; ∆VVo ∼ 0.2 l; curved plot, ∆E ∼ 0

a. H. Mehrer, T. Zumkley, M. Eggersmann, R. Galler, and M. Salamon, Mater. Res. Soc. Proc., 527:3 (1998) b. R. Galler, R. Sterzel, Wolf Assmus, and H. Mehrer, Defect Diffusion Forum, 194–199:867 (2001) c. T. Zumkley, H. Nakajima, and T. A. Lograsso, Philos. Mag., A80:1065 (2000) d. C. Khoukaz, R. Galler, M. Feurerbacher, and H. Mehrer, Defect Diffusion Forum, 194–199:873 (2001) e. W. Sprengel, H. Nakajima, and T. A. Lograss, Proc. 6th Int. Conf. Quasicrystals ICQ6 (S. Takeuchi and T. Fujiwara, eds.), World Scientific, Tokyo, Japan (1998), p. 429 f. D. Gupta, K. N. Tu, and K. W. Asai, Phys. Rev. Lett., 35:796 (1975) g. F. Faupel, P. W. Hüppe, and K. Rätzke, Phys. Rev. Lett., 65:1219 (1990) h. A. Heesmann, K. Rätzke, F. Faupel, J. Hoffmann, and K. Heinmann, Europhys. Lett., 29:221 (1995) i. P. Klukgist, K Rätzke, and F. Faupel, Phys. Rev. Lett., 81:614 (1998) j. A. Grandjean and Y. Limoge, Defect Diffusion Forum, 143–147:711 (1997) k. Y. Loirat, Y. Limoge, and J. L. Bocquet, Defect Diffusion Forum, 194–199:827 (2001) l. T. Zumkley, V. Naundorf, M.-P. Macht, and G. Frohberg, Defect Diffusion Forum, 194–199:801 (2001)

magnitude of the activation volumes of Zn (0.74Ω at 776K) and Mn (0.74Ω at 1023K) in quasicrystals[99
100] are similar to those observed in metals (see Chapter 2, Table 2.4). These observations suggest diffusion by a vacancy mechanism in quasicrystals as well. Because quasicrystals are essentially intermetallic compounds, the diffusion mechanism may have similar characteristics in view of maintaining the long-range degree of

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order. Divinski and Larikov[101] have reviewed the mechanisms of diffusion in quasicrystals at length. They have favored a diffusion mechanism of collective cyclic movement mediated by a vacancy similar to the sixjump cycle in B2 intermetallic compounds (see Chapter 4), with the difference that a sequence of 10 or 6 jumps may be involved in the A6B type of atomic packing in quasicrystals. While cyclical atomic diffusion is selfcorrecting to maintain order, it is an inefficient process of mass transport. Consequently, diffusion coefficients are likely to be 2 to 4 orders of magnitude smaller by virtue of a smaller pre-exponential factor rather than a larger activation energy. The experimental data support such a hypothesis. Kalugin and Katz[102] have suggested the alternate mechanism of diffusion by phason flips of the atomic packing in quasicrystals. Such a mechanism has not been ruled out for noble metal diffusion at low temperatures. There does not appear to be a universal diffusion mechanism in quasicrystal. The mechanism can change from case to case depending on the type of bonding, composition, temperature, pressure, and so forth.

1.6.2 Diffusion in Amorphous Alloys: Metallic Glasses This discussion of diffusion and related kinetic is confined to amorphous metallic alloys. The other groups of amorphous solids – the amorphous semiconductors and silicate glasses – will not be covered. The amorphous metallic alloy group itself has become very large: It now comprises metal-metalloid glasses, metal-metal glasses, the recently discovered bulk metallic glasses, and super-cooled liquids. We provide here a few examples of diffusion measurements and their implications in terms of the diffusion mechanisms that best explain the results. Long-range diffusion in meta-stable glasses was considered unlikely in the early years following their discovery because it would initiate crystallization in these meta-stable alloys. Only short-range diffusion was considered possible. However, in 1975, the existence of long-range diffusion of 110mAg radioactive tracer in the Pd19Si81 metallic glass produced by the Duwez group was demonstrated.[103] The x-ray diffraction patterns taken before and after diffusion showed that the metallic glass remained amorphous except for a little phase separation at high temperatures approaching the glass transition temperature (Tg). The diffusion coefficients, while small, could be extracted from the broadening of the 110mAg profiles at the respective temperatures and yielded the activation energy and the pre-exponential factor with reasonable precision. In addition to the thermal broadening, the 110mAg profiles were also found to be displaced, indicating structural relaxation of the host material. Based on the observation of an unusually small pre-exponential factor, 2  10
10 m2/sec,

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compared to the normal value of 10
5 m2/sec in close packed metals (see Table 1.1), a cooperative atomic diffusion mechanism involving a group of atoms was proposed without the benefit of any point defect. It is similar to the relaxion diffusion mechanism shown in Fig. 1.2(e). The proposed mechanism was consistent with the dense random packing of the metal atoms analogous to the hard-sphere packing model of Bernal, with metalloid species filling the larger voids. It was envisioned that an atomic volume, approximately the size of a vacancy, would be distributed among a group or cluster of an unspecified number of atoms. It was hypothesized that thermal vibrations below the glass transition temperature would result in continual redistribution of the free volume, thereby permitting atomic diffusion. The small probability of such events taking place was reflected in a small pre-exponential factor. At the same time, diffusive jumps would be easier because of the looser atomic packing and lower saddle-point energy, which would result in larger diffusion coefficients than what could be construed from a similar crystalline alloy or the viscosity data. Diffusion data in some metallic glass systems, consisting of the metalmetalloid, the metal-metal, and the bulk metallic glasses, are shown in Fig. 1.25 and listed in Table 1.6. In view of very small diffusion length (∼100 nm), ingenious experimental techniques were used. These included radioactive tracer profiling by ion-sputtering, use of stable isotopes and detection by secondary ion mass spectroscopy (SIMS), Rutherford back scattering (RBS), and monitoring of the x-ray satellite in a compositionally modulated amorphous film package similar to that described in Sec. 1.2.5. Rothman has described these experimental techniques.[30] Diffusion coefficients and the attendant parameters (Q and Do) have been determined with good precision in most cases, and Arrhenius behavior is observed except for the bulk metallic glass and undercooled liquids. In addition to the determination of the diffusion parameters, activation volumes (∆VVo) and the isotope effects (∆E) have been measured in some systems to enable full characterization of the diffusion mechanism in the respective host amorphous alloy. These parameters are listed in Table 1.6. Frank,[104] Mehrer and Hummel,[105] and Faupel et al.[106] have critically reviewed diffusion in a variety of amorphous metallic alloys. There is general agreement that (1) diffusion coefficients follow Arrhenius behavior in the all-metallic glasses with the exception of the bulk metallic glasses, and (2) there is a change from a vacancy diffusion mechanism in the as-produced metallic glasses, by techniques such as splat-quenching, melt-spinning or thin-film deposition, to a cooperative atomic process mentioned above in the relaxed state.[103] Diffusion coefficients in the unrelaxed state are about one order of magnitude larger compared to the relaxed state, which is obtained within a few hours, depending on the temperature of annealing. Contribution of a higher diffusion coefficient in

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Figure 1.25 Diffusion of radioactive tracers in amorphous metallic alloys and a bulk metallic glass. (See Table 1.6 for compositions and diffusion parameters.)

the unrelaxed state is likely to be seen in techniques such as the decay of the satellite peaks in the compositionally modulated films and RBS, requiring short anneals, rather than in the radioactive tracer sectioning technique where annealing periods may extend to hundreds of hours. The

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activation volumes ∆VVo ∼ 0.0 and ∆E ∼ 0 in the relaxed glasses, as listed in Table 1.6. There is, however, an ambiguity in the transition metal-metal glasses. While diffusion of early transition metals (such as Co, Cu, and Fe) in the metal-metal glasses is consistent with a cooperative mechanism, diffusion changes to a vacancy mechanism for late-transition tracers (Hf and Zr), as evidenced by sizable values of ∆E and ∆VVo (see Table 1.6). The difference is largely attributed to the larger atomic size prevalent in the late transition metals. Diffusion coefficients in the bulk metallic glasses are apparently nonArrhenius, with a knee present at about the glass transition temperature (Tg) (see Figure 1.25). This effect has been explained by Faupel et al..[106] There are actually two atomic diffusion mechanisms operating in bulk metallic glasses: Below Tg, diffusion occurs by a cooperative mechanism in the relaxed glasses or early transition metal-metal glasses that is supported by (a) isotope effect ∆E ∼ 0, (b) small lifetimes of positrons, indicating little free volume, and (c) molecular dynamics simulations. Above Tg, the diffusion occurs by a vacancy mechanism as the frozen structure attempts to revert to the undercooled liquid state. Each has its own Arrhenius dependence, resulting in a curved plot.

1.7

Summary

In this chapter, we first provided important mathematical bases of diffusion for a variety of experimental situations and diffusion mechanisms in crystalline solids. The thermo-chemical effects encountered in homogeneous and inhomogeneous alloys were then discussed within linear and nonlinear diffusion regimes. Diffusion processes operating in common engineering metals and alloys with extended defects such as the grain boundaries and dislocations were discussed and illustrated by experimental results in some model alloys. In thin films, which are of great importance to the information technology discipline, grain boundary diffusion plays a vital role in reliability at low temperatures. Therefore, the diffusion process in grain boundaries was described in some detail and its interrelationship with the diffusion in the lattice and the grain boundary energy was established. Finally, diffusion in the recently discovered quasicrystalline and amorphous metallic alloys was briefly discussed. Tabulated data of diffusion coefficients in a number of materials under varying physical and chemical conditions were provided for ready reference. This chapter was confined to diffusion in relatively model materials. The following chapters discuss materials of technological importance individually. It is hoped that the topics covered will be useful to materials scientists in general without getting them deeply involved in the mathematical and analytical complexities of the science of diffusion.

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Acknowledgment The author is grateful to Dr. S. J. Rothman of Argonne National Laboratory for critically reviewing this chapter and for many stimulating discussions.

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90. P. Wynblatt and R. C. Ku, in Proc. ASM Sci. Seminar Interfacial Segregation (W. C. Johnson and J. M. Blakely, eds.), ASM, Metals Park, OH (1979), p. 115 91. J. Smithells and E. A. Brandes (eds.), Metals Reference Book, vol. 15, Butterworths, London (1976), p. 2 92. E. D. Hondros, M. P. Seah, S. Hoffman, and P. Lecek, in Physical Metallurgy (R. W. Cahn and P. Haasan, eds.), North-Holland Physics Pub. Co. (1996), pp. 1201–1288 93. E. D. Hondros, in Proc. Interfaces Conf., Australian Inst. of Metals, Butterworths, London (1969), p. 77 94. D. Schechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett., 53:1951 (1984) 95. P. Duwez, R. H. Williams, and Klement, “Non-crystalline structure in solidified gold-silicon alloys,” Nature (London), 187:869 (1960) 96. A. Peker and W. L. Johnson, Appl. Phys. Lett., 63:17 (1993) 2342 97. J. Philibert, Atom Movements and Mass Transport in Solids, Les Editions de Physique (1991) 98. H. Nakajima and T. Zumkley, “Diffusion in quasicrystals,” Defect Diffusion Forum, 194–199:789 (2001) 99. C. Khoukaz, R. Galler, M. Feuerbacher, and H. Mehrer, “Self-diffusion of Ni and Co in decagonal Al-Ni-Co quasicrytals,” Defect Diffusion Forum, 194–199:873 (2001) 100. R. Galler, R. Sterzel, W. Assmus, and H. Mehrer, “Diffusion in icosahedral Zn-Mg-RE and Al-Pd-Mn quasicrystals,” Defect Diffusion Forum, 194–199:867 (2001) 101. S. V. Divinski and L. N. Larikov, “Mechanisms of diffusion in quasicrystals,” Defect Diffusion Forum, 143–147:861 (1997) 102. P. A. Kalugin and A. Katz, “A mechanism for self-diffusion in quasicrystals,” Europhys. Lett., 21:921 (1993) 103. D. Gupta, K. N. Tu, and K. W. Asai, “Diffusion in the amorphous phase of Pd-19 at.% Si metallic alloy,” Phys. Rev. Lett., 35:796 (1975) 104. W. Frank, “Diffusion in amorphous solids-metallic alloys and elemental semiconductors,” Defect Diffusion Forum, 143–147:695 (1997) 105. H. Mehrer and G. Hummel, “Amorphous Metallic Alloys,” in Diffusion in Amorphous Materials (H. Jain and D. Gupta, eds.), The Minerals, Metals and Materials Soc. (1994), p. 163 106. F. Faupel, H. Ehmler, C. Nagel, and K. Rätzke, “Does the diffusion mechanism change at the caloric glass transition?,” Defect Diffusion Forum, 194–199:821 (2001)

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2 Solid State Diffusion and Bulk Properties Gyanendra P. Tiwari and Radhey S. Mehrotra Bhabha Atomic Research Centre, Trombay, India Yoshiaki Iijima Department of Materials Science, Tohoku University, Japan

2.1

Introduction

The basic equation for self-diffusion in cubic crystalline lattice via a monovacancy mechanism[1] is as follows: D
fa2u exp(∆S*
R) exp(∆H*
RT)

(1)

where D
diffusion coefficient f
correlation factor a
lattice parameter u
jump frequency ∆S*
activation entropy ∆H*
activation enthalpy R
universal gas constant T
temperature, in degrees Kelvin Furthermore, the enthalpy and free energy (∆G*)of activation for diffusion are given by: ∆H*
∆H*f  ∆H*m ,

(2.1)

DG*
DH*  TDS*.

(2.2)

The terms ∆H*f and ∆H*m are the activation energies for formation and migration of vacancies. Equation (1) was derived by Wert and Zener[2] on the basis of absolute reaction rate theory. This equation is applicable for self-diffusion in all crystalline lattices. For diffusion in complex lattices and changes in the diffusion mode, introduction of a numerical factor is necessary. The present

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discussion is focused on the self-diffusion in metallic elements, where the migration of a monovacancy is the dominant mode of diffusive motion. Section 2.2 surveys the interrelationships between diffusion parameters and some bulk properties of the solids. These relationships, which can be qualified as the “enlightened empiricism,” have contributed significantly to the growth in the field of diffusion in solids by serving as the reference point for the rationalization of the diffusion data. The most important developments in this area have been (a) the establishment of a firm basis for the correlation between diffusion and melting parameters, (b) the Zener’s hypothesis relating the activation entropy of diffusion (∆S*) with the temperature dependence of elastic constants, and (c) a formal theory relating the vacancy formation energy with the Debye temperature. The main feature of such relationships is a constant, which directly relates the concerned diffusion parameter, most often the activation energy for self-diffusion, with a bulk property. For any particular class of solids, the magnitude of the constant depends on the crystal structure and the mode of diffusion. Starting with Van Leimpt,[3] there exists a fairly long history of empirical relationships between the diffusion parameters and bulk properties of the solids.[2–13] The formation of a vacancy results in the breaking of bonds between the atoms in the matrix. Similarly, the migration of an atom from its equilibrium position to the next site causes distortions in the bonds between the diffusing atom and its nearest neighbors. The kinetics of both of these steps in diffusion are governed by the nature and strength of the cohesion in the matrix. Therefore, in trying to relate the diffusion parameters with the bulk properties, we are, in fact, seeking to correlate them, albeit indirectly, with the cohesive energy. This explains, to some extent, why the correlations of the diffusion parameters with melting point and elastic constants have been sought extensively in the lit-

Notes: 1. Activation energies were obtained from Smithells’ Metal Reference Book, vol. VII (E. A. Brandes and G. B. Brook, eds.), Butterworths Pub. (1992). 2. Cohesive energies were obtained from C. Kittel, Introduction to Solid State Physics, VII ed., Wiley (1986).

Figure 2.1 Plots of activation energy for self-diffusion in metals against their cohesive energies; slopes 0.57 (FCC), 0.65 (HCP), and 0.60 (BCC).

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erature. Figure 2.1 shows plots of ∆H* for metals against the cohesive energy, E. The slopes vary only marginally between FCC, HCP, and BCC structures. Figure 2.1 demonstrates a general rule that activation energy for selfdiffusion in any matrix depends basically on its cohesive strength. From cohesive energy considerations, diffusion is easier in metals and polymers because the breaking and bending of bonds in them requires proportionately lesser energy as compared to that in ionic and covalent solids. In other words, energy fluctuations needed for diffusion to occur are smaller in metals and organic solids than in ionic and covalent solids. Following this, a thermodynamic equation relating the activation volume to the activation entropy is derived in Sec. 2.3 for pressure-dependent diffusion measurements. This equation is further used to obtain estimates of the activation entropy of diffusion, the frequency of diffusion, and the specific heat of the activated complex. It is shown that the equation derived here gives reliable values of all associated parameters. Disregarding the chronological sequence, the important relationships between diffusion parameters and bulk properties discussed here have helped (a) in the growth of the subject, through being a reference point for the rationalization of diffusion data in the absence of a formal theory, and (b) in the estimation of diffusion rates when the experimental data are lacking. Indeed, when diffusion measurements are attempted in a new solid, these empirical rules greatly help in choosing the experimental technique and the subsequent thermal treatments.

2.2

Correlations with Bulk Properties

2.2.1 The Melting Parameters Melting of a solid leads to the destruction of its long-range periodicity. Therefore, melting point can be considered, on a relative scale, to be a measure of the pair-wise binding energy of the matrix.[14, 15] Around a vacancy, pair-wise bonds are broken and, at the same time, there is an increase in the overall volume of the matrix. Hence, vacancy formation energy has been linked intuitively to the melting point.[16, 17] Figure 2.2 shows plots between the vacancy formation energy and the melting point. The data on the activation enthalpy for the formation of monovacancies in various metals are satisfactorily represented by a straight line passing from the origin and are given by: DH*f
80.2 Tm

(3)

for FCC and HCP metals, and DH*f
97 Tm for BCC metals.

(4)

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(a)

(b) Notes: 1. Energies for vacancy formation were obtained from Atomic Defects in Metals, vol. 25 (H. Ullmaier, ed.), Landolt-Börnstein Series, Springer-Verlag (1991). 2. Melting temperatures were obtained from Thermochemical Data of Pure Substances, vols. I and II (Ihsan Barin, ed.), VCH, Weinheim, FRG (1995).

Figure 2.2. (a) Linear correlation between the vacancy formation energy and melting temperature for FCC and HCP close-packed metals; slope
80.2 J mol
1 K
1. (b) Linear correlation between the vacancy formation energy and melting temperature for BCC metals; slope
97 J mol
1 K
1.

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In vacancy formation, the coordination number decides the number of bonds that are broken. The data points for FCC and HCP metals lie on the same line because the coordination number is same in either case. Equations (3) and (4) also provide an example to show that the proportionality constant between the diffusion parameter and the bulk properties depends on the crystal structure. The data on ∆Hf* plotted in Fig. 2.2(a) and (b) are listed in Tables 2.1 and 2.2. The tables also give the vacancy migration energy, ∆H m* , as well as the ratio between the two quantities. Several interesting facts emerge out of the data in the two tables: • The vacancy formation energies are generally higher than the migration energies. However, the ratio of vacancy formation to the migration energy shows significant variations from metal to metal. Despite this, the correlation with the melting point is very well maintained. It shows that the subtle differences in the process of diffusion itself among different metals do not affect the relationship between the melting point and the vacancy formation energy. • The ratio of vacancy formation energy to the migration energy in FCC and HCP lattices, in comparison to BCC, is generally smaller. • The same feature is seen in the ratio of vacancy formation energy to the melting point; namely, Eqs. (3) and (4). The proportionality constant for closed packed lattices is smaller, indicating that the vacancy formation in these cases is energetically easier than the BCC structures. • The effect of electronic band structure on the ratio of vacancy formation energy to the migration energy can also be discerned from Tables 2.1 and 2.2. This ratio is higher for normal metals, where s and d bands are separated. This rule is followed without exception by BCC metals. The FCC and HCP metals show the same trend. However, nickel is one exception for which the ratio ∆H*f
∆Hm* is nearly the same as that for copper. Diffusion parameters have been linked to the melting parameters in several other ways. Historically, as well as from practical considerations, the correlations between the melting and diffusion parameters are very important. The activation enthalpy for diffusion has been related[3–5] to the melting point (Tm) and the enthalpy of fusion (Hm) as: ∆H*
K1Tm

(5)

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and ∆H*
K2Hm .

(6)

Here, K1 and K2 are constants for any given class of solids. The plots for Eqs. (5) and (6) are shown in Figs. 2.3 and 2.4. In both cases, the relationship Table 2.1. Vacancy Formation and Migration Energies in FCC and HCP Metalsa Melting Vacancy Formation Vacancy Migration ∆H*f
Tm Point (K) Energy (∆H*f
k J mol
1) Energy (∆Hm*
k J mol
1) (J mol
1 K
1) ∆H*f
∆Hm*

Metal FCC

Ag Al Au Cu Ni Pb Pd Pt Th

1233.8 933.1 1336 1356 1726 600.4 1825 2042 2024

107.12 64.65 89.75 123.52 172.74 55.97 178.52 130.27 123.52

63.69 58.86 68.51 67.55 100.36 41.50 99.40 138.00 196.86

86.79 69.29 67.16 91.08 100.06 93.27 97.81 63.79 61.02

1.682 1.098 1.310 1.829 1.721 1.350 1.796 0.944 0.627

HCP Cd Mg Tl Zn

594 923 576 692.5

44.39 77.20 44.39 52.11

38.60 48.25 55.97 40.53

74.7 83.6 77.1 75.2

1.150 1.600 0.793 1.286

a. P. Erhart, in Atomic Defects in Metals, Landholt-Bornstein New Series, vol. 25 (H. Ullamier, ed.), Springer-Verlag (1991), p. 88

Table 2.2. Vacancy Formation and Migration Energies in BCC Metalsa

Metal

Melting Point (K)

Li Na K V Nb Ta Cr Mo W

453 371 336 2173 2690 3269 2148 2885 3695

Vacancy Formation Vacancy Migration ∆H*f
Tm Energy (∆H*f
k J mol
1) Energy (∆Hm*
k J mol
1) (J mol
1 K
1) ∆H*f
∆Hm* 46.32 32.81 32.81 202.65 260.55 299.15 202.65 299.15 347.4

3.67 2.90 3.67 48.25 53.10 67.55 91.68 130.28 164.10

102.2 88.4 97.6 93.2 96.8 91.5 94.3 103.7 94.0

12.62 11.31 8.940 4.000 4.907 4.429 2.210 2.296 2.117

a. H. Schultz, in Atomic Defects in Metals, Landholt-Bornstein New Series, vol. 25 (H. Ullamier, ed.), Springer-Verlag (1991), p. 115

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Notes: 1. Activation energies were obtained from Smithells’ Metal Reference Book, vol. VII (E. A. Brandes and G. B. Brook, eds.), Butterworths Pub. (1992). 2. Melting temperatures were obtained from Thermochemical Data of Pure Substances, vols. I and II (Ihsan Barin, ed.), VCH, Weinheim, FRG (1995).

Figure 2.3 Correlation between the activation energy for self-diffusion in metals and melting point; slope
146 J mol1 K1.

is linear. The linearity of these plots is validated by the high value of their regression coefficients. The values of K1 and K2 are the same, even if FCC, HCP, and BCC metals are considered separately. From Figs. 2.3 and 2.4, K1
146 and K2
14.8. The validity of Eq. (5) has been demonstrated for alkali halides as well by Barr and Lidiard.[6] For inert gas solids and molecular organic solids, the validity of Eqs. (5) and (6) has been established by Chadwick and Sherwood.[18] The diffusivity plots for any group of solids having identical physical and chemical properties scale inversely with the magnitude of the entropy of fusion.[19, 20] In Figs. 2.5 through 2.10, the diffusivity plots for metals as well as other classes of solids are shown. In the case of alkali halides, intrinsic conductivities, which are mediated by the ionic diffusion, have been used. The two quantities are related by the Nernst-Einstein equation.

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Notes: 1. Activation energies were obtained from Smithells’ Metal Reference Book, vol. VII (E. A. Brandes and G. B. Brook, eds.), Butterworths Pub. (1992). 2. Melting temperatures were obtained from Thermochemical Data of Pure Substances, vols. I and II (Ihsan Barin, ed.), VCH, Weinheim, FRG (1995).

Figure 2.4 Correlation between the activation energy for self-diffusion in metals and the latent heat of fusion; slope
14.8.

Numbers in parentheses represent the entropy of fusion. In each group or a class of solids, a low value of entropy of fusion is an indication of high diffusion rates, and vice versa. The relationship between the magnitude of the entropy of fusion and the relative rates of the diffusion within a group of solids holds, in general, irrespective of the nature of chemical bonding.[19, 20] Similar plots for a larger number of systems are shown elsewhere.[20] This phenomenon can be explained on the basis of the assumption that the free energy of activation for diffusion, ∆G*, is in direct proportion to the free energy of the liquid state. Hence, as first suggested by Dienes,[9] we may write:[21] ∆G*
k Gl,

(7)

where k is a constant and Gl is the free energy of the matrix in the liquid state. Differentiation of Eq. (7) yields: ∆S*
k Sm ,

(8)

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Note: Numbers in parentheses represent the entropy of fusion, in units of Jmol1 K1, from Thermochemical Data of Pure Substances, vols. I and II (Ihsan Barin, ed.), VCH, Weinheim, FRG (1995).

Figure 2.5 Logarithmic plots of self-diffusion coefficients in metals plotted against the homologous temperature (Tm
T), showing that the self-diffusion rates vary inversely with the entropy of fusion. Numbers in parentheses represent the entropy of fusion in J mol1 K1.

where Sm is the entropy of fusion. Further, Gl
Hm  TSm .

(9)

Therefore, from Eqs. (7) and (9), we get: ∆G*
kHm  k TSm .

(10)

A comparison of temperature-independent parameters between Eqs. (2.2) and (10) shows that: ∆H*
k Hm .

(11)

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Note: Numbers in parentheses represent the entropy of fusion, in units of Jmol1 K1, from Thermochemical Data of Pure Substances, vols. I and II (Ihsan Barin, ed.), VCH, Weinheim, FRG (1995).

Figure 2.6 Logarithmic plots of self-diffusion coefficients in lanthanide metals plotted against the homologous temperature (Tm
T ), showing that the self-diffusion rates vary inversely with the entropy of fusion. Numbers in parentheses represent the entropy of fusion in J mol
1 K
1.

It is obvious that Eqs. (6) and (11) are identical. Hence, k
K2. By substitutions from Eqs. (8) and (11) in Eq. (1), we have:



K2Sm D
fa2n exp   R

T  1. Tm

(12)

According to Eq. (12), D should vary inversely with the entropy of fusion at a constant value of Tm
T. A plot of Eq. (12) for some common

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Note: The entropy of fusion is indicated by the numbers in parentheses, in units of Jmol1 K1, from Thermochemical Data of Pure Substances, vols. I and II (Ihsan Barin, ed.), VCH, Weinheim, FRG (1995).

Figure 2.7 Logarithmic plots of ionic conductivity (s ) multiplied by temperature (T ) as a function of homologous temperature (Tm
T ) for silver and lithium halides. Numbers in parentheses represent the entropy of fusion in J mol1 K1. The product sT is directly proportional to the self-diffusion rates. Ionic conductivity data for alkali halides are from Uvarov et al.[12]

metals shown in Fig. 2.11 bears out this expectation. The linearity of the plots in Fig. 2.11 validates Eq. (12) and the assumption made in its derivation. The diffusion coefficient is pressure-dependent in view of the contribution of the term P∆V* to the Gibbs free energy (∆G*), which is the controlling thermodynamic factor in its entirety. Consequently, in the pressure-dependent diffusion measurements at constant temperature, the activation volume, ∆V*, becomes an analogous parameter to ∆H*. Nachtrieb et al.[22] have correlated these two parameters with the pressure dependence of the melting temperature (dTm
dP) as: ∆V*
(∆H*
Tm)(dTm
dP).

(13)

Equation (13) predicts that ∆V* is controlled by the sign and magnitude of dTm
dP. In fact, it shows excellent agreement with the experimental data.[23] In general, dTm
dP is positive for metals. However, it is negative

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Note: The entropy of fusion is indicated by the numbers in parentheses, in units of Jmol1 K1, from Thermochemical Data of Pure Substances, vols. I and II (Ihsan Barin, ed.), VCH, Weinheim, FRG (1995).

Figure 2.8 Logarithmic plots of ionic conductivity (s) multiplied by temperature (T ) as a function of homologous temperature (Tm
T ) for potassium halides. Numbers in parentheses represent the entropy of fusion in J mol
1 K
1. The product sT is directly proportional to the self-diffusion rates. Ionic conductivity data for alkali halides are from Uvarov et al.[12]

for plutonium and, as expected from Eq. (13), ∆V* is negative for this element.[24] A critical test of this equation was performed by Zanghi and Calais,[25] who measured the activation volume for self-diffusion in the Pu-Zr system and showed that the magnitude and sign of ∆V * are controlled by dTm
dP. The work of Zanghi and Calais[25] provides support for extending the correlation between the diffusion and melting parameters to alloys as well. As pointed out earlier by Vignes and Birchenall[26] for systems

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Note: The numbers in parentheses represent the magnitude of entropy of fusion, in units of J mol1 K1, from Thermochemical Data of Pure Substances, vols. I and II (Ihsan Barin, ed.), VCH, Weinheim, FRG (1995).

Figure 2.9 Logarithmic plots of self-diffusion coefficients in organic plastic solids as a function of homologous temperature (Tm
T ). Numbers in parentheses represent the magnitude of entropy of fusion in J mol1 K1. On a relative scale, a low value of the entropy of fusion indicates a higher self-diffusion rate, and vice versa. Self-diffusion data for organic solids are from Chadwick and Sherwood.[18]

exhibiting extended solubility, the variations in the activation energy for interdiffusion scale uniformly with the solidus temperature. More interestingly, Roux and Vignes[27] and Ablitzer[28] have shown that the solute diffusion rates vary systematically with the slope of the solidus curve. Close relationships between diffusion and melting parameters are depicted in Eqs. (3) through (6). Furthermore, the inverse scaling of the diffusion coefficient with the entropy of fusion and the magnitude of the activation volume are satisfactorily predicted by Eqs. (12) and (13). Although not yet fully understood, these features are common to all types of solids, irrespective of the type of crystal structure and nature of chemical bonding.[12, 18, 19, 20] The configuration consisting of a vacancy and its

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Note: The numbers in parentheses represent the magnitude of entropy of fusion, in units of J mol1 K1, from Thermochemical Data of Pure Substances, vols. I and II (Ihsan Barin, ed.), VCH, Weinheim, FRG (1995).

Figure 2.10 Logarithmic plots of self-diffusion coefficients in rare gas solids as a function of homologous temperature (Tm
T ). Numbers in parentheses represent the magnitude of entropy of fusion in J mol
1 K
1. On a relative scale, a low value of the entropy of fusion indicates a higher self-diffusion rate, and vice versa. Selfdiffusion data for rare gas solids are from Chadwick and Sherwood.[18]

neighboring atoms formed at the saddle-point in the path of diffusion has been designated as relaxion by Nachtrieb and Handler.[5] Structurally, this configuration is hypothesized to be similar to the molten state of the matrix. Within the relaxion, ions can roll over or squeeze past one another, and the diffusive jump is facilitated by the absorption of phonons. The relaxion concept provides a simple explanation for the linear relationship between diffusion and the melting parameters.

2.2.2 Elastic Constants The correlation between the elastic constants and diffusion parameters was pioneered by Wert and Zener.[2] They first suggested that a fraction, l, of the free energy difference between the equilibrium position and the saddle-point configuration of a diffusing atom (∆G*) arises from the straining of the surrounding lattice. Consequently, the entropy of activation

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Figure 2.11 Logarithmic plots of self-diffusion coefficients in some common metals against the parameter [(Tm
T )  1]. The linearity of the plots demonstrates the validity of the relationship between self-diffusion and the entropy of fusion as given in Eq. (12).

in Zener’s equation[1] is represented as: H* ∆S*
lb  , Tm

(14)

b
d(E
E0)
d(T
Tm).

(15)

where b is defined as

In this formulation, E and E0 are the values of appropriate elastic constants at a temperature T and 0 K, respectively. Although this equation was originally proposed for interstitial diffusion,[2, 29] it has been extended for selfdiffusion via a vacancy mechanism by Zener,[30] LeClaire,[7] Buffington and Cohen,[8] and Flynn.[31] The extension of Zener’s hypothesis to the substitutional diffusion is based on the assumption that the free energy associated with the straining of the lattice during atomic migration bears

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a constant ratio to the total free energy of diffusion. Le Claire,[7] as well as Buffington and Cohen,[8] assumed that the appropriate elastic modulus involved in the straining of the lattice during the diffusion jump is C44  1
2(C11  C12). According to Flynn,[31] however, the appropriate modulus () is given differently, as follows: 15 3 2 1 
     . C C11 C11  C12 C44

(16)

The activation energy for vacancy migration, ∆H*m , for FCC and BCC metals is related to C[32] as: ∆H*m
K3Ca3,

(17)

where a is the lattice parameter and K3 is another constant. The values of K3 for FCC and BCC are 0.022 and 0.020, respectively. According to Erhart et al.,[32] these values, though slightly different from the one given by Flynn,[31] provide a better agreement between the experimental values of ∆Hm and Eq. (17). In the application of Eq. (14), we are concerned with the temperature dependence of C, not with its actual value. This is usually taken as equal to the temperature dependence of Young’s modulus, determined at low temperatures to avoid errors that may be caused by the grain boundary viscous flow at higher temperatures. Zener[1] suggested l to be equal to 0.55 and 1.0, respectively, for FCC and BCC metals. Using these parameters, Lazarus[33] showed that the experimental values of the temperatureindependent pre-exponential factor in Eq. (1) agreed reasonably well with the values calculated with the help of Eq. (14). In practice, the product lb varies between 0.15 and 0.35 for selfdiffusion in metals via a vacancy mechanism, if we take Zener’s value for l[1] and Koster’s value for temperature dependence of Young’s modulus.[34] This narrow range of the product lb allows only a small variation in the frequency factors for the self-diffusion of metals. This is the most important conclusion from Zener’s hypothesis. For any group of solids, its value depends on the mode of diffusion and the crystal structure.[35] A violation of Zener’s equation for ∆S* and melting point correlations for ∆H*, namely, Eqs. (3), (4), and (14), in any system is characterized anomalous and considered worthy of explanation.[4] Beke et al.[36] have extended Zener’s hypothesis to the impurity diffusion in metals using dimensional analysis. Their equation is given as: H*0i  H*0 D0i ln 
F  , Tm D0

  



(18)

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where F is a constant and other terms have the same meaning as before. The subscripts 0 and 0i refer to self-diffusion and impurity diffusion, respectively. Beke et al.[36] showed that the impurity diffusion data in aluminum, copper, and silver are in accord with Eq. (18).

2.2.3 Bulk Modulus Toth and Searcy[37] were first to correlate the activation energy for vacancy migration with bulk modulus. They stated that an adequate analysis of the elastic distortions undergone by the diffusing atoms is not available. The elastic distortions associated with the saddle-point configuration are neither isotropic, as implied by the use of bulk modulus, nor unidirectional, as implied by the use of shear modulus. They are multidirectional. If only one of them is to be used, then any one can be used. Toth and Searcy preferred to use bulk modulus because the data are much more easily available. Following Le Claire[7] and Nachtrieb and Handler,[5] Toth and Searcy used Eq. (6), with a slightly different constant, to evaluate vacancy formation energy. The equation Toth and Searcy developed for self-diffusion in FCC and HCP metals is: ∆H*
22.6BV  0.27Hm ,

(19)

where B is the bulk modulus and V is the specific volume. Room-temperature values were used for bulk modulus and specific volume. The corresponding equation for BCC metals is similar except that the multiplier for the term BV is changed to 25.4. Leibfried[38] has shown that the quantity BV is proportional to the melting point. Using this proportionality, Toth and Searcy also gave the following equation for self-diffusion in FCC and HCP metals: ∆H*
16.0Tm  0.27Hm .

(20)

For BCC metals, the multiplier for Tm is altered only marginally to 15.8 in Eq. (20). Satisfactory agreement with the actual data was obtained for Eqs. (19) and (20). Using median values, identical expressions were also developed for substitutional alloys. Varotsos and Alexopoulos[39] suggested that ∆G* is a function of the product of bulk modulus and the specific volume of the matrix as follows: ∆G*
cBV ,

(21)

where c is an arbitrary constant found to be independent of temperature and pressure. However, it is a function of the mechanism of diffusion. The value of c also varies from one system to another. With the help of Eq. (21),

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Eq. (1) for the diffusion coefficient can be written as:

 

cBV ln D
ln( fa2u)   . RT

(22)

Further, the activation entropy and energy for self-diffusion can be expressed as:





∂(BV) ∆S*
c  , ∂T P

(23.1)

and





∂(BV) ∆H*
cBV  Tc  . ∂T P

(23.2)

Equations (22), (23.1), and (23.2) form the basis of the model proposed by Varotsos and Alexopoulos.[40] In this model, all the information about the process of diffusion is contained in the parameter cBV. The constant c varies with the change of host lattice as well as with the change in the mode of diffusion. Its magnitude is estimated with the help of Eqs. (21) and (22) at one particular temperature, usually at T = 0 K. The temperature and pressure variations of the diffusion coefficient or any other diffusion parameter are governed by the corresponding variations in the product BV. The model has been applied extensively to self-diffusion in metals, alkali halides, and rare gas solids.[40-43] It has also been applied to the estimation of the formation and migration volumes of vacancies, solute diffusion in metals, and nonlinear/curved diffusivity plots for self-diffusion in FCC and BCC metals.[43] There seems to be little doubt that a reasonable numerical agreement exists between the model and the experimental results, and these equations can be used to estimate the diffusion data when actual data are lacking. However, one basic difficulty with this method of data treatment is that Eq. (21) equates free energy with enthalpy. This thermodynamic approximation is strictly valid only at absolute zero temperature. In fact, this condition is used by the authors to evaluate the constant c. Secondly, the basis of this model is the same as Zener’s hypothesis.[1,29,30] However, in Zener’s model, the constant l has a simple and direct interpretation. A similar interpretation cannot be attributed to the constant c in the model proposed by Varotsos and Alexopoulos.[39]

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Patil and Tiwari have given an equation correlating the vacancy formation energy with the compressibility.[44] Applying thermodynamics to the continuum models of lattice, the equation: a S*
  * b V

(24)

was independently derived by Keys[45] and Lawson.[46] Here, a is the coefficient of volumetric thermal expansion, and b is the compressibility.

2.2.4 The Debye Temperature From semi-empirical considerations, Mukherjee[10] derived a relationship between the vacancy formation energy and Debye temperature (q) as: H*f q
K4  MV 2
3





1
2

,

(25)

where M is the atomic weight, V is the specific volume, and K4 is another constant. This equation connects vacancy formation energy with a dynamic property of the perfect lattice. The Mukherjee’s equation was later derived by March,[47] somewhat more rigorously, on the basis of screening theory. Subsequently, Tewary[48] gave a formal derivation based on Fourier-transform of two-body pair potential in a crystal. Vacancy formation energy is treated as equal to half the sum of pair potential at a lattice site. It is shown that in an isotropic Debye approximation, this sum is proportional to the Debye temperature. According to Tewary,[48] the constant in Eq. (25) is defined by: K4
(h
k)(9N
4p)1
3(2 J)1
2,

(26)

where h is Planck’s constant, N is Avogrado’s number, and J is the mechanical equivalent of heat. The relationship between q and ∆H* as given in Eq. (25) is shown graphically in Fig. 2.12. The value of K4 given by Eq. (26) is equal to 34.2, which is the same as that obtained from Fig. 2.12. The agreement between the theory and the experiment is indeed very satisfactory. Equation (25) has been extended to alkali halides by Sastry and Mulimani.[49]

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Notes: 1. Debye temperatures were obtained from C. Kittel, Introduction to Solid State Physics, VII ed., Wiley (1986). 2. Vacancy formation energies were obtained from Atomic Defects in Metals, vol. 25 (H. Ullmaier, ed.), Landolt-Börnstein Series, Springer-Verlag (1991).

Figure 2.12 Debye temperatures versus vacancy formation energies; slope
34.7.

2.2.5 Valence Bond Parameter Valence bond parameter for a metal is defined as the ratio of cohesive energy (E) to its most prominent chemical valence (Z). An interesting relationship between the valence bond parameter and ∆H* is discussed here.[50] The kinetic energy, Uo, of an atom of mass m vibrating in a crystal is given by: 1 Uo
 m g v 2. 2

(27)

Here, g and v are the maximum displacement and the corresponding vibration frequency, respectively. Assume that Uo is some constant fraction,

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r, of cohesive energy per valence bond, that is, the valence bond parameter, E
Z.[51] Hence, we may write:

   

1 v
 mg

1
2

2rE  Z

1
2

.

(28)

Further, in the harmonic approximation, v is the central frequency of the atomic vibration and is equal to the Debye-frequency evaluated from the specific heat. Hence, Eqs. (25) and (28) can be combined to yield:

 

E ∆H*f
A1  , Z

(29)

where A1 is a proportionality constant. A more useful relationship for predicting the activation energy for self-diffusion can be derived from Eq. (29) on the basis that the ratio between the activation energies for vacancy formation and self-diffusion is a constant for elements that belong to the same crystal class.[52] Therefore, we have:

 

E ∆H*
A2   A3, Z

(30)

where A2 and A3 are other constants. Figure 2.13 shows a plot of ∆H*f versus E
Z. As in the case of Eqs. (3) and (4), the data points for FCC and

Note: Vacancy formation energy was obtained from Atomic Defects in Metals, vol. 25 (H.Ullmaier, ed.), Landolt-Börnstein Series, Springer-Verlag (1991).

Figure 2.13 Vacancy formation energy shown as a function of valence bond parameter (E /Z); see Eq. (30).

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Figure 2.14 Activation energy for self-diffusion in metals as a function of valence bond parameter (E/Z ). This correlation makes a distinction between allotropic (dotted lines) and nonallotropic metals; see Eq. (30).

HCP metals lie on the same line. For BCC metals, the data points for transition and the alkali metals lie on different lines. This is in contrast to Fig. 2.2(b), where the data points for all BCC metals lie on the same line. This feature is also repeated in Fig. 2.14, which shows the plots between the valence bond parameter and the activation energy for self-diffusion. The variations from one group to another in Figs. 2.13 and 2.14 can be attributed to the term r in Eq. (28). Thus, the constants A1, A2, and A3 vary with the nature of chemical bonding and the crystal structure. A novel feature of correlation depicted in Figs. 2.13 and 2.14 is that it distinguishes between the allotropic and nonallotropic matrices among FCC, HCP, and BCC metals. Incidence of allotropy enhances the overall diffusion rates in a matrix. The influence of allotropy on the diffusion characteristics is discussed elsewhere.[53] An interesting aspect of Figs. 2.13 and 2.14 is that Cu, Ag, Au, and Pt correlate satisfactorily with other metals only when their Z values are taken as 2, 2, 2, and 3, respectively. This suggests that in some cases, the number of electrons contributing to the cohesion are different from those effective in chemical reaction. Considering the configuration of their outermost shell, Cu, Ag, and Au are regarded as monovalent metals. Their most prominent chemical valencies are 2, 1, and 3, respectively. Similarly, the prominent valencies of Pt are 2 and 4. It has been clearly demonstrated for copper and silver that the agreement between calculated and experimental values of cohesive energy is poor if only a single electron is allowed to take part in the bonding.[54] In the noble metals group, s-d hybridization contributes to the bonding and alloying behavior in a significant way, and thus indirectly controls the number of effective bonding electrons.[55] This number could be different from their prominent chemical

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valencies. This factor seems responsible for the behavior exhibited by these elements in Figs. 2.13 and 2.14.

2.2.6 Electron-to-Atom Ratio Fumi[56] has shown that the vacancy formation energy is a simple function of Fermi energy [EF] as: ∆H*f
t x EF ,

(31)

where t is a constant less than unity and x represents the valence number of the matrix element. In the rigid-band approximation, this equation can be extended to alloys, if x is identified as the electron-to-atom ratio. Then, x is the average number of electrons contributed by each atom to the conduction band. Fermi energy is a function of the number of electrons per unit volume: EF
(h2
8m*)(3
p)2
3(n*
V)2
3.

(32)

Here, m* and n* are the electronic mass and number of free electrons present in a crystal of volume V. If c is the volume occupied by a single atom, then x is given by: x
(n*
V)c.

(33)

c
a3
4.

(34)

In an FCC lattice,

Combining Eqs. (31) through (34), and assuming, as before, that the ratio of the activation energies for vacancy formation and the self-diffusion is the same for any one system, we get:[57] ∆H*
K5 x 5
3 a2,

(35)

where K5 is a constant. A plot of solvent diffusion in silver-base alloys based on Eq. (35) is shown in Fig. 2.15. Although the plots are linear, the slopes are different for different solutes. If ∆H* depends only on x, then the slopes for different solutes should be identical. It is generally presumed that the variations in the solvent diffusion rates at a fixed value of x for different solutes are caused by the differences in the electro-negativity of the solutes. This parameter controls transfer of electrons from electropositive

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Notes: 1. The diffusion data for Ag-Ge alloys were obtained from R. E. Hoffman, D. Turnbull, and E. W. Hart, Acta Metall., 3:417 (1955). 2. The diffusion data for Ag-Cd and Ag-In alloys were obtained from A. Schoen, Ph. D. thesis, University of Illinois (1958). 3. The diffusion data for Ag-Sb alloys were obtained from E. Sonder, Phys. Rev., 100:1662 (1955). 4. The diffusion data for Ag-Tl alloys were obtained from R. E. Hoffman, Acta Metall., 6:95 (1958).

Figure 2.15 ∆H *a2 versus (e/A )5
3 plots for silver-base systems: Ag-Ge, Ag-Cd and Ag-In, Ag-Sb, and Ag-Tl alloys.

component to electronegative component within the matrix. However, the correct value of electronic charge associated with any solute is difficult to establish in most cases. Therefore, it is difficult to account properly for this effect. All the same, Eq. (35) remains valid for each single-alloy system. Equation (35) can be extended to solute diffusion as well.[57]

2.2.7 Summary of Empirical Correlations To summarize Section 2.2, the correlations between the diffusion and melting parameters give the best results compared to any other relationship involving diffusion parameters and bulk properties. In this way, these correlations are unique for three reasons: 1. Simplicity 2. Different possible ways in which the two phenomena correlate with each other 3. Their universal character in the sense that their validity is not restricted by the crystal type and nature of cohesive forces.

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The correlations with the elastic constants and the bulk properties have somewhat restricted applicability, mainly because the nature of elastic deformation associated with the saddle-point configuration is not known with any degree of certainty. The success of Zener’s hypothesis in metallic systems is due to the fact that the temperature dependence of Young’s modulus and other elastic constants is nearly the same. However, the applications of Zener’s hypothesis have so far been restricted mainly to the metals. Lack of appropriate data in other types of solids may be responsible for this situation. The correlations with Debye temperature, valence bond parameter and electron-to-atom ratio should be viewed as first-order effects such that no violations of the proposed relationships have been observed. A reasonable degree of scatter in data points in these cases is not unexpected when we consider matrices with different crystal structure and the subtle variations in the nature of cohesion. The equations given in this section to estimate the values of ∆H* and ∆S* can be used to obtain the magnitudes of the diffusion parameters from the knowledge of bulk properties. Their substitution in Eq. (1) will give the diffusion coefficient. Finally, it should be emphasized that because the correlations discussed in this section are generally empirical or semi-empirical in nature, their success should be judged more by the systematics rather than by the numbers only. From fundamental considerations, it is more important to consider the diverse nature of solids for which any particular relationship is valid. Further, any case of nonconformability within a group should be closely examined to ascertain the reasons for its deviation from the pattern followed by others within the same group. In this context, two examples are relevant. When Wert and Zener[2, 29] first proposed their correlations between Do and ∆H*
Tm for interstitial diffusion in BCC metals, some of the results deviated from the predicted behavior. Subsequently, these data were found to be spurious, and the experiments on the well-characterized material yielded the expected agreement.[1] Much later, Barr and Dawson[58] showed that the deviations in the activation energy for cationic diffusion in alkaline earth oxides from the value expected from the melting point correlations were due to the extrinsic factors arising from the presence of impurities in the matrix.

2.3

Equilibrium Thermodynamic Calculation of Diffusion Parameters

In this section, we derive an equation relating the activation volume ∆V* with the activation entropy ∆S*. The derivation of this equation is based on equilibrium thermodynamics.

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2.3.1 The Activation Volume Vaidya showed[59] that the logarithm of the diffusion coefficient exhibits a regular variation with the specific volume of the matrix. Vaidya noted that, despite the discontinuous change in the volume upon melting, the extrapolated value of the diffusion coefficient agrees satisfactorily with the experimental value for the self-diffusion in liquid state at the melting point. Earlier, a Taylor series expansion of ∆G* as a function of temperature was used by Vineyard and Dienes[60] to analyze the effect of thermal expansion on the formation energies of vacancies and interstitials. These authors assumed that the crystal vibrational frequencies as well as its potential energies depend on crystal volume alone and on no other thermodynamic variable. This view is supported by Borelius,[61] who showed that the vibrational entropies of liquids are the same as those of solids, and the difference in the entropies of solid and liquid arises mainly from the changes in the volume. It has also been shown that the elastic constants[62, 63] and the compressibility[64] vary in a regular way with specific volume. Mott and Gurney[65] used a similar approach to estimate the temperature variation of activation energy for self-diffusion in ionic crystals. Following the same approach, we expand here the free energy of activation for self-diffusion as a function of specific volume in a Taylor series.[66] A prime requirement of Taylor series expansion is the fixing of origin. The value of the variable parameter at the origin constitutes the first term of the expansion. We choose T
0 K as the origin, and Eq. (2.2) gives the value of ∆G* at origin as ∆H*. Hence, ∆H* becomes the first term in the expansion. Therefore, the expansion of ∆G* in a Taylor series as a function of specific volume (V) yields: ∆G*
∆H*  [∂(∆G*)
∂V](∆V)  [∂2(∆G*)
∂V 2](∆V)2
2! ......... (36) Equation (36) is a general equation for ∆G* as a function of V. It is implicitly assumed in writing this equation that ∆G* is completely defined by change in the magnitude of the specific volume, which can be brought about by changing temperature and pressure, either individually or collectively. It is also implied that when we try to evaluate the derivative of ∆G* with respect to specific volume, it is of little consequence which one of the two parameters, T or P, is held constant. At any particular value of V, ∆G* has a fixed value. Accordingly, the slope of the plot of ∆G* with respect to V at any given point is controlled only by the magnitude of V (see Appendix 2A). In evaluating the first derivative of ∆G* in Eq. (36), it would therefore be proper to use the basic equation, which defines the activation volume as: ∆V*
[∂(∆G*)
∂P]T
[∂(∆G*)
∂V]T bV.

(37)

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As before, b represents compressibility. Ignoring second and higher order terms in Eq. (36) and combining it with Eq. (37), we get: ∆G*
∆H*  (∆V*
b) (∆V
V).

(38)

The specific volume of the matrix at any temperature is related to that at 0 K, V0, by V
V0(1  aT ),

(39)

where a is the volumetric coefficient of thermal expansion. Furthermore, ∆V
V  V0
Vo aT.

(40)

From Eqs. (39) and (40), we have: ∆V
V
(aT)
(1  aT).

(41)

Substituting for ∆V
V from Eq. (41) in Eq. (38), we obtain: ∆G*
∆H*  (∆V*
b) (aT)
(1  aT ).

(42)

Comparison of Eq. (42) with Eq. (2.2) gives the desired relationship: ∆V*
∆S*(a
b)1 (1  aT ).

(43)

The thermodynamic principle used in the derivation of Eq. (43) is contained in Eq. (37). Equation (43) represents a formal relationship between the diffusion parameters and bulk properties. The calculation of activation volume with the help of Eq. (43) involves the parameter a
b, which is given by: (∂S
∂V)T
a
b.

(44)

Here, S denotes the entropy of the matrix. Thus the magnitude of a
b controls the variation of entropy with the specific volume of the matrix. This, in turn, is dependent on the nature of chemical bonding between the atoms. Using a
b values of the matrix implies that the variation of entropy with specific volume follows identical behavior for the matrix as well as for the activated complex. This is a reasonable assumption because in both situations, the chemical species involved are identical and therefore the nature of bonding remains unaltered.

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Table 2.3. Comparison of Experimental and Calculated Values of the Activation Volume for Self-Diffusion in Zinc, Cadmium and Sodiuma Activation Volume (106 m3 mol1) Serial No. Matrix Temp (K) Experimental Calculated Differences (%) 1 2 3 4 5 6 7 8 9

Zn

Cd

Na

573 623 673 524 549 574 592 287.8 364.3

4.28 3.97 3.72 6.90 7.14 7.65 7.49 11.46 12.40

3.20 3.22 3.44 4.84 5.08 5.49 5.87 10.23 17.25

25.2 18.9 7.5 29.8 28.8 28.2 21.6 10.7 39.1

a. G. P. Tiwari, Scripta Mater., 39:991 (1998)

Table 2.3 compares activation volumes estimated from Eq. (43) with the experimental values. Data for the activation volume for self-diffusion as a function of temperature are available only for zinc,[67] cadmium,[68] and sodium.[69] The differences between the experimental and the calculated values range from 7.5 to 39.1%. Two opposite trends are seen in Table 2.3. For zinc and cadmium, the agreement improves with the temperature, while the opposite is true for sodium. With sodium, the measured compressibility values are affected by the extreme plasticity of the matrix at high temperature. Bridgman[70] pointed out that the plasticity of metals like sodium at T  0.5 Tm introduces error in the compressibility measurements. Therefore, the agreement is poorer at higher temperatures. With zinc and cadmium, the available compressibility data have been obtained on polycrystalline specimens, whereas the diffusion experiments were performed on single crystals. The presence of grain boundaries will lower the compressibility. The difference between the compressibility of the matrix and grain boundary will decrease with the increase of temperature. In accordance with this suggestion, the agreement between calculated and experimental values improves with the increase in temperature. It is concluded that discrepancies in the estimated value of ∆V* arise mainly due to the uncertainties in the values of parameters used in its calculation. The relationship given in Eq. (43) is otherwise perfectly valid as it does not contain any empirical or semi-empirical constant. Therefore, as

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shown in the following section, it can be used to obtain estimates of other diffusion parameters.

2.3.2 Activation Entropy and Diffusion Frequency Temperature variation of diffusion coefficients usually follows an Arrhenius relationship and is written as: D
D0 exp(∆H*
RT).

(45)

If D0 and ∆H* are both constant in Eq. (45), a logarithmic plot of the diffusion coefficient as a function of reciprocal of absolute temperature will be a straight line. The slope of such a plot yields ∆H*, whereas D0 is given by the intercept on the y-axis. Thus, D0 and ∆H* are both experimentally determinable quantities. A comparison between Eqs. (1) and (45) shows that the frequency factor, D0, is: D0
fa2u exp(∆S*
R).

(46)

Equation (46) gives the value of ∆S* with the help of the experimentally determined value of D0. The value of ∆S* estimated from Eq. (46) depends, however, on the assumed value of the diffusion frequency (usually the Debye frequency, νD). Equation (43) permits us to obtain an independent value of ∆S* without the input of the assumed value of diffusion frequency. Substituting for (1  aT) from Eq. (39) in Eq. (43) and rewriting, we obtain: ∆S*
(a
b) (∆V*
V)V0 .

(47)

The only inputs needed to calculate ∆S* from Eq. (47) are the experimental values of a
b, ∆V*
V, and V0. Substituting the above value of ∆S* in Eq. (46), the diffusion frequency u may be independently estimated because f and a are known and D0 is an experimentally determined quantity. The estimated values of ∆S* and u as well as the data used in their evaluation are recorded in Table 2.4 for a number of metals. Values of ∆S* thus estimated are compared with those computed traditionally from Eq. (46) by taking Debye frequency as the diffusion frequency derived from the Debye temperatures. The differences between the two values are also recorded in the table. Similarly, independent estimates of diffusion frequencies computed with the help of independently estimated ∆S* are compared with Debye frequencies. Differences between them are also listed in Table 2.4.

a, c

b

9.993 7.092 10.207 6.597 8.852 10.284 9.095 18.272 7.097 7.097 7.097 23.672 45.465 55.869 12.996 9.170 9.170 13.977 13.977 12.955 12.955 c

0.87 0.90 0.85 0.80 1.01 0.86 1.09 0.80 0.77 0.77 0.77 0.41 0.55 0.40 0.26 0.65 0.65 0.77 0.77 0.63 0.63 d

0.78146 0.78146 0.78146 0.78146 0.78146 0.78146 0.78146 0.78146 0.78146 0.72722 0.72722 0.72722 0.72722 0.72722 0.72722 0.78146 0.78121 0.78146 0.78121 0.78146 0.78121 b

4.0491 3.6153 4.078 3.5238 3.8902 4.086 3.931 4.9489 2.8664 2.8644 2.8644 4.289 5.334 5.63 3.5089 2.6649 2.6649 3.2088 3.2088 2.9787 2.9787 e

0.047 0.16 0.04 0.92 0.21 0.04 0.33 1.37 0.49 2 1.9 0.004 0.16 0.23 0.038 0.13 0.18 1 1.5 0.12 0.18 f

428 343 165 450 274 225 240 105 470 470 470 158 91 56 344 327 327 400 400 209 209

47.24 42.80 61.12 35.31 58.88 50.91 73.47 53.44 32.43 32.43 32.43 12.98 17.43 11.60 5.47 42.48 42.48 31.01 31.01 34.18 34.18

11.76 25.67 18.22 38.38 28.58 15.61 33.26 48.14 36.21 48.51 48.08 0.80 30.83 36.99 14.79 29.41 32.12 41.61 44.98 30.62 33.99

R. E. Hanneman and H.C. Gatos, J. Appl. Phys., 36:1794 (1965) H. E. Boyer and T.L. Gall (eds.), Metals Handbook, Desk Edition, ASM, OH (1985) A. M. Brown and M. F. Ashby, Acta Metall., 28:1094 (1980), table 3 A. D. LeClaire, in Physical Chemistry: An Advanced Treatise (H. Eyring et al., eds.), vol. 10, Academic Press, New York (1970), chap. 15 N. L. Peterson, J. Nucl. Mater., 69–70:3 (1978) C. Kittel, Introduction to Solid State Physics, Wiley Eastern, New Delhi (1977), chap. 5, table 1, p. 126

a,b

Ref.

1.28  1016 7.51  1017 6.00  1017 5.69  1017 5.28  1017 1.00  1016 3.60  1017 2.37  1016 5.94  1017 5.94  1017 5.94  1017 1.56  1015 3.57  1015 5.20  1015 8.70  1016 1.67  1016 1.67  1016 2.82  1016 2.82  1016 2.14  1016 2.14  1016

f

*

75.1 40.0 70.2 8.7 51.5 69.3 54.7 9.9 11.7 49.6 48.2 106.1 76.9 219.0 170.2 30.8 24.4 34.2 45.1 10.4 0.6

1.25  1011 9.10  1011 1.97  1010 1.36  1013 1.49  1011 6.72  1010 3.97  1010 1.16  1012 1.54  1013 6.77  1013 6.43  1013 6.27  1011 9.50  1012 2.47  1013 2.20  1013 1.41  1012 1.96  1012 2.98  1013 4.48  1013 2.84  1012 4.26  1012

8.92  1012 7.15  1012 3.44  1012 9.37  1012 5.71  1012 4.69  1012 5.00  1012 2.19  1012 9.79  1012 9.79  1012 9.79  1012 3.29  1012 1.90  1012 1.17  1012 7.17  1012 6.81  1012 6.81  1012 8.33  1012 8.33  1012 4.35  1012 4.35  1012

nD (
kθ
h) (sec
1)

98.6 87.3 99.4 44.6 97.4 98.6 99.2 47.1 57.6 95.1 84.8 94.8 80.0 95.3 67.4 79.2 71.2 72.0 81.4 34.8 2.2

Difference (%)

Diffusive Frequency

∆S ∆S n D0 Debye T, (calculated) (from nD) Difference (calculated) (10
10 m) (10
4m2/s) θ (K) (J/mol/K) (J/mol/K) (%) (sec
1) a

Activation Entropy

8:45 AM

a. b. c. d. e. f.

6.96 5.04 4.23 3.81 3.48 5.76 2.67 8.67 3.53 3.53 3.53 20.9 24.9 27.0 14.1 11.9 11.9 8.13 8.13 8.94 8.94

Al Cu Au Ni Pd Ag Pt Pb g-Fe a-Fe d-Fe Na K Rb Li Zn- ‘ c Zn⊥c Mg ‘ c Mg⊥c Cd ‘ c Cd⊥c

V0 (10
6 m3/mol) ∆V*
V

*

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a b Metal (10
5 K
1) (m2/dyne)

Basic Data

Table 2.4. Calculated Values of Activation Entropy and Diffusive Frequency for Self-Diffusion in a Number of Metals

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2.3.3 Specific Heat of the Activated Complex This section further explores Eq. (43) to arrive at the value of specific heat of the activated complex formed during the diffusive jump. The specific heat of the activated complex signifies the difference in the specific heats between two states of the diffusing atom: activated and normal. The Grüneisen relationship is: a
b
g CV
V,

(48)

where g and CV are the Grüneisen constant and the specific heat at constant volume, respectively. Applying this relationship to the activated state, we obtain: ∆C*V
(∆V*
g) (a
b).

(49)

Here, ∆CV* is the specific heat of the activated complex at constant volume. Substituting for a
b from Eq. (43) in Eq. (49), we obtain: ∆C*V
∆S*(1  aT)
g .

(50)

The importance of Eq. (50) lies in the fact that it yields a relationship between two diffusion parameters and can be used to calculate the specific heat of the activated complex in diffusion, since all the parameters in the right-hand side of this equation can be obtained experimentally. The corresponding value of specific heat at constant pressure, ∆CP* , is obtained from the thermodynamic relationship: ∆CP*
∆C*V  {(a2T∆V*)
b}.

(51)

Table 2.5 lists the values of ∆CV* and ∆C*P estimated from Eqs. (50) and (51) for a number of metals, along with the data used in their calculation. The variation of ∆CP* with temperature for some metals such as Cu, Pt, Na, K, and Fe is given in Table 2.6.

2.3.4 Magnitude of Estimated Values The Entropy of Activation ∆S*. Table 2.4 shows that the two estimated values of activation entropy, one from Eq. (47) and the other from Eq. (46), using Debye frequency in the latter, are of the same order: the differences in these values range from 0.6% for Cd to 75.1% for Al. In the case of alkali metals, however, the disparity between the two values is

700 700 700 900 900 700 1000 400 1400 700 1700 350 320 312.64 400 400 600 600

Al Cu Au Ni Pd Ag Pt Pb g-Fe a-Fe d-Fe Na K Rb Li Zn Mg Cd c

0.87 0.90 0.85 0.80 1.01 0.86 1.09 0.80 0.77 0.77 0.77 0.41 0.55 0.40 0.26 0.65 0.77 0.63 a, b

6.96 5.04 4.23 3.81 3.48 5.76 2.67 8.67 3.53 3.53 3.53 20.9 24.9 27.0 14.1 11.9 8.13 8.94 g

2.19 2.00 3.04 1.88 2.28 2.36 2.56 2.62 1.66 1.66 1.66 1.31 1.37 1.67 1.18 2.01 1.48 2.23

g

b

9.993 7.092 10.207 6.597 8.852 10.284 9.095 18.272 7.097 7.097 7.097 23.672 45.465 55.869 12.996 9.170 13.977 12.955

V0 (10
6 m3/mol)

a, h

1.28  1016 7.51  1017 6.00  1017 5.69  1017 5.28  1017 1.00  1016 3.60  1017 2.37  1016 5.94  1017 5.94  1017 5.94  1017 1.56  1015 3.57  1015 5.20  1015 8.70  1016 1.67  1016 2.82  1016 2.14  1016 22.620 22.156 20.702 19.427 26.635 22.440 29.467 21.103 20.502 20.020 20.709 10.633 13.735 7.529 4.889 22.140 21.971 15.874

25.034 23.719 22.565 20.679 28.537 24.576 31.481 23.021 22.183 20.840 22.771 11.651 15.235 8.591 5.225 24.260 23.558 17.140

R. E. Hanneman and H.C. Gatos, J. Appl. Phys., 36:1794 (1965) H. E. Boyer and T.L. Gall (eds.), Metals Handbook, Desk Edition, ASM, OH (1985) A. M. Brown and M. F. Ashby, Acta Metall., 28:1094 (1980), table 3 A. D. LeClaire, in Physical Chemistry: An Advanced Treatise (H. Eyring et al., eds.), vol. 10, Academic Press, New York (1970), chap. 15 N. L. Peterson, J. Nucl. Mater., 69–70:3 (1978) C. Kittel, Introduction to Solid State Physics, Wiley Eastern, New Delhi (1977), chap. 5, table 1, p. 126 K. A. Gschneider, Jr., in Solid State Physics, vol. 16 (F. Seitz and D. Turnbull, eds.), Academic Press (1964), table XXIV, p. 412 C. Kittel, Introduction to Solid State Physics, Wiley Eastern, New Delhi (1977), chap. 3, table 4, p.85 H. M. Gilder and D. Lazarus, Phys. Rev., B11:4916 (1975)

a–f

47.24 42.80 61.12 35.31 58.88 50.91 73.47 53.44 32.43 32.43 32.43 12.98 17.43 11.60 5.47 42.48 31.01 34.18

a ∆V*
V (10
5 K
1)

35.5

82.3 65.5

31.0

38.074

65.898 44.183

16.736

i

38.3

14.644

Difference ∆C*P(%)

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a. b. c. d. e. f. g. h. i.

∆S (J/mol/K)

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Ref.

Temp (K)

Metal

*

∆C*V ∆C*P b (calculated) (calculated) ∆C*P 2 (m /dyne) (J/mol/K) (J/mol/K) (J/mol/K)

Table 2.5. Calculated Values of ∆C*V and ∆C*P of the Activated Complex for a Number of Metals

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Table 2.6. ∆C*P as a Function of Temperature for a Few Common Metals ∆C*P Difference Temp ∆C (ref. a) in ∆C*P Temp ∆C*P Metal (K) (J/mol/K) (J/mol/K) (%) Metal (K) (J/mol/K) * P

Cu

Na

K

a-Fe

600 700 800 900 1000 1100 1200 1300 298.15 350 371 298.15 320 336.35 600 700 800 850 900 950 1000 1100 1184

23.381 23.716 24.059 24.401 24.745 25.092 25.440 25.791 11.384 11.651 11.760 15.056 15.235 15.369 20.652 20.840 21.030 21.125 21.220 21.315 21.411 21.603 21.765

12.552 14.644 16.736 18.828 20.920 23.012 25.104 27.196 56.137 65.898 69.852 41.166 44.183 46.442

46.6 38.3 30.4 22.8 15.5 8.3 1.3 5.5 79.7 82.3 83.2 63.4 65.5 66.9

g-Fe

d-Fe

Pt

1184 1200 1300 1400 1500 1600 1665 1665 1700 1800 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

21.765 21.795 21.989 22.183 22.378 22.574 22.702 22.702 22.771 22.968 30.356 30.636 30.917 31.198 31.481 31.765 32.050 32.336 32.623 32.911 33.200 33.490 33.781 34.073 34.366

a. H. M. Gilder and D. Lazarus, Phys. Rev., B11:4916 (1975)

higher and may be attributed to the uncertainty in the magnitude of compressibility values for these metals.[70] The Diffusion Frequency u. Table 2.4 also shows the comparison between the calculated values of diffusion frequency and the Debye frequency. In a large majority of cases, the differences between them are below 100%. The calculated values are smaller than the Debye frequency for Cd, Pb, Zn, Cu, Pd, Ag, Al, Pt, Au, etc., whereas these values are larger for Ni, Fe, Mg, Li, K, and Rb. Surprisingly, the value is smaller for Na than for the other alkali metals. The implication is that the calculated diffusion frequency values are somewhat different from the Debye frequencies.

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The Specific Heat ∆CP* . Table 2.5 shows the calculated values of ∆CP* for some of the metals. The maximum estimated errors in ∆CV* and ∆CP* are 10 and 20%, respectively. The significance of ∆CP* in diffusion studies arises from the relationship: ∆CP*
[∂(∆H*)
∂T]P .

(52)

Thus, the parameter ∆CP* governs the variation of ∆H* with respect to temperature, which is responsible for curvature in the diffusivity plot [
n(D) vs. 1
T plot]. The change in ∆H* in going from temperature T1 to T2 is given by Eq. (52) as: ∆H*T2  ∆H*T1
(T2  T1)∆C*P .

(53)

∆CP* is assumed to be constant in Eq. (53). As shown by Eq. (52), ∆C*P is indeed a function of temperature. However, justification for assuming the invariance of ∆CP* with respect to temperature is provided by the data in Table 2.6, where the maximum variation in ∆CP* over a range of more than 500 K is less than 0.3 R. This is significantly smaller than the inherent error in the measurement of ∆H*. Therefore, in the temperature range of T1 to T2, an inherent curvature in the diffusivity plot will be observed only when the magnitude of the right-hand side of Eq. (53) is significantly larger than the sum of systematic errors in the determination of ∆H*. Furthermore, it may be indicative of a change in the mechanism of diffusion only when the deviation in ∆H* exceeds the limit given by Eq. (53) over the temperature range under consideration. Several earlier attempts to estimate ∆CP* have been reported in the literature. Using the Debye theory of specific heat and thermodynamic relationships between ∆CP* and ∆C V* , Levinson and Nabarro[71] derived the following expression for the temperature variation of the entropy for vacancy formation in noble metals: (∂∆S*f
∂T)P  {(V0a 2B)(∆V*
V)} 21  104 J mol 1 K 1. (54) Equation (54) was then used to examine the possibility of curvature in the Arrhenius plot for vacancy formation in metals. Over a temperature range of 400 K, the maximum variation expected in the vacancy formation entropy is only about 2 J mol1 K1. Levinson and Nabarro concluded that at this magnitude, no departures from linearity are expected in the Arrhenius plots of vacancy concentration. Nowick and Dienes[72] first write ∆CP* as: ∆C*P
(∆C*P  ∆C*V )  ∆C*V .

(55)

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103

For temperatures higher than Debye temperature, Nowick and Dienes argue that ∆CV* is very nearly zero and, therefore, the contribution to ∆C P* comes only from the first term on the right-hand side of Eq. (55). Further, using Eqs. (48) and (51), they arrive at the following equation for ∆CP*: ∆C*P
0.5 R[(∆b
b0)  (∆V
V0)].

(56)

Here, b0 and V0 are the compressibility and specific volume of the matrix, and ∆b and ∆V are the changes in these parameters per mole of defects, respectively. Nowick and Dienes further observe that both the quantities within the bracket are positive, and the bracketed term will closely approximate unity. Therefore, ∆CP* ≈ 0.5R.

(57)

The value of ∆CP* given by Eq. (57) is small, and the observed deviations in the diffusivity plots at low as well as high temperatures cannot be accounted for by the inherent temperature variation of ∆H*. Nowick and Dienes concluded that curvatures in the diffusivity plots might be manifestations of the multiplicity of diffusion mechanisms. Gilder and Lazarus[73] used Grüneisen Eq. (48) and the Debye theory of specific heat for estimation of ∆CP* . In their analysis, the contributions to ∆CP* arising from the change in the phonon spectrum during the formation of an activated vacancy complex were considered responsible for the diffusion process. An additional contribution to ∆CP* arises from the generation of new vibrational modes around the vacancy. According to Gilder and Lazarus, ∆CP* is given by: ∆C*P
∆C*V  a 02 T∆V *{(2aV
a0)  (bV
b0)}
b.

(58)

Here, a0 represents the volume thermal expansion coefficient of the matrix. bV and aV are the compressibility and volume thermal expansion coefficient, respectively, for activated vacancy. According to Gilder and Lazarus, ∆CP* for metals can be as large as 15R and can account, in principle, for the curvature in their diffusivity plots. However, in the absence of precise data on the high-temperature values of a0, aV, b0, and bV, Eq. (58) cannot be applied directly. They further state that to a good approximation: ∆C*P
mV T,

(59)

mV
2(aV ∆V*)g0 ∆C0V
V.

(60)

and

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Here, g 0 is the Grüneisen constant and ∆CV0 is the specific heat of the matrix at constant volume. Gilder and Lazarus determined mV through the curve fitting of the diffusion data by: T ln D
(∆H*
R)  (ln D0)T  1
2(mV
R)T 2 .

(61)

Tables 2.5 and 2.6 compare the values of ∆CP* for some cases given by Eqs. (50), (51), and (59). The two sets of values differ by 31 to 82% for different metals; the difference narrows to 1% for copper at 1200 K. With the exception of Na, K, b-Zr, and b-Ti, the experimentally observed n D vs. 1
T Arrhenius plots for other metals do not exhibit a continuous curvature. Instead, they may be broken, indicating contributions of more than one Arrhenius dependence and diffusion process. Therefore, in the case where the n D versus 1
T plot is only broken, its representation by a continuous curve can compromise the accuracy of mV and the corresponding ∆CP* values. In contrast, the expression for ∆CP* derived here depends on the relationships among the various diffusion parameters. It is validated by the appropriate values for ∆S*, u, and ∆V* obtained from one basic equation, which is Eq. (43).

2.3.5 Reliability of Estimated Parameters The reliability of the calculated values of various parameters can be judged by two considerations. (1) Their magnitude should be close to the normally acceptable values. We consider this important because this kind of independent estimate of the diffusion parameters has not been carried out before. (2) The errors associated with the calculated values should lie within reasonable limits. The evaluation of errors in the calculated values of , ∆S*, u, ∆CV*, and ∆CP* has been carried out using standard procedures;[74] the details are given in Appendix 2B. The estimated values are indeed close to the normally accepted values, and the errors in their estimation arising from those in the experimental data are also within reasonable limits. An error of not more than 10% is expected in the evaluation of ∆S*, about 90% in the estimation of u, a maximum of 10% in ∆CV* , and a maximum of 20% in ∆C*P . Therefore, with respect to the two considerations mentioned earlier, the method of calculation proposed here has proved to be correct and yields reliable values of the diffusion parameters. However, improvement in the precision of the concerned parameter can lower the error limits.

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2.4

105

Summary

Two different approaches to estimating diffusion parameters from the bulk properties have been discussed in this chapter. First, empirical as well as semi-empirical correlations between diffusion and the bulk properties are discussed. These include the correlation of activation energy for self-diffusion with the cohesive energy, melting point, latent heat of fusion, bulk modulus, and the Debye temperature. The Zener’s hypothesis, which paved the way for development of relationships between the diffusion parameters and the elastic modulus, is also discussed. Finally, a correlation between the activation energy for self-diffusion in metals with the valence bond parameter and that for solvent diffusion in dilute alloys with electron-to-atom ratio is presented. Secondly, the empirical treatment of diffusion data is followed by the derivation of a thermodynamic relationship between the activation volume and activation entropy for self-diffusion. These two diffusion parameters are shown to be related to each other through the compressibility and the volume expansion coefficient. Because this equation does not contain any arbitrary constant, it is used to obtain the values of other diffusion parameters. Satisfactory agreement is obtained between the calculated and experimental values for all the parameters. The correlations between the bulk properties and the diffusion parameters discussed here can be used to rationalize new data and estimate the diffusion rates in the absence of experimental values.

Acknowledgment The authors thank Dr. R. Nakamura for help in preparing the figures.

Appendix 2A. Taylor Series Expansion of ∆G* The general form of the Taylor series for f (x) about x
h is: f (x)
f(h)  f (h)(x  h)  f (h)(x  h)2
2!  f (h)(x  h)3
3!  ...........

(A1)

The Taylor series permits us to obtain an algebraic expression for any parameter in terms of a desired variable. The condition necessary is that the derivatives of f(x) should exist at f (x)
f (h). Equation (36) is obtained from Eq. (A1) when f(x)
∆G*, x
V, and h
V0. In order to

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justify the procedure to derive Eq. (36), we try the expansion of ∆G* as a function of T and P. Like V, both T and P are thermodynamic state properties of the matrix; therefore, the expansion of ∆G* as a function of these two variables is valid. We first expand ∆G* as a function of temperature to obtain: ∆G*
∆G*0  [∂(∆G*)
∂T](∆T)  [∂2(∆G*)
∂T2](∆T)2
2!  ......... (A2) Here, ∆G*0 is the value of ∆G* at T
0. From the basic thermodynamic equation: ∆G*
∆H*  T∆S*,

(A3)

∆G*0
∆H*

(A4)

[∂(∆G*)
∂T)]P
∆S*,

(A5)

∆T
T.

(A6)

we obtain:

for T
0. Further, by definition:

and

Substituting from Eqs. (A4), (A5), and (A6) into (A2), after ignoring second and higher order terms, we obtain: ∆G*
∆H*  T∆S*, which is the same as Eq. (A3). We now expand ∆G* as a function of pressure: ∆G*
∆G*0  [∂(∆G*)
∂P](∆P)  [∂ 2(∆G*)
∂P2](∆P)2
2! ........ (A7) Here, ∆G*0 is the value of ∆G* at P
0. When pressure is a variable, the standard equation for ∆G* is: ∆G*
∆H*  P∆V *  T∆S*,

(A8)

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107

where ∆V* is the activation volume. Therefore, at P
0: ∆G*0
∆H*  T∆S*.

(A9)

[∂(∆G*)
∂P)]T
∆V*,

(A10)

∆P
P.

(A11)

By definition,

and

Substituting from Eqs. (A9), (A10), and (A11) into (A7), after ignoring second and higher order terms, we obtain: ∆G*
∆H*  P∆V*  T∆S*, which is the same as Eq. (A8). The results obtained from the expansion of ∆G* as a function of T and P allow us to draw the following conclusions: 1. We retrieve the basic equations for ∆G* in both cases. Therefore, the procedure used is justified. 2. The truncation of the Taylor series after first term, and ignoring the second and higher order terms, are justified because the results obtained are true and correct. 3. The results obtained by Taylor series expansion of ∆G* as a function of T and P are mathematically trivial because they do not yield any new result. As shown in this appendix, only the expansion of ∆G* as a function of V yields a new result.

Appendix 2B. Evaluation of Errors in Estimated Parameters 2B.1 ∆S* Using Eq. (47), the error in the estimation of ∆S* is given by: ∆(∆S*)
∆S*
∆(a
b)
(a
b)  ∆(∆V *
V)
(∆V *
V).

(B1)

Equation (B1) shows that the error in estimating ∆S* originates from the errors in a
b and the experimental determination of activation volume

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∆V*
V. In contrast, the error in V0 can be neglected, because this is usually determined to an accuracy of better than 1 in 1000. According to Hanneman and Gatos,[75] the a
b values are approximately pressure- and temperature-independent at room temperature and above. We can therefore conclude that the error in ∆S* is limited to the error in the experimental measurement of ∆V *
V. A survey of the experimental data for a number of metals shows that the error in measurement of ∆V*
V varies from 1 or 2%, as in the case of Zn and Cd,[65, 66] to ∼7%, as in case of tin.[76] Consequently, the maximum possible error in the ∆S* value estimated from Eq. (47) does not exceed 10%.

2B.2 u From Eq. (46), the expression for estimation of u is given as: u
D0
[ fa2 exp(∆S*
R)],

(B2)

and the corresponding equation for estimation of error is: ∆u
u
(∆D0
D0)  (∆f
f )  2(∆a
a)  [exp(∆S*
R)
exp(∆S*
R)].

(Β3)

The error in u is dependent on the errors in the determination of frequency factor D0, lattice parameter a, and activation entropy ∆S*. Because the correlation factor f is a pure number for self-diffusion in metals, it does not contribute to the error. In a conventional measurement of self-diffusion in metals through radioactive tracer technique, the absolute error in the magnitude of D0 can vary from 50 to 80%.[77, 78] The error in the determination of the lattice parameter rarely exceeds 5%.[79] With regard to the error in ∆S*, the preceding paragraph shows that the maximum possible error in its magnitude may amount to 10%. A summation of all these errors shows that use of Eq. (B3) to estimate u could be in error by a maximum of about 95%.

2B.3 ∆C*V Because g is a constant, use of Eq. (50) gives the error in estimation of ∆C*V as: ∆(∆C*V )
∆CV*
[∆(∆S*)
∆S*]  [∆(1  aT)
(1  aT).

(B4)

Since aT is usually of the order of 103, contribution by the last term to the error in ∆C*V can be neglected. Thus, the error in ∆C*V can be put at a maximum of 10%, the same as the error in ∆S*.

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109

2B.4 ∆CP* From Eq. (51), the error in ∆C*P is given by: ∆(∆C*P)
∆CP*
[∆(∆C*V)
∆C*V]  [∆(aT)
aT]  [∆(a
b)
(a
b)]  [∆(∆V *)
∆V *].

(B5)

It has already been stated that aT is usually of the order of 103 and a
b is practically temperature-independent. Therefore, the error in the estimation of ∆C*P is the sum of the errors in ∆C*V and ∆V *, and can be put at a maximum of 20%. The preceding analysis shows that the error in the estimated diffusion parameters comes from the error inherent in measurement of ∆V* and D0. Sometimes the diffusion parameters are measured with a precision of ∼5%. We have indicated here the maximum possible error limits for all the parameters. These could be significantly lower if the diffusion data are measured with greater precision. In other words, the accuracy of the values calculated by the present method basically depends on the precision of diffusion parameters.

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18. A. V. Chadwick and J. N. Sherwood, in Diffusion Processes, vol. 2, Proceedings of Thomas Graham Memorial Symposium, University of Strathclyde (J. N. Sherwood et al., eds.), Gordon and Breach Publishers (1971), p. 472. 19. G. P. Tiwari, Trans. Jpn. Inst. Metals, 19:125 (1978). 20. G. P. Tiwari, Trans. Indian Inst. Metals, 47:125 (1994). 21. G. P. Tiwari, Z. Metallkd., 72:211 (1981). 22. N. H. Nachtrieb, H. A. Resing, and S. A. Rice, J. Chem. Phys., 31:135 (1959). 23. A. M. Brown and M. F. Ashby, Acta Metall., 28:1085 (1980). 24. J. A. Cornet, J. Phys. Chem. Solids, 32:1489 (1971). 25. J. P. Zanghi and D. Calais, J. Nucl. Mater., 60:145 (1976). 26. A. Vignes and C. E. Birchenall, Acta Metall., 16:1117 (1968). 27. F. Roux and A. Vignes, Rev. Phys. Applique, 5:393 (1970). 28. D. Ablitzer, Philos. Mag., 35:1239 (1977). 29. C. Wert, Phys. Rev., 79:601 (1950). 30. C. Zener, J. Appl. Phys., 22:372 (1951). 31. C. P. Flynn, Phys. Rev., 171:682 (1968). 32. P. Erhart, K.-H. Robrock, and H. R. Schober, in Physics of Radiation Effects in Crystals (R. A. Johnson and A. N. Orlov, eds.), North-Holland Publishing Co., Amsterdam (1986), chap. 1. 33. D. Lazarus, Solid State Phys., 10:71 (1960). 34. W. Koster, Z. Metallkd., 39:1 (1948). 35. P. G. Shewmon, Diffusion in Solids, 2nd ed., TMS AIME, Warrandale, PA (1989), chap. 2. 36. D. Beke, T. Getszti, and G. Erdelyi, Z. Metallkd., 68:444 (1977). 37. L. E. Toth and A. W. Searcy, Trans. AIME, 230:690 (1964). 38. G. Leibfried, Z. Phys., 127:344 (1950). 39. P. Varotsos and K. Alexopoulos, Phys. Rev., B15:4111 (1977). 40. P. Varotsos, W. Ludwig, and K. Alexopoulos, Phys. Rev., B18:2683 (1978). 41. P. Varotsos and K. Alexopoulos, Phys. Rev., B24:904 (1981). 42. P. Varotsos and K. Alexopoulos, Phys. Rev., B22:3130 (1980). 43. P. Varotsos and K. Alexopoulos, in Thermodynamics of Defects and Their Relations with Bulk Properties, North–Holland Publishing Co., Amsterdam (1986). 44. R. V. Patil and G. P. Tiwari, Indian J. Pure Appl. Phys., 42:206 (1977). 45. R. W. Keys, J. Chem. Phys., 29:467 (1958). 46. A. W. Lawson, J. Phys. Chem. Solids, 3:250 (1957). 47. N. H. March, Phys. Rev. Lett., 20:231 (1966). 48. V. K Tewary, J. Phys., F3:704 (1973). 49. P. V. Sastry and B. G. Mulimani, Philos. Mag., 20:859 (1960). 50. G. P. Tiwari and R. V. Patil, Trans. Jpn. Inst. Metals, 17:476 (1976). 51. J. N. Plendl, Phys. Rev., 123:1172 (1961). 52. S. I. Ben-Abraham, A. Rabinovitch, and J. Pelleg, Phys. Status Solidi, 84:435 (1977). 53. G. P. Tiwari and K. Hirano, Trans. Jpn. Inst. Metals, 21:667 (1980). 54. L. Brewer, in Phase Stability in Metals and Alloys (P. S. Rudman et al., eds.), McGraw-Hill Co. (1967), p. 113.

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55. R. E. Watson and L. H. Bennett, in Noble Metals and Alloys (T. B. Massalski et al., eds.), TMS, Warrandale, PA (1986), p. 3. 56. F. G. Fumi, Philos. Mag., 46:1007 (1955). 57. G. P. Tiwari, K. Hoshino, Y. Iijima, and K. Hirano, Scripta Metall., 14:735 (1980). 58. L. W. Barr and D. K. Dawson, Proc. Brit. Ceramic Soc., 19:151 (1971). 59. S. N. Vaidya, J. Phys. Chem. Solids, 42:621 (1981). 60. G. H. Vineyard and G. J. Dienes, Phys. Rev., 93:265 (1954). 61. G. Borelius, in Advances in Research and Applications, Solid State Physics, vol. 6 (F. Seitz and D. Turnbull, eds.), Academic Press, NY (1958), p. 65. 62. J. R. Neighbours and C. S. Smith, Acta Metall., 2:591 (1954). 63. J. L. Tallon, Philos. Mag., A39:151 (1979). 64. J. L. Tallon and W. H. Robinson, Philos. Mag., 36:741 (1977). 65. N. F. Mott and R. W. Gurney, in Electronic Processes in Ionic Crystals, The Clarendon Press, Oxford (1946), p. 30. 66. G. P. Tiwari, Scripta Mater., 39:931 (1998). 67. L. C. Chhabildas and H. M. Gilder, Phys. Rev., B5:2135 (1972). 68. B. J. Buescher, H. M. Gilder, and N. Shear, Phys. Rev., B7:2261 (1961). 69. J. N. Mundy, Phys. Rev., 3:2431 (1971). 70. P. W. Bridgman, Collected Experimental Papers of Bridgman, vol. III, Harvard University Press (1964), paper 43. 71. L. M. Levinson and F. R. N. Nabarro, Acta Metall., 15:785 (1967). 72. A. S. Nowick and G. J. Dienes, Phys. Status Solidi, 24:461 (1967). 73. H. M. Gilder and D. Lazarus, Phys. Rev., B11:4916 (1975). 74. H. S. Mickley, T. S. Sherwood, and C. E. Reed, Applied Mathematics in Chemical Engineering, Tata McGraw-Hill, New Delhi (1975), Chap. 2. 75. R. E. Hanneman and H. C Gatos, J. Appl. Phys., 36:1794 (1965). 76. N. H. Nachtrieb and C. Coston, in Physics of Solids at High Pressures (C. T. Tomizuka and R. M. Emrick, eds.), Academic Press (1956), p. 336. 77. T. Heumann and R. Imm, J. Phys. Chem. Solids, 29:1613 (1968). 78. Y. Iijima, K. Kimura, and K. Hirano, Acta Metall., 36:2811 (1988). 79. W. B. Pearson, A Handbook of Lattice Spacing and Structure of Metals and Alloys, Pergamon Press (1958).

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3 Atomistic Computer Simulation of Diffusion Yuri Mishin School of Computational Sciences George Mason University, Fairfax, Virginia

3.1

Introduction

The past decades have seen a tremendous increase in the application of computer modeling and simulation methods to diffusion processes in materials. Along with continuum modeling aimed at describing diffusion processes by differential equations, atomic-level modeling is playing an increasingly important role as a means of gaining fundamental insights into diffusion phenomena. One of the reasons for the growing interest in the atomistic modeling of diffusion is the recognition that experimental methods only deliver effective diffusion coefficients, i.e., coefficients averaged over the diffusion zone and over many atomic jumps contributing to the diffusion flux. Recovering information relating to individual diffusion mechanisms, their activation barriers, correlation factors, and other atomic-level characteristics is extremely difficult, if not impossible. As a result, many experimental measurements produce useful reference numbers for handbooks and immediate technological applications (which is very important too) but do not add much to our basic understanding of diffusion processes. In the long run, however, it is highly desirable to be able to predict diffusion coefficients by calculation. This ability is critical for the ongoing effort to reduce the dramatic costs associated with the development of new technologically advanced materials by the traditional empirical approach. Atomistic modeling appears to offer the only viable way of gaining insights into mechanisms of complex diffusion processes and thus creating a fundamental framework for predictive diffusion calculations. A second reason for this growing interest is that drastically increased computer speeds have enabled computer simulations that could only be dreamed about just two decades ago. New methods have been developed that provide a realistic description of atomic interactions in materials, accelerate molecular dynamics (MD) and Monte Carlo simulations, and allow more reliable calculations of transition rates. The new methods provide

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access to diffusion processes in larger and more complex systems and allow us to observe diffusion over more extended periods of time. Some of the recent atomistic calculations of diffusion coefficients show an encouraging agreement with experimental data. This chapter presents an overview of atomistic simulation methods currently available for diffusion modeling in materials, focusing on metals and metallic systems. When examining each method or class of methods, some background, leading literature references, and a brief overview of the most recent developments are provided. Although various examples of applications are discussed along the way, two areas are emphasized where much progress has recently been achieved due to computer simulations. The first area is diffusion in ordered intermetallic compounds. The atomic order in such compounds imposes strong selection rules on possible diffusion mechanisms by favoring mechanisms that either preserve the order or destroy it only locally and temporarily. A glimpse of understanding of such mechanisms is now emerging through computer simulations. The other area is diffusion in grain boundaries, in which new collective mechanisms have been discovered that involve both vacancies and interstitials as equal partners. The common thread of these and other examples is the notion that atomistic computer simulations offer a powerful tool for gaining deeper insights into diffusion phenomena, and that this tool is applicable not only to simple systems but also to complex materials of technological importance.

3.2

Atomic Interaction Models

3.2.1 Embedded-Atom Method The first step in any atomistic simulation is to establish a model that describes atomic interactions. Because diffusion processes involve the motion of atoms over considerable distances and require statistical averaging, diffusion simulations inevitably deal with relatively large ensembles of atoms. This explains why the overwhelming majority of such simulations are based on classical interatomic potentials, also called force fields. Interatomic potentials allow fast MD and Monte Carlo simulations to be performed for systems containing up to millions of atoms. Early diffusion simulations used pair potentials of the Morse or Lennard-Jones type. Although useful insights were obtained, the agreement with experiment was only qualitative at best. The problem with pair potentials is that they do not capture the nature of atomic bonding even in simple metals, not to mention transition metals or covalent solids. The invention of the embedded-atom method (EAM)[1, 2] in the 1980s opened a new page in atomistic simulations. Due to the incorporation, in an approximate manner, of

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many-body interactions between atoms, EAM potentials have enabled a semi-quantitative and, in good cases, even quantitative description of interatomic bonding in metallic systems. In the EAM model, the total potential energy Etot of a collection of atoms is given by the expression: 1  ). Etot
 Σ Φij(rij)  Σ Fi(r i 2 i,j i

(1)

Here, Φij is the pair-interaction energy between atoms i and j at positions rj
→ ri  → r ij , respectively. Fi is the embedding energy of atom i, and r and →  ri is the host electron density at site i induced by all other atoms in the system. The latter is given by the sum:

→ i

 ri
Σ rj(rij),

(2)

j≠i

where rj(r) is the electron density function assigned to atom j. The second term in the right-hand side of Eq. (1) represents many-body effects, which are responsible for a significant part of bonding in metallic systems. The EAM has an excellent record of describing basic properties of simple and noble metals,[3] but is less accurate for transition metals. The latter reflects the intrinsic limitation of the EAM, which is essentially a central-force model and is, therefore, unable to capture the covalent component of bonding due to d-electrons present in transition metals. The EAM has also been applied, with reasonable success, to several intermetallic compounds.[4–6] The functional form of Eq. (1) was originally derived as a generalization of the effective medium theory[7] or the second moment approximation to tight-binding theory.[2, 8] Later, however, Eq. (1) lost its close ties with the original physical meaning and came to be used as a semi-empirical expression with adjustable parameters. The potential functions Φij(r),  ) are typically parameterized with three to five fitting rj (r), and Fi (r i parameters each and are fit to selected properties of the material. While early EAM potentials were fit to experimental properties only, the current trend is to incorporate into the fitting database both experimental and firstprinciples data. The experimental properties traditionally include the lattice constant, cohesive energy, elastic constants, thermal expansion factors, and phonon frequencies, as well as the vacancy formation energy and the stacking fault energy. The first-principles data usually come in the form of energy-volume relations for the ground-state structure and a few hypothetical structures of the material. Furthermore, first-principles energies along uniform deformation paths between the structures were calculated on

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several occasions, although they were used primarily for testing rather than fitting the potentials.[5, 6, 9, 10] An alternative way is to include a set of interatomic forces drawn from snapshots of first-principles MD simulations for the solid and liquid phases of the material (force matching method).[11] The incorporation of first-principles data into the fitting database improves the reliability of the potential due to the sampling of atomic configurations away from those represented by experimental data.[9–12] For a binary system A-B, the cross-interaction function ΦAB(r) is optimized by fitting to experimental and
or first-principles data for the experimentally observed and
or hypothetical compounds, usually with several stoichiometric compositions to cover a broader region of configuration space.

3.2.2 Angular-Dependent Potentials Baskes and co-workers[13–16] developed a non-central-force extension of the EAM, called the modified embedded-atom method (MEAM). In the MEAM, the electron density is treated as a tensor, and the host electron  is expressed as a function of the respective tensor invariants. In density r i the simplest approximation,  ri is given by the expansion:  )2
(r  (0))2  (r  (1))2  (r  (2))2  (r  (3))2, (r i i i i i

(3)

where  (0))2
(r i

Σj≠i r



2

(0) j

 (1))2
(r i Σ a

(rij) ,

Σj≠i r

Σ r a,b j≠i

 (r ri(3))2
Σ

a,b,g



raij 2 (rij)  , rij

(1) j

 (2))2
(r i Σ

j≠i

(5)



ra rbij (rij) ij r2ij

(2) j

Σ

(4)

2

1  3



raij rbij rgij 2  r(3) (r ) . j ij r3ij

Σj≠i r



2

(2) j

(rij) ,

(6)

(7)

Here a, b, and g
1,2,3 are summations over Cartesian coordinates of the ri(k) (k
0,1,2,3) represent contributions radius-vector → rij. The terms 

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corresponding to s, p, d, and f electronic orbitals, respectively. The regular EAM is recovered by including the electron density of s-orbitals only and neglecting all other terms [compare Eq. (2)]. In comparison with the regular EAM, the MEAM introduces three (2) (3) additional parametric functions, r(1) i (r), ri (r), and ri (r), for each species, which are fit to the same type of database as in the EAM. Furthermore, while EAM potential functions are typically long-ranged and are smoothly truncated on a sphere encompassing several coordination shells, the MEAM uses short-range functions (1-2 coordination shells) but introduces a many-body screening procedure described in detail by Baskes.[15, 17] Computationally, MEAM simulations are slower than EAM simulations, but the MEAM is more suitable for transition metals and can even be applied to covalent solids, such as Si and Ge.[15] MEAM potentials have been constructed for many FCC, BCC, and HCP metals,[15, 17, 18] as well as for several compounds, including Mo silicides[19] and TiAl.[20] Pasianot et al.[21] proposed a slightly different way of incorporating angular-dependent interatomic forces into the EAM. In their so-called embedded-defect method (EDM), the total energy is written as: 1 )  G Y, Etot
 Σ Φij(rij)  Σ Fi(r i Σi i 2 i,j i

(8)


r (r ), r i Σ j ij

(9)

where j≠i

Yi
Σ

a,b

Σ j≠i



ra rbij rj(rij) ij  r 2ij

2

1  3

Σ j≠i



2

rj(rij) .

(10)

Equation (8) was originally derived from physical considerations different from those underlying the MEAM. However, mathematically, Eqs. (8) through (10) represent a particular case of Eqs. (1) through (7), in which  ) is approximated by a linear ri(3) are neglected, F(r the terms  ri(1) and  i expansion in terms of the small perturbation  ri(2), and the latter is expressed through the unperturbed electron density function rj(r). In comparison with the regular EAM, the EDM contains only one additional adjustable parameter, G. Like the EAM, the EDM is based on cutoff functions and does not involve any screening procedure. EDM potentials have been constructed for HCP[22] and BCC transition metals.[21, 23–25] A number of specific, usually angular-dependent, interatomic potentials have been constructed for silicon, carbon, and other semiconductor

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materials.[26–29] Likewise, accurate empirical and semi-empirical potentials have been developed for many ionic systems.[30, 31]

3.2.3 More Accurate Methods An encouraging trend in recent years is the appearance of diffusion studies using first-principles methods based on the density-functional theory.[32] Such methods have been applied to calculate vacancy formation energies in a number of metals and intermetallic compounds.[33–41] While early first-principles calculations used relatively small supercells with 16 to 32 atoms, the sizes of systems accessible by such methods rapidly increase with computer power. For example, the Vienna Ab initio Simulation Package (VASP),[42, 43] in conjunction with ultra-soft plane-wave pseudopotentials,[44] already gives access to systems containing a few hundred atoms. Clearly, the role of first-principles methods in diffusion will continue to grow rapidly in the years to come. Presently, however, such methods are limited to molecular statics calculations of point defect formation energies and, in rare cases, of migration energies. Meaningfully long MD or Monte Carlo simulations are not yet feasible. Another group of methods worth mentioning comprises semi-empirical methods based on tight-binding theory, particularly bond-order potentials (BOPs)[8, 45–48] and the tight-binding method developed at the U.S. Naval Research Laboratory (NRL-TB).[49–51] Both methods contain adjustable parameters that are optimized by fitting to experimental and first-principles data (BOP), or to first-principles data only (NRL-TB). Besides properties that depend on the total energy, the NRL-TB Hamiltonian is also fit to the electronic band structure of the material.[49–51] In contrast to classical potentials, these methods are based on quantum mechanics and explicitly address the electronic structure of the material. An appealing feature of these methods is their ability to handle both simple and transition metals, intermetallic compounds, and semiconductors. Although BOP and NRLTB methods are almost two orders of magnitude slower than EAM calculations, they are much faster than first-principles methods. In a sense, they serve to bridge the gap between semi-empirical and first-principles methods. For example, it has been demonstrated that the NRL-TB method can readily implement thousands of MD steps on a system containing about a thousand atoms.[52–54] The computational efficiency of such methods invariably improves as processors become faster and new parallel computer codes are developed. Such methods can start playing a role in diffusion simulations in the future.

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3.3

119

Molecular Statics

3.3.1 Simulation Block and Boundary Conditions The most common procedure in atomistic simulations is the static relaxation, i.e., minimization of the total energy of a system with respect to atomic positions as well as other degrees of freedom available to it. The system is represented in the computer by a finite set of atoms, called a simulation block (or simulation cell), which is subject to suitable boundary conditions.[30, 31, 55–57] To accelerate the relaxation of a large block, the computer program generates and stores in the memory a list of appropriately close neighbors of each atom. This makes the computation an effectively linear-N process, N being the number of atoms in the block. If the relaxation is accompanied by large atomic displacements, the neighbor list should be promptly updated. The construction of a neighbor list for a large block is a computationally expensive procedure by itself, but it can be accelerated by applying the link-cell method and other tricks of the trade that are well documented in the literature.[56, 57] If the boundary conditions are periodic in all directions, the block is often called a supercell, the term borrowed from the area of first-principles calculations. Often, the movable (free, or dynamic) atoms are embedded in a mantel of fixed atoms, i.e. atoms frozen in their perfect-lattice positions relative to one another (fixed-boundary condition). The thickness of the mantel is made larger than the distance at which atoms can “see” each other, so that the free surfaces separating the mantel from vacuum would not affect the dynamic atoms. Different boundary conditions can be combined with each other, i.e., be fixed in some directions and periodic in other directions. For example, in typical grain boundary (GB) simulations, the boundary conditions are periodic in directions parallel to the GB plane and fixed in the direction normal to the GB plane. In this geometry, the fixed atoms represent lattice regions far away from the GB (the grains). The choice of boundary conditions for simulating lattice dislocations is a more delicate matter because the elastic strain field around a dislocation is long-ranged and boundary conditions may have a substantial effect on the dislocation core structure.[55] For an isolated dislocation, simulations usually use a cylindric geometry with the dislocation line aligned parallel to the cylinder axis (periodic direction). Prior to beginning the relaxation, the fixed atoms in the cylindrical mantel are displaced according to the elasticity theory solution for the elastic strain field of a straight dislocation with the chosen Burgers vector. Recently, Rao et al.[58] proposed a more accurate boundary condition, in which the atomic displacements in the fixed region are determined in a

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self-consistent manner through the lattice Green function (flexible, or Green-function boundary condition). Furthermore, it has been demonstrated that this boundary condition makes it possible to reduce the size of the free region (containing the dislocation core) to the extent that it can be simulated by first-principles methods.[59] This boundary condition could be applied in future atomistic simulations of dislocation pipe diffusion.

3.3.2 Point-Defect Formation Energy Point defects also produce long-range elastic fields, and their computed properties generally also depend on the boundary condition. As a simple example, we will consider the calculation of the formation energy efv of a single vacancy in a metal. This energy is determined as the total energy of a relaxed simulation block containing a vacancy minus the energy of the perfect lattice containing the same number of atoms. A part of efv is associated with strong atomic distortions within the vacancy core, but a significant part is stored in the elastic deformation field around the vacancy. It can be shown that the elastic energy stored within a sphere of radius R around a point defect converges to its value in an infinite system as 1
R3 when R → ∞.[60, 61] Since simulations are performed in a finite-size block, the elastic field is disturbed by the block boundaries, and the resulting efv value is either underestimated or overestimated, depending on the boundary condition. Two types of boundary conditions are typically used in point-defect simulations. Under the constant-volume condition, the volume of the block is fixed at its initial, perfect-lattice value, and only atomic positions are allowed to vary during the energy minimization. Under the zero-pressure condition, both the volume and atomic positions are relaxed simultaneously. It can be shown that the vacancy formation energy converges as 1
N (N → ∞) to the same value, regardless of the boundary condition.[61] Furthermore, it turns out that the constant-volume condition always overestimates the true vacancy formation energy, whereas the zero-pressure condition always underestimates it.[61] This fact is illustrated in Fig. 3.1 using EAM calculations of the vacancy formation energy in copper as an example. The zero-pressure branch of the plot has a slightly smaller slope, which agrees well with the elasticity theory analysis within a simple spherical model.[61] However, even though the smaller slope means a faster convergence, the difference in the slopes of the two branches is small and should be weighted against much longer computation times required for zero-pressure calculations.

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800

N 200

400

121

125

1.261 Vacancy formation energy (eV)

Ch_03.qxd

1.260

V=const

1.259 1.258 p=0

1.257 1.256 0.0

0.2

0.4

0.6

0.8

1.0

2

10 /N Figure 3.1 Vacancy formation energy in copper as a function of the inverse number of atoms(N) in a cubic simulation block under constant-volume (V
const) and zero-pressure (p
0) boundary conditions.[61] Calculations were performed with an embedded-atom potential. The vacancy formation energy in a macroscopic crystal is evaluated by a linear extrapolation to 1
N → 0. The extrapolated value, 1.258 eV, does not depend on the boundary condition.

This issue is especially acute in first-principles calculations, where embarking on expensive zero-pressure calculations may not be the best strategy. Instead, the computational resources can be used for making a constant-volume calculation on a larger supercell. If a highly accurate efv value is desirable, a number of constant-volume calculations can be performed, followed by a linear extrapolation of the results to 1
N → 0. It should be remembered that, above a certain block size, most atoms are residing in nearly perfect lattice positions. Therefore, any further increase in the block size will only result in a better estimate of the elastic strain energy of the slightly deformed lattice regions. Instead of modeling elastic strains by first principles or atomistic calculations, it can be more efficient to apply an extrapolation based on elasticity theory. These considerations apply equally to interstitial atoms as well as any other point defects in materials.

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Harmonic Approximation

3.4.1 Harmonic Entropy of Point Defects Besides point defect formation energies, their formation entropies are another important ingredient required for calculating equilibrium defect concentrations. The defect formation entropy is traditionally attributed to perturbations of atomic vibrations in the defect core as well as around it. The electronic entropy and other entropy effects can also make contributions, but their role has not yet been examined in much detail. As far as the vibrational entropy is concerned, its calculations are commonly based on the harmonic approximation to atomic vibrations.[62, 63] In the harmonic approximation, the potential energy of a statically relaxed simulation block is approximated by a Taylor series up to the second-order terms with respect to small atomic displacements from equilibrium. For a system containing N atoms, a 3N  3N dynamical matrix, ∂2E 1 Diajb
 , m  imj ∂xia ∂xjb

(11)

is constructed, where E is the potential energy, xia is the displacement of atom i in Cartesian direction a, and mi is the atomic mass. The normal ,i where li vibration frequencies of the block are determined as ni
l are positive eigenvalues of the dynamical matrix. The total number M of normal modes depends on the boundary conditions. A block with fixed boundary conditions in all three directions has M
3N normal modes, while for a supercell M
3N  3 because three vibrational modes are replaced by translations of the center of mass.[63] Knowing the normal vibration frequencies, the free energy G associated with atomic vibrations at temperature T can be found from the standard quantum-mechanical expression: M





hni G
kBT Σ ln 2sinh  2k i
1 BT

 ,

(12)

where h is Planck’s constant and kB is Boltzmann’s factor. The vibrational entropy S is readily obtained from this expression as the temperature derivative, S
∂G
∂T. In the commonly used classical approximation

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123

(all hni  kBT), we have: M

hn S
kB Σ ln i  MkB. k i
1 BT

(13)

The entropy sfv of vacancy formation in an elemental crystal is determined as the entropy of a relaxed simulation block with a single vacancy minus the entropy of a perfect lattice block containing the same number of atoms. Importantly, because the two entropies refer to the same number of vibrational degrees of freedom, the factors kBT and MkB in Eq. (13) cancel out, and sfv turns out to be a temperature-independent quantity that only depends on the relevant vibration frequencies: M

n i
1

i

. sfv
kBln  M  ni0

(14)

i
1

Here, ni and ni0 represent normal vibration frequencies in the defected and perfect crystal, respectively. This expression holds true for any point defect as long as the classical harmonic approximation is at work. In the quantum-mechanical regime (low temperatures), the defect formation entropy does depend on the temperature.

3.4.2 Effect of Boundary Conditions on Point-Defect Entropy Like with the defect formation energy, the defect formation entropy also depends on the choice of boundary conditions. In particular, the constantvolume and zero-pressure relaxation schemes result in two different defect formation entropies, which we denote as (s f )V and (s f )p, respectively. We are usually interested in (s f )p in the context of diffusion calculations. It can be shown that both (s f )V and (s f )p tend to their infinite-system values linearly in 1
N as we increase the number of atoms in the block.[61] However, in contrast to the defect formation energy (which converges to the same value regardless of the boundary condition), the two extrapolated entropies are generally different. This fact is illustrated in Fig. 3.2 for a vacancy in copper. Thus, even in an infinitely large system, the defect formation entropy depends on the external conditions, and in particular is different under the constant-volume and zero-pressure conditions. The difference between the

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N 500

108

3.0

Vacancy formation entropy / kB

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32

FHA V=const

2.5

LHA 2.0

1.5

FHA 1.0

p=0

LHA

0.5 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

102/N Figure 3.2 Vacancy formation entropy in copper as a function of the inverse number of atoms (N) in a cubic simulation block under constant-volume (V
const) and zero-pressure (p
0) boundary conditions.[61] The vacancy formation entropy in a macroscopic crystal is evaluated by a linear extrapolation to 1
N → 0. Calculations were performed with an embedded-atom potential in the full harmonic approximation (FHA) and the local harmonic approximation (LHA).

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two entropies can be evaluated from well-known thermodynamic relations,[64] which lead to:[60, 61, 65, 66] (s f )p  (s f )V
bB∆V,

(15)

where ∆V is the relaxation volume of the defect formation, B is the bulk modulus, and

 

1 ∂V b
  V ∂T

(16)

p
0

is the thermal expansion coefficient of the material at zero temperature and pressure. The latter can be readily determined by independent computations.[61] Notice that for defects whose formation is accompanied by a significant volume effect, the difference (s f )p  (s f )V can be quite large. For example, it equals 1.7kB for a vacancy and 13.9kB for a self-interstitial in copper.[61] To deduce the correct defect formation entropy, it is essential to perform the full (i.e., both atomic and volume) relaxation of the defected block prior to the harmonic calculations. Alternatively, the volume can be kept constant and (s f )V determined, but then (s f )p should be recovered from Eq. (15). For point defects in Cu, the second scheme has proved to be almost as accurate as the first.[61]

3.4.3 Embedded Cluster Method A major problem associated with harmonic calculations is their significant computational cost. The calculation of the dynamical matrix for a simulation block large enough to represent a defect still remains beyond the reach of first-principles methods. Using interatomic potentials, the dynamical matrix can be calculated relatively fast, but then the bottleneck of the computation shifts to its diagonalization. The diagonalization of a dynamical matrix is a computationally demanding procedure which, for very large systems, is associated with serious numerical problems. Because of this limitation, full harmonic calculations become impractical beyond a certain block size (typically, N ≈ 103). While for a single vacancy or other simple defect an accurate formation entropy can be obtained with a few hundred atoms (Fig. 3.2), more complex defects or defect clusters require more atoms. A number of approximate harmonic methods have been developed for dealing with large systems, such as the embedded-cluster method,[61, 67] the local harmonic approximation,[68] the second-moment approximation,[69–71] and other methods.[72]

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In the embedded-cluster method, only some of the free atoms, namely those forming a cluster centered on the defect core, are treated as dynamic; all other atoms are treated as static. Thus, the defect is created and statically relaxed in a large simulation block, but the dynamical matrix is only constructed and diagonalized for a relatively small cluster embracing the defect. The defect formation entropy is then computed using the dynamic atoms only. The elasticity theory analysis, as well as atomistic simulations, show that the defect entropy S* delivered by the embedded-cluster method equals approximately:[61] N* S*
Sc   S . N el

(17)

Here, Sc is the so-called defect core entropy, Sel is the entropy associated with the elastic strain field around the defect, N* is the number of atoms in the cluster, and N is the total number of atoms in the simulation block. Given that the true formation entropy of the defect is s f
Sc  Sel, we see that the cluster entropy S* is incomplete: it includes only a part of the elastic strain entropy Sel in proportion to N*
N. By calculating S* for several N*
N values and extrapolating this function to N*
N → 0, we can determine the defect core entropy Sc. The latter turns out to be a well-defined physical quantity that does not practically depend on the boundary conditions and characterizes the degree of local distortions within the defect core. For example, in copper Sc
1.73kB for a vacancy and 5.75kB for an interstitial.[61] On the other hand, the linear extrapolation to N*
N → 1 should give us sf. In practice, the accuracy of the obtained sf value is not very high because Eq. (17) relies on strong approximations, including a spherical symmetry of the elastic field around the defect. A more accurate scheme, called the “elastically corrected” embeddedcluster method, has been proposed.[61] In this method, the cluster entropy S* is augmented by a quasi-continuum term Sq such that the sum of S* and Sq equals the true formation energy of the defect: s f
S*  Sq.

(18)

The quasi-continuum term represents the entropy associated with the elastic strain field outside the cluster. The calculation of Sq is based on the thermodynamic fact that the entropy density of a continuum (entropy per unit volume) equals bp, where p is the local hydrostatic pressure (negative of hydrostatic stress). This fact follows from the standard

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theromodynamic relation:[64]
bV,  ∂p  ∂S

(19)

T

where S is the system entropy and V is its volume. Based on this relation, Sq is represented by the sum: Sq
b Σ piΩi,

(20)

i

where pi and Ωi are hydrostatic pressures and atomic volumes associated with individual lattice sites. The summation is performed over all atoms outside the cluster. The product piΩi can be readily calculated within the EAM scheme, as well as with any other semi-empirical potential. We emphasize that Eqs. (18) and (20) are more general than Eq. (17). They are not restricted to any symmetry and only require that the lattice strains beyond the cluster be within the linear elasticity range. This general character makes the method applicable to point defects in GBs and dislocations, to defect clusters, and to other complex situations. This method again appeals to the idea that, in a large simulation block, most atoms belong to elastically strained lattice regions whose direct atomistic treatment is computationally inefficient. Instead, properties of such regions (entropy, in this case) can be well represented in a continuum approximation, while the highly distorted regions comprising the defect cores are treated atomistically. We can notice a parallel with similar ideas currently emerging in the area of multiscale modeling of materials. Because the calculation of Sq from Eq. (20) is a fast linear-N procedure, the quasi-continuum term comes at almost no extra cost in comparison with the cluster term S*. This computational advantage makes the method applicable to very large systems, including those involving multiple length scales. Simple tests[61] have demonstrated a high accuracy and excellent computational efficiency of the method.

3.4.4 Local Harmonic Approximation An even greater acceleration of entropy calculations can be achieved by adopting the so-called local harmonic approximation (LHA).[68] In this approximation, the off-diagonal terms of the dynamical matrix, which are responsible for the coupling of vibrations of different atoms, are neglected: Diajb
0, i  j. This reduces the dynamical matrix to a block-diagonal form consisting of 3  3 blocks, each representing the local dynamical

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matrix DiL with elements Diaib. The problem thus reduces to diagonalizing the local dynamical matrix for each atom, or, equivalently, calculating its determinant DLi . For example, the classical entropy expression, Eq. (13), in LHA becomes: N hDLi 1
6  3NkB. S
3kBΣ ln  kBT i
1

(21)

In the second-moment method proposed by Sutton,[69, 70] the summations over the density of vibrational states are expressed through the local second moments, mi
∑a Diaia . A slight modification of this method was proposed by Foiles.[71] Although the derivation of the basic equations in both cases proceeds along different lines in comparison with the LHA, this boils down to neglecting even the diagonal elements of the local dynamical matrices. This obviously means a more drastic approximation than LHA. An attractive feature of LHA and the second-moment methods, besides their computational efficiency, is the feasibility of finding the thermodynamic equilibrium of a system at a finite temperature by minimizing its free energy with respect to all atomic displacements. This idea has been implemented in the free-energy minimization method by LeSar et al.[68] and through temperature-dependent interatomic forces by Sutton.[69, 70] Due to the relatively simple form of the free energy, analytical expressions can be derived for its derivatives with respect to individual atomic coordinates (thermodynamic forces), which enables the use of gradient methods for minimizing the free energy. Unfortunately, the crude approximations underpinning LHA and the second-moment methods inevitably result in a significant loss of accuracy compared with the full harmonic scheme.[71–73] Taking a Cu vacancy as an example, LHA gives the vacancy formation entropy (at zero pressure) a factor of two lower than the full harmonic value (Fig. 3.2). Thus, when the free energy is minimized within LHA, the optimized atomic coordinates can differ significantly from their true equilibrium positions.

3.4.5 Quasi-Harmonic Approximation For a perfect crystal, the full harmonic free energy can be readily minimized with respect to the crystal volume, thus giving the thermal expansion factor at a chosen temperature. This minimization is easy to implement due to its one-dimensional character. Indeed, the dynamical matrix and thus the free energy can be computed for several volumes, and the free

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energy minimum can be located by a polynomial interpolation. This scheme is often referred to as the quasi-harmonic approximation[71, 72] to emphasize that the anharmonicity-caused thermal expansion is evaluated through harmonic calculations. Unfortunately, an extension of this scheme to a simulation block containing a defect is problematic. Since thermal expansion is then nonuniform, the free energy should be minimized with respect to all atomic positions, a problem too ambitious for today’s computers. As a rough approximation, a statically relaxed simulation block can be simply expanded uniformly by the thermal expansion factor of the perfect lattice, followed by the construction of the dynamical matrix and calculation of the defect formation entropy.[71] This way of including the thermal expansion into the defect formation entropy has proved to work well in some cases[71, 74] but fails in other cases where some of the eigenvalues of the dynamical matrix after the expansion become negative. Such failures are not surprising since the neglect of the local thermal expansion in the defect core introduces an uncontrollable approximation. There are methods of thermodynamic calculations beyond the harmonic approximation. Perhaps the most common of them are various thermodynamic integration schemes[56, 72] and the adiabatic switching method,[75, 76] which can be implemented within either Monte Carlo or MD simulations. Such methods have been applied to calculate the formation entropy of point defects in metals[71, 76] and intermetallic compounds.[77] Although very powerful, such methods require extremely extensive computations for achieving a reasonable accuracy. They are also useful as a reference for assessing the applicability range of harmonic methods. As expected, deviations between harmonic and “exact” methods are observed at high temperatures, often already above half of the melting temperature. The onset temperature and degree of such deviations depend on the particular defect type, the atomic interaction model, and other factors.[72]

3.5

Equilibrium Defect Concentrations

3.5.1 Elemental Solids The knowledge of the formation energies and entropies of point defects is required for calculating their equilibrium concentrations. For example, the equilibrium vacancy concentration (number of vacancies per lattice site) in an elemental solid at temperature T is given by:





evf  Tsvf  . cv
exp  kBT

(22)

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A similar equation holds for self-interstitials and other simple defects, except that a pre-exponential factor z should be introduced to reflect the entropy arising from the multiplicity of possible orientations of the defect. For example, a self-interstitial in fcc metals typically exists as a split dumbbell centered on a lattice site. The dumbbell possesses an additional, “rotational” entropy kB ln z, z being the number of its symmetrically equivalent orientations. Thus, the concentration of a general point defect can be written as:





ef  Tsf c
z exp   . kBT

(23)

Factor z is often eliminated by including the term kB ln z into the defect formation entropy sf. Another way of writing Eq. (23) is:





gf c
exp   , kBT

(24)

gf
ef  Tsf

(25)

where

is the free energy of defect formation. At this point, we assume that the external pressure is zero. The effect of high pressures on defect concentrations is discussed in Sec. 3.5.3.

3.5.2 Non-Stoichiometric Compounds The situation is more complex in ordered compounds due to the inherent multiplicity of point defects.[39, 78–80] Vacancies can occupy different sublattices and thus have different formation energies and entropies. Compounds are also able to support antistructural atoms, or antisites (atoms occupying the wrong sublattice), and in some cases also interstitials. Since all point defects should be in equilibrium with each other, the vacancy concentration cannot be determined separately without considering all other defects. In other words, even if we are only interested in the vacancy concentration, we still have to solve the problem of global dynamic equilibrium in the entire defect system. As an example, consider an equiatomic intermetallic compound AB. We assume that the structure of this compound consists of two sublattices

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a and b, which in the perfectly ordered state are occupied by atoms A and B, respectively. The compound can support four types of point defects: vacancy on sublattice a (Va), vacancy on sublattice b (Vb), antisite on sublattice a (Ba), and antisite on sublattice b (Ab). Interstitial atoms are neglected for simplicity. The creation of each point defect slightly changes the chemical composition of the compound. The latter, therefore, depends on all four defect concentrations and can deviate from the exact stoichiometric composition. Statistical models describing the defect equilibrium in compounds normally treat point defects as a lattice gas of noninteracting particles.[39, 78–80] In our case, we deal with a four-component gas. The particles of the gas can only interact through defect reactions, which are similar to chemical reactions in a real gas. For example, an exchange of a regular atom B with a neighboring vacancy Va is represented by the defect reaction: Va ∆ Ba  Vb.

(26)

The equilibrium in the defect gas can be described by writing the mass action law for three independent reactions.[80] The fourth equation expresses the constraint that the defect concentrations should match a particular chemical composition of the compound. These four simultaneous equations are solved numerically for four equilibrium defect concentrations. That way, the defect concentrations can be computed as functions of the chemical composition and temperature. The input data for this calculation are the free energies of the chosen defect reactions. These can be expressed by linear combinations of the respective formation free energies gf of the defects involved in the reactions. Instead of dealing with individual point defects, it is more convenient to introduce hypothetical composition-conserving defect complexes.[39, 78] For example, the formation of a divacancy Va  Vb or an exchange defect Ba  Ab does not change the chemical composition of the compound. Such complexes are assumed to be totally dissociated, so that interactions between their constituent defects can be neglected. The advantage of dealing with composition-conserving complexes is that all reference constants involved in the energies and entropies of individual defects always cancel out when combined into a complex energy or entropy.[78] This allows direct comparison of results obtained by different calculation methods, for example by EAM and first-principles calculations. The language of composition-conserving complexes also has certain conceptual advantages that have been well discussed in the literature.[39] Point defects can be thought of as appearing and disappearing in the form of such complexes. Furthermore, the complexes can be viewed as elementary excitations of thermal disorder that follow Boltzmann’s distribution.

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One of the complexes usually dominates all others, although this leading complex may change with temperature and/or pressure. Dealing with composition-conserving complexes means making a judicious choice of the defect reactions. The reactions should conserve the chemical composition and describe the creation (annihilation) of the chosen defect complexes. For example, the three reactions can be chosen as the divacancy (Dv) formation: Va  Vb ∆ 0,

(27)

the exchange defect (Ex) formation: Ba  Ab ∆ 0,

(28)

and the triple-defect formation on the a sublattice (TA), 2Va  Ab ∆ 0.

(29)

In these reactions, 0 denotes the ground state of the compound (perfect lattice). The free energies of these reactions represent the formation free energies of the respective defect complexes (Dv, Ex and TA). The free energy of each complex is found by combining the formation free energies of individual defects, each obtained by a separate calculation, according to the formula of the reaction. For example, the free energy of an exchange defect equals gEx
gAf  gfB . Similar rules apply to energies and entropies of defect complexes. Thus, the energy of a triple defect, TA, equals eTA
2e Vf  e fA . Remember that the formation energy and entropy of an individual defect are defined here under the condition of a constant number of atoms. As an illustration of such calculations, Table 3.1 presents the formation energies and entropies of several defect complexes in the intermetallic compounds TiAl and NiAl.[6, 81] The crystal structures of these compounds are L10 (CuAu prototype) and B2 (CsCl prototype), respectively. These compounds demonstrate two different mechanisms of non-stoichiometry that are discussed next. They also represent promising structural materials for advanced high-temperature applications, particularly in the aerospace industry.[82, 83] The entropies listed in Table 3.1 were computed in the classical harmonic approximation. Figure 3.3 shows the calculated composition dependencies of the point defect concentrations in these compounds (number of defects per lattice site). We observe that TiAl is strongly dominated by antisites TiAl and AlTi for all compositions, whereas the vacancy concentrations are relatively small. In contrast, in the NiAl compound, the antisites NiAl and vacancies VNi are the dominant defects; other defect b

a

b

a

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Table 3.1. Energies and Entropies of Composition-Conserving Point Defect Complexes in the Intermetallic Compounds TiAl[81] and NiAl[6] Calculated with Embedded-atom Potentials. Defect Complexes: Ex, Exchange Defect; Dv, Divacancy; TA, Triple Defect on the a Sublattice; TB, Triple Defect on the b Sublattice.

Notation

Complex

Ex Dv TA TB

Ab  Ba Va  Vb 2Va  Ab 2Vb  Ba

TiAl Energy (eV) Entropy (kB) 0.765 3.168 3.525 3.576

1.420 2.804 3.952 3.075

NiAl Energy (eV) Entropy (kB) 2.765 2.396 2.281 5.276

4.903 2.965 3.588 7.245

concentrations are orders of magnitude smaller. These observations are in agreement with previous calculations performed by neglecting defect entropies but using first-principles defect energies.[39, 84] Constitutional point defects in compounds are an important concept. At low temperatures, a deviation of the chemical composition from the stoichiometry is accommodated by a certain type of point defect, which is called constitutional. Constitutional defects are conceptually different from thermal defects. The latter are thermally excited and only exist at elevated temperatures due to the configurational entropy of the compound. In contrast, constitutional defects exist at any temperature and serve to maintain the particular off-stoichiometric composition of the compound. If we gradually decrease the temperature at a fixed composition, the concentrations of all thermal defects will go to zero, while the concentration of constitutional defects will remain constant and will only depend on the composition. In other words, constitutional defects are what is left when all thermal defects freeze out. Clearly, constitutional defects only exist in off-stoichiometric compositions. Importantly, the type of constitutional defect can be different in different compounds, and in each compound can be different on either side of the stoichiometry. For example, in TiAl, the off-stoichiometry is accommodated by TiAl antisites on the Ti-rich side and AlTi antisites on the Al-rich side. Thus, the antisites are the constitutional defects. The antisites are also the dominant thermal defects in this compound [Fig. 3.3(a)]. Both facts have one common origin: the exchange defect TiAl  AlTi is the most favorable compositionconserving defect complex in TiAl (Table 3.1). This explains why antisites are excited most easily by thermal fluctuations and are readily available to accommodate the off-stoichiometry. Such compounds are classified as antisite-disorder type.

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100

134

10-1 -2

Defect concentration

10

AlTi TiAl

-3

10

-4

10

-5

10

10-6 -7

VAl

-8

VTi

10 10

44

46

48

(a) TiAl

10

50 at.% Ti

52

54

10

56

0

VNi

Defect concentration

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NiAl

-2

10-4

AlNi VNi

10-6

10-8

10-10

VAl

NiAl VAl AlNi

10

(b) NiAl

-12

44

46

48

50 52 at.% Ni

54

56

Figure 3.3 Equilibrium point-defect concentrations in TiAl[81] and NiAl[6] at 1000 K. These plots illustrate two different mechanisms of atomic disorder in compounds: the antisite mechanism (TiAl) and the triple-defect mechanism (NiAl).

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In contrast, in NiAl, it is the triple defect 2VNi  NiAl that is the most favorable defect complex (Table 3.1). This makes VNi vacancies and NiAl antisites the dominant thermal defects [Fig. 3.3(b)]. Furthermore, being most easily available, these defects also serve as constitutional defects, with NiAl antisites appearing on the Ni-rich side and VNi vacancies appearing on the Al-rich side. The existence of constitutional Ni vacancies in Al-rich NiAl was confirmed experimentally.[85] Intermetallic compounds with this mechanism of disorder are called triple-defect type. Besides NiAl, they are represented by the B2 compounds CoAl and CoGa. To further demonstrate the convenience of the defect-complex approach, we will derive approximate expressions for point defect concentrations in stoichiometric TiAl and NiAl. For the antisite-disorder compound TiAl, in a first approximation we can neglect the vacancies and consider antisite defects only. Then, the concentrations of both antisites are equal. The mass action law for the exchange defect reaction (28),





gEx , cTi cAl
exp   kBT Al

Ti

(30)

immediately gives us





gEx cTi
cAl
exp   . 2kBT Al

Ti

(31)

At the next step, we write the mass action law for the triple-defect reaction (29),





gTA . c2V cTi
exp   kBT Ti

Al

Substituting cTi from Eq. (31), we obtain the Ti vacancy concentration: Al





2gTA  gEx . cV
exp   4kBT Ti

(32)

By simply swapping the species in this equation, we immediately obtain the Al vacancy concentration:





2gTB  gEx , cV
exp   4kBT Al

(33)

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where, gTB is the free energy of the triple defect TB
2VAl  AlTi on the Al sublattice. Using the obvious relation gTA  gTB
2gDv  gEx, we can rewrite Eq. (33) in a different form:





4gDv  gEx  2gTA . cV
exp   4kBT Al

(34)

These equations allow us to calculate all four defect concentrations given three complex free energies: gDv, gEx and gTA. Turning to NiAl, we proceed along similar lines. We first consider the dominant defect reaction, Eq. (29), and neglect the concentrations of VAl and AlNi. Then, we have cV
2cNi , and the mass action law, Ni

Al





gTA c2V cNi
exp   , kBT Ni

Al

(35)

gives us





(36)





(37)

gTA cV
21
3 exp   3kBT Ni

and gTA . cNi
22
3 exp   3kBT Al

Knowing the major defect concentrations, we determine the minor ones. From the divacancy reaction, Eq. (27), we easily obtain:





3gDv  gTA . cV
21
3 exp   3kBT Al

(38)

Similarly, the exchange reaction, Eq. (28), gives us





3gEx  gTA . cAl
22
3 exp   3kBT Ni

(39)

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Again, the four defect concentrations are expressed through three complex free energies gDv, gEx, and gTA. Remember that all defect concentrations in Eqs. (30) through (39) are measured relative to the respective sublattices. They should be divided by a factor of two to obtain the concentrations relative to the total number of lattice sites. The approximate solutions that we have derived predict the defect concentrations in the form of the Arrhenius relation. This allows us to define socalled effective formation energies of individual defects as the respective exponents. For example, Eq. (36) shows that the effective formation energy of Ni vacancies in stoichiometric NiAl equals eTA
3. Although the exact solutions of the problem cannot be expressed in a closed form, effective formation energies of defects can still be derived by fitting the numerical solutions to the Arrhenius Law in a reasonable temperature range. Since the Arrhenius Law is not followed exactly, the result may depend on the chosen temperature range. Even so, the effective formation energies are useful characteristics that reflect basic trends and facilitate a comparison with experiment. Many of these considerations apply to ionic solids as well. However, the charge neutrality requirement imposes a new constraint that has to be incorporated in the statistical-mechanical models of the defects. Besides vacancies and antisites, many ionics contain interstitials and (almost inevitably) impurity atoms, as well as electrons and holes.

3.5.3 Effect of Pressure Until this point, we neglected the pV term in the Gibbs free energy, p being the external pressure and V the system volume. If the external pressure is high, this term cannot be neglected, and the free energy of defect formation, Eq. (25), should be written in a more general form: gf
e f  Tsf  pV f.

(40)

Here, V f is the defect formation volume, which can be defined as the volume of a relaxed defected block minus the energy of a perfect lattice block containing the same number of atoms. Furthermore, the free energy of a composition-conserving defect complex can be expressed in a form similar to Eq. (40), with the complex volume given by a linear combination of the respective defect formation volumes. The linear coefficients are identical to those for the complex energy and entropy.[79, 86, 87] For example, the free energy of a triple defect TA equals: gTA
eTA  TsTA  pVTA, where VTA
2V fV  V fA . a

b

(41)

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Defect complexes whose formation is accompanied by a larger volume effect are more susceptible to external pressure. Furthermore, the relative stability of defect complexes at low temperatures now depends on their enthalpies g
e  pV and not just energies. It is, therefore, conceivable that high pressures can alter the type of the most stable defect complex. This, in turn, would change the types of constitutional defects. This scenario is especially plausible for NiAl in which all volume effects are large due to the significant atomic size difference between Ni and Al.[86, 88] It was predicted that high pressures could change the disorder mechanism in Al-rich NiAl from the triple-defect type to the antisite type, and that this change would occur as a first-order isostructural phase transformation.[86, 88] The possible transition between the vacancy-disordered and antisite-disordered phases of NiAl can be represented by a coexistence line on a p-T phase diagram terminating at a critical point. Equations (24) and (40) are also valid for point defects in elemental solids, both in the lattice and in the core of extended defects. The way the external pressure alters the defect concentration depends on the magnitude and sign of the defect formation volume. Again, high pressures can affect the relative importance of defects with different formation volumes, which in turn can reflect on diffusion rates by different mechanisms.

3.6

Transition Rate Calculations

3.6.1 Transition State Theory Diffusion in solids is mediated by point defect mechanisms. Under such mechanisms, point defects walk through the solid by random, thermally activated jumps that displace atoms, thus inducing atomic diffusion. Each defect jump can induce either a single-atom displacement (for example, an atom exchanging with a vacancy) or a collective displacement of two or more atoms (for example, collective mechanisms on the surface and in grain boundaries). In most cases, point defect jumps fall under the category of rare events.[89] Namely, the atom moved by a defect makes a huge number of thermal vibrations around its equilibrium position before making a successful transition to a new state. It is convenient to discuss this process by considering a dynamic trajectory of the system on the potential energy surface in its configuration space. The latter has a dimension, M, varying between 3N and 3N  3 (N being the number of atoms in the system), depending on the boundary conditions. The initial and final states of the system (i.e., the states before and after the point defect jump) correspond to local minima on the energy surface. The two states are separated by an (M  1)-dimensional dividing

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1

1 Dividing surface

2 Saddle point

2

MEP

(a) Initial Elastic Band

(b) Relaxed Elastic Band

Figure 3.4 Schematic illustration of the nudged-elastic band method. The lines show contours of potential energy, 1 and 2 are the initial and final states of a diffusion jump. The circles mark “images” of the elastic band.

surface that passes along the ridge top between the two energy basins (Fig. 3.4). We assume that the system follows classical dynamics and is coupled to a heat bath at a temperature T (canonical ensemble). After spending a long time wandering within the initial energy basin, the system eventually accumulates enough thermal energy to reach the dividing surface. Once the dynamic trajectory has crossed the dividing surface, there is a high probability that it will slide down to the neighboring basin. Once there, the system will thermalize (establish Boltzmann’s distribution) and spend a long time in the new state before making another transition back to the initial or some other state. During the thermalization, the system forgets how it came to the current state, so that the new jump probabilities do not depend on the previous jump. The scenario just described reflects the idea of the transition state theory (TST),[90] a powerful tool that enables the calculation of absolute rates, or rate constants, of thermally activated transitions. The rate constant is defined as the transition probability per unit time and, in TST, represents an equilibrium quantity. The critical assumptions of TST are that (1) a dynamic trajectory crossing the dividing surface inevitably leads to a transition (i.e., re-crossing events are neglected), and (2) after the transition, the system has enough time to thermalize before making a next transition. Other important concepts associated with reaction rate theory are those of the minimum-energy path (MEP) and the saddle point (Fig. 3.4). The MEP connects the two energy minima and has the property that the energy is a minimum in all directions normal to the MEP, but not necessarily along the MEP. The energy along the MEP reaches a maximum at the saddle point, which coincides with the intersection of the MEP and the dividing

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surface. At the saddle point, one of the eigenvalues of the dynamical matrix is negative, and the corresponding eigenvector is parallel to the MEP. Thus, the system at the saddle point has M  1 normal vibration modes. The missing vibration has converted to a translational degree of freedom associated with the motion along the reaction path. The importance of the MEP and the saddle point is that they have the highest statistical weight. Although a particular dynamic trajectory does not have to coincide exactly with the MEP, it typically follows it rather closely and crosses the dividing surface near the saddle point. This makes the saddle point the bottleneck of the transition. An approximate formulation of TST, called the harmonic TST, is obtained by applying second order expansions of the potential energy in the initial state and at the saddle point. Then, the rate constant Γ of the transition is given by the simple exponential expression:[91]





em Γ
n0 exp   . kBT

(42)

Here, em is the energy barrier of the transition, also called the defect migration energy (hence the superscript m). e m equals the saddle-point energy minus the initial energy. The pre-exponential factor n0 is called the attempt frequency; it is expressed through normal vibration frequencies of the system at the saddle point (n*i ) and in the initial state (ni): M

n i
1

i

n0
M1 .  ni*

(43)

i
1

This expression originates from the entropy difference between the saddle point and the initial state evaluated in the harmonic approximation. A remarkable feature of Eqs. (42) and (43) is that they do not contain any information about the final state of the transition. The MEP is not involved either, except that its direction at the saddle point can be recovered from the eigenvector corresponding to the negative eigenvalue of the dynamical matrix. Equation (42) is the cornerstone of all diffusion rate calculations in solids. Migration energies of vacancies have been computed for many elemental solids using either semi-empirical potentials or first-principles methods. The calculation of n0 is a slightly more difficult task, but this has also been done on many occasions. It is finding the saddle point of the defect jump that becomes the central problem of the rate constant calculation in many systems.

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3.6.2 Finding the Saddle Point For a vacancy jump in a monoatomic crystal, the saddle point can often be located from symmetry considerations. However, the required symmetry is not always available. For example, vacancy jumps in a partially disordered compound are influenced by other point defects residing next to the jumping vacancy, so that the MEP does not have a symmetry that allows us to locate the saddle point. Furthermore, self-interstitial jumps in metals are collective events in which the two atoms forming a split dumbbell move in a concerted manner to produce the dumbbell rotation with a simultaneous translation of the center of mass. This process actually involves three atoms moving simultaneously, the third one being the atom forming the new dumbbell. As another example, GB diffusion demonstrates a rich variety of collective mechanisms mediated either by vacancies or interstitials.[92, 93] In these and other situations, finding the saddle point becomes a challenging problem. Early diffusion calculations employed the “drag” method, in which the atom exchanging with a vacancy was moved between its two equilibrium positions by small steps and the simulation block was partially relaxed after each step.[80, 94–97] The partial relaxation includes arbitrary displacements of all atoms except for the jumping atom. The latter is only allowed to move in directions normal to the jump vector (i.e., vector connecting the initial and final states). Since no symmetry is imposed on the MEP, the latter can readily be a curve connecting the two states. The saddle point is identified as the energy maximum along the curve. This method has proved to work satisfactorily in many applications, including diffusion in intermetallic compounds and GBs. A serious limitation of the method is that only single-atom jumps can be simulated. More advanced methods of saddle-point search have recently appeared in the literature.[98] Perhaps the most common and best tested of them is the nudged elastic band (NEB) method, which was successfully applied to a variety of rate processes in solids.[98, 99] To implement the NEB method, the initial and final states of the transition must be known. The method starts by generating a number of replicas, or “images,” of the system. The images initially represent some arbitrary intermediate configurations between the initial and final states [Fig. 4.4(a)]. For example, they can be created by a linear interpolation of atomic coordinates. This set of images is referred to as an elastic band. The total energy of the elastic band is defined as the sum of actual potential energies of all images plus the sum of fictitious elastic deformation energies of imaginary springs connecting neighboring images. This total energy is minimized with respect to atomic displacements in all images. To improve the convergence, the elastic band is nudged by modifying some components of atomic and spring forces during the energy minimization process. Namely, the atomic forces parallel to the elastic band and

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elastic forces normal to it are eliminated at each step of the relaxation. It turns out that the relaxed elastic band positions itself approximately along the MEP and thus passes through the saddle point[98, 99] [Fig. 4.4(b)]. The image with the highest potential energy is, therefore, taken as an approximation to the saddle point. The latter can be located more precisely by applying an interpolation of atomic coordinates between the highest energy image and its neighbors. More sophisticated schemes of refining the saddlepoint position have been developed, such as the climbing-image NEB method.[100] The crucial advantage of the NEB method is that it imposes no restriction on the number of atoms participating in the transition, which makes it suitable for collective jumps. The elastic band typically contains 10 to 15 images. In some cases, as few as 5 to 7 images were sufficient for obtaining accurate results.[100] On the other hand, if the energy landscape is complex and it is important not to miss a hidden minimum along the transition path, as many as 25 to 30 images can be used.[101] Figure 3.5 illustrates two NEB calculations with two different numbers of images. Both calculations were made for the same collective [110] vacancy jump of a Ni vacancy in NiAl[101] (Fig. 3.6). The two 3.0 2.5 2.0 Energy (eV)

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1.5 1.0 0.5 0.0 0

1

2

3

4

5

6

7

o

Reaction path (A) Figure 3.5 Energy along the reaction path of the collective [110] nearest-neighbor jump of a Ni vacancy in NiAl. The curves were calculated with 7 images (•) and 27 (
) images.[101] The energy is measured relative to the initial state of the vacancy and the distance along the reaction path is the Euclidian distance between neighboring images in the configuration space of the system. The simulation block contains 1024 atoms, and the EAM potential[6] was used.

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2''

143

Ni Al

2

2' 1

Vacancy

NNN Figure 3.6 Geometry of Ni vacancy jumps in NiAl. NNN: next-nearest neighbor jump; 1-2, 1-2 and 1-2 : collective jumps by vectors [110], [100], and [111], respectively.[101]

jump barriers are rather close to each other, as are the respective saddlepoint structures. The NEB method has proved to be highly robust in recent studies of diffusion in intermetallic compounds[101] and grain boundaries.[92, 93] Other, more sophisticated methods of saddle-point search have also been proposed.[98] In particular, the dimer method is capable of finding a saddle point without knowing the final state of the transition.[102] So far, most of such new methods have not been well tested for bulk diffusion in solids, but applications to surface diffusion show promising results.[98, 102, 103]

3.7

Kinetic Monte Carlo Simulations

3.7.1 Basic Idea of the Method Kinetic Monte Carlo (KMC) simulations present a very powerful method for modeling diffusion processes.[104, 105] We will explain the idea of the method by considering diffusion mediated by the random walk of a single point defect. The potential energy surface of the system (Fig. 3.4) has a large number of local energy minima, each corresponding to a dif-

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ferent position of the defect. Suppose for every equilibrium state i, we know the transition rates Γij to all other states j. The set {Γij } is called the rate catalog of the diffusion process. We then create a single defect in the simulation block and let it walk by jumping from state to state. For each state i, we determine the residence time, 1 ti
 ,

Γ Σ j≠i

(44)

ij

and the probabilities of all transitions from that state, Pij
Γijti.

(45)

One transition is selected by dividing a unit interval into segments of lengths Pij and generating a random number with a uniform distribution on this interval. The transition is implemented by updating atomic coordinates according to the diffusion mechanism, the clock is advanced by ti, and the process continues from the new state. Periodic boundary conditions are usually applied to the simulation block, so that if the defect attempts to jump out of the block, it reappears on its opposite border. As the defect walks through the system, it moves atoms around and thus induces their diffusion. After a long defect walk, the diffusion coefficient Dx in a chosen direction x is calculated from the average squared displacement of all atoms. Namely, using Einstein’s relation, we have:

Σk X
N 2 k

Dx
x  . 2t

(46)

Here, Xk is the x-component of the displacement of atom k from its initial position, N is the number of atoms in the block, and t is the accumulated simulation time. Notice the important factor x in Eq. (46). This factor makes a correction for the fact that during the simulation, we always have exactly one defect in the block, whereas in equilibrium conditions, the block would contain, on average, x
Nc defects, c being the equilibrium defect concentration (Sec. 3.5). We assume that the latter is small and that each point defect moves atoms independently of other defects. Then, the diffusion coefficient calculated with one defect should be

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multiplied by the factor x. Eq. (46) can be conveniently rewritten in the form

Σk X

2 k

Dx
c  . 2t

(47)

To improve the accuracy of the calculation, the defect walk is repeated many times starting from different configurations, and the results are averaged. Other precautions include making sure that the average value of Xk is close to zero (no drift in the absence of external forces) and checking the convergence with respect to other factors such as the block size. The KMC method is extremely fast and accurate. Since every jump attempt results in a transition, millions of Monte Carlo steps per atom can be readily accumulated, resulting in a high accuracy of the diffusion coefficient. The jump correlation effects are automatically incorporated in the calculation, which removes the necessity of knowing the correlation factors. In fact, KMC simulations are often used as a means of calculating jump correlation factors.[104, 106] However, the speed and accuracy of the method come at a price. (1) The KMC method only applies to systems that can be mapped onto a lattice. This leaves amorphous solids, complex defect structures in crystals, and many other disordered or long-period structures beyond the reach of the method. (2) The diffusion mechanisms operating in the system should be known in advance. In many situations, diffusion mechanisms are either unknown or very complex. GBs and surfaces offer a wealth of examples where multiple diffusion mechanisms can be operative, including collective events that can hardly be guessed a priori. In such systems, compiling a reasonably complete rate catalog can be a challenging problem. (3) The KMC method is not easy to apply to binary or multicomponent alloys and compounds. Even though diffusion in such alloys is likely to occur by the simple vacancy mechanism, the vacancy jumps are environmentally dependent. Creating a catalog of jump rates for all possible atomic permutations even within a few coordination shells around a vacancy would be an overwhelming task, not to mention the possible effect of long-range interactions.[107]

3.7.2 Grain Boundary Diffusion Grain boundary diffusion in metals is an area where KMC simulations have been particularly useful. GBs provide high-diffusivity paths in materials and control many processes at elevated temperatures, such as

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microstructural evolution, phase transformations, Coble creep, and sintering.[108] The difference between the GB and lattice diffusion coefficients in the same material can easily reach 6 to 8 orders of magnitude. This interesting phenomenon has been known and extensively studied for five decades.[95, 108–111] While a large amount of experimental data has been accumulated over the years, an atomistic understanding of GB diffusion has been in a rudimentary state until very recently. Significant progress in gaining such understanding has been achieved through computer modeling.[92–95, 112–120] It has been established that GBs readily support both vacancies and interstitials. The interaction of these defects with GBs is short-range and is limited to a few atomic layers around the GB plane. Within the GB core region, the vacancy and interstitial formation energies are on average lower than in the bulk, but their variations from site to site are remarkably large. For example, the vacancy formation energy can be as low as 10% of the bulk value at some sites and above the bulk value at other sites.[92, 93, 112, 113] The low average values and large site-to-site variations of point defect formation energies reflect specific features of the GB structure. The latter is on average more open than the regular lattice[121] but contains alternating regions of compression and expansion. It is those regions that give rise to the strong variations in the defect formation energy. Furthermore, in contrast to the bulk situation, the formation energies of vacancies and interstitials are very close to one another, which makes both defects equally important for diffusion and other properties of GBs. Atomistic simulations[92, 93, 112, 113] have revealed new interesting structural forms of point defects in GBs. Vacancies can be localized at certain GB sites (as they are in the lattice), but in many cases they can delocalize over a relatively large area. Many GB sites do not support a stable vacancy at all: the vacant site is spontaneously filled by a neighboring atom during the static relaxation process. The vacancy delocalization and instability can be explained by the existence of strong stress gradients between adjacent compression and tension regions within the GB core.[112, 113] Likewise, interstitial atoms can be localized in interstices, but they often form split dumbbell configurations or highly delocalized displacement zones reminiscent of crowdions.[92, 93, 112, 113] Given this multiplicity of structures and energies of point defects, it is not surprising that GBs support a large variety of diffusion mechanisms, many of which are collective displacements of several atoms. Such mechanisms, as well as the methods by which they were found, are discussed in Sec. 3.8.2. Figure 3.7 shows the temperature dependence of GB diffusion coefficients calculated by the KMC method for a few symmetrical tilt GBs in Cu.[92] The KMC code used in the calculations was general enough to accommodate diffusion mechanisms of any complexity, including collective

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Figure 3.7 Arrhenius plot of grain boundary diffusion in copper obtained by KMC simulations.[92] Only diffusion in the fastest direction in each boundary is shown in this plot. The fastest direction (⊥, normal to the tilt axis; ;, parallel to the tilt axis) and the dominant point defect are indicated in the legend. Experimental diffusion coefficients measured on polycrystalline samples[122] are shown for comparison.

transitions induced by either vacancies or interstitials, transitions between localized and delocalized states of point defects, and other events. To obtain each diffusion coefficient, up to 1014 Monte Carlo steps were made to guarantee the convergence of the results. Because the vacancy and interstitial mechanisms were simulated separately, the relative importance of the two defects could be evaluated. Figure 3.7 demonstrates that the Arrhenius Law is followed by the diffusion coefficients very accurately, despite the involvement of many diffusion mechanisms with different activation energies. The effective activation energy was determined by numerically fitting the Arrhenius relation to the calculated diffusion coefficients. The effective activation energy was often found to correlate with the activation energy of a particular defect jump, suggesting that this jump dominated the overall diffusion process. Interestingly enough, the dominant diffusion jump was not necessarily the one with the lowest activation

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energy. The latter is quite understandable. In order to contribute to longrange diffusion, the jumps should form a penetrating network through the GB structure. It was not uncommon to find jumps that had a low activation energy but were isolated from each other. Such jumps, then, did not effectively contribute to diffusion, but rather slowed it down by acting as traps. Furthermore, under the vacancy mechanism, isolated chains of easy jumps did not contribute to diffusion. This is consistent with the known fact that the correlation factor of one-dimensional diffusion is zero.[118] Figure 3.7 also demonstrates that diffusion in some GBs is dominated by vacancies and in others by interstitials. Thus, there is no unique mechanism of GB diffusion: either vacancies or interstitials can dominate, depending on the GB structure. The diffusion coefficients are anisotropic, which reflects the structural anisotropy of tilt GBs. Note that the fastest diffusion direction can be either parallel or perpendicular to the tilt axis. The latter observation seems to contradict the experimental measurements suggesting that diffusion parallel to the tilt axis is always faster.[108] Note, however, that the calculations were made for ideal singular boundaries with relatively low Σ values (Σ being the reciprocal density of coincidence sites). The real GBs studied in experiments are likely to be vicinal and contain secondary GB dislocations running parallel to the tilt axis. Such dislocations are able to enhance diffusion in the parallel direction. Another remarkable effect evident from Fig. 3.7 is the broad range of GB diffusivities found in the same material. Depending on the GB structure, the diffusivity varies by almost four orders of magnitude even at high temperatures. Notice that the experimental data measured on polycrystalline samples[122] lie well within the calculated range, closer to its upper part. We are led to conclude that the concept of an average, or general, GB, whose diffusivity is measured in polycrystalline materials, is quite loose. This conclusion emphasizes the importance of experimental measurements on bicrystals containing individual, well-characterized GBs.[95, 108, 109] On the computational side, an interesting problem for future research would be to calculate GB diffusion as a function of misorientation around a low-Σ boundary and compare the results with experimental data. Another interesting topic is the effect of segregated atoms on GB diffusion. It is very likely that GB segregation can alter not only the diffusion rates but also the diffusion mechanisms. This possible effect has not been studied in much detail. The KMC method has been widely applied to surface diffusion, an area where the multiplicity and complexity of diffusion mechanisms is a wellestablished notion.[89, 98, 103] Qin and Murch[123] simulated diffusion along a dislocation core by a postulated vacancy mechanism. Although their dislocation model was simplistic and the simulations did not address any specific material, useful insights into possible jump correlation effects were obtained.

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3.7.3 Diffusion in Alloys and Compounds Another area of KMC simulations is diffusion in disordered and ordered bulk alloys. As far as disordered alloys are concerned, the simplest KMC model, which has been widely used since the 1970s, is based on the two-frequency approximation. This model describes vacancy diffusion in a random binary alloy A-B with a rigid lattice (Ising model) in terms of two parameters: the vacancy exchange rates with atoms A (wA) and B (wB). Dozens, if not hundreds, of KMC simulations within this model have been published. We will not review that large area since we want to focus on models based on realistic atomic interactions in materials. We should mention, however, that this simple model has been extremely useful in clarifying many general issues of diffusion kinetics in solids, including jump correlations and their relationship to phenomenological coefficients.[104, 124] An appealing feature of this model is that it underpins many analytical models, which typically also treat alloy diffusion in terms of two-jump frequencies, wA and wB.[104, 124–126] The only difference is that KMC simulations offer an exact numerical solution of the problem, whereas analytical models introduce further approximations aimed at simplifying the problem to the extent that it can be solved in a closed form. This makes KMC simulations a useful testing ground for analytical theories, including the classical Manning theory of diffusion in a random alloy.[126] Extensive KMC simulations within the two-frequency model, together with the concurrent development of analytical models, were pioneered by Murch[104] and are now indisputably led by Belova and Murch.[127–130] For diffusion in ordered compounds, the two-frequency model can be extended to include more frequencies representing vacancy exchanges with atom A or B depending on whether the vacancy and atom are residing on the same or opposite sublattices. In the simplest case of a B2ordered compound, only four jump frequencies are introduced: waA, wbA, waB, and wbB, where the superscript indicates the sublattice occupied by the vacancy. By varying ratios of these frequencies, qualitatively different situations can be reproduced, including strongly and weakly ordered compounds, the vacancy preference to one sublattice or another, and different off-stoichiometries. Extensive KMC simulations within this model have largely contributed to the understanding of the anti-structural bridge mechanism,[131] six-jump vacancy cycles,[132, 133] and the effect of longrange order on the ratio of the tracer diffusivities of atoms A and B.[134] A more advanced model, which was applied to both ordered and disordered alloys, addresses the alloy energetics within the Bragg-Williams approximation. It describes atomic interactions on a rigid lattice in terms of nearest neighbor bonds and is based on three energetic parameters: eAA,

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eBB, and eAB. It is more convenient, however, to analyze the model in terms of two reduced parameters: e
kBT and u
kBT, where e
eAA  eBB  2eAB is the ordering energy and u
eAA  eBB is the asymmetry energy. The configurational entropy is treated within the mean-field approximation. By adjusting the parameters e
kBT and u
kBT, various types of phase diagrams can be simulated, from one with a miscibility gap to a series of ordered compounds.[135] For the modeling of diffusion processes, the vacancy formation energy is identified with the energy of broken bonds when removing an atom from the lattice. The rate of vacancy exchange with an atom is postulated in the Vineyard form, Γ
n0 exp(em
kBT), where em is an environmentally dependent activation energy. The latter is represented in the form: em
es  eb,

(48)

where eb is the total energy of all bonds of the atom exchanging with the vacancy (i.e., bonds that will be broken by the jump), and es is the energy at the saddle point. For simplicity, es and n0 are usually assumed to be constant. This approximation for em can be traced back to Kikuchi and Sato,[136] but it was probably Martin and co-workers[137–139] who put it into a KMC algorithm and applied to a variety of alloy simulations. Using this or similar approaches, Athènes et al.[138] and, more recently, Belova and Murch[132, 140, 141] examined diffusion mechanisms in ordered intermetallic compounds, particularly six-jump cycles and anti-structural bridges. Note that the KMC scheme based on Eq. (48) does not require any rate catalog since the jump rates are computed at each step of the simulation, depending on the local environment of the vacancy. This scheme, therefore, can be viewed as a particular case of on-the-fly Monte Carlo simulations discussed in Sec. 3.7.4. Although the previously discussed KMC schemes have largely contributed to solving general issues of diffusion kinetics, the drastic approximations underlying these schemes prevent them from producing quantitatively meaningful results for specific materials. The BraggWilliams approximation does not capture alloy thermodynamics on a quantitative level. Perhaps more importantly, Eq. (48), especially with es
const and n0
const, hardly has a sound physical justification. Further improvements of such schemes can be sought. For example, the broken-bond energy of the atom in the final state can be added to Eq. (48) with some weight. The attempt frequency can be correlated with the activation energy using Zener’s relation ln n0  em.[142] Such improvements, however, are unlikely to elevate these calculations to a quantitative level. Bocquet[107] has recently examined many of the approximations and empirical correlations commonly used in KMC simulations. He performed

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extensive calculations of vacancy formation and migration energies, as well as the relevant attempt frequencies, in disordered Au-Ni alloys. His calculations were based on molecular statics and the harmonic approximation, with atomic interactions described by many-body potentials of the second-moment approximation (similar to EAM). He found no strong correlation between em and site occupations within the first four coordination shells around a vacancy. Keeping the composition of that region fixed and changing site occupations beyond that region, significant variations (up to a factor of two) in em could be readily produced. This observation emphasizes the critical importance of atomic relaxations and long-range interactions, which are totally neglected in existing KMC schemes. The es
const approximation was also examined by Bocquet[107] and was found not to work with any reasonable accuracy. Zener’s correlation was found to hold as a general trend, but n0 values could easily deviate from the correlation line by two orders of magnitude on either side. Finally, we should mention the brief interest in KMC simulations of diffusion and ionic conductivity based on a large rate catalog generated by the static relaxation method.[143, 144] The catalog was constructed by including a variety of possible environments of a vacancy. To our knowledge, after the initially successful calculations of ionic conduction in Y-doped CeO2,[144] the work in that direction was not continued.

3.7.4 On-the-Fly Monte Carlo Simulations The main limitation of the KMC method arises from the necessity of having a rate catalog before starting a simulation. Even after putting a lot of effort into constructing a large catalog, we can never be sure that all important diffusion mechanisms have been included. Therefore, the atomistic simulation community has been motivated to search for more flexible Monte Carlo schemes, particularly self-learning algorithms in which the possible escape paths from every equilibrium state are identified in the course of the simulation. A few such algorithms have been proposed and are often referred to as on-the-fly Monte Carlo (OFMC). Taking diffusion in alloys as an example, the most straightforward (although not necessarily the most efficient) OFMC scheme could be described as follows.[107, 145] We postulate that (1) the crystal structure of the alloy is preserved during the diffusion process; (2) diffusion takes place by the vacancy mechanism; and (3) vacancy jumps follow harmonic TST. Then, for each vacancy position, we try all z possible exchanges with the nearest neighbors, z being the coordination number of the lattice. For each possible exchange, we find the saddle point (using the NEB or the drag method), the jump barrier em, the attempt frequency n0, and thus the

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transition rate Γ. Knowing all z transition rates, we selected one of them at random, using the standard KMC procedure (Sec. 3.7.1). The clock is advanced by the vacancy residence time (the inverse of the sum of all Γs), and the simulation continues. Thus, the rigid-lattice assumption underlying the standard KMC scheme is replaced by a relaxed lattice at each vacancy position. More importantly, this scheme does not require a global rate catalog that attempts to enumerate all possible environments of a vacancy. Instead, a local catalog is constructed for each vacancy position and used for making a decision on the next jump. However, as in regular KMC simulations, the structure and the diffusion mechanism are postulated prior to beginning the simulation. The computational efficiency of this scheme is poor because the calculation of n0 is an extremely demanding procedure. This problem could be alleviated by using the embedded-cluster method (Sec. 3.4.3), which restricts the dynamical matrix calculation to a local area around the defect. Further improvements could be achieved by combining this scheme with the regular KMC approach. For example, the program could keep in the memory the structures and local rate catalogs for a number of steps back. If the vacancy returns to a previously seen configuration, the transition rates are simply drawn from the memory, not calculated. This can help the vacancy escape from traps more efficiently by switching to the KMC mode. Bocquet[107] recently proposed a more sophisticated and presumably more efficient detrapping algorithm that guarantees a quick escape of the vacancy from a pair of sites separated by a low barrier. A more serious problem, which does not seem to have a solution within this scheme, is the possibility of unstable vacancy configurations. Namely, a configuration arising after a vacancy exchange with an atom can turn out to be mechanically unstable. In that case, the atom would return to its initial position during the static relaxation. In terms of TST, the barrier of the return jump is zero or negative, meaning that Vineyard’s Eq. (42) does not apply. The occurrence of the vacancy instability signals the existence of a cooperative vacancy jump involving a concerted displacement of two or more atoms. Such unstable vacancies and cooperative vacancy jumps were found in the intermetallic compound NiAl.[101] Unstable vacancies were also observed in the simulations of Au-Ni alloys using this scheme.[107] Since the search for alternative diffusion mechanisms is not a part of this scheme, the jumps leading to unstable configurations had to be removed from the catalog, which we hope left the overall diffusion process unaffected.[107] Henkelman and Jónsson[103] proposed a more advanced OFMC scheme, which they refer to as long-timescale KMC. From each equilibrium position of the system, possible escape paths are explored by performing multiple saddle-point searches using the dimer method.[102]

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Although this procedure can never guarantee that all possible transitions have been found, the assumption is that the most important transitions can be identified by a reasonable number of dimer searches. The relevant transition rates are calculated within the harmonic TST. All other steps are the same as in the regular KMC. In contrast to the previous scheme, no diffusion mechanism is imposed on the system. The system is allowed to find the diffusion mechanisms by itself, which can be single-atom jumps or collective events of any complexity. Moreover, the atoms do not even have to occupy lattice positions, which makes the method suitable for studying amorphous and other structurally disordered systems. This method has been successfully applied to model metal surface diffusion, island ripening, and surface growth.[103] Application to bulk diffusion in alloys and compounds would be an interesting topic for future research.

3.8

Molecular Dynamics

3.8.1 Calculation of Diffusion Coefficients Molecular dynamics has been a prime simulation tool in solid-state diffusion for a few decades.[146, 147] In the MD method, the exact dynamic evolution of a collection of atoms is followed by integrating the Newtonian equations of motion. The classical forces acting on the atoms are computed using either interatomic potentials or first-principles methods. Typically, a canonical ensemble is simulated, in which the system is coupled to a thermostat under a temperature T. The temperature control is implemented using the Nose-Hoover method or the Langevin equation with random forces.[56] In most diffusion simulations, the system volume is held fixed, but it can also be allowed to fluctuate to model constantpressure conditions. More sophisticated ensembles, such as the variable block shape implemented in the Parrinello-Rahman method,[148, 149] are also available but have rarely been applied to diffusion problems. The most straightforward way of using MD is to calculate the diffusion coefficient from mean-squared displacements of atoms. For a diffusion process mediated by a single type of point defect, the procedure is very similar to the KMC method (Sec. 3.7.1). A single defect is created in the simulation block, and its random walk is implemented by making a long MD run. The diffusion coefficient is then deduced from meansquared atomic displacements using Eq. (47). The equilibrium defect concentration c appearing in Eq. (47) should be known from separate calculations. [In some publications, the diffusion coefficient was deduced directly from Eq. (46) with x
1, which is incorrect.]

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An advantage of this method over KMC simulations is that diffusion mechanisms do not have to be known in advance. The system evolves naturally under interatomic forces, and the defect has all the freedom to choose the most favorable diffusion mechanisms. A significant drawback, however, is the limited computational efficiency of MD. Most of the simulation time, atoms idly vibrate around equilibrium positions without making any contribution to diffusion. Consequently, accumulating good statistics of jumping, even at high temperatures, is highly problematic. However, estimates of diffusion coefficients of both vacancies and interstitials in metals have been obtained by this method.[150] For diffusion along extended defects, such as surfaces or GBs, the jump barriers of point defects are lower than in the bulk, and extensive diffusion can be readily observed already at medium temperatures. But other problems that come into play are discussed next. Consider GB diffusion as an example. Due to relatively low transition barriers of point defects, mean-squared atomic displacements sufficient for a reliable calculation of the diffusion coefficient can be obtained at temperatures above 0.7Tm, where Tm is the bulk melting point.[120, 151, 152] With massive parallel computations, the lower bound can be pushed down to 0.6Tm ,[153] and in the future, it can perhaps be pushed even lower. However, the energy barriers for the generation of new point defects in GBs are also relatively low, and generation events inevitably happen during the MD simulation. The Frenkel pair formation is the most common mechanism of defect generation, but other mechanisms can also operate, depending on the GB structure.[92, 93] Under such conditions, the number of point defects in the simulation block is no longer conserved, and the reasoning that led us to the correction factor x in Eq. (46) no longer applies. One possible solution could be to perform a very long MD run prior to calculating the diffusion coefficient, with the hope that the GB would generate the equilibrium amount of defects. Then Eq. (46) could be applied without any prefactor. The success of this scheme would depend on the characteristic time required for the particular GB to arrive at point defect equilibrium. It can easily be beyond the time scale accessible by MD simulations. Furthermore, even if the defect generation process is fast enough, the simple GB structures that are typically simulated do not contain any nonconservative sinks or sources of point defects. The total number of atoms in the GB core is always conserved, meaning that the boundary can only support an equal number of vacancies and interstitials. This constraint can prevent the GB from ever arriving at the true point defect equilibrium, which in turn can affect the diffusion coefficient. Optimistically, we can hope that some of the point defects would migrate away from the GB to the adjacent lattice regions, bringing the GB closer to equilibrium. However, this migration also takes time, which again can be beyond the reach of regular MD simulations. Clearly, this problem needs to be examined carefully.

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3.8.2 Diffusion Mechanisms in Grain Boundaries Calculating diffusion coefficients is not the only way of using MD for diffusion. MD also offers an ideal tool for the exploration of unknown diffusion mechanisms in complex structures. As an illustration, we will discuss the recent studies of diffusion mechanisms in GBs. It has been the basic assumption for many years that atoms move in GBs by simple exchanges with vacancies.[108, 121, 154] More recent simulations of GB diffusion in Ag[116, 152] and Cu[93, 114, 115, 155] pointed to a possible role of self-interstitials along with vacancies. In the most detailed study of diffusion mechanisms in GBs,[92] diffusion in six symmetrical tilt GBs in Cu was modeled using an accurate EAM potential[9] fit to both experimental data and first-principles calculations. A simulation block containing about 1200 dynamic atoms was used through the work. A single point defect (vacancy or interstitial) was created at an arbitrary position in the GB core, and a long MD run was implemented at a constant temperature of 1000 K. The computer program automatically generated a snapshot of the block whenever a significant atomic displacement relative to the previously stored configuration pointed to a possible jump. The diffusion mechanisms were analyzed a posteriori by examining the whole set of snapshots produced by the MD run. All snapshots were relaxed in a static mode, and if two consecutive snapshots were found to relax to different states, they were assumed to be separated by a defect jump. The relevant jump barrier and attempt frequency were then determined by the NEB method (Sec. 3.6.2) by taking the two snapshots as the initial and final states. These data were later fed into the KMC model for the calculation of GB diffusion coefficients (Sec. 3.7.2). The NEB calculation also served to verify that the two states were indeed separated by one saddle point, which was always found to be the case. Using this laborious but robust procedure, hundreds of defect jumps were examined and ranked according to their rates in a chosen temperature range. The results of this analysis can be summarized as follows. Vacancies can move in GBs by single-atom exchanges, as they do in the lattice, but they can also move by collective jumps involving a simultaneous displacement of several atoms. Such collective jumps always involve GB sites that do not support a stable vacancy (Sec. 3.7.2). For example, site 6 in the Σ
5 (310) GB [Fig. 3.8(a)] is unstable with respect to vacancy formation: an atom removed from that site is filled by atom 1 during a static relaxation. This instability is responsible for the∼ collective ∼ ∼ jump 1 → 6 → 4 of two atoms, as well as for the 1 → 6 → 2 → 1 jump involving three atoms. These jumps can be viewed as attempts of vacancy 1 to jump to site 6. Because the latter does not support a stable vacancy, the ∼ ∼ vacancy is immediately filled by atom 4 or 2. As a result, the vacancy jump

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~ 1

6

~ 2

2

1 ~ 4

4 1' 6'

[3 1 0]

(a) Vacancy Diffusion Mechanism [0 0 1] 2

1

~ 6

[1 3 0]

~ 2 ~

I

~ 3

I

2'

(b) Interstitial Diffusion Mechanism Figure 3.8 Vacancy and interstitial diffusion mechanisms in the Σ
5 (310)[001] grain boundary in Cu.[92] Squares and circles mark atomic sites in alternating (002) planes. Selected sites are labeled for reference. Symbol I labels interstitial positions. The prime and tilde signs mark symmetrically equivalent sites on either side of the boundary plane or in neighboring structural units. The arrows show point defect jumps. ∼

∼

continues to sites 4 or 2, without landing at site 6. Collective vacancy ∼ jumps∼ were found to be especially popular in the Σ
7(2 3 1)[111] and Σ
13(3 4 1)[111] GBs, in which only a few sites support a stable vacancy.[92] Interstitials can move by the direct or indirect mechanisms. Under the direct mechanism, an interstitial atom is wandering along the GB by jumping between neighboring interstitial positions. While energetically unfavorable in Σ
5 GBs,[93] this mechanism was found to operate in other GBs.[92] Under the indirect mechanism, an interstitial atom displaces a neighboring regular atom to another interstitial position and takes its

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place. This process occurs by a simultaneous displacement of both atoms. Furthermore, an interstitial atom can initiate a chain of atomic displacements and push the terminal atom of the chain into an interstitial position, which can be separated from the initial position. It is important to realize that all atoms taking part in this process move in concert and not one after another. Figure 3.8(b) illustrates this mechanism for diffusion perpendicular to the tilt axis in the Σ
5(310)[001] GB. Notice that the same inter∼ stitial jump (e.g., I → I) can be implemented in several different ways, some of them involving more atoms than others. Interstitial dumbbells in GBs always move by collective jumps of three or more atoms, as they do in the lattice. The delocalized crowion-type interstitial configurations found in the Σ
9 and Σ
11 GBs[92, 113] move in a highly collective way with a very low migration barrier. At each step, individual atoms within the displacement zone each move by a very small amount but the overall displacement zone translates by one period along the tilt axis. This motion looks more like a dislocation glide than an atomic jump. Besides the vacancy and interstitial mechanisms, ring mechanisms were also found in GBs. A ring can include up to six atoms and can either be induced by an existing point defect or happen spontaneously without any pre-existing defects. On the whole, diffusion in GBs is profoundly different from lattice diffusion. It occurs by a variety of different mechanisms, most of which are collective. Considering also that the collective events often happen by a displacement of atomic chains, we can notice an analogy with diffusion mechanisms in bulk metallic glasses.

3.8.3 Diffusion Mechanisms in Intermetallic Compounds Diffusion mechanisms in ordered intermetallic compounds have been an attractive subject for modeling and simulation for a few decades.[34, 106, 156] (See Chapter 4.) It is generally recognized that diffusion in such compounds is mediated by vacancies, but the question is, how exactly do the vacancies move in a structure with long-range order? A random vacancy walk would inevitably involve inter-sublattice jumps, which would soon produce a large amount of antisite defects and eventually cause a complete disorder of the compound. Vacancies should, therefore, move in a highly correlated way to preserve the equilibrium degree of long-range order. Identifying such correlated mechanisms is the key issue in this area. The simplest possible mechanism is one in which vacancies migrate on their own sublattices without any inter-sublattice jumps. This mechanism obviously preserves the crystal order. Alternatively, a number of cyclic mechanisms have been proposed in which the order is

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destroyed only locally and temporarily and is fully recovered once the cycle is complete. All mechanisms that have been proposed so far are based on geometric considerations and have been studied primarily by KMC simulations within simple models (Sec. 3.7.3). Instead of attempting to review this whole area, we will focus on recent evaluations of diffusion mechanisms in the B2-NiAl compound by means of atomistic calculations. Our aim is to demonstrate the capabilities of atomistic simulations rather than to make final judgments regarding the role of particular mechanisms. Similar atomistic calculations of diffusion mechanisms have been performed for B2-FeAl[157] and Ti aluminides TiAl and Ti3Al.[80] NiAl is a triple defect compound in which the off-stoichiometry on the Ni-deficient side is accommodated by structural vacancies on the Ni sublattice, whereas Ni-rich compositions are dominated by NiAl antisites [Fig. 3.3(b)]. Given this asymmetry of the disorder mechanism, we should anticipate an asymmetry and multiplicity of diffusion mechanisms in this compound. An important feature of the B2 structure is that Ni atoms are surrounded by Al atoms only and vice versa. Thus, this structure does not support sublattice diffusion by nearest neighbor jumps. Several diffusion mechanisms have been proposed for NiAl, including sublattice diffusion by next-nearest neighbor (NNN) vacancy jumps on the Ni sublattice,[94, 158] six-jump vacancy cycles (6JCs),[97, 159, 160] the anti-structural bridge (ASB) mechanism,[161] and the triple defect mechanism.[96, 162] Some of the mechanisms were examined by atomistic computer simulations, with controversial results. For Ni diffusion in NiAl, the lowest activation energy (2.76 eV in the stoichiometric composition) was found for the NNN vacancy mechanism, with slightly higher activation energies for the ASB and 6JC mechanisms.[94, 163] It is tempting to conclude that Ni diffuses in NiAl predominantly by NNN jumps along its own sublattice. However, if the NNN vacancy mechanism were indeed dominant, this would lead to a very high Ni diffusivity in Al-rich compositions containing a large amount of Ni vacancies. This is contrary to the experimental observations by Frank et al.,[96] who found the Ni diffusivity to be about the same in the stoichiometric and Al-rich compositions. Frank et al. suggested that the NNN vacancy mechanism is not the dominant one, at least not in the temperature range of their measurements. Instead, they proposed a triple defect mechanism. The estimated activation energy of the triple defect mechanism (3.18 eV, regardless of the composition) compares reasonably well with the experimental activation energy in Al-rich and near-stoichiometric compositions (3.00 ± 0.07 eV), but fails to explain the drop in the experimental activation energy in Ni-rich compositions ( 53 at.% Ni). Frank et al. have attributed this drop to a contribution of the ASB mechanism arising when the concentration of antisites on the Ni sublattice reaches the percolation threshold.

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Some of these mechanisms have recently been revisited with more advanced simulation techniques.[101, 164] Using an accurate EAM potential, the migration energy of the NNN jump of a Ni vacancy was found to be em
2.33 eV, a value consistent with concurrent first-principles calculations.[101] This value is low enough to make NNN jumps a plausible diffusion mechanism in NiAl. Since this mechanism presents a standard case of diffusion on a simple cubic lattice (i.e., Ni sublattice), the activation energy and pre-exponential factor of diffusion can be readily found, provided we know the effective formation energy and entropy of Ni vacancies. For the stoichiometric composition, these are given by eTA
3 and sTA
3, respectively, where eTA and sTA are the energy and entropy of the triple defect on the Ni sublattice (Sec. 3.5.2). The values of eTA and sTA calculated with the same EAM potential are listed in Table 3.1. The resulting expressions for the Arrhenius parameters are:

 

sTA D0
21
3 a2 f0 n0 exp  , 3kB

(49)

eTA Q
  em. 3

(50)

Here, a is the lattice parameter of NiAl, f0
0.653 is the geometric correlation factor for the simple cubic lattice, and n0 is the attempt frequency that was also calculated by Mishin et al.[101] Calculations from these equations give Q
3.09 eV and D0
1.6  106 m2/s, which compare reasonably well with the experimental values Q
3.0 eV and D0
3  105 m2/s.[96] This agreement suggests that NNN vacancy jumps in NiAl are capable of contributing to Ni diffusion significantly and cannot be totally excluded from the possible mechanisms. Using static calculation, the 6JC mechanism in NiAl was also evaluated.[101] Geometrically, the B2 structure is able to support a [110] cycle as well as straight and bent [100] cycles (Fig. 3.9). As a result of a cycle, the Ni vacancy makes a [100] or [110] jump and simultaneously switches positions of two Al atoms. Each cycle starts with an exchange of the Ni vacancy with a neighboring Al atom. Calculations have shown that, contrary to the commonly accepted picture of the 6JC mechanism, the configuration arising after this first jump is mechanically unstable. Because of this instability, the vacancy exchanged with an Al atom returns to its initial position during static relaxation. This important fact was verified by independent first-principles calculations.[101] As we mentioned before, mechanical instability of a vacancy is always a sign

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2 3

2

4

1 5 6 3

6 5 1

(a) [110] Cycle

4

(b) [100] Straight Cycle

Ni Al Vacancy

4

3 6 2

(c) [100] Bent Cycle

5 1

Figure 3.9 Schematic illustration of six-jump vacancy cycles in the B2 structure of NiAl. The arrows show vacancy jumps; the numbers label the jump sequence.

of a collective jump. Indeed, it was found by NEB calculations that the first jump of a cycle merges with the second one and forms a collective, two-jump transition. Three types of such collective jumps are geometrically possible in the B2 structure, as shown schematically in Fig. 3.6. Jumps 1-2 and 1-2 initiate [110] and [100] 6JCs, respectively, while jump 1-2 is not involved in the 6JC mechanism and simply creates two antisites. Due to the inherent symmetry of 6JCs, we can conclude that their last two jumps should also merge and form collective transitions. Furthermore, NEB calculations have shown that the central configuration of the [110] cycle is also mechanically unstable, so that the whole cycle happens as three collective transitions. For the [100] cycles, the central configuration is stable, and the cycles happen by two collective transitions separated by two singleatom jumps. Thus, contrary to the existing paradigm, 6JCs in NiAl happen

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by either three or four vacancy jumps, not six. This conclusion may have an impact on future calculations of correlation effects associated with this mechanism. The collective transitions may also need to be incorporated into future KMC simulations. NEB calculations also revealed that the [110] cycle is energetically substantially more favorable than [100] cycles, suggesting that it is the [110] cycle that should be studied more carefully in the future. Diffusion mechanisms in stoichiometric NiAl were also studied by MD.[164] Since the same EAM potential was used, the results could be directly compared with the static calculations of Mishin et al.[101] MD simulations of NiAl present a serious computational challenge. Due to the high migration barrier of vacancy jumps (~2.5 eV), the vacancy residence time is on the order of a nanosecond or longer, even near the melting point. Consequently, producing even a few hundred vacancy jumps in a sizable simulation block is already a formidable task for today’s computers. The simulations were performed at 1900 K (slightly below the experimental melting point) on a 1024-atom simulation block containing a single vacancy. The vacancy was initially created on a Ni site and was observed to remain on the Ni sublattice through most of the simulation time. This observation is consistent with the predominance of Ni vacancies over Al vacancies in this compound [compare Fig. 3.3(b)]. A snapshot was saved each time the atomic displacements exceeded a preset critical value that could signal vacancy jumps. By analyzing the whole set of statically relaxed snapshots after the MD run, a few hundred sequential vacancy jumps were found, and their barriers were determined by NEB calculations. The following diffusion mechanisms were deduced from this analysis: (1) NNN vacancy jumps on the Ni sublattice; (2) [110] 6JCs implemented by three collective transitions; and (3) collective [111] jumps (1-2 in Fig. 3.6). The latter jumps created short-lived configurations that were annihilated by a reverse jump after less than a picosecond. Such distracting jumps often occurred during the 6JCs but did not change their course. They also happened very often between NNN jumps or 6JCs. Collective 1-2 jumps were observed several times but were soon reversed, and none of them resulted in a complete [100]-type 6JC. These simulations present what appears to be the first direct observation of the NNN and 6JC diffusion mechanism in NiAl by MD, along with a proof of the collective nature of many diffusive events in this compound. It would be premature to conclude that these two mechanisms totally dominate diffusion in NiAl. This work did not address other geometrically possible mechanisms whose simulation would require more than a single vacancy. For example, to study the triple defect mechanism, two vacancies and an antisite would need to be created in the simulation block. Although the question of the dominant diffusion mechanism in NiAl

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remains open, these preliminary results give hope that continued atomistic simulations of this type could soon provide a satisfactory answer. It is both possible and desirable to apply the same simulation methods to other intermetallic compounds that can reveal different diffusion mechanisms. The fact that NiAl supports NNN vacancy jumps is due to its relatively open (bcc-based) structure and a large atomic size difference, with Ni atoms being significantly smaller than Al. The same factors are likely to be responsible for the collective nature of many diffusive jumps in this compound. Preliminary MD simulations for L10-TiAl (close-packed structure with a small atomic size difference) found no NNN vacancy jumps or collective transitions and presented a relatively simple picture of first-neighbor vacancy jumps in a manner reasonably consistent with current KMC models.

3.8.4 Accelerated Molecular Dynamics Many diffusion applications of MD are hampered by the time-scale limitation inherent in this method. Indeed, atomic vibration frequencies in solids are typically on the order of 10 THz. To resolve such vibrations properly, the integration time step of MD cannot be larger than a few femtoseconds. Therefore, even after a million MD steps, the accumulated time is only about a nanosecond. For relatively small systems (a few hundred atoms), the time can be pushed as far as to the microsecond range. In many systems, however, the number of diffusion jumps that can happen over this time is not enough for a reliable evaluation of the diffusion coefficient, even at high temperatures. It could be argued that this time scale can still be suitable for the exploration of unknown diffusion mechanisms. This is true, but again, only at relatively high temperatures. We cannot exclude, however, the possibility that the dominant diffusion mechanisms at low temperatures can be different from those at high temperatures. High temperatures favor mechanisms with relatively high barriers and large attempt frequencies, while low temperatures are typically dominated by low barrier mechanisms with small attempt frequencies. Moreover, many structures only exist at low temperatures, due to structural transformations, atomic disorder, melting, or any other factors that may prevent the structure of interest from being studied at high temperatures. In such situations, even the identification of diffusion mechanisms by MD becomes a difficult computational task. In recent years, a number of new methods have been developed that expand the time scale accessible by MD.[89] Such methods, referred to collectively as accelerated dynamics methods, include parallel replica dynamics,[165] hyperdynamics,[166] and temperature-accelerated dynamics.[167]

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The common idea of these methods is to accelerate the rate at which the system escapes from the energy basins corresponding to its equilibrium states, but to do so in a way that preserves the relative probabilities of different escape paths. In parallel replica dynamics,[165] the system is replicated on M parallel computer processors used in the simulation. The states of the replicas are randomized, and independent MD runs are implemented on the processors in a parallel mode. The M independent dynamic trajectories simulated this way explore the phase space a factor of M faster than a single trajectory would, which is the key to the accelerated escape rates. Whenever a processor detects a transition, simulations on all other processors are stopped, and the clock is advanced by the time summed over all trajectories. The trajectory that experienced a transition continues to run for some time to allow for possible recrossing events, and the clock is advanced by that time. The final state is then replicated on all M processors, randomized, and the simulation continues. This method is very general; in particular, it does not require that the system follow TST. The only assumption made is that the transitions follow first-order kinetics with an exponential decay of the transition time. This assumption guarantees an unbiased state-to-state evolution of the system. Although the ideal acceleration factor M is never achieved, the boost can be very significant.[89] The efficiency of the method is reduced by the randomization time as well as the time required to include possible dynamic correlations after each transition. There is also an overhead associated with the automated detection of transitions. The latter is implemented by relaxing the system at regular time intervals and comparing the relaxed state with the initial one. The hyperdynamics method[166] takes a different approach. Namely, the potential energy surface within each energy basin is modified by adding a specially designed boost potential. The latter should be positive near the equilibrium state and go to zero near the dividing surfaces. By raising the energy around the equilibrium, the boost potential reduces all transition barriers and accelerates the system’s evolution from state to state. The boost potential distorts the atomic vibrations in each equilibrium state and makes them totally unphysical. However, it can be constructed in such a way that the probabilities of different escape paths from each state, determined within TST, are not affected. As a result, the system will follow the same state-to-state evolution as it would without the boost potential, but will do so much faster. In contrast to parallel replica dynamics, the hyperdynamics method relies on TST. The method has been used in a number of simulation studies relating to metal surfaces, and impressively high boost factors (~104) have been demonstrated.[89, 166] An attractive idea is to combine hyperdynamics with parallel replica dynamics, which could lead to even higher boost factors.[89] The critical

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step and the major challenge of the method is the construction of the boost potential. Ideally, the latter should not rely on any prior information regarding possible transitions, so that the method could be applied to an exploration of unknown structures and diffusion mechanisms. An optimum should also be found between the acceleration of transition rates and the computational overhead associated with the construction of the boost potential. This is a nontrivial problem, even in relatively simple systems.[89, 166] In complex situations, the efficiency of the method can be impeded by difficulties in constructing a boost potential for a wrinkled energy surface with multiple shallow minima. The temperature-accelerated dynamics (TAD) method[167, 168] relies on more approximations than the two previous methods, but the boost factors can be much higher. The idea is that an MD simulation is run at a high temperature, Th, but the system is advanced from state to state with transition probabilities corresponding to a lower temperature, Tl. At each step of the procedure, an MD simulation starts from an equilibrium state and is run at temperature Th until a transition is detected. As mentioned previously, the algorithm checks for possible transitions by periodically relaxing the system and comparing the relaxed structure with the initial one. Once a transition is detected, its energy barrier em is evaluated by the NEB method. Assuming that the harmonic TST is valid at both temperatures (Th and Tl), the transition rate should follow the Arrhenius temperature dependence, Eq. (42). Then the escape time th detected at temperature Th can be extrapolated to the escape time tl at temperature Tl:





em em tl
th exp    . kBTl kBTh

(51)

Notice that only the transition barrier, not the attempt frequency, is needed for this extrapolation. The MD trajectory is then reflected back to the original basin and the simulation continues. This procedure produces a set of transitions that can happen from the initial state at temperature Th, together with the extrapolated escape times tl at temperature Tl. The transition corresponding to the shortest value of tl is considered to be the one that would happen if MD simulations were run at temperature Tl. Assuming also that there is a lower bound on possible attempt frequencies, the time can be estimated at which the high-temperature simulation can be safely stopped. At that point, the transition with the shortest tl is implemented, the clock is advanced by tl, and the simulation continues from the new state. As in the previous methods, no prior information on possible diffusion mechanisms is used in TAD, which makes it suitable for the exploration of new structures and unknown diffusion mechanisms.

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The procedure carries a significant computational overhead, since a certain number of possible transitions from each state needs to be detected before one of them is accepted. This overhead limits the number of transitions that are actually implemented in comparison with running regular MD at temperature Th. However, if the difference between the temperatures Th and Tl is large enough, the extrapolated MD time at Tl can reach spectacularly large values (for example, up to several months[169]) and lead to boost factors as high as 109.[169, 170] To alleviate the possible trapping effect arising from low transition barriers, the repeated transitions can be treated in the KMC mode.[167, 168] Overall, the accelerated dynamics methods have a potential to make a significant difference to modern atomistic studies of diffusion in solids. So far, they have mainly been applied to surface diffusion and surface growth of metals. However, the coming years may see other interesting applications, including diffusion in grain boundaries, dislocation cores, ordered intermetallic compounds, and other systems.

3.9

Conclusions

Atomistic computer simulations present a powerful approach to gaining new knowledge on diffusion processes in materials. The capabilities of such simulations have drastically improved in recent years due to the development of new simulation methods reinforced by increased computer power. More reliable models of atomic interaction in materials have been developed, allowing a description of point defect properties on a quantitative level. The impact of first-principles calculations on diffusion science is already quite significant and will increase in the future. At the same time, the use of semi-empirical interatomic potentials will continue to be the mainstream of diffusion simulations. Since such simulations require statistical averaging and involve rare events, fast methods that provide access to increasingly larger systems and longer simulation times will always be in demand. The traditional MD and KMC methods are and will remain the main tools of diffusion simulations. At the same time, such simulations may greatly benefit from adopting the new approaches developed in recent years, such as the accelerated dynamics methods and the emerging on-the-fly Monte Carlo schemes. We believe that two trends are playing a fundamental role in shaping modern diffusion science. First, diffusion theory is turning more towards specific materials and more complex systems. Although the traditional analysis of vacancy diffusion in an abstract alloy A-B is still an important and respectable subject that well deserves further efforts, the new challenge is to understand, describe, and calculate diffusion coefficients in a particular

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metal, alloy, or compound. In some (but not all) of the rare cases when this was done, the agreement with experiment was encouraging. There is notable progress in the modeling of surface and GB diffusion, but diffusion along interphase boundaries and dislocations remains an almost unexplored area. The second trend is the growing interest in understanding diffusion mechanisms on the atomic level. While most of the traditional diffusion theory conveniently assumes the simple vacancy mechanism, the recent studies of surface and GB diffusion reveal a rich variety of other mechanisms, which can be mediated either by vacancies or interstitials and can involve collective atomic displacements. Many of the new mechanisms originate from previously unknown structural forms of point defects, such as delocalized vacancies and interstitials, and unstable vacancies. In fact, unstable vacancies and collective jumps have now been found, by atomistic simulation, even in bulk alloys and compounds. Why should we be so interested in diffusion mechanisms? Clearly, gaining knowledge of the relevant atomic mechanisms is the first step in any predictive calculation of diffusion coefficients in a particular material. Besides this practical aspect, however, there is a basic question lying in the very foundation of diffusion science: how exactly do atoms move in solids? It is the inexhaustible desire to find a satisfactory answer to this question that motivates the pursuit of diffusion mechanisms, a pursuit that will never be over.

Acknowledgment Work on this chapter was supported by the U.S. Air Force Office of Scientific Research (Metallic Materials Program) under Grant No. F49620-01-0025.

References 1. 2. 3. 4. 5. 6. 7. 8.

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4 Bulk and Grain Boundary Diffusion in Intermetallic Compounds Christian Herzig and Sergiy Divinski Institut für Materialphysik Universität Münster, Münster, Germany

4.1

Introduction

According to a simple definition,[1] the intermetallics are compounds of two and more metals with structures that differ from those of the constituent metals. The intermetallic compounds exist in a variety of lattice structures: from the simplest, such as B2 (NiAl, FeAl) and L10 (TiAl, CuAu), to very sophisticated configurations, such as quasicrystalline i-AlCuFe. The short- and long-range order are important phenomena of the intermetallic compounds that notably affect their mechanical and physical properties.[2] Thus, sublattices and preferential site occupation are important terms that describe microscopic features of the atomic distribution in these compounds. Intermetallic compounds are widely used in different technological applications and still have an enormous potential for future applications.[3] The development of intermetallic compounds for application as structural materials inevitably requires the knowledge of the relevant bulk and grain boundary (GB) self-diffusion and solute diffusion data. During the last decade, defect and diffusion phenomena in intermetallics have attracted much attention, not only from the point of view of accumulating experimental data, but also with the purpose of gaining a deeper insight into the underlying microscopic diffusion mechanisms. Recently, general overviews on diffusion in intermetallic compounds were published.[4, 5] The diffusion behavior of some technologically important compounds was also comprehensively analyzed, namely, the L12-structures in the Ni–Al, Ni–Ge, and Ni–Ga systems,[6] in the Ti–Al system,[7, 8] and in the Fe–Al system.[9] Diffusion in pure materials is mostly mediated by the nearly random motion of vacancies. Due to the ordered structure of intermetallic compounds and the different probabilities of finding a vacancy on a particular sublattice, diffusion in intermetallics occurs through more sophisticated jump sequencies involving several atoms. It is very important to realize that even in a completely ordered intermetallic compound at stoichiometric composition, a given amount of thermal defects (vacancies and

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anti-structure atoms) exists at the given temperature T
0. This implies that the order parameter is not really unity, which has an important consequence: While the given state of order has to be maintained on average, local deviations are possible. Thus, the relevance of a specific diffusion mechanism in an intermetallic compound as a well-defined sequence of atom jumps becomes vague, especially for compounds and compositions with a large deviation from the perfectly ordered state. It is still attractive, however, to classify the vacancy motion in terms of particular diffusion mechanisms that maintain the order locally. Several mechanisms of this kind, such as the six-jump-cycle mechanism,[10] the triple-defect mechanism,[11] and the antistructure bridge (ASB) mechanism,[12] were suggested for ordered structures. The concept of particular diffusion mechanisms is attractive not only because it provides a simplified physical picture of a complex diffusion behavior, but also because it allows calculation of effective activation energies and entropies for these processes (energies and entropies of defect formation, migration, and binding, as well as correlation effects), as has recently been done, e.g., for the six-jump-cycle diffusion mechanism in NiAl.[13] Using elaborate approaches, such as ab initio calculations or molecular static calculations with embedded-atom (EAM) potentials, we can compare the theoretical predictions with experimental data and select the most plausible diffusion mechanism. At present, however, large-scale molecular dynamic calculations including a large number of isolated vacancies are still impossible. This chapter focuses on diffusion in ordered binary aluminides of Ni, Ti, and Fe. In these cases, a lot of new, reliable experimental data exist, and the diffusion mechanisms are already elaborated in some detail. These aluminides form different structures, such as B2 (NiAl, FeAl), L12 (Ni3Al), D019 (a2-Ti3Al), L10 (g-TiAl), and D03 (Fe3Al). Some fundamental insight into the interdependence of diffusion behavior and diffusion mechanisms on structure and ordering are provided. The impact of order, thermal and structural defects, and composition are discussed.

4.2

Crystal Structures and Point Defects in Ni, Ti, and Fe Aluminides

Phase diagrams for the Ni–Al,[14] Ti–Al,[15, 16] and Fe–Al[17] systems are shown in Fig. 4.1. Self-diffusion and solute diffusion in ordered aluminides with XAl and X3Al compositions are reviewed here, with X  Ni, Ti, or Fe. These ordered phases exist in wide compositional ranges of the corresponding phase diagrams. Whereas the TiAl phase field extends predominantly on the Al-rich side of the stoichiometric composition and FeAl on the corresponding Fe-rich side, NiAl can accommodate a remarkable

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(b) Ti-Al System[15,16] Figure 4.1 Phase diagrams of Ni-Al, Ti-Al, and Fe-Al.

(continued)

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Figure 4.1 (Continued)

excess of both Al and Ni atoms (Fig. 4.1). The Ni3Al phase field also exists on both sides of the stoichiometric composition, but in a narrower compositional interval. In contrast, both Ti3Al and Fe3Al exist as Al-rich ordered phases in limited temperature intervals. Thus, diffusion studies, especially in the two nickel aluminides, are very promising from a fundamental point of view for explaining the influence of composition and defect structure. In Fig. 4.2, the ideally ordered crystalline structures of Ni, Ti, and Fe aluminides are schematically presented, that is, the structures at zero temperature and at perfect stoichiometric compositions. As the temperature increases and/or the composition deviates from stoichiometry, point defects are inevitably generated. Four types of point defects can be introduced in two-atomic intermetallics AB; namely, the vacancies on both sublattices, VA and VB, and the atoms on unlike sublattices AB and BA (the antistructure atoms). Both structural (constitutional) and thermal point defects exist in an off-stoichiometric intermetallic compound. In a strict definition, the structural defects are those defects that remain in thermal equilibrium in the intermetallic compound, even at T  0 in its maximally ordered state,[18] to accommodate the deviation from the stoichiometric composition. The difference between the real concentration of defects at T
0 and the concentration of the structural defects presents the concentration of thermal defects. In such a definition, the concentration of thermal defects

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(a) Ni3Al

(b) NiAl

(c) Ti3Al

(d) TiAl

(e) Fe3Al Figure 4.2 Lattice structures of the aluminides: Ni3Al, NiAl, Ti3Al, TiAl, and Fe3Al. Ni, Ti, and Al atoms are represented by black, grey, and white spheres, respectively. In the D03 structure of Fe3Al (e), black and grey spheres distinguish the two sublattices for Fe atoms.

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can even be negative. The Al-rich phase NiAl seems to present such an example.[18] A further difference between the structural and thermal defects stems from the fact that one type of structural point defect is generally sufficient to accommodate the deviation from the stoichiometry, whereas at least two types of thermal point defects have to be simultaneously created to satisfy the mass-balance conditions (to preserve the given composition, that is, the given ratio between the constitutional elements). Moreover, the point defects do not have to be uniformly distributed over the different sublattices in an ordered intermetallic compound; this was established experimentally[19–21] and by theoretical analysis.[22, 23] In Fig. 4.3, the concentrations of different defects in the intermetallic compounds under consideration are compared at T  0.75Tm. This temperature corresponds to T  1252 K for Ni3Al, 1434 K for NiAl, 1457 K for Ti3Al, 1294 K for TiAl, and 1195 K for FeAl. Tm is the melting temperature of the stoichiometric composition of the given compound. The defect concentrations were calculated according to the chemicalreaction approach.[8] The concentration of point defects depends on the formation energies of all four types of defects because, in an intermetallic compound, point defects are created in a correlated way to preserve the cAl [at.%] -132 10

30

26

24

AlTi

-2

TiAl

10 Defect concentration

28

-3

10

-4

10

VTi -5

10

-6

10

VAl -7

10

68

70

72 cTi [at.%]

74

76

(a) Ti3Al Figure 4.3 Vacancy and antistructure atom concentrations on transition metal and Al sublattices in Ti3Al, TiAl, Ni3Al, NiAl, and FeAl as functions of the composition at T  0.7Tm.

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cAl [at.%] -154 10

52

AlTi

-2

10 Defect concentration

50

TiAl

-3

10

-4

10

VTi -5

10

-6

10

VAl -7

10

47

46

48 cTi [at.%]

49

50

(b) TiAl cAl [at.%] -128 10 -2

10

27

26

25

24

AlNi

23

22

NiAl

-3

Defect concentration

10

-4

10

-5

10

VNi

-6

10

-7

10

-8

10

VAl

-9

10

-10

10 72

73

74

75 76 cNi [at.%]

(c) Ni3Al Figure 4.3 (Continued)

77

78

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52

10

50

48

46

NiAl

VNi

-2

10

44

-3

Defect concentration

10

-4

10

-5

10

VAl

-6

10

-7

10

AlNi

-8

10

-9

10

48

46

50

52 cNi [at.%]

54

56

(d) NiAl cAl [at.%] 50

-1

10

-2

10

48

46

AlFe

FeAl

-3

10

-4

Defect concentation

10

-5

10

VFe

-6

10

-7

10

-8

10 -14 10 =

=

-15

VAl

10

-16

10

-17

10

50

52 cFe [at.%]

(e) FeAl Figure 4.3 (Continued)

54

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given composition. The defect formation energies for Ti3Al and TiAl have been calculated[7, 8] and are discussed in this chapter. The EAM potentials developed by Voter and Chen[24] for Ni3Al, and the potentials of Mishin and Farkas[22] for NiAl, have been applied to Ni aluminides. The results of ab initio calculations of defect formation energies by Mayer et al.[25] were used for B2-FeAl. Formation entropy effects were neglected. Figures 4.3(a) through (e) demonstrate a few important features of defect behavior. Both the Ti aluminides, Fig. 4.3(a) and (b), and Ni3Al, Fig. 4.3(c), obviously belong to the anti-structure-defect type of intermetallic compounds, since anti-structure atoms are predominantly generated to accommodate the deviation from the stoichiometry. In contrast, NiAl reveals a triple-defect type of point defect disorder, and constitutional Ni vacancies exist in NiAl on the Al-rich side, as shown in Fig. 4.3(d). Moreover, the Ni vacancy concentration is very large on the Ni-rich side, for example, CV
104 at T  0.75Tm. In the other intermetallics under consideration, the vacancies are also mainly concentrated on the transition-metal sublattice, and their concentration amounts to about 106 to 105 at T  0.75Tm. These are also the typical vacancy concentrations in close-packed pure metals at the same reduced temperature. The vacancy concentration on the Al sublattice is remarkably smaller, especially in B2-FeAl, as shown in Fig. 4.3(e). According to Mayer and co-workers,[25, 26] B2-FeAl is neither a compound with pure anti-site disorder nor a compound with pure triple-defect disorder. FeAl demonstrates a hybrid behavior in which the relation between the Fe vacancy concentration and that of the anti-structure atoms depends crucially on temperature. The concentration of Ti anti-structure atoms in the Ti aluminides is generally larger than that of Ni anti-structure atoms in the Ni aluminides of the same composition (Fig. 4.3). This corresponds to a higher degree of thermal disorder inherent in Ti aluminides at similar reduced temperatures. These features play a decisive role in the analysis of the respective self-diffusion behavior. The important question is how the particular crystal structure of the given intermetallic compound can affect the self-diffusion properties. It is generally accepted that self-diffusion in closed-packed structures occurs via nearest neighbor jumps of vacancies. Since random vacancy jumps between different sublattices would generally produce disorder, and since there is a strong tendency to accomplish a reverse, ordering jump after a given disordering jump, the correlated jumps of vacancies will clearly play a decisive role in the long-range diffusion process. Comparing the structures in Fig. 4.2, we can observe an important feature that largely determines the self-diffusion properties of these compounds. It is obvious that in Ni3Al, Ti3Al, TiAl, and Fe3Al, the transition Ni

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metal sublattice presents a connected network for nearest neighbor jumps of vacancies, whereas the nearest neighbor jumps in NiAl and FeAl are exclusively the jumps between different sublattices. Since vacancies are concentrated on the transition-metal sublattices, the sublattice diffusion mechanism may be proposed for Ni3Al, Ti3Al, TiAl, and Fe3Al from this general consideration. We can expect, a priori, that the structural point defects appreciably affect the self-diffusivity in intermetallic compounds and produce a compositional dependence of the diffusivity. From this consideration, several questions arise: (1) Are the structural vacancies the defects that most remarkably enhance self-diffusion? (2) To what extent do the anti-structural atoms affect the diffusivity? (3) How is this effect related to the given crystalline structure? (4) Does the absence of the connected network for nearest neighbor jumps of Ni vacancies slow down self-diffusion in NiAl in comparison with the presence of such a network in the more closed structure of Ni3Al? Discussions in Sections 4.3 through 4.5 aim to provide insight into the diffusion behavior and the diffusion mechanisms with respect to the crystalline structure and the type of disorder in the Ni, Ti, and Fe aluminides.

4.3

Diffusion Mechanisms in Intermetallics

The ordered structure of intermetallic compounds imposes certain limitations on geometrically possible vacancy-mediated diffusion mechanisms. The most important diffusion mechanisms relevant to the diffusion behavior of the aluminides under consideration are discussed in this section. Six-Jump-Cycle Mechanism. This mechanism was originally proposed for B2 compounds;[10] later, it was elaborated for other ordered structures. This mechanism is shown in Fig. 4.4. The correlated jumps of atomic species during execution of the cycles impose certain limitations on the quantitites that can be measured in a diffusion experiment. Firstly, in a highly ordered state, the ratio of tracer diffusivities of both components, D*AD*B, can adopt only the values within the interval: D* 1   A*  q, DB q

(1)

where q was calculated to be 2[27] and was later corrected to q  2.034 by including the correlation effects.[28] The ratio of diffusivities of Ag and Mg in b-AgMg,[27] Zn and Au in b-AuZn,[29] and Cd and Au in b-AuCd[30] fall

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2 3 1

6 5

A

B

AB

4

BA

V

Figure 4.4 A scheme of the six-jump-cycle mechanism in a B2-compound AB.

into these limits. This was considered to be strong support for the sixjump-cycle mechanism in these compounds. As the composition deviates from the stoichiometric one, a large amount of constitutional anti-structure defects appear. Interaction of the six-jump cycles with these anti-structure atoms remarkably changes the limits in Eq. (1).[31] Thus, in a less ordered state, experimental values of DADB larger than 2 can no longer be considered as an indication that the six-jump-cycle mechanism does not operate. Diffusion by the six-jump cycles is a highly correlated process. Thus, the correlation factor is supposed to be rather small. However, we should generally distinguish two types of correlations that characterize the sixjump cycles. Considering the individual cycles as effective vacancy jumps occurring withçthe given frequency, we can calculate the resulting correç [28] and f . For B2 NiAl, the Monte Carlo calculations lation factors f A B ç ç resulted in f Ni  0.782 and f Al  0.860.[13] In contrast, the tracer correlation factors for Ni and Al atoms in that case were calculated to be fNi  0.445 and fAl  0.022, respectively. Note that fNi is not as small as it was usually anticipated to be for the six-jump-cycle mechanism. This should be taken into account when interpreting the results of experiments such as the Mössbauer effect experiments, which allowed the geometry of individual

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atomic jumps to be established and by which the corresponding correlation factor can be estimated by comparing the local jump rates with the long-range diffusion data.[32] The isotope effect has been measured for both Au and Zn in the B2ordered b-AuZn alloys, with EAu (and, correspondingly, fAu) considerably larger than EZn (fZn) in Zn-rich alloys.[33] (For example, EAu  0.35 and EZn  0.05 in the Au-51.85 at.% Zn alloy.) This resembles the relation between fNi and fAl in B2-NiAl for the six-jump-cycle mechanism[13] and can be explained by the predominant vacancy concentration in the Au sublattice and an increased probability of a reverse jump of a Zn atom that has initiated a six-jump cycle. Thus, the relationship fA  fB in an AB compound cannot be considered an argument against the six-jump-cycle mechanism. Sublattice Diffusion Mechanism. When one of the components forms a lattice structure that enables nearest neighbor jumps through the respective sublattice, random jumps of a vacancy on this sublattice will not affect the order in the compound. An example of this mechanism for the L12 structure is shown in Fig. 4.5. It is important that this mechanism can dominate diffusion of the minority component as well as the majority component. In such a case, a minority atom jumps into the “wrong” sublattice and continues its migration through this sublattice. The sublattice diffusion mechanism has been extensively analyzed.[34, 35] It is obvious that the diffusivities of both components are not coupled by a relation similar to Eq. (1) if the sublattice diffusion mechanism operates. The correlation factors for the sublattice diffusion mechanism in the L12 structure of Ni3Al have been calculated.[35] It was found that fNi  0.689 and fAl can be expressed in a usual way via the vacancy Al atom-exchange frequency w2 and the vacancy escape frequency H:[35] H fAl   . w2  H

(2)

The expression for H in terms of the modified five-frequency model is given by Numakura.[35] Triple-Defect Diffusion Mechanism. This mechanism was proposed by Stolwijk et al. for the B2 compound CoGa.[11] It specifies the migration of a triple defect, which represents a bounded entity composed of two transition metal vacancies and one transition metal atom in an antistructural position. The triple-defect mechanism in CoGa was described to correspond to two nearest neighbor jumps of a Co atom

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Figure 4.5 A scheme of the sublattice diffusion mechanism in the L12 structure of an A3B compound.

and one next-nearest neighbor jump of a Ga atom.[11] The detailed calculations for NiAl predict that the Al atom performs two nearest neighbor jumps instead of one next-nearest neighbor jump.[36] Figure 4.6 shows the triple-defect mechanism with this modification to NiAl, where an inverse triple defect (2VAl  AlNi) was found to exist as an intermediate stage.[36] As a result of the indicated sequence of four jumps of atoms, the triple defect moves, leaving the order in the compound unchanged. Since a correlated sequence of atomic jumps is involved, the diffusivities of both components in the perfectly ordered state are coupled by Eq. (1) with q  13.3.[11] The correlation factors are supposed to be small for the triple-defect diffusion mechanism. They were calculated for NiAl and fNi  0.05 at T  1300 K.[36] fNi was found to depend remarkably on temperature, and the contribution of this temperature dependence to the overall activation enthalpy of Ni diffusion by the triple-defect mechanism amounts to 17 kJ/mol.[36] The triple-defect mechanism is closely related to a divacancy diffusion mechanism, which in its sequence corresponds to the configurations apprearing after the jumps 1 or 3 in Fig. 4.6. Antistructure Bridge Mechanism. This mechanism was originally proposed by Kao and Chang for the B2 structure[12] and was later extended to L12 structures.[37] The ASB mechanism is schematically presented in

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Figure 4.6 A scheme of the triple-defect diffusion mechanism in a B2 structure AB. The modification of this mechanism that is specific for NiAl[36] is shown.

Fig. 4.7(a). As a result of the two indicated jumps, the vacancy and the antistructure atom effectively exchange their positions. Since the vacancy can jump up to the fourth or fifth coordination shell from its initial position (depending on the lattice structure[37]), the resulting large geometrical factor of the ASB mechanism increases its contribution to the diffusivity. It is important to note that the contribution of this mechanism has a percolation effect in the sense that long-range diffusion by the ASB mechanism will occur only if the concentration of the antistructure atoms is sufficiently high. A relatively high critical concentration for a B2 structure was initially estimated from purely geometrical arguments.[12] The Monte Carlo simulation of this process resulted, however, in a smaller value of the percolation threshold, ∼5%.[37] Such an antistructure atom concentration can indeed exist in intermetallics, and the ASB mechanism becomes important for explaining the observed diffusion behavior in Ni aluminides.[36, 38]

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Figure 4.7 A scheme of the antistructure bridge mechanism in a B2 structure and a variant of the ASB mechanism in an L10 structure. As a result, the vacancy VB and the antistructure atom AB exchange their initial positions. In the B2 structure (a), the antistructure atom AB can be situated at any B-atom position from 26 unit cells that neighbor the unit cell with the VB vacancy. In (b), the vacancy VA moves along the “bridges” formed by the antistructure atoms AB.

In the L10 structure of the phase TiAl, other types of the ASB mechanism are of prime importance.[7, 8] One such variant is presented in Fig. 4.7(b). After the indicated two jumps (1  2), the A vacancy moved perpendicular to the A atom layers using an antistructure A atom as a bridge. If a further antistructure atom in a suitable nearest neighbor position is available for the vacancy after its second jump, the next ASB

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sequence may start, as indicated in Fig. 4.7(b). The Monte Carlo calculation of the percolation threshold for the long-range diffusion by this variant of the ASB mechanism yields about 11% of the antistructure atoms as the critical concentration. During the ASB sequence of jumps, only one species of atoms moves (see Fig. 4.7). Therefore, the diffusivities of the two components are not coupled. In a strict sense, the genuine ASB mechanism operates only after the percolation threshold is reached. However, in combination with another mechanism (usually the sublattice diffusion mechanism), the ASB mechanism [for example, jump sequence 1 → 2 in Fig. 4.7(b)] can substantially contribute to long-range diffusion without any percolation threshold.[7, 8, 39] We will therefore use the term ASB mechanism in such cases as well, referring to the specific sequence presented in Fig. 4.7. Other Diffusion Mechanisms. Several other mechanisms, which may be relevant in some specific cases, were proposed for ordered intermetallic compounds. The next-nearest neighbor jump mechanism was shown to correspond to the lowest activation energy of single Ni vacancy migration in Al-rich NiAl.[22] The divacancy mechanism, with both vacancies belonging to the same sublattice in NiAl, was considered by Divinski et al.[40] After the given sequence of four atomic jumps, the initial order is completely restored and the divacancy has moved by one step.

4.4

Experimental Results on Bulk Diffusion in Ordered Aluminides

A common feature of all aluminides is that there are practically no direct tracer measurements of Al self-diffusion in these compounds. This problem is related to the very high price and the low specific activity of the only available 26Al radioisotope and to the difficulty of avoiding oxidation of this isotope during a diffusion study. The only directly measured Al-diffusion data[41] are considered to be unreliable. There exist, however, two approaches to overcome this difficulty. First, by combining interdiffusion and transition metal tracer diffusion ∼ data (D and D*TM, respectively), the diffusivity of the Al component D*Al in a binary system can be evaluated by applying the Darken-Manning formalism,[42, 43] neglecting the volume effects: ∼

D  (NAlD*TM  NTMD*Al)ΦS.

(3)

Here, NTM and NAl are the mole fractions of the transition metal element and Al in the compound, respectively; Φ is the thermodynamical factor;

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and S is the vacancy wind factor. Φ is determined by independent activity measurements, and S is usually assumed to be about unity. For the random alloy model, Manning has shown that: 1 1  S  , f0

(4)

where f0 is the geometrical correlation factor for the given lattice.[43] (Chapter 1 of this book presents a detailed discussion of interdiffusion and the Darken-Manning equation.) The applicability of Eq. (3) was experimentally varified for several compounds.[44, 45] Thus we also apply Eq. (3) to interdiffusion in alu* is much smaller than D*TM, the determination minides. However, when DAl * due to the experimental uncertainties in the of DAl becomes unreliable ∼ determination of D and Φ. This fact should be taken into account when analyzing interdiffusion data. A series of theoretical papers derived that the Darken-Manning approach is applicable to intermetallic compounds with the B2, L12, and D03 lattices at a smaller degree of order.[46, 47] Simultaneously, it was found that the correlated diffusion mechanisms in well-ordered intermetallic compounds result in values of S that do not necessarily fall into the limits imposed by Eq. (4). For example, S can be as low as 0.42 for the B2 compounds when only the six-jump-cycle mechanism operates.[48] Likewise, [49] Therefore, the upper value of S is not bound to f 1 0 in the L12 structures. each factor has to be thoroughly analyzed when applying Eq. (3) to a given intermetallic compound. In an alternative approach, we can use tracers, which substitute Al in the intermetallic compound (such as Ga, In, or Ge), as a surrogate to the Al tracer. Both approaches are considered in the present investigation.

4.4.1 Ni3Al 4.4.1.1 Self-Diffusion Ni Diffusion. Ni self-diffusion in Ni3Al was intensively investigated in several experimental studies.[50–54] The corresponding Arrhenius dependencies are presented in Fig. 4.8(a). There is good consistency between the experimental results for the temperature interval T  1100 K. At lower temperatures, the results of Hoshino et al.[51] show an upward deviation from the otherwise linear Arrhenius dependence. Note that an upward deviation was also indicated by Shi et al.,[52] although at a somewhat

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T [K] 1600

1400 1300 1200

1100

1000

Ni diffusion: Frank et al. Bronfin et al. Shi et al. Hoshino et al.

-13

10

-14

10

-15

~ * 2 -1 D , D [m s ]

10

Al (Fujiwara et al.)

-16

10

-17

10

Al (Ikeda et al.) -18

10

interdiffusion: Watanabe et al. Fujiwara et al.

-19

10

Ikeda et al.

-20

10

6

7

8 T

-1

9 -4

10

-1

[10 K ]

(a) Self-Diffusion in Stoichiometric Ni3Al Figure 4.8 Arrhenius plot of self-diffusion and solute diffusion in stoichiometric Ni3Al. In (a), the results of interdiffusion measurements[57–59] as well as the calculated D*Al (dashed line[57] and full circles[59]) are compared with Ni diffusion.[50–53]

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T [K] 1600 -13

10

1400 1300 1200

1100

1000

Al Ge2

-14

10

Fe

-15

Mo

-16

10

Ge1

D

*

2 -1

[m s ]

10

-17

10

Ga

-18

10

-19

10

Ni Ti

Nb

-20

10

6

7

8 T

-1

9 -4

10

-1

[10 K ]

(b) Solute Diffusion in Stoichiometric Ni3Al Figure 4.8(b) Solute diffusion is presented for Ga,[38] Ge (Ge1[38] and Ge2[61]), Nb,[63] Ti,[63] Mo,[61] and Fe.[64]

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higher temperature, about 1200 K. This feature was explained[52] by the contribution of short-circuit diffusion, since polycrystalline material was used. Frank et al.[53] thoroughly measured Ni self-diffusion, combining the tracer technique at higher temperatures and the SIMS analysis at lower ones in poly- and single-crystalline Ni3Al, respectively. No curvature of the Arrhenius plot was established [see Fig. 4.8(a)], confirming the explanation given by Shi et al.[52] These data indicate that only one diffusion mechanism operates in the temperature interval under consideration. This mechanism is commonly identified with the Ni-sublattice diffusion mechanism.[52, 53] According to this mechanism, a Ni vacancy performs random jumps in the Ni sublattice with the coordination number 8 (see Fig. 4.5). Some controversy still exists about the compositional dependence of Ni self-diffusion in Ni3Al (Fig. 4.9). Hoshino et al.[51] have found a shal* at the stoichiometric composition below T  1100 K. low minimum of DNi [52] Shi et al. also observed such a minimum in D*Ni, but around 76 at.% Ni and, again, at the lowest temperature of their investigations, about 1200 K. At higher temperatures, the variation of D*Ni with composition in all investigations was within experimental uncertainities. These fine features resulted in a maximum activation enthalpy QNi at about 76 at.% Ni[52] (Fig. 4.9). Since Ni grain boundary diffusion in Ni3Al shows a deep minimum at the stoichiometric composition, Frank and Herzig[55] suggested that short-circuit diffusion via grain boundaries affected the low-temperature data of Hoshino et al.[51] and Shi et al.,[52] and produced a slight minimum in the bulk diffusion data. Comparing results of different investigations, Fig. 4.9 suggests that the change of the activation enthalpy QNi with composition in Ni3Al is within experimental uncertainties. In contrast, a minimum in D*Ni was found in a recent Monte Carlo study of Ni diffusion in Ni3Al at the stoichiometric composition[56] due to the existence of Al and Ni antistructure atoms in Ni- and Al-rich alloys, which enhance Ni selfdiffusion by lowering the energy barriers for the Ni vacancy jumps. In that approach, pair interaction potentials were used and no lattice relaxation was taken into account. For additional insight into this problem, we simulated self-diffusion in Ni3Al using the Voter and Chen EAM potentials.[24] The calculations show that there is a negligible effect of adjacent Ni and Al antistructure atoms on the energy barriers for Ni vacancies performing jumps via the Ni sublattice. (The energy barriers change by about 10 kJ/mol, depending on the particular configuration.) Thus, the vacancy jump rates are nearly the same, regardless of the composition of Ni3Al. Calculations of the Ni vacancy concentration [Fig. 4.3(c)] and the effective formation energies also resulted in very similar values for Al-rich, stoichiometric, and Ni-rich compounds. Simultaneous Monte Carlo simulations of the correlation effects demonstrated that the correlation factor for the Ni

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Figure 4.9 Activation enthalpy QNi of Ni self-diffusion in Ni3Al as a function of composition (open circles,[52] triangle up,[53] triangles down,[51] and squares[54]).

sublattice diffusion mechanism across the stoichiometry changes negligibly and equals about 0.689. These calculations support the conclusion that Ni self-diffusion reveals only a marginal compositional dependence in Ni3Al.[53] From the experimental point of view, Ni diffusion in the Ni3Al phase can be presented by a unique Arrhenius line with the parameters given in Table 4.1. Al Diffusion. 26Al radiotracer diffusion in Ni3Al was measured[41] at temperatures between 1273 and 1473 K. Unfortunately, the absolute values of Al diffusion are considered to be unconvincing because there is a

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Table 4.1. Arrhenius Parameters of Self-Diffusion and Solute Diffusion in Ni3Al

Tracer Ni Ni Ni Al* Ga Ge Ge Nb Ti Mo Fe *

Composition (at.% Ni) 75 74.7 75.9 75 75.9 75.9 75 75.9 75.9 75.1 75.1

D0 (m2 s
1) 2

1.5 10 1.0 104 3.1 104 8 103 7.8 102 1.0 101 3.6 101 2.6 101 8.6 101 2.3 102 1.3 102

Q (kJ/mol) 347 303 302 340 363 369 379 476 425 493 335

Ref. 51 54 53 63 63 61 63 63 61 64

Estimation from the Al intrinsic diffusion data of Fujiwara and Horita[59]

substantial deviation of the simultaneously measured Ni self-diffusion results from the data of Frank et al.[53] Incomplete consideration of shortcircuit diffusion in polycrystalline samples may be one possible reason for such deviations. However, the ratio of Al and Ni diffusivities measured by Larikov et al.[41] is likely to be more reliable, and we can anticipate that Al and Ni diffuse with similar rates in Ni3Al in this temperature interval. ∼ Al was The interdiffusion coefficient D accross the phase field of Ni ∼ 3 measured by two groups.[57, 58] The temperature dependence of D for stoichiometric Ni3Al is presented in Fig. 4.8(a). Intrinsic diffusivities of Al and Ni were determined between 1425 and 1523 K.[59] Analyzing the relative positions of the Kirkendall and Matano planes, Fujiwara and Horita concluded that the ratio of Al and Ni tracer diffusivities is very close to unity in this temperature interval,[59] supporting the finding of Larikov et al.[41] In Fig 4.8(a), the estimated Al diffusion coefficients are compared with the Ni self-diffusivity. On the other hand, the Al diffusivity was estimated by Ikeda et al.[57] using the Darken-Manning equation [Eq. (3); see Fig. 4.8(a)]. Their smaller values of D*Al with respect to Fujiwara and Horita’s results[59] may stem from some uncertainty involved in the calculations of the product Φ S. Similar to Ni tracer diffusion, the interdiffusion coefficient in Ni3Al reveals only a marginal compositional dependence, which remains within

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the experimental uncertainties.[59] From this we conclude that the Al diffusivity does not markedly depend on composition in Ni3Al.

4.4.1.2 Solute Diffusion Since Ge and Ga occupy the Al sublattice in Ni3Al,[60] these tracer atoms can be used as the substitute for the Al diffusion.[38, 61] The available data are presented in Fig. 4.8(b). Their diffusivities follow very similar and straight Arrhenius dependencies in Ni3Al, with an activation enthalpy of about 60 kJ/mol larger than that of Ni self-diffusion. It is important that Ga and Ge diffusion in Ni3Al is very close to the Al diffusivity estimated from the interdiffusion data[59] [Fig. 4.8(b)]. We conclude that these solute atoms are indeed useful to simulate Al diffusion in Ni3Al. Figure 4.8(b) suggests D*Ga D*Ni at lower temperatures. This relation is reversed, however, at higher temperatures, T  1350 K, where D*Ga  * . This trend can also be anticipated for Al diffusion [see Fig. 4.8(b)]. DNi Such behavior cannot be understood in terms of the Ni sublattice diffusion mechanism of the minority component, as suggested by Numakura et al.[35] Indeed, the effectiveness of diffusion by this mechanism strongly depends on the probability of finding an Al (or Al-substitute) atom on the wrong Ni sublattice. Consequently, Al should diffuse slower than Ni, in agreement with the so-called Cu3Au rule.[62] However, it was shown by Divinski et al.[38] that the ASB diffusion mechanism provides an additional and strong contribution for Al atoms comparable to the Ni sublat* can approach the D*Ni values at high tice diffusion mechanism and that DAl temperatures, in accordance with the experimental findings of Fujiwara and Horita[59] [Fig. 4.8(b)]. Tracer diffusion of Ti and Nb was investigated[63] [Fig. 4.8(b)]. In contrast to Ge and Ga diffusion, Ti and Nb reveal a distinctly smaller diffusivity with a larger activation enthalpy. From the direction and extent of the homogeneity range of the L12 phase in a ternary region, the site preference for Ti and Nb in Ni3Al was estimated.[60] Because Ti and Nb atoms mainly dissolve on Al positions in the L12 lattice, a diffusion process similar to that of Al is to be expected. The difference in the activation enthalpies of selfdiffusion and diffusion of such solutes is related to the formation energy of the solute atom at an antistructural position in the Ni sublattice.[53] In Fig. 4.8(b), the tracer diffusivities of Mo[61] and Fe[64] in Ni3Al are also plotted, as functions of the inverse temperature. The solute diffusion behavior in Ni3Al can be well systemized in terms of an interplay of intrasublattice (the Ni sublattice mechanism) and inter-sublattice (the ASB mechanism) jumps.[61] The Arrhenius parameters of self- and solute diffusion in Ni3Al are summarized in Table 4.1.

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4.4.2 Ti3Al 4.4.2.1 Self-Diffusion Ti Diffusion. Self-diffusion and solute diffusion in Ti3Al was comprehensively analyzed in a recent review.[8] In an investigation of Ti selfdiffusion in the phase a2-Ti3Al,[65] several compositions were studied: 75, 72, 68, and 65 at.% Ti. Within experimental error, the diffusivities were very similar, with a slight increase of D*Ti with increasing Al content at the lowest temperature investigated. The change of Al content produced only a marginal effect in the activation enthalpy of self-diffusion, QTi, although the tendency of an increase of QTi with increasing Al content was indicated.[65] As a result, one master Arrhenius dependence for Ti self-diffusion in a2-Ti3Al was suggested [Fig. 4.10(a)] with the Arrhenius parameters given in Table 4.2. Ti self-diffusion in Ti3Al can be interpreted within the framework of the Ti-sublattice diffusion mechanism.[65] The experimental data suggest that the compositional Ti or Al antistructure atoms change the energy barriers for vacancy jumps on the Ti sublattice only slightly. Since the Ti vacancy formation enthalpies are also very similar for the stoichiometric and Al-rich alloys,[8, 19] the weak compositional dependence of D*Ti in Ti3Al is not unexpected. Al Diffusion. Interdiffusion in the a2-Ti3Al-phase field was studied,[65, 66] with good agreement of the deduced diffusion data, which practically overlap in Fig. 4.10(a). Applying the Darken-Manning analysis with S  1 [Eq. (3)], Fig. 4.10(a) presents the calculated Al diffusion coefficient D*Al as a function of temperature. The thermodynamic factors required for such calculations were calculated by the CALPHAD Table 4.2. Arrhenius Parameters of Self-Diffusion and Solute Diffusion in Ti3Al

Tracer Ti Al* Ga Fe Ni Nb *

D0 (m2 s
1) 5

2.24 10 2.3 101 6.3 105 2.0 103 1.8 105 3.2 104

Calculation from the Darken-Manning equation

Q (kJ/mol)

Ref.

288 394 315 277 195 339

65 65 68 70 70 70

-15

-17

-16

7 -1

-4

-1

[10 K ]

9

Ti

Sprengel et al.

(a) Self-Diffusion in Ti3Al

T

8

Al

1100

Rüsing et al.

interdiffusion:

1200

T [K]

10

-20

10

-19

10

-18

10

10

-17

-16

10

-15

-14

10

-13

10

-12

10

8

T

-1

9

Ti

-1

10 [10 K ]

-4

Ga

Fe

1000

T [K] 1100

11

∼

Ni

900

(b) Solute Diffusion in Ti3Al

7

Nb

Al

1400 1300 1200

12

Figure 4.10 The Arrhenius plots of self-diffusion and solute diffusion in Ti3Al. In (a), the interdiffusion data D [66] and the estimated Al diffusivity[65] are plotted in comparison to Ti diffusion.[65] In (b), diffusion data for Ti,[65] Al,[66] Ga,[68] and Nb, Fe, and Ni[70] diffusion are compared.

10

-19

10

-18

10

~ * 2 -1 D , D [m s ]

10

1300

2 -1

[m s ]

10

1400

6:36 PM

*

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197

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DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

method.[67] The weak compositional dependence of D*Al in Ti3Al is within experimental error, but a slight increase with increasing Al content is indicated.[8] The activation enthalpy QAl of Al diffusion ≈400 kJ/mol (Table 4.2) turned out to be fairly large. The analysis of the experimental data suggests that the large value of QAl may be related to uncertainties in the D*Al determination via the Darken relation, especially at low temperatures.[68]

4.4.2.2 Solute Diffusion Since Ga occupies the Al sublattice in Ti3Al,[69] the study of Ga diffusion is meaningful to obtain additional information on diffusion of an Alsubstituting solute in Ti3Al.[68] The measured Ga diffusion coefficients are presented in Fig. 4.10(b) as a function of temperature. The figure indicates that the absolute values of D*Ga are similar to those of D*Al, but the activation enthalpy of Al diffusion is distinctly larger (see Table 4.2). Ga diffusion in Ti3Al was interpreted in terms of the Ti-sublattice diffusion mechanism of the minority component.[68] Thus, the excess of QGa with respect to QTi is related to the formation enthalpy of Ga atoms in antistructure positions on the Ti sublattice.[68] Due to the predominant vacancy concentration in the Ti sublattice [Fig. 4.3(a)], Al is also expected to diffuse predominantly as anti-structure atoms via the Ti sublattice. Thus we regard the measured QGa value as realistic to represent QAl in Ti3Al. Information is limited on diffusion of other solutes in a2-Ti3Al. Diffusion of Nb in Ti3Al, studied[70] within the composition range from 27 to 33 at.% Al, was found to be practically independent of composition. The corresponding temperature dependence is presented in Fig. 4.10(b). Nb atoms occupy Ti sites in Ti3Al.[69] Therefore, their diffusion is dominated by exchanges with vacancies on the Ti sublattice (Ti-sublattice diffusion mechanism). In view of the extremely fast diffusivity in a-Ti, diffusion of Fe and Ni was investigated in Ti3Al.[70] Only a negligible compositional dependence was observed, similar to the diffusion of Ga and Nb. The temperature dependencies obtained are presented in Fig. 4.10(b). This diagram shows that Fe and Ni are also fast diffusers in Ti3Al, since their diffusivities exceed that of Ti self-diffusion by several orders of magnitude. Fast diffusion of Fe and Ni in Ti3Al was explained[8, 70] by the dissociative diffusion mechanism; that is, the diffusing atoms occupy both interstitial and substitutional sites. In such a case, the effective diffusion coefficient D is given by: ci cs c D   Di   Ds  i Di. ci  cs ci  cs cs

(5)

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Here Di (Ds) is the interstitial (substitutional) diffusion coefficient, and ci (cs) is the interstitial (substitutional) solubility of the solute atoms. Equation (5) was simplified under the plausible assumptions that ci

cs and ciDi  csDs. Since ci and cs are temperature-dependent, the activation enthalpy of the dissociative diffusion can be remarkably larger than that of pure interstitial diffusion. This conclusion agrees with the observation that the activation enthalpies of Fe and Ni diffusion (277 and 195 kJ/mol, respectively), in comparison with the Ti self-diffusion activation enthalpy (288 kJ/mol), are not as small as expected for the pure interstitial diffusion mechanism.

4.4.3 TiAl 4.4.3.1 Self-Diffusion Ti Diffusion. The self-diffusion behavior in Ni3Al and Ti3Al follows almost ideal linear Arrhenius dependencies (see Figs. 4.8 and 4.10, respectively), a behavior that is typical for the majority of intermetallic phases. From a simplest point of view, such a behavior is somewhat unexpected in view of the variety of point defects and the number of possible diffusion mechanisms in ordered intermetallics. However, the experimental data suggest that in many typical intermetallic phases, there is a given diffusion mechanism that produces the main contribution to self-diffusion and operates dominantly over a wide temperature interval. Astonishingly, Ti self-diffusion in g-TiAl clearly revealed a non-Arrhenius character[7] [see Fig. 4.11(a)]. The Ti diffusivity follows an Arrhenius line up to about T  1470 K, with a subsequent upward deviation to larger D*Ti values at higher temperatures. This effect was found to be almost independent of the composition in TiAl.[7] Table 4.3 presents the activation enthalpy QTi of Ti-self diffusion in TiAl calculated by the Arrhenius fitting within the temperature interval 1150 to 1470 K. The pronounced curvature of Ti self-diffusion at T  1470 K is explained by the additional effect of a second diffusion mechanism that operates parallel to the Ti-sublattice mechanism.[7] In view of the high concentration of Ti antistructure atoms, even in Al-rich compositions [see Fig. 4.3(b)], the antistructure bridge mechanism [Fig. 4.7(b)] was proposed to produce this additional contribution. The most important feature of the ASB mechanism, when operating in the L10 structure of TiAl, is that it reveals a nonpercolative nature and thus contributes at any concentration of the Ti antistructure atoms. This effect results from the interdependence of the ASB mechanism with the simultaneous contribution of the Ti-sublattice diffusion mechanism. Thus, in TiAl, the situation is very different from that in the B2 compounds, where the percolation threshold

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T [K] 1700 1600 1500

1400

1300

1200

-12

10

Ti in Ti44Al56 Ti in Ti46Al54 Ti in Ti47Al53

-13

10

-14

-15

10

*

2 -1

[m s ]

10

D

||

D Ti

-16

10

-17

10

-18

⊥ Ti

10

D 6

7 T

-1

8 -4

9

-1

[10 K ]

(a) Ti Component in TiAl Figure 4.11 The Arrhenius presentation of Ti and Al component and solute diffusion in TiAl. In (a), Ti self-diffusion in polycrystalline TiAl alloys of the indicated compositions[7] is plotted in comparison to Ti diffusion along (D
Ti) and perpendicular (D Ti⊥ ) to the Ti layers in the L10 structure of TiAl.[71] In (b), the interdiffusion data,[72] estimated Al[7] and Ga[68] diffusivities are shown. In (c), Ti;[7] Nb, Zr, Fe, and Ni;[39] and Co and Cr[74] diffusion is presented.

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201

T [K] 1700 1600 1500

1400

1300

1200

-12

10

Ti in Ti44Al56 -13

10

-14

* ~ 2 -1 D , D [m s ]

10

~

D (Sprengel et al.) -15

10

Ga

-16

10

-17

10

Al -18

10

6

7 T

-1

8 -4

-1

[10 K ]

(b) Al Component in TiAl Figure 4.11 (Continued)

9

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T [K] 1600 1500 1400 1300

-12

10

Zr -13

10

1200

1100

Cr (Herzig et al, 2001)

-14

10

Ni -15

Nb Ti

2 -1

[m s ]

10

-16

10

D

*

Co -17

10

Fe

-18

10

-19

10

Cr (Lee et al,1993) -20

10

6

7

8 T

-1

-4

9 -1

[10 K ]

(c) Solute Diffusion in TiAl Figure 4.11 (Continued)

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203

Table 4.3. Arrhenius Parameters of Self-Diffusion and Solute Diffusion in TiAl*

Tracer Ti Al** Ga Co Cr Fe Ni Nb Zr Cr

D0 (m2 s
1) 6

3.2 10 2.1 102 4.4 105 1.1 103 4.4 103 2.7 106 1.8 102 1.5 104 6.4 103 5.2 103

Q (kJ/mol)

Ref.

261 358 293 318 350 252 340 324 348 332

7 7 68 74 73 39 39 39 39 39

* Activation

enthalpies of Ti And Fe diffusion were calculated by the fit of experimental points below T  1470 K. ** Calculation from the Darken-Manning equation

for the ASB mechanism inevitably has to be reached to produce a longrange diffusional contribution from this mechanism, as shown by Divinski and Larikov.[37] Note that in TiAl we need to distinguish between different types of ASB mechanisms that start from either a Ti or an Al vacancy.[8] TiAl has the layered L10 structure that, generally, should result in an anisotropy of self-diffusion. Diffusion measurements were carried out on polycrystalline samples that concealed the anisotropy effects.[7] The anisotropy of Ti self-diffusion in single-crystalline TiAl alloys was recently studied;[71] results are presented in Fig. 4.11(a). As can be expected, Ti diffusion along Ti layers in the L10 structure is faster than perpendicular to this direction. The geometrical consideration of the L10 structure [see Fig. 4.2(d)] suggests that the Ti-sublattice diffusion mechanism should result in a strong anisotropic contribution, whereas the ASB mechanism, which occurs by intersublattice jumps of the Ti atoms, corresponds to an almost isotropic mass transport. As the temperature increases, the contribution of the ASB mechanism increases and the anisotropy of Ti diffusion decreases.[71] This increase in the diffusional contribution of the ASB mechanism with increasing temperature was already manifested in the nonlinear Arrhenius dependence of Ti selfdiffusion.[7] The relevant Arrhenius parameters are listed in Table 4.3. Al Diffusion. Interdiffusion in g-TiAl was measured using singlephase diffusion couples.[72] These data were used to deduce the Al

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diffusion coefficients D*Al.[7] The corresponding Arrhenius dependencies are presented in Fig. 4.11(a). Using the thermodynamic factor Φ calculated by the CALPHAD method[67] and applying the Darken-Manning equation [Eq. (3)] with S  1.14, the Al tracer diffusion coefficients were deduced.[7] The resulting activation enthalpy of Al diffusion is fairly large, QAl  358 kJ/mol.

4.4.3.2 Solute Diffusion Al diffusion in TiAl was examined using Ga as an Al substitute.[68] The corresponding diffusion data are presented in Fig. 4.11(b). It is observed that D*Ga D*Ti in the whole temperature interval of the investi* turns out to be smaller than D*Ti only at lower temgations, whereas DAl peratures. However, the absolute values of D*Al are very similar to the Ga diffusivity. The observed difference in QAl and QGa was related to the uncertainty in the determination of D*Al from the interdiffusion data via the Darken relation.[68] Since Ga occupies exclusively the Al sublattice in TiAl,[69] the diffusion process of Ga atoms consists of jumps from the Al to the Ti sublattice, migration through it, and reverse jumps on the Al sublattice.[68] The difference between QGa and QTi in the low-temperature interval, where the Ti-sublattice diffusion mechanism dominates for Ti atoms, is therefore mainly attributed to the formation enthalpy of Ga atoms as antistructure atoms on the Ti sublattice. Results of tracer solute diffusion in TiAl are also available for Cr,[39, 73] Co,[74] Nb,[39] Zr,[39] Fe,[39] and Ni.[39] These data are plotted in Fig. 4.11(c). Diffusion of the solutes studied deviates by less than one order of magnitude from Ti self-diffusion. This behavior corresponds to the substitutional character of these solutes and to the vacancy mechanism of diffusion. Nb preferentially occupies Ti sites and is a slow diffuser in TiAl. The Ti-sublattice diffusion mechanism was suggested for Nb atoms,[39] and the slow diffusivity is most likely related to a certain repulsive vacancy-Nb atom interaction. Fe diffusion in TiAl presents a very interesting case because it demonstrates a strong nonlinear Arrhenius dependence [see Fig. 4.11(c)]. The magnitude of the Fe diffusion coefficients is similar to that of Ti selfdiffusion. Moreover, the curvatures of the Ti and Fe Arrhenius plots and the activation enthalpies in the lower temperature intervals are also quite similar. The behavior of Fe diffusivity was explained[39] in relation to the equal preference of Fe atoms to occupy Ti or Al sites in TiAl.[69] In such a case, Fe atoms located on the Ti sublattice mainly dominate Fe diffusion at lower temperatures, and the ASB mechanism, which becomes progressively

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205

important at higher temperatures, produces a curvature of the Arrhenius dependence.[39] Similarly to Fe, the Ni diffusivity in TiAl is close to the Ti selfdiffusivity in the temperature range investigated. This finding contrasts with their diffusion behavior in Ti3Al. When changing from a-Ti to a2-Ti3Al and, finally, to g-TiAl, the mechanism of Fe and Ni diffusion obviously changes from interstitial to dissociative, and then to substitutional diffusion. This change is accompanied by a remarkable decrease in the diffusion rates: While Fe and Ni mobilities are seven to eight orders of magnitude faster than Ti in a-Ti,[75, 77] they exceed Ti self-diffusion in a2-Ti3Al only by two to four orders of magnitude [Fig. 4.10(b), and all of them diffuse with similar magnitude in g-TiAl (Fig. 4.12). This behavior was explained by the different type of atomic arrangements forming interstitial sites in the materials under consideration.[39] In the D019 structure of Ti3Al, there are two different interstitial sites: One of them is built up exclusively by Ti atoms; the second type is formed by two Al and four Ti atoms. In the L10 structure of TiAl, all interstitial sites have both Al and Ti atoms as environment. The observed diffusion behavior suggests that the presence of Al atoms strongly decreases the interstitial solubility of Fe and Ni atoms at such positions.[39]

4.4.4 NiAl 4.4.4.1 Self-Diffusion Ni Diffusion. The B2 structure of NiAl is very interesting from the theoretical point of view, since all nearest neighbor jumps are the jumps between different sublattices in this structure [see Fig. 4.2(b)]. Unlike the other transition-metal-rich Ni and Ti aluminides, NiAl exhibits a tripledefect type of disorder. [The triple defects (two Ni vacancies and one Ni antistructure atom) reveal the lowest formation energy between the point defects, the thermal formation of which does not change the composition.] Moreover, the structural Ni vacancies, which are available in NiAl on the Al-rich side of compositions, are generally thought to affect significantly Ni self-diffusion in this compound.[78, 79] Ni diffusion in NiAl was measured on polycrystalline NiAl alloys[78] and on single crystals.[36] These results are presented in Fig. 4.13. Note that at T  1500 K, an upward deviation from the otherwise straight Arrhenius dependencies was observed in all compositions.[36] (This is not indicated in Fig. 4.13 to avoid overloading of the figure. This curvature is analyzed below.) A qualitatively very different diffusion behavior was observed in

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Figure 4.12 Comparison of Fe and Ni diffusion with Ti self-diffusion in a -Ti,[75–77] a2-Ti3Al,[65, 70] and g -TiAl.[7, 39] Tm is the melting temperature of corresponding material.

these two investigations, especially in Al-rich compositions. First, Ni dif* measured by Frank et al.[36] are generally smaller by a factor fusivities DNi of two to five than those measured by Hancock and McDonnell.[78] Second, * at the stoichiometric composition was postua deep minimum in DNi [78] * in Al-rich and stoichiometric NiAl lated, whereas similar values of DNi alloys were measured by Frank et al.[36] [see Fig. 4.14(a)]. Finally, a small activation enthalpy of Ni diffusion in Al-rich compositions was determined

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Figure 4.13 The Arrhenius diagram of Ni diffusion in different NiAl alloys (in at.% Ni). The results of Hancock and McDonnell[78] (dashed lines) and Frank et al.[36] (solid lines) are compared.

by Hancock and McDonnell to be about 180 kJ/mol,[78] whereas Frank et al. obtained a value of about 290 kJ/mol.[36] The activation enthalpy Q of Ni self-diffusion in NiAl calculated by the Arrhenius fit in the interval 1050 K T 1500 K[36] is given in

(m s )

2 -1

-1354

48

1273 K

1523 K

52 44

42

58

60

200

250

300

350

48

b)

46

50

52

54

In (Minamino et al.)

Ni (Frank et al.)

48

56

44

(b) Activation Enthalpies

56

40

(a) Diffusion Coefficients

54

Ni (Frank et al.)

In (Minamino et al.)

46

Composition (at.% Al) 50

Composition (at.% Ni)

52

48

52

Composition (at.% Ni)

50

50

Q (kJ/mol)

58

42

Figure 4.14 Composition dependence of the diffusion coefficients and activation enthalpies of Ni[36] and In[87] diffusion in NiAl.

10 46

-17

-16

10

-15

10

10

-14

10

Composition (at.% Al)

6:36 PM

*

208

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Table 4.4. Arrhenius Parameters of Self-Diffusion and Solute Diffusion in NiAl

Tracer Ni

In

Composition at.% Ni 46.8 50.0 51.8 54.6 56.6 47.7 49.5 50.6 52.0 55.3

D0 (m2 s
1) 5

2.3 10 3.5 105 4.8 105 4.4 105 1.0 106 1.6 104 4.6 104 2.0 103 9.8 104 1.1 103

Q (kJ/mol)

Ref.

287 290 288 278 231 271 311 336 322 311

36 36 36 36 36 87 87 87 87 87

Fig. 4.14(b) as a function of composition. Astonishingly, Q results in the constant value of about 290 kJ/mol at compositions lower than 53 at.% Ni, but substantially decreases in alloys with larger Ni concentrations approaching the value of 230 kJ/mol in Ni56.6Al43.4. The Arrhenius parameters of Ni self-diffusion in representative Al-rich, stoichiometric, and Ni-rich NiAl alloys are summarized in Table 4.4. Hancock and McDonnell’s results[78] agree qualitatively well with recent calculations of the activation enthalpy of Ni diffusion by nextnearest neighbor jumps of Ni vacancies over the Ni sublattice.[22] In the case of Al-rich compositions, where constitutional Ni vacancies exist, the activation enthalpy QNi should be equal to the migration enthalpy of Ni vacancies, which was calculated to about 200 kJ/mol.[22] It was shown that along with the small activation enthalpy, the next-nearest neighbor jumps of Ni atoms entail a large and negative migration entropy, which decreases remarkably the contribution of this mechanism at elevated temperatures.[36] Moreover, the next-nearest neighbor jumps of Ni atoms would produce a continuous increase of D*Ni with increasing Al content on the Alrich side, which was not observed in the recent experiments on singlecrystalline NiAl alloys. The difference between results[36, 78] was proposed[36] to stem from the difference in the thermal equilibration procedure of the samples and/or from short-circuit diffusion, which, according to our recent experiments on grain boundary diffusion of Ni in polycrystalline NiAl samples,[80] could have affected Hancock and McDonnell’s low-temperature results.[78]

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Possible diffusion mechanisms in NiAl were analyzed[36, 40] for dependence on the composition. The results suggest that the triple-defect mechanism, which is characterized by a constant activation enthalpy of about 300 kJ/mol independent of the composition (Fig. 4.6), most likely is the dominating diffusion process in NiAl.[36] This result agrees perfectly with the recent experiments [see Fig. 4.14(b)]. The observed increase of D*Ni and the decrease of QNi at compositions with larger Ni content correspond to the activation of the ASB mechanism after reaching the percolation threshold at about 55.5 at.% Ni.[36] These results do not exclude an additional and strong contribution of next-nearest neighbor jumps of Ni atoms at lower temperatures in Al-rich compositions expected by Mishin and Farkas.[22] The relatively small activation enthalpy would favor such a diffusion process.[22] The upward deviation of D*Ni at T  1500 K from the otherwise linear Arrhenius dependence reported by Frank et al.[36] correlates with recent results of differential dilatometry measurements in NiAl alloys,[81] which reveal a nonlinearity in the Arrhenius plot of the vacancy concentration in the high temperature range. Al Diffusion. At present, there exist no directly measured Al tracer diffusion data in NiAl. However, interdiffusion in NiAl was investigated.[82, 83] A∼ substantial compositional dependence of the interdiffusion coefficient D was deduced,[83] characterized by a deep minimum around the stoichiometric composition, [see Fig. 4.15(a)]. Using the thermodynamic data of Steiner and Komarek[84] and the Darken-Manning equation, the contribution of Ni tracer diffusion to the interdiffusion coefficient can be estimated, rewriting Eq. (3) as: ∼

∼

∼

D  D(Ni)  D(Al),

(6)

where ∼

D(Ni)  NAlD*Ni Φ S

and

∼

D(Al)  NNi D*Al Φ S. ∼

(7)

Assuming S  1, the results of this estimation of D(Ni) are also presented in Fig. 4.15(a). A paradoxical difference between the measured inter∼ ∼ and D is observed, especially around stoichiodiffusion coefficient D (Ni) ∼ ∼ than D(Ni) metric NiAl. D is by about two orders of magnitude smaller ∼ in Ni50Al50. Any contribution of the Al tracer diffusion, D(Al), further increases this difference. As already noted, the value of the vacancy wind factor S may not fall into the limits predicted by Manning [Eq. (4)], especially in well-ordered compounds. Bearing this in mind, we can tentatively estimate the values

(m s )

2 -1

52

54

44

56

Kim et al. ~ D(Ni)

Composition (at.% Ni)

50

46

(a) Interdiffusion in NiAl Alloys

48

48

T = 1273 K

50

Composition (at.% Al)

58

42

S 46

-2

10

-1

10

0

10

54

50

52

54

Composition (at.% Ni)

50

46

56

44

(b) Vacancy Wind Factors in NiAl Alloys

48

48

Composition (at.% Al)

T = 1273 K

52

58

42

Figure 4.15 Comparison of experimentally measured[83] and calculated [by Eq. (7)] interdiffusion coefficients in NiAl (a). In (b), the vacancy wind factor S estimated from Eq. (8) (dashed line) is compared with the values of S calculated for the triple-defect diffusion mechanism[85] (full circles) and for the next-nearest neighbor jumps of Ni atoms (dotted line).

10 46

-17

-16

52

3:28 PM

10

-15

10

10

-14

54

11/30/04

~ D

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of S that would formally satisfy Eq. (3): ∼

D S ∼ . D(Ni)

(8) ∼

Although such calculations neglect the contribution of D(Al), a rough estimate is obtained.[85] In Fig. 4.15(b), this hypothetical value of S is plotted against composition. On the other hand, the vacancy wind factor S was calculated by the Monte Carlo approach for a number of possible diffusion mechanisms in NiAl.[85] The results for the next-nearest neighbor jump mechanism and the triple-defect mechanism, for example, are presented in Fig. 4.15(b). The unusually small values of S deduced from Eq. (8) are qualitatively reproduced by the triple-defect diffusion mechanism in NiAl. In the perfectly ordered B2 structure of NiAl, the triple-defect mechanism would result in S  0 at stoichiometry. It is the existence of thermal defects that gives rise to S
0 in such conditions.[85] Correspondingly, S is temperaturedependent in Ni50Al50 and also contributes to the activation enthalpy of interdiffusion to about 70 kJ/mol.[85] In the nonstoichiometric alloys, the triple-defect mechanism gives rise to normal values of the vacancy wind factor S ≈ 1. In summary, the estimate of S [Eq. (8)] supports the conclusion about the dominance of the triple-defect diffusion mechanism, at least in nearstoichiometric NiAl alloys.

4.4.4.2 Solute Diffusion Indium diffusion in NiAl was measured as an Al-substituting solute.[86, 87] In Fig. 4.14(a), the compositional dependence of the In diffusivity DIn* measured at T  1523 K and calculated for T  1273 K from the corresponding Arrhenius plots[87] are compared with the Ni tracer diffusion results of Frank et al.[36] Note that In diffusion is faster than Ni at higher temperatures in Ni-rich compositions, and that this relation is reversed at lower temperatures. However, the absolute values of D*In and D*Ni remain similar. This indirectly supports the application of Eq. (8) as a rough estimate of S in NiAl. In Al-rich compounds, however, In diffuses remarkably faster than Ni at all studied temperatures; the difference amounts to an order of magnitude. Such behavior does not contradict the triple-defect diffusion mechanism in near-stoichiometric compositions. The abrupt increase of D*In in Al-rich compositions can be related to the corresponding type of ASB mechanism that involves Ni vacancies and Al antistructure atoms. The appearance of Al atoms in antistructure positions in Al-rich compositions of NiAl was already indicated.[18, 23]

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This diffusion behavior in NiAl is somehow paradoxical: Because Ni atoms in antistructure positions are necessary for long-range diffusion, and they are only supplied by thermally created triple defects, a large amount of structural Ni vacancies does not enhance Ni self-diffusion. It does, however, facilitate diffusion of the Al component. Co[88, 89] and Pt diffusion[90] in NiAl was studied, revealing a pronounced compositional dependence with a minimum value near the stoichiometric composition. Simultaneously, the activation enthalpies QCo and QPt have a maximum near the stoichiometry. The observed diffusion behavior of Co and Pt favors the triple-defect diffusion mechanism, especially in compositions around stoichiometric NiAl.[89, 90] The Arrhenius parameters of self-diffusion and solute diffusion in some compositions of NiAl alloys are presented in Table 4.4.

4.4.5 Fe-Al System 4.4.5.1 Self-Diffusion Fe Diffusion. Tracer measurements of self-diffusion in the Fe-Al system were carried out in several investigations.[41, 91, 92] The results for FeAl alloys with compositions of approximately Fe3Al, Fe2Al, and FeAl are presented in Fig. 4.16. According to Fig. 4.1(c), the phase diagram, Fe3Al reveals sequentially ordered A2, B2, and D03 structures with decreasing temperature. Thus, the temperature dependence of diffusion in Fe3Al may elucidate the effect of order on diffusion. Fe self-diffusion was measured in a very extended temperature interval that comprises all three possible structures in Fe3Al[92] [Fig. 4.16(a)]. The increase of the degree of order resulted in an increase in the activation enthalpy: (B2) )

Q(D0 Q(A2) Fe QFe Fe .

(9)

3

The Arrhenius parameters of diffusion in FeAl alloys are summarized in Table 4.5. The effect of the A2  B2 transition on Fe self-diffusion was studied in a Fe3Al alloy with a slightly different composition (corresponding to a slightly different A2  B2 transition temperature).[91] It is observed that a more perfect order results in a larger activation enthalpy for Fe diffusion. This effect was analyzed[91] in terms of Girifalco’s model,[93] which predicts: Q (1  g S2) Dv  D0 exp   . RT





(10)

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T [K] -111500

10

1300

900

1100 1000

-12

800

Fe in Fe3Al:

10

-13

Eggersmann et al.

-14

Larikov et al. Tôkei et al.

10 10

-15

2 -1

[m s ]

10

-16

D

*

10

A2

-17

10

D03

B2

-18

10

Al in Fe3Al:

-19

Larikov et al.

10

-20

10

-21

10

8

10 T

-1

12 -4

14

-1

[10 K ]

(a) Fe3Al Figure 4.16 Self-diffusion[41, 91, 92] and interdiffusion[99] in Fe3Al, Fe2Al, and FeAl. In (c), Al(1) and Al(2) refer to Al tracer diffusion data estimated from the DarkenManning equation with the thermodynamic factor calculated according to two different theoretical models.[99]

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T [K] 1500

1300

-11

10

900

1100 1000

800

Fe in Fe67Al33: Eggersmann et al.

-12

10

Larikov et al. -13

* ~ 2 -1 D , D [m s ]

10

~

D (Salamon et al.)

-14

10

A2

B2

-15

10

-16

10

Al -17

10

-18

10

-19

10

7

8

9 T

-1

10 -4

11 -1

[10 K ]

(b) Fe2Al Figure 4.16 (Continued)

12

13

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T [K] -11 1500

10

1300

-12

800

Fe in Fe52Al48:

10

Eggersmann et al. Larikov et al.

-13

10

~

-14

10 * ~ 2 -1 D , D [m s ]

900

1100 1000

D (Salamon et al.)

-15

10

Al(2)

-16

10

-17

Al(1)

10

-18

10

Al in Fe52Al48: Larikov et al.

-19

10

-20

10

7

8

9 T

-1

10 -4

-1

[10 K ]

(c) FeAl Figure 4.16 (Continued)

11

12

13

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Table 4.5. Arrhenius Parameters of Self-Diffusion and Solute Diffusion in

FeAl

Approximate Composition

D0 (m2 s
1)

Q (kJ/mol)

Ref.

A2 A2 B2 D0 A2/B2 A2/B2 A2 A2/B2 A2/B2 A2/B2 A2/B2

8.1 105 3.2 105 3.8 104 3.3 101 1.9 104 1.8 104 4.7 104 2.9 101 5.4 102 8.6 104 1.1 103

217 204 232 278 214 221 240 310 290 233 240

92 91 92 92 92 92 91 102 102 102 102

Fe In Zn

A2/B2 A2/B2 A2/B2

1.1 103 2.1 103 4.3 103

241 239 240

92 92 92

Fe In Zn

B2 B2 B2

4.1 103 5.4 103 1.2 102

262 257 251

92 92 92

Tracer

Structure

Fe3Al

Fe Fe Fe Fe In Zn Ni Ni Co Cr Mn

Fe2Al

FeAl

Here, Dv is the bulk diffusivity, D0 and Q are the Arrhenius parameters of diffusion in the disordered state, S is the long-range-order parameter, and g is a constant. The value of g was found to be g  0.1.[91] In such an approach, the change in the correlation factor below the transition temperature is neglected, an effect that was shown later to be very important.[94] If we simply approach Tôkei et al. data[91] by two independent Arrhenius dependencies for the A2 and B2 regions, we arrive at the values 3.2 105 and 1.4 101 m2/sec, and 204 and 274 kJ/mol, for D0 and Q, respectively. Fe self-diffusion in Fe2Al and FeAl was investigated almost exclusively in the B2 phase region. In both cases, perfectly linear Arrhenius dependencies were obtained [Fig. 4.16(b) and (c)].[92] Since only two points lie in the A2 phase region for the Fe2Al composition,[92] it was not possible to detect the effect of order on diffusion reliably. However, a relation similar to Eq. (9) can be expected. Fe diffusion was studied by Larikov et al.[41] in much more restricted temperature intervals. It was found that the absolute values of the diffusivity are within the same order of magnitude. However, the deduced activation enthalpies are considerably higher than those in the recent investigation.[92]

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A systematic change of the absolute values of the Fe diffusivity and the corresponding activation enthalpies with increasing Al content in the B2 phase region was observed:[92] Al) Al) D(FeAl)

D(Fe

D(Fe Fe Fe Fe

(11)

Al) Al)  Q(Fe  Q(Fe . Q(FeAl) Fe Fe Fe

(12)

2

3

and 2

3

The absolute vacancy concentration was measured in FeAl alloys by means of differential dilatometry and positron annihilation techniques.[95] These results suggest that the B2 phase field has to be split into several regions: B2, B2(l), and B2(h). [These are shown in Fig. 4.1(c).] It was concluded that single vacancies are the main vacancy-type defects in the B2 region, whereas triple defects are the main defects in the B2(l) region and additional divacancies are produced in the B2(h) region.[95] These changes in the defect behavior are not manifested in the self-diffusion behavior. We can relate this to some specific mechanism of diffusion in B2 FeAl. Note that in NiAl, the large concentration of structural vacancies does not increase the Ni diffusivity. This was explained by the tripledefect diffusion mechanism. A similar effect may exist in FeAl, if the triple-defect mechanism dominates self-diffusion. The dominating diffusion mechanism in FeAl is unclear.[92] Large values of the activation volume of Fe self-diffusion have been reported,[96] which indicates a possible effect from a composed defect as a diffusion vehicle. Results of Mössbauer spectroscopy and quasielastic neutron scattering measurements were interpreted in terms of nearest neighbor jumps of Fe atoms.[32, 97] However, the type of diffusion vehicle could not be identified in such measurements. Summarizing the data of the differential dilatometry study,[95] we can conclude that single vacancies are the main defects in all possible structures of the Fe3Al alloy. Formally, the Fe sublattice diffusion mechanism can be suggested for Fe diffusion in the D03 structure [Fig. 4.2(e)]. Since the formation enthalpy of Fe vacancies on the a sublattice is smaller than that on the g sublattice,[25, 26] the jumps a → g are likely to be the rate-determining step, which results in a relatively high activation enthalpy of Fe diffusion in this structure (Table 4.5). The decrease of the activation enthalpy QFe of Fe diffusion in the B2 and then in the A2 structure in comparison with the D03 structure is likely to be related to a relative easiness for a Fe atom to explore the g and then the b sublattice at higher temperatures. Al Diffusion. 26Al diffusion was measured for several FeAl alloys[41] [Fig. 4.16(a) and (c)]. The main result was that the Al and Fe diffusivities

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are of the same order of magnitude. However, since the measured activation enthalpy of Fe self-diffusion[41] is remarkably larger than that observed in other studies [Fig. 4.16(a) and (c)], these tracer data of 26Al diffusion are suspect. Interdiffusion in the Fe–Al system was investigated in several studies.[98, 99] From the measured value of the Kirkendall shift, corrected for volume contraction, it was concluded that the ratio of intrinsic diffusivities D*AlD*Fe is always larger than unity. It is about 1.2 to 1.5 in B2 FeAl and about 1.8 in the A2 phase reagion for a composition roughly correcouples sponding to Fe3Al.[95] Interdiffusion was studied using diffusion ∼ between Fe50Al50 and Fe70Al30.[99] A weak dependence of D on the composition was stated, with no significant influence of the A2  B2 transition.[99] The results for compositions of approximately Fe2Al and FeAl are presented in Fig. 4.16(b) and (c), respectively. Using the thermodynamic factors Φ calculated from the x-ray scattering data,[100] and assuming S  1 for the vacancy wind factor, the Al tracer diffusivity was estimated for these two compositions[99] [Fig. 4.16(b) and (c)]. In conclusion, both components diffuse in Fe2Al with very similar rates. In FeAl, this situation remains less clear, since no experimental data for the thermodynamic factor are available for such compositions. Different theoretical models for the calculation of the thermodynamic factor result in Al diffusivities that differ by a factor of 5 to 10[99] [see Fig. 4.16(c)]. At present, the uncertainty in the Al diffusivity does not allow definite conclusions about the relevant diffusion mechanism to be drawn.

4.4.5.2 Solute Diffusion Recent comprehensive information on solute diffusion in Fe3Al is presented in Fig. 4.17(a). A pronounced effect of the A2  B2 transition on Ni diffusion in Fe3Al was determined.[101] The high-pressure measurements indicated that the diffusion process is controlled by single-vacancy motion in both structures.[101] The slower Ni diffusion and the larger activation enthalpy with respect to that of Fe self-diffusion were explained by a predominant Ni solubility on the a sublattice [Fig. 4.2(e)]. Diffusion of In and Zn as Al-substituting solutes was studied.[92] Both solutes diffuse faster than Fe [Fig. 4.17(a)]. The A2  B2 transition does not practically influence diffusion of Al substitutes, whereas some small effect from the B2  D03 transition is indicated in the case of In diffusion [Fig. 4.17(a)]. Recently, diffusion of several transition metals was studied in Fe3Al in the temperature interval corresponding to the A2 and B2 structures.[102] An effect of the A2  B2 transition remained within limits of the experimental uncertainties. The activation enthalpies and frequency factors of

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Figure 4.17 Solute diffusion of In and Zn[92] and of Ni, Co, Cr, and Mn[102] in Fe3Al and FeAl. In (a), Ni1 and Ni2 represent Ni diffusion in Fe3Al measured by Peteline et al.[102] and Tôkei et al.,[101] respectively.

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the combined data for the A2B2 region are listed in Table 4.5. The diffusion behavior of the transition metal atoms can be correlated with the preferential solubility of these atoms on different sublattices.[102] Co and Ni, which reveal preferential solubility on the a sublattice,[103] are slow diffusers in Fe3Al, whereas Cr and Mn, which occupy mainly Al sites,[103,104] diffuse slightly faster than Fe.

4.5

Discussion of Lattice Diffusion in Intermetallics

Section 4.4 analyzed self-diffusion in Ni, Ti, and Fe aluminides with different structures. The systematic investigations demonstrate that the antistructure-atom systems, that is, the systems with the antistructuredefect type of disorder on both sides of the stoichiometry (Ni3Al, Ti3Al, and TiAl), reveal only a weak compositional dependence of self-diffusivity. In these systems, the effect of the compositional antistructure defects is only marginal and is within experimental error. The analysis shows that there are a number of reasons for such behavior: (1) The transition-metal sublattice forms a connected network for intrasublattice nearest neighbor jumps of atoms, which does not change the state of the order in the compound [Fig. 4.2(a)–(d)]. (2) The vacancy concentration on the transitionmetal sublattice is larger by several orders of magnitude than that on the Al sublattice [Fig. 4.3(a) and (b)]. (3) The sublattice diffusion mechanism produces the dominant contribution in these compounds. Note that in TiAl, the third argument is relevant only at temperatures below 1470 K, where a linear Arrhenius dependence is observed [Fig. 4.11(a)]. Al diffusion in Ni3Al and Ti3Al, calculated from the interdiffusion data, reveals a slight but clear increase of DAl with increasing Al content on the Al-rich side of the compositions. This is related to the transition metal sublattice diffusion mechanism for Al atoms and the appearance of compositional Al antistructure atoms in addition to the thermal ones in these compositions. The most prominent compositional dependence of the activation enthalpy Q of self-diffusion in Ni and Ti aluminides is observed in the phase NiAl exhibiting triple defects. The most remarkable change in Q, however, occurs on the Ni-rich side in NiAl, while the formation of the Ni structural vacancies does not practically affect Q [see Fig. 4.14(b)]. The latter behavior is explained by the triple-defect diffusion mechanism producing a dominant contribution in the Al-rich, stoichiometric, and slightly Ni-rich compositions of NiAl. The distinct decrease of Q at larger Ni content (above 54 at.% Ni) is explained by an additional contribution of the ASB mechanism, which operates in NiAl with the percolation threshold at about 55 at.% Ni.

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TiAl and NiAl are the only phases that reveal a nonlinear Arrhenius behavior of self-diffusion at high temperatures; that is, above 1470 K in TiAl[7] and above 1500 K in NiAl.[36] The curvature in the Arrhenius plot of TiAl is explained by an additional contribution of a specific ASB mechanism [Fig. 4.7(b)] and is related to the large concentration of thermal Ti antistructure atoms at high temperatures, even in the Al-rich TiAl alloys.[7, 8] The observed upward deviation of D*Ni from the otherwise straight Arrhenius line in NiAl at T  1500 K seems to have another origin. The concentration of thermal Ni antistructure atoms in NiAl is notably lower than in TiAl [see Fig. 4.3(b)], which corresponds to a higher degree of order in NiAl. Moreover, the upward deviation of D*Ni is observed in NiAl at about the same temperature, independent of the composition.[36] These features correlate with recent measurements of the vacancy concentration in NiAl alloys.[81] The regular change of Fe diffusion in the B2 FeAl alloys of different compositions (Fe3Al, Fe2Al, and FeAl) was related to the change in the long-range order parameter S.[92] Using the long-range order parameter for the B2 order, which is related to the Al content by the relation Smax  2XAl (XAl is the atomic fraction of aluminium), the activation enthalpy of Fe selfdiffusion in the B2 region can be described by the following relation:[92] QFe  (220  46 S2max) kJmol.

(13)

In Fig. 4.18, Ti self-diffusion and Ga solute diffusion in different titanium aluminides are compared with respect to the homologous temperature, TmT. Here Tm is the melting temperature of the corresponding stoichiometric compound. Systematic changes in the diffusion behavior of the Ti aluminides are clearly observed: Ti and Ga diffusion decreases gradually from pure a-Ti to Ti3Al and finally to TiAl. This phenomenon most likely originates from: (1) the sublattice diffusion mechanism in these materials, (2) prevailing vacancy concentrations on the transition metal sublattices, and (3) a systematic decrease of the coordination number of the transition metal sublattice. Figure 4.18 reveals that the self-diffusivity follows the following relationship: DTi*  D*Ti Al  D*TiAl, in a line with the coordination number of the transition metal sublattices in the corresponding structures: zTiTi  12  zTiTi Al  8  zTiTiAl  4. Figure 4.3(a) and (b) indicates that the vacancy concentrations on the transition metal sublattices are similar in Ti3Al and TiAl. The ratio QTm also lies in the common limits of vacancy diffusion: 17.4k for Ti3Al and 17.5k for TiAl. These values are close to that in pure a-Ti, QTm  18.8k. The main difference in the absolute values of the diffusivities therefore stems from the pre-exponential factors D0  gfa2n0 exp{S f/k}. Here, f is the 3

3

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Figure 4.18 Ti and Ga diffusion in a-Ti (solid lines[68, 77]), Ti3Al (dashed-dotted lines[65, 68]), and TiAl (dashed lines[7, 68]) as functions of the reduced temperature Tm
T.

correlation factor; a is the lattice constant; n0 is the attempt frequency; Sf is the formation entropy; the numerical factor g reflects the smaller coordination number of the sublattice with respect to the whole lattice: g
1, 2
3, and 1
3; f
0.781, 0.467,[105] and 0.71;[80] and a
2.95, 2.91, and 2.83 Å for the a-Ti, Ti3Al, and TiAl phases, respectively. Comparative ratios of the products g
gfa2 for the three phases are g Ti :gTi3Al:

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g TiAl  5.4:3.2:1. From the actual diffusion measurements shown in Fig. 4.18 and Tables 4.2 and 4.3, the corresponding ratios of 70:5:1 are obtained. While the ratios gTi3Al:g TiAl are reasonable, the one for Ti is rather large. The large discrepancy in a-Ti may be attritubed to the entropy factor, which should be taken into account, especially for pure Ti. Thus, the observed systematics alone does not result from the structural limitations, although the change in D follows the constraints imposed by the given sublattice structures. Note that the Ga diffusivity D*Ga in these compounds follows a similar tendency (Fig. 4.18). Such behavior was also established for the Al diffusion data extracted from the interdiffusion measurements in the Ti aluminides.[68] This can be explained by the same arguments as for transition metal self-diffusion and further indicates that the sublattice diffusion mechanism operates for Ga and Al in the Ti aluminides.

4.6

Grain Boundary Diffusion

Grain boundary (GB) diffusion in intermetallic compounds was investigated to a much smaller extent than bulk diffusion. Some experimental information is available already for Ni,[55, 106–108] Ti,[109] and Fe[110] aluminides. The primary problem, however, is to improve our understanding of GB diffusion in intermetallic compounds on the atomistic level. For example, diffusion mechanisms in GBs, and the effects of the order and of structural multiplicity, are still not well understood.[111] It is not clear if the local disorder at GBs occurs by the same mechanism as in the bulk lattice, that is, by means of antistructure atoms or structural vacancy formation.[111] Therefore, in this overview, experimental results will be described and unresolved problems will be highlighted. GB diffusion measurements in intermetallic compounds were performed in the Harrison B-regime conditions.[112] Schematically, in such a case, the tracer atoms diffuse fast along GBs. Then, at some depth, they penetrate into the bulk of the grains and diffuse further, at a slower rate, over a distance that is distinctly larger than the GB width d .[112] As a result, the so-called triple product P  sdDgb can be determined from the detected diffusion profile. Here s is the segregation factor and Dgb is the GB diffusion coefficient. In GB self-diffusion experiments in pure metals, s  1 and P represents a double product P  dDgb. In intermetallic compounds, especially in off-stoichiometric alloys, we can expect a certain segregation of a constituent component to the GBs, resulting in s
1. The total effect, however, is likely to be small and can be neglected in a first approximation.

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4.6.1 Ni3Al Ni GB diffusion in the Ni3Al phase field was systematically investigated in the temperature interval 968 to 1190 K.[55] Although only a marginal compositional dependence of Ni bulk diffusion is observed in this compound, a distinct V-type dependence of the product PNi on the composition is established, with a minimum in PNi about the stoichiometric composition[55] [see Fig. 4.19(a)]. The resulting activation enthalpies reveal a maximum at about the stoichiometric composition and decrease on both sides as the Ni content deviates from stoichiometry [Fig. 4.19(b)]. The Arrhenius parameters of the GB diffusivity PNi in Ni3Al are summarized in Table 4.6. Zulina et al. studied Ni GB diffusion in near-stoichiometric Ni3Al and several Ni3Al-based alloys.[108] The resulting values of PNi are smaller than those measured by Frank and Herzig[55] at the given composition, and the activation enthalpy of Ni GB diffusion in Ni74.8Al25.2 is larger [Fig. 4.19(b)]. However, note that Zulina et al. determined the activation enthalpy QNi gb by fitting only three experimental points in the Arrhenius diagram. It thus includes a larger uncertainty. As shown in Fig. 4.19(a), Cermak et al.[106] observed a maximum in the PNi versus composition plot instead of a minimum measured by Frank and Herzig[55] at temperatures 950 K. On the other hand, at T 900 K, a minimum of PNi around stoichiometry was observed, in agreement with Frank and Herzig.[55] The deduced activation enthalpies of Ni GB diffusion[106] are remarkably larger than the values determined by Frank and Herzig.[55] This behavior may be explained by GB segregation of residual impurities in the alloys and/or by the applied experimental procedure. Table 4.6. Arrhenius Parameters of Grain Boundary Self-Diffusion in Ni, Ti, and Fe Aluminides

Phase

Tracer

Ni3Al

Ni

NiAl

Ni

Ti3Al

Ti

TiAl Fe3Al

Ti Fe

Composition at.% Al 26.6 24.8 22.4 50.0 46.5 25 33 56 25

P0 (m3 s
1) 14

7.3 10 4.4 1014 2.2 1015 4.6 1015 9.0 1014 4.8 1013 3.2 1011 4.6 1015 4.0 109

Qgb (kJmol)

QgbQv

Ref.

153 154 115 152 182 195 252 123 227

0.51 0.51 0.38 0.53 0.65 0.68 0.88 0.47 0.8

55 55 55 80 80 109 109 109 110

10

74

1012 K

1190 K

26 24

75 76 Composition [at.% Ni]

25

77

968 K

1066 K

23

78

22

(a) Triple product P for Ni3Al alloys. Open circles[55] and solid squares[106] at T = 1013 and 1073 K, respectively.

10 73

-22

10

-21

-20

27

Composition [at.% Al]

( )

25 24

74

23

75 76 Composition [at.% Ni]

77

Ni in Ni (Mishin et al.)

Ni in Ni (Frank et al.)

26

78

22

(b) Activation enthalpy Qgb for Ni3Al alloys. Open circles are from Frank and Herzig[55]; dashed lines represent activation enthalpy of grain boundary diffusion of Ni in pure Ni[55,113]; the full triangle is from Zulina et al.[108]

73

100

125

150

175

200

225

27 250

Composition [at.% Al]

6:36 PM

Qgb [kJ/mol]

Figure 4.19 The triple product P and the activation enthalpy Qgb of Ni GB diffusion in Ni3Al as functions of composition.

3 -1

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Frank and Herzig established[55] that by increasing the Ni content on the Ni-rich side of Ni3Al, both PNi and QNi gb values approach the values measured for pure Ni [see Fig. 4.19(b)]. This can be explained by the segregation of Ni, which was experimentally observed[114] at GBs in Ni3Al on the Ni-rich side. On the other hand, the increase of the PNi values and the decrease in QNi gb on the Al-rich side can be explained by an excess of Al segregating to GBs.[114] These Al atoms increase the free volume at GBs; thus the formation enthalpy of the Ni vacancies at the GBs is decreased.[115] These features most likely contribute to the enhancement of Ni GB diffusion in these compositions.[55] Atomistic simulations are necessary to explain the effect of GB segregation on diffusion behavior. Polycrystalline Ni3Al is well known to reveal grain boundary brittleness, which is successfully suppressed by microalloying with boron.[116, 117] The effect of boron addition on Ni GB diffusion in Ni3Al was carefully studied.[106, 118] The doping of Ni3Al with 0.24 at.% B and the segregation of B decrease Ni GB diffusion by a factor of 2 to 3 and increase the activation enthalpy Qgb slightly with respect to pure Ni3Al.[55, 118] This effect is explained by an increase in GB cohesion and Ni–Ni atom bonding upon boron alloying.[118] Moreover, B segregation is likely to block otherwise energetically favorable diffusion paths along GBs and increases the vacancy formation enthalpies in the boundary core.

4.6.2 Ti3Al Ti GB self-diffusion was studied as a function of composition in Ti3Al within the temperature interval 940 to 1316 K.[109] As shown in Sec. 4.4.2.1, Ti bulk self-diffusion is practically independent of composition in the Ti3Al phase.[65] However, Ti GB self-diffusion reveals a distinct compositional dependence: As the Al content on the Al-rich side of the compound increases, the PTi values systematically decrease [see Fig. 4.20(a)]. This tendency is opposite to that observed in the other A3B compound investigated, Ni3Al [Fig. 4.19(a)]. With increasing deviation from the stoichiometric composition, the GB diffusivity in Ti3Al is decreased and the activation enthalpy increases to an unusual large value in comparison with that of bulk diffusion in this compound. In Fig. 4.20(b), the compositional dependence of the activation enthalpy of Ti GB diffusion in Ti3Al is compared with that of Ti GB diffusion in pure a-Ti.[109] QTigb in a-Ti is similar within experimental accuracy to the value of QTigb in stoichiometric Ti3Al. The absolute value of the product PTi in a-Ti, however, is larger by one order of magnitude than that in Ti75Al25.[109] This correlates with the geometric limitations imposed by the particular structure of the Ti sublattice in GBs of Ti3Al.[119]

-23

7.5

32

8 T

35

-1

-4

-1

[10 K ]

8.5

1200

T [K]

28

25

9

33

Ti3Al

1100

9.5

(a) Arrhenius dependence of triple product P in Ti3Al alloys. Compositions in at.% are marked on each plot.

10

10

-22

10

-21

-20

10

1300

66

34

30

26

74

Ti in α2-Ti3Al

28

70 72 68 Composition [at.% Ti]

Ti in α-Ti

32

Composition [at. % Al]

76

24

(b) Activation enthalpy Qgb of Ti grain boundary diffusion in α-Ti and α2-Ti3Al alloys.[109]

150 64

175

200

225

250

275

36 300

6:36 PM

Qgb [kJ/mol]

Figure 4.20 Triple product P and the activation enthalpy Qgb of Ti GB diffusion in a2-Ti3Al. The dashed line is for grain boundary diffusion in a-Ti.

3 -1

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The Arrhenius parameters of Ti GB diffusion in several representative Ti3Al alloys are given in Table 4.6. While in stoichiometric Ti75Al25 the ratio QTigbQTiv adopts a typical value for GB diffusion, QTigbQTiv  0.68, this value increases remarkably in Al-rich compositions. In Ti67Al33, QTigb hardly deviates from QTiv , the activation enthalpy of Ti bulk self-diffusion in this alloy. Note that such larger values of QgbQv were already observed in other cases: QgbQv  0.9 for Co diffusion in CoSi2[120] and QgbQv  0.8 for Fe diffusion in Fe3Al,[110] as discussed in Sec. 4.6.5.

4.6.3 TiAl Ti GB self-diffusion was recently measured in several TiAl alloys.[109] The data suggest negligible compositional dependence of Ti GB diffusion in TiAl. The Arrhenius dependence of Ti GB diffusion in two representive compositions of a2-Ti3Al and a-Ti is compared in Fig. 4.21. Ti diffuses very fast along GBs in TiAl, even in comparison with a-Ti. The ratio QTigbQTiv  0.45 for TiAl is typical for pure close-packed metals. The comparison of the absolute values of Ti GB self-diffusion in Ti3Al and TiAl shows that the GB self-diffusivity in TiAl is larger by about one order of magnitude than in Ti3Al with stoichiometric composition, and even larger by two orders of magnitude than PTi in Ti68Al32 at similar temperatures. Ti diffusion was measured along a2g phase boundaries between a2Ti3Al and g-TiAl phases.[121] Material, so-called polysynthetically twinned (PST) single crystals, with a uniformly oriented lamellar structure, was investigated.[121] In Fig. 4.21, these results are compared with Ti GB selfdiffusion in both a2 and g phases. Diffusion in a2g phase boundaries is very slow and is in line with the GB diffusion data in the Al-rich Ti3Al phase. These low values of Ti GB diffusivity in the a2g interphase boundaries reflect their compact structure and were explained by vacancymediated nearest neighbor jumps in the Ti sublattice of the phase boundary in correspondence with its special atomic arrangement.[121]

4.6.4 NiAl Ni GB self-diffusion in NiAl was measured within the temperature interval 958 to 1194 K.[80] No distinct compositional dependence of Ni GB self-diffusion has been established in the limited range of compositions investigated. The results are given in Fig. 4.22. A slight tendency for an increase of the activation enthalpy for Ni GB diffusion in Ni-rich NiAl alloys is shown in Fig. 4.22(b). The Arrhenius parameters of Ni GB selfdiffusion in representative NiAl alloys are listed in Table 4.6.

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Figure 4.21 Comparison of grain boundary diffusion in pure a-Ti (solid line), a2-Ti3Al (dashed lines), g-TiAl (dotted-dashed line), and TiAl polysynthetically twinned (PST) interphase boundaries (circles).[109] The Al content of the corresponding alloys is indicated in at.%.

48

47

53

1004 K

1115 K

51 52 Composition [at.% Ni]

958 K

50

49

1052 K

1194 K

50

54

46

(a) Triple product P for Ni grain boundary diffusion in NiAl[80]

49

-23

10

10

-22

10

-21

51

Composition [at.% Al]

Qgb [kJ/mol]

50

50

48

51 52 Composition [at.% Ni]

49

53

47

54

46

(b) Activation enthalpy Qgb for Ni grain boundary diffusion in NiAl[80]

120 49

140

160

180

200

51 220

Composition [at.% Al]

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Figure 4.22 Triple product P and the activation enthalpy Qgb of Ni grain boundary diffusion in NiAl alloys as a function of composition.[80]

3 -1

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Note that the absolute values of the GB diffusivity PNi in NiAl are appreciably smaller than those in Ni3Al and pure Ni at the same temperatures.

4.6.5 Fe3Al Fe GB diffusion in the Fe3Al compound was investigated in the extended temperature interval 733 to 1065 K.[110] The temperature dependence of PFe is presented in Fig. 4.23. The compositional dependence of Fe GB diffusivity has not been studied. Tokei et al. found[110] that the A2  B2 transition does not remarkably affect the GB diffusion, although a negative deviation of the PFe values from the otherwise linear Arrhenius dependence is indicated below the B2  D03 transition temperature (see Fig. 4.23). These deviations are not caused by a hypothetical transition from the B to the C GB diffusion regime, since the B regime conditions were realized even at the lowest temperature.[110] The smaller diffusivities of GBs in the D03 phase field suggest a higher degree of order in the GB region of the corresponding phase in comparison with the B2 structure. Atomistic calculations of defect formation and migration enthalpies would give more insight into this problem. It was also proposed[110] that the GB segregation of Al in Fe3Al can modify the order-disorder transition temperature in GBs as compared to the bulk transition temperature, especially for the A2  B2 transition.

4.7

Summary

Self-diffusion of the transition metal component in Ni3Al, Ti3Al, and TiAl mainly occurs by nearest neighbor jumps via vacancies in the transition metal sublattice (sublattice diffusion mechanism). This mechanism (that is, next–nearest neighbor jumps of Ni atoms) does not dominate in NiAl. The triple-defect mechanism produces the main contribution to diffusion in the near-stoichiometric NiAl alloys. Under specific conditions (high temperatures in TiAl, T  1450 K, or large Ni content in NiAl, cNi  55 at.%), the antistructure bridge mechanism also becomes important in these intermetallics. The transition metal sublattice diffusion mechanism superimposed by the ASB mechanism governs Al diffusion in Ni3Al, Ti3Al, and TiAl. Ga, which dissolves exclusively on the Al sublattice in Ni and Ti aluminides, is a suitable solute to simulate Al diffusion in these compounds because it diffuses by the same mechanisms as Al. Al diffusion in near-stoichiometric NiAl alloys mainly occurs by the triple-defect mechanism, and the ASB mechanism dominates Al diffusion in Al-rich compositions.

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T [K] -191100

10

900

1000

800

-20

10

-21

3 -1

P [m s ]

10

A2

D03

B2

-22

10

-23

10

-24

10

-25

10

9

10

11 T

-1

12 -4

13

14

-1

[10 K ]

Figure 4.23 Arrhenius plot of Fe grain boundary diffusion in Fe3Al.[110] The temperature intervals of the A2, B2, and D03 phases are indicated. The solid line represents an Arrhenius fit of the data measured in the A2 and B2 phase regions.

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Bulk self-diffusion in the antistructure-defect-type Ni and Ti aluminides (Ni3Al, Ti3Al, and TiAl) reveals only a marginal compositional dependence, whereas a pronounced compositional dependence of DNi is observed in the triple-defect-type NiAl phase. This behavior is related to the fact that the sublattice diffusion mechanism operates mainly in Ni3Al, Ti3Al, and TiAl. Self-diffusion of the transition metal component in the closed-packed Ni and Ti aluminides, if measured at the same reduced temperature, decreases systematically when changing from pure metal to A3B and to AB compound. Grain boundary self-diffusion in closed-packed A3B compounds (Ni3Al and Ti3Al) reveals a distinct compositional dependence, despite a marginal compositional dependence of the bulk self-diffusivity in these compounds. The deviation from the stoichiometric composition in these alloys produces different effects on the transition metal GB diffusion in Ni3Al and Ti3Al: An increase of PNi and a decrease of QNi gb in Ni3Al, and the opposite in Ti3Al (a decrease of the PTi values and an increase of QTigb in the Al-rich compositions).

Acknowledgments The present investigation was supported by funding from the Deutsche Forschungsgemeinschaft (Projects He89825-1 and 25-3).

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14. H. Okamoto, J. Phase Equilibria, 14, 257 (1993). 15. C. McCullough, J.J. Valencia, C.G. Levi and R. Mehrrabian, Acta Metall., 37, 1321 (1989). 16. R. Kainuma, M. Palm and G. Inden, Intermetallics, 2, 321 (1994). 17. O. Kubaschewski, Iron Binary Phase Diagrams, Springer Verlag, Berlin (1982). 18. P.A. Korzhavy, A.V. Ruban, A.Y. Lozovoi, Y.K. Vekilov, I.A. Abrikosov and B. Johansson, Phys. Rev. B, 61, 6003 (2000). 19. H.-E. Schaefer, K. Frenner and R. Würschum, Intermetallics, 7, 277 (1997). 20. M. Kogachi, Y. Takeda and T. Tanahashi, Intermetallics, 3, 129 (1995). 21. H.-E. Schaefer and K. Badura-Gergen, Defect Diffusion Forum, 143–147, 285 (1997). 22. Yu. Mishin and D. Farkas, Philos Mag. A, 75, 177, 187 (1997). 23. B. Meyer and M. Fähnle, Phys. Rev. B, 59, 6072 (1999). 24. A.F. Voter and S.P. Chen, Mat. Res. Soc. Symp. Proc., Vol. 82, 175 (1987). 25. J. Mayer, C. Elsässer and M. Fähnle, Phys. Status Solidi, B, 191, 283 (1995). 26. J. Mayer and M. Fähnle, Defect Diffusion Forum, 143–147, 285 (1997). 27. H.A. Domian and H.I. Aaronson, in Diffusion in Body-Centered Cubic Metals, American Society for Metals (1965), p. 209. 28. M. Arita, M. Koiwa and S. Ishioka, Acta Metall., 37, 1363 (1989). 29. D. Gupta and D.S. Lieberman, Phys. Rev. B, 4, 1070 (1971). 30. D. Gupta, D. Lazarus and D.S. Lieberman, Phys. Rev., 153, 863 (1967). 31. I.V. Belova and G.E. Murch, Philos Mag. A, 82, 269 (2002). 32. G. Vogl and B. Sepiol, Acta Mater., 42, 3175 (1994). 33. R. Hilgedieck and Chr. Herzig, Z. Metallkd., 74, 38 (1983). 34. H. Wever, Defect Diffusion Forum, 83, 55 (1992). 35. H. Numakura, T. Ikeda, M. Koiwa and A. Almazouzi, Philos Mag. A, 77, 887 (1998). 36. St. Frank, S.V. Divinski, U. Södervall and Chr. Herzig, Acta Mater., 49, 1399 (2001). 37. S.V. Divinski and L.N. Larikov, J. Phys.: Condens. Matter., 9, 7873 (1997). 38. S.V. Divinski, St. Frank, U. Södervall and Chr. Herzig, Acta Mater., 46, 4369 (1998). 39. Chr. Herzig, T. Przeorski, M. Friesel, F. Hisker and S.V. Divinski, Intermetallics, 9, 461 (2001). 40. S.V. Divinski, St. Frank, Chr. Herzig and U. Södervall, Solid State Phenomena, 72, 203 (2000). 41. L.N. Larikov, V.V. Geichenko and V.M. Falchenko, Diffusion Processes in Ordered Alloys, Amerind, Oxonian Press, New Delhi (1981). 42. L.S. Darken, Trans. AIME, 175, 184 (1948). 43. J.R. Manning, Acta Metall., 15, 817 (1967). 44. T. Shimozaki, Y. Goda, Y. Wakamatzu and M. Onishi, Defect Diffusion Forum, 95–98, 629 (1993) . 45. H. Numakura, T. Ikeda, H. Nakajima and M. Koiwa, Defect Diffusion Forum, 194–199, 337 (2001). 46. G.E. Murch, Philos Mag. A, 46, 575 (1982). 47. I.V. Belova and G.E. Murch, Philos Mag. A, 78, 1085 (1998).

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5 Diffusion Barriers in Semiconductor Devices/Circuits Shyam P. Murarka Rensselaer Polytechnic Institute Troy, New York

5.1

Introduction

When two different materials are brought in contact with each other, there is a tendency for them to mix. This mixing, a naturally driven process that leads to a lowering of the free energy of the system comprising these materials, occurs by a transfer of matter from one part of the system to another as a result of the random atomic or molecular motion. This is diffusion, which is measured in terms of a quantity called diffusivity or diffusion coefficient D. D is defined, by Fick’s first law of diffusion in one dimension, as: J
D [∂c(x, t)
∂x].

(1)

Here c is the concentration of the diffusing species; it is a function of the position x in the sample and the diffusion time t. D is the proportionality constant relating the flux (J) of the diffusing species to the concentration gradient (∂c
∂x). The magnitude of D determines the extent of mixing and is dependent on the materials type (its melting point, structure, and the defect type and content), the temperature, and the surroundings of the materials in contact with each other. Such diffusion and mixing lead to changes in the properties of the materials, sometimes to a better material and other times to an undesirable material behavior. Most solid-state transformations are the result of interdiffusion. In microelectronic devices and circuits, semiconductors, metals, and insulators coexist with each other (generally in a multilayer scheme), leading to an ongoing concern about the intermixing and resultant loss of the desired functional behavior. There are several known examples of failures associated with this solid-state diffusion or mixing. Interaction of Al with underlying Si in the contact windows of the devices has led to considerable current leakages and contact failures. Diffusion of Cu into SiO2 is known to cause increased leakage through the oxide and thus to cause an insulator failure.[1] In these and similar circumstances, we would like to inhibit the diffusion

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and intermixing, and thus obtain the expected performance and reliability of the materials involved. In the simplest way, we can find and interpose a layer of a third material between the two interacting materials to achieve a successful inhibition of the interaction. The material used for this diffusion barrier must have certain characteristics and must be compatible in properties with the materials it joins. For example, an electrically conducting barrier is required between two metals; it is also preferred between an insulator and a metal. On the other hand, an insulating barrier is needed between two insulating materials. The desired characteristics of an ideal diffusion barrier material, X, between two materials, A and B, have been listed by Nicolet:[2] 1. The material transport rate of A and B across X should be negligible. 2. The material transport of X into A and B should be negligible. 3. X should be thermodynamically stable against A and B. 4. X should adhere strongly to A and B. 5. The specific contact resistance of A to X and of X to B should be small. 6. X should be laterally uniform in thickness and structure. 7. X should be resistant to mechanical and thermal stress. 8. X should be a good electrical and thermal conductor. With the advent of subquarter mm dimensions, high aspect ratio vias and trenches, and Cu as the interconnection metal in the silicon integrated circuits, a few new requirements have emerged: 9. X should be as thin as needed (for example, the projected need for the years 2005 through 2007 for the high-resistivity barriers, such as TaN, is a barrier less than 5 nm thick, and preferably 2 to 3 nm thick) or should have electrical conductivity similar to that of Cu (for example, alloys of Cu, if they work). 10. Deposition of X should be carried out in such a way that it coats the via/trench walls and bottoms conformally. 11. X can be an insulator, but it should not be thicker than a few monolayers, not contribute significantly to the capacitance of the interlayer dielectric (ILD), and be anisotropically dry-etchable by reactive ions from the bottoms of the vias and trenches. It is obvious that the defined requirements for the diffusion barrier characteristics are changing with the application, although Nicolet’s criteria 1

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through 4, 6, and 7 hold for all uses of the diffusion barriers. Criteria 5 and 8 are specific to applications in a metallic system where metallic behavior is essential. Criteria 9, 10, and even 11 are specific to advanced subquarter mm design rule microelectronic applications. With the advent of nanoelectronics and nanostructured materials, these requirements are expected to change, and we may be looking for very unconventional ways of inhibiting or eliminating the interactions caused by the diffusion/interdiffusion. This chapter focuses on the diffusion barriers for application in microelectronics, and especially between semiconductor and insulator or metal, insulator, and metal, and between different metal systems used. It is important to note that with use of new materials in these systems of semiconductor devices/circuits, we invariably find a need for a suitable adhesion promoter between two materials, thus making Nicolet’s criterion 4 essential. Our goal, therefore, is to consider thin films that will function as adhesion promoters and diffusion barriers (APDBs) to optimize material systems with respect to the layer thicknesses along with the process costs. The ultimate objective of any research focused on APDBs is to identify the ideal APDB for a given specific application, in our case for use in a multilevel metallization scheme for advanced microelectronics. Like everything else in microelectronics, the requirements of the so-called ideal APDB films are changing with time. Thickness is reducing; so is the temperature at which the films should be reliable. The introduction of newer materials is sought to achieve better performance and reliability of the device/circuit. However, an ideal APDB becomes feasible in a real situation only if it can be fabricated without incurring disproportionate expenses, and if the process of fabrication is reproducible and reliable. This chapter briefly reviews the past history of diffusion barriers, followed by a short discussion of the available theories of diffusion, relation to structure of materials, manipulation of the structures, and related considerations. The future requirements of the specific applications, and various approaches taken to satisfy these requirements, both conventional and unconventional, are discussed. Finally, specific examples of new approaches are reviewed.

5.2

Diffusion Barriers from the 1960s Through the 1990s

Historically, the interaction between metals and the surrounding environment has been mitigated by alloying with selected impurities, which reduced the diffusion coefficient D and the tendency to react. A few classic examples are the making of steel, stainless steel, and copper with a small amount of boron. Yet in microelectronic applications, thin films of

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various types have been placed as APDBs between two reacting films. During the early years of the microelectronics era (the 1960s and early 1970s), gold-beam leaded interconnection technology led to very reliable devices and circuits.[3] In this technology, the interconnections to the outside world were made by using Au, Pd, or sometimes a bilayer of Ti and Pd between the Au and Si contacts. Contacts to Si were made using a silicide of Pd or Pt, although some work was reported on the use of electroless- plated Ni.[4] Au could not be used directly on Si because of (1) its very high diffusivity in Si, (2) its inability to bond with SiO2, which is always present as a native oxide layer on the surface of silicon, and (3) the existence of a low temperature (370°C) eutectic in the Au-Si system.[5] A direct extension of the gold-beam leaded technology, when Al was introduced as the contact to Si, was not possible because of the metallurgical interactions between Al and Au and between Al and Pd. Such interactions, occurring at relatively low temperatures, made the metallization mechanically and electrically unstable. A suitable barrier was needed. Ti/Pt bilayer and Ti/TiN/Pt trilayer APDBs were found to satisfy this need and to be reliable under actual fabrication and use conditions.[6] Note that a study of the thermodynamics of the metal system (individual metals and their reaction products), appropriate phase diagrams, and interdiffusion was essential for making the right choice of the APDB between Al and Au. The need for an effective Ti-Pt diffusion barrier between Al and Au, with a thickness of at least 200 nm of each individual metal, was recognized.[6] To investigate their relative effectiveness as diffusion barriers, several two-metal, three-metal, fourmetal, and five-metal layers were fabricated and tested for interdiffusion, failures, and preventing a reaction between Al and Au. Interaction between two materials is a result of fast interdiffusion. In thin films, a large number of fast diffusion mechanisms are operating, leading to an obvious suggestion to use high-melting-point metals/alloys. For diffusion through a lattice (so-called lattice diffusion), the self-diffusion coefficient Dself is indirectly related to the melting point (Tm) of the material by an empirical relation between the activation energy (Qself) of selfdiffusion and the melting point: Qself
34 Tm,

(2)

where Tm is given in degrees Kelvin and Qself is given in calories per mole. In most cases, D and Q are related through the Arrhenius relationship: D
Do exp(Q
RT),

(3)

where R is the gas constant. Thus the higher the melting point, the higher the activation energy and the lower the self-diffusion coefficient. However,

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in thin films, which are generally deposited by vapor deposition techniques, the diffusion is dominated by the grain boundaries. For self-diffusion through grain boundaries, we can approximate the activation energy to be about half that for the lattice diffusion given by Eq. (2). Thus, most thinfilm interactions are the result of grain boundary diffusion and other fast diffusion paths, such as dislocations. If these fast diffusion paths can somehow be blocked (or stuffed[2]), the interaction rate can be significantly reduced. These relationships provided the concept of using refractory metals with stuffed grain boundaries for application as APDB thin films and led to the use of W, Ta, Nb, Mo, Hf, Zr, V, Cr, and Ti metals. The electrically conducting nitrides, carbides, and borides of these metals have very high melting points. In early applications, these materials were suggested as APDBs, and some were used, not only because they have high melting points, but because the presence of a slight excess of the nitrogen, carbon, and boron, respectively, in these materials stuffs the fast diffusion paths. With decreasing device size in microelectronics (which caused decreasing junction widths and depths, leading to the concern about the use of Al contacts on Si) of the late 1970s and early 1980s, a diffusion barrier was needed between Al and Si. The most commonly suggested barriers (which have one common factor common: the high melting point) have been the following: 1. 2. 3. 4. 5.

Silicides Refractory nitrides, carbides, or borides Ti-W alloy Refractory metal-noble (or near noble) metal alloy A refractory metal alone

The most commonly used barriers between Al and Si have been the Ti-W alloy, TiSi2
TiN, TiSi2
W
Ti
TiN, CoSi2
TiN, CoSi2
Ti
TiN, W, and W
TiN. In most advanced Si integrated circuits, a CoSi2
Ti
TiN
W thinfilm structure is used in the Al or Cu interconnect technology.Unlike the Al interconnects on SiO2, which do not require any APDB material due to their natural adhesion property without interdiffusion, APDB material is needed between Cu and SiO2. This has led to a renewed search. Presently, a nitride of Ta (TaNx) is being used between SiO2 and Cu, and between W (in the contact windows) and Cu.[7] The barrier thickness is in the range of 5 to 30 nm, and the resistivity is high (200 to 300 mΩ cm). For future applications, either the thickness has to be reduced by an order of magnitude or the resistivity of the barrier has to be similar to that of Cu. These requirements have led to a flurry of activities in self-forming barriers/adhesion promoters[8] and in a newer group of materials called self-assembled molecules (SAMs)[9] in which a monolayer of molecules,

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generally about 1 nm thick, is formed on a given surface and could act as an APDB layer between the ILD and Cu. In the future, a lower dielectric constant (k) interlayer dielectric (ILD) film,[10] which may be a polymer, fluorinated/carbonated SiO2, or a porous material, may replace pure SiO2 as an ILD. Such replacements may also place new constraints on the properties of APDB materials. These new concepts and their applications are reviewed in Sec. 5.6.

5.3

Brief Review of Diffusion and the Influencing Material Factors

5.3.1 Diffusion in the Lattice and Grain Boundaries Equation (1) simply states that the rate of transfer of a diffusing species across a unit area perpendicular to the x direction is proportional to the concentration gradient in that direction. If we consider diffusion across an element of volume and use the law of conservation of matter, we reach a conclusion that the rate at which the concentration changes in this volume must equal the local decrease of the diffusion flux, leading to the one-dimensional Fick’s second law of diffusion: ∂c(x, t)
∂t
∂{D [∂c(x, t)
∂x]}
∂x.

(3)

For concentration-independent D, Eq. (3) simplifies to: ∂c(x, t)
∂t
D [∂2c(x, t)
∂x2].

(4)

To determine D, we must carefully determine the concentration of the diffusing species as a function of the depth in the sample for a given diffusion treatment (that is, time, temperature, and ambient). Equation (3) or (4) is then solved using the experimentally placed boundary conditions. A variety of solutions fitting various boundary conditions are available.[11] Two of the most commonly used solutions of Eq. (4) are Gaussian and error-function distributions given, respectively, by the following equations: c(x, t)
M
√(pDt)  exp(x2
4Dt)

(5)

c(x, t)
cs  erfc[x
2√(Dt)].

(6)

and

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Here M is the total amount of the diffusing substance that is constant, cs is the time-independent surface concentration of the diffusing species, and erfc is the complimentary error function related to the error function by: erfc (z)
1  erf (z).

(7)

Note that the solutions given by Eq. (5) or (6) are for concentrationindependent D and are useful in most cases of interest. For concentrationdependent D, several solutions (and approximations) of Eq. (3) have been examined and presented.[11] To obtain a reasonable estimate of the penetration by a diffusing species, we can calculate the diffusion depth, xD, for a diffusion time t and at a temperature for which a value of D has been obtained. Theoretically, xD should be infinite, since xD will be determined by the condition c
0 at t  0. For most practical purposes, xD is defined as: XD2
4Dt.

(8)

Use of this definition to determine a typical diffusion depth simply indicates that c
cs has a predetermined value: for Gaussian solution (Eq. 5), c
cs
1/e
0.37; for the error-function solution (Eq. 6,) c
cs
0.1573. In this context, we also define the transport velocity, V(cm/sec), across a material of thickness d, which is given by: V
D
d.

(9)

Thus, to minimize the mass transport, V should be reduced by minimizing D across the APDB and/or by increasing the thickness, d, of the APDB. The latter is difficult to achieve, leaving the choice of selecting a very thin film of APDB material with the lowest possible D of the impurity. To use Eqs. (5) and (6) rigorously, diffusion measurements are generally made in the lattice of ideal solids that contain only point defects in thermal equilibrium that provide energetically favored migration paths. In real solids and especially in thin films, their microstructures become very important and a large number of fast-diffusion paths, such as grain boundaries, surfaces, and dislocations, exist and contribute to enhancing the low-temperature diffusion coefficient by several orders of magnitude. At diffusion temperatures of less than half the melting point (in degrees Kelvin), surface, grain boundary, and dislocation diffusion coefficients are higher than the lattice diffusion coefficient by about 8, 7, and 5 orders of magnitude, respectively.[12] We can derive relationships between Jgb and JL that represent grain boundary and lattice fluxes, respectively, when diffusion

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from the grain boundary into the adjoining grains is permitted: Jgb
JL
2dDgb
DL.

(10)

Dgb and DL are diffusion coefficients in the grain boundary and in the lattice, respectively, and d is the grain boundary width. This relationship is obtained by considering a situation in which the concentration on the front and back surfaces of the slab of the polycrystalline material is maintained constant, and a steady state is assumed because all concentration gradients parallel to the plane of the slab are eliminated. Note that in these considerations, grain boundaries offer unrestricted diffusion. Any segregation of the diffusing species and others present in the films will seriously affect both fluxes. Gupta et al.[13] have explained the solute (impurity) effect in changing the diffusion of solvent atoms (for example, Al in an Al-Cu alloy) phenomenologically. They considered interactions of the solute (impurity) atoms with defects in the lattice, in the grain boundaries and the equilibrium solute adsorption at the grain boundaries. The solute effect on the grain boundary diffusivity has been shown to depend not only on the associated solute-binding free-energy difference (∆Ga) between the grain boundary and the lattice sites, but also on the changes in the lattice diffusion, as given by: [(Dgbp
Dgba)  (DLa
DLp]1
2
1  co exp(∆Ga
RT),

(11)

where subscripts p and a refer to pure and alloyed metal, respectively, and co is the solute concentration in the lattice. For co  1, Eq. (11) predicts diffusion enhancement at high temperatures and retardation at low temperatures.

5.3.2 Interdiffusion Between Two or More Materials in Contact There are no universal models of diffusion/interdiffusion in thin films of various materials in contact with each other. Balluffi and Blakeley[14] have examined the complexities in the details of diffusion processes in thin films and point out the following: “Diffusion in thin films may be expected to have special characteristics for a number of reasons, which include the following: (1) thin films are invariably diffused at relatively low temperatures because of their poor thermal stability; (2) all volume elements are in close proximity to either a free surface or interphase boundary of some kind; (3) films generally contain high densities of low-temperature short circuits for diffusion

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such as grain boundaries and dislocations; (4) large biaxial stresses in the plane of the film are often present; (5) relatively high concentration of uncontrolled impurities may be present as a result of special fabrication or diffusion conditions; (6) disordered or metastable structures may be present; (7) diffusion often occurs over short distances under the influence of large concentration or electrostatic potential gradients; and (8) steep-chemical gradients and low temperature may affect the possible maintenance of local equilibrium at phase boundaries during multiphase diffusion.” It is then expected that at the typical post-metal-anneal temperatures of 300 to 450°C, grain boundary diffusion will dominate in films of the high-melting-point metals of interest, that is, Ti, Ta, Pt, W, metallic silicides, and the refractory nitrides, carbides, and borides. On the other hand, in metals such as Al and Mg, lattice diffusion will dominate at the same temperatures. Diffusion behavior in copper is somewhere in between. In developing an APDB film, our goal is therefore to eliminate or minimize the interaction or reaction between materials of interest and the APDB film. The reactivity can generally be estimated by evaluating the interdiffusion coefficients in thin films. Interdiffusion coefficients are strongly related to the microstructure of these films; the mutual solid solubilities; the intermetallic or compound formation, as suggested by the phase diagrams; the free energies or heats of fomation of these intermetallics and compounds; and the temperature. For the devices/circuits in use, the ILDs sandwiched between two interconnect layers are exposed to very high electric fields (of the order of 1 MV/cm), which causes diffusion of metal ions into the ILD. Thus, an APDB material is required between the metal and the ILD. The interdiffusion coefficient (Dij) reflects the average overall movement of both constituents, say, i and j (representing two solids in contact), diffusing in a concentration gradient. For an ideal case of a binary system, Dij is given in terms of the individual diffusion coefficients, Di and Dj, and their respective concentrations, ci and cj, in the diffused alloy at a depth x: Dij (x)
Di cj (x)  Dj ci (x).

(12)

Dij is then used to replace D in Eq. (3) or (4). In thin-film interactions, invariably, Dij is obtained even when the binary system does not form the ideal solid solution. Also in such cases, because of the lack of crystallinity, the experimentally determined Dij is an averaged grain boundary interdiffusion coeficient, at best. For the interaction between an APDB film and another film in contact, Dij must be negligibly small or the thickness of the barrier film large so that the reaction does not reach the other surface/interface.

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5.3.3 Role of Material Properties Controlled Microstructure Barrett et al. have stated:[15] “Practically all the useful properties of materials are strongly dependent on their internal structure,” and “for a material of given chemical composition, the internal structure is not constant.” The effect of the internal structure or microstructure on the bulk properties of materials is known. The microstructure of thin-film interconnections plays a very important role in determining two important reliability factors: (1) the reactivity with the surroundings, and (2) the electromigration and stress voiding. As mentioned in Sec. 5.3.1, real materials are used for metallization applications such as thin films of copper. Information on the lattice parameter, density, and so forth for Cu can be readily found in handbooks. All defects are extremely important in thin films. Chemical reactivity, atomic diffusion, and electrical and mechanical properties are affected by the presence of defects. Also, the properties of the materials are not isotropic and depend on a specific crystallographic direction under consideration. For example, in the face-centered cubic Al or Cu, (111) planes have the highest density of atoms with low energy, and therefore behave differently than (100) planes. For bulk polycrystalline metal with random texture, isotropic behavior is obtained when the individual cystallites are oriented throughout space with equal directional probability. The effective macroscopic properties can be calculated by considering the directionally dependent values averaged over all orientations in space. For bulk material, especially the metals, the practical ways to control the microstructures to yield the desired and useful properties are well known to metallurgists and materials scientists. The concepts are also important for thin films to be used as APDB, but are difficult, if not impossible, to be realized. Preparation of the substrate surfaces, surface forces, and deposition methods, and the attendant parameters, yield a microstructure variability that is not well understood. It is well known that the substrate-film interface, the film surface (which may be covered by a native oxide or other compounds), the impurities that segregate at defect sites, and the grain boundaries are effective barriers to the motion of dislocations, thus affecting both the film strength and film microstructure. In Al and Cu thin films, texturing leads to deviation from the isotropic polycrystalline thin-film case. Texture is defined as the tendency of the individual crystallites of a material to acquire a preferred crystallographic orientation. The texture affects materials properties, and thus affects performance. For example, the texture of electroless copper film has been shown to affect the resulting oxidation behavior.[16]

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The correlation of room-temperature stress with texture and the subsequent influence on film resistivity have also been noted.[17] A recent review discussed the observed texture responses of copper thin films deposited by a variety of techniques.[18] In general, (111), (200), and random components of texture dominate the response, although in some cases, (220) and (520) texture components have been observed. From surface energy considerations, the close-packed (111) oriented grains should be favored. Frequent occurrence of an extensive fraction of the (200) texture, which has been found the most stable, is not well understood. Texturing and microstructure are affected by thermal treatments, after or during deposition, and are influenced considerably by the annealing ambient. In certain cases, thermal annealing of copper films is found to induce the formation of giant grains.[19, 20] The strain energy has been suspected of playing a role in this regard. Impurities significantly influence the microstructural evolution due to (1) diffusion in and out of grain boundaries, (2) interaction with the host metal, and (3) segregation in grain boundaries and dislocations, and at surfaces and interfaces. Impurity fluxes in the grain boundaries have contributed to grain boundary motion[21] and diffusion-induced recrystallization,[22] and thus to grain growth. On the other hand, impurity segregation at the grain boundaries and other short-circuiting paths such as dislocations, surfaces, and interfaces is known to suppress the diffusion in or on the material. Solute segregation at the grain boundaries and interfaces is discussed later in this section. The lower the solid solubility is in the grain, the higher the concentration is in the grain boundary.[23] For example, the segregation of Cu in the grain boundaries of Al, either as metallic Cu or as a Cu-Al alloy, leads to a retardation of the Al migration caused by the imposed electric field (that is, electromigration is retarded, leading to improved mean time to failure). Grain boundary stuffing has been known to produce effective thin-film diffusion barriers.[2] The effect of Mg or Al added to Cu, in concentrations less than the solid solubility limit, on the microstructure has been investigated to reveal the metallurgical variables of the APDB effectiveness of these alloys used between Cu and SiO2.[24, 25] The effectiveness of Mg or Al is attributed to their thermodynamically favorable abilities to reduce SiO2, leading to the formation of an interfacial layer, between Cu and SiO2, that acts as an APDB layer. The microstructures of annealed Cu, Cu-Al, and Cu-Mg structures, however, show very different behaviors. Pure copper has moderate to heavy twinning with relatively uniform microstructure. Cu-Al has a clear bimodal distribution of small (∼25 nm) and large (∼160 nm) grains and heavy twinning. Cu-Mg, like pure Cu, shows a bimodal microstructure with considerably less twinning.

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Figure 5.1 compares the (111) and (200) x-ray diffraction peak integrated intensities of three films as a function of annealing temperature. Cu and Cu-Al films have a preferred texture [with (111) to (200) ratio in annealed films as high as 15] that agrees with the observed twinning typical of (111) orientations. Cu-Mg shows a complete reversal of preferred texture, with (200) stronger than (111) [a (111) to (200) ratio of 0.04 after 400°C and 0.01 after 600°C anneal]. Temperature also has a large influence on the growth of (200) texture compared to that of (111). It is apparent that there is a remarkable difference in the behavior of Al and Mg alloying elements in controlling the microstructure of Cu films, especially when the added impurity concentrations are below the solid solubility limit. These differences could have a large impact on the diffusion in the metal and thus on the electromigration behavior. Note that in the case of Al metal, the measured mean time to failure for the electromigration was found to increase with the increasing grain size and the degree of (111)-preferred orientation; decreasing the spread of grain size distribution is beneficial.[26] Electron microscopic studies of the narrow lines have shown that the grain structure takes on a so-called bamboo appearance, with grain boundaries generally running perpendicular to the direction of the current flow (that is, along the interconnection length).[27] Similarly, the superiority of the Al-Cu alloys[28] and of e-gun evaporated metals or alloys is associated with the resulting preferred (111) texture and uniform grain size distribution.

Mutual Solid Solubilities and the Phase Diagrams Solid solubilities of i into j and of j into i play an important role in determining the outcome of the reaction between i and j. If the solid solubilities are large, the interdiffusion leads to a solid solution prior to formation of any compound. For example, titanium can dissolve significant amounts of oxygen prior to the oxide formation. The advantage of such a system is that no new phase appears in early periods of reaction. On the other hand, when the mutual solid solubilities are low, compound formation occurs immediately. For example, oxides of W form readily because W does not dissolve any significant amount of oxygen. In the absence of a compound formation, second-phase (i or j) precipitation has been seen on cooling from high temperatures where solid solubilities were higher. For example, aluminum dissolves larger amounts of silicon at 450°C, the typical metal anneal temperature. On cooling to room temperature, silicon precipitates out as the second phase in aluminum film and at the siliconaluminum interface. Solid solubility does play a role in the second-phase formation, but it does not play a direct role in influencing the diffusion

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Figure 5.1 Area under x-ray diffraction peak vs. annealing temperature of Cu and Cu alloys. All annealing times were 30 minutes except annealing done at 600°C, which lasted 120 minutes.

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coefficient where neutral species are involved. However, if electric fields are involved and charge buildup can occur, this factor may play a significant role in determining the diffusion coefficient. However, as described above, the solid solubility will determine the amount of material that can be dissolved or rejected by the film (or the substrate), the compound formation, if any, and the segregation at the film-substrate interface and at the grain boundaries of the polycrystalline films.[29]

Segregation at the Grain Boundaries and Interfaces When a solute is added to a polycrystalline solid, the concentration of the solute in the grain (or the bulk of the crystal) is controlled by the solid solubility (that is, the solubility maximum) at the given temperature. However, the concentration maximum of the solute in the grain boundaries depends on several factors.[23, 30] We define an enrichment factor Egb as: Solute concentration in the grain boundaries Cgb
 . Egb
 Solute concentration in grains Cb Several investigators[31–33] have concluded that the lower the atomic solid solubility is in the grains (or bulk), the larger Egb is. Figure 5.2 shows the classical plot of the measured Egb as a function of the atomic solubility.[31] Lower bulk solubility leads to a higher rejection (of solute) into the grain boundaries. Note that at such high concentrations of impurities (in grain boundaries), a thin layer (or small volume) of a new phase may form in the grain boundary. Impurities of one type can also control the concentration of some other undesirable impurities. An example of this is the addition of Al to steels. Although Al primarily controls the oxygen in solution in liquid steel, it also controls grain size and shape through the formation of aluminum nitride particles that retard the motion of selected grain boundaries.[32] Similarly, small amounts of Cu added to Al are known to segregate in the grain boundaries and form q phase-particles, thus enhancing the electromigration lifetime of such interconnects. The conclusion is that the impurity with the lowest atomic solid solubilities should be used if we want to “stuff” grain boundaries.[2] Experience, described in the preceding examples, has shown that grain boundary stuffing leads to considerably reduced diffusion and metallurgical interactions, which are predominantly controlled by grain boundary diffusion, especially at low temperatures.[33] A theoretical framework exists to rationalize the observations and to characterize broadly the behavior observed. Examine the problem as

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253

Note: From E. D. Hondros, J. Phys. 36: coll. C-117 (1975); M. P. Seah and E. D. Hondros, Proc. R. Soc. London, A335:191 (1973); and D. Gupta, Metall. Trans., 8A:1431 (1977)]

Figure 5.2 Correlation of grain boundary enrichment (ratio of concentration of the grain boundary to the grain) factor with the atomic solid solubility.

composed of two, two-phase systems: the thin-film/boundary system and the APDB/boundary system. Segregation of the impurity to the boundary region is driven by the free energy of segregation that has been derived by McLean.[34] Consider a lattice made up of N undistorted surface sites with P solute (impurity) atoms distributed on them. Also consider n distorted surface sites with p solute atoms distributed among them. If the energy of the solute on the lattice is E, and the energy of the solute on the surface is e,

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then the free energy of the system is: G
pe  PE  kt [ln(n!)N!  ln(n  p)!p!(N  P)!P!].

(13)

The minimum in G can be derived by differentiation with respect to p. This results in the equation: P
(n  p)
[P
(N  P) exp[(E  e)
kt],

(14)

or the more familiar: Xb
(1  Xb)
[Xc
(1  Xc)] exp(∆E
kt),

(15)

where Xb is the adsorption level as a mole fraction of a monolayer, Xc is the solute mole fraction, and ∆E is the heat of adsorption of the segregant at the boundary. The extent of segregation at the interface boundary is thus determined by ∆E and by the concentration of the solute (impurity) in the solvent. For an ideal situation, for example, epitaxially grown silicon on silicon, ∆E will be near zero and no segregation will occur at the interface. However, dislocations have been found experimentally at this interface, resulting from lattice parameter differences between the deposited layers and the substrate arising from differences in impurity concentrations. In such a case, ∆E is a positive quantity and will lead to some impurity segregation even at the epitaxial interface. The problem of a boundary separating two different materials presents significant difficulties in modeling segregation behavior. One approach is to treat each material separately with a common boundary. The thin film has a segregation coefficient ΘTF that is defined: ΘTF
Xb
XTF,

(16)

where XTF is the fraction of impurity dissolved in the thin film. Similarly, a segregation coefficient for the given underlying (or overlying) film, such as an APDB film, can be defined: Θulf
Xb
Xulf ,

(17)

where Xulf is the fraction of impurity in the given underlying film. Therefore, the amount of impurity in the boundary between the thin film and the underlying film can be viewed as the competition between the two segregation coefficients. At equilibrium, the impurity concentration in the boundary reflects the equilibration of the impurity levels in the two films

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in contact. In such a case, a segregation coefficient can be defined as a ratio of the two interfacial segregation coefficients: Kseg
ΘTF
Θulf
Xulf
XTF,

(18)

where Kseg, usually known as the segregation coefficient between the two thin-film materials, is truly a lattice segregation coefficient, in contrast to the interfacial segregation coefficients Θulf and ΘTF. The segregation coefficient will thus control the equilibrium concentration across the phase boundary. Precipitation of a new phase at the phase boundary will, however, change the segregation behavior drastically. In thin films, the grain boundaries will contribute very significantly, and possibly erratically, to the segregation phenomenon and thus to desired APDB behavior. Information on the solid solubilities can be obtained from the binary phase diagrams. In cases where such information is not available, we can make estimates based on a similar binary system or by using HumeRothery rules.[35] One of these rules states that the extent of primary solid solution is limited whenever the differences in the atomic radii of the solvent and solute atoms exceed 15%. The larger the electronegativity difference is, the more stable the solid solution is. The crystalline structure of the solids also influences the solubility limits. Experience has shown that materials with similar crystal structures have extended mutual solid solubility. These observations and rules on solid solubility apply to crystalline materials, where the lattice solubility greatly exceeds the contribution of defects, phase boundaries, or surfaces to the solute behavior. In thin films, where grain boundaries and dislocations are abundant, the segregation of impurities to these sites will, in general, control the total amount of solute dissolved in such films. For example, large amounts of arsenic and phosphorus could be present in the grain boundary regions of polysilicon without affecting the electrical resistivity. Upon annealing at high temperatures, larger amounts of dopant can dissolve into the silicon grains, leading to a lower resistivity in subsequently quenched samples. On slow cooling, the excess dopant is rejected to grain boundaries, leaving behind a higher resistivity material.[36] Thus, the grain structure of the films and the cooling rate will strongly affect the amount of solute retained in the lattice and will thus determine the segregation and redistribution behavior. As far as thin-film reactions leading to compound formation are concerned, the thermodynamic considerations correctly predict the outcome of reactions. If the reaction leads to a decrease of total free energy, then under suitable kinetic conditions, a reaction will proceed. On the other hand, if the reaction does not lead to a lowering of free energy, it may not

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occur. Thus, by use of the free energy of formation data from thermodynamics data in handbooks, we can safely predict the possibility of a reaction occurring. In the absence of the free-energy data, we can use the heats of formation to obtain a similar answer. When a reaction between two metals occurs, the outcome generally is a solid solution or one or more intermetallic compounds. Established binary phase diagrams show the existence of such intermetallics. Even with this knowledge, a realistic modeling of the interfacial reactions and compound formation (and their impact on the properties) is very difficult and complicated. An interfacial reaction can usually be treated as a binary diffusion couple. For components of such a couple that exhibit a series of equilibrium intermetallic phases across the alloy system, the prediction of equilibrium thermodynamics is that the resultant diffusion zone will include a series of bands with sharp boundaries, each band corresponding to one of the intermetallics.[37] The width of a given band will depend on a variety of competitive factors, such as the solubility range across the phase and component diffusion coefficients within the band in relation to those in adjacent phases. In this way, the chemical potential of each component, the gradient of which is the fundamental driving force for diffusion, will be continuous across the entire diffusion zone. Thus, at the interface separating any pair of bands, say g and e, the chemical potentials are equal for each component; that is, the phases at the interface may be in quasiequilibrium. However, the advanced state of reaction is, at best, only a reference point in relation to the initial reaction conditions. Figure 5.3(a) presents a portion of a binary equilibrium phase diagram at constant pressure for a system A, B that forms a single intermetallic b. The solid solubility limits for the thermal solid solutions a and g at temperature T1 are c and f, respectively. The intermetallic phase b exhibits a range of solubility, d-e. The corresponding Gibbs free-energy-composition diagram for this temperature is shown in Fig. 5.3(b) and (c), where the common tangent construction, equivalent to the chemical potential equalities of defining equilibrium, is included as solid straight lines. Note that c, d, a, and f are points of tangency, with the metastable equilibrium a  g indicated by the dashed common tangent. For this system, the reaction a  g → b is a peritectoid reaction. Qualitatively, however, Fig. 5.3 would be unchanged if the intermetallic b were congruently or peritectically melting, rather than associated with a peritectoid reaction as shown. In any event, this situation results from deposition of component B (for example, copper) onto substrate A, at low enough temperature to prevent any reaction during deposition. When a reaction does occur, limited interdiffusion that creates terminal solutions very near the interface precedes the formation of any phase or phases. In such systems, a state of metastable equilibrium involving a  g tends to be established first, a

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Note: From C. W. Allen and A. G. Sargent, Mat. Res. Soc., 54:97 (1986).

Figure 5.3. (a) Binary system A-B with one intermetallic; (b) and (c) molar freeenergy composition diagrams at T2 and T1, respectively. Stable equilibrium is shown by solid tangent and metastable equilibria by dashed lines.

process that may be intercepted by nucleation of the intermetallic. This is in contrast to the situation at the temperature T2 (Fig. 5.3), which represents the stable state. With the formation of an intermetallic, as in the typical peritectoid case, the reacting components are physically separated as the product phase forms. When the interface is covered with a new phase, the thickening growth reaction requires diffusion through the intermetallic layer, which may still be controlled by mass transport kinetics or by interface reaction kinetics.[37] The actual situation, however, is usually quite different from such a classical model. Realistic modeling of interfacial reactions and compound formation require an analysis of experimental results obtained under different conditions and for different combinations of layered diffusion systems. Note that generally, not all the intermetallics reported in a phase diagram are formed in thin-film couples. For example, in one of the complex binary systems of Pt and Al, in thin-film reaction couples annealed in the temperature range 200 to 500°C, only 6 of 12 possible phases were detected.[38] In a relatively simple system of Ti reacting with silicon, only three of the possible five phases have been reported.[39] On the other hand, for the simple system of the thin-film couple of Mg and Cu, both reported intermetallics have been detected.[40] Several factors determine the absence of certain phases in thin-film couples: limited range of thickness, atomic diffusivities in various phases, concentration gradients, temperature, free energies, impurities, nucleation and growth, and the amount of material available during the interaction. The available amount of the reacting material, measured by its thickness in films, also determines the

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extent of intermetallic compound formed in the final product. Goselle and Tu[41] have examined this aspect of the reaction between thin films, considering some of the factors mentioned here. The absence of the phases in thin-film interaction products is explained on the basis of nucleation and growth probabilities and diffusion across very thin phases that may not be detected by the analytical techniques. It is pointed out that the above-mentioned principles and discussions also apply to ternary, quaternary, and other systems. However the phenomena become very complicated, and reaction products difficult to predict. Only experimental results, obtained under a defined set of conditions, provide an immediate answer.

Free Energy and Heat of Formation Thermodynamic considerations predict the stability of a system under given conditions. We are concerned about the adhesion of the APDB film with the films it is separating under the conditions of forming such structures and in actual use. A film is said to adhere well to the substrate if all film-substrate interfaces are not physically affected during the fabrication or service, even when exposed to a reasonably high level of stress. Thus to promote adhesion, we must ensure (1) excellent physical and chemical bonding, preferably the latter, since it is energetically more favorable, across the substrate-metal film interface, and (2) low levels of stress in the film arising from the device/circuit fabrication processes. Reasonable assumptions have led to the conclusion that for good adhesion, chemical interactions leading to the interatomic bonding at the desired interface are essential Such interactions must, however, be self-limiting so that only a very thin layer (preferably a monolayer or two) of the interfacial reaction product results. In addition, stresses must be kept below a level of ∼0.5 Gpa (5  109 dyn/cm2).[42] To ensure adhesion, we must also ensure the absence of easy deformation/fracture modes, reactive environments that produce stress, and long-term degradation modes. For example, Al adheres well to oxides, nitrides, carbides, silicides, and Si. On the other hand, copper does not bond well with the surfaces of the same materials. The difference is reflected in the energies of formation of the metal oxide, nitride, carbide, and silicide, or in the metal-nonmetal diatomic bond strengths. It is thus apparent that the deposition of Cu directly on SiO2 or on a typical polymer interlayer dielectric will not provide the needed adhesion at the interface. Note, however, that the energies of formation of the Cu fluoride and sulphate are high and are comparable to those of silicon compounds, but are not as high as those of Al compounds. We can then postulate that if the ILD surface is terminated in a (F) and/or (SO4) species, Cu may

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form bonds with such species and thus form an interfacial layer to provide the necessary adhesion between the ILD and Cu. Thicker interfacial layers of these types should, however, be avoided to eliminate the tendency of crystallization of these compounds at the interface.

Temperature Temperature is perhaps the most important parameter in determining the need and usefulness of the APDB between two films it is separating. Our goal is to process sets of films under optimized process conditions, then to use the product under so-called operating conditions. We can either choose the process and use a temperature/time combination to suit the given APDB, or select the APDB to survive the given process and temperature/time combination. Classically, a 450°C, 30- to 60-minute annealing in hydrogen-containing ambients has been used for Al metallization schemes. Presently, Cu has replaced A, and new low-dielectric-constant ILDs are being developed to replace SiO2. These changes are leading to post-metal anneals as low as 300°C. In multilevel metallization schemes, such anneals are repeated every time a new layer of metal interconnects is created. Thus, the first APDB layer at the contact level may experience several of these anneals plus those (around 300 to 450°C for about 30 minutes) used during packaging of these chips. It is safe to say that concerns related to the stability of the APDB and metal layers at the process temperatures are becoming fewer. The temperature rise during the device or circuit in use is increasing with the continued miniaturization of the devices and circuits. However, the operating device/circuit temperature is still expected to be lower than the process temperature. Now we are concerned with the combined effect of applied field and temperature on the stability of the APDB used. Because lower temperatures will reduce the impact of increasing electric fields, an efficient way to dissipate heat away from the active circuits is becoming a major challenge.

Electric Field The final applicability of the APDB films will be determined by the stability of the electrical properties of the devices/circuits in which such films are used. In most cases, the metallurgical stability, first determined by the experiments that perform diffusion analyses and identify phase changes, is found to determine the effectiveness of the APDB. These observations, however, do not guarantee the electrical stability when the films

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are subjected to high electric fields and current densities. It is thus necessary to determine the two stabilities independently, especially now when new materials such as Cu and low-k ILD are being introduced and film thickness, particularly that of the APDB, is being reduced. When using the APDB films on semiconductors, such as during the formation of contacts, the electrical stability of a p-n junction and/or the Schottky-type contacts is examined as a function of the annealing temperature, time, and ambient. Changes from the idealized (or control) currentvoltage (I-V) behavior are noted and analyzed to determine the stability and reliability under the actual fabrication and use conditions. When the APDB layers are formed between the dielectric (ILD) and the interconnections (Cu or Al), their electrical stability has to be fully determined. This is done by examining their capacitance-voltage (C-V), current-voltage (I-V), capacitance-time (C-t), and current-time (I-t) characteristics. Furthermore, triangular voltage sweep characteristics, charge and voltage-to-breakdown, in-plane and out-of-plane k, and dielectricloss measurements of a metal-insulator semiconductor and/or a metalinsulator-metal capacitor need to be measured. Bias-temperature stressing (BTS) during C-V and I-V measurements determines the electrical stability under simulated use conditions. Exposure to invading environments (high humidity and temperature) during such tests challenges the reliability. The I-V measurements illustrate the dielectric strength, charge trapping, and conduction mechanisms. Any metallic penetration, in general, will change these characteristics and thus determine the usefulness of the APDB used.

5.4

Diffusion Barrier Materials

Many investigations of diffusion barrier materials have been carried out over the past four decades. This section reviews materials used as the APDBs between silicon or silicide and Al, and between SiO2 and Cu.

5.4.1 Metal Nitrides, Carbides, and Borides as APDB Used with Al Table 5.1 summarizes the reported effectiveness of various refractory metal nitrides, carbides, and borides as APDB materials between Si or silicide and Al. All the refractory materials have very high melting points and are effective barriers up to at least 500°C. They all qualify, therefore, for silicon integrated circuit applications in the thicknesses used in these qualifying experiments (in the range of approximately 40 to 200 nm).

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Table 5.1. High-Melting Diffusion Barriers Between a Substrate and Al Metallization: Nitrides, Carbides, and Borides

Barrier Material TiN

ZrN HfN TaN Ta N TiC

ZrB

Substrate

Temperature* (°C)

Si TiSi2 PtSi CoSi2 NiSi Si Si NiSi Pd Si CoSi2 Si PtSi TiSi2 CoSi2 Si

550 550 600 550 500 550 500 600 550 550 500 600 500 500 600

Intermetallics Detected

AlN, Al3Ti

Al4Zr3Si5 AlN, Al3Ta

Al4C3, Al3Ti Co2Al9

References a a, b, c d, e, f d g h i e j j k f k, l l m

*

Temperature at which the reaction is first observed. a. C. Y. Ting, J. Vac. Sci. Technol., 21:14 (1982) b. N. Cheung, H. Von Seefeld, and M.-A. Nicolet, Proc. Symp. on Thin Film Interfaces and Interactions (J. E. E. Baglin and J. M. Poate, eds.), Electrochem Soc., Princeton, NJ (1980), p. 323 c. C.-Y. Ting and M. Wittmer, Thin Solid Films, 96:327 (1982) d. R. J. Schutz, Thin Solid Films, 104:89 (1983) e. M. Wittmer, Appl. Phys. Lett., 37:540 (1980) f. M. Wittmer, J. Appl. Phys., 53:1007 (1982) g. M. Finetti, I. Suni, and M.-A. Nicolet, J. Electron. Mater., 13:327 (1984) h. L. Krusin-Elbaum, M. Wittmer, C.-Y. Ting, and J. J. Cuomo, Thin Solid Films, 104:81 (1983) i. I. Suni, M. Maenpaa, and M.-A. Nicolet, J. Electrochem. Soc., 130:1215 (1983) j. M. A. Farooq, S. P. Murarka, C. C. Chang, and F. A. Baiocchi, J. Appl. Phys., 65:3017 (1989) k. M. Eizenberg, S. P. Murarka, and P. A. Heimann, J. Appl. Phys., 54:3195 (1983) l. A. Applebaum and S. P. Murarka, J. Vac. Sci. Technol., A4:637 (1986) m. J. R. Shappiro, J. J. Finnegan, and R. A. Lux, J. Vac. Sci. Technol., B4:1409 (1986)

Besides high melting point, what makes these nitrides, carbides, and borides so effective as an APDB? These films are generally deposited by reactive sputtering at or near room temperature and thus have a very small grain size and a high density of fast diffusion paths. Deposited films are, in general, in a state of compressive stress, possibly as a result of a small excess of nitrogen, carbon, boron, and/or oxygen. This excess of nitrogen,

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carbon, boron, and/or oxygen leads to the so-called stuffing of the fast diffusion paths, such as grain boundaries and dislocations, and concomitantly to the barrier effectiveness. Note that the presence of oxygen in these materials is related to extremely high affinity of the refractory metals that make these compounds. Invariably the oxygen, trapped in such films from contamination in the deposition chamber, has been detected but ignored from discussions. Such small amounts of oxygen can stuff the grain boundaries and/or form the oxide on the surface or on the grain boundary surfaces. For example, a Ti film deposited on air-exposed Al was stable during a 86-hour, 350°C anneal.[6] When both layers were deposited during one pump-down of the evaporator, a reaction between Ti and Al occurred readily.[43] The effect of nitrogen as an impurity in Mo used as a barrier between Ti and Au and in the case of Ti-W as the effective barrier has been clearly demonstrated.[44–46] In many such triple-layer barriers, an exposure to air was found to be the key to the success of these barriers.[6, 47] Similarly, all carbon-rich carbides were effective barriers, whereas titanium-rich carbides were not.[48] Presently, all technologies using Al as the interconnect metal use a Ti
TiN bilayer or a Ti
TiN
Ti trilayer as the APDB layer. Ti is used to provide enhanced adhesion and protection from nitrogen exposure of the underlying material. These layers are also now used as antireflective coating to facilitate lithography on Al.

5.4.2 Barriers Between the ILD and Cu Tables 5.2 and 5.3 list resistivities and reported effectiveness (with Cu) of various electrically conducting barrier films. Table 5.2 lists the compounds, and Table 5.3 lists the elemental metals. The resistivity and thickness of APDB material will determine its applicability in schemes of the ULSI
GSI interconnection technologies using Cu. Although the resistivities of these materials in pure bulk form could be low, the resistivities of the same materials in thin-film form are considerably higher, depending on the techniques used for deposition and purity. Both the resistivity and thickness of APDB material should be as low as possible. A simple computation of the total interconnect resistance, as illustrated in Fig. 5.4, shows the need for a barrier thinner than 10 nm[49] if the resistivity advantage of copper over aluminum or its alloys is to be fully exploited. The higher the resistivity of the barrier film, the lower the thickness required. As the metal width is reduced, the barrier thickness must also be reduced to maintain the resistance advantage of a given scheme of metallization. Simple calculations of the line resistance of copper with and without barriers of 150 and 5 mΩ cm resistivities,

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Table 5.2. Resistivities and Reported Barrier Effectiveness (with Cu) of Various Barrier Films (Refractory Compounds)

Barrier Material TiN

TiCN Nitrogentreated TiCN Ti34 Si23N43 TaN Ta2N Ta36Si14N50 WN/W W23B49N28 Wsi0.6N *Reported

Reported Resistivity (mΩ cm) 20* to 1000**

Film Thickness (nm)

Stable to (°C) 500 450 400 to 450 400 600

References

2000

100 to 110 40 10 10

a b c d e

200

10

400

f

660/410*** 200 200 200 500 – 2000 –

∼100 ∼8 50 50 40 2
5 100 ∼30

850 700 700 500 700 650 700 600

g h a a g, i j, k l m

approximate value for the pure and bulk material. films of reported resistivity as high as this number. ***As-deposited/annealed l hour at 700°C. a. N. Awaya, H. Imokawa, E. Yamamoto, Y. Okazaki, M. Miyake, Y. Arita, and T. Kobayashi, in Conf. Proc. VMIC, Cat. No. 951ISMIC-104, VMIC, Tampa, FL (1995), p. 17 b. A. Berti and S. P. Murarka, Mat. Res. Soc. Symp. Proc., 318:451 (1994) c. S. Kumar, M. S. thesis, Rensselaer Polytechnic Institute, Troy, NY (1995) d. N. Agrawal, M. S. thesis, Rensselaer Polytechnic Institute, Troy, NY (1998) e. M. Eizenberg, Mat. Res. Soc. Symp. Proc., 427:325 (1996) f. D. Smith, Applied Materials, Santa Clara, CA, private communication (1993) g. X. Sun, J. S. Reid, F. Kolawa, and M.-A. Nicolet, in Conf. Proc. VLSI XI, MRS, Pittsburgh, PA (1996), p. 401 h. T. Oku, M. Uekubo, E. Kawakami, K. Nii, T. Nakano, T. Ohta, and M. Mrakami, in Conf. Proc. VMIC, Cat. No. 951ISMIC-104, VMIC, Tampa, FL (1995), p. 182; see also J. O. Olowolafe, C. J. Mogab, R. B. Gregory, and M. Kettke, J. Appl. Phys., 72:4099 (1992); K. Holloway, P. M. Fryer, C. Cabral, Jr., J. M. E. Harper, and K. H. Kelleher, J. Appl. Phys., 71:5433 (1992); and L. A. Clevenger, N. A. Bojarczak, K. Holloway, J. M. E. Harper, C. Cabral, Jr., R. G. Schad, F. Cardone, and L. Stolt, J. Appl. Phys., 73:300 (1993) i. E. Kolawa, J. S. Chen, J. S. Reid, P. J. Pokala, and M.-A. Nicolet, J. Appl. Phys., 70:1369 (1991) j. T. Nakano, H. Ono, T. Okta, T. Oku, and M. Murakami, in Conf. Proc. VMIC, VMIC, Tampa, FL (1994) ; see also Kailasam et al.[69] k. J. G. Fleming, E. Roherty-Osman, J. Custer, P. MartinSmith, J. S. Reid, and M.-A. Nicolet, in Conf. Proc. VLSI XI, MRS, Pittsburgh, PA (1996), p. 369 l. T. Iijima, Y. Shimooka, and K. Suguro, in Conf. Proc. VLSI XI, MRS, Pittsburgh, PA (1996), p. 325 m. S.-Q. Wang, in Conf. Proc. VLSI IX, MRS, Pittsburgh, PA (1994), p. 31 **Thin

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Table 5.3. Resistivities and Reported Barrier Effectiveness (with Cu) of Various Barrier Films (Elemental Metals)

Barrier Material

Resistivity (µΩ cm)

Reported Film Thickness (nm)

Stable to (°C)

References*

Cr Ta W Nb Mo Co Ni Pd Pt α-C Al**

∼15 ∼15 to 200 ∼7 to 10 ∼15 to 200 ∼6 to 10 ∼10 ∼10 ∼11 to 12 ∼11 to 12 High 3

20 to 60 20 to 60 50 to 60 60 60 – – 100 – 6.5 to 13.5 5 to 7 10

200 to 650 200 to 650 200 to 600 500 500 250 to 450 250 to 450 200 250 to 450 500 350 350

a a, b a a a a a a a b, c, d e, f e, f

Original references cited in Wanga. Barrier effectiveness determined by electrical tests of p-Si/SiO2/Al/Cu capacitors. Reaction does occur in Al/Cu sandwich leading to higher resistivity if Al concentration in Cu exceeds 0.3 at.%. a. S.-Q. Wang, in Conf. Proc. VLSI IX, MRS, Pittsburgh, PA (1994), p. 31 b. N. Awaya, H. Imokawa, E. Yamamoto, Y. Okazaki, M. Miyake, Y. Arita, and T. Kobayashi, in Conf. Proc. VMIC, Cat. No. 951ISMIC-104, VMIC, Tampa, FL (1995), p. 17 c. R. G. Purser, J. W. Strane, and J. W. Mayer, Mat. Res. Soc. Symp. Proc., 309:481 (1993) d. Jan M. Neirynck, thesis, Rensselaer Polytechnic Institute, Troy, NY (1996) e. E. Kirchner, S. P. Murarka, E. Eisenbraun, and A. Kaloyeros, Mat. Res. Soc. Symp. Proc., 318:319 (1994) f. E. Kirchner, Ph. D. thesis, Rensselaer Polytecnic Institute, Troy, NY (1996)

*

**

Figure 5.4 The resistance ratio of the various interconnection schemes as a function of the TiN barrier thickness.

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(a) 150 µΩ cm resistivity

(b) 5 µΩ cm resistivity Figure 5.5 Total resistance of a 1-mm-thick interconnect line as a function of the interconnect linewidth. Comparison of pure Cu with a Cu with 20-nm-thick diffusion barrier/adhesion promoter of 150 mΩ cm (a) and 5 mΩ cm (b) resistivity. The ratio of Cu plus barrier resistance to pure Cu resistance is also shown.

respectively, are shown in Fig. 5.5[50] to show the need for lower resistivity APDB materials. Conventional deposited barrier materials may not be useful in devices with minimum feature sizes of 50 to 120 nm unless a technique is found to make these films robust at thicknesses 5 nm. Note that “stuffing” the fast-diffusion paths, dislocations, and grain boundaries (see Sec. 5.4.1) does not help because stuffing leads, in general, to an increase in resistivity and is the primary cause of highly variable resistivities reported in the literature. The use of pure metal as an APDB can be incorporated using a different approach.[51] Part of the APDB is allowed to interact with the dielectric

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and the top metal in such a way that the reaction products stabilize the dielectric-APDB-metal stacked layers against further interactions during thermal or bias-temperature excursions. This approach, which can also be taken using Cu alloys, is discussed at the end of Sec. 5.5.1. Wang (73) has reviewed the effectiveness of these barriers for Cu technologies. In all cases, the barrier effectiveness was determined using the barrier between silicon and copper (that is, Si/barrier/Cu) and examining the interactions between Si and Cu, Si and the barrier, and the barrier and Cu. The barrier thicknesses used were larger than 20 nm (generally much larger). A variety of techniques were used to evaluate the interactions and performance of barriers as a function of annealing in different conditions. Both adhesion promoter and diffusion barrier are needed between an ILD and Cu; the material interactions may be different and the methodology required to determine barrier performance (such as leakage current) may be application-dependent. However, Wang’s conclusions regarding various barriers are very instructive, and we reproduce these in the following paragraphs.[52] (Note that the references cited in the quotation have been renumbered, to follow those in this chapter.) “The reported stable temperatures (Ts) for transition metal barriers between thin Cu films and Si substrates range from 200 (Pd, Cr and Ti) to 650°C (Ta). Contradictory results were reported for W barrier (Ts varies from 200 to 500°C) and for Ta barrier (Ts varies from 200 to 650°C). Cu reacts with near noble metals such as Cr, Co, Ni, Pd, Pt in the temperature range of 250 to 450°C but is non-reactive and immiscible with refractory metals such as Mo, Ta and W (Ts  750°C)[53]. It should also be noted that the metals from the first group react with Si to form silicides at lower temperatures (100–450°C) than those in the second group (525–650°C).[54] Therefore, it is not surprising that the metals from the first group fail as diffusion barriers at lower temperatures than those in the second group. The failure mechanism is usually due to the high reactivities of barrier metal-Cu or Si-barrier metal, followed by ultimate Cu-Si reaction for the metals from the first group and diffusion of Cu through grain boundaries of the polycrystalline barrier films at relatively low temperatures for the metals from the second group. In general, transition metals are not stable diffusion barriers between Cu and Si. “Transition metal alloy barriers studied usually consist of one nearnoble metal (reactive with Cu to form compounds) and one refractory metal (non-reactive and immiscible with Cu) element. They can be deposited in amorphous states, hence free of grain boundaries. However, the crystallization temperatures of these amorphous alloy films are not as high as those of some other binary or ternary amorphous systems which consist of at least one non-metal element (see below). Grain boundaries form in these films at elevated temperatures. Nevertheless, Ts of transition metal alloy

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diffusion barriers are all above 500°C, higher than those of polycrystalline transition metal diffusion barriers. Metal alloy-Cu reaction is suppressed due to the presence of the non-reactive and immiscible refractory metal element. The temperature of reaction at the Si/metal alloy interface depends on the composition and crystal structure of the metal alloys. It is between the reaction onset temperature of the system of Si/near noble metal component of the metal alloy, and that of the system of Si/refractory metal component of the metal alloy. Failure occurs as a result of Si-metal alloy reaction or Cu diffusion through grain boundaries of polycrystalline metal alloy films at elevated temperatures. “The effectiveness of silicide or transition metal-silicon systems as diffusion barriers follows the similar trend of their corresponding transition metal diffusion barriers. The diffusion barriers formed by refractory metal (Ta and W)-Si are more stable (Ts ranges from 600 to 700°C) than near noble metal (Cr and Co-Si) barriers (Ts ranges from 200 to 300°C). Adding Si to W to form an amorphous W-Si diffusion barrier improves barrier performance significantly (Ts
600–700°C) as compared to polycrystalline W barrier (Ts
250–500°C). The barriers fail by the reaction of Cu with transition metal silicide to form Cu-silicide, by Cu diffusion through polycrystalline transition metal silicides or by Cu-induced premature crystallization of the amorphous metal-silicon barrier films. Ti-N, Ta-N and W-N are three transition metal-nitrogen systems studied for diffusion barrier applications. High stability of these barriers is achieved due to non-reactivity of Cu with N, Ta and W. Barrier failure is caused by diffusion of Cu along grain boundaries or through weak spots generated at elevated temperatures in barrier films, which are relatively intact, or by the reactions between barrier films and Si to form metal-rich silicides. The conducting metal-oxygen diffusion barrier (Ru2O and Mo-O) can survive thermal anneals up to the temperature range of 500 to 600°C. Cu-oxide tends to form at the metal-oxygen/Cu interface at higher temperatures. The amorphous TiB2 barrier breaks down above 750°C. “Some amorphous ternary systems consisting of either one or two nonmetal components such as Ta36Si14 N50 and TiPN2, exhibit highly stable barrier properties due to their high crystallization temperatures. Although their crystallization process is somewhat accelerated by the presence of Cu overlayer, Ts are still among the highest. The barrier property of amorphous C is marginal, which shows a best Ts of no more than 500°C. “Polycrystalline compound TiN and polycrystalline TiW (usually W-rich) solid solution are the two most commonly used diffusion barrier materials for Al metallization on Si substrates and have been successfully integrated into current VLSI devices. TiW can prevent interaction between Cu and Si at temperatures up to 775°C in an RTP anneal time of

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30 sec, as reported by Wang et al.[55] Ts for TiN barriers range from 400 to 700°C (30 min anneal) or to 900°C (30 sec RTP anneal) The failure mechanism for these two diffusion barriers between Cu and Si depends on the reactivity of Cu and Si with each constituent of the barriers, or interdiffusion of Cu and Si through grain boundaries of the barriers. Cu does not react with either W or N. Ti is bonded with N in the TiN compound and tied up with W in the TiW solid solution Therefore, free Ti is not available and reactions between Cu and either the TiN or TiW do not occur up to the lower temperature between the Si/diffusion barrier reaction temperature and the interdiffusion temperature of Cu and/or Si through grain boundaries of the barriers. TiN is stable in contact with Si to high temperatures. However, TiW reacts with Si to form binary and ternary silicides at 800°C.[55] Up to this temperature, no significant Cu diffusion through TiW grain boundaries occurs. Therefore, the failure mechanism for TiN in the Si/TiN/C structure is not barrier dissociation by chemical reaction but rather by diffusion of Cu in TiN films along grain boundaries or through weak spots generated at elevated temperatures.[56] TiW, on the other hand fails as a diffusion barrier between Cu and Si by dissociation reaction with Si substrates.[55] “The studies conducted by Kolawa et al.[57–60] using polycrystalline Ta, amorphous Ta74Si26 and amorphous Ta36Si14N50 as diffusion barriers between thin Cu films and Si substrates can be used here to illustrate the process of choosing a stable diffusion barrier. Ta is non-reactive immiscible with Cu, and reacts with Si to form TaSi2 around 650°C.[54] Therefore, Ta is a good diffusion barrier candidate compared to other polycrystalline transition metal films which usually react with both Cu and Si at relatively low temperatures. However, Ta fails to prevent Cu from diffusing through its grain boundaries to react with Si at 500°C in the Si/Ta/Cu structure, as detected by leakage current measurement. To prevent Cu grain boundary diffusion, Si is added to Ta to create a grain-boundary-free amorphous Ta74Si26 diffusion barrier with a crystallization temperature of about 850°C. Unfortunately, the crystallization temperature of Ta74Si26 film drops to 650°C when in contact with Cu. As a result, Cu diffuses through grain boundaries of the crystallized Ta74Si26 film at 650°C and reacts with Si, resulting in barrier failure. In order to improve the performance of Ta-Si barrier Ta36Si14N50 barrier is made by reactively sputtering Ta5Si3 in N2. This film is amorphous and has higher crystallization temperature (1100°C) than that of Ta74Si26. Consequently, the Ta36Si14N50 barrier does not fail until 950°C. By using amorphous Ta74Si26 and Ta36Si14N50 barriers, the stable barrier temperature is improved by 150°C and 450°C respectively, over that of polycrystalline Ta barrier. As a matter of fact Ta36Si14N50 has been reported to be a stable diffusion barrier between A1 and Si at 700°C (above the Al melting temperature) and is also the most

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stable diffusion barrier ever reported for Cu metallization. Further work needs to concentrate on integration issues such as patterning, electromigration and stress voiding behavior of the metallization on this barrier material.” Most of the published work on this topic has paid attention to the barrier needed between Cu and SiO2 as the dielectric. Polymers offer lower dielectric constants and are being researched for their application as the interlayer dielectric material, replacing deposited SiO2-based materials. An APDB will be needed for the Cu-polymer interconnection scheme. A variety of polymers could find application as an ILD, with one APDB probably not optimum for all polymers of interest. Each polymer presents a different chemical composition and surface structure to which adhesion must be achieved and into which diffusion must be inhibited. Since all polymer surfaces are carbon-rich, bonding to carbon may lead to the desired adhesion. Most transition metals bond well with C; for example, the diatomic bond strengths of C-Ti, C-Nb, and C-Ir are 101, 136, and 151 kcal/mol, respectively.[61] In this regard, a carbide like TiC may be a good choice.[48] Such barriers might be a universal APDB material for polymers. As an alternative to conducting barriers, we can use insulating dielectric materials like Si3N4 or SiOxNy as APDB materials. Cho et al.[62] compared 4% phosphosilicate glass, plasma-enhanced CVD silicon dioxide, low-temperature CVD silicon dioxide, plasma-enhanced silicon oxynitride, and CVD silicon nitride subjected to bias temperature aging at 250°C with a copper electrode on these dielectrics. Capacitors with silicon nitride showed no shifts, indicating that this material is an excellent barrier. However, use of thicker Si3N4 layers is undesirable because of the high dielectric constant (approximately 7), which contributes significantly to the effective dielectric constant of the ILD. Also, as mentioned earlier, such APDB material will require a clean etch process to remove it from the metal surfaces in contact windows.

5.5

New Concepts in Affecting APDB Behavior at the Interfaces

5.5.1 Alloying of Cu to Form an APDB at Interfaces/Surfaces Two approaches, different from the conventional ones, have been used recently to address the APDB materials needs. The first approach explores alloying Cu with small amounts of another element.[8, 63, 64] The

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criteria for selecting this alloying element include (1) low solubility and low ∆r
∆c in Cu, (2) the capability to form chemical bonds through a thermodynamically favorable self-limiting interaction with the ILD, and (3) high diffusivity in Cu and low diffusivity in the ILD and in the interfacial reaction product on the ILD surface. Here ∆r
∆c refers to the resistivity change ∆r for a concentration change (∆c) of the alloying element in Cu. The role of the solid solubility and the interfacial reactivity, as determined by thermodyanamic factors, were discussed in Sec. 5.3.3. A large variety of alloying elements, selected arbitrarily or using the criteria given, have been examined: Al, B, Be, Ga, In, Sn, Mg, Ti, Nb, Ni, Pd, Ti, V, W, Zn, Ca, Cd, Zr, Ag, and Au. Some of them (Ag, Au, Ca, Cd, Zn, Pd, Be, Ga, and In) do not satisfy one or more of the criteria. For example, Ag, Au, and Pd offer low ∆r
∆c but will not provide enhanced adhesion as defined by the second criterion. There are others, such as Sn, Zr, and Ti, that have a very high ∆r
∆c ratio. We had carefully examined these criteria and possible alloying elements for Cu in the late 1980s before selecting Mg, Al, and B. Once ∆r
∆c was established by experiments, Ta was included in the list. However, it was found that for Ta-Cu alloys, very high temperatures (higher than 500°C; usually 800°C) and long anneals are needed to bring the alloy resistivity down to an acceptable range of 2 to 2.5 mΩ cm. Similar but unreported results were obtained with Nb additions to Cu.[65] Such high temperatures are impractical in modern chip fabrication. Interactions of Mg or Al with Cu and the effectiveness of’ Mg and Al alloying additions in Cu in concentrations below the solid solubility limit, including the evaluation of the electrical behavior of metal-oxide semiconductor (MOS) capacitors made with these alloys, have been thoroughly investigated and reviewed.[1, 63–67] Copper alloys containing either 2 at.% Mg or approximately 0.5 ∼ 2 at.% Al with electrical resistivities of about 2 or 2 to 5 mΩ cm, respectively, seem to provide all the necessary properties for use as APDB materials on the SiO2 used as an ILD. In the case of Mg, high-resolution microscopy has demonstrated that Mg does react with SiO2 to form interfacial bonding or a barrier layer a few monolayers thick.[66, 67]

5.5.2 Self-Assembled Molecular Monolayers The second approach is radically different: self-assembled molecular (SAM) monolayers are used as an APDB interfacial layer between Cu and SiO2.[9] Because the thicknesses of SAMs are near zero, they will occupy an insignificant fraction of the total via/hole volume, thereby maximizing the room for filling in low-resistivity Cu metal. Because of their high

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sticking probability on the substrate and low probability of depositing on themselves, they offer very good step coverage, which is necessary for effective APDB behavior. Krishnamoorthy et al.[68] investigated SAMs where, characteristically, trimethoxysilane (Si with three OCH3 groups) is tethered to the SiO2 substrate and the fourth Si bond is attached to a tail group that consists of aliphatic and/or aromatic groups, for example, a 3-[2-(trimethoxysilyl)ethyl] pyridine and n-propyl trimethoxysilane. MoS devices of Cu/SiO2/Si (001), with and without SAMs, were subjected to biased thermal stressing (BTS) at 200°C with a nominal electrical field of 2 MV/cm. The current-voltage and capacitance-voltage measurements showed that SAMs with an aromatic terminal group were effective adhesion promoters and barriers to Cu diffusion into SiO2. Figure 5.6 shows typical normalized capacitance C/Cmax versus voltage plots, one each in the as-deposited state, that is, tBTS
0, and one

Figure 5.6 Typical normalized capacitance C/Cmax vs. voltage plots of MOS capacitors with and without the self-assembled molecular (SAM) monolayer between Cu and SiO2, bias annealed at 200°C for duration tBTA with an electric field of 2 MV cm1. Open symbols without SAM; closed symbols with SAM.

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each at the electrical fields for both the control sample and the sample with a SAM between Cu and SiO2. The SAM-coated sample had a very small shift in capacitance-voltage plots, and a shift of only about 4 to 5 V was seen after 650 minutes, at which time the capacitor failed. On the other hand, capacitance-voltage plots shifted considerably with time for the control sample, and after 150 minutes, when it failed, the capacitance-voltage shift was greater than 18 V. In the control sample, the leakage current increases rapidly and continuously with time. On the other hand, in the SAM-coated sample, a relatively constant current leakage of about 10 to 30 nA/cm2 persists until failure (at 650 minutes), when current shoots up to 100 mA/cm2. The shifts in capacitance-voltage and the increased leakage currents are indicative of’ the diffusion of ionic Cu under electrical bias. These results seem to establish the effectiveness of SAMs as APDB materials. The barrier properties of SAMs were explained in terms of the size and configuration of the terminal group and the molecular chain length.[9] The larger volume occupied by the aromatic rings (compared with, for example, aliphatic groups) sterically hinder Cu diffusion between the molecules through the SAM layer. Self-assembled monolayers with long chain lengths apparently screen Cu atoms from the influence of the underlying SiO2, thereby preventing ionization and consequent acceleration by the externally applied electric field, as happens in the absence of an APDB layer between Cu and SiO2. Both approaches, the use of Mg-Cu or Al-Cu alloy and the use of SAMs, are very promising but will need validation in actual devices or circuits fabricated, aged, and tested in actual manufacturing conditions. Both offer APDB layers in thickness ranges of 1 to 2 nm or less. The Cu alloys offer near-Cu resistivities and will not act as an insulator between two vertical interconnections. The SAM layer may interfere with electrical conduction, and thus may require selective removal from the via bottom or an externally induced change in conduction behavior.

5.5.3 Zero-Flux Diffusion Zones (Multicomponent Diffusion Effects) My co-workers and I have taken a completely new approach in our research to address diffusion-related instabilities in metal interconnects.[69, 70] The concept of multicomponent diffusion effects is new to the area of metal interconnections. The diffusion behavior in multicomponent alloys (three or more elements) could be significantly different from what is observed in single-component and binary diffusion. In multicomponent alloys, the flux of each element depends on the concentration gradient of

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all the elements in the alloy. Hence, certain special effects such as uphill diffusion and zero flux planes are likely to be observed. Such effects could be used to limit and even inhibit the diffusion of a certain species within the alloy. One example of a multicomponent effect would be the occurrence of a zero-flux plane (ZFP). A ZFP is a region in the reaction zone of the diffusion couple where the flux of a component goes to zero.[71] The occurrence of ZFPs for Al or Cu in an Al or Cu alloy would imply that diffusion of Al or Cu is held in check by virtue of interactions with the interdiffusing alloying elements. We could thus limit the diffusive spreading of Al or Cu, thereby improving the resistance to electromigration and diffusioninduced interactions. A careful approach has been taken to select alloying elements to aluminum to achieve the desired characteristics.[69, 70] The research methodology is based on choosing homovalent alloying elements that will have negligible solid solubility in Al (a) to minimize the undesirable increases in resistivity associated with the scattering behavior of the solute atoms in Al, and (b) to cause “grain boundary stuffing”.[2] The search for these ternary (or higher component) alloys of homovalent metals (such as Al, In, and La) that exhibit so-called ZFPs is also, therefore, undertaken to explore the use of multicomponent diffusion effects in Al alloys. Note that the multicomponent concepts are not unique to Al alloys and can easily be extended to and thus explored in the Cu alloys. To study the existence of ZFPs and other multicomponent effects, a PC-based software program called Profiler is used. This program calculates the variation, with time and distance, of the concentrations of n species diffusing across the interface of the single-phase diffusion couple.[72] It requires the investigator to input the diffusivity [D] matrix. In the case of the ternary system, [D] is a 2  2 matrix. The elements of this matrix are interaction terms that describe the pair-wise interactions of elements in the presence of a third element. In general, the flux of component i is linked to the concentration gradients of all the elements in the alloy via the pair-wise interaction coefficients, Dij, Ji
Dij (∂cj
∂x).

(19)

If the off-diagonal terms of the [D] matrix are comparable in magnitude to the diagonal terms, we can expect strong interactions between the pair of elements for which the [D] matrix is considered. Such strong interactions among the elements lead to the occurrence of multicomponent effects such as the ZFP. We would have to conduct several interdiffusion experiments to determine the [D] matrix. A procedure has been developed to evaluate

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Figure 5.7 Schematic showing the terminal compositions in composition space.

all four elements of the [D] matrix by conducting a single interdiffusion experiment. Figure 5.7 schematically shows the terminal compositions in composition space. Terminal alloy compositions are represented by points P and Q in the unrotated orthogonal composition space. Schut and Cooper[73] point out that as long as the line PQ, joining P and Q, is not parallel to the eigenvector of the [D] matrix, an S-shaped diffusion path will result. Profiler is used to generate the penetration profile and the diffusion path for a given diffusion couple. The procedure developed for calculating [D] involves rotation of the axes to align one of the two axes parallel to PQ. Using the diffusion path from the actual experiment, the penetration profiles for alloys in the rotated composition space are determined. Appearance of extrema in the penetration curves for components in the rotated space enables the determination of all elements of the [D] matrix from three equations. These equations are: * c1 (l)



dc * dc*2 2ldc1*
D1*1 1  D1*2 , * dl dl c ( )

(20)

1

* c 21(l)



* c 2( )

dc * dc*2 2ldc*2
D*21 1  D*22 , dl dl [D*]
[B][D][B]1,

(21) (22)

where





Cosa Sin a [B]
Sin a Cos a . [B] is the rotation matrix, C*1,2 is the composition in the rotated space, and [D*] is the diffusivity matrix in the rotated space. At the extremum for a

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given component in the concentration profile, the derivative of concentration at that point will become zero. Thus two out of the four Dij will be determined. The other two diffusivity elements can be determined at the Matano interface. Clearly, the number of interdiffusion experiments required to generate concentration profiles that would be needed to determine the [D] matrix is minimized. Not only does this method save time and cost, it also avoids time-consuming calculations. The Profiler program can be used to simulate the concentration profiles that would develop during diffusion in several diffusion couples, eschewing the need to perform “real” experiments. Subsequently, the concentration profiles from each of the simulations are analyzed for ZFPs. If the ZFP for Al (or Cu) is predicted for a certain diffusion couple, the couple could be deposited and tested for interdiffusion of Al (or Cu) to verify the occurrence of the ZFP.

5.6

Brief Discussion of an APDB for Low-k ILD Materials

We are concerned with the metal diffusion into and metal adhesion to low-k dielectrics and vice versa. Low-k materials can be inorganic, polymers, polymer-inorganic blends, or porous materials. Thus there is a serious concern related to finding an APDB material that may be unique for a given low-k ILD or universal for all low-k materials. This discussion focuses on polymer and polymer-type materials only. Although there are differences in the adhesion behavior of a metal on a polymer and a polymer on a metal, the adhesion is generally weak. As mentioned in Sec. 5.1, we expect chemical bonding across the interface to provide a good adhesion. This, however, may be challenged by stress in the films and substrate and by the process parameters, such as shear stress during chemical-mechanical polishing. Interface stability also plays an important role in keeping adhesion properties constant for a reasonable time. Near-interface changes of polymer properties due to diffusion and/or precipitation processes can affect adhesion. Small atoms, like Cu, can form different precipitates and clusters in polymers. The metal/polymer near-interface regions can contain metal-related defects after thermal treatment, and metal can diffuse into the bulk of polymers (see the discussion below). Short-range diffusion (tens of nanometers) is, however, not expected to affect polymer properties and adhesion unless strong clusters form. There are many examples of process-affected and chemistry-dependent metal-polymer adhesion, such as interfacial oxide formation, metal-oxygen complex formation, carbide formation, and polymer cross-linking.

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Polymer and metal interactions are strong and lead to metal diffusion in most polymers.[74] Besides externally induced forces (thermal, electrical, chemical, and occasionally mechanical), the structure, crystallinity, porosity, and surface functional groups play very important roles in these interactions. The effect of an applied electric field on the polarization and charge generation within the dielectric is expected to influence the diffusion and interactions. With respect to diffusion in socalled crystalline polymers, the microcrystallites in polymers can be regarded as impermeable islands embedded in a continuum of permeable, amorphous media, and a nonuniform distribution of diffusant can be expected during thermal processes. In amorphous phases, diffusion and viscoelastic properties are expected to be strongly influenced by the amount of free volume. However, not enough experimental data are available on void distribution in most polymers as a function of preparation conditions. It is not surprising, therefore, that many theories have been developed describing diffusion in polymers. These theories can be classified as either molecular or free-volume models.[75, 76] Fractal models for describing diffusion in polymers are particular promising. While there are other theories of diffusion in polymers,[77] metallic diffusion cannot be described in terms of these models. Discussion of the relationship between polymer structure and diffusivity of species is beyond the scope of this chapter. The entire Chapter 7 in this book is devoted to this topic. Generally, metal atoms diffuse at the metal-polymer interface. However, there are cases where reactive groups of ions present on the polymer chain-react with the metal and produce species that diffuse in the metal. The reactivity is highest for oxygen containing moieties (for example,
C
O, COOH, CHO, COC), followed by those containing nitrogen (such as CN, NH2, and
NH). Alkyl groups have the least reactivity, although phenyl-type groups have comparatively higher reactivity.[78] Fluorine and other halogens also have considerable reactivity; a classic example is the interactions between Al films and fluorinated polymers.[79] The interactions sometimes lead to interfacial reaction products that form barriers to further interactions and diffusion. The literature is full of metal-polymer interactions, organometallic formations, and decompositions, and metal’s chemical reactivity has been reported to play a role in the diffusion (or in the drift under electrical bias) in polymers.[80–82] For example, Mallikarjunan et al.[80] studied a hybrid organosiloxane (OS) polymer with k
2.5 in conjunction with different gate metals such as aluminum, tantalum, and platinum, using C-V and BTS measurements on the metal/polymer/SiO2/Si capacitors. The results obtained with Cu and Al metals could not be explained satisfactorily with existing understanding; therefore, gate metals Ta and Pt were also investigated. Tantalum, pure or

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as a nitride, is useful as an adhesion promoter/diffusion barrier layer and is similar to aluminum. Copper is, comparatively, much less chemically active. Platinum, on the other hand, is a noble metal and is not expected to diffuse or drift into, or interact readily with, the dielectrics. The total number of charges obtained from measured C-V shifts under BTS increases in the order Pt, Cu, Ta, and Al, with no charges detected for Pt and maximum charge detected with Al as the gate metal. This is because both aluminum and tantalum react strongly with SiO2 and could be trapped by dangling bonds at the SiO2/OS interface. From the plot of the number of charges as a function of the heat of formation for the oxides of these metals per oxygen atom, it was concluded that the number of charges (or metal ions)/cm2 that reach the OS/SiO2 interface is directly related to the metal’s oxidation tendency as measured by the oxide’s heat of formation. Also note that the first ionization energies of Pt, Cu, Ta, and Al are 9.0, 7.726, 7.89, and 5.986 eV, respectively. These numbers are reasonably in line with the oxidation tendency. For metals such as Al, Ti and Ta that show strong reactions with SiO2, the formation of a stable metal-oxide diffusion barrier limits diffusion or drift into SiO2. However, in polymers with significant organic content, such a reaction may not occur. Instead, oxygen-containing groups may increase the ionization tendency of the metal and aid the drift process. Loke et al.[83] have used such a model to qualitatively explain Cu drift kinetics in different polymers. Low-k dielectric/metal technologies need a low-resistivity metal or a thin, low-k insulator as the APDB material suitable for the chosen ILD material. This is a real challenge, in view of the complexity of such ILD materials.

5.7

Summary

This chapter begins with a brief review of the early history of the diffusion barrier thin films used in the vintage semiconductor devices and circuits of the last century. It provides a short overview of the diffusion processes and the various influencing material factors that must be considered in the development of diffusion barriers. Special aspects of diffusion barrier materials recently developed to contact interlayer dielectric films with the main conductor are covered for modern circuits, notably the Cu metallization introduced in the industry during the past few years. In the context of Cu, an entirely new concept of using alloys with zero flux planes is introduced. Finally, recent innovations and novel concepts used in the adhesion-promoting diffusion barrier films for applications in Al and Cu metallization and low-k dielectrics technologies are discussed.

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25. T. Suwwan de Felipe, S. P. Murarka, S. Bedell, and W. A. Lanford, Thin Solid Films, 335:9 (1998) 26. S. Vaidya, D. Fraser, and A. K. Sinha, in Proc. 18th Annual Symposium on Reliability Physics, IEEE, Piscataway, NJ (1980), p. 165 27. S. Vaidya, T. T. Sheng, and A. K. Sinha, Appl. Phys. Lett., 36:464 (1980) 28. F. M. d’Heurle, Metall. Trans., 2:683 (1971) 29. G. V. Kidson, J. Nucl. Mater., 3:21 (1961) 30. R. J. Borg and G. J. Dienes, The Physical Chemistry of Solids, Academic Press, Boston (1992), chap. 11 31. E. D. Hondros, J. Phys., 36, coll. C4–117 (1975); also see M. P. Seah and E. D. Hondros, Proc. R. Soc. London, A335:191 (1973) 32. H. W. Paxton, in Alloying (J. L. Walter, M. R. Jackson, and C. T. Sims, eds.), ASM Int., Materials Park, OH (1988), p. 199 33. D. Gupta, Metall. Trans., 8A:1431 (1977) 34. D. McLean, Grain Boundaries in Metals, Oxford University Press, London (1957) 35. W. M. Hume-Rothery and R. B. Coles, Atomic Theory for Students of Metallurgy, Institute of Metals, London (1969), p. 120 36. M. M. Mandurah, K. C. Saraswat, C. R. Helms, and T. J. Kamins, J. Appl. Phys., 51:5755 (1980) 37. C. W. Allen and A. G. Sargent, Mat. Res. Soc. Symp. Proc., 54:97 (1986) 38. S. P. Murarka, I. A. Blech, and H. J. Levinstein, J. Appl. Phys., 47:5175 (1976) 39. S. P. Murarka and D. B. Fraser, J. Appl. Phys., 51:342 (1980) 40. B. Arcot, S. P. Murarka, L. A. Clevenger, Q. Z. Hong, W. Ziegler, and J. M. E. Harper, J. Appl. Phys., 76:5161 (1994) 41. V. Goselle and K. N. Tu, J. Appl. Phys., 53:3252 (1982) 42. D. S. Campwell, in Handbook of Thin Film Technology (L. I. Maissel and R. Glang, eds.), McGraw-Hill, NY (1970), chap. 12 43. R. W. Bower, Appl. Phys. Lett., 27:99 (1973) 44. J. M. Harris, E. Lugujjo, S. U. Campisano, M.-A. Nicolet, and R. Shima, J. Vac. Sci. Technol., 12:524 (1975) 45. R. S. Nowicki and I. Wang, J. Vac. Sci. Technol., 15:235 (1978) 46. M.-A. Nicolet and M. Bartur, J. Vac. Sci. Technol., 19:786 (1981) 47. V. Hoffman, Solid State Technol., 26:119 (1983) 48. M. Eizenberg, S. P. Murarka, and P. A. Heimann, J. Appl. Phys., 54:3195 (1983) 49. A. Berti and S. P. Murarka, Mat. Res. Soc. Symp. Proc., 318:451 (1994) 50. Jan M. Neirynck, thesis, Rensselaer Polytechnic Institute, Troy, NY (1996) 51. E. Kirchner, Ph. D. thesis, Rensselaer Polytechnic Institute, Troy, NY (1998) 52. S.-Q. Wang, in Conf. Proc. VLSI IX, MRS, Pittsburgh, PA (1994), p. 31 53. J. Li, Y. Schacham-Diamand, and J. W. Mayer, Mater. Sci. Rep., 9:1 (1992) 54. J. M. Poate, K. N. Tu, and J. W. Mayer, in Thin Films – Interdiffusion and Reactions, Wiley, NY (1978), p. 376 55. S.-Q. Wang, S. Suthgar, C. Hoeflich, and B. J. Burrow, J. Appl. Phys., 73:2301 (1993)

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56. S.-Q. Wang, I. J. Raaijmakers, B. J. Burrow, S. Suthgar, S. Redkar, and K.-B. Kim, J. Appl. Phys., 68:5176 (1990) 57. E. Kolawa, J. S. Chen, J. S. Reid, P. J. Pokala, and M.-A. Nicolet, J. Appl. Phys., 70:1369 (1991) 58. J. S. Reid, E. Kolawa, T. A. Stephens, J. S. Chen, and M.-A. Nicolet, in Abstracts of Advanced Metallization for ULSI Applications, 1991, MRS, Pittsburgh, PA (1992) 59. E. Kolawa, P.J. Pokela, J. S. Reid, J. S. Chen, and M.-A. Nicolet, Appl. Surf. Sci., 53:373 (1991); see also the same authors in Trans. 1st Int. High Temp. Electronics Conf., Albuquerque, NM, June 16–20 (1991), and in Abstracts of Advanced Metallization for ULSI Applications, 1991, MRS, Pittsburgh, PA (1992) 60. E. Kolawa, P. J. Pokela, L. E. Halperin, Q. T. Vu, and M.-A. Nicolet, in ASM International’s 3rd Electronic Materials and Processing Congress, ASM Int., Materials Park, OH (1990), p. 243 61. Handbook of Chemistry and Physics, 72nd ed., CRC, Boca Raton, FL (1991–1992) 62. J. S. H. Cho, H.-K. Kang, C. Ryu, and S. Wong, Proc. IEEE-IEDM, Cat. No. 93Ch3361-3, IEEE, Piscataway, NJ (1993), p. 265 63. B. Arcot, Y. T. Shy, S. P. Murarka, C. Shepard, and W. A. Lanford, Mat. Res. Soc. Symp. Proc., 203:27 (1991) 64. S. P. Murarka, Mater. Sci. Eng. (Reports), R19:87 (1997) 65. S. Hymes and S. P. Murarka, unpublished results 66. T. Suwwan de Felipe, S. P. Murarka, P. M. Ajayan, and J. Bonevich, in Proc IITC, IEEE, Piscataway, NJ (1999), p. 293 67. T. Suwwan de Felipe, Ph. D. thesis, Rensselaer Polytechnic Institute, Troy, NY (1998) 68. A. Krishnamoorthy, K. Chanda, S. P. Murarka, and G. Ramanath, Appl. Phys. Lett., 78:2467 (2001) 69. S. K. Kailasam, S. P. Murarka, M. E. Glicksman, and S. M. Merchant, Mater. Res. Soc. Symp. Proc., 514:113 (1998) 70. S. K. Kailasam , Ph.D. thesis, Rensselaer Polytechnic Institute, Troy, NY (1999) 71. M. A. Dayananda and C. W. Kim, Metall. Trans. A, 10A:333 (1979) 72. M. S. Thompson and J. E. Morral, Acta Metall., 34:2201 (1986) 73. R. J. Schut and R. A. Cooper, Acta Metall., 30:1957 (1980) 74. A.K. Chakraborty, in Polymer-Solid Interfaces (J.J. Pireaux, P. Bertrand, and J. L. Bredas, eds.), Institute of Physics Publishing, Philadelphia, PA (1992) 75. R.J. Pace and A. Datyner, J. Polym. Sci.: Polym. Eng., 20:51 (1980) 76. D. Ehrilich and H. Sillescu, Macromolecules, 23:1600 (1990) 77. S. P. Murarka, in Interlayer Dielectrics for Semiconductors (S. P. Murarka, M. Eizenberg, and A. K. Sinha, eds.), Elsevier, Kidlington, UK (2003), chap. 6 78. P. K. Wu, G.-R. Yang, X. F. Ma, and T.-M. Lu, Appl. Phys. Lett., 65:508 (1994) 79. G. C. Herdt, D. R. Jung, and H. W. Czanderna, J. Adhesion, 60:197 (1997)

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80. A. Mallikarjunan, S. P. Murarka, and T.-M. Lu, Appl. Phys. Lett., 79:1855 (2001) 81. T. P. Nguyen, H. Ettaik, S. Lefrant, and G. Leising, in Polymer-Solid Interfaces (J. J. Picreaux, P. Bertrand, and J. L. Bredas, eds.), Institute of Physics Publishing, Philadelphia, PA (1992) 82. Y. Hirose, A. Kahn, V. Aristov, P. Soukiassian, V. Bulovic, and S. R. Forrest, Phys. Rev., B54:748 (1996) 83. A. L. S. Loke, J. Wetzel, P. Townsend, T. Tanale, R. Vrtis, M Zussman, D. Kumar, C. Ryu, and S. Wong, IEEE Trans. Electron Devices, 46:2178 (1999)

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6 Reactive Phase Formation: Some Theory and Applications François M. d’Heurle and Christian Lavoie IBM T. J. Watson Research Center, Yorktown Heights, New York Patrick Gas Université d’Aix-Marseille, Marseille, France Jean Philibert Laboratoire des Matériaux, Université de Paris-Sud, Orsay, France

6.1

Introduction

In general, the expression “reactive phase formation” encompasses all phenomena whereby the reaction between two adjacent phases leads to the formation of one or several product phases. However, it is usually limited to cases where at least one of the reactants is solid, and that reactant acts as a substrate for the formation of a new solid phase. In current electronic technology, examples of such reactions would be the thermal oxidation of Si, the reaction of a metal film with single-crystal or polycrystalline Si to form one or several silicides, and the reaction of Cu, Ni, Au or some alloy thereof with Sn during soldering. Such phenomena are not only important in electronic technology; they are encountered in an endless variety of conditions, as in the high-temperature reactions between finely divided oxide particles to form cements. Limoge and Boquet[1] provide excellent general information about such phenomena. This chapter is composed of two main sections. In Section 6.2, the modalities of reactive diffusion are considered from a more or less theoretical point of view. The conditions leading to linear or linear-parabolic kinetics are analyzed, as well as those leading to nucleation-controlled reactions. A distinction is made between the formation of intermediate phases from reacting elements, which seems dominated by kinetic effects (diffusion), and subsequent reactions, when new phases are born from already formed phases. Then nucleation can play the dominant role in the process of phase formation and, in some cases, in the absence of such formation. Section 6.2 considers the process of phase formation with silicides that are of current technical importance in the electronics industry: TiSi2, CoSi2, and NiSi. This discussion either illustrates some of the points made in the earlier theoretical section, or complements such points, as when considering ternary effects with alloyed CoSi2. Most of the effects reported are well established. However, in order to be up to date, some

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tentative explanations are advanced in considering the nucleation of the C49 phase of TiSi2, or enthalpy effects in solutions of CoSi2 and NiSi2. This chapter is limited to binary systems, except for short references to ternary ones such as the solid solutions just mentioned.

6.2

Theoretical Considerations

6.2.1 One Phase Growing, Diffusion Controlled This chapter assumes that all phases are solid, so that, as sketched in Fig. 6.1, the reaction of A with B results in the growth of a compound AmBn according to: mA
nB  AmBn

(1)

that is driven by a decrease of free energy ∆G per mole of AmBn. It is often true that the growth of such compounds is dominated by the motion of only one atomic species, here assumed to be A. The atomic flux of A atoms jA

Figure 6.1 Schematic representation of a single phase (here, for example, AB) growing from the reaction of two elements, A and B. It is assumed that only the A atoms are mobile in the growing phase.

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across the growing layer (that is responsible for its growth) can be written as: jA  NA  (DA
kT)  (dmA
dx),

(2)

where NA is the density of A atoms in the compound (number of A atoms
cm3); DA is the diffusion coefficient of A atoms in AmBn, so that according to the Einstein formulation, DA
kT is the mobility of these A atoms; and dmA
dx is the driving force on these atoms, the chemical potential gradient. (Be careful; it is not the self-diffusion coefficient of A atoms in an A matrix. Although this is obvious, mistakes are often made.) If DA is constant across the whole thickness of the growing layer, or is a value averaged over the thickness L of the layer, Eq. (2) can be written as: jA  NA  (DA
kT)  (∆GA
L),

(3)

where ∆GA  ∆G
m, the change of free energy accompanying the transfer of one mole of A atoms from one interface to the other and resulting in the formation of 1
m mole of AmBn. If V is the volume per mole of AmBn, the transfer of a mole of A atoms across the layer causes an increase in the volume of the growing layer of VA, equal to V
m. Since the rate of growth of the layer dL
dt is equal to the product jA  VA, and NA  m
V, Eq. (3) yields: dL
dt  (DA
kT)  (∆GA
L).

(4)

Hence, L is proportional to t1
2, implying parabolic growth controlled by diffusion. To be accurate, we should pay attention that the equivalence between dmA
dx and ∆GA
L in going from Eqs. (3) to (4) is precise only when the solubility limit of B in A is very low, and the range of composition of AmBn is very narrow. In general, if the reactant and the product phases are solid, the usual relation ∆G  ∆H is valid, so that we can use enthalpy values (enthalpy of formation) if free energies are not available. Writing Eqs. (1) through (4) required making a number of assumptions that may not be valid in all cases: (1) that the solubility limits in both A and B are small (negligible), and (2) that the concentrations of A or B atoms do not vary significantly from one interface to the other. The treatment here follows the Nernst-Einstein Eq. (2) rather than the more usual Fick’s law. One advantage is that in Eq. (2), the various factors that contribute to jA are clearly separated and easily identifiable. Another advantage is that if additional forces are applied to the moving atoms, for example, in electrical, stress, or magnetic fields, they can simply be added to ∆GA
L, provided that the forces are expressed in the correct units. That cannot be done, except with extreme care, using Fick’s formulation, and carelessness may lead to extremely erroneous conclusions. Using Fick’s law, jA  N  DFA  (dcA
dx),

(5)

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and using the same procedure as for the derivation of Eq. (3) from Eq. (2), we obtain: jA  N  DFA  (∆cA
L),

(6)

where DFA is the Fick’s coefficient of diffusion for species A; ∆cA (dimensionless) is the change in concentration from one interface to another; and N is the atomic density in the compound. One of the problems with the use of Fick’s law is that generally ∆cA is poorly known, particularly when it is very small. Unfortunately, reliance on Eq. (6) has led to statements in the literature to the effect that in comparing the rates of growth of two intermediate layers, the one with the greatest ∆cA (or the equivalent ∆cB) will have the fastest rate of growth, which is absurd. For example, for a phase with composition limits very close to each other, namely, with ∆c approaching zero, a finite rate of growth would be possible only if DF became nearly infinitely big, which is clearly a physical impossibility. In reality, D may increase significantly with deviations from stoichiometry (because of the formation of defects, vacancies, and antisite atoms), as in AlNi[2] or AlCo,[3, 4] but then the increased growth rate results from the extremely high value of D, not from the minimally increased value of ∆cA. When writing Fick’s relation, Eq. (5), the diffusion coefficient was written with the superscript F to distinguish it from D, truly DNE, in the Nernst-Einstein formulation [Eqs. (2) through (4)]. The two coefficients, DNE and DF, are not equal at all, except in the case of ideal solutions, which have little to do with the intermediate compounds presently under consideration. The NernstEinstein coefficient is the coefficient measured if we follow the diffusion of a radioisotope of A in AmBn, and is thus truly representative of the diffusion of that element in the compound. Years ago, Herring[5] pleaded for expressing diffusion relations in terms of free energy rather than concentration: “It has long been recognized that it is more appropriate to write the diffusion equations for multicomponent systems with chemical potentials, or their equivalents, rather than with concentrations, and this fact has recently been the subject of a number of papers in the metallurgical literature. The most recent, and probably the most satisfactory, of these is that of J. Bardeen.[6]” In spite of the justly earned reputation of the two authors, what they had to say on this subject remains largely ignored, 50 years later. If both atomic species are mobile simultaneously in the growing compound, we would simply add the contribution of the B atoms to the overall growth rate: dLB
dt  (DB
kT)  (∆GB
L).

(7)

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With ∆GA  ∆G
m and ∆GB  ∆G
n, the sum of Eqs. (4) and (7) yields: dL
dt  [1
(kTL)]  (1
cA
1
cB)  (cBDA
cADB)  [∆G
(m
n)],

(8)

where the concentration cA (not per unit volume, but from 0 to 1) of A atoms is m
(m
n), that of B atoms n
(m
n), cBDA
cADB is the wellknown Darken coefficient of diffusion, and ∆G
(m
n) is the free energy change per g atom (Avogadro’s number of atoms, not per mole of compound) upon reaction 1.

6.2.2 Two Phases Growing Simultaneously, Diffusion Controlled A fairly complete analysis of this subject is presented in the literature[4] for readers who want a more thorough presentation than the one offered here. In reacting two elements A and B together, we might anticipate the simultaneous growth of all the phases present in the equilibrium diagram at the temperature at which the reaction is being carried out. For such a general case, we usually refer to two treatments,[7, 8] but there are others, for example, specifically on Mo-Si[9] and V-Si[10] reactions. Here we consider only two phases, AmBn and ApBq, growing simultaneously, as illustrated in Fig. 6.2. In this case, the growth of either one of the two phases is dependent (again assuming that only the A atoms are mobile) not only on the flux of atoms in the one phase, but also on the flux in the other phase. This is because each phase simultaneously grows at the expense of the other phase and is consumed by that phase. Hence: dL1
dt → f(
jA1  jA2),

(9)

where the subscripts 1 and 2 refer to the two growing phases. In writing the expression for jA1, care must be taken that the force on the moving atoms is no longer ∆GA obtained from Eq. (1), but ∆GA1 corresponding to Eq. (10), since the A-rich phase 1 does not grow from B but from the relatively A-poor phase ApBq. (n
q)ApBq
[m  (np
q)]A  AmBn.

(10)

The diffusion coefficient DΑ1 (DNEA1) remains the same as previously, DA, since phase 1 remains identical to itself, although not entirely. Were we to use Fick’s law, care should be taken that ∆cA1 for two-phase growth is somewhat smaller than ∆cA for single-phase growth, because

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Figure 6.2 Schematic representation of two phases (here, for example, A2B and AB) growing simultaneously from the reaction of A with B. It is assumed that only the A atoms are mobile.

in this latter case, AmBn would be taken as in equilibrium with B, and in the other with ApBq. However, DFA1 would still be different from DFA. From Eq. (10), ∆GA1  (1
m)  [∆GfAmBn  (n
q)  ∆GfApBq)],

(11)

where ∆GfAmBn is the free energy of formation of one mole of AmBn, the same as ∆G for Eq. (1), and ∆GfApBq is the free energy of formation of ApBq. In general, ∆GA1 will not be too different from ∆GA, perhaps a factor of 2 or 3, which is small with respect to variations in D’s; for example, DA1 versus DA2, which may differ by several orders of magnitude. But that is not always true. A brief mental estimation shows that if the difference in composition between AmBn and ApBq is small, for example, m  n  1, and p and q to 4 and 5, respectively, ∆GA1 can be very small, and the product DA1  ∆GA1 is very small also. Thus thermodynamics may affect the rate of growth of a phase, depending on whether it is growing alone or in competition with other phases; that is true even if such growth effects are dominated by variations in D values from phase to phase. In calculating ∆GA2 for the flux of A atoms in phase 2, we should consider that the motion of A atoms results in the formation of ApBq at the AmBn
ApBq interface by removal of A atoms from AmBn, as well as at the

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ApBq
B interface by the addition of A atoms to B. With proper care, the simultaneous growths of the two phases can be expressed as: dL1
dt  a
L1  b
L2

(12a)

dL2
dt  d
L2  g
L1.

(12b)

and

It is easy to show that L1, L2, and L1
L2 are all proportional to t1
2; growth remains parabolic, and the phases start to grow at time zero. The terms a and d are equal to, or of the form, DA1  ∆GA1 and DA2  ∆GA2, respectively. If w is the ratio of the volumes per A atom in phases 1 and 2, equal to VA1
VA2, then b and g contain, in addition to the terms found in a and d, the factors w and 1
w, respectively. This is because b and d correspond to the effects of diffusion in phases 2 and 1 on the growths of phases 1 and 2. The simultaneous growths of phases 1 and 2 require that Eqs. (12a) and (12b) should be positive, which leads to: a
b  L1
L2  g
d.

(13)

If the differences between a, b, g, and d are entirely dominated by differences between DA1 and DA2, or in general between D1 and D2 (including terms for the diffusion of B atoms), then L1
L2 should be a function of D1
D2. If that ratio is big, at the beginning when L1 remains small, L2, although mathematically greater than zero, would remain physically meaningless if smaller than about 1 nanometer. Actually, phases may not start to grow simultaneously, even in the case of pure diffusioncontrolled growth. That may not always be due to purely kinetic factors, since thermodynamics may also dictate occasionally that some of the terms a, b, g, and d differ widely from one another. Calculations[6] show that the ratio L1
L2 is a constant from time zero if we start with pure A and B, and tends toward this constant if the sample is artificially modified, for example, by the insertion of some phase 1 or 2 between A and B. Physically, things may behave differently for a number of reasons. Consideration of Eqs. (l2a) and (12b) reveals that, if such a sample were prepared, the rate of growth of the inserted phase at time zero should be negative. It also follows from these two equations that, mathematically, an existing phase cannot be made to disappear, because when the thickness of a phase tends towards zero, its rate of growth tends toward infinity.

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6.2.3 Linear Parabolic Kinetics, One-Phase Growth, Oxides, Equilibrium, Plotting of Data One problem with diffusion kinetics is that for thin layers, the kinetic rates tend toward infinity, which is a physical impossibility. With thin metal films reacting with silicon substrate, experimentally determined kinetics follow the pattern in Fig. 6.3. A plot of the square of the thickness versus the time[11] is correct for the growth of Ru2Si3, the only silicide observed to grow in the reaction Ru-Si from a thickness of Ru smaller than 100 nm. The data points for the diffusion-controlled growth do not extrapolate to the origin. What occurs during the initial, so-called incubation period is often not known; when known, it tends to vary with the mode of preparation of the samples (for example, deposition via evaporation or sputtering). For the phenomenological description of interest here, the details do not matter greatly. It is sufficient to state that initially, the layer grows as a function of tn, where in order to have a finite rate of

Figure 6.3 Plot of the growth of Ru2Si3, correctly done as thickness squared versus time. Note the positive time interception near the origin of the horizontal axis. This corresponds to a period when overall growth proceeds as tn, with n equal to or greater than 1. Unless precise information is obtained during that initial period, we cannot know whether it corresponds to rate limitation from interface reaction. From Petersson et al.[11]

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growth dL
dt for t close to zero, n must be greater or equal to 1. In the absence of either precise information or a proper description of the kinetics during this initial period, it is simpler to take n equal to 1. As will be shown in Sec. 6.2.4, this leads to linear-parabolic kinetics. In cases not so well-known or defined, when n is indeed equal to 1, the initial growth kinetics would be dictated by the rate of reaction. By reference to Eq. (4), the growth law is then expressed as: dL
dt  a  [1
(L
a
K)].

(14)

If L is large with respect to a
K, Eq. (14) becomes identical to Eq. (4); if L is small, the rate becomes equal to K. This latter parameter is referred to as the reaction rate constant. It is indeed meaningful when the initial stage of growth is limited by such an interface reaction. However, as understood here, it should be taken more as a phenomenological constant that provides a justifiable account of what happens at the beginning of a phase formation when the rate cannot be infinite, so that experimental observations can be plotted as in Fig. 6.3. The quantity a
K has the dimension of length, so that commentaries about Eq. (14) include statements to the effect that this equation corresponds to two growth stages: the first with linear growth, until a thickness a
K, and a second stage with parabolic kinetics. Such statements are erroneous, on L and t coordinates. Eq. (14) defines a parabolic curve with a shift of origin so that at time zero, dL
dt assumes a finite value, K; there is no discontinuity at L  a
K.[12] Nevertheless, Eq. (14) leads to what is known as linear-parabolic kinetics, often expressed as in the well-known formula for the growth of silicon oxide:[13] L2
AL  Bt,

(15)

where A
B  1
K and B  2a. When K has a physical meaning, in solid state reactions, we may think of it as referring to diffusion across an interface, somewhat akin to the situation during recrystallization, with relaxed phase growth driven by the difference in free energy between the two regions. Then diffusion across an interface of constant thickness is also constant, hence the finite value of K. But in this sense, it is some sort of diffusion constant and does not have the precise chemical meaning that an interface reaction rate implies. Things are different when one of the reactants is a gas, for example, oxygen during oxidation. Then one of the components of K is the rate of dissociation of the molecular gas, from O2 to atomic oxygen, that is necessary to incorporate oxygen in a solid oxide (or H2 and hydrides, and so forth).[14] It is paradoxical that the kinetics of oxide growth, which are often identified with the thermal oxidation of Si,[13] and should therefore be

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plotted as in Fig. 6.3 (namely L2 versus t), and extrapolate to some finite value of t for L  zero, do not in fact satisfy the conditions for such a plot. Instead, if we want to extract proper values for long-range diffusion constants, they should be plotted as L versus t1
2, because extrapolation gives a positive value of L for t  0. The physical cause of this apparent paradox is that, initially, long-range oxidation is preceded by a period of anomalously fast oxidation, the source of which is usually not analyzed in detail.[12] There is little doubt, however, that this complexity arises as follows: The usual analysis of phase formation kinetics, as above, is based on the assumption that the reactants remain in their equilibrium state. However, this assumption is invalid during the initial stage of oxidation, which is driven by very high free-energy changes. When the oxide becomes thick enough and the rates slow enough, the reacting surfaces (interfaces) return to equilibrium. Initially, however, the rate of oxidation is too high to allow the substrate atoms to relax to equilibrium positions and states before being consumed by the growing oxide. This results in supersaturation of defects, mostly vacancies during metal oxidation, and vacancies as well as interstitials during Si oxidation, that indeed manifest their presence in different ways, for example, dislocation loops, inside the substrate itself. During that period, the equilibrium values of ∆G are not correct. Excess values lead to enhanced growth rates and to the necessity of plotting the data as L versus t1
2.

6.2.4 Linear-Parabolic Kinetics: Sequential Phase Growth, Grain Boundary Versus Lattice Diffusion If several phases are likely to grow, then Eqs. (12a) and (12b) must be written as: dL1
dt  a
[L1
(a
K1)]  b
[L2
(b
K2)]

(16a)

dL2
dt  d
[L2
(d
K2)]  g
[L1
(g
K1)].

(16b)

and

If phase 1 forms first, the growth of phase 2, from Eq. (16b), requires that: d
[L2
(d
K2)]  g
[L1
(g
K1)],

(17)

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which, for L2  0, yields: L1c  d  (1
K2  1
K1).

(18)

Phase 2 cannot grow until the first phase has reached a critical thickness L1c, which increases as d (proportional to D1  ∆G1) increases, and decreases as the initial rate of growth of phase 2, K2, increases. (If that rate is infinite, as in the case of pure diffusion control, then L1c  0.) It becomes zero already if K1  K2; namely, there is no critical thickness for second-phase growth if the maximum growth rates of the two phases are equal, which is indeed as anticipated. An increase in K1, like an increase in d, increases L1c. The literature on silicide formation relating to the reactions of thin metallic films deposited either on single-crystal or polycrystalline Si bears ample witness to the fact that phases do indeed form sequentially.[15, 16] With a 200-nm-thick film, growth of the first phase consumes all of the metal present before a second phase begins to grow, for example, Ni2Si or Pt2Si, then NiSi and PtSi, implying that L1c is larger than or of the order of 300 nm.[17–19] As was emphasized above, such evidence is no confirmation of a true reaction rate limitation at the beginning, so we should not argue too much that no apparent activation energies have been determined for this initial stage of silicide growth. What is certain is that when plotted as in Fig. 6.3, the data show time intercepts that come closer to the origin as the temperature increases. Thus, if we compare observations made on thin films, at relatively low temperatures, with observations on bulk specimens, usually examined on a larger scale and consequently at considerably higher temperatures, in the latter case, the sequential formation of phases is usually ignored. This is probably more the result of the mode of observation than of a true difference in behaviors between thin film and bulk samples, which indeed obey the same laws except in extremely rare cases (generally with extremely thin layers, below 50 nm). With thin films, the interfaces tend to be quite clean. Retardation of phase formation in bulk samples because of interface impurities, such as oxide with Al or Ti samples, is not discussed here. In discussing diffusion, the question of whether we refer to lattice or grain boundary diffusion has been eluded. That was true also when writing the corresponding equations, which remain the same except for a change in the value of D. Thus from the kinetics alone, it is difficult to decide whether growth is due to one or the other of these two diffusion mechanisms, since if the grain size of the growing layer remains constant, the time exponent 1/n of the growth law remains 1
2. This ceases to be true, however, if grain boundary diffusion is dominant and the grain size increases during compound growth; then 1/n becomes smaller

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than 1
2 (because diffusion slows down as the grain size increases). An example is found in Sn-Ni reactions (for soldering) where a time exponent smaller than 1
2 for the growth of Ni3Sn2 is attributed to grain growth.[20] The exponent reduces to 1
3 if the grain size is proportional to the thickness of the layer, and even to 1
4 for normal grain growth, where the driving force for grain growth is inversely proportional to the grain size.

6.2.5 First Phase Formed A lot of ill-spent ink has been poured on the question of what is the first phase to form. Although it would be prudent to avoid the question altogether here, it is nearly impossible to do so. According to what has been written in Sec. 6.2.4, when a new phase is formed from the reaction between two elements, it is generally true that the free energies of formation of the different phases do not differ much from each other: factors of 2, 3, or perhaps 4, but rarely more than that. Thus the competition between phases as to which would grow first depends on the product Di  ∆Gi. It would be dominated by Di’s, which differ much more widely than ∆Gi, so that the first phase to grow would be the one with the highest D,[21] or to be more precise, the highest value for D  ∆G. An extension of that statement based on some observations about the relative diffusion of the two atomic species in binary compounds, somewhat systematized as the “ordered Cu3Au rule,” predicts that when there is a strong difference in melting points between the reactants A and B, the first phase to form would be a phase rich in the element with the low melting point, probably the one richest in that element.[22] A more sophisticated approach,[23] based on irreversible thermodynamics, implies that the first phase to grow should be the one with the highest rate of free energy dissipation, namely, D  ∆G2. However, inasmuch as the ∆G’s vary little from phase to phase in the initial reaction, the phase with the highest D  ∆G2 product is likely to be that with the highest D. Regarding the scale at which observations are made, the observations above would seem to be valid for layers about 20 nm thick. At a smaller scale, we might need to look into nucleation phenomena, about which little has been said until now. With silicides, one can examine the process of formation of amorphous phases reviewed by Sinclair.[24]

6.2.6 Nucleation of the First or Second Phases In Sec. 6.2.6, the description of phenomena that occur during what can loosely be called the incubation time has remained deliberately vague,

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except that the overall kinetics may be described as proportional to tn, with n usually chosen for the sake of simplicity as 1, regardless of what is physically happening during this period. Considerable progress was made with the discovery that the actual phase growth is preceded by an earlier process related to nucleation;[25] the key to experimental success here was the use of multilayered films that magnify the initial stage of a reaction in comparison with the second stage, which is growth. During heating at a constant heating rate, the evolution of heat occurs as shown in Fig. 6.4. The first peak at low temperature results from the nucleation of the new phase (in the case of silicides, often an amorphous one), which nucleates first, then spreads along the original interface while maintaining a small constant thickness. It is to be noted that the magnitude of the first exothermic peak in Fig. 6.4 is too large to be due entirely to nucleation, which

Figure 6.4 Heat evolution as a function of time during heating at a constant rate of a reacting diffusion couple in a differential scanning calorimeter. There are usually two peaks. The first, at low temperature, corresponds to a nucleation step and often a lateral growth of the compound along the interface. The second peak corresponds to what is usually considered growth itself. Very interesting results have been obtained using multilayers that allow the relative magnitudes of the two peaks to be affected almost at will. Does the picture thus obtained provide a balanced view of what happens in an ordinary simple diffusion couple?

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contributes only a fraction of the total heat evolved in this first stage. Such thermal behavior is not limited to silicides; it is also encountered with other systems, such as Al-Co.[26] Although the literature should be studied on a case-by-case basis, it is probably correct to say that we still need a clear analysis of the transition between what occurs in stage 1 and the normal phase growth in phase 2, as described by Eqs. (4), (8), (12a), and (12b). Considerable work has been published regarding the nucleation that occurs during peak 1 (Fig. 6.4). This research suffers from a number of difficulties, many of which are common to all studies of nucleation phenomena. The activation energy for nucleation ∆G* is the sum of two terms: ∆G*  ∆G*th
∆G*kin,

(19)

where ∆G*kin is the activation energy for the growth of a nucleus of critical size. In some cases, it may be equal or close to the activation energy for growth that occurs later, but conceptually it is different. The other term ∆G*th has the form: ∆G*th → ∆s3
∆G2,

(20)

where ∆s is the increase in interface energy on the formation of a nucleus of critical size, and ∆G is the decrease in free energy (which should be expressed here per unit volume) on the formation of the new phase. The signs of ∆s and ∆G are opposite, and for small nuclei, the first term is dominant. Thermodynamics dictate that the equilibrium density of nuclei of critical size should be proportional to exp(∆G*th
kT). The rate of nucleation will necessarily be proportional to that quantity, yet nucleation does not occur unless the critical nuclei grow. Therefore, as written in Eq. (19), the activation energy for nucleation is the sum of two terms. Diffusion theory states that the activation energy for diffusion via a vacancy mechanism is also the sum of two terms: the energy for the formation of a vacancy, plus the activation energy for the motion of these vacancies. The case of nucleation considered here very closely parallels diffusion. Conceptually, the first term for the activation energy for nucleation ∆G*th is the easiest to manipulate, at least on paper. In principle, we know what surface and interface energies are, so that we have an idea what ∆s is. In principle also, we can assume that ∆G [in Eq. (20)] is the equilibrium value for the formation of a new phase, as written in Eq. (1). Or, going

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further, we can avail ourselves of more precise calculations of phase diagrams (CALPHAD) that will take into account the shape of the free energy versus composition curves for the product and reactant phases. The situation for ∆G*kin is more ambiguous, because the formation of the nuclei may occur along several competing paths with several different activation energies, especially when considering the formation of a compound new phase. In fact, the conditions for any reliable calculations are often quite desperate. Information about surface energies are usually lacking, and we are unsure of the extent to which bulk or equilibrium free energies are valid for submicroscopic particles. Nevertheless, it is quite current (although wrong) to analyze activation energies for nucleation purely with respect to ∆G*th only, neglecting ∆G*kin, or assuming that in considering the rate of nucleation, ∆G*kin is simply lost in the form of exp(∆G*kin
kT) among the preexponential terms in front of exp(∆G*th
kT). There may be some justification for neglecting the kinetic term for nucleation, for example in the study of the precipitation of Co atoms in a supersaturated solid solution of Co in Cu. That may occur via homogeneous or heterogeneous nucleation, and in comparing the two, we can assume that ∆G*kin remains constant, so that experiments and calculations can be conducted with focus on ∆s (and therefore on ∆G*th) only. When ∆G in Eq. (20) is very small, ∆G*th becomes very large, so that qualitatively no harm is being done if ∆G*kin in Eq. (19) is neglected.[27] However, when ∆G*kin is neglected in attempting to determine which phase forms first, for example, in the Ni-Al reaction,[28] which will be used presently to illustrate difficulties encountered in the analysis of competitive nucleation between phases of widely different compositions. The Ni-Al system is almost ideal, because it is nearly completely symmetrical, with NiAl in the middle, Ni3Al and NiAl3 on opposite sides of the diagram, plus two intermediate phases, Ni5Al3 and Ni2Al3. The enthalpies of formation of the two extreme phases Ni3Al and NiAl3 are very close: 153 and 150 kJ/mol, respectively.[29] (This is surely within the limits of accuracy of the measurements.) To demonstrate that NiAl3 should form preferentially to Ni3Al, the difference of enthalpies between the terminal solid solutions is used: Al is somewhat soluble in Ni; Ni in Al is not. It follows that the free energy of the Al in Ni solid solution is somewhat lower than Ni in Al. Therefore, the ∆G (or ∆H) for the formation of Ni3Al is somewhat smaller than ∆G for the formation of NiAl3. This leads to a somewhat greater value for ∆G*th, and the preferred formation of NiAl3. However, considering that in one case the phase is mostly composed of Ni and in the other mostly of Al, it is likely that the differences in ∆G*kin would be 1 eV or more (because of the large difference in melting points and consequent large difference in the

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respective diffusion coefficients). Moreover, the enhanced thermodynamic stability of a solid solution of Al in Ni, as opposed to that of Ni in Al, implies that in the first case Al atoms will favor Al-Ni bonds, that is, surround themselves with Ni atoms. Thus, since Ni3Al has the L12 structure, each Al atom in solid solution in Ni would constitute a potential Ni3Al unitary cell. It is hard to understand that such a configuration would not favor the nucleation of Ni3Al, independent of what the mathematics of ∆G*th imply. We must also consider the fact that the nucleation of a new phase is accompanied by sharp concentration and therefore also free energy gradients because the very first phenomenon to occur is the superficial saturation of A with B and B with A.[30, 31] Figure 6.5 shows that in such concentration gradients, nucleation would happen preferentially in the phase, for example, A, where the concentration gradient is minimum, because the diffusion coefficient is maximum. Thus the phase to be nucleated would likely be rich in A. From either Eq. (19) or Fig. 6.5, we conclude that the question of the first phase to be nucleated is dominated by kinetic considerations rather than by variations in ∆G*th. Moreover, if a phase with small diffusion coefficients is nucleated first, it is likely to be immediately consumed by another phase with faster kinetics, which has appropriately been called vampire phase.[32] The discussion is carried out in terms of nucleation,[32] but very much the same conclusions were reached earlier by simple considerations of competitive phase growth.[33] Thus it does not seem that the analysis of nucleation is very helpful in understanding much of the process of phase formation from reacting solid elements. Whether this statement is generally valid or not, it is supported by a systematic analysis of the reaction of Ti with Al, where experimental observations could be understood in terms of competitive phase growth alone.[34] As anticipated, the first phase formed is rich in the element with the lowest melting point, Al. Not much more can be said about nucleation in the present context. It was emphasized above that whether we consider competitive phase growth alone or nucleation in a concentration gradient (the usual condition), we are likely to predict that the first phase to grow will be rich in the element with the lowest melting point. Further, in the reaction of Al with Ni, NiAl3 and Ni2Al3 are rich in Al, so that in Eq. (19), the ∆G*kin for these two phases might be quite similar. Then in comparing NiAl3 versus Ni2Al3, ∆G*th might play a significant role, quite different from the role it would have in comparing NiAl3 with Ni3Al. But in comparing differences in ∆G*th, we are obliged to factor in the role of elastic energy ∆Helas, which can significantly affect the value of ∆G and about which extremely little is known. In the absence of firm information about ∆s, that term also remains somewhat unknown. We are left with experimental

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Figure 6.5 Schematic representation of the initial step in a reaction between two elements, A and B. The elements are assumed to have symmetrical thermodynamic properties, as shown (the same mutual solubilities, ceA and ceB, at equilibrium, or cmA and cmB, with the two elements in contact with each other), and similar compounds (for example, A3B and AB3, not yet nucleated in the graph above), but different diffusion coefficients. (The diffusion coefficient of B in A, DBA, is approximately equal to the coefficient of self-diffusion, DAA. Both are much greater than the corresponding coefficients in the other element, B.) The driving force G [Eq. (20)] for the nucleation of an A-rich or B-rich compound is proportional to the blackened areas on each side of the interface. Nucleation would then occur first on the side of the equilibrium diagram where the element has a high diffusion coefficient. It follows that nucleation tends to be dominated by kinetic factors similar to those that control growth. The reason for this is that, according to the Cu3Au rule,[22] if DAA is much greater than DBB in general, DAA3B would be much greater than DBAB3, and these latter coefficients control growth.

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observations, which show that the reaction of Ti with Al produces metastable forms of TiAl3 with structures at variance with the equilibrium structure,[35] while Ni-Al reactions can also lead to a metastable phase with a composition not encountered in the equilibrium diagram Ni2Al9.[36, 37] (This appears to be in accordance with the Cu3Au rule[22] since, other things being equal, Al should be more mobile in Al9Ni2 than in Al3Ni.) Yet it is likely that nucleation processes play a role in determining which one of these phases is formed. However, conceptual analysis has been of little help in resolving this problem. Small differences in sample preparation and heat treatment affect the observed results, which does not simplify any realistic analysis. The complexity is increased in a system such as Al-Pd, where many aluminum-rich compounds are stable; reports of the first phase to form tend to change with the investigators.[38] Whatever role nucleation plays in the selection of the formation of these first or second phases, the relatively large ∆G in Eq. (1) [although it is not the precise value that should be entered in Eq. (20)] results in a relatively easy nucleation process and in a great abundance of nucleation sites. A practical effect of the numerous nucleation sites is that for layers at least 10 nm thick, the interfaces are planar and parallel, as in Figs. 6.2 and 6.3. This is particularly noticeable with silicide reactions, where the Si substrates are perfectly specular to begin with, and the samples remain equally specular after the deposition of some 100 nm of Ni, as well as after annealing to form Ni2Si and then NiSi. There are two general aspects to be considered: (1) Reactive phase formation is usually dominated by diffusion considerations, rather than nucleation. (2) In the same way, kinetic factors, notably the growth modulated by diffusion, dominate over thermodynamic ones. The fact that the first phase to grow does not seem to depend significantly on the state of the reactants (solid, liquid, or gas) provides experimental support for the first point. For example, the disilicides NbSi2 and MoSi2 are obtained from the reaction of thin metal films with Si substrates.[39] However, the same products are obtained from the reaction of the metal with molten salt[40] or with SiH4 (a gas),[41] respectively. The compound Cu6Sn5 is obtained indifferently whether Cu reacts with molten or solid Sn.[20, 42, 43] Although thermodynamics is not unimportant, it indicates mostly what cannot happen: A system cannot evolve in such a way as to increase the overall free energy level. In the reaction of Mo with SiH4, MoSi2 cannot be formed if the pressure of SiH4 and, consequently, the activity of Si are too low. Gagliano et al.[43] provide a very good example of an experimental study of nucleation in the reaction of Cu with liquid Sn. The potential role of nucleation at the start of a reaction was theoretically investigated by Gusak et al.[44]

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6.2.7 Nucleation-Controlled Reactions and Consequences: Sequence of Phase Formation, Bulk Samples, Stresses If we follow the example of the formation of Ni silicides, we find that with excess Si, the equilibrium phase should be NiSi2. However, the formation of the equilibrium phase does not occur smoothly after the formations of the two previous phases. These processes were diffusion-controlled and occurred at about 350 and 400°C, respectively. But once NiSi is formed, further heating does not cause any significant change until a temperature above 750°C is reached. Then suddenly, and apparently explosively, the sample transforms to NiSi2, with the sample assuming a new frosted-glass appearance that is quite visible to the naked eye. The nucleation of the new phase in this case was so difficult that it required a relatively high temperature, where for samples of the order of 400 nm thick, diffusion was no longer rate-limiting. The reason is simple: The overall ∆G (∆H) for the formation of Ni2Si from Ni and Si is 143 ± 11 kJ/mol; that for the formation of NiSi from the reaction of Ni2Si with Si is 18 ± 9 kJ/mol;[27] but for the reaction: NiSi
Si  NiSi2 ,

(21)

the ∆H (∆G) is only 5 ± 20 kJ/mol. It is so small that it is not easily accessible from experimental measurements. When the ∆H’s and ∆G’s become small, it is no longer correct to consider that the two sets of values are identical.[45] However, what matters presently is that both values are indeed quite small, so that from Eq. (20), ∆G*th becomes very large. The practical consequences of the high temperature required for nucleation to overcome this problem, the paucity of nuclei, and the rough surface of the silicide layers are considered in Sec. 6.3. Even without knowing the quantitative details of the nucleation process, games can be played: Replacing monocrystalline Si with amorphous Si does not significantly change the formations of either Ni2Si or NiSi, but the formation of NiSi2 becomes diffusion-controlled.[46] The increase in ∆G, for Eq. (21), equal to the difference in free energy between crystalline and amorphous Si, is enough to cause a change in the mode of formation of NiSi2. Errors have been made in interpreting reactions such as the formation of NiSi2, where the sudden occurrence leads to the conclusion of a reactive explosion. This is wrong. The nature of the reaction is dominated by a small ∆G and ∆H, and cannot therefore be truly explosive, especially on Si substrates with a high coefficient of thermal conductivity, which limits any self-

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heating that may occur. The modalities of nucleation-controlled formation in the silicides are detailed in the literature;[26] some consequences are summarized below. With thin films, and experiments conducted at relatively low temperatures, difficult nucleation can affect the order of phase formation. A case in point is offered by comparing the sequences of phases in Rh and Ir reactions with Si. The two metals are quite similar and are found just below one another in the same column of the periodic table. Hence the respective equilibrium diagrams display remarkable similarities yet are not totally identical. With either metal, one step in the sequence of phase formation is the monosilicide (either RhSi or IrSi); then the similarity ends. With Ir, we observe the formation of Ir3Si5, bypassing two other phases: Ir4Si5 and Ir2Si3. In this case, the ∆G for the formation of Ir3Si5 is sufficiently large to allow easy nucleation and the diffusion-controlled formation of that phase at temperatures lower than necessary to nucleate the two missing phases, with smaller composition changes with respect to IrSi, and therefore also smaller ∆G changes. With Rh, however, the list of silicon-rich silicides ends before Rh3Si5, which does not exist. Then, after the formation of RhSi, we observe the nucleationcontrolled formation of Rh3Si4, curiously with very few nucleation sites, some 2 or 3 mm apart, visible with the naked eye. Thus we can conclude that the existence of Ir3Si5 prevents the formation of the other phases, Ir4Si5 and Ir2Si3. A somewhat similar effect is observed in the sequence of phases during the reaction of Ni films with Si. The sequential formations of Ni2Si and NiSi were considered at the beginning of this section. Another phase exists, Ni3Si2, that does not appear in the sequence of phases experimentally observed. In the absence of an explanation for the failure of that phase to form after Ni2Si, it might be tempting to assign this failure to some nucleation process; that would at least please nucleation afficionados. But we do not know. However, once NiSi has began to form from the reaction of Ni2Si with Si, Ni3Si2 would form from the reaction: Ni2Si
NiSi  Ni3Si2,

(22)

with a ∆H  0 ± 9 kJ/mol,[24] nearly infinitely small. To the extent that the phase is thermodynamically stable, the value of ∆G for Eq. (22) has to be negative. By artificially preparing thin-film samples with an overall composition corresponding to Ni3Si2, but composed of adjacent layers of Ni2Si and NiSi, it has been shown[47] that the nucleation of Ni3Si2 under the conditions where the two adjacent phases preexist occurs only

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Figure 6.6 Cross section of a Si-Ni diffusion couple after 6 hours at 800°C. Note the very irregular interface between Ni3Si2 and NiSi. From Gülpen.[48]

at temperatures above those necessary to form NiSi. In bulk samples where observations are carried out at temperatures on the order of 800°C, rather than 400°C with thin films, nucleation effects are hardly noticeable. All anticipated phases are observed to grow, more or less simultaneously, at least at times sufficiently long to allow observations in the optical microscope. Yet singularities are observed even then.[48, 49] Figure 6.6 is the cross section of a bulk Ni-Si sample heat treated at 800°C for 6 hours. A number of phases are distinguishable, but attention should be focused on Ni3Si2. The other phases have more or less planar interfaces, as anticipated in binary systems, but not Ni3Si2, which displays a morphology similar to that of the jaws of Tyrannosaurus rex. It might be tempting to attribute such a singular appearance to anisotropic diffusion, but that is quite unlikely since this would require extremely large anisotropy factors. Tentatively, we may propose the attractive alternative that the cause of the extremely irregular interface is stress and stress relaxation that occur in very different conditions, whether at the valley between teeth or at the peak of the teeth. Because the equilibrium ∆G is so small, even minor changes in elastic (and possibly plastic) energy would make considerable differences in the

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driving force for growth; hence the unexpected morphology. Initially, differences in orientation and consequently, differences in anisotropic stress or diffusion, might be the cause of irregularities that increase in magnitude as growth proceeds, with stress assuming the primary role. Because of the small ∆G’s, we anticipate that stress would affect these nucleation-controlled reactions, so that one of us (d’Heure) has been on the lookout for such effects for some 30 years, in vain. The bulk growth of Ni3Si2 might have finally provided the smoking gun. We may wonder whether some strong crystalline anisotropy is necessary for the development of interfaces such as those observed with Ni3 Si 2 . Otherwise, we might expect the same behavior for NiSi2, although in this case, the chemical (equilibrium) ∆G could be too large to be affected significantly by variations in ∆Helas. In multiphase growth, however, as difficult as it might be to know small values of ∆G, it is clear that the value is zero at a eutectic or peritectic point, even if, as a result, the exact eutectic or peritectic temperature is not known precisely. In such cases, the nucleation of the corresponding phase (or phases) is likely to be very difficult. In an otherwise excellent discussion of V-Si reactions,[10] that point is not so clearly outlined with respect to the formation of V6Si5.

6.2.8 Amorphous and Other Metastable Phases, Quasicrystals, Ternary Systems So far, nothing in this discussion has excluded the formation of metastable phases as a result of reactive diffusion. It has been emphasized that in the competition for the growth of different phases from reacting elements, the respective ∆Gf’s vary little from each other, at least in comparison with the respective D’s. This is also true, for example, of amorphous phases, with free energies of formation ∆G only slightly higher than the corresponding values for the crystalline phases. As a first approximation, the excess free energy ∆Gexcs  ∆T  ∆Smelt, where ∆T is the difference between the temperature T and the melting temperature Tmelt, and ∆Smelt is the entropy of melting ∆Hmelt
Tmelt. Samwer et al.[50] and Herr et al.[51] review this subject. Two criteria are usually put forward for the formation of amorphous phases. The first appears to be thermodynamic, a high ∆Gf of formation for the corresponding crystalline phases; the second is kinetic, a high difference in the diffusion coefficients of the two or three species present in the phase. The second criterion seems to be the most important and the most direct because crystallization in a binary system requires that the two elements be mobile. If one element is not

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mobile, the phase can grow via the motion of the mobile element, but once grown, it will remain in an amorphous state until the sample is brought to a temperature high enough to allow the motion of the slow-moving element. This is well typified by the growth of thermal SiO2, which grows via the very fast interstitial diffusion of molecular oxygen in the SiO2 network, while the Si atoms with an activation energy for motion of the order of 5 eV are totally immobile at temperatures below 1100°C. That is simple enough. The first criterion, a high ∆Gf of formation, has a direct thermodynamic implication: The sum ∆Gf
T∆Smelt has to remain negative for the phase to grow, even in a metastable amorphous condition. But this high ∆Gf of formation has two important kinetic implications as well. A high ∆Gf of formation implies, in general, a high Tmelt, so that at ordinary temperatures (much below Tmelt), the atomic mobilities will be small, making any crystallization difficult. To be sure, a high Tmelt implies a high driving force for crystallization at a low temperature T, as the force increases linearly with the difference Tmelt  T, while the atomic mobility necessary for crystallization decreases as some exponential function of that difference. Moreover, there is a second aspect to this: A high ∆Gf of formation implies almost necessarily a large difference in atomic mobilities in any compound that is not equiatomic (type AB). This is so because, while the majority of atoms can usually diffuse along their own network without undue interference, the motion of the minority atoms often requires that these occupy the sites of majority atoms. This requires the local destruction of the structure of the growing phase and an excess activation energy for motion equal to at least a fraction of the free energy of formation; hence, the higher the latter, the bigger the difference in mobilities between the two atomic species.[21] This is the prime criterion for the growth of an amorphous phase. A brief exploration of the literature shows that the mostly kinetic conditions that govern the reactive formation of amorphous phases do not apply for the formation of other metastable phases. The formation of the metastable Ni2Al9 already discussed[36, 37] seems to depend not on kinetic details but on the complex details of the nucleation process. That a compound with a corresponding composition is a stable phase in the Co-Al system confirms suspicions that Ni2Al9 is not far from being stable also, so that even a minor decrease in surface or elastic energy would be enough to stabilize that phase in preference to the stable NiAl3. The formation of NiSi with the NiAs structure epitaxial on (111) Si is clearly dictated by the low interfacial energy,[52] not by considerations about the relative mobilities of Ni and Si. The same is true of the formation of FeSi2 and other Fe silicides with the CaF2 structure.[53] These phases

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tend to transform to the stable form beyond a critical thickness where the constant “gain” due to a low interface energy becomes lower than the increasing “loss” (proportional to the thickness) due to the metastable phase. Some of these metastable phases do not form initially, but after other phases have already formed, for example, Ni2Si before NiSi. Two metastable phases were observed in the reaction of Al with Pt, after an amorphous phase was formed initially.[54] We have already alluded to the formation of quasicrystalline phases.[25] Further information is contained in the proceedings of symposia.[55, 56] In Al-Pd reactions,[37] the stable phase Al3Pd2 was observed to grow prior to the decagonal quasicrystalline Al3Pd. During the reaction of Co with Al, the stable Co2Al9 grows first, then a decagonal phase Co4Al13.[57] For this phase, it has been shown[57–59] that 1) the enthalpy released during its formation is within experimental errors equal to that expected for the stable phase; 2) changes in thickness do not affect the phase sequence and do not suppress its formation; 3) an increase in temperature causes the formation of the stable phase. A low interface energy and, therefore, preferential nucleation are presumed to be at the origin of the formation of such quasicrystalline phases.[55, 56] Few words are more randomly misused in the literature than the word “stable.” In the same vein, care should be taken with the expression metastable: In the reaction of Au with amorphous Si,[60] we can form compounds that are metastable with respect to thermodynamic equilibrium, yet are stable with respect to amorphous Si, since they form with a negative enthalpy of formation. Reactions in ternary systems deserve more attention; they constitute a subject in themselves. Reactions between Ni and SiC are considered by Lien et al.[46] and Gas et al.[47] Aspects of the reaction of Ti with Si-Ge are analyzed by Thomas et al.[61] and Aldrich et al.[62] Kodentsov et al.[63] and Loo[64] discuss ternary reactions in general. All that has been written, of a more or less theoretical character, implies that researchers are working with very pure materials. Impurities even in small quantities may change the character of the reactions from that of a binary to a ternary or higher order system, with serious consequences. Impurities may affect kinetics by forming more or less porous diffusion barriers, or by reducing grain boundary diffusion. In the reaction of Si with commercial Fe, different results are obtained with different grades of Fe (see Shimozaki et al.,[65] Fig. 2). Single-phase growth (Fe3Si) is observed in one case, and multiple-phase growth in the other. (The explanation given by the authors may or may not be correct.) Similarly, the reaction between Mo and Si should lead to multiphase growth, contrary to the report of the formation of MoSi2 only.[9]

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6.2.9 Other Effects: Grain Boundaries and Impurities, Diffusion Coefficients Varying with Composition In general, with binary systems, the interfaces remain planar and parallel to themselves. This has been observed to be true with silicide phases such as NiSi and PtSi, and less true with phases whose formation is nucleation-controlled, such as NiSi2. Effects due to grain boundary diffusion can often be neglected, but not always. A study of Au-Cu interactions at about 200°C in thin films[66] did not lead to publication because the interactions were dominated by the very fast intergranular diffusion of Cu inside the Au film, which made any kinetic interpretation extremely ambiguous. A similar effect with Al films had serious practical implications for electromigration. We can delay electromigration failures in Al thin-film conductors, by forming a continuous intermetallic layer[67, 68] of Al with some transition metal in parallel. Because of the considerably smaller diffusion coefficients in such layers, they remain as an electrical bridge in those regions where failure has already occurred in the adjacent Al film with its relatively high diffusion. When attempting to form a thin continuous layer of an intermetallic compound between, for example, Al and Hf, it was observed that the compound formed preferentially along the grain boundaries of the Al film, and thus could not act later as the desired electrical bridge (see Fig. 6.7). Continuous intermetallic layers, however, could be formed if, instead of using pure Al, one uses Al alloyed with Cu, which is known to reduce grain boundary diffusion in Al. Indeed, this alloy addition is used to retard electromigration failure in Al thin film conductors by a factor of about 50.[69, 70] The grain boundary impurity effect being considered here is quite general: Impurities segregate to grain boundaries, thereby lowering the grain boundary energy, and simultaneously increasing the activation energy for grain boundary diffusion and decreasing the diffusion coefficient.[71] This effect is mentioned again later. There is also an interesting experimental mode where diffusion does not proceed in a direction normal to the interface, as in ordinary diffusion couples; instead, it occurs along the lateral dimension of a thin film. A very good example referring to Al-Ni reactions is provided by Liu et al.[72] This discussion of the reactive phase formation has assumed that the diffusion coefficient in the growing layer is either constant in each growing layer from one interface to the other, or that some average value provides a valid approximation. In some cases, this assumption leads to an erroneous interpretation. In the thin-film reaction of Al with Ni at about 100°C,[73] it was observed that the central phase, NiAl, did not appear as a

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(A)

(B)

Figure 6.7 Schematic representation of the effect of a layer of transition metal compound on electromigration in Al thin-film conductors. From left to right: In (A) , the presence of a layer of intermetallic compound provides an electrically continuous path, even after failure of the Al conductor itself. In (B) , with pure Al the product of the reaction between Al and a transition metal tends to be scattered along the grain boundaries of the Al film, but a continuous layer is obtained if the Al is alloyed with Cu, which reduces grain boundary diffusion in Al. Not to scale.

single phase. Instead, it was divided into two regions, each with a composition corresponding to the extreme limits of stability, about 45 and 55 at.% Ni. This effect is not actually due[74] to some thermodynamically mandated phase separation; it is the simple result of kinetics, because the diffusion coefficients at these two extreme compositions are immensely higher than at the central composition (50 at.%). Since phases grow only at interfaces, the flux of atoms, proportional to D  dm
dx, must be constant from one interface to the other. If in some region of the growing phase D is very small, a constant flux is achieved by making the gradient dm
dx very large. For example, in the case of NiAl, the phase does not grow at the stoichiometric composition where the diffusion coefficient is

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the smallest. Another example, Fe-Si, has been discussed by d’Heurle et al.[75] At least in some phases with very narrow composition limits, the diffusion coefficient is likely to be a very strong function of composition, but the resulting effect may not be observable. Following considerations of a more or less theoretical character, some very practical effects in silicide thin films are examined in Section 6.3. Before doing so, let us recall some topics of general interest. Thin-film studies of interactions of Al or Au with transition metals[76, 77] demonstrate the following: (1) The first phase to form (there as well as elsewhere) is one that is rich in the metal with low melting point (either Al or Au). (2) This phase grows by the motion of the species with the low melting point, as mandated by the ordered Cu3Au rule.[21] This rule (1) does not apply so clearly to silicides because during reaction, Si undergoes a state transformation (from covalent with coordination number 4 to a much more metallic state with coordination number 10). For example, in TiSi2, Si has little similarity with elemental Si (unlike, for example, Ni in Ni3Al). For more information, it would be worthwhile to consult Schmalzried’s general book,[78] although it is a little heavy on ionic solids for our purpose. The general equations for multiple phase growth have already been mentioned.[7, 8] There are several good reviews of thin-film silicides formation.[21, 79, 80] For reactions in bulk samples, there are some earlier investigations on the formation of Al-Ti[81] and Ni-Al[82] intermetallics, and some recent ones on Ni-Si[49] and Co-Si[83] silicides. Diffusion in silicides[84] and in intermetallic compounds in general[85–87] have also been discussed.

6.3

Practical Problems in Electronic Technology

This section discusses some practical aspects of the use of compounds formed via reactive diffusion in the electronics industry. Quite a few of the problems currently encountered with these materials directly illustrate some of the issues discussed in Section 6.2. Others complement aspects that were not fully developed. For example, for VSi2 or WSi2, the mode of reaction does not fit neatly in the other two categories of diffusion-controlled and nucleation-controlled reactions. Considerations about the formation of TiSi2 will allow examination of a class of silicides once called “ill-behaved”.[16] Although the appellation has not met with great success, there are good reasons to believe that it remains valid.

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6.3.1 Titanium Disilicide, Activation Energy for the Motion of the Interface Between the C49 and C54 Phases, Nucleation of the C49 Structure For many years, the interconnections between devices in integrated circuits were made of highly doped polycrystalline Si. This material has a number of serious advantages. For one thing, it was ideally suited for processing on Si substrates from which active devices are fabricated. Moreover, it was also quite ideally suited as electrode material over the SiO2 gate of insulated gate field effect transistors (FETs). About 25 years ago, with device dimensions approaching 1 mm, the resistance of even highly doped polycrystalline Si became too high, causing unacceptable IR (current × resistance) drops and RC (resistance × capacitance) time delays. Substitute materials must also be compatible with existing fabrication processes, which ideally might include exposure to high temperatures (of the order of 1000°C) for diffusion, or the activation of dopants in the active part of the devices. Silicides fulfilled most of the requirements: resistance to oxidation, a conductivity considerably higher than that of polycrystalline Si, resistance to some etching solutions, and the ability to withstand exposure to high temperatures without suffering serious damage to their shape and physical properties. One of the first silicides used was WSi2. However, following the discovery that the resistivity of TiSi2 (20 mΩ.cm) was about one-third to one-fourth that of WSi2,[88] this compound became and remained for many years the favored silicide for use in integrated circuits. (The resistivity values referred to here are those that are encountered in thin-film applications, not necessarily the values obtained with single crystals of high purity.) Because of the technical applications, the literature on TiSi2 films is enormous. Yet in spite of much effort, a good understanding of the behavior of TiSi2 remains wanting. Two aspects of this behavior are analyzed here. Unfortunately, when forming TiSi2 from the reaction of Ti films with Si, or during the crystallization of amorphous TiSi2 obtained from the codeposition of Ti and Si, it is not the low-resistivity structure of TiSi2, C54, that forms first, but the C49 structure, with a resistivity about three or four times higher. The reason for this is not clear, but it may be due to the fact that the mobile species in TiSi2, Si (in agreement with the ordered Cu3Au rule), has a higher mobility in the C49 than in the C54 phase.[89] Also, the elastic constants of the C49 phase have been reported[90] to be lower than those of the C54 phase, which would lead to smaller stress effects (∆Helas). It is likely that the C49 structure is metastable, although it is the stable form of the closely related ZrSi2 and HfSi2. The low resistivity phase is the desired one, and it requires heat treatment at a temperature sufficiently high to allow the nucleation and growth of the C54 phase via

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a paramorphic transformation (the term allotropic being reserved for the transformations of elemental phases) driven by a small difference in free energy. The experimentally determined enthalpy change for this transformation is reported to be 1.5 kJ/mol.[91] This is commensurate with the driving force for the nucleation of the C54 phase, and then for its growth. The point has been made that in the reaction between phases, if the composition difference is small, for example, Eq. (22), the corresponding ∆G is also small. Here, the limit is reached when, in a paramorphic transformation, there is no change in composition at all. With respect to nucleation, the condition remains the same as before; namely, from a small ∆G it ensues that nucleation is difficult and the density of nuclei small. That is no problem with blanket films that transform at temperatures in the vicinity of 700°C, about 150 K above the temperature required for the formation of C49. However, when the width of the thin-film conductors decrease below about 0.3 mm, it is observed that for a given heat treatment, some conductors do not transform at all, while others may transform only partially.[92] That is not tolerable. A partial remedy is to increase the heat treatment temperature, but engineering constraints prevent this approach. One of the constraints, which is built into the conductors themselves, is that with films 10 nm thick, exposure to high temperatures causes the film to agglomerate into individual islands to minimize the surface energy. The temperature at which this occurs may fall below that required for nucleation. Many recipes have been used to try to alleviate these difficulties. Since a good account of these is given elsewhere,[93] they are not considered here. Rather, some quantitative aspects of the nucleation and growth process are discussed. The increase in temperature (akin to superheating) necessary to nucleate the C54 form of TiSi2 has a close equivalent in the classical example of the solidification of liquid metals, where the solidification of mercury requires increased undercooling as the liquid metal is divided into droplets of decreasing size.[94] Many attempts have been made to understand the intricacies of the nucleation and growth of the C54 phase of TiSi2. Different authors arrive at the same conclusion: a very high activation energy of about 4 eV. The most recent and complete study is based on mapping the new grains via micro-Raman scattering as well as electrical resistance measurements.[95] (The two coincide quite nicely.) The analysis was carried out according to a modified form of the Johnson-Mehl-Avrami method because the direct JMA method did not provide a good account of the evolution of the nucleating centers during the process of nucleation and growth. The study led to values of apparent activation energies, ∆G*, ∆G*kin, and ∆Gmot, for growth (the motion of the interface between C49 and C54) of the same order of magnitude, about 4 eV. Therefore, ∆G*th should be quite small, perhaps about 0.5 eV. A value of ∆Gmot of 4 eV for a material with a melting

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point of about 1500°C is quite high. So, not trusting the computer fitting of a large quantity of data, the value of ∆Gmot was obtained by direct observation of the evolution of the largest grain size as a function of time and temperature. The activation energy thus obtained was the same: about 4 eV.[96] In a pure material, the activation energy for the motion of the C49
C54 boundary should be defined, as in recrystallization or grain growth, by the grain boundary diffusion of the slow element; here, the slow element was Ti. In a recent paper,[97] a maximum value of about 3 eV was estimated for such a process. The key to this discrepancy between anticipated and experimentally derived activation energies would be that the material is not pure. Some investigators[98] observing the motion of the C49
C54 boundaries in an electron microscope equipped with a residual gas analyzer discovered a quite strong hydrogen signal occurring simultaneously with the growth. It too had an apparent activation energy of about 4 eV, the same as the other(s). This cannot be the activation energy for the evolution of hydrogen from the grain boundaries of TiSi2, which is most probably of the order of 1 eV, half of which would be for diffusion and the other half for desorption (perhaps 1.5 eV if we assume hydrogen traps with a trapping energy of 1 eV). It is hard to guess, but for diffusion of hydrogen in Ni (with a melting point close to that of TiSi2), the activation energy is 0.4 eV.[99] That value is for lattice diffusion, but it seems that the grain boundary diffusion of hydrogen is hardly any larger than that in the lattice.[100] The analysis of boundary motion during recrystallization[101] of impure materials (with impurity contents in the range of parts per million) depends on the impurity drag effect.[102] At low temperatures, the segregation of impurities to the boundaries and the attraction between these impurities and a moving boundary cause the activation energy to have a higher value, equal to that for lattice diffusion plus the energy of segregation (or rather desegregation) of the impurities. At high temperatures, where entropy dictates that the impurities should “evaporate” from the boundaries, the activation energy equals the value for grain boundary diffusion. In an intermediate temperature region between “low and high,” the apparent activation energy assumes meaninglessly very high values that cannot be defined analytically.[103] In the present case of the C49 to C54 transformation of TiSi2 in thin films, available data are unfortunately limited to a very narrow range of temperatures (30 K in the most extensive study[92]), so that we do not have any knowledge of its behavior in low, high, or intermediate temperatures. Thus, the apparent activation energy value of 4 eV could be meaningless (intermediate range) or correspond to the sum of the activation energy for the lattice diffusion of Ti (not measured, but probably about 3 eV) plus those for the desorption and diffusion of hydrogen. The effect of hydrogen has been discussed as an example of the

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impurity drag effect;[102] yet it may also result from the hindering effect of impurities on grain boundary diffusion,[71] usually thought of as occurring in fixed grain boundaries, although the two effects should not be considered to be exclusive of one another. Support for such a view is found in the important retardation of electromigration failure in Al thin-film conductors in the presence of hydrogen.[104] Furthermore, it was observed that the formation of purple plague (Au2Al) at Al-Au contacts was almost completely suppressed in samples annealed in hydrogen,[103] and similar results were obtained in the reactions of Cu, Ag, and Ni with Sn.[105] This impurity effect on grain boundary diffusion is also illustrated in the selfaligned technology, where the goal is to obtain a Ti disilicide over source, drain, and gate, but not over the short spaces that separate source and drain from the gate. With a continuous Ti film annealed in a vacuum or an inert atmosphere, a most unwanted bridging occurs via the diffusion of Si either in Ti itself or in one of the precursor phases to C49 TiSi2. That phenomenon is prevented[106] by annealing in nitrogen, seemingly because the small interstitial nitrogen atoms diffuse rapidly in the grain boundaries where they successfully block any subsequent diffusion of Si atoms. With very thin films (10 nm) in current devices, the question of the roughness of the layers used becomes important, which led to the systematic study of that roughness by optical means,[107] used in situ during the heating of samples at constant ramp rates. Such a study[108] reveals a strong roughening at about 600°C (see Fig. 6.8), corresponding to the formation of the C49 phase of TiSi2, and no roughening at the temperature of the C49 to C54 transition. The latter should not be surprising. (The small light-scattering peak for distances of 5 mm at about 800°C for that transformation is undoubtedly due to the difference in refractive indices for the two phases, not to topological irregularities.) The paramorphic transformation implies small changes in volume and therefore small changes in the surface aspect of the film. The signal obtained at 600°C for lateral distances of the order of 5 mm reveals a nucleation-controlled reaction, with bumps on the samples for each nucleation center. (A study as a function of scattering angles or laser wavelengths would indicate the average distance between such sites.) As the film transforms completely, the bumps disappear, and so does the scattering signal. With the new phase, C49 TiSi2 grows in “cauliflower” fashion from the nucleation centers. Scattering corresponding to short distances (0.5 mm) reveals the local roughness at the surface of the cauliflowers, which does not change significantly with the completion of the process. Before we attempt to analyze a possible nucleation process in the initial formation of TiSi2, we should consider what happens before that stage in Fig. 6.8. In the diffraction pattern, at low temperatures, the line at about 45 degrees is the (101) line of Ti and at 44 degrees is the (002) line. The

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Figure 6.8. (A) Simultaneous measurements of resistance and roughness at two different lateral scales during heating at 3 K/sec of a Ti film (20 nm thick) on polycrystalline Si. (B) Partial x-ray diffraction diagrams showing distinctly the formation of the C49 and C54 phases of TiSi2. Note the correspondence between the features in the two sets of pictures.

latter reveals what may be assumed to be an abnormally high expansion coefficient, but that would be incorrect. The somewhat surprising behavior is due to the dissolution of oxygen, which increases the c axis of hexagonal Ti without significantly affecting the other axis, a. Values obtained by Villars and Calvert[109] are a  0.2951 nm and c  0.4685 nm for Ti, a  0.296 nm and c  0.483 nm for Ti2O, and intermediate values for lower oxygen content. The oxygen intake causes the corresponding increase in resistivity (Fig. 6.8), an increase that does not occur if the film is protected against possible oxidation.[110]

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It had long been known[26] that the so-called ill-behaved silicides, including TiSi2, became rough in appearance (to the naked eye) during preparation, and that this roughness implies some nucleation-controlled reaction. However, while in the case of NiSi2, for example, the preceding phase NiSi was clearly identified, and the small ∆G driving the reaction was thereby identified also, the same has not been true of TiSi2, or MoSi2, and so forth. It was thus difficult to know at what stage of the reaction the samples became rough. With the help of Fig. 6.8, we now know that roughness occurs simultaneously with the nucleation of the C49 phase. In the continuous absence of firm information about what precedes the C49 phase, we hazard here some educated guesses. It is probable that the preceding phase or phases have not been clearly identified for a number of reasons: Amorphous phases are not clearly seen in x-ray diffraction. In the case of Ti-Si reactions, a number of phases exist (TiSi, Ti3Si4, and Ti5Si3) that are hard to identify because they have somewhat similar x-ray diffraction patterns, often very poorly displayed from thin-film samples. Finally, while the rough appearance of the C49 phase is constantly annoying in all samples, it probably varies with the sample preparation such as the mode of film deposition and the doping of the substrates. A possible reaction would be: TiSi
Si  TiSi2(C49).

(23)

(Because the phase considered here is C49, we must take into account the small enthalpy difference between C49 and C54.) According to Hodaj and Dumas,[28] the standard enthalpy change for this reaction would be 3.5 ± 36 kJ/mol, which is extremely small. Although crystalline TiSi does not seem to form during the reaction of Ti with Si, there are reports of the formation of an amorphous phase with a composition close to TiSi.[111, 112] With TiSi in an amorphous state, the enthalpy change for Eq. (23) would be increased (in absolute value) by an amount corresponding to the crystallization of that phase; that may not be enough for the formation of the TiSi2 to cease to be nucleation-controlled. Since Ti5Si3 has been observed in some experiments,[113, 114] we can consider: 1
5 Ti5Si3
7
5 Si  TiSi2(C49).

(24)

The standard enthalpy change[28] for this would be 13.5 ± 36 kJ/mol. This seems bigger than the preceding value (as anticipated, considering the greater change in composition), but could still be small enough to cause the reaction to be nucleation-controlled. [Compare this with Eq. (21).] According to a recent publication, the phases Si, Ti5Si3, and the

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two forms of TiSi2 can coexist, which implies some sort of equilibrium among them.[115] Moreover, the phase Ti5Si3, like other such transition metal silicides with the same formula, should be stabilized by interstitial impurities: oxygen, carbon, and nitrogen.[116] A considerable amount of research[117–119] has been devoted to this subject. The impurity atoms, including hydrogen, occupy the center of trigonal antiprisms, with binding energies that can be high enough to decrease the overall lattice parameters. At high temperatures, oxygen impurities in Zr5Si3 completely destabilize adjacent phases such as Zr2Si, implying a considerable decrease in the free energy of the 5
3 phase. Yet quantitative data for the lowtemperature reaction of interest here could not be found. In practice, with unavoidable contamination, the ∆H and (∆G) for Eq. (24) could be considerably smaller than evaluated here. Thus, whether considering Eq. (23) or (24), we can find some theoretical justification for the nucleationcontrolled formation of the C49 phase of TiSi2. Nothing has been said about interface energy and stress effects, although we know that these factors are very important physically. Experimentally, more effort should be spent to identify the precursor phase in TiSi2 formation, as well as for other such phases, for example, MoSi2. We should note that in the reaction of Ti with crystalline Ge, the reaction: Ti6Ge5
Ge → TiGe2(C54)

(25)

is clearly nucleation-controlled.[61] When considering small driving forces, as in any form of nucleation-controlled reactions or transformations, remember that the ∆G’s and the ∆H’s cease to be comparable. Moreover, we should not use the table values for the stoichiometric compounds, but should operate from the actual curves for the various ∆Gf’s as a function of composition. So the message is that the approach used here, based on table values of ∆H’s, provides a simple tool to differentiate diffusion-controlled from nucleation-controlled reactions. Where possible, refer to tables that give the free energies of compounds (and elements). According to the most recent thermodynamic evaluation of the Ti-Si system,[120] the free energy change at 900 K for Eq. (23) would be 20.8 kJ, including the correction for the C49 to C54 transition.[91] The same reference explicitly mentions that the phase Ti5Si3 can contain significant amounts of excess Si substitutionally located on one of the two different Ti sites, so that for our purpose (in the presence of excess Si), its formula should be (Ti,Si)2Si3Ti3. For Eq. (24), assuming stoichiometric Ti5Si3, ∆G would be 53.5 kJ. This quantity would be much reduced, in absolute value, for Ti5Si3 saturated with Si and/or contaminated with interstitial impurities.

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6.3.2 Cobalt Disilicide, Entropy of Mixing, (Possible) Enthalpy and Density-of-State Effects in Ternary Reactions With conductor dimensions reduced below about 0.3 mm, the operating window between the temperatures necessary to nucleate the C54 form of TiSi2 and that at which surface energy effects cause the films to break into separate islands has become intolerably small. Thus attention was turned to CoSi2, with a similar or even somewhat smaller resistivity, which does not suffer from the kind of paramorphic transformation that plagues users of TiSi2. Because f.c.c Co and Ni are so similar from a metallurgical point of view, it is not surprising that the sequence of metalsilicon reactions is the same with both metals: first Co2Si, then CoSi and CoSi2. Although it is not especially relevant to what follows, we should note that the Co-Si phase diagram is much simpler than that of Ni-Si, with no phase corresponding to Ni3Si2, and without several metal-rich phases. Although the phase sequences are similar with the two metals, on a temperature scale the formation of CoSi2 is not so distinct from that of CoSi as the formation of NiSi2 from that of NiSi. Yet early measurements showed that during isothermal heat treatments, the growth of CoSi2 ceased to be parabolic as the temperature was lowered towards 500°C,[121] and indeed the formation of CoSi2 becomes practically impossible below that temperature. By comparison with NiSi2, the effect was correctly attributed to a difficult nucleation. Recent results give a ∆Hf of 95 ± 4 kJ/mol for the formation of CoSi and 99 ± 6 kJ/mol for CoSi2,[122] resulting in a difference of 4 ± 10 kJ/mol for the nucleation of that phase, a value commensurate with 5 ± 20 kJ/mol for the nucleation of NiSi2 [Eq. (21)]. The two values are nearly equal, as anticipated within the limits of accuracy. One problem with CoSi2 is that on account of its nucleation-controlled formation, its surface is rough. Attempts made to reduce the roughness by alloying reveal some interesting aspects of the control of ∆G, the driving force that enters into ∆G*th (∆s3
∆G2) and thus controls the density of critical nuclei. The effect is well illustrated in Fig. 6.9, where the nucleation temperature for the formation of CoSi2 is seen to increase when the Co film is alloyed with different amounts of Ge. (See Lavoie et al.[123] for this and other alloying elements.) An opposite effect had been observed with the use of Co additions to Ni, where alloying with 50 at.% reduced the nucleation temperature to about 350°C,[124] below what is required to nucleate pure CoSi2. In that case, the two preceding phases, NiSi and CoSi, with different structures (respectively, MnP type: orthorhombic, and FeSi type: cubic, complex) did not mix and were clearly separated. In

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Figure 6.9 Elastic scattering of light for two different lateral scales, (A) 5 mm and (B) 0.5 mm, for a Co film with various amounts of Ge during heating on Si (100) at 3 K/s. The sharp features correspond to the nucleation and formation of CoSi2. There is very little evolution of roughness below 600°C during the formation of Co2Si and then CoSi. Both of these reactions are diffusion-controlled.

Fig. 6.10, the NiSi and CoSi monosilicides are seen to form in the same sequence near the Si substrate and at the surface, regardless of the order of the metal film depositions, Co then Ni, or vice versa, or alloyed solid solution. The new solid solution (Co, Ni)Si2 forms at the interface between the two phases, clear evidence of the role of nucleation control. The key

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Figure 6.10 Backscattering spectra showing the formation of (Ni, Co)Si2 at the interface between NiSi and CoSi, where entropy of mixing effects increase the free-energy change for the formation of the disilicide solid solution. The position of the initial transformation underlines the importance of nucleation in opposition to diffusion as the controlling mechanism.[91] The spectra correspond to samples (Si/Co/Ni) heated at 475°C for 1.5, 3, 7, 12, and 34 hours, respectively. Backscattering alone does not enable us to distinguish between Co and Ni; that was done separately by means of Auger spectroscopy.

point here is the new phase, a single-disilicide phase, made possible by the nearly total similarity of NiSi2 with CoSi2, which have the same structure (CaF2) and the same lattice parameters; namely, a solid solution of the two. The reaction then becomes: 1
2 CoSi
1
2 NiSi
Si  (Co, Ni)Si2.

(26)

For such a reaction, in comparison with Eq. (21) for the formation of NiSi2 and identically for CoSi2, the ∆G is increased by a quantity proportional to the entropy of mixing. For one mole of (Co, Ni)Si2, that quantity would be kT[xlnx
(1x)ln(1x)], where x is the metal composition [from 0 to 1, but 1
2 in Eq. (26)]. The increased ∆G increases the density of critical nuclei and thereby decreases the temperature at which nucleation is observed. For the additions of Ge to Co (Fig. 6.9), the effect is opposite

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because, in that case, the monosilicide and monogermanide are soluble (with the same structure and compatible lattice parameters). The digermanide and disilicide with different structures appear not to be very miscible. The reaction has the form: Co[Si(1x), Gex]
Si → (1x)CoSi2
1
2 xCoGe2.

(27)

Compared to Eqs. (21) and (26), Eq. (27), going from left to right, entails a decrease in entropy, a decrease in the absolute value of ∆G, a decrease in the density of nuclei, and, as observed, an increase in the nucleation temperature. A number of alloying additions are observed to behave according to either Eq. (26) or (27); these are nicely systematized by Detavernier et al.[125] If excess Si is present with the product phases in 27, as is often the case, the digermanide can be reduced to CoSi2 and Ge subsequent to its formation. This is due to the higher stability of the silicides in comparison to the germanides (higher heats of formation in absolute values). The latter effect, however, is not germane to this discussion. These qualitative considerations are fundamentally correct, and with Co-Ni alloys, they apply fully with compositions near 50%. However, we must be careful. A ternary section of the Co-Ni-Si equilibrium phase diagram, with Si at the top, is dominated by two parallel (horizontal) domains of complete solubilites for the dimetal silicides and the disilicides and in the middle a third region for the monosilicides, with solid solution domains extending to about 20% on each side separated by an immiscibility gap.[126] Between 500 and 800°C, where CoSi2 or (Co, Ni)Si2 nucleates, no drastic change is anticipated. Then why should small amounts of Ni additions, as low as 2%, cause detectable reduction in the nucleation temperature? Inasmuch as both the parent monosilicide and the product disilicide phases are solid solutions, an explanation based on the entropy of mixing fails. It is probable that the effect is due to an additional enthalpy effect in the disilicide. In general, phases are most stable when the density of state (DOS) at the Fermi level is low. Other things being equal, a structure with the Fermi level at a minimum in the DOS curve implies a maximum in the number of bonding states below the Fermi level, and low energy levels for those states. This is exemplified by semiconductors where the density of states falls to zero in the gap: Compare diamond to nitrogen, adjacent in the Periodic Table, or Si to Al, or Ge to Ga. More subtle examples can be found. For the transition metals with BCC structure, the calculated density of states at the Fermi level is found in a trough (referred to as pseudo gap) in the BCC but not in the FCC structure. It is the BCC structure that is stable [127, p. 135]. For HCP metals (Zr, Hf, Ru, and Os), the Fermi level is also found at troughs in the DOS curves.[128] In the Hume-Rothery phases, the b and g brasses in the

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Figure 6.11 Density of states versus energy for CoSi2, with the Fermi level marked by a vertical line near zero on the energy scale. The numbers on the vertical scale are eV per formula unit. For silicides with the same CaF2 structure, the Fermi levels would be found in the antibonding states for NiSi2 at the arrow on the right, and among the bonding states for FeSi2 at the arrow on the left. The density of states curves are essentially the same for all three disilicides but would be slightly displaced on the energy scale. Here the positions of the Fermi levels are given in relation to that of CoSi2. According to Jepsen et al.[128]

Cu-Zn system, the density of states at the Fermi level is also at a minimum.[129] Figure 6.11 displays the DOS versus energy curve for CoSi2.[130] The Fermi level is found with bonding states on the low-energy side of a pseudogap. For NiSi2, with one more electron, and for FeSi2 (in the CaF2 structure), with one less electron, the curves are nearly identical although somewhat shifted on the energy scale. However, the Fermi level for NiSi2 is found on the high-energy side of the pseudogap, in the antibonding states. For FeSi2, on the contrary, the Fermi level moves to low energies and lies precisely at the peak found at about 0.5 eV in Fig. 6.11. As a further illustration, the high density of states at the Fermi level constitutes an unstable condition, so that in equilibrium FeSi2, a distortion

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akin to a Jahn-Teller effect occurs that destroys the degeneracy of the structure and opens an energy gap, making FeSi2 a semiconductor with zero states at the Fermi level. Christensen[130] does not give the curve for CuSi2, but for such a compound the Fermi level would move toward higher energies and higher density of states among the antibonding states; such a structure would be highly unstable, so that the highest stable silicide of Cu is CuSi. It is very likely therefore that for silicides with the CaF2 structure, the cohesive energies as a function of the number of electrons are distributed along some parabolic curve, descending from FeSi2 to CoSi2 and then ascending again with NiSi2 and CuSi2, with a minimum for an alloy of (Co, Ni)Si2, for which the Fermi level falls at or near the bottom of the pseudogap.[131] Experimentally, such a model would agree with an increased value of ∆G and enhanced nucleation of CoSi2, even with small alloying additions of Ni. Such considerations about DOS curves and Fermi levels have a rather qualitative character because they do not give a complete account of the total cohesive energy of a crystal. However, they find strong support in the fact that Fe additions to Co have precisely the opposite effect of Ni additions: With 2 at.% Fe, and a heating rate of 3 K/s, the temperature of nucleation of CoSi2 is increased by 50 K. The monosilicides with the same structure are totally soluble, and at the level of alloying, Fe is also soluble in CoSi2. This eliminates the possibility of significant entropy of mixing effects, so that what is observed must be due to enthalpy. Further confirmation of this DOS effect is found in the ternary phase diagrams,[132] where it is seen that FeSi2 is more soluble (by a factor of 10 or more) in NiSi2 than in CoSi2 because, in the first case, it lowers the density (of NiSi2) at the Fermi level, and in the second case, it increases it. As a first approximation, we would have expected the disilicides to behave like the monosilicides: The monosilicides of Fe and Co (near neighbors in the Periodic Table) have the same structure and are totally soluble. However, in going over to Ni, there is a change in structure and only limited solubility (of FeSi and NiSi). As in many such considerations about nucleation, nothing has been said here about surface energy. Considering the similarity between Co and Ni, segregation effects would tend to be small, and almost nonexistent for the solid solution NiSi2-CoSi2. Some interface segregation of Ni might be expected in CoSi (because of the limited solubility), but if such segregation stabilized the monosilicide, because of a reduced surface energy, it would result in an effect opposite to what is observed. The concern with silicide contact is not so much the upper surface as seen by elastic light scattering but the interface with Si. Epitaxy and reduced interface roughness can be obtained by the initial intercalation of some Ti[133] between the Si substrate and the Co film. Similar results have been achieved with other transition metals,[134] where the effect of such

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intercalated layers is attributed to kinetics: reduction in the rate of growth of the disilicide would enhance epitaxy.

6.3.3 Nickel Monosilicide Although CoSi2 has been used with considerable success for selfaligned silicide technology (usually misnamed “salicide,” which brings to mind not Si technology but some form of aspirin), this compound suffers from serious defects: The reaction between Co and Si is quite sensitive to the presence of surface impurities, including oxide, that are difficult to avoid in actual device fabrication. So attention has turned to studies of NiSi, which presents a number of advantages. In the sequence of phase formation, it follows after the formation of Ni2Si, but the transition does not involve any nucleation-controlled step, so it is quite smooth. Because of its low resistivity and composition for conductors of equal resistance, it requires a reduced consumption of Si, which is important with ever more shallow junctions (n/p or p/n at source and drains, for example). This problem of shallow depth is emphasized further in the current silicon-on-insulator (SOI) technology. In classical studies, namely with Ni films about 100 nm thick and isothermal annealing, NiSi forms after Ni2Si, with the two reactions diffusion controlled and the layers remaining extremely smooth. However, during rapid thermal processing of thin layers, other nickel-rich silicides have been observed,[135] which will require further investigations. In a recent publication,[136] NiSi is reported to have a serious advantage over its competitors (TiSi2 and CoSi2): lower stresses, both in the silicide itself and in the adjacent Si. However low these stresses might be, they have been the cause of a serious error; the lattice parameters derived from diffraction measurements on thin films[52] and incorporated in the reference literature[137] are not equilibrium values, but correspond to stressed films.[138] The correct, equilibrium unit cell dimensions should be close to a  0.5175 nm, b  0.3321 nm, and c  0.5609 nm.[139, 140] The expansion coefficients along the a and c axes are extremely large,[138–140] which would lead to high residual tensile stresses in the films at room temperature; that this is not so implies low elastic modulii. Curiously, however, the expansion coefficient along the b axis is negative. NiSi forms rather easily via two diffusion-controlled steps, and would be quite satisfactory as such. However, NiSi reacts with Si to form NiSi2, via a nucleation-controlled reaction that normally occurs at about 750 to 800°C. Such a step is definitely undesirable when working with NiSi. In practical applications, with possibly strange topography and various dopants and levels of doping, the danger exists that the disilicide would

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form at temperatures below 750°C. So with this compound, efforts are directed toward making the nucleation of NiSi2 more difficult. Considerable success has been met in this regard by the addition of Pt to Ni.[141] It turns out that PtSi, with the same structure as NiSi (MnP), is relatively soluble in NiSi, but the solubility of Pt in NiSi2 is almost nil. Thus with solubility of Pt in the monosilicide and lack of solubility in the disilicide, the entropy of mixing would increase the stability of NiSi with respect to NiSi2. Addition to Ni of some 5 at.% Pt increases the nucleation temperature of NiSi2 to above 900°C. The authors of the original article[141] are careful to point out that surface effects can also enter into these considerations. To this we can add that (1) entropy of mixing effects would appear to be sufficient to explain the results observed, and (2) if there is surface segregation of Pt, it is more likely to happen with the disilicide because of the low solubility than in the monosilicide. As a first approximation we can anticipate such an effect to stabilize the disilicide, in opposition to what occurs. Be that as it may, experimental evidence that Pt additions delay the agglomeration of NiSi implies that Pt is surface active in the system Si-NiSi and presumably in NiSi2 as well.[142] Yet NiSi is not entirely problem-free. It has been reported that films of NiSi on (100)Si develop at the interface little pyramids of NiSi2 with (111) faces that penetrate into the substrate.[143] Assuming equality of the free energies of formation of the monosilicides and disilicides, such a phenomenon will be energetically favored if the interface energy between NiSi and (100)Si is greater than that of epitaxial NiSi2 with (111)Si by a factor greater than √3 (according to the ratio of areas). Dopants are likely to influence such pyramidal formations. We may be encountering here truly thinfilm effects (not operative in bulk samples): Differences in surface energy may enhance the formation of a thin epitaxial film of NiSi2 (not necessarily smooth but faceted), under conditions that would cause the formation of NiSi with thicker samples. NiSi suffers from another defect that almost directly results from its own composition. The compound has been quite elegantly shown to form via the diffusion of Ni atoms,[144] However, because of the equiatomic composition, the less mobile Si atoms should diffuse quite readily at temperatures not much higher than those required for Ni motion. Such motion of Si makes it possible for films of NiSi to agglomerate even before the formation of NiSi2, and that, more than the nucleation of the disilicide, may limit the use of NiSi.

6.4

Conclusions

Phases formed as a result of reactive diffusion will grow according to parabolic kinetics if the growth is controlled entirely by diffusion, or

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according to linear parabolic kinetics if the initial stage of growth can be expressed in the form L proportional to tn with n greater or equal to 1. Mathematically, in the first case, if several phases should grow, they would start to grow at time zero, although physically, things would be different with the phase characterized by the highest diffusion coefficient growing first, followed by the others. In the other case, where growth can be approximated by linear-parabolic kinetics, the phases should appear sequentially. Ideally, the linear effect could be due to kinetics limited by an interface reaction rate, but there is little experimental evidence to prove that this is indeed physically correct. In metal-silicon reactions, at least, the kinetics clearly obey linearparabolic equations, even if it is difficult to obtain precise information about any interface reaction. In the reaction between A and B, where A and B are elements, the driving force for the reaction will often be high enough to make nucleation relatively easy, so that the kinetics are dominated by diffusion phenomena. Nucleation is characterized by an activation energy ∆G*, which is the sum of two terms (∆G*  ∆G*th
∆G*kin). The first of these, ∆G*th, proportional to ∆s3
∆Geff2, provides the density of nuclei of critical size; the second term, ∆G*kin, gives a measure of the rate at which these nuclei can grow, which is necessary for nucleation to occur. ∆Geff is the sum (in actual value difference) of the chemical driving force (negative) ∆G and the elastic energy (positive) ∆Helas. Diffusion-type phenomena will usually dictate what is the first phase to form. In reactions between metallic elements that have different melting points, the first phase to form will usually be a phase richest in the element with the lowest melting point. Si should be excluded from this scheme as it undergoes a state transformation during a reaction with a metal. However, nucleation phenomena dictated by interface energy and elastic energy considerations (∆G*th) may account for a selection between phases with close compositions, or stable and metastable forms of a phase with a given composition. We usually lack sufficient information to even offer plausible explanations for such natural choices; except in some cases, for example, when a metastable phase is clearly the result of epitaxy between the phase and the substrate. Quite often, the selection of the first phase to form will depend in some obscure way upon the mode of sample preparation. The change in free energy in going from one phase to the next may become very small. This is because, given the fundamental chemical fact that in the series such as ABn to ABz (with the subscripts increasing monotonically from n to z), the gain in free energy for additional B’s decreases continuously, consequently in going from one phase to the next

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the change in free energy becomes smaller. At some point, nucleation becomes the rate-controlling mechanism for the formation of a new phase. This effect will be most remarkable at low temperatures, for example, in thin-film reactions, and may play only a minor role in bulk reactions at high temperatures. It is easy to understand this qualitatively, but unfortunately, we often lack sufficiently precise thermodynamic information to make possible any significant quantitative evaluation. One consequence of a difficult nucleation, and therefore a low density of nucleation sites, is that the reaction products, in thin films at least, will tend to display a very rough surface. That is quite undesirable from a practical point of view but provides means of pinpointing nucleation phenomena by relatively simple optical means. Such means revealed that in the formation of TiSi2, with the desired C54 structure, initially the C49 structure itself is nucleation-controlled. It is hoped that this recent work (Sec. 6.3) will lead to a better understanding of the mechanisms of formation of not only TiSi2, but of other transition metal silicides such as VSi2 or WSi2. Even if the precise thermodynamic data about the nucleation of phases such as CoSi2 and NiSi2 are not available, it is known that the freeenergy changes are small and can therefore be modified by alloy additions, altering the ∆G’s due to the entropy of mixing. Furthermore, in the case of effects observed with small alloying additions, where mixing effects would not be significant, we can tentatively explain observed effects from an analysis of alloying effects on the density of state curves.

Acknowledgments The authors wish to express their appreciation to a number of colleagues who contributed to this chapter in various ways: J. Tersoff, D. Migas, and L. Miglio for discussions about density of state curves; and R. Vandermeer for comments about phase boundary motion. Many people contributed in some ways or another to our understanding of the phenomena analyzed in this chapter: C. Cabral, Jr., J. Dempsey, V. Dybkov, T. Finstad, R. Ghez, D. Gupta, A. Gusak, J. Harper, E. A. Irene, S. La Placa, C. S. Petersson, S. Privitera, O. Thomas, S.-L. Zhang, and many others too numerous to list here. Several friends read the manuscript in various states of completion; we are grateful to them all for their kind help. This chapter covers the work done by many investigators over a long period of time; inevitably, some references that should have been given were accidentally ignored. We offer our sincere apologies to all those who should have been quoted but were not.

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110. R. Cocchi. D. Guibertoni, G. Ottaviani, T. Marangon, G. Mastracchio, G. Queirolo, and A. Sabbadini, J. Appl. Phys., 89:6079 (2001) 111. E. J. van Loenen, A. E. Fischer, and J. F. van der Veen, Surf. Sci., 155:65 (1985) 112. R. J. Kasica and E. J. Cotts, J. Appl. Phys., 82:1488 (1997) 113. M. Nathan, J. Appl. Phys., 63:5534 (1988) 114. A. Qintero, M. Libera, C. Cabral, Jr., C. Lavoie, and J. M. E. Harper, J. Mater. Res., 14:4690 (1999) 115. I. Matko, B. Chenevier, O. Chaix-Pluchery, R. Madar, and F. La Via, Thin Solid Films, 408:123 (2002) 116. H. J. Goldschmidt, Interstitial Alloys, Plenum Press, NY (1989), p. 303 117. L. Brewer and O. Krikorian, J. Electrochem. Soc., 103:38 (1956) 118. Y.-K. Kwon, M. A. Rzeeznik, A. M. Guloy, and J. D. Corbett, Chem. Mater., 2:546 (1990) 119. J. D. Corbett, E. Garcia, A. M. Guloy, W.-M. Hurng, Y.-U. Kwon, and E. A. Leon-Escamilla, Chem. Mater., 10:2824 (1998) 120. H. J. Seifert, H. L. Lukas, and G. Petzow, Z. Metallkd., 87:2 (1996), p. 8 and table 6 121. F. M. d’Heurle and C. S. Petersson, Thin Solid Films, 128:283 (1985) 122. D. Lexa, R. J. Kematick, and Clifford E. Myers, Chem. Mater., 8:2636 (1996) 123. C. Lavoie, C, Cabral, Jr., F. M. d’Heurle, J. L. Jordan-Sweet, and J. M. E. Harper, J. Electron. Mater., 31:597 (2002) 124. T. G. Finstad, D. D. Anfiteatro, V. R. Deline, F. M. d’Heurle, P. Gas, V. L. Moruzzi, K. Schwartz, and J. Tersoff, Thin Solid Films, 135:229 (1986) 125. C. Detavernier, R. L. Van Meirhaeghe, F. Cardon, and K. Maex, Phys. Rev., B62:12045 (2000) 126. F. J. J. van Loo, J. Alloys Comp., 297:137 (2000) 127. D. J. Singh, in Intermetallic Compounds, Principles and Practice, vol. 1 (J. H. Westbrook and R. L. Fleischer, eds.), Wiley, NY (1995), p. 127 128. O. Jepsen, O. K. Andersen, and A. R. MacIntosh, Phys. Rev., B12:3084 (1975) 129. A. T. Paxton, M. Methfessel, and D. G. Pettifor, Proc. R. Soc. London, A453:1493 (1997) 130. N. E. Christensen, Phys. Rev., B42:7148 (1990) 131. D. Migas and L. Miglio, private communication 132. Handbook of Ternary Alloy Phase Diagrams, American Society for Metals, Metals Park, OH (1995) 133. M. Lawrence, A. Dass, D. B. Frazer, and C. W. Wei, Appl. Phys. Lett., 58:1308 (1991) 134. C. Detavernier, R. L. Van Meirhaeghe, F. Cardon, K. Maex, H. Bender, B. Brijs, and W. Vandervorst, J. Appl. Phys., 89:2146 (2001) 135. C. Lavoie, F. M. d’Heurle, C. Detavernier, C. Cabral, Jr., Microelectron. Eng., 70:144 (2003) 136. A. Steegen and K. Maex, Mater. Sci. Eng., R38, no. 1 (June 4, 2002) 137. Joint Committee on Powder Diffraction Standards 38-0844, Int. Center for Diffraction Data, Newtown Square, PA

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138. C. Detavernier, C. Lavoie, and F. M. d’Heurle, J. Appl. Phys., 93:2510 (2003) 139. M. Kh. Rabadanov and M. B. Ataev, Inorg. Mater., 38:120 (2002) 140. D. F. Wilson and O. B. Cavin, Scripta Metall. Mater., 26:85 (1992) 141. D. Z. Chi, D. Mangelinck, S. K. Lahiri, P. S. Lee, and K. L. Pay, Appl. Phys. Lett., 78:3256 (2001) 142. P. S. Lee, K. L. Pey, D. Mangelinck, J. Ding, D. Z. Chi, and L. Chan, IEEE Electr. Dev. Lett., 22:568 (2001) 143. V. Theodorescu, L. Nestor, H. Bander, H. Steegen, A. Lawers, K. Maex, and J. V. Landuyt, J. Appl. Phys., 90:167 (2001) 144. T. G. Finstad, Phys. Status Solidi, 63:223 (1981)

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7 Metal Diffusion in Polymers and on Polymer Surfaces Franz Faupel and Vladimir Zaporojtchenko Lehrstuhl für Materialverbunde, Technische Fakultät der Universität Kiel, Germany Axel Thran Philips Research Hamburg, Hamburg, Germany Thomas Strunskus Ruhr-Uni Bochum, Lst.f. Physikalische Chemie I, Bochum, Germany Michael Kiene AMD Saxony LLC&Co.KG, Dresden, Germany

7.1

Introduction

Metal diffusion plays a crucial role in polymer metallization, which is indispensable in applications ranging from food packing to microelectronics.[1, 2] Diffusion of metal atoms, and sometimes also of small clusters, along the polymer surface determines nucleation and growth of metal films on polymers and hence has a strong effect on the resulting microstructure.[3] Recently, there is also great interest in nano-size metal clusters, which form by surface diffusion in the initial stage of polymer metallization if the metal is not too reactive. This interest not only arises from quantum size effects and single-electron tunneling phenomena[4] but is also triggered by applications in medicine[5] and as substrates for biomolecules.[6] Above the polymer glass transition, which can be depressed at the surface,[7, 8] metal clusters are generally embedded in the polymer and, depending on their size and the polymer viscosity, may perform a Brownian motion.[9, 10] In microelectronics, diffusion of metal atoms into the polymer bulk is also a concern. This field has stimulated intensive research throughout recent decades.[1–3, 11–14] Polymers have been used extensively as lowpermittivity (low-k) dielectrics in packaging and are now even used in onchip interconnects.[13–16] Both processing and operation of microelectronics components involve exposure to elevated temperatures. In chip applications, diffusion of even very small amounts of copper, acting as a deeplevel impurity, from the polymer into silicon would lead to device failure.[14] Transport may be further enhanced by the very strong electrical fields resulting from the small feature sizes. Therefore, much effort has been made to control the microstructure and thermal stability of metal-polymer

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interfaces, especially with the aim to block diffusion by suitable barriers, improve adhesion, and prevent degradation.[1–3, 9, 11–14, 17–24] Meanwhile, polymers are not only used as passive dielectrics in conventional microelectronics; they are increasingly being used as active semiconducting components in low-cost organic electronic devices such as light-emitting diodes and displays, solar cells, and field effect transistors.[25–27] Here direct contact between the organic semicoductor and the metal contacts generally cannot be avoided because the metals, which are selected to adjust their work functions to the highest occupied and the lowest unoccupied molecular orbitals, are needed for charge injection. Metal diffusion into the organic semiconductor has particularly been observed for metal-on-polymer deposition, whereas the polymer-on-metal interface seems to be quite sharp.[27] Massive diffusion can be expected for alkali metals and other metals like In that are able to reduce the organic molecules and to diffuse as highly mobile cations.[28–30] The repulsion of the ions prevents aggregation, which usually gives rise to strong immobilization of diffusing neutral metal atoms.[9] From the fundamental point of view, metal diffusion in polymers can only be understood by taking into account the strongly contrasting properties of both materials. While metals are densely packed crystalline solids with a high cohesive energy, noncrosslinked polymers are made up of large covalently bonded macromolecules held together by very weak, mostly only van-der-Waals-type interactions in an open structure. The cohesive energy of metals is typically two orders of magnitude higher than the cohesive energy of polymers. Furthermore, the interaction between moderately reactive metals and polymers is generally much weaker than the strong metal-metal binding forces. As a consequence, these metals are expected to exhibit a strong aggregation tendency, and their solubility in polymers should be extremely low under equilibrium conditions. Hence, practically no intermixing should occur when a piece of metal of low reactivity is brought into close contact with a polymer surface.[9] For the same reasons, metals of low reactivity do not wet untreated polymer surfaces. They form clusters during polymer metallization, which finally coalesce and form a continuous film (Volmer-Weber growth). In view of the extremely low solubility of metals in polymers, it is obvious that the aforementioned embedding of metal clusters in polymers above the glass transition temperature is a process being entirely different from ordinary dissolution, for example, of gas molecules in polymers. Apparently, there is a driving force for embedding of metal clusters; that is, the Gibbs free energy of a metal particle inside the polymer is lower than that of the particle at the surface. Once more this is related to the high cohesive energy of metals, which gives rise to a correspondingly high surface Gibbs free energy of metal particles. The surface Gibbs free energy can be reduced by

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embedding if the surface tension gM of the metal particles exceeds the sum of the interfacial tension gMP and the polymer surface tension gP:[10, 31] gM
gMP  gP.

(1)

Since the cohesive energy of polymers is so much lower than that of metals, gP is very small in comparison to gM. When a metal particle is covered by polymer, there is still a net van der Waals force driving it deeper into the bulk, and an entropic force, due to the confinement of the polymer chains near the metal particle, pushing it back to the surface.[10, 31] This results in a size-dependent potential minimum pinning large particles below the surface. Clusters of the order of 10 nm or less can overcome the potential barrier by thermal activation.[9, 10, 31, 32] While the above considerations on metal solubility and the absence of significant metal diffusion into polymers are based on equilibrium thermodynamics, the conditions during the initial stage of polymer metallization are far from thermodynamic equilibrium. Here, the virgin polymer surface is exposed to isolated metal atoms that do not have to overcome the strong metallic cohesive force to become mobile. Therefore, significant diffusion of metal atoms into polymers is only expected to take place during the early metallization process or when the metal is deposited at a very low rate. Unfortunately, despite the many investigations that have been carried out on metal diffusion in polymers, the reported conclusions are still strongly conflicting.[9, 24] The origin of the controversial views lies in the complicated interplay of metal atom diffusion and aggregation. This behavior contrasts diffusion of metals in polymers strongly with the ordinary diffusion and may give rise to gross misinterpretations of diffusion experiments carried out by surface analytical tools. This chapter is mainly concerned with diffusion of metals in fully cured polymers and on their surfaces. The general characteristics of the polymer-on-metal interfaces are discussed by Kowalczyk et al.[33] and Godbey et al.[34] Emphasis throughout is placed on investigations performed by the Kiel group, but other important investigations[1–3, 9–12, 14, 17] are also addressed, although no attempt is made to give a fully comprehensive literature review. We treat metal diffusion during the early deposition stages, involving metal condensation, nucleation, and growth. In Sec. 7.3, metal-polymer interaction is addressed as a key factor determining atomic mobility. Section 7.4 is devoted to metal diffusion in the bulk of polymers. Here, the interplay of atomic diffusion and aggregation, which is observed for metals of not-too-high mobility, is discussed. This interplay leads to a strong aggregation-induced metal immobilization. We will see that the degree of aggregation and hence the extent of diffusion strongly depend on the metal deposition conditions. Diffusion profiles of

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metal atoms in polymer are generally non-Gaussian, and different ranges are related to diffusion of clusters of different sizes. Therefore it is difficult to extract quantitative diffusion data that can be attributed unambiguously to single atoms or clusters of known size. Section 7.4 also briefly covers the diffusion behavior of reactive metals whose mobility is usually blocked by strong chemical interactions with the polymer. Moreover, alkali and other metals that are able to form highly mobile ions in polymers are addressed. In this connection, the role of interfacial oxygen in ion formation is also discussed for copper and other metals. The chapter closes with a summary and important conclusions.

7.2

Diffusion During Nucleation and Growth of Metal Films on Polymers

As discussed in Sec. 7.1, the early stages of polymer metallization are far from thermodynamic equilibrium since isolated metal atoms impinge on the polymer surface. Various competing processes, illustrated in Fig. 7.1, have to be considered.[35] Following deposition, the arriving atoms (a) may perform a random walk on the surface (b), or diffuse into the polymer (e). Metal atoms encountering each other on their diffusion path may form aggregates at the surface (c) and in the polymer bulk (f). These aggregates are stable if their size exceeds the size of a critical

Figure 7.1 Processes taking place during the initial deposition stage where isolated metal atoms impinge on the polymer surface.

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nucleus. Above the glass transition temperature, metal clusters may also be embedded into the polymer. Moreover, metal atom reemission into the vacuum (d) has to be taken into account.[36] The sticking or condensation coefficient of metal atoms on metal surfaces is generally very close to unity, even at elevated temperatures. In contrast, using two novel techniques based on radiotracer measurements[36] and x-ray photoelectron spectroscopy (XPS),[37, 38] we have recently demonstrated that the tendency of metals of low reactivity not to wet polymer surfaces can be accompanied by a very low condensation coefficient. The condensation coefficient C is defined as the ratio of the number of adsorbed metal atoms to the total number of metal atoms arriving at the surface. Depending on polymer-metal combination and temperature, C varies by several orders of magnitude. For example, condensation coefficients close to unity are observed for Cu on the PMDA-ODA (pyromellitic dianhydride-oxydianiline) and SiLK® up to temperatures as high as 200 and 300°C, respectively.[39] Above these temperatures, however, C decreases drastically. On a polystyrene surface, the condensation coefficient of Cu is only 0.26 at room temperature and drops further at higher temperatures.[38] Finally, for Teflon AF®, C is as low as 0.02.[38] This means that only 2 of 100 atoms impinging on the surface of this important low-k polymer stick to the polymer even at room temperature. At elevated temperature, the fraction is much smaller still. The condensation coefficients for Ag and Au are smaller than those for Cu on all polymers. The differences are most prominent for strongly incomplete condensation. For example, we found a C value as low as 0.002 for Ag on Teflon AF®,[36, 37] which is about an order of magnitude smaller than the value for Cu. The extreme differences in the metal condensation coefficients of the polymers are the subject of ongoing investigations. We note, however, that C follows the trend of the surface energies of the polymers. Since a low surface energy impedes macroscopic wetting of the surface, it does not appear unreasonable that a low surface energy also impedes atomic condensation. The increase of the condensation coefficient from Ag and Au to Cu seems to be related to the higher reactivity of Cu. A detailed discussion of condensation of metals on polymers and a compilation of C values for various metal-polymer systems is given by Zaporojtchenko et al.[37–40] In connection with diffusion, it is interesting to note that results from the radiotracer technique, which measures the integral condensation coefficient, and photoelectron spectroscopy, which exhibits a high surface sensitivity, generally proved to be in good agreement.[37] This shows that even for the low evaporation rates used in the measurements of C (typically 1 monolayer per min.), most of the metal atoms do not diffuse into the polymer bulk. This can be attributed to the general behavior[41] that surface

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diffusion is much faster than bulk diffusion. Therefore, most metal atoms get trapped in metal clusters at the surface before they are able to “escape” into the polymer bulk, as discussed in Sec. 7.4. Consequently, to a good approximation, metal diffusion into the polymer bulk can be neglected during nucleation and growth of metal films. This is not expected to hold for the aforementioned alkali metals and other metals that reduce polymers, thus forming highly mobile and mutually repelling ions. At higher metal coverages, metal atoms increasingly impinge on metal clusters, and finally the condensation coefficient approaches unity, which is typical for condensation of metals on metals at moderate temperatures.[36] Note that even in the case of extremely low sticking coefficients, the metal atoms are not backscattered directly but perform a random walk prior to reemission. This conclusion is based on the measured cos Φ angular distributions of reemitted atoms, which is a “fingerprint” of random emission and turned out to be independent of the angle of incidence.[36] Apparently, a fraction (1  C) of the metal atoms diffusing on the surface desorbs thermally before it finds a nucleation site. Two possibilities have to be taken into account for the nucleation. In so-called preferred nucleation, metal atoms are trapped at preferred sites; in random nucleation, nuclei are formed by metal atom encounters. Both processes have been observed in polymer metallization.[36, 37, 40] Preferred nucleation was shown for Ag on trimethylcyclohexane (TMC) polycarbonate, for example.[36] Here the Ag condensation coefficient is independent of the metal evaporation rate. This clearly allows random nucleation to be ruled out.[36] Apparently, nucleation takes place at special surface sites. The nature of these sites is not known yet; one can, for example, think of terminal groups of the polymer chains, impurities, or attractive local arrangements of the chains. The number of these surface defects, and hence the condensation coefficient, can be strongly increased by even moderate ion-beam treatment.[36, 40] Preferred nucleation at defect sites is particularly expected for low-condensation coefficients and at the initial deposition stage. In all other cases, random nucleation dominates. For a detailed quantitative treatment of metal nucleation on polymers in terms of nucleation theory, we refer to Zaporojtchenko et al.[38, 40] Here we only outline an approach for random nucleation, which also yields values for the activation enthalpy of surface diffusion. The approach is based on measurements of the deposition rate and temperature dependence of the maximum cluster density Nmax in the regimes of incomplete and complete condensation. The latter regime is always accessible at sufficiently low temperatures, and cluster densities can be determined from transmission electron microscopy (TEM) measurements.[37] The cluster density quickly reaches a maximum during the nucleation period, shows little changes thereafter, and finally drops as a result of cluster coalescence.

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The dependence of the maximum cluster density on the deposition rate R can be used to evaluate the size of the critical cluster, which is important to understand the interplay of diffusion and aggregation. For random nucleation in the regime of complete condensation, nucleation theory predicts:[42] i 

Nmax  (R
Ds)i2.5,

(2)

where the integer quantity i is the critical cluster size and Ds is the diffusivity of metal atoms on the surface. A fit to experimental data for noble metals was only possible with i  1.[37] This means that noble metal dimers already form stable clusters on polymers. This is in accord with the high cohesive energy of metals and the calculated binding energies for dimers, for example, 77 kJ/mol for Ag.[43] With i  1 and the Arrhenius law for surface diffusion, Eq. (3) immediately yields the temperature dependence of the maximum cluster density: Nmax  exp[Ed
(3.5kT)].

(3)

Thus, measurements of the maximum cluster density as a function of temperature allow us to determine the activation enthalpy Ed of surface diffusion. In the regime of incomplete condensation, there is a competition between thermal desorption with a rate t  exp(Ea
(kT)) and diffusioncontrolled nucleation. Here nucleation theory predicts:[42] Nmax  exp[(2
3)(2Ea  Ed)
(3.5kT)].

(4)

Based on Eqs. (3) and (4), we have determined adsorption enthalpies Ea and activation energies of surface diffusion Ed for various noble metal polymer systems. Representative plots of ln (Nmax) against 1/T are shown in Fig. 7.2. A compilation of Ea and Ed values from Zaporojtchenko et al.[40] is given in Table 7.1. The adsorption enthalpies of Cu on polyimide, 58 ± 10 kJ/mol, and on SiLK®, 67 ± 10 kJ/mol, turned out to be surprisingly high.[39] They reflect a substantial interaction of the noble metal with these polymers, in accord with conclusions drawn from bulk diffusion studies discussed in Sec. 7.4. Ag and Au exhibit a much weaker interaction (29 kJ/mol for Ag on polyimide, for example). The adsorption energies are also much lower for noble metals on polymers like polystyrene and Teflon AF®. The activation enthalpies of surface diffusion are substantially lower than the adsorption enthalpies, for example, 19 ± 5 kJ/mol for Cu on polyimide and 29 ± 5 kJ/mol on SiLK®.[39] This reflects nonlocalized binding forces, which make it much easier to displace an atom at the surface than to remove it completely. Similar to the adsorption enthalpies (Ea), Ed also

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Table 7.1. Activation Enthalpies of Surface Diffusion Ed and Adsorption Enthalpies Ea for Metals on Polymers

Metal/Polymer

Ed (kJ/mol)

Ea (kJ/mol)

Reference

Ag/PMDA-ODA Ag/TMC-PC Cu/PMDA-ODA Cu/ SiLK®

8 ± 3.0 6 ± 3.0 20 ± 5.0 29 ± 5.0

29 ± 10 17 ± 10 58 ± 10 67 ± 10

[37] [36] [39] [39]

Figure 7.2 Maximum cluster density Nmax as a function of reciprocal absolute temperature on a semi-logarithmic scale for Cu evaporated onto the polyimides PMDA-ODA, DuPont PI-2545, and DuPont SiLK®. The regimes of complete condensation at low temperatures and incomplete condensation at higher temperatures are clearly distinguished by two linear ranges of different slopes. The slopes are related to Ea and Ed via Eqs. (3) and (4), and allow these important microscopic quantities to be determined.

decreases following change of the metal species (from Cu to Ag and Au or the polymers themselves) from polyimide and SiLK® to polystyrene and Teflon AF®. Ed values are also given by Zaporojtchenko et al.[40] Based on the activation enthalpies of surface diffusion, we also estimated the ratio of the surface diffusivities Ds of Ag, Au, and Cu, assuming

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that the pre-exponential factors do not differ much. For polyimide at room temperature, Ds ratios for Ag/Au/Cu are 100/2/1.[37] The larger surface diffusivity of Ag compared to Au is unexpected, judging from the metal reactivity. Apparently, the general trend of the reactivity does not account for the interaction with the polymer. Deviations from the general trend are also seen in the complex formation behavior of Ag and Au.[44] The higher surface diffusivity of Ag compared to Cu also calls for further comment, because the opposite behavior has been observed in radiotracer measurements of bulk diffusion.[9, 45] The slower diffusion of Ag in the bulk was attributed to the larger size of Ag atoms, which should reduce the bulk diffusivity substantially in a glassy polymer. However, diffusion at the surface should not be affected significantly by size effects because factors like availability of free volume and distortion of the polymer do not come into play. This supports the view that the reported Ag bulk diffusion coefficients do not reflect atomic diffusion but are diffusivities of very small clusters (see Sec. 7.4).

7.3

Metal-Polymer Interaction

It is obvious that the mobility of metal atoms in polymers and on their surfaces is correlated with the extent of chemical interaction. Surprisingly, irrespective of extensive research throughout recent decades, our present knowledge of metal-polymer interaction is still rather incomplete, particularly with respect to the details of the interaction mechanisms and the early deposition stages.[17, 28, 46, 47] This is partly due to the strong aggregation tendency of moderately reactive metals on polymers. Here, the chemical interaction at room temperature and above occurs between metal clusters and the polymer. This not only leads to a significant drop of the detectable interfacial area but also may change the mode of interaction significantly because the chemistry of clusters and single atoms is not expected to be the same. The observations of chemical interactions of isolated noble metal atoms and polymer surfaces require the deposition at much lower temperatures.[48] An additional complication arises because of the frequently low sticking coefficients discussed in Sec.7.2.[36–38] In most studies, the sticking coefficient was assumed to be unity, and the absolute metal coverage was determined by means of a quartz balance. This procedure strongly overestimates the metal coverage if the sticking coefficient on the polymer surface is low. Nevertheless, consensus exists that Au, Ag, Cu, and Pd interact weakly with polymers.[48, 50–58] For highly reactive metals such as the transition metals Cr[17, 46, 52, 58, 59] and Ti,[60–62] the rare-earth metal Ce,[63] and Al,[64–67] the available experimental data show clear signs of strong chemical

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reactions with polymers, involving the formation of new compounds, at coverages of about one monolayer and higher. It has been argued that the data for polyimide do not imply reactive metals like Cr and Ti to be bonded directly to the polymer, but rather suggest a disruption of the polyimide and the formation of oxidic, nitridic, and carbidic compounds.[28, 47] Co was also found to react strongly with PMDA-ODA.[68] Ni appears to be much less reactive, but more reactive than Cu.[19, 58, 69] The alkali metals K[28] and Cs[29] have been demonstrated to reduce PMDA-ODA by transferring an electron to the PMDA unit of the polymer. Dianion formation was also observed at high alkali metal concentrations. The high reactivity of Cr, Ti, Al, and Co in the high-coverage regime does not necessarily imply that individual metal atoms are strongly bound to polymer chains. Based on measurements of the near-edge x-ray absorption fine structure in the very early stages of interface formation, Strunskus et al.[28] ruled out purely physical interactions between Cr and polyimide and discussed p-complex formation[17, 48] as one possible interaction mechanism. On the other hand, the fact that an intermixing layer was observed at the Al-polyimide interface in cross-sectional TEM studies[17, 19] points to relatively weak interactions of this reactive metal at very low coverages. Our XPS and TEM experiments carried out in the Crpolyimide system at very low Cr coverages also show short-range intermixing at the interface. However, the absence of long-range diffusion even after very slow deposition at elevated temperatures in conjunction with the lack of any influence of the deposition rate on the extent of intermixing (compare Sec. 7.4) clearly points to strong chemical interactions.[70]

7.4

Diffusion in the Polymer Bulk

Surface spectroscopy experiments indeed confirmed a strong correlation between reactivity and mobility of metal atoms. In these experiments, the metal was deposited at room temperature, and the drop of the metal intensity was measured after annealing. While substantial intensity drops were observed for noble metals,[50, 69, 71–73] reflecting appreciable metal mobility, interfaces of polymers with Cr and Ti proved to be thermally stable.[17, 60, 69] Al showed features of some mobility,[17, 65] and the mobility of Ni turned out to be somewhere between that of Al and Cu. Note that the metal mobility reflected in a drop of the metal intensity in surface analytical techniques cannot necessarily be taken as evidence of metal diffusion into the polymer bulk; it is often caused by metal clustering at the surface, as discussed below. The ability of K and Cs to reduce polymers proved to have drastic consequences on the diffusion behavior. Since positive ions repel each

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other, their diffusion is not impeded by the formation of immobile clusters. Moreover, positive ions are much smaller than the neutral atoms. As a result, these ions were found to be highly mobile in polymers, and almost uniform K and Cs distributions were observed by means of angularresolved XPS[29] and Fourier-transform infrared reflection-absorption spectroscopy[28] in polyimide films of appreciable thickness. A similar behavior was seen for In in films of the organic semiconductor perylenetetracarboxylic dianhydride (PTCDA).[30] Here even Al was seen to diffuse into the organic film quite extensively, which was attributed to the relatively low ionization energy. In contrast, Ti, Sn, Ag, and Au turned out to form interfaces that display evidence of overlayer metallicity at coverages as small as 5 to 10 nm. The first direct evidence of noble metal diffusion and aggregation in the polymer bulk has been provided by cross-sectional TEM studies of interface formation between Cu and polyimide (PMDA-ODA). In these studies, LeGoues et al.[19] observed marked clustering of Cu at considerable distances below the polyimide surface after metal deposition at elevated temperatures (but still well below Tg) and low deposition rates. However, no clustering in the bulk was observed at high deposition rates and even elevated temperatures. After room-temperature deposition and subsequent annealing, no metal particles were detected either. Investigations carried out by Kiene et al.[22] have essentially confirmed these early results and have shown that other noble metal-polymer systems, for example, Ag/PMDA-ODA polyimide[21, 74, 75] and Au/TMC polycarbonate,[32, 76] exhibit a similar behavior. Figure 7.3 gives a striking example that demonstrates the crucial role of the metallization conditions on the interfacial structure. While deposition of Cu at a very low rate at 350°C produces a rather spread-out interface, implying pronounced Cu diffusion (Fig. 7.3, bottom), deposition at room temperature and subsequent annealing at the same temperature results in a sharp interface without cluster formation inside the polyimide (Fig. 7.3, top).[22] Note that the metal films in Fig. 7.3 are still not continuous, despite the relatively large nominal metal coverage of about 30 monolayers. The impression of a continuous film is a consequence of the finite thickness of the samples of 40 to 100 nm, depending on the cutting procedure. Upon tilting, isolated and connected clusters are clearly visible. The results depicted in Fig. 7.3 and corresponding results obtained for other noble metal systems[9] show that metal clustering at the surface effectively impedes metal diffusion into the bulk and strongly suggests that no significant metal diffusion into the polymers is expected from a continuous film or an arrangement of large clusters. This conclusion is also corroborated by the pioneering medium-energy ion scattering experiments of Cu diffusion in polyimide by Tromp et al.[18] On the other hand,

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Figure 7.3 Cross-sectional TEM micrographs showing the striking differences in the copper diffusion characteristics between Cu deposition at room temperature (8 nm at 0.16 nm/min.) followed by subsequent annealing at 350°C for 30 min. (top) and very slow evaporation of Cu (4 nm at 0.16 nm/min.) at 350°C (bottom).

the absence of clusters in cross-sectional TEM images does not allow diffusion of metal atoms into polymers and the formation of small clusters that are not detectable in TEM to be ruled out. We have used a very sensitive radiotracer technique to study atomic diffusion. Diffusion profiles of the radiotracer atoms in the polymer films

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are measured by means of ion-beam microsectioning in a high-vacuum chamber. A very high depth resolution of typically 3 to 4 nm can be achieved by use of argon or krypton ions of energies as low as 100 to 200 eV[77] (as opposed to several keV in secondary ion mass spectrometry). Surface charging of the insulating polymer film is avoided by use of an electron-emitting neutralizer filament. The sputtered-off material of each section is collected on a catcher foil, which is advanced like a film in a camera. After sputtering, the catcher foil is cut into pieces corresponding to the individual sections, and the radioactivity of each section, which is proportional to the tracer concentration, is counted. After depth calibration, a penetration profile is obtained by plotting the radioactivity of the individual sections against the penetration depth. Using a calibrated source, absolute metal concentrations can be determined. Representative penetration profiles are shown in Fig. 7.4. Here the logarithm of the activity of 110mAg is plotted versus the square of the penetration depth x. The total amount of metal is extremely small. On this scale, ordinary diffusion of the tracer according to the thin-film solution of Fick’s second law,





x2 c(x)  const. exp   , 4Dt

(5)

leads to a straight line of slope 1
(4Dt). In Eq. (5), D is the tracer diffusivity and t is the diffusion time. It is obvious from Fig. 7.4 that silver diffuses deeply into polyimide during deposition at elevated temperatures. However, only the upper profile, obtained after evaporation of silver onto the hot polymer sample at an extremely low rate, is nearly Gaussian. At high and moderate deposition rates, profiles often resemble the lower profile in Fig. 7.4, where most of the metal deposit is located at or very close to the polymer surface. The fraction of atoms that diffuse over large distances and contribute to the linear range is very small. (Note the logarithmic scale.) Metal diffusion into the bulk is only detectable because of the high sensitivity of the radiotracer technique. Nevertheless, diffusion of even trace amounts of Cu through the polymer into Si devices may cause damage in ultra-large-scale integrated chips if it is not effectively blocked by a barrier. Penetration profiles of a similar shape have been recorded in various measurements for noble-metal tracers in polyimides[9, 24, 78, 79] and polycarbonates.[80, 81] Only the ratio of metal in the near-surface region and in the tail varied from case to case. In view of the characteristic shape of the penetration profiles and the well-known metal aggregation tendency, special attention was paid to sputtering artifacts.[9, 23, 80, 82] For example, it was shown that the sputter

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Figure 7.4 Effect of the deposition rate on penetration profiles of 110mAg into PMDA-ODA polyimide. The tracer was deposited continuously at a rate of about 0.1 ML/min. during annealing at 315°C and flash evaporated at the annealing temperature of 328°C.

rate of noble metals is even slightly higher than that of polycarbonates.[23, 82] Hence retarded sputtering of metal clusters, which would mimic metal diffusion, can be excluded. In all cases, a resolution function was recorded by sputtering a very thin tracer layer on a polymer film without diffusion annealing. Moreover, sputtering was performed not only in the direction of diffusion but also in the opposite direction after removing the polymer film from the substrate.[23, 82] Finally, at a given temperature, we determined diffusivities from the Gaussian tails of the diffusion profiles (see Fig. 7.5) and showed that the resulting diffusivities were independent of the annealing time.[75, 81] Monte Carlo simulations have shown that the dominant role of the deposition rate is a direct consequence of the interplay of atomic diffusion

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Figure 7.5 Simulated penetration profiles for the total metal concentration and for metal clusters of different size ranges, indicated by the number n of atoms, after deposition during (a) 2,500 and (b) 50,000 jumps per free atom (jpa) and subsequent annealing. The total annealing time of 5  105 jpa, the nominal metal coverage of 0.2 monolayer, and the ratio of surface and bulk diffusivity of Ds
D  60 are equal for both runs. Cross-sectional views at the bottom show clusters that have formed in a section of the simulated volume. The calibration of the deposition rates depends on the experimental diffusion temperature. For diffusion of noble metals in polyimides and polycarbonates, a deposition rate of the order of 1 ML/min. typically corresponds to 100 jpa in the vicinity of the glass transition temperature.[9]

and aggregation.[9, 83, 84] Such effects are not observed in ordinary diffusion experiments. Simulated metal concentration profiles and simulated TEM images were found to exhibit the characteristics observed in the experiments if we incorporated the condition that stable clusters form whenever metal atoms encounter each other on their diffusion path. This condition is implied by our observation of a critical cluster size of i  1 (see Sec. 7.3). An example is given in Fig. 7.5. The Monte Carlo simulations also reflect the experimental observation that the total metal concentration found at large concentration depth decreases strongly with increasing deposition rate due to the metal immobilization through clustering.[9] This is illustrated in Fig. 7.6, where the fraction of metal atoms at depths of 3 and 30 nm is plotted against the deposition time (1
rate). We can

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Figure 7.6 Total metal concentration found in the computer simulations at penetration depths
3 and
30 nm, as a function of the metal deposition time in units of jumps per free atom.

conclude that at industrial metallization rates, only traces of noble metal are to be expected in the polymer bulk. Because trace amounts of metals can cause device failures in chip interconnects involving organic low-k dielectrics, diffusion barriers must be used. Based on the notion that metal diffusion into polymers is an interplay between atomic diffusion and aggregation, suggested by the Monte Carlo simulations and the experiments described in this chapter, the extended linear tails in the low-concentration range of the radiotracer profiles (see Fig. 7.4) were attributed to diffusion of isolated metal atoms. Arrhenius plots of the tracer diffusivities for Cu and Ag noble metals in polyimides (PMDA-ODA and BPDA-BDA) and polycarbonates, determined from the linear tails of penetration profiles similar to those in Fig. 7.4, are shown in Fig. 7.7. Diffusivities of O2, CO2, and H2O are also shown. Our results from ion-beam depth profiling in conjunction with x-ray photoelectron spectroscopy on Cu diffusion in SiLK® indicate that Cu diffusion is more than an order of magnitude slower in SiLK® than in the polyimide PMDA-ODA at 315°C.[39] This is in accord with the higher adsorption

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Figure 7.7 Diffusion coefficients versus reciprocal temperature for diffusion of 67Cu and 110mAg in PMDA-ODA polyimide, Cu in Kapton®, and 67Cu in BPDA-PDA polyimide. Arrhenius plots for diffusion of O2, CO2, and H2O in Kapton®[85] are shown for comparison. Legend: open circles,[75, 78] open squares,[79, 86] solid squares,[45] solid circles,[87] open triangles from RBS combined with ion implantation,[88] and crosses.[79, 86]

energy of Cu on SiLK®, discussed in Sec. 7.2, which indicates stronger interaction. Diffusivities for Au and Ag in polycarbonates are displayed in Fig. 7.8. Note in Fig. 7.7 that the diffusivities of noble-metal atoms in polyimide are many orders of magnitude smaller than those of simple gas molecules of comparable size. This was interpreted in terms of significant contributions from metal interactions with the polymer even for noble metals. Substantial interactions between noble metal atoms and polymers are also reflected in the relatively high adsorption energies discussed in Sec. 7.2 and listed in Table 7.1. Based on a detailed analysis of the metal diffusion behavior in BPA polycarbonate,[9] which turned out to be completely decoupled from the polymer dynamics, for example, we have attributed the drastic slowing down of the atomic mobility of metals in comparison to noninteracting gas molecules to metal-atom-induced temporary

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Figure 7.8 Arrhenius plots for atomic diffusion of 198Au (solid symbols) and 110mAg (open symbols) in the polycarbonates BPA-PC and TMC-PC. The dashed lines indicate the glass transition temperatures measured by means of DSC at a heating rate of 20°C/min. Diffusion in BPA-PC is well described by linear Arrhenius fits with activation energies of Q  98 and 109 kJ/mole and pre-exponential factors of D0  0.0016 and 0.022 cm2/s, respectively.[9, 80, 87] The data for TMC-PC exhibit a downward curvature.[81, 87]

cross-linking.[24, 80] It was reported, for example, that polyimidepolyamide copolymers could no longer be dissolved in typical solvents after a fine dispersion of Ag and Pd had been incorporated (by reduction of the corresponding metal salts).[89] Despite the available evidence of substantial interaction of noble metal atoms with polymers, we cannot exclude the possibilty that the diffusivities determined from the linear tails of the radiotracer profiles reflect diffusion of small clusters rather than atomic diffusion. This particularly holds for the measured Ag diffusivities because the tracer concentration in

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Figure 7.9 Penetration profiles obtained after 110mAg deposition and annealing at 35°C, with a predeposited layer of 0.08 ML Cr, and with co-deposition of ≈3 ML of nonradioactive silver after 3 min. of tracer evaporation (12 min. total deposition time). The resolution function, obtained from a thin layer of 110mAg evaporated on a Fe-Si-B glass without annealing, is shown for comparison. The depth scale is linear.

the experiments with Ag typically corresponds to several monolayers (while only ≈1
4 monolayer was used in the Cu tracer studies). As discussed in Sec. 7.2, the differences in the ratios of DCu
DAg for surface and bulk diffusion also point to this possibility. Our recent radiotracer measurements of 110mAg in TMC polycarbonate[23] indeed suggest that the diffusivities reported by Willecke and Faupel[81] are those of small clusters and that single Ag atoms, which are only present in trace amounts, are mobile even at room temperature. In these experiments, several diffusion profiles were measured in samples where a thin layer of 110mAg was deposited at 35°C over periods of 12 to 15 minutes, and annealing was performed for up to 11 days.[82] A representative 110mAg profile is shown in Fig. 7.9 (open squares). As expected from the aggregation-induced immobilization, the Ag concentration drops by several orders of magnitude over very short penetration

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depths below the polymer surface. However, traces of Ag, which are still harmful to semiconductor devices, can be detected even at a depth of 400 nm. The annealing time did not significantly affect the silver distribution, which appears to be almost stationary. Fig. 7.9 also shows the effect of 0.08 monolayer (ML) of Cr deposited prior to 110mAg evaporation. The penetration profile largely resembles the resolution function, which was obtained from a thin layer of 110mAg deposited onto a Fe-Si-B glass at room temperature. (The Fe-Si-B glass was chosen because the absence of diffusion at room temperature had been checked before.) The small differences are most likely due to the dependence of the resolution function on the matrix material. Comparison of the profiles with and without Cr predeposition provides clear evidence of Ag diffusion at room temperature. The experiments also demonstrate that a Cr layer as thin as 0.08 ML is an excellent barrier against silver diffusion into the polymer.[23] The barrier function of Cr, which also acts as an adhesion promoter, has been reported before.[90, 91] However, much thicker continuous films well above 1 nm were used. We attribute the excellent barrier function of a 0.08 ML film to the well-known fact that Cr is very reactive and forms strong bonds to most polymers (see Sec. 7.3). This results in a very fine (probably atomic) dispersion of the Cr atoms, which act as effective traps to the Ag atoms. Results from our Monte Carlo simulations shown in Fig. 7.10 confirm the excellent barrier function of atomic traps on a surface well below monolayer coverage. Experiments carried out at 35 and 187°C show that predeposited nonradioactive silver layers in the ML range also act as barriers.[82] The barriers are less effective, however, despite their much larger thickness, because Ag forms larger clusters with a lower cluster density at the polymer surface.[9] Co-deposition of ≈3 ML of nonradioactive silver after 3 minutes of tracer evaporation (12 minutes total deposition time) also leads to a considerable barrier effect (Fig. 7.9). The barrier function of the Ag clusters, which are much larger than the clusters formed without predeposition of nonradioactive Ag, allows us to rule out the argument that the barrier function of Cr is only a sputtering artifact caused by the finer dispersion of Ag in the presence of the Cr layer. This artifact can also be excluded because the sputtering rate of Ag is slightly higher than that of the polymer. Therefore, preferential sputtering of the polymer matrix cannot occur. Additional striking evidence for the reliability of the present ion-beam depth profiling technique comes from experiments mentioned above, where the polymer film was removed from the substrate after diffusion annealing and sputtering was performed from the backside through the entire film.[23]

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Figure 7.10 Monte Carlo simulations showing the trapping of evaporated mobile silver atoms by an atomic dispersion of immobile chromium traps corresponding to about 0.01 nm Cr. The top and bottom figures are top and cross-sectional views, respectively. (a) Without traps, silver clusters form at the surface by random nucleation, and a small amount of silver diffuses into the bulk. (b) The chromium atoms act as nucleation centers and lead to a much finer dispersion of silver clusters. Due to trapping in the surface clusters, diffusion into the bulk is effectively blocked. The simulations were carried out with the program described by Faupel et al.[9] and Thran[82] using a ratio of 100 for the surface and bulk diffusivities.

Figure 7.11 shows a penetration profile from a sample annealed at 187°C, which is plotted on a semilogarithmic scale versus the square of the penetration depth as suggested by the ordinary thin-film solution of Fick’s 2nd law (Gaussian curve). On this scale, we detect a linear range with a diffusivity D  8  1015 cm2/s, which is very close to that reported by Willecke and Faupel[81] (8.6  1015 cm2/s; see Fig. 7.8). In light of the present results, the range is attributed to diffusion of clusters. The inset also shows additional linear ranges attributed to the reduced mobility of larger clusters. The diffusion of trace amounts of isolated atoms, which are orders of magnitude faster than the clusters investigated,[81] was obviously not detected in the experimental window chosen. The almost stationary nature of the Ag110m penetration profiles obtained in TMC polycarbonate under the conditions is described by the empirical law exp[(z
z0)]1
x0 - with x0 close to 10 (z  penetration depth). Physically, it is attributed to the immobilization of the silver

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Figure 7.11 Penetration profiles obtained after 110mAg deposition and annealing at 185°C for 70 min. The data are plotted on a semi-logarithmic scale versus the square of the penetration depth. Diffusivities estimated from straight lines fitted to the data in different depths ranges according to the thin-film solution of Fick’s second law are displayed.

atoms by cluster formation and trapping of the remaining highly mobile single atoms deeper in the polymer bulk. This is in accord with calculations reported by Dabral et al.[90] and with our Monte Carlo simulations, which also confirm the exp[(z
z0)]1
x0 -law as well as the magnitude of x0.[23, 82] Moreover, the simulations directly show how a penetration profile gradually becomes stationary, starting from the surface where immobilization by clustering sets in first (see Fig. 7.12). Hence, the simulations strongly support the present view of aggregation-induced immobilization. In contrast to the numerous investigations that demonstrate the very effective impediment of metal diffusion into polymers by metal aggregation (see also Faupel et al.,[9, 24] there are many publications reporting strong diffusion from continuous metal films into polymers. For example, the drop of the metal intensity in XPS[60, 92, 93] and the broadening of metal spectra in Rutherford backscattering[92, 94] upon annealing have been taken

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Figure 7.12 Monte Carlo simulations showing how a penetration profile becomes gradually stationary [KA1] due to immobilization of metal atoms by aggregation. Note that the immobilization starts from the surface where clustering sets in first. The time scale is jpa. The simulations were carried out as described by Faupel et al.[9] and Thran.[82] Here 0.08 monolayer of silver was evaporated during the first 104

as evidence for metal diffusion. Furthermore, it has been concluded from angle-resolved XPS measurements that Cu diffuses strongly into the low-k fluoropolymers Teflon-AF1600TM[95] and FLARETM.[93] We have shown, however, that XPS results on metal diffusion into polymers can be rather misleading without complementary microstructural observations and careful investigation of the long-time behavior of the metal intensity.[22] Concerning the Rutherford backscattering results, it was recently demonstrated that metal clustering at the polymer surface might mimic metal diffusion into the polymer bulk.[96] Clustering is driven by the reduction in surface energy and may even take place after annealing of continuous thin metal films.[96] To demonstrate the absence of significant diffusion from a continuous metal film, we have evaporated Cu, with a nominal coverage of several monolayers, onto polyimide at room temperature. Subsequently, the samples were annealed at high temperatures, and the Cu 2p3
2
C 1s peak

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Figure 7.13 Evolution of the Cu 2p3 2/C 1s peak intensity ratio in XPS upon annealing at 350°C after deposition of 1.6 nm Cu (7 ML nominal coverage) onto polyimide at room temperature at rates of 1.6 and 0.03 nm/min. The inset shows results from a separate experiment involving room-temperature deposition (1.6 nm of Cu[KA2] at 0.16 nm/min.) and prolonged annealing for 30 hours at 350°C. The leveling off is a clear indication of Cu immobilization.

intensity ratio was measured as a function of the annealing time. An example is given in Fig. 7.13. Similar measurements were also performed for Au on TMC polycarbonate.[76] Without further quantitative evaluation, the occurrence of the plateau in the intensity ratio unambiguously demonstrates Cu immobilization. Any significant copper diffusion into polyimide would lead to a drop of the Cu 2p3
2 intensity because of the very short inelastic mean free path of the photoelectrons. Moreover, the C 1s intensity would increase if Cu disappeared from the surface. The inset in Fig. 7.13 demonstrates that no deviations from the plateau value were detected, even after prolonged annealing for 30 hours at 350°C. TEM investigations show that the Cu immobilization is due to the formation of Cu clusters, which form during metal deposition at room temperature. Annealing at 350°C causes the clusters to grow. As discussed in Sec. 7.3, Cu clustering at polymer surfaces arises from the high cohesive energy of the metal and the low metal-polymer interaction energy. Apparently, the cohesive energy is high enough to immobilize the Cu atoms by preventing their dissociation from the clusters at temperatures far below the metal boiling point. Cluster growth does not appear to occur by ordinary Ostwald ripening, but by a temperature-induced mobility of smaller clusters and cluster coalescence, thus causing a plateau in the Cu 2p3
2
C 1s peak

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intensity due to the decrease in the cluster mobility and the increase in the cluster distance with increasing cluster size.[71] Simple calculations[22, 97] show that the cluster growth observed in TEM is the main reason for the small drop of the Cu 2p3
2
C 1s peak intensity ratio of about 40% during annealing (see Fig. 7.13). Even in the initial annealing stages prior to complete Cu immobilization, Cu diffusion into the polymer obviously plays a minor role at best. On the other hand, capacitance-voltage measurements involving Cu/oxide/polymer/oxide/Si capacitors suggest drift of Cu ions into poly (arylene ether) and fluorinated polyimide at elevated temperatures in strong electric fields of the order of 106 V/cm, although the drift of Cu ions through the polymers was not demonstrated directly.[98, 99] The drift current dropped rapidly within several minutes and was not observed after incorporation of thin-nitride barrier layers.[99] Benzocyclobutene proved to be much less prone to Cu ion drift.[98–104] Mallikarjunan et al.[100] studied interfaces of Cu, Al, Ta, and Pt with hybrid organosiloxane polymer (HOSP) using bias-temperature stressing. Surprisingly, they found an increasing extent of ion transport in the order Cu  Ta  Al. No ions were seen in Pt, in accord with its very high ionization potential. It was demonstrated that the metal-polymer interface chemistry, particularly the presence of interfacial oxygen, plays a key role in aiding metal ionization and subsequent mobile ion penetration into the polymer. While Ta and Al form stable oxide or metal-silicon oxide layers on SiO2, and Ta is even used as a diffusion barrier between Cu and SiO2, it was suggested that insufficient surface oxygen content of HOSP leads to the formation of metal-rich unstable and discontinuous oxides with unbonded metal ions. Even the conventional barrier material TaxN was shown to fail in contact with HOSP due to the incorporation of oxygen impurity or the reaction with surface oxygen, which can lead to tantalum ionization and drift into HSPO. This clearly shows that conventional barrier materials may not be suitable for the novel low-k dielectrics. Cu can also oxidize readily. The interfacial oxides do not form protective barriers and were suggested to be the source for mobile metal ions.[101] More ion penetration was in fact observed when copper was deposited on highly oxidized surfaces.[100–103] This implies that eliminating interfacial oxygen can suppress the formation of mobile Cu ions. The interpretation of the drift experiments does not conflict with the present view of aggregation-induced metal immobilization in polymers. No diffusion was observed at zero and low electric fields. Moreover, the oxide layer between Cu and the polymer in the drift experiments apparently acts as a catalyst for the formation of Cu ions, which are injected into the polymer in the electric field. These small Cu ions are expected to be highly mobile and, unlike neutral Cu atoms, unable to form immobile

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clusters because of their repulsion. This interpretation is supported by the strong diffusion of K and Cs ions into the polyimid PMDA-ODA[28, 29] and of In ions into PTCDA,[30] as mentioned earlier. Reactive metals like Cr and Ti generally form strong bonds to the polymer chains (see Sec. 7.3). Consequently, long-range diffusion appears to be blocked completely. As mentioned earlier in this section, no indications of Cr diffusion into polyimide were found in XPS studies, even after very slow evaporation near the glass transition temperature.[70] Interfaces of polymers with Cr and Ti proved to be sharp and thermally stable,[17, 60, 69, 90] and these metals were shown to act as diffusion barriers. In the case of Cr, a coverage as low as 0.08 monolayer proved to be very effective.[23] Al showed features of some mobility.[17, 65] The mobility of Ni turned out to be somewhere in between that of Al and Cu. The substantial mobility of Ni atoms was also seen in the formation of Ni clusters containing nanocomposites with a matrix of polyimide[105] or Teflon AF®[106] produced by vapor phase co-deposition of the organic and metallic component and subsequent annealing.

7.5

Summary and Conclusions

This chapter reviews the diffusion behavior of metals on polymer surfaces and in the bulk. We focus on work carried out by our group in Kiel involving radiotracer measurements, surface spectroscopy, electron microscopy, Monte Carlo simulations, and other techniques. The adsorption energy of noble metals on polymers is much smaller than the metal-metal interaction energy, although it can exceed 0.5 eV, for example, for Cu on polyimide and SiLK®. The activation energy for surface diffusion is considerably lower. This reflects undirected bonding and gives rise to a high surface mobility down to temperatures well below room temperature. The high metal cohesive energy compared to the relatively weak interaction energy of noble metals with polymers and to the much lower polymer cohesive energy leads to a very strong aggregation tendency. Therefore, noble metals exhibit a Volmer-Weber surface growth mode (cluster formation) with a critical nucleus consisting of only one atom. The same holds for other metals of low reactivity. The strong aggregation tendency effectively impedes metal diffusion into polymers. In particular, no significant diffusion of noble metal into polymers is expected from continuous metal films unless interfacial reactions, particularly the presence of interfacial oxygen, promote the formation of metal ions. These cations are highly mobile and repel each other, thus preventing aggregation. In some cases, substantial ion drift has been observed under bias temperature stress conditions of microelectronics

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devices where low voltages may already give rise to very strong electrical fields due to the small feature sizes. Isolated neutral metal atoms of low reactivity are also highly mobile and diffuse into polymers at very low deposition rates, where immobilization by aggregation is less effective. In the initial deposition stage, isolated metal atoms impinge on the polymer surface. Therefore, trace amounts of noble metal atoms always diffuse into polymers and can penetrate several hundreds of nm even at room temperature. This calls for using barriers in microelectronic chip applications. Reactive metals like Cr and Ti form relatively sharp interfaces with polymers and act as diffusion barriers. We have demonstrated excellent barrier function for a Cr layer as thin as 0.08 monolayer. The behavior of Al and Ni is somewhere between that of noble metals and reactive metals like Cr. Both metals exhibit some mobility on and in most polymers, but Al reacts strongly with fluoropolymers. The latter case shows that general trends must be treated with caution since the diffusion behavior for a given metal-polymer combination is strongly governed by the individual chemical interactions.

Acknowledgments We would like to thank Jörn Kanzow and Jörn Erichsen for stimulating discussions and critical reading of the manuscript, and Haile Takele for technical assistance. Financial support by the Volkswagen Foundation and German Science Foundation (DFG) is gratefully acknowledged.

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35. See, for example, K. D. Rendulic, Surf. Sci., 272:34 (1992) 36. A. Thran, M. Kiene, V. Zaporojtchenko, and F. Faupel, Phys. Rev. Lett., 82:1903 (1999) 37. V. Zaporojtchenko, K. Behnke, A. Thran, T. Strunskus, and F. Faupel, Appl. Surf. Sci., 144–145:355 (1999) 38. V. Zaporojtchenko, K. Behnke, T. Strunskus, and F. Faupel, Surf. Interf. Anal., 30:439 (2000) 39. V. Zaporojtchenko, J. Erichsen, T. Strunskus, K. Behnke, F. Faupel, M. Baklanov, and K. Maex, MRS Proceedings, L8.1 (2001) 40. V. Zaporojtchenko, J. Erichsen, J. Zekonyte, A. Thran, T. Strunskus, and F. Faupel, in Metalization of Polymers, vol. 2 (E. Sacher, ed.), ACS Symposium Series, Kluwer Academic/Plenum Publishers, NY (2002), p. 107 41. M.A. Dayanada, K.N. Tu, A.D. Roming, and D. Gupta, Diffusion in High Technology Materials, Trans. Tech. Pubs. (1988) 42. J. A. Venables, Surf. Sci., 299–300:798 (1994) 43. R. Poteau, J. L. Heully, and F. Spiegelmannn, Z. Phys., D40:479(1997) 44. H. Wieberg, Lehrbuch der Anorganischen Chemie, Walter de Gruyter (1995) 45. F. Faupel, Adv. Mater., 2:266 (1990) 46. N. J. DiNardo, in Metallized Plastics, 1: Fundamental and Applied Aspects (K. L. Mittal and J. R. Susko, eds.), Plenum Press, NY (1989) 47. L. J. Matienzo and W. J. Unertl, in Polyimides: Fundamental Aspects and Technological Applications (M. Ghosh and K. L. Mittal, eds.), Marcel Dekker, NY (1996) 48. T. Strunskus, M. Kiene, R. Willecke, A. Thran, C. v. Bechtolsheim, and F. Faupel, Mater. Corrosion, 49:180 (1998) 49. F. Faupel, A. Thran, V. Zaporojtchenko, M. Keine, T. Strunskus, and K. Behnke, in Stress-Induced Phenomena in Metallization, Stuttgart, 1999 (O. Kraft, E. Arzt, C. A. Volkert, P. S. Ho, and H. Okabayashi, eds.), AIP Conf. Proc. (1999), p. 491 50. H. M. Meyer, S. G. Anderson, L. Atanasoska, and J. H. Weaver, J. Vac. Sci. Technol., A6(30):1002 (1988) 51. N. J. DiNardo, J. E. Demuth, and T. C. Clarke, Chem. Phys. Lett., 121:239 (1985) 52. J. L. Jordan-Sweet, in Metallization of Polymers (E. Sacher, J. Pireaux, and S. P. Kowalczyk, eds.), ACS Symposium Series 440, Am. Chem. Soc., Washington, DC (1990) 53. R. Rossi and B. D. Silverman, in Polymeric Materials for Electronic Packaging and Interconnection (J. H. Lupinski and R. S. Moore, eds.), ACS Symposium Series 407, Am. Chem. Soc., Washington, DC (1989) 54. L. J. Gerenser, J. Vac. Sci. Technol., A8:3682 (1990) 55. J. Pertsin and Y. M. Pashunin, Appl. Surf. Sci., 47:115 (1991) 56. G. D. Davis, B. J. Rees, and P. L. Whisnant, J. Vac. Sci. Technol., A12:2378 (1994) 57. A. V. Walker, G. L. Fisher, A. E. Hooper, T. Tighe, R. Opila, N. Winograd, and D. L. Allara, in Metallization of Polymers, vol. 2 (E. Sacher, ed.), ACS Symposium Series, Kluwer Academic/ Publishers, NY (2002), p. 117 58. P. Bebin and R. E. Prud’homme, Chem. Mater, 15 (4):965 (2002)

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59. J. J. Pireaux, C. Grégoire, M. Vermeersch, P. A. Thiry, M. R. Vilar, and R. Caudano in Metallization of Polymers (E. Sacher, J. Pireaux, and S. P. Kowalczyk, eds.), ACS Symposium Series 440, Am. Chem. Soc., Washington, DC (1990), p. 47 60. F. S. Ohuchi and S. C. Freilich, J. Vac. Sci. Technol., A4:1039 (1984); A6:1004 (1988) 61. W. N. Unertl, High Performance Polymers, 2:15 (1990) 62. S. C. Freilich and F. S. Farnsworth, Polymer, 28:1912 (1987) 63. J. G. Clabes, J. Vac. Sci. Technol., A6:2887 (1988) 64. L. Atanasoska, S. G. Anderson, H. M. Meyer, Z. Lin, and J. H. Weaver, J. Vac. Sci. Technol., A5:3325 (1987) 65. P. S. Ho, P. O. Hahn, J. W. Bartha, G. W. Rubloff, F. K. LeGoues, and B. D. Silverman, J. Vac. Sci. Technol., A3:739 (1985) 66. J. J. Pireaux, M. Vermeersch, C. Grégoire, P. A. Thiry, R. Caudano, and T. C. Clarke, J. Chem. Phys., 88:3353 (1988) 67. S.-J. Ding, V. Zaporojtchenko, J. Kruse, J. Zekonyte, and F. Faupel, Appl. Phys., A76:851 (2003) 68. S. G. Anderson, H. M. Meyer, and J. H. Weaver, J. Vac. Sci. Technol., A6:2205 (1988) 69. P. O. Hahn, G. W. Rubloff, J. W. Bartha, F. K. LeGoues, and P. S. Ho, Mater. Res. Soc. Symp. Proc., 40:251 (1985) 70. M. Kiene, H. Kiesbye, T. Strunskus, and F. Faupel, unpublished 71. M. Kiene, T. Strunskus, and F. Faupel, in Metallized Plastics: Fundamentals and Applications (K. L. Mittal, ed.), Marcel Dekker, NY (1998) 72. R. G. Mack, E. Grossman, and W. N. Unertl, J. Vac. Sci. Technol., A8:3827 (1990) 73. M. Fontaine, J. M. Layet, C. Grégoire, and J. J. Pireaux, Appl. Phys. Lett., 62:2938 (1993) 74. A. Foitzik and F. Faupel, Mater. Res. Soc. Symp. Proc., 203:59 (1991) 75. F. Faupel, Phys. Stat. Solidi, (a)134:9 (1992) 76. C. V. Bechtolsheim, V. Zaporojtchenko, and F. Faupel, J. Mater. Res., 14:9 (1999) 77. F. Faupel, P. W. Hüppe, K. Rätzke, R. Willecke, and T. Hehenkamp, J. Vac. Sci. Technol., A10:92 (1992) 78. F. Faupel, D. Gupta, B. D. Silverman, and P. S. Ho, Appl. Phys. Lett., 55:357 (1989) 79. D. Gupta, F. Faupel, and R. Willecke, in Diffusion in Amorphous Materials (H. Jain and D. Gupta, eds.), The Minerals, Metals and Materials Society (1994), p. 189 80. R. Willecke and F. Faupel, Macromolecules, 30:567 (1997) 81. R. Willecke and F. Faupel, J. Polym. Sci., Polym. Phys. Ed., 35:1043 (1997) 82. A. Thran, Ph.D. thesis, University of Kiel (2000) 83. A. Thran and F. Faupel, Defect Diffusion Forum, 143–147(1):903 (1997) 84. D. Silverman, Macromolecules, 24:2467 (1991) 85. F. Faupel and G. Kroll, in Landolt-Börnstein, Numerical Data and Functional Relationships in Science and Technology, New Series, vol. III/33B1 (W. Martienssen and D. L. Beke, eds.), Springer-Verlag, Berlin, Heidelberg, NY (1999)

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86. D. Gupta, in Thin Films, Encyclopedia Appl. Phys., vol. 5, VCH Pub. Inc., NY (1993), p. 75, Diffusion 87. R. Willecke, Ph.D. thesis, University of Göttingen (1993) 88. J. H. Das and J. E. Morris, J. Appl. Phys., 66:5816 (1989) 89. L. Tröger, H. Hünnefeld, S. Nunes, M. Oehring, and D. Fritsch, J. Phys. Chem., B101:1279 (1997) 90. S. Dabral, G. Yang, H. Bakhru, T.-M. Lu, and J. F. Mc.Donald, Proceedings of the VMIC, IEEE, NY (1991), p. 408 91. N. Agmon, J. Chem. Phys., 81:2811 (1984) 92. N. Y. Kim, H.-S. Yoon, S. Y. Kim, C. N. Whang, K. W. Kim, and S. J. Cho, J. Vac. Sci. Technol., B17:380 (1999) 93. M. Du, R. L. Opila, V. M. Donnelly, J. Sapjeta, and T. Boone, J. Appl. Phys., 85:1496 (1999) 94. J. H. Das and J. E. Morris, IEEE Transactions on Components, Packaging and Manufacturing Technology, B17:620 (1994) 95. D. Popovici, K. Piyakis, M. Meunier, and E. Sacher, J. Appl. Phys., 83:108 (1998) 96. N. Marin, Y. Serruys, and P. Calmon, Nucl. Instr. Methods in Phys. Res., B108:179 (1996) 97. M. Kiene, Ph.D. thesis, University of Kiel (1997) 98. S. S. Wong, A. L. S. Loke, J. T. Wetzel, P. H. Townsend, R. N. Vrtis, and M. P. Zussman, in Low-Dielectric Constant Materials and Applications in Microelectronics IV (C. Chiang, P. S. Ho, T.-M. Lu, and J. Wetzel, eds.), Mater. Res. Soc. Symp. Proc., vol. 511 (1998), p. 317 99. A. L. S. Loke, J. T. Wetzel, C. Ryu, W.-J. Lee, and S. S. Wong, Symposium on VLSI Technology, Honolulu (1998) 100. A. Mallikarjunan, J. Juneja, G. Yang, S. P. Murarka, and T.-M. Lu, Mater. Res. Soc. Symp. Proc., 734:B9.60 (2003) 101. D. Kapila and J. L. Plawsky, Chem. Eng. Sci., 50:2589 (1995) 102. S. Rogojevic, A. Jain, W. Gill, and J. L. Plawsky, J. Electrochem. Soc., 149:F122 (2002) 103. T. Fukuda, H. Nishino, A. Matsuura, and H. Matsunaga, Jpn. J. Appl. Phys., 41:L537 (2002) 104. J. M. Neirynck, R. J. Gutmann, and S. P. Murarka, Proc. CMP-MIC Conference IMIC (1999), p. 192 105. K. Behnke, T. Strunskus, V. Zaporojtchenko, and F. Faupel, in Proc. 3rd Int. Conf. MicroMat 2000 (B. Michel, T. Winkler, M. Werner, and H. Fecht, eds.), Verlag ddp goldenbogen, Dresden (2000), p. 1052 106. A. Biswas, Z. Marton, J. Kanzow, J. Kruze, V. Zaporojtchenko, F. Faupel, and T. Strunskus, Nanoletters, 3(1):59 (2003)

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8 Measurement of Stresses in Thin Films and Their Relaxation Oliver Kraft Institut für Materialforschung II, Forschungszentrum Karlsruhe und Institut für Zuverlässigkeit von Bauteilen und Systemen Universität Karlsruhe, Germany Huajian Gao Max-Planck-Institut für Metallforschung Stuttgart, Germany

8.1

Introduction

Thin films are used in many technological applications because they provide special chemical, mechanical, and electrical properties, as and where needed, using only small amounts of materials. For example, the high density and fast performance of modern computers have been possible due to the incorporation of multilayer thin-film structures coupled with submicron photolithography on the back of the active devices formed in Si chips. Because thin-film structures are always deposited onto relatively rigid substrates, they are heavily stressed and undergo relaxation when subjected to fabrication processes involving higher temperatures than those used for the deposition. This chapter discusses techniques for measuring stresses in thin metal films attached to substrates and their relaxation by plastic deformation and diffusional creep. Figure 8.1 shows the origin of stresses in thin films as described by Nix.[1] As a start, we consider a film/substrate composite that is free of stress [Fig. 8.1(a)]. If we remove the film from the substrate, its lateral dimensions will match those of the substrate [Fig. 8.1(b)] and the film can be reattached without causing any stress. However, if we assume that the film is subject to a dilatational strain relative to the substrate, then the lateral dimensions of the film no longer match those of the substrate [Fig. 8.1(c)]. A stress must be applied to the film in order to make it fit to the substrate again [Fig. 8.1(d)]. Now the film can be rigidly reattached to the substrate and the applied stress removed. As a result, shear stresses are produced near the edge of the film/substrate composite, which maintains a biaxial stress in the film [Fig. 8.1(e)]. In addition, the shear

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Figure 8.1 Illustration origin of biaxial stress in a thin film: (a) stress-free film on the substrate; (b) remove film from the substrate; (c) film dimension change relative to the substrate; (d) external stress is applied to the film to return to the substrate dimension; (e) film is reattached to the substrate and external stresses are removed. As a result, (1) an internal biaxial stress remains in the film; (2) the inhomogeneous stress state near the edges leads to a bending moment, which causes the substrate to bend; and (3) a homogeneous stress in the substrate is developed, which can be neglected for the thin-film limit. After Nix.[1]

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stresses cause a bending moment that will lead to a curvature of the substrate. As discussed below, the measurement of this curvature is a common technique to determine the average stress in the film. It should be noted that, under the assumed condition that the film thickness tf is much smaller than the thickness ts of the substrate (100 tf
ts), stresses in the substrate are of minor importance. For pure dilatational strain, that is, a volume change of the film with respect to the substrate, with the strain components being ∆exx  ∆eyy  ∆ezz  ∆e and considering only elastic deformation for an isotropic material, the stress state in the film is given by: Ef s
sxx  syy   e 1nf

and

szz  0,

(1)

where Ef is Young’s modulus and nf is Poisson’s ratio of the film material. The stress component szz perpendicular to the film surface equals zero because the film is free to expand or contract in this direction. There are many possible reasons for a relative volume change of the film material with respect to the substrate. The most important is change of temperature if the thermal expansion coefficients of film and substrate are different: e  (af  as)T  aT,

(2)

where ∆T is the imposed temperature change, and af and as are the thermal expansion coefficients of the film and the substrate, respectively. Other mechanisms that cause dilatational strains include the annihilation of lattice defects such as excess vacancies, dislocations, and grain boundaries; phase transformations; and composition changes. These mechanisms typically occur during film growth or during an initial heat treatment until a stable microstructure has evolved. For the special case of heteroepitaxial films, the film is strained to accommodate the mismatch of the lattice parameters. For crystalline materials with an elastic anisotropy, the film stress depends on the orientation of a single crystalline film or the texture of a polycrystalline film. For cubic materials, grains oriented with both (111) and (100) directions parallel to the film normal are biaxially isotropic in the film plane. For single crystal films with one of these orientations, a uniform elastic strain leads to a uniform stress state, and Eq. (1) becomes: s111  Y111e

and

s100  Y100e,

(3)

with 6C44(C11  2C12) Y111    C11  2C12  4C44

(4)

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and 2C 212 (5) Y100  C11  C12   , C11 where Y111 and Y100 are the biaxial moduli for (111) and (100) out-of-plane oriented crystals, respectively, and C11, C12, and C44 are the anisotropic stiffness coefficients of the cubic material. For instance, for Al and Cu at room temperature, the biaxial moduli calculate to Y111  116 and 261 GPa, and Y100  100 and 115 GPa, so that Y111Y100  1.2 and 2.3, respectively. From Eq. (3), it is evident that for a given ∆e, the ratio of stresses for the two orientations is s111s100  Y111Y100. For a polycrystalline film, however, stress continuity requires s111  s100 at the grain boundaries.[2] Let us consider stress and strain states in a film with a mixed texture of (100) and (111) grains as shown in Fig. 8.2. In large grains, the stress in the interior of the grains reaches the expected value for the respective orientation [Fig. 8.2(a)]. Near the grain boundaries, the required stress variation is confined to roughly the film thickness.[3] Thus, in the limit of d  tf (d is the grain size), the volume fraction of the film with inhomogeneous strain is small (isostrain condition). In finegrained films [Fig. 8.2(b)], nearly complete stress accommodation is required, the stress will approach a uniform value everywhere (isostress condition), and the strains will follow e100e111  Y111Y100. Note that near the interface, if perfect adhesion is assumed, a stress singularity would arise and shear stresses must develop on the grain boundaries. In a real polycrystalline metal film, the grain size is typically on the order of the film thickness and, as a result, the average stresses in each orientation are expected to lie between the isostrain and isostress limits. Finally, note that the considerations above are valid for purely elastic behavior and, once plastic deformation of the film material is involved, an even broader stress distribution can be expected.

8.2

Measurement Techniques

Stresses can never be measured directly. Any experimental determination of a stress involves the measurement of an elastic strain and the use of appropriate elastic constants to calculate the stress. Two main methods are used to determine the stresses or strains in thin films. The first method is based on the measurement of the curvature of the substrate, which is bent as a result of the film stress. The second method is based on the measurement of the elastic strain directly in the film by x-ray diffraction. For the first technique, the elastic properties of the substrate material need to be known to determine the film stress; for the second, those of the film material itself need to be known.

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Figure 8.2 Schematics showing the stress and strain distributions during elastic deformation for a film microstructure consisting of (a) large (111) and (100) grains (d W tf), and (b) alternating narrow (111) and (100) grains (d ≈ tf). In the limit of large grains, the average strains in both components are the same (isostrain case). In the limit of small grains, we assume that the grains are narrower than the film thickness such that the stress is the same in both orientations (isostress case). After Baker et al.[2]

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8.2.1 Substrate Curvature The substrate curvature technique makes use of the fact that the radius of curvature, R, of the substrate is proportional to the stress s in the film, as quantitatively expressed by Stoney’s equation:[4]





Es t2s 1 1 s      , 1  ns 6tf R Ro

(6)

where Es and ns are the Young’s modulus and Poisson’s ratio of the substrate material; tf and ts are the thickness of the film and the substrate, respectively; and tf must be much less than ts for Eq. (6) to be valid. It should also be noted that the substrate is usually not perfectly flat and Ro is the radius of curvature of the substrate without the film. As Eq. (6) shows, an advantage of this method is that the only film property that must be known to evaluate the stress is the film thickness; knowledge of the elastic properties of the film is not required. However, the value measured for s represents the average stress in the film, and a statement on the uniformity of the stress is not possible. Another sample geometry, which is also frequently used, consists of a cantilever beam with one fixed end. Then the film stress is given by: t 2s u, s  Es  3L2tf

(7)

where L is the length and u is the deflection of the free end of the cantilever beam. To determine stresses accurately, the radius of curvature needs to be measured over a range from a few meters to several kilometers, or the displacement of the cantilever beam must be determined in the micrometer regime. This has been achieved experimentally in many different ways; for example, by optical methods;[5] profilometric, interferometric, or capacitive methods;[6] and, for different conditions, during film deposition,[7, 8] in vacuum, or at elevated temperatures.[9–12] Figure 8.3 shows an experimental setup that has been used by a number of authors[5, 12, 13] to measure film stresses as a function of temperature for different thin-film materials. A laser beam is used to scan the surface of the sample while it is heated. The curvature is determined from the orientation of the sample surface as measured by the angle of reflection of the laser beam using a position sensitive detector. Figure 8.4 shows some typical results for the stress as a function of temperature for a thin metal film (a) and a thin glass

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Figure 8.3 Schematic showing an experimental setup for wafer curvature measurements.

Notes: 1. Data for (a) from Keller et al.[13] 2. Data for (b) from C. A. Volkert, unpublished work (1997)

Figure 8.4 Film stress as a function of temperature as determined from wafer curvature measurements for (a) a 1.0-µm-thick Cu film and (b) SiO2 films made by chemical vapor deposition on Si substrates.

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film (b) on Si substrates. At room temperature, the metal film (500-nm-thick Cu film) is in a biaxial tensile stress state. On heating, the film tends to expand more than the substrate and the tensile stress is reduced. At roughly 150°C, the stress equals zero, and on further heating, the film stress becomes compressive. Up to a temperature of about 250°C, the film is deformed elastically. The slope of the curve is given by ∆aEf(1  nf), which is obtained by combining Eqs. (1) and (2). Above 250°C, the film is deformed plastically and the compressive stress does not increase further. On cooling from the maximum temperature of 500°C, the film contracts more than the substrate and, as a result, the film stress becomes tensile. Again, the film is first deformed elastically and then plastically at temperatures below 400°C. The general trends of the described behavior are typical for many metal films (for example, Al,[10] Cu,[14] or Ag[15]) because metals usually have a much higher thermal expansion coefficient compared to substrate materials such as Si or glass. In contrast, glass films on Si show a different behavior because the relation of the thermal expansion coefficients is reversed. Figure 8.4(b) shows the cooling curves of two different glass films on a Si substrate. At high temperature, the films cannot support any stress because the glass transition temperature is exceeded and the film flows viscously. Reaching the glass transition temperature, the films are deformed elastically and stresses are produced. Here, the films tend to contract less than the substrate and compressive stresses arise. Note that the stress temperature curve is not linear because the thermal expansion coefficient and Young’s modulus are not constant in the wide range of temperatures investigated. This example shows nicely that the measurement of film stresses is a powerful tool to study film properties. For example, from the curves in Fig. 8.4(b), the glass transition temperature for the two CVD-SiO2 films with different dopants can be determined. The elastic or reversible portions of the thermal cycling data can be used to estimate the product ∆aEf (1  nf). If, for instance, two different substrates with known but different thermal expansion coefficients are used, the values of af and Ef(1  nf ) can be determined independently for the film.[16] The interpretation of the stress temperature behavior to obtain information about plasticity is not straightforward. Because the temperature and strain are changed simultaneously, the measured stress is a complicated function of film strength, time- and temperature-dependent creep processes, and possibly strain hardening, voiding, or cracking events. Nevertheless, useful information has been gained from thermal cycling experiments by simply approximating the measured stress as the film strength at a given temperature.[9] Another common experiment is to interrupt the temperature cycle at a certain temperature and to observe the stress relaxation.[10, 11] In this situation, the total film strain is held constant but elastic strain is changed into plastic strain corresponding to a stress relaxation. Figure 8.5 illustrates such an experiment, showing the stress relaxation at three dif-

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Figure 8.5 Stress relaxation in a 0.6-µm-thick Cu film on a Si substrate at various temperatures as measured by the wafer curvature technique. Data from Keller et al.[11]

ferent temperatures for the same film/substrate specimen after cooling from a maximum temperature of 500°C. It can be seen that a strong initial stress drop is observed, followed by a much slower further decrease in stress. Mechanisms for plastic deformation and stress relaxation in metal thin films are discussed in Sec. 8.3.

8.2.2 X-Ray Diffraction The most common x-ray technique to determine stresses in thin films is the so-called sin2ψ method.[17–19] This method is based on the measurement of lattice spacings with a resolution better than 104 Å. This is illustrated in Figure 8.6, which shows a thin crystalline film stressed in tension [Fig. 8.6(a)]. The spacing of lattice planes that are perpendicular to the film plane is increased, due to this tension, while the spacing of planes parallel to the film is decreased, due to Poisson’s contraction. For an equi-

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Figure 8.6 Schematic representation of the sin2ψ method for a textured thin film.

biaxial stress state, which is often present in thin films, the lattice spacing dψ depends linearly on sin2ψ [Fig. 8.6(b)], where ψ is the angle between the normals of the measured lattice planes and the film surface. Comparing the strained lattice spacings dx and dz for ψ  1 and ψ  0, respectively, to the unstrained lattice spacing do allows us to determine the strains exx  eyy and ezz in and out of the plane of the film: dx  do exx   do

(8a)

dz  do ezz   . do

(8b)

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Furthermore, it is possible to calculate the stress state of the film using the elastic constants of the film material. This method works particularly well for single-crystalline or textured films with an orientation that results in an isotropic biaxial modulus of the film. This is the case for cubic materials with (111) and (100) planes parallel to the surface. For the latter, the stresses are given by: sxx  C11exx  C12eyy  C12ezz

(9a)

syy  C12exx  C11eyy  C12ezz

(9b)

szz  C12exx  C12eyy  C11ezz.

(9c)

Note that for a (100)-oriented crystal, the Cartesian coordinate systems of the crystal and the sample have the same orientation. This is no longer the case for a (111)-oriented crystal or texture. Therefore, the stiffness coefficients need to be transformed from the crystal to the sample coordinate system. Then the stress components in the sample coordinate system are calculated to be: sxx  C11exx  C12eyy  C13ezz,

(10a)

syy  C12exx  C11eyy  C13ezz,

(10b)

szz  C13exx  C13eyy  C33ezz,

(10c)

where C11,C12,C13, and C33 are the transformed stiffness coefficients.[19] It is obvious from Eq. (8) that the accuracy of the measurement depends very strongly on the value used for the unstrained-lattice spacing do. Since this value depends sensitively on temperature and material impurities, it is advantageous to determine do for each stress measurement. This leads also to a reduction of the influence of systematic errors in the experimental setup related to the alignment and calibration of the goniometer and detector. For a thin film with a strong texture, it is reasonable to assume that the out-of-plane stress component szz equals zero. Assuming an equi-biaxial stress state, it can be shown that the unstrained lattice spacing is given for sin2ψo  2C13(C33  2C13), where the appropriate stiffness coefficients for the sample coordinate system are used. This is illustrated in Fig. 8.7(a), which shows a plot of d versus sin2ψ from measurements on a thin Al film on a Si substrate[20, 21] at temperatures of 20, 200, and 440°C. The curves with the negative slopes were obtained on heating, and the ones with positive slopes on cooling, indicating compressive and tensile film stresses, respectively. The stress as a function of temperature obtained from these measurements is shown in Fig. 8.7(b).

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Figure 8.7. (a) Spacing of the (422) lattice planes in a 0.7-µm-thick Al film with a (111) texture as a function of sin2ψ for three different measurements on heating and cooling. On heating, the film is under compression (lines with negative slope); on cooling, it is under tension (lines with positive slope). The intersection of the lines is associated with the unstrained lattice spacing for each temperature. (b) Stress as a function of temperature as obtained from the sin2ψ method. Data from Kraft and Nix.[21]

For films with a mixed texture, such as (111) and (100), as often observed in Cu thin films, measurements for each texture component become necessary.[2, 14, 22] Assuming only minor stress interactions between the components, the average stress for each orientation can be determined. Figure 8.8 compares results of wafer curvature and x-ray measurements on identical samples.[2, 13] A passivated Cu film with 1.0 µm thickness and a mixed (111) and (100) texture was subjected to a temperature cycle up to 600°C. As shown, the x-ray method allows the average stress for the two texture components to be measured, while the wafer curvature techniques give the average film stress. The details of the stress analysis for films that have other mixed textures or other crystal structures can be more complicated. A more detailed treatment of the sin2ψ method including grain interaction and elastic anisotropy is given in Clemens and Bain[18] and Van Leeuwen et al.[23] Finally, it should be pointed out that x-ray measurements can be used to determine more complex stress states such as those present in narrow conductor lines.[19, 20, 24] The x-ray measurements described so far typically use x-ray beams with a size on the order of mm. As a result, the measured stress is averaged over many grains. This allows us to study the stress behavior

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Figure 8.8 Stress-temperature evolution in a 1-µm-thick passivated Cu film: Comparison between the film stress as measured by the wafer curvature technique and the stress as measured by x-rays in (111)- and (100)-oriented grains. Data from Baker et al.[2] and Keller et al.[13]

of different texture components, but not stress variations from grain to grain. More recently, several techniques have been developed to measure stresses with very high spatial resolution.[25–27] For some of those measurements, performed with a submicrometer resolution, white synchrotron x-ray radiation was used. In order to determine stresses, the deterioration of the Laue diffraction patterns obtained was analyzed. As an example of such a measurement, Fig. 8.9 shows the stress distribution in an area of approximately 15 by 15 µm2 in a thin Al film as a function of temperature.[28] It can be seen that there are very strong variations of more than 100 MPa in stress from grain to grain. These variations may correspond to different orientations, but even for grains with the same orientation, significant differences in stress are observed.

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Figure 8.9 Thermal cycling results of a 15 15 µm2 area of an Al(Cu) bond pad (blanket film) showing an averaged biaxial stress component vs. temperature. The insets show details of the stress distribution in the film at different temperatures. (The two-dimensional maps are plots of measurements of the in-plane stress.) Note the large stress inhomogeneities from grain to grain and even within individual grains. From Tamura et al.,[28] with permission.

8.3

Stress Relaxation

As described above, the elastic behavior of thin films is basically understood and does not differ from bulk behavior. In contrast, the plastic behavior of thin metal films is not well understood and is fundamentally different compared to that of bulk materials. This section summarizes the current understanding of thin-film plasticity. Contributions from dislocation plasticity and creep deformation to the relaxation of stresses are discussed, as well as the experimental evidence.

8.3.1 Experimental Observations Figure 8.10 shows the stress evolution in Cu thin films on a Si substrate and exemplifies typical behavior of metal films on low-thermal-expansion-

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Figure 8.10 Wafer curvature experiments have shown stress temperature curves that are strongly dependent on the surface passivation of Cu thin films on substrate. Data are given for pure Cu and Cu-1 at.% Al films where Al tends to segregate toward the surface and form an oxide cap. The film stress was measured by the wafer curvature technique. Data from Weiss et al.[31]

coefficient substrates. These films have a silicon nitride underlayer and are either capped by an aluminum oxide layer, which was formed by selfpassivation of a Cu-1 at.% Al film,[29–31] or are uncapped. The stresses were measured by the wafer curvature technique. At the beginning of the experiment, after sputter deposition and annealing at 600°C, the films are under tensile stress at room temperature. Because of the larger thermal expansion coefficient of the film compared to the substrate, the stress decreases linearly with increasing temperature. The film undergoes both elastic and plastic deformation during thermal cycling. In general, the stresses in the capped film are much higher than in the uncapped one throughout the experiment. At high temperature, this difference has been attributed to diffusional creep,[13, 30, 31] which is active in the uncapped film and is described in detail in Sec. 8.3.3. The plastic deformation in the capped film can be assumed to be mediated only by dislocation motion since the presence of the passivation layer suppresses surface diffusion, which is required for diffusional creep (Fig. 8.10 and Sec. 3.3). Often, the stress-temperature evolution is regarded as a measure of film strength as a function of temperature.[9] However, this picture is not completely accurate because the film stress relaxes quite substantially when temperature is held constant.[10, 32] Therefore, the time dependence of the plastic deformation needs also to be taken into account. Furthermore, the stresstemperature evolution is influenced by strain hardening[13, 33, 34] as the film

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is plastically strained; for instance, the total thermal strain for the films shown in Fig. 8.10 is 0.7% and the plastic strain is about 0.5%. Beside thermal straining, mechanical properties of thin metal films have been measured by micro-tensile testing; for example, results have been reported for Ni,[35] Al,[36] Cu,[35, 37] and multilayered films.[38, 39] These experiments confirmed that thin films exhibit very high yield strength, which usually increases with decreasing film thickness, grain size, and/or layer thickness in multilayered thin films. As an example, Fig. 8.11(a) shows the stress-strain curve of freestanding 1.1-µm-thick Cu film. However, in such tests on freestanding films, the deformation is not constrained by a substrate and, furthermore, testing in compression is not possible. It has been shown that this can be overcome by testing thin films on a polyimide substrate.[40] Figure 8.11(b) shows the stress-strain curve of such an experiment on a 0.7-µm-thick Cu film obtained using x-ray measurements. On loading, the film deforms first elastically until yielding at about 250 MPa is observed. Then, on straining to 0.5%, the flow stress increases to 400 Mpa, indicating the presence of strain-hardening effects. On unloading of the specimen, the contracting elastic substrate compresses the film, which then undergoes plastic deformation in the opposite direction. It appears that on reverse loading, the yield strength is somewhat smaller compared to the initial loading. For interpreting mechanical

Notes: 1. Data for (a) from Read.[37] 2. Data for (b) from M. Hommel, O. Kraft, and E. Arzt, “A new method to study cyclic deformation of thin films in tension and compression,” J. Mater. Res., 14(6):2373–2376 (1999).

Figure 8.11 Stress evolution as a function of applied strain during tensile tests for (a) a freestanding Cu film with a thickness of 1.1 µm and (b) for a 1.0-µm-thick Cu film on a polymer substrate. The film stress for (b) was measured by x-ray diffraction.

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behavior, the advantage of this test method over thermal straining is related to the fact that the temperature is not altered during the experiment. In particular, this allows us to attribute the observed asymmetry in yield stress to hardening effects since any influence from the temperature change in the thermal cycling experiments is eliminated.

8.3.2 Dislocation Plasticity As a start, we consider the motion of a single dislocation in a singlecrystalline thin film attached to a substrate, as shown in Fig. 8.12(a) and

Figure 8.12 Schematic representation of the motion of a single dislocation in a single-crystalline film, according to the model of Nix and Freund[1, 45] for a film with (a) a free surface and (b) a cap layer. It indicates in (a) that a dislocation segment with the length dx needs to be created when the dislocation moves by dx.

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(b) for a free and an impenetrable surface, respectively. As the dislocations advance in the film by “channeling,” they create additional dislocation segments along the bottom and the top surface if it is impenetrable. This mechanism has been observed for the motion of dislocation in largegrained polycrystalline Al films on amorphous substrates[9] and in epitaxial Al films on Si[41] or Al2O3 substrates.[42] For this last example, Fig. 8.13 shows a dislocation, which has channeled over a distance of several micrometers. It appears that there is a repulsive force from the interface on the dislocation as it stands off by as much as 100 nm. The contrast close to the interface indicates the presence of other dislocations, which may be related to the epitaxial misfit. In contrast, it was observed that in polycrystalline films with grain size of the order of the film thickness, dislocations did not channel. Kobrinsky and Thompson[32] describe the dislocation motion below 150°C in thin Ag films (200 nm in thickness) as jerky. As observed during in situ transmission electron microscopy (TEM), dislocations are pinned by obstacles and do not move most of the time, until a sudden jump of a dislocation segment between two pinning points is observed. The typical pinning point and jump distances were both found to be between 50 and 100 nm, which are significantly smaller than the film thickness or grain size. These observations are confirmed by in situ TEM, both in cross section as well as in plan view, on polycrystalline Cu films that were deposited onto amorphous SiNx layers on Si substrates.[42, 43] Figure 8.14 shows the typical dislocation distribution after cooling from 600 to 130°C in such a film. At tempera-

Figure 8.13 Dislocation in a 350-nm-thick Al film that was grown epitaxially on a (0001)-oriented Al2O3 substrate. Glide of a dislocation on the Al plane, which is inclined ∼70 degrees to the (111)Al G (0001)Al2O3 interface, created a dislocation segment nearly parallel to the interface. Contrast near the film/substrate interface indicates the presence of other, possibly misfit, dislocations. The cross-sectional TEM image is from Dehm et al.[42] with permission.

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Figure 8.14 Dislocations in Cu grains at (a) 600°C and (b) 130°C. No interfacial dislocations deposited by advancing threading dislocations are discernable in the images. At elevated temperatures (a), dislocations appear to be longer and more mobile than at lower temperatures (b), where dislocation motion became jerky and dislocation tangles formed. Plan-view TEM images are from Dehm et al.,[42] with permission.

tures below 220°C, short segments of tangled dislocations and a jerky dislocation motion were observed. It was also pointed out that no interfacial dislocation segments were found, which would be expected for the dislocation motion described by Fig. 8.12. Furthermore, cross-sectional in situ TEM revealed that dislocations are attracted rather than repelled by the Cu/SiNx interface and, again, no evidence for interfacial dislocation segments was found.[43] In contrast, Shen et al. [34] and Weihnacht and Brückner [44] observed interfacial dislocations and dislocation pileup in plan-view and cross-sectional TEM, respectively. Nix[1] and Freund[45] have derived the following formalism for the motion of a single dislocation in a thin film, as shown in Fig. 8.12. As the dislocation channels through the film, an interface dislocation segment is created. As a result, the critical resolved shear stress ty to move a dislocation in a thin film depends on its thickness h and is given by: sin j bGeff , ty, Nix   2p(1  n) h

(11)

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where

 

GGs xh Geff   ln s ; G  Gs b

(12)

j is the angle between the glide plane normal and the film normal; b is the magnitude of the Burgers vector; n is Poisson’s ratio; G and Gs are the shear moduli of the film and the substrate, respectively; and x s is a numerical constant close to unity that defines the cutoff radius of the stress field of the dislocation at the film/substrate interface. Extending this model to polycrystalline thin films, Thompson[46] suggested that additional dislocation segments must be created at the grain boundaries. This reasoning results in the following critical shear stress: Wd 2 sin j ty,Th      , b d h





(13)

 

(14)

with d Gb2 Wd   ln  , b 4p(1  n)

where d is the grain size and Wd is the line energy of the dislocation segments along the grain boundaries and the film/substrate interface where, for the sake of simplicity, Thompson assumed that the effective modulus equals the film modulus (G  Geff). As can be seen from Eqs. (11) and (13), these models predict nearly a 1/h and/or a 1/d dependence of the yield strength. A similar result has been also obtained by Chaudhari.[47] An experimental validation of these models has been given by Venkatraman and Bravman[9] for coarse-grained Al films or by Dehm et al.[48] for epitaxial thin Al films on Al2O3 substrates. However, for most polycrystalline fine-grained films, the model tends to underestimate the yield strength.[13, 22] Kobrinsky and Thompson[32] pointed out that it is not adequate to use the simple picture shown in Fig. 8.12 for dislocation motion in thin films to predict their strength. This is because a much more complex dislocation behavior is observed in TEM, including the jerky motion of dislocations at low temperatures and the absence of interface dislocations. Also, time-dependent stress relaxation in thin films is not addressed by this model. Therefore, Kobrinsky and Thompson suggest that dislocation-mediated plasticity in thin metal films is significantly

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affected by thermal activation of the dislocation glide. Note that this mechanism was previously discussed by Flinn et al.[5] and Volkert et al.[10] to control the stress-temperature behavior of Al thin films. The thermally activated glide of dislocations implies that the applied shear stress is not large enough to drive a dislocation through an array of obstacles, such as forest dislocations or particles. As a result, dislocations are pinned at these obstacles. However, thermal activation may provide the additional energy to help a dislocation segment to overcome the obstacle. This results in the observed jerky motion as a dislocation moves step by step from obstacle to obstacle. A constitutive law for the thermally activated glide can be given if a certain obstacle “shape” is assumed. For rectangular obstacles, the plastic deformation rate e is given by Frost and Ashby [49, p. 8]: F ss , e  e o exp   1   kT t^







(15)

where e o is a characteristic constant, ∆F is the activation energy at zero stress, t^ is the critical shear stress without thermal activation, and s is the Schmid factor. Figure 8.15 compares this approach and experimental data for selfpassivated 0.5 and 1.0 µm thick Cu 1 at.% Al films. The starting points on

Figure 8.15 Stress evolution as a function of temperature for Cu films with a cap layer and a thickness of (a) 1.0 mm and (b) 0.5 µm. The film stress was measured by the wafer curvature technique.[29] The dashed lines represent modeled curves for thermally activated dislocation glide. Parameters used were for bulk Cu:[49] e o  1 106 s1, ∆F  3.5 1019 J.The Schmid s  0.27 is for (111)-oriented grains. t^ was adjusted to 150 and 235 MPa for the 1.0- and 0.5-µm-thick films, respectively.

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heating were adjusted to the experimental value. It can be seen that a reasonable agreement is obtained, although the compressive plateau is not well represented. The agreement for the two film thicknesses was obtained by adjusting t^ to 150 and 235 MPa, respectively. Using Orowan’s classical result that the critical shear stress depends on the pinning point distance L as t^  GbL, the corresponding pinning point distances are 80 and 50 nm, respectively. These values are indeed much smaller than the film thickness and of the same order of magnitude as the ones determined by in situ TEM on Ag films.[15, 43] However, the approach presented here does not take into account that the pinning point distance changes during heating and cooling (see Fig. 8.14) as a result of strain hardening and possibly recovery processes. Strong kinematic strain hardening is a competing approach to model the stress-temperature curves for explaining some of the observed effects. Shen et al.[34] suggested that the increase in flow stress on cooling is related to the buildup of a back stress sb, which lowers the yield strength on reverse loading. This behavior is represented in Fig. 8.16:

Figure 8.16 Stress-strain curve given by the kinematic strain-hardening model in Shen et al.[34] under cyclic loading, compared to the experimental data from Fig. 8.2.

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After the stress reaches sy on loading, plastic deformation occurs with a linear increase in flow stress. Due to this increase, a back stress sb builds up. On reverse loading, the yield strength in compression is lowered by this back stress and the film yields at sreverse  sy  sb. Figure 8.16 also shows that this predicted behavior is in agreement with the cyclic stress-strain behavior of the Cu films on polyimide substrates. Shen et al. have shown in their original work that their model can be applied fairly well to model thermal stress cycles of Cu films when a temperature-dependent yield strength is assumed. The incremental increase in stress, that is, the hardening rate, in the elastic-plastic regime is given by: a∂T , ∂s   1 e*    M sy(T)





(16)

where M is the biaxial modulus of the film, e* is the hardening parameter, and sy(T) is the temperature-dependent yield strength. The yield strength was assumed to decrease linearly with temperature, that is, sy(T )  so(1  TT *), where so is the yield strength at T  0 K and T* is the temperature at which the yield strength becomes zero. Hence, e*, so, and T* can be used to fit the model empirically to experimental data. Nix and Leung[50] have adapted this approach and tried to reduce the empirical fit parameters. They suggested that the yield strength is related to the dislocation motion in the film, as given by Eq. (11). Therefore, its temperature dependence is related to the temperature dependence of the shear modulus, which can be assumed to be G  Go(1  c*(T  300)Tm), where Go is the shear modulus at room temperature, c* is a constant close to 0.5 for most FCC metals, and Tm is the melting temperature. Figure 8.17 shows a quantitative comparison of this approach to the data from Fig. 8.15. Again, a reasonable agreement between experiment and model can be seen. In particular, the plateau in compression is well reproduced. Note that the hardening rate, which is adjusted to fit the experimental data, is of the order of the shear modulus of Cu, which is about two orders of magnitude larger than the hardening rate in bulk Cu. Furthermore, this approach does not allow us to account for any time-dependent deformation; therefore, it is not possible to account for strain rate effects or stress relaxation. More recently, Kraft et al.[51] suggested that both thermally activated dislocation glide and strain hardening be included by modifying

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Figure 8.17 Experimental data from Fig. 8.15 compared to the kinematic strainhardening model.[34] The solid line represents the model using the following parameters: G  45 GPa, e*  0.01, M  220 GPa. (This value was adapted to fit the thermoelastic region.) The dashed lines indicate the yield strength as a function of temperature.

Eq. (15) as: F s(s  sb) e  e o exp   1  ^ t kT





,

(17)

where sb is the back stress acting on dislocations in the active slip system, which can be, for example, described to depend linearly on the plastic t^ e l strain epl: sb   p, as suggested by Shen et al. [see Eq. (16)]. The influence s e* of the back stress is also illustrated in Fig. 8.18, where the behavior leading to the two stress-temperature curves is identical except for the presence (solid curve) or absence (dashed line) of the back stress. Similar to Fig. 8.17, the effect of the back stress leads to the asymmetry in the magnitude of the stresses on heating and cooling, and the high stresses are obtained by adjusting L to 50 nm for t^  GbL. (Compare to Fig. 8.15.) However, the approach presented here does not take into account that the pinning point distance changes during heating and cooling (see Fig. 8.14) as a result of strain hardening or recovery processes. Baker et al.[52] pointed out that channeling of a single dislocation, as shown in Fig. 8.12, is completely reversible. As a result, a reduction of an

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Figure 8.18 Stress vs. temperature change for a film on a substrate, illustrating the difference between response according to the standard plastic rate equation for thermally activated dislocation glide [Eq. (15), dashed line] and a rate equation modified by inclusion of a back stress, depending on plastic strain [Eq. (17), solid line]. The parameters used were the same as in Fig. 8.15 and e*  0.005.

applied stress below the critical stress, given by Eq. (11), should lead to spontaneous removal of the dislocation. This phenomenon leads to a plastic deformation in the reverse direction to the applied stress. Such a behavior has been observed in a variety of face-centered-cubic metal films, which were contaminated by oxygen, with the role of the oxygen not yet understood. This effect should also influence dislocation interaction mechanisms, which may lead to the strong (kinematic) hardening effects in thin films. Figure 8.19 shows three scenarios that lead to hardening behavior comparable to the experimental observations: (a) A channeling dislocation has to overcome interfacial dislocations.[53] (b) The interfacial dislocation of a channeling dislocation is deposited in an existing array of parallel interface dislocations.[44] (c) The nucleation of dislocation occurs at a Frank-Read source located in the interior of a grain. The repeated

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Figure 8.19 Schematic representation of possible dislocation arrangements leading to strain hardening in a thin film on a substrate.

emission of dislocation loops leads to a back-stress onto the dislocation source,[54, 55] making the nucleation event more difficult from time to time. So far, the models presented, which are intended to describe the strength and hardening behavior of thin metal films, are based on the concept of misfit dislocations; they imply that the misfit dislocation at the interface to the amorphous substrate still supplies a stress field.

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Dislocation core structure is known to play an important role in the strength of solids. The classical dislocation core models are based on a line defect in a crystalline structure. Such models do not apply in the case of amorphous solids. Since, in many cases, misfit dislocations are located at an interface between a crystalline film and an amorphous substrate (or an adhesive or functional layer between film and substrate), a fundamental question arises: What is the equilibrium core structure of dislocations at an interface between crystalline and amorphous materials? At a crystallineamorphous interface, as in the case of Al or Cu films deposited on an aSiOx or aSiNx substrate, the core of a misfit dislocation may spread along the interface, as illustrated in Fig. 8.20. The substrate has no simple crystallographic relationship to the film and hence may be thought of as a continuum. Dashed lines have been drawn on the substrate to mark the original positions of the atomic planes in the film. Figure 8.20 depicts an edge dislocation with Burgers vector b parallel to the interface, climbing toward the substrate under an applied load and then spreading its core along the interface. The dispersed dislocation core may be modeled as an

Figure 8.20 Schematic of dislocation core spreading at an incoherent interface between a crystalline and an amorphous solid. (a) The dislocation climbs down toward the substrate but cannot penetrate it. (b) If it is possible for sliding to occur at the interface, the dislocation core may spread into the interface to a width 2c.

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array of infinitesimal dislocations, as described by Eshelby.[56] The extent of spreading can be characterized by the half-width, c, of the slipped region. The dislocation core spreading illustrated in Fig. 8.20 requires that the material on one side of the interface be able to slide by small amounts (
b) relative to the material on the other side of the interface. This cannot occur over large distances in real coherent interfaces. However, for incoherent interfaces between crystalline and amorphous solids, small relative displacements are possible, and dislocation core spreading may occur over large distances. When a shear strain is applied across the interface, we can imagine that some film-substrate bonds are broken and new, energetically more favorable bonds are formed in their place. Since only a few bonds move, the mean position of the plane can move by an amount that is much less than b. A wide range of film-substrate systems have incoherent interfaces and may be susceptible to dislocation core spreading following a mechanism like that illustrated in Fig. 8.20. Transmission electron microscopy (TEM) investigations have shown dislocations disappearing at the interface between an Al film and its native Al2O3 passivation[57] and at the interface between a Cu film and a SiNx barrier layer.[43] Gao et al.[58] have developed a mathematical model to describe the time-dependent process of dislocation core spreading along a crystallineamorphous interface. The model is briefly described here. We consider edge dislocations spreading along an interface characterized by a shear adhesive strength, t0, below which no core spreading occurs and above which spreading takes place in a viscous manner. The equilibrium core width and the rate of core spreading for single dislocations and a periodic array of dislocations can be calculated as a function of interface adhesion. We consider an edge dislocation spreading along an interface between a crystalline and an amorphous solid, as depicted in Fig. 8.20. At time t  0, the dislocation core structure is compact. Subsequently, deformation near the film-substrate interface is assumed to be described by a viscous model. This problem has been treated in detail by Gao et al.[58] for a single or planar array of dislocations in a homogeneous bulk material or at the interface between a thin film and a semi-infinite substrate, where the film and substrate may have the same, or different, elastic constants. Some of the solutions are analytic and others are based on an implicit finite difference method. For an edge dislocation spreading along an interface between a crystal and an amorphous material with the same elastic constants, the solution shows that the equilibrium core width for a single dislocation is: Eb , ce   8(1  n2)t0

(18)

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and for a period array of edge dislocations with spacing L is:

 



L pEb . ce   sin1 tanh  p 8(1  n2)t0L

(19)

We observe that the equilibrium core width scales inversely with the interface adhesion t0. The stronger the interface, the more compact the core. The analyses by Gao et al.[58] and Baker et al.[59] show that the viscous spreading of a single dislocation core can be described by the change in the half width,





9t0ti Eb 1  0.3634 exp   t , c(t)   2 hb 8(1  n )t0



(20)

of the slipped zone, where h and ti are the viscosity and the thickness of the interface, respectively. At t  0, the half-width Eb c(0)   4p(1  n2)t0

(21)

defines a region over which viscous slip occurs immediately because the shear stress associated with a compact dislocation core exceeds the interface adhesion t0. Although the slipping width c(t) is somewhat ad hoc in representing the core width, the evolution Eq. (20) provides the insight that the time scale over which the core spreading occurs is largely determined by the ratio ht0. Since viscosity usually decreases with temperature, we conclude that core spreading occurs faster at higher temperatures and stronger interfaces, and slower at lower temperatures and weaker interfaces. It can be further argued that dislocation core spreading reduces the effectiveness of hardening mechanisms, as discussed in Figs. 8.12 and 8.19.[58] As a result, dislocation-mediated plasticity in thin films is immediately related to diffusional processes as well as to the properties of surfaces and interfaces. However, a complete understanding of dislocation plasticity is still not achieved and an incorporation of thermally activated core spreading in the dislocation models described above has not yet been attempted.

8.3.3 Diffusional Creep As discussed in Sec. 3.1 and shown in Fig. 8.10, wafer curvature experiments have shown that the thermal mechanical behavior of Cu

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thin films on substrates depends strongly on the passivation of the film surface. This phenomenon has recently been explained by a special kind of Coble-creep in thin films on substrates, often referred to as the constrained diffusional creep.[31, 60] The continuum model for constrained diffusional creep developed has adopted the hypothesis that uncapped, pure Cu films are subject to active surface and grain boundary diffusion, which causes transport of material between the film surface and the grain boundaries.[60] Diffusion in thin films on substrates is fundamentally different in nature from that in bulk materials. A constraint is imposed by the usually perfect bonding between film and substrate (for example, material sequence Cu-aSiN x -aSiO x -Si, [61–63] which implies that no sliding occurs at the film-substrate interface. In contrast to previously proposed models of diffusion in thin films,[64] the constraint that no sliding occurs at the film-substrate interface renders the diffusion in thin films an inherently transient problem. Steady-state solutions frequently used to describe grain boundary diffusion may not be used. An additional constraint is that material transport cannot proceed in the substrate and diffusion has to stop at the film-substrate interface. Constrained diffusional creep leads to formation of a new material defect called the grain boundary diffusion wedge, which also plays an important role in dislocation nucleation processes in thin films.[60] This process corresponds to the classical Coble creep in a thinfilm structure, except that the roots of the grain boundaries are constrained not to slide with respect to the substrate. In the remainder of this section, we briefly describe the theory of constrained diffusional creep as well as the grain boundary diffusion wedge and its associated dislocation mechanisms. In the model of Gao et al.,[60] constrained diffusional creep under a constant nominal stress in the film is considered and diffusion is modeled as dislocation climb in the grain boundary. The solution for a single edge dislocation is used as the Green’s function to construct a solution with infinitesimal Voltera edge dislocations. For an inserted grain boundary wedge of width 2u(z,t) in a film with thickness h(0
z
h), the stress along the grain boundary is given by:



h

G ∂u(z,t) sgb(z,t)  s0   S(z, z) dz, 2p(1  n) 0 ∂z

(22)

where S(z,z) is a Green’s function kernel for the continuous dislocation problem (Cauchy kernel), G is the shear modulus, n is Poisson’s ratio, and s0 is the stress in the absence of diffusion. Balancing the diffusive atomic flux with the displacement rate (mass conservation), the diffusion equation

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based on Coble creep,[65] dgbDgbΩ ∂2sgb(z,t) ∂u(z,t)      , 2kT ∂z2 ∂t

(23)

is woven into the problem. Here, dgbDgb denotes the product of grain boundary thickness and atomic diffusivity, t is time, Ω is the atomic volume, and kT has the usual meaning. Combining Eqs. (22) and (23) yields an integro-differential governing equation for constrained diffusional creep:



h

GdgbDgbΩ ∂sgb(z,t) ∂3s (z,t)  dz,   S(z,z) gb 2p(1  n)kT 0 ∂t ∂z 3

(24)

which describes the time evolution of grain boundary stress under the following boundary conditions: sgb(0,t)  0,

sgb(h,t)  sgb(h,t)  0

(25)

and an initial condition sgb(z,0)  s0.

(26)

The boundary conditions in Eq. (25) correspond to continuity of chemical potential at the junction between the film surface with the grain boundary, as well as no mass flux and no sliding at the film-substrate interface. The initial condition in Eq. (26) defines a Green’s function solution to a step loading on the thin film. The solution can be used later to construct more general solutions to an arbitrary time-dependent loading s0(t) via a convolution scheme.[31] The solution to Eq. (26) shows that the insertion of a material wedge into the grain boundary causes exponential stress decay with a characteristic time scaling with the cube of the film thickness, 2p(1  n)kTh3 t0   . GdgbDgbΩ

(27)

The diffusion wedge represents columns of atoms corresponding to an array of climb-edge dislocations that have been transported into the grain

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boundary. With respect to the lattice distortion around the diffusion wedge, the dislocations in the grain boundary exemplify geometrically necessary dislocations[66, 67] that are usually associated with nonuniform plastic deformation. Gao et al.[60] used an eigenmode expansion to obtain an integro-differential eigenvalue problem that is subsequently transformed into a standard Cauchy-type singular equation to be solved by a Gauss-Chebyshev quadrature scheme. Gao et al. provide details of the numerical procedure.[60] A simplified solution, l sgb(z,t)   s0eltt f (zh), 3p 0

(28)

can be used for all practical purposes, where the function f(x) is defined as:  x2 f (x)  3 1 ,

(29)

the parameter l is related to the film thickness h and grain size d as: l



11.72  1.69(hd)  73.09(hd)2 8.10  30.65(hd)

0
hd 0.2 0.2
hd 10.

(30)

Averaging Eq. (28) over the film thickness shows that the average stress on the grain boundary decays exponentially as: sgb(t)  s0eltt .

0

(31)

Therefore, on the scale of a characteristic time t0l, the traction along the grain boundary is fully relaxed and the diffusion wedge exhibits the same traction-free condition as a crack, indicating a singular stress field near the root of the grain boundary. This leads to extraordinarily large resolved shear stresses on planes parallel and close to the film-substrate interface and, in the (111)-textured films, can cause emission of parallel glide dislocations. Although the resolved shear stresses on the parallel glide planes induced by a diffusion wedge are similar to those caused by a crack, differences in the dislocation nucleation process may arise. The mechanisms involved are still not well understood and are currently being investigated. Experimentally, diffusional creep as a major stress relaxation mechanism in uncapped thin copper films has been verified in a number of studies.[11, 15, 31, 61, 63] Experiments have indicated that, while plastic deformation

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Figure 8.21 During the cooling portion of the second thermal cycle of the Cu-aSiNxaSiOx-Si(100) film-on-substrate system, parallel glide dislocations were emitted from the grain boundary source. (a) At 170°C, the first dislocation has been emitted and is pinned in the middle. (b) The second emitted dislocation has quickly overcome the pinning point and has pushed the first dislocation forward by the time 150°C is reached. (c) At 135°C, the third dislocation is pinned, while the first two have advanced far into the grain (180 and 300 nm). From Dehm et al.,[63] with permission.

in passivated or relatively thick copper films is characterized by the “classical” threading dislocations, parallel glide dislocations become increasingly dominant in unpassivated thin films with thicknesses of less than 400 nm.[61–63] The experimental observations of parallel glide dislocation emission strongly indicate that constrained grain boundary diffusion is a dominant stress relaxation mechanism for unpassivated thin films on substrate. Figure 8.21 shows an in situ TEM observation of parallel glide dislocations being emitted from a grain boundary source during the cooling portion of the second thermal cycle of the Cu-aSiNx-aSiOxSi(100) system. The experiments have also indicated that nucleation of parallel glide dislocations is very selective to the type of grain boundary. Grain boundaries with 6- and 66-degree tilt angles seem to be preferred nucleation sites for parallel glide dislocations. The smallest film thickness investigated is about 35 nm. In modeling a thermal cycling experiment, both the nominal stress s0 and temperature (hence the characteristic time t0 also) vary with time, and Eq. (31) cannot be directly used. Weiss et al.[31] have developed an integral convolution scheme to calculate the grain boundary stress and have used it to determine the average stress over the entire film using an equation

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by Xia and Hutchinson:[68]



 

4tf d s  s0  (s0  sgb)  tanh  . d 4tf

(32)

Weiss et al.[31] took the experimentally measured stress in passivated Cu 1 at.% Al film as the nominal stress s0 and found that the stress s calculated from Eq. (32) is in reasonable agreement with the measured stress in unpassivated pure Cu film. Therefore, they concluded that the observed difference in thermomechanical behavior between passivated and unpassivated films can be attributed to constrained diffusional creep. By implicitly assuming that surface diffusivity is much faster than grain boundary diffusivity, the model of Gao et al.[60] does not explicitly consider mass transport along the surface of the film. Depending on materials, this assumption may not always be valid. Zhang and Gao[69] recently studied coupled surface and grain boundary diffusion and concluded that the analyses by Weiss et al.[31] and Gao et al.[60] remain valid even when surface diffusivity is comparable to grain boundary diffusivity. Specifically, Zhang and Gao[69] have shown that the results in[31] and[60] become generally valid when an effective diffusivity 12pds Ds dgb Dgb deff Deff   12pds Ds  dgb Dgb

(33)

is used to replace the grain boundary diffusivity dgbDgb, where dsDs denotes the surface diffusivity. Note that, due to the large constant coefficient of 12p, the effect of surface diffusion becomes significant only when the grain boundary diffusivity is much larger than the surface diffusivity.

8.4

Conclusion

In this chapter, techniques for the measurement of stresses in metallic thin films held on substrates and their relaxation by plastic deformation and diffusional creep are described. All studies mentioned indicate clearly that the motion of dislocations is constrained in thin metal films by the presence of interfaces and grain boundaries. As a result, the mechanical behavior of thin metal films can be qualitatively summarized by saying that smaller is stronger. On the other hand, diffusional processes play a more important role with decreasing dimensions as the density of fast diffusion pathways increases. Therefore, it would be helpful to

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describe the deformation behavior of thin films using deformation mechanism maps[70] similar to the ones established for bulk materials by Frost and Ashby,[49] which include both dislocation-mediated and diffusionmediated mechanisms. Unfortunately, clear, quantitative descriptions have not yet been established for thin films, neither empirically nor theoretically, and the use of quantitative descriptions obtained from bulk materials does not appear to be adequate. This is further accentuated by the fact that subtle changes in the chemistry of the interfaces, which might be beyond experimental control, can dramatically influence the thin-film mechanical behavior. The models discussed for diffusional creep and dislocation core spreading will contribute to a framework that will lead eventually to a more complete description of thin-film mechanical behavior that takes into account the interaction of dislocation plasticity and diffusional processes. For this, a unified view of the experimental observations, including mechanical testing as well as in situ electron microscopy, and of the theory is indispensable. In most technological applications, thin films are rigidly attached to a substrate that is typically much thicker than the film itself. As a result of this configuration, the thin films are commonly subject to high internal stresses, which may endanger the operation of small-scale devices. Failure mechanisms discussed in the literature include the following. (1) Thermal-stress-induced voiding has been a problem for many years for Al or Cu conductor lines.[71, 72] This phenomenon is driven by high hydrostatic tensile stresses within interconnect lines under a capping layer during cooling. Even in blanket films, where two-dimensional inplane stresses are observed, a few cases of stress-induced voiding have been reported.[73, 74] (2) Stresses may also cause the delamination or cracking of thin films.[75] It has been shown that the plasticity in thin metal layers affects the interface fracture resistance in thin-film interconnect structures.[76] Specifically, the TaN/SiO2 interface fracture energy was measured in thin-film Cu/TaN/SiO2 structures in which the Cu layer was varied over a wide range of thickness. It was found that in a regime of 0.25 to 2.5 µm, the delaminating resistance is dominated by the contribution of plastic deformation in the metallic layer. (3) More recently, the behavior of thin metal films under cyclic loading conditions has been given experimental as well as theoretical consideration. In particular, cracking of a brittle layer caused by ratcheting in an adjacent ductile layer has been observed in a thin-film system under cyclic thermal loading.[77] Moreover, cyclic mechanical or thermomechanical loading can also lead to the formation of voids and extrusions in thin metal films.[78–80] From these observations, it has been concluded that the role of point defects and diffusion is much more important for fatigue in thin films than in bulk materials.

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In summary, continuum theories of plasticity or lifetime models that incorporate dislocation plasticity and diffusional processes with respect to a physical or microstructural length scale may turn out to be very helpful if they can be validated by experiments. The predictive capability of such theories will be very useful in device simulation codes for microelectronics, MEMS, and other applications.

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15. M. J. Kobrinsky and C. V. Thompson, “The thickness dependence of the flow stress of capped and uncapped polycrystalline Ag thin films,” Appl. Phys. Lett., 73:2429 (1998) 16. A. Lahav, K. A. Grim, and I. A. Blech, “Measurement of thermal expansion coefficients of W, WSi, WN and WSiN thin film metallizations,” J. Appl. Phys., 67(2):734–738 (1990) 17. P. A. Flinn and C. Chiang, “X-ray diffraction determination of the effect of various passivations on stress in metal films and patterned lines,” J. Appl. Phys., 67(6):2927–2931 (1990) 18. B. M. Clemens and J. A. Bain, “Stress determination in textured thin films using x-diffraction,” MRS Bull., 12(7):46–51 (1992) 19. P. R. Besser, S. Brennan, and J. C. Bravman, “An x-ray method for direct determination of the strain state and strain relaxation in micron-scale passivated metallization lines during thermal cycling,” J. Mater. Res., 9:13 (1994) 20. O. Kraft and W. D. Nix, “Thermomechanical behavior of continuous and patterned Al thin films,” Materials Reliability in Microelectronics VIII, MRS Symp. Proc., vol. 516, Warrendale, PA (1998), pp. 201–212 21. O. Kraft and W. D. Nix, “Measurements of the lattice thermal expansion coefficient of thin films on substrates,” J. Appl. Phys., 83:3035–3038 (1998) 22. M. Hommel and O. Kraft, “Deformation behavior of thin copper films on deformable substrates,” Acta Mater., 49:3935–3947 (2001) 23. M. Van Leeuwen, J.-D. Kamminga, and E. J. Mittemeijer, “Diffraction stress analysis of thin films: modeling and experimental evaluation of elastic constants and grain interactions,” J. Appl. Phys., 86(4):1904–1914 (1999) 24. W.-M. Kuschke and E. Arzt, “Investigations of stresses in continuous thin films and patterned lines by x-ray diffraction,” Appl. Phys. Lett., 64(9):1097–1099 (1994) 25. B. C. Larson, W. Yang, G. E. Ice, J. D. Budai, and J. Z. Tischler, “Threedimensional structural x-ray microscopy with submicrometer resolution,” Nature, 451:887–890 (2002) 26. R. Spolenak, N. Tamura, B. C. Valek, A. A. MacDowell, R. S. Celestre, H. A. Padmore, W. L. Brown, T. Marieb, B. W. Batterman, and J. R. Patel, “High resolution microdiffraction studies using synchrotron radiation,” 6th International Workshop on Stress-Induced Phenomena in Metallization, AIP Conf. Proc. 612, Melville, NY (2002), pp. 217–228 27. N. Tamura, R. S. Celestre, A. A. MacDowell, H. A. Padmore, R. Spolenak, B. C. Valek, N. Meier Chang, A. Manceau, and J. R. Patel, “Submicron x-ray diffraction and its applications to problems in materials and environment,” Rev. Sci. Instrum., 73(3):1369–1372 (2002) 28. N. Tamura, A. A. MacDowell, R. S. Celestre, H. A. Padmore, B. C. Valek, J. C. Bravman, R. Spolenak, W. L. Brown, T. Marieb, H. Fujimoto, B. W. Batterman, and J. R. Patel, “High spatial resolution grain orientation and strain mapping in thin films using polychromatic submicron x-ray diffraction,” Appl. Phys. Lett., 80(20):3724–3726 (2002) 29. D. Weiss, unpublished data (2001) 30. M. D. Thouless, K. P. Rodbell, and C. J. Cabral, “Effect of surface layer on the stress relaxation in thin films,” J. Vac. Sci. Technol., A14(4):2454–2461 (1996)

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31. D. Weiss, H. Gao, and E. Arzt, “Constrained diffusional creep in UHVproduced copper thin films,” Acta Mater., 49:2395–2403 (2001) 32. M. J. Kobrinsky and C. V. Thompson, “Activation volume for inelastic deformation in polycrystalline Ag thin films,” Acta Mater., 48:625–633 (2000) 33. M. Ronay, “Yield stress of thin fcc polycrystalline metal films bonded to rigid substrates,” Philos. Mag., A40(2):145–160 (1979) 34. Y.-L. Shen, S. Suresh, M. Y. He, A. Bagchi, O. Kienzle, M. Rühle, and A. G. Evans, “Stress evolution in passivated thin films of Cu on silica substrates,” J. Mater. Res., 13(7):1928–1937 (1998) 35. J. A. Ruud, D. Josell, F. Spaepen, and A. L. Greer, “A new method for tensile testing of thin films,” J. Mater. Res., 8:112–117 (1993) 36. D. T. Read and J. W. Dally, “A new method for measuring the strength and ductility of thin films,” J. Mater. Res., 8(7):1542–1549 (1993) 37. D. T. Read, “Tension-tension fatigue of copper thin films,” Int. J. Fatigue, 20(3):203–209 (1998) 38. D. Josell, D. van Heerden, D. Read, J. Bonevich, and D. Shechtman, “Tensile testing low density multilayers: aluminum/titanium,” J. Mater. Res., 13(10):2902–2909 (1998) 39. H. Huang and F. Spaepen, “Tensile testing of free-standing Cu, Ag and Al films and Ag/Cu multilayers,” Acta Mater., 48(12):3261–3269 (2000) 40. M. Hommel, O. Kraft, S. P. Baker, and E. Arzt, “Micro-tensile and fatigue testing of copper thin films on substrates,” Materials Science of Microelectromechanical System (MEMS) Devices, MRS Symp. Proc., vol. 546, Pittsburgh, PA (1999), pp. 133–138 41. E. A. Stach, U. Dahmen, and W. D. Nix, “Real time observations of dislocationmediated plasticity in the epitaxial Al (011)/Si (100) thin film system,” MRS Symp. Proc., vol. 619, Warrendale, PA (2000) 42. G. Dehm, B. J. Inkson, T. J. Balk, T. Wagner, and E. Arzt, “Influence of film/substrate interface structure on plasticity in metal thin films,” MRS Conf. Proc., vol. 673, Warrendale, PA (2001), pp. 2.6.1–2.6.12 43. G. Dehm, D. Weiss, and E. Arzt, “In situ transmission electron microscopy study of thermal-stress-induced dislocations in a thin Cu film constrained by a Si substrate,” Mater. Sci. Eng., A309–310:468–472 (2001) 44. V. Weihnacht and W. Brückner, “Dislocation accumulation and strengthening in Cu thin films,” Acta Mater., 49:2365–2372 (2001) 45. L. B. Freund, J. Appl. Mech., 43:553 (1987) 46. C. V. Thompson, “The yield strength of polycrystalline films,” J. Mater. Res., 8(2):237–238 (1993) 47. P. Chaudhari, Philos. Mag., A39(4):507–516 (1979) 48. G. Dehm, B. J. Inkson, T. Wagner, T. J. Balk, and E. Arzt, “Plasticity and interfacial dislocation mechanisms in epitaxial and polycrystalline Al films constrained by substrates,” J. Mater. Sci. Technol., 18(2):113–117 (2002) 49. H. J. Frost and M. F. Ashby, Deformation-Mechanism Maps, Pergamon Press, Oxford, UK (1982) 50. W. D. Nix and O. S. Leung, “Thin films: plasticity,” Encyclopedia of Materials: Science and Technology (K. H. J. Buschow, R. W. Cahn, U. C. Flemings, B. Ilschner, E. J. Kramer, and S. Mahajan, eds.), Elsevier Science Ltd., Oxford, UK (2001)

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51. O. Kraft, L. B. Freund, R. Phillips, and E. Arzt, “Dislocation plasticity in thin metal films,” MRS Bull. 27(1):30–38 (2002) 52. S. P. Baker, R.-M. Keller, and E. Arzt, “Energy storage and recovery in thin metal films on substrates,” Thin Films - Stresses and Mechanical Properties VII, MRS Symp. Proc., vol. 505, Warrendale, PA (1998), pp. 605–610 53. W. D. Nix, “Yielding and strain hardening of thin metal films on substrates,” Scripta. Mater., 39(4–5):545–554 (1998) 54. B. von Blanckenhagen, P. Gumbsch, and E. Arzt, “Discrete dislocation simulation of thin film plasticity,” Dislocations and Deformation Mechanisms in Thin Films and Small Structures, MRS Symp. Proc., vol. 673, Pittsburgh, PA (2001), p. 2.3 55. B. von Blanckenhagen, P. Gumbsch, and E. Arzt, “Dislocation sources and the flow stress of polycrystalline thin metal films,” Philos. Mag. Lett., 83(1):1–8 (2003) 56. J. Eshelby, Dislocations in Solids, vol. 1, F. R. N. Nabarro, North-Holland Publ. Co., Amsterdam (1979), p. 167 57. P. Müllner and E. Arzt, “Observation of dislocation disappearance in aluminum thin films and consequences for thin film properties,” Thin Films Stresses and Mechanical Properties VII, MRS Symp. Proc., vol. 505, Pittsburgh PA (1998), pp. 149–154 58. H. Gao, L. Zhang, and S. P. Baker, “Dislocation core spreading at interfaces between metal films and amorphous substrates,” J. Mech. Phys. Solids, 50:2169–2202 (2002) 59. S. P. Baker, L. Zhang, and H. Gao, “Effect of dislocation core spreading at interfaces on the strength of thin-films,” J. Mater. Res., 17:1808–1813 (2002) 60. H. Gao, L. Zhang, W. Nix, C. Thompson, and E. Arzt, “Crack-like grain boundary diffusion wedges in thin metal films,” Acta Mater., 47:2865–2878 (1999) 61. T. J. Balk, G. Dehm, and E. Arzt, “Observations of dislocation motion and stress inhomogeneities in thin copper films,” MRS Symp. Proc., vol. 673, Warrendale, PA (2001), pp. 2.7.1–2.7.6 62. T. J. Balk, G. Dehm, and E. Arzt, “A new type of dislocation mechanism in ultrathin copper films,” MRS Symp. Proc., vol. 695, Warrendale, PA (2002), pp. 2.7.1–2.7.6 63. G. Dehm, T. J. Balk, B. von Blanckenhagen, P. Gumbsch, and E. Arzt, “Dislocation dynamics in sub-micron confinement: recent progress in Cu thin film plasticity,” Z. Metallkd. 93:383–391 (2002) 64. M. D. Thouless, “Effect of surface diffusion on the creep of thin films and sintered arrays of particles,” Acta Metall. Mater., 41:1057–1064 (1993) 65. R. Coble, “A model for boundary controlled creep in polycrystalline materials,” J. Appl. Phys., 41:1679–1682 (1963) 66. M. F. Ashby, “The deformation of plastically non-homogeneous materials,” Philos. Mag., 21:399–424 (1970) 67. W. D. Nix and H. Gao, “Indentation size effects in crystalline materials: a law for strain gradient plasticity,” J. Mech. Phys. Solids, 46(3):411–425 (1998) 68. Z. C. Xia and J. W. Hutchinson, “Crack patterns in thin films,” J. Mech. Phys. Solids, 48:1107–1131 (2000)

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69. L. Zhang and H. Gao, “Coupled grain boundary and surface diffusion in a polycrystalline thin film constrained by substrate,” Z. Metallkd., 93:417–427 (2002) 70. M. D. Thouless, J. Gupta, and J. M. E. Harper, “Stress development and relaxation in copper-films during thermal cycling,” J. Mater. Res., 8:1845–1852 (1993) 71. P. Børgesen, J. K. Lee, R. J. Gleixner, and C.-H. Li, “Thermal-stress-induced voiding in narrow, passivated Cu lines,” Appl. Phys. Lett., 60:1706 (1992) 72. J. A. Nucci, R. R. Keller, D. P. Field, and Y. Shacham-Diamand, “Grainboundary misorientation angles and stress-induced voiding in oxide passivated copper lines,” Appl. Phys. Lett., 70:1242–1244 (1997) 73. T. M. Shaw, L. Gignac, X.-H. Liu, R. R. Rosenberg, E. Levine, P. Mclaughlin, P.-C. Wang, S. Greco, and G. Biery, “Stress voiding in wide copper lines,” 6th International Workshop on Stress-Induced Phenomena in Metallization, AIP Conf. Proc., vol. 612, Melville, NY (2002), p. 177–189 74. D. Weiss, O. Kraft, and E. Arzt, “Grain-boundary voiding in self-passivated Cu-1 at.% Al alloy films,” J. Mater. Res., 17(6):1363–1370 (2002) 75. A. G. Evans and J. W. Hutchinson, “The thermomechanical integrity of thin films and multilayers,” Acta Metall. Mater., 43(7):2507–2530 (1995) 76. M. Lane, R. H. Dauskardt, A. Vainchtein, and H. Gao, “Plasticity contributions to interface adhesion in thin-film interconnect structures,” J. Mater. Res., 15(12):2758–2769 (2000) 77. M. Huang, Z. Suo, Q. Ma, and H. Fujimoto, “Thin film cracking and ratcheting caus temperature cycling,” J. Mater. Res., 15(6):1239–1242 (2000) 78. O. Kraft, R. Schwaiger, and P. Wellner, “Fatigue in thin films: lifetime and damage formation,” Mater. Sci. Eng., A319–321:919–923 (2001) 79. R. R. Keller, R., Mönig, C. A. Volkert, E. Arzt, R. Schwaiger, and O. Kraft, “Interconnect failure due to cyclic loading,” 6th International Workshop on Stress-Induced Phenomena in Metallization, AIP Conf. Proc., vol. 612, Melville, NY (2002), pp. 119–132 80. R. Schwaiger and O. Kraft, “Size effects in the fatigue behavior of thin Ag films,” Acta Mater., 52:195–206 (2003)

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9 Electromigration in Cu Thin Films Chao-Kun Hu, Lynne M. Gignac, and Robert Rosenberg IBM T. J. Watson Research Center, Yorktown Heights, New York

9.1

Introduction

Cu wiring in integrated circuits for high-performance chips has been pursued for many years.[1–4] The production of IC chips with Cu interconnections has increased each and every year since its initial commercialization in 1997 by IBM.[5] Chips with Cu wiring have improved conductivity, which has resulted in reduced RC time delays for wiring, where R is resistance and C is capacitance. When IC chip technology is extended below 0.2-mm dimension, the RC delays from interconnections will constitute a large percentage of the total chip time delay.[5] In addition to reduced resistance, Cu interconnections have longer electromigration lifetimes compared to chips with the conventional Al(Cu) metallization.[5–13] Electromigration arises from the motion of atoms under the influence of intense electric field and current. The imbalance of atomic flux at some critical sites in metal lines results in the formation of voids or extrusions, resulting in failure of the chips. Electromigration in Al(Cu) thin films has been a subject of extensive studies for several decades.[14–16] Investigators have reported that electromigration in Al thin-film lines is related to grain boundary diffusion,[16] interface diffusion between Al and TiN,[17] TiAl3,[17, 18] AlOx,[19] crystallographic texture,[20, 21] solute (Cu) effects,[22, 23] and, to a lesser extent, bulk diffusion.[24] However, electromigration rules for Cu lines cannot be drawn from the Al(Cu) experience. In the Al(Cu) case, lifetime increases as linewidth decreases, because the drift velocity at Al(Cu) dielectric surfaces/interfaces is lower than that at grain boundaries, and the number of grain boundary paths are reduced in a bamboo-like microstructure of fine lines.[25] But electromigration of Cu atoms is mainly dominated by interface transport,[26, 27] which is usually faster than grain boundary diffusion. So the electromigration lifetime in Cu interconnects decreases as the cross section of line area becomes smaller.[10] Modern commercial IC chips, which usually contain four to eight levels of Cu interconnections, are generally fabricated by a damascene process,[5, 6] which is described in Sec. 9.2. Cu interconnections require metal liner and insulator adhesion/diffusion barrier layers to form the

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multilevel lines and vias.[28] The metal barrier layer or liner creates material dissimilarities at level-to-level interfaces and is often the cause for electromigration flux divergence, as long as mass transport through the liner is negligible. Furthermore, to increase device density and performance, each generation of IC chips has smaller line/via sizes and higher current densities. Interconnects with smaller via size require less time to grow voids that result in failure. Consequently, the electromigration lifetime for each generation of interconnections is expected to decrease.[10] Therefore, electromigration in these newly emerging Cu on-chip interconnections becomes increasingly important and has been a popular topic for research in recent years.[29, 30] In damascene interconnection structures, the top surface of a Cu line is generally covered by an insulator diffusion barrier layer, such as SiNx, SiCxHy, SiCxHyNz, or SiC, and the bottom and sides of the line are covered with a metal liner, such as TaN/Ta[31] or TiN.[32] Typically, mass transport along Cu grain boundaries is not dominant.[10] The fast diffusion paths in Cu interconnects strongly depend on the materials and the fabrication processes used, which vary for each laboratory.[10–12, 33–37] Fast diffusion along the Cu/liner[11, 12, 33–36] or Cu/dielectric[10, 27, 38] interfaces has been reported. The variation in the reported “fast paths” is probably due to significant differences in the various integration processes, such as degas, chemical-mechanical polishing (CMP) slurry, and precleaning steps before depositing a capping layer on Cu. The fraction of the total Cu atoms that are present at interfaces increases as the dimensions of the interconnections are scaled down and cause further reduced chip lifetime in every new generation. In addition, the Cu electromigration lifetime distributions in dual-damascene lines have been found to have multiple failure modes[33–39] and are further complicated by the reservoir effect in structures of Cu line/Cu via/Cu line or Cu via to Cu line.[38–40] The reservoir effect occurs when the thin liner at the bottom of the via cannot completely block Cu diffusion from a high electromigration-induced compressive stress. This chapter focuses on understanding the differences in test data as a function of the test structure used, instead of the usual approach of presenting a functional fit to the change in resistance of lines to obtain electromigration lifetime distribution. Furthermore, the relationship of electromigration lifetimes to (1) the barrier layer at the Cu line/via interface, (2) the via size, and (3) the linewidth and diffusion paths is elucidated. Correlations of electromigration lifetimes with void growth/sites and the underlying diffusion mechanisms in Cu thin-film lines are emphasized. This form of analysis is preferred because of an enhanced understanding of the failure physics, which allows a much greater generalization of the experimental findings and a more precise projection of IC Cu chip reliability.

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This chapter is divided into 14 sections. Section 9.2 reviews basic interconnection integration. Section 9.3 describes the test structures used in the experiments and discusses the experimental setup and failure analyses. Section 9.4 discusses the microstructure of the thin-film interconnects. Section 9.5 describes the theoretical background information of electromigration physics. Section 9.6 presents the lineresistance change and void growth data. Section 9.7 discusses the dominant diffusion paths for electromigration. Sections 9.8 and 9.9 present the lifetime distribution and current density dependencies, respectively. Section 9.10 presents the relationship between lifetime and metal linewidth. Section 9.11 presents the electromigration scaling rules and lifetime prediction. Section 9.12 presents the short-length effect in Cu interconnections. Section 9.13 discusses the reduction of interface diffusion using surface coatings. Section 9.14 summarizes the chapter.

9.2

Cu Interconnection Integration

On-chip Cu interconnections can be fabricated by two different schemes.[28] The first approach is similar to conventional Al interconnect processing and generally entails first patterning the metal lines, either by dry etching, liftoff, or selective deposition, followed by dielectric deposition. Among dry-etching techniques, reactive ion etching (RIE)[41] and ion milling[28] have been shown to pattern Cu lines. The second approach is to pattern the dielectric level first, then fill metal into the patterned trenches and holes, followed by CMP to remove the surface layer and planarize the structure, leaving material in the holes and trenches. This approach is known as damascene processing and has been commonly accepted by IC manufacturers. Since the tested chips discussed in this chapter used both approaches, we will briefly describe Cu interconnections fabricated by liftoff, dry etching, and damascene processing. For the liftoff process, a negative image of the metal line was patterned in a resist layer. Then the metal was evaporated into the opening of the resist. The metal on the resist was lifted off when the resist was immersed in a resist solvent, leaving the desired metal lines on the substrate. In the dry-etching process, the photoresist was patterned on the Ta/Cu/Ta/SiNx multilayer structure. The photoresist was used as an etching hard mask to etch Ta/SiNx by RIE. Then the Cu lines were patterned by argon and nitrogen plasma using photoresist, and silicon nitride/Ta as the etch mask. Damascene processing was first used by IBM for fabricating W-vias,[42] copper-polyimide,[43, 44] and Cu-SiO2[7, 45] wiring structures. For advanced Cu interconnection structures, Fig. 9.1(a)

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Figure 9.1 Process sequence used to fabricate a single-damascene [(a) through (d)] and dual-damascene [(e) through (h)] Cu interconnect.

through (d) and Fig. 9.1(e) through (h) depicts the single- and dualdamascene sequences,[46–48] respectively. A typical single-damascene level is fabricated by the deposition of a planar dielectric stack, which is then patterned and etched using standard lithographic and dry-etch techniques to produce the desired wiring or via pattern. This is followed by physical vapor deposition (PVD) of a metal liner, such as TaN/Ta, and a Cu seed, then filled with Cu using an electroplating deposition technique.[7] In dualdamascene processing, both the vias and the trenches are patterned in the dielectric before depositing the metal. The excess metal in the field region is then removed using a Cu CMP process, leaving planarized wiring and vias imbedded in an insulator. Subsequent levels are fabricated by repeated application of this process. In the damascene process, all wiring levels are planar at every level, which typically results in enhanced wafer yield over a nonplanar structure. A seven-level damascene interconnection

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Figure 9.2 SEM cross-section image of an IC chip with seven levels of Cu interconnects.

structure is shown in Fig. 9.2.[9] This structure demonstrates excellent planarity and low contact resistance between the levels. Each level of Cu
SiO2 contains a dielectric trilayer of chemical vapor deposition (CVD) SiNx
SiO2
SiNx. The thin silicon nitride film coated on the Cu interconnection structures serves several functions: an adhesion layer between the levels, Cu and H2O diffusion barrier, RIE mask for trench/via definition, RIE stop (prevents exposure of lower Cu levels to O2 RIE), and capping layer to suppress Cu hillock formation and solvent absorption during subsequent low-dielectric-constant (e) material cures.[28] However, in regions where the high-dielectric-constant (e
7) SiNx remains between the dielectric layers, the interline capacitance is increased. Other materials,[49] such as SiC (e  5), SiCxNy (e
4.9), and a-SiCxHy (e
4.5), with lower dielectric constant serving the same function as the silicon nitride have been investigated in recent years.

9.3

Test Structure and Experiment

Figure 9.3(a) through (e) shows schematic diagrams of the test structures discussed in this chapter. Figure 9.3(a) is a Blech-type drift velocity test structure[50] that consists of a series of Cu line segments on a continuous W underlayer. Figure 9.3(b) through (d) and (e) shows three- and twolevel interconnect test structures, respectively. M0, M1, M2, and M3 are the W local, the first Cu level, the second Cu level, and the third Cu level interconnects. M1 connects to M0 and M2 through the CA and V1 vias, respectively. M2 connects to M1 and M3 through the V1 and V2 vias, respectively. In Fig. 9.3(b), the W M0 lines, W CA via, and Cu M1 were fabricated by a single-damascene process. In Fig. 9.3(c) through (e), Cu

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Figure 9.3 Test structures discussed in this chapter. (a) Blech-type drift velocity setup on W; (b) and (c) three-level-metal structure of M0/CA/M1/V1/M2; (d) threelevel-metal structure of M1/V1/M2/V2/M3; (e) two-level-metal structure M0/CA/M1.

CA and Cu M1 were fabricated by a dual-damascene process. The upper levels of V1
M2 and V2
M3 were also fabricated by dual-damascene processes. The Cu interconnects were embedded in a SiNx
SiO2 dielectric or amorphous a-SiCxHy
SiLK™ dielectric. (SiLK semiconductor dielectric is a trademark of the Dow Chemical Co.) The final Cu lines were passivated with SiO2
SiNx, and the Al(Cu) metallization was used to coat the bonding pads. However, some samples with no passivation on top of Cu lines were also studied. The Cu line/via microstructure was analyzed by both focused ion beam (FIB) microscopy and scanning transmission electron microscopy (STEM). The thickness and width of the metal lines and vias were measured from either FIB, scanning electron microscopy (SEM), or TEM images. Void growth and extrusion of the tested lines were also examined by FIB and SEM on cross sections prepared by FIB. Other investigators have used transmission X-ray microscopy[51, 52] or highvoltage (120 keV) SEM[53] to observe void size and growth. A dc current was applied to the test lines. Failure time, t, was determined by

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the time to increase the line resistance significantly from the initial value Ro by 2%, ∆R
Ro  2%. The failure time distribution was analyzed as a log-normal[22] or a multi-log-normal distribution, depending on the test structures used.

9.4

Microstructure

Cu microstructure is an important factor in determining the diffusion and mass transport rate. The grain size distribution, crystallographic texture, surface plane orientation, and impurity distribution are all related to Cu diffusivity. The diffusion will determine the electromigration drift velocity, and variation of local microstructure can cause an electromigration flux divergence. The texture of lines can be controlled by the texture of the underlying Ti and TiN,[32] TiW,[54] Ta,[32, 54] or TiN,[54] and a highly 111 textured Cu showed improved electromigration resistance in a one-level line structure. The grain distribution of PVD Cu films has been reported previously and is generally found to be bimodal with a significant twinned grain volume fraction.[55] Texture components of 111, 200, random, 220, and 511 in PVD Cu have been reported.[55, 56] Grain growth, texture, and microstructure of electroplated Cu films have gained much attention in recent years because the electroplating technique has been widely chosen as the main process for Cu filling in damascene lines. This is because the electroplating process of filling Cu in trenches/vias from the bottom up is far superior to the other processes. A bimodal grain size distribution, considerable random texture components, and a high degree of twinning have been found in electroplated Cu thin films.[57] In Cu damascene lines, the 111 texture decreases as the linewidth decreases from 1 to 0.3 mm, and the twinned grain volume fraction increases after annealing.[29] Further, it is known that texture in electroplated films is sensitive to a large number of deposition parameters, such as liner/Cu seed-layer microstructure, electroplating current density, and bath chemistry.[58] Figure 9.4(a) through (d) shows plan-view FIB images of 3-, 2-, 0.8-, and 0.18-mm-wide electroplated Cu lines, respectively, taken at an ion beam angle of 10 degrees. Figure 9.4(e) is a cross-section view FIB image of a 0.18-mm-wide electroplated Cu dual-damascene line, taken at an ion beam angle of 45 degrees. In these images, large grains and twins are seen. The interfacial energies of a Cu twin boundary and a normal grain boundary are reported to be 44 erg/cm2[59] and 650 erg/cm2,[60] respectively. The small interfacial energy at twin boundaries and reasonable match between twin and parent lattices across a twin plane imply that Cu diffusion along a twin boundary should be close to Cu bulk diffusion. Thus, twin boundaries are not

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Figure 9.4 FIB images of Cu lines. (a) through (d) Plan view of 3-, 2-, 1-, and 0.18-mm-wide Cu lines taken at an ion beam angle of 10 degrees; (e) cross-section of a 0.18-mm-wide line taken at an ion beam angle of 45 degrees.

considered to be fast diffusion paths. Across the linewidth, polycrystalline grains intermixed with single-crystal grain segments were observed for the 3- and 2-mm-wide lines, while bamboo-like grain structures were observed in the submicron Cu lines and vias. The microstructure of the 5-mm-wide line was very similar to that observed in Fig. 9.4(a). The bamboo-like and polycrystalline grain structures are defined as single grain per linewidth or per via and two or more grains per linewidth, respectively. All Cu grains in the Cu lines analyzed occupied the entire line thickness. The initial fine-grain structure of the Cu seed and electroplated Cu layers as shown in Fig. 9.5 are converted to large grains during abnormal grain growth, which can occur at room temperature or by annealing.[61–63] Figure 9.5 shows a STEM cross-section of an as-plated, 0.7-mm-thick plated Cu film.[64] The sample was prepared within 1.5 hours of warming to room temperature, and the grains did not recrystallize during FIB milling. Once the sample was prepared, the grain structure did not undergo any change over time. The cross section in Fig. 9.5 shows that the plated film has multiple small grains stacked in the film thickness. A mean grain size and standard distribution of 0.05 ± 0.03 mm was calculated from a Gaussian fit to a log-normal distribution of grain areas assuming circular-shaped grains. Both twins and dislocations

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Figure 9.5 STEM cross section of an as-plated Cu microstructure. The sample was prepared by FIB and liftoff within 1.5 hours of warming to room temperature.

are visible in the as-plated grains. The barrier layer can be seen as the dark contrast material underneath the Cu film. However, the PVD Cu seed layer is not distinguishable from the plated Cu film. The transformation time from fine Cu grains to large grains is strongly dependent on deposition parameters such as plating current, bath chemistry, and layer thickness. It has been reported that plating Cu on a 0.15-mm-thick PVD Cu seed has to be larger than the critical thickness of 0.25 to 0.35 mm for the abnormal grain growth to occur.[29, 65] The large Cu grain sizes in the damascene lines and vias are due to the dual-damascene process which has a thick Cu film (overburden) over the trenches before CMP and thus abnormal grain growth in the electroplated Cu. The bamboo-like structure in damascene lines and the single crystal nature of Cu in vias are shown in Fig. 9.4(c) to (e). Figure 9.6 shows TEM cross sections of V1/M1 and V2
M2 interfaces.[38] These images show a thin liner at the interface. The thicknesses of the liners at the bottom of the V1 and V2 vias for these samples measured 3 and 10 nm, respectively. The variation in the liner thickness is due to the variation of the deposited film thickness and the PVD step coverage in the various structures.

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Figure 9.6 TEM cross-section micrographs of V1/M1 and V2/M2. (a) Thin, 3.1-nm liner between V1 and M1; (b) 10-nm-thick liner at V2/M2.

9.5

Theory

9.5.1 Drift Velocity The observations of mass transport under an electric field in lead-tin and mercury-sodium condensed phases was first recorded in 1861.[66] The

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atom flux produced by an electromigration driving force Fe is given by Je
n vd, where n and vd are the atomic density and drift velocity, respectively. The drift velocity is expressed by the Nernst-Einstein relation, vd
(Deff
kT)Fe,

(1)

where Fe
Z*eE
Z*erj, E is the electric field, e is the absolute value of the electronic charge, Z* is the apparent effective charge number, r is the metallic resistivity, Deff is the effective diffusivity of atoms diffusing through a metal line, T is the absolute temperature, and k is the Boltzmann constant. The quantity of Z* represents the strength of the electromigration effect and ranges in value from 102 to 102.[67] It is customary to divide Z* into two parts, Z*
Z*el  Z*wd, where Z*el arises from the direct force of the pure electrostatic nature and Z*wd is the contribution from the so-called “electron wind” force that arises from the momentum exchange between charge carriers and the diffusing atom. The wind force suggested by Skaupy[68] can usually be expressed by A/r(T)[69] or ne lese  nh l hs h,[70, 71] where A is a constant; ne and nh are the electron and hole densities, respectively; le and lh are the mean free paths of the electrons and holes, respectively; and se and s h are the atom’s intrinsic cross section for collision with the electrons and holes, respectively.

9.5.2 Diffusivity The effective diffusivity in a given line at one cross section can be written as: 0

Deff
nGBDGB  Σ niDi ,

(2)

0

where the subscripts GB and i refer to the grain boundary and the ith interface (atom diffusion along metal/insulator or metal/metal interfaces), respectively. nGB and DGB, ni and Di are the fractions of atoms and diffusivities in grain boundaries and the ith interface, respectively. The diffusivity D is expressed in terms of Do exp(Q
kT), where Do and Q are the pre-factor and activation energy, respectively. In Eq. (1), diffusivity is the dominant factor for the mass transport. Only atoms diffusing along the fast diffusion paths will control the atomic movement. Several types of possible fast diffusion paths are considered: dislocations, the Cu/dielectric interface, the Cu/metal liner interface, free surfaces, and grain boundaries. The Cu bulk diffusivity with a high activation energy of 2.2 eV[72] is the slowest and is many orders of magnitude less than the fast diffusion paths. For bulk diffusion, we can estimate that the

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time to grow a 0.1-mm void at 300°C using Eq. (1) with j
2  106 A
cm2 and Z*
5[73, 74] is about 50,000 years. Thus the contribution from the Cu bulk diffusion is negligible. A range of activation energies for Cu dislocation, Cu/SiNx interface, free surface, and grain boundary diffusion have been reported as 1.53 eV,[75] 0.8 to 1.1 eV,[37, 76, 77] 0.5–2 eV,[75, 78–80] and 0.8–1 eV,[81–84] respectively. The grain boundary structure can vary widely within a given line, which causes considerable variability in diffusivity from boundary to boundary.[84] Impurities on or in the fast path can also play an important role in determining the Cu diffusivity.[82, 83] Grain boundary diffusivity is often an average value of the measurements taken over a large number of grains that have a variety of orientations. Dislocation pipe diffusion refers to atomic motion along dislocations. However, the cross-sectional area of a single dislocation is small, and the net diffusivity depends on the density of dislocations. Interfacial diffusion refers to atom motion along the interfaces such as between the metal/insulator (Cu
SiNx) or the metal/metal (Cu
Ta) and is highly dependent on the chemistry, bonding, impurity, and structure at the interfaces. The observations of a dominant fast diffusion path along Cu
SiNx,[10] Cu
TaN,[11] Cu
Ta,[11, 12] and Cu
TiN[32] interfaces have been reported. These differences indicate that interface diffusion is related to the interface property and materials, which are very sensitive to the sample preparation, such as processing. The surface diffusion is also strongly influenced by ambient. The surface diffusion on clean Cu or in a Cl2 ambient is faster than on air-exposed Cu surfaces or in a H2 ambient.[79]

9.5.3 Effective Diffusivity and Microstructure Let us consider a polycrystalline line structure with a Cu grain across the metal line thickness, h. The number of fast paths for grain boundary fGB are (w
d  1) for w  2d, or 1 for 2d  w  d, where w is the linewidth and d is the grain size. The fraction of atoms diffusing through grain boundaries is nGB ≈ (dGB
w)fGB, where dGB denotes the width of the grain boundary. The effective diffusivity for the case of an unpassivated liftoff Cu
Ta line is the sum of the top and the two sidewalls of free Cu surfaces, the Cu/bottom Ta interface, and the Cu grain boundary diffusion. Assuming transport paths are independent, the product of Z*eff and Deff along a thin-film line of a given cross section can be described as:[85] Z*effDeff
Z*IDId I
h  Z*SD*Sd S(2
w  1
h)  Z*GBDGBnGB,

(3)

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where the subscripts S, I, and GB refer to the uncoated, free Cu surface (at two sidewalls plus top of the line), the Ta
Cu interface, and the Cu grain boundary, respectively; d I, d S and d GB denote the width of the interface, surface and grain boundary, respectively. d I
h, d S(2
w  1
h) and nGB are the fractions of atoms diffusing through the interface , the surface and the grain boundary in the line, respectively. Finally, d is the grain size of the Cu line. For the Cu damascene test structures, the top surface of a line is covered by an insulator, typically silicon nitride, and the bottom and sides of the line are covered with a liner, such as Ta. The fast-diffusion paths are along grain boundaries, the Cu/silicon nitride, and the Cu/Ta interfaces. The effective diffusivity can be written as: Z*effDeff
Z*IDId I (2
w  1
h)  Z*NDN* d N(1
h)  Z*GBDGBnGB,

(4)

where Z*N and DN are the effective charge number and diffusivity at the Cu/silicon nitride interface, respectively, and d N denotes the effective width of the Cu/silicon nitride interface. For the bamboo-like grain structure, the contribution of mass transport by electromigration along the grain boundary (GB) is negligible because of the absence of a continuous GB path and an electromigration driving force that is perpendicular to the GBs. The drift velocity can be written as: vd
dN(1
h)DN Z*N erj
(kT)  d I (2
w  1
h)DI Z*Ier j
(kT).

(5)

Equation (5) states that the drift velocity in the bamboo-like grain damascene line is a function of the metal line thickness if the Cu/silicon nitride interface diffusion is dominant. The drift velocity is a function of metal line thickness and width if the Cu/Ta interface diffusion is dominant. For test structures with completely blocking boundaries at both ends of the line, the boundary condition for the atomic Cu fluxes at the contact interface is: JCu(Cu)  JB(Cu)
JCu(Cu)
n vd ,

(6)

where JCu(Cu) and JB(Cu) are the atomic Cu fluxes in the Cu lines and the blocking boundary, respectively, and JB(Cu)
0 because no Cu can diffuse through the blocking boundary. Edge displacement (void growth), ∆L, at the cathode end of the line causes the conductor line resistance to increase by ∆R. The void growth rate is directly related to electromigration drift velocity by: ∆L
∆t
vd .

(7)

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For a constant drift velocity, the lifetime t can be obtained as follows: t
∆Lcr
nd,

(8)

where ∆Lcr is the critical void size for the lifetime t.

9.5.4 Electromigration-Induced Backflow Under electromigration test conditions, two opposing transport mechanisms operate simultaneously: atom migration due to the electromigration force, and atom backflow due to an electromigration-induced stress gradient.[50] The stress gradient occurs because atoms, which are driven out of the cathode end of the conductor, causing tensile stresses, accumulate at the anode end, where the atomic density becomes higher, causing compressive stresses. This gradient results in a backflow of material (Blech effect).[50] Combining the electromigration force and backflow effects produces a net drift velocity: vd
ve  vb
(D
kT)(Z* erj  ∆s Ω)
∆x,

(9)

where ve is the electromigration drift velocity and vb is the average mechanical backflow velocity. An important implication of this effect is that for sufficiently short lines or low current densities, stress can completely suppress mass transport. We can define threshold values: a given j and a critical linelength L c [∆x in Eq. (9)] below which net mass transport vanishes (vd
0) and jLc ∝ ∆s. The magnitude of the electromigration-induced stress ∆s is dependent on the electromigration force. The electromigration-induced stress ∆sι in the line has to be less than the fracture strength ∆sc of the passivation layer and has a maximum value of ∆sc. In addition to the mechanical strength of the dielectric material, the anode end of the line has to connect to a complete blocking boundary to generate the short-length effect. As will be discussed in Sec. 9.12, for Cu interconnections below 0.25-mm technology, the thickness of the liner at the via and line interface is often less than 10 nm and the Cu current density at the liner interface is often more than 3 mA/mm2. Thin liners at the anode end may not withstand the incoming Cu flux and the compressive stress is relieved. In addition, the critical current densities obtained by using a very wide underlying or overlying line connected to a fine test line will not be the same as those from an interconnect structure with similar linewidth. Therefore, applying the short-length effect in Cu interconnections should be performed with caution.

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9.5.5 Partial Blocking Boundary In the case of partial blocking boundaries, such as thin liner at the Cu via/Cu line interface, voids and extrusions will not necessarily be formed at the via/line interface but will occur whenever there is an imbalance of Cu fluxes at a certain location, by the equation: (Jout  Jin) ∂h  
 , ∆x ∂t

(10)

where Jin and Jout represent the Cu flux entering and leaving at that location. The calculation of void or hillock growth rates related to drift velocity in the partial blocking boundary case becomes rather complicated since it is difficult to estimate the drift velocity in this continuous equation. No void or extrusion can grow if Jin and Jout are equal.

9.6

Resistance and Void Growth

This section discusses the line resistance changes as a function of current-stress time in metal lines connected to blocking boundaries, such as W or thick diffusion barrier liners at a contact interface. Here the local flux divergence at the contact interface is the dominant failure mode. The diffusion flux of Cu atoms in the Cu line at the contact interface is directly correlated to the electromigration drift velocity. Material depletion at the cathode end causes the conductor line resistance to increase. For layered interconnections (such as TaN/Ta/Cu), the relationship between the rates of material depletion and the line resistance change, ∆L
∆t and ∆R
∆t, can be generally obtained as follows: ∆R(t)
(rTa ∆L)
ATa  rCu (L  ∆L)
ACu.

(11a)

∆R
∆t
(rTa
ATa  rCu
ACu)∆L
∆t is proportional to vd, since ∆L
∆t
vd. The subscripts Ta and Cu refer to Ta liner and Cu conductor, respectively; r is the electrical resistivity; A is the cross-sectional area of the specific metal; L is the initial conductor length; and ∆L is the void growth length. The change in line resistance is simply a linear function of the electromigration drift velocity. However, the joule heating generated from the thin liner, the first item of the right-hand side of Eq. (11a), sometimes cannot be neglected; a ∆T of 100°C,[86] or even melted TaN/Ta liner, have been observed.

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Figure 9.7 Test line resistance vs. stressed time. ∆Lo and ∆Ld are the void growth within and beyond the Cu/W overlapping area, respectively.

The line resistance change curves as a function of time for a sample temperature of 296°C are shown in Fig. 9.7. The void formation in a damaged line is shown in Fig. 9.8, which is a cross-sectional view SEM micrograph of a 0.28-mm-wide line, illustrating the typical degradation mode of void growth at the cathode end of the line. The lifetimes are rather uniform, varying within 30% sample-to-sample and illustrating that the mass transport rate is an average measurement through a large number of grain surfaces. Initially, the line resistance changes slowly, followed by an abrupt step of resistance change and a period of rapid constant resistance rise. We can take the data points with a dotted line as an example for correlating void growth and line resistance change. The initial period of the slow resistance change rate for the testing time 60 hours was caused by void growth within the W/Cu overlap length (∆Lo), because the large voltage change will be sensitive only to a void that grows beyond the Cu/W overlap area. Therefore, ∆R(t) ∼ 0, if ∆L  (∆Lo). The abrupt line resistance step is believed to occur when the void grows just beyond the W stud. It corresponds to a change in the contact resistance between

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Figure 9.8 SEM micrograph of the electromigration damage in a 0.28-mm-wide line. Arrows show the electron flow direction.

W studs/TaN/Ta liner/Cu and also to current flowing over a thinned liner region covering a step between the end of the W stud and the line, as shown in Fig. 9.8. The final period of resistance change is attributed to void formation and growth where the current has to pass through the thin, high-resistance liner underlayer to connect the remaining Cu line to the W via. For this period, Eq. (11a) should be modified to: ∆R(t)
(rTa ∆Ld)
ATa  rCu [(L  ∆Lo)  ∆Ld]
ACu ∼ (rTa ∆Ld)
ATa, (11b) where ∆Ld is the void length beyond ∆Lo. Equation (11b) shows the relationship between the edge displacement ∆L and line resistance change. The resistance change rate in the final period should be a constant, if the void growth rate ∆Ld
∆t is constant. However, a very high current density of 107 A
cm2 will be applied to the liner with typical resistivity of 200 mΩ-cm in the 20-nm-thick Ta in the region of ∆Ld. Joule heating could be generated, especially in a defective liner. We can roughly estimate a ∆R of 120 Ω for ∆Ld
0.3 mm. Thus for a high current density test, a faster void growth rate for ∆L  ∆Lo is expected compared to the region of ∆L  ∆Lo. This explanation is in agreement with the upward

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curvature of the resistance change curves shown in Fig. 9.7. Consequently, we would expect the void growth rate for high current density test cases not to be linearly proportional to the current density, which is in contrast to Eq. (1). We can, of course, make the liner joule heating less of a factor by increasing the liner thickness at the expense of the line conductivity.

9.7

Fast Diffusion Paths

9.7.1 Free Surface and Grain Boundary Diffusion A surprisingly wide range of activation energies, from 0.5 to 2 eV, for electromigration in thin-film Cu lines has been reported,[76–78, 87–97] and many diffusion mechanisms (bulk, grain boundary, surface, and interface) have been proposed to explain these results. It is important to identify the dominant diffusion paths in Cu lines. Once the fast diffusion path is identified, we may find a way to reduce the fast diffusivity therein and increase the Cu reliability. This section discusses the electromigration studies of liftoff, unpassivated Cu lines of widths varying from 0.15 mm (bamboo) to 10 mm (polycrystalline), to differentiate the relative contributions of surface from grain boundary transport.[80] The test structure[98] consisted of Cu lines connected to two sputtered Ti (10 nm)
W (200 nm) bar electrodes, the purpose of which was to simulate blocking contacts and to measure the drift velocity of Cu in the tested lines. The Ti (10 nm)
Ta (15 nm)
Cu (300 nm) test lines were all deposited by e-gun evaporation at a base pressure of 107 torr and fabricated by a liftoff process using e-beam lithography.[98] The bottom Ti layer in the Ti
W bar served as an adhesion layer to SiO2 and in the Ti
Ta
Cu line was used to reduce the contact resistance between Ta and W. The Ta film was a diffusion barrier layer between Ti and Cu. The Cu test lines were 0.15 to 10 mm wide and overlapped the Ti
W bar by 0.85 and 1.0 mm for 0.15- to 0.25-mm-wide lines and 0.75- to 10-mm-wide lines, respectively. The samples were annealed in helium at 400°C for 3 hours before testing to stabilize the microstructure. Bamboo-like, near-bamboo, and polycrystalline structures were found in the metal linewidths between 0.15 to 0.5 mm, 0.75 to 1 mm, and 2 to 10 mm, respectively. The samples were tested in a vacuum furnace at temperatures ranging from 255 to 405°C with a current density of 15 mA/mm2 in a chamber pressure of 15 torr of nitrogen. Mass transport of Cu in the Cu lines as a function of temperature was measured using both drift-velocity (void growth rate) and resistance measurements. Void size was measured by scanning electron microscopy. For these test structures, complete blocking boundaries exist at both ends of the line. Edge displacement void growth, ∆L, at the cathode

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end exposed the Ti/Ta underlayer and caused the conductor line resistance to increase by ∆R. The drift velocity of Cu was directly related to the rate of edge displacement by Eq. (11b) as vd
∆L
∆t and can be estimated by vd
∆Lf
t, assuming a constant drift velocity, where t is the amount of time required to grow a void to critical size ∆Lf. (Incubation times were considered negligible.) Let us assume the Cu/Ta interface diffusion is significantly slower than GB and surface diffusion. The evidence of the slow Cu/Ta diffusion is given in Sec. 9.7.3. For the case of a bamboo-like grain structure, the contribution of mass transport by electromigration along GBs is negligible because there is no continuous GB path and the electromigration driving force is perpendicular to the GBs. The surfaces at the sidewalls and on top of the lines become the fast diffusion paths in liftoff structures. The combination of Eqs. (1) and (3) can be written as: vd
dS(2
w  1
h)DS Z*S erj
(kT).

(12a)

Equation (12a) states that the marker velocity (or void growth rate) in the bamboo-like line structure will be increased as the metal linewidth or thickness is decreased, at a fixed sample temperature and current density. For the case of a polycrystalline line structure, the effective drift velocity becomes: vd
{(dGB
d)(1  d
w)D0GB exp(QGB
kT)Z*GB  dS(2
w  1
h)D0S exp(QS
kT)Z*S}erj
kT,

(12b)

0 and D0S are the pre-exponential factors, and QGB and QS are the where DGB activation energies for grain boundary and surface diffusion, respectively. In this case, we would expect the drift velocity to be less sensitive to linewidth for large w, because (1  d
w)  1. Surface and grain boundary diffusivities are known to be dependent on purity, the orientation of the surface, and the plane of the grain boundaries;[75, 79, 84, 99] however, the measured values were generally average values obtained by sampling drift velocities along many fast paths. For example, the number of grain surfaces sampled in a 0.25-mm-wide line for a void size of 1 mm is estimated to be more than 300, using ∆Lhw
(dSd (2h  w)), where ∆L
1 mm, w
0.25 mm, h
0.3 mm, dS
0.5 nm, and d
0.8 mm. The size of the void ∆Lf at the cathode end of the line (mass depletion) was measured from SEM micrographs. Hillock growth at the anode end of the line was also observed. Figure 9.9(a) to (c) shows the SEM micrographs of Cu depletion at the cathode ends of 0.15-, 1-, and 5-mm-wide lines for t
6.8, 30, and 18 hours, respectively, with j
15 mA
mm2 at 314°C.

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Figure 9.9 SEM microgaphs of tested Cu lines with j
15 mA/mm2 at 314oC for various linewidths. (a) 0.15-mm-wide line for t
6.8 hours; (b) 1-mm-wide line for t
30 hours; (c) 5-mm-wide line for t
18 hours.

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Figure 9.10 Plot of drift velocity as a function of linewidth stressed at sample temperatures of (a) 405 and (b) 255oC. The lines are least-squares fits.

The Cu drift velocity was estimated by ∆L
t, where ∆L and t are the mean void size and mean lifetime, respectively. Figure 9.10 shows the drift velocity vd as a function of w. A careful inspection of Fig. 9.10 shows that the velocity decreases monotonically with linewidth w but goes through a minimum at w
1 mm and then increases to a constant value. The lines are the least-squares fits of the data to Eq. (12a) for w 1 mm and to Eq. (12b) for w  1 mm, respectively. In the case of the bamboo grain structures for w 1 mm, grain boundary transport is eliminated and only surface diffusion of the Cu lines is considered. The film thickness, the profiles of the cross-sectional area (trapezoid shape) of the line, and the values of erj
kT are known. The only adjustable parameters in Eq. (12a) are d and dSDSZ*S. The best values of fitting parameters for the data w 1 mm were used as the constrained values for analyzing the data points for w  1 mm in Eq. (12b). The solid lines shown in Fig. 9.10 are the least-squares fits. The contribution of Cu drift velocity in grain boundaries is also pointed out in Fig. 9.10. The ratio of vS to vGB for a 10-mmwide line decreases from 2 to 1 as T increases from 255 to 405°C. This indicates that major void growth is due to surface migration even in a polycrystalline 10-mm-wide line for T  250°C. The activation energy of grain boundary diffusion is 0.2 eV higher than that of surface diffusion, and the ratio of (dSDS0 Z*S)
(dGBD0GB Z*GB) is about 0.02. The extracted values of fitting parameters from the data in Fig. 9.10, d GBDGBZ*GBerj and d SDSZ*Serj, as a function of 1
T are plotted in Fig. 9.11. The activation

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Figure 9.11 Plot of dSDSZ*SES and dGBDGBZ*GBEGB vs. 1/T. The straight lines are calculated from the best-fitting values of Z* to the data using the known values of dSDSES and dGBDGBEGB.

energies of surface and grain boundary diffusion are 0.9 and 1.1 eV, respectively. The derived values of activation energies are in good agreement with the reported values of 0.78 to 0.90 eV for the activation energy of surface diffusion[75, 99] and 0.88 to 0.95 eV for grain boundary diffusion.[81–83] The apparent effective charge number Z* can be estimated by using the diffusivities and data in Fig. 9.11. From the above relationship, after substituting the surface diffusivities[79, 99] computed either from D0S
0.15 cm2
s and QS
0.78 eV, or D0S
0.26 cm2
s and QS
0.90 eV, at 400°C, r
5.0 mΩ-cm, and d S
0.5 nm, we obtain Z*S
0.1 or 0.8, respectively. The negative sign in Z* means that Cu atoms drifted in the direction of the electron wind force. Using the grain boundary diffusivities,[83] we obtain Z*GB
14. The values of Z* estimated in this way

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depend on the accuracy of published values of D0 and Q. Although the absolute value of Z*S  Z*GB appears to be consistent with a theoretical prediction, the electron wind force decreases as an atom moves from the bulk to a grain boundary and to a surface.[100] In summary, a systematic study of Cu electromigration drift velocity shows that the paths for fast migration are the surfaces in bamboo-like and near-bamboo structures, and a mixture of surface and grain boundaries in polycrystalline films. The activation energy for grain boundary diffusion is approximately 0.2 eV higher than that of surface diffusion, and the ratio of (dS D0S Z*S ES)
(dGB D0GB Z*GBEGB) is approximately 0.02.

9.7.2 Ambient Effect The effect of impurity on Cu surface diffusion has been reported.[75, 79, 101, 102] The measured surface diffusivities are strongly dependent on the atmosphere and absorbed impurity on the surface of the diffusion experiments. The Cu surface self-diffusion increases with the vapors of solute Pb, Tl, Bi, or Cl over the Cu surface[75, 79] and is reduced with the absorption of C, Ca, Mn, or O2 on the Cu surface.[79, 101] The kink and ledge sites of the Cu surface can absorb impurities that reduce the concentration of diffusing defects and thus suppress the rate of surface self-diffusion.[75] The Cu surface diffusivity is higher in an oxygen atmosphere or vacuum than in a hydrogen atmosphere.[75, 79] This result suggests that oxygen may remove the inhibiting effect of surface impurities.[75] To this end, the electromigration lifetime of unpassivated 0.18-mm-wide bamboo-like Cu lines with a Fig. 9.3(e) structure was measured. The samples were tested in a vacuum furnace with a chamber pressure of 10 to 30 torr of high-purity nitrogen or forming gas (N2  5% H2) in the temperature range 180 to 362°C. The total impurity concentration in the nitrogen gas was less than 1 ppm, with H2O the major impurity at 0.5 ppm. The concentration of oxygen was found to be less than 10 7 ppm from an oxygen detector. With a test structure shown in Fig. 9.3(e), the Cu void growth rate at the cathode end of the line was the same as the Cu drift velocity because the end of the M1 line was connected to a completely blocking boundary of W M0. The migration fast path is the top surface in the bamboo-like damascene line. Thus the extracted activation energy from these samples is the electromigration activation energy of Cu surface diffusion. The mean lifetime t as a function of 1/T is plotted in Fig. 9.12. The electromigration activation energies of Cu surface diffusion in nitrogen and forming gas are found to be 0.7 ± 0.1 and 0.9 ± 0.1 eV, respectively. The derived values of electromigration activation energies are in good agreement with the reported values of the activation energy of surface diffusion of 0.78 and 0.95 eV measured

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Figure 9.12 Plot of mean lifetime t vs. 1/T. The straight lines are least-squares fits. The data points are obtained from the samples using a Fig. 9.3(e) test structure.

in oxygen (105 torr) and hydrogen (760 torr) ambients, respectively.[75, 79] Higher values reflect the slowing of surface diffusion by the presence of gases compared to the value of 0.5 eV measured in situ in ultrahigh vacuum on a clean Cu surface.[103]

9.7.3 Alloying Effect This section discusses the results of electromigration in pure Cu and Cu alloys using standard drift velocity test structures [Fig. 9.3(a)]. The tested samples were the Ta/Cu or Cu alloys/Ta line segments on top of a W underlayer line. The Cu line segments were patterned by an ion milling technique.[28] The top and bottom Ta layers served as an etch mask and etch stop. The bottom Ta layer was removed by RIE using CF4 chemistry. During RIE processing, CuFx may have formed on the surfaces of two sidewalls of the Cu lines. All the samples were annealed in He at 400°C for 3 hours. The final metal structure is Ta (20 nm)/Cu or Cu alloys (300 nm)
Ta (20 nm). The top Ta layer thickness was estimated from

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Figure 9.13 SEM micrographs of 5-mm-wide lines after electromigration stressing with 2  106 A/cm2 at 250°C for (a) 55 hours with Cu(Mg); (b) and (c) 128 hours with pure Cu and Cu(Zr), respectively.

Augh electron spectroscope (AES) analysis with Ar sputtering profiles. The SEM micrographs in Figure 9.13 show the depleted cathode ends after electromigration stressing of 5-mm-wide lines at 250°C for 55 hours of a Ta
Cu(Mg)
Ta structure, and for 128 hours of a Ta/pure Cu/Ta structure and a Ta/Cu(Zr)/Ta structure. Voids developed under the top, immobile Ta layer and along the sidewalls of the line in this sandwich structure. The dominant Cu electromigration paths would be along the Cu grain boundary plane and/or along the two Cu sidewalls. These Ta layers are not seen in Fig. 9.13 because of the high penetration depth of the 30 KeV electron beam in the SEM. Many fine islands are seen in the depleted Cu(Mg) sample. The edge displacement ∆L of Ta
Cu(1 wt.% Mg), pure Cu, and Cu(0.7 wt.% Zr)
Ta isolated lines as a function of stress time is plotted in Fig. 9.14.[104] The linear behavior of Cu(Mg) and pure Cu indicates that drift velocities of Cu in Cu(Mg) and pure Cu are independent of time, with little incubation time for a sample temperature of 250°C. In the case of Cu(Zr), a nonlinear behavior was observed. The combination of a small

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Figure 9.14 Edge displacement in Cu and various Cu alloys as a function of stress time at 250°C.

Cu grain size in Cu(Mg), the formation of fine Cu islands seen in Fig. 9.13, and the lack of a pinning effect of Cu by Mg along the fast diffusion paths resulted in enhancing the Cu line damage. The void growth rate on the Cu line in Cu(Mg) was found to be 5 and 35 times faster than in pure Cu and Cu(Zr), respectively. This is partly due to the fine islands left behind the depletion front. The islands can be formed if Cu on the trailing surface of the grains has not been completely fed into grain boundary depletion before the voids encircle the grain. The activation energy for electromigration in pure multigrained Cu is found to be 0.77 ± 0.04 eV, which is less than the Cu grain boundary diffusion.[80, 82] The value of 0.77 eV should represent the surface diffusion activation energy along the two sidewalls of the line.

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Figure 9.15 Line resistance change vs. electromigration stressed time for Cu and Cu(Sn) alloys with j
2.6  106 A/cm2 at temperatures of (a) 250 and (b) 203°C.

The time for an equivalent resistance change ∆R corresponding to electromigration damage in pure Cu and Cu(Sn) alloys is progressively increased as a function of Sn content in the Cu at 250 and 203°C for 2.0-mm-wide, 0.3-mm-thick lines, as shown in Fig. 9.15(a) and (b), respectively. A deviation from linear behavior was observed in Cu(Sn). This nonlinear behavior of the line resistance change is similar to that observed during electromigration in Al(Cu),[105] in which Cu depletion precedes

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damage in the Al. The variation of line resistance in the Cu alloy lines can be roughly divided into three damage stages: an incubation stage, during which resistance does not increase, followed by slow-increasing and steady-state stages. During the incubation stage, an initial reduction in resistance is observed, due to the depletion of the Sn solute in the grains, which decreases the contribution of solute scattering to resistivity. Once the void forms, the resistance of the lines starts to increase. Similar slopes for pure Cu and Cu(0.5%Sn), as shown in Fig. 9.15, in the final steadystate stage suggest that the Cu depletion rate in the Cu(Sn) sample is the same as in pure Cu. This was also observed in our previous results for sample temperatures above 250°C.[28, 91] The effect of Sn solute in bulk Cu is different from that in thin films. In bulk Cu(Sn) samples, Sn solute enhances Cu and Sn diffusion in Cu(Sn),[106] while solute Sn decreases Cu diffusion in Cu(Sn) grain boundaries of thin films.[83] These observed behaviors are similar to Pd in Cu(Pd)[93] and Au in Au(Ta)[107] studies. The effect of Sn in Cu is similar to Ta in Au and Pd in Cu. A 0.5 wt.% Sn addition can cause the time required for ∆R
1Ω (equivalent to a 1.6-mm edge displacement) to increase by a factor of 10 at 203°C. The nature of solute and solvent interaction at grain boundaries and surfaces is not clear. However, the observations of reducing Cu grain boundary diffusion in Cu(Sn) alloys can be qualitatively interpreted in terms of the solute Sn reducing the grain boundary energy[108] and/or acting as a trapping site[109, 110] for Cu. Both models predict D(Cu)/D(Cu(Sn)) ∼ 1  ZCo exp((∆E  T∆S)
kT), where D(Cu) and D(Cu(Sn)) are the Cu diffusivities in pure Cu and Cu(Sn) alloy, Co is the solute concentration, Z is the coordination number for the solute atom, and ∆E and ∆S are the corresponding binding energy and entropy for grain boundary and solute interaction, respectively. The free Cu atoms or vacancies are drastically reduced in the fast paths because of the Sn-Cu atom or Sn-vacancy interactions, which depend on the diffusion mechanisms[111] in the grain boundaries. If we assume that the ratio of time for ∆R
0.5Ω(∆L
0.8 mm) for pure Cu to Cu(Sn) alloys is due to changes in effective diffusivity (that is, changes in Z*r are small), then a binding energy ∆E of the order of 0.5 ± 0.3 eV between Sn-Cu atoms and/or Sn-vacancy at grain boundaries is obtained from Fig. 9.15. The binding energy ∆E ∼ 0.5 eV is the same order as the increase in the activation energy for grain boundary diffusion observed in Cu upon addition of 2 wt.% Sn.[83] In summary, electromigration mass transport rates in Blech-type test structures have been observed. The drift velocity of Cu can be greatly enhanced or reduced by solute addition. The electromigration surface activation energy in pure PVD-ion-milled Cu lines measured by void edge displacement in nitrogen ambient was found to be 0.77 ± 0.04 eV. The

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corresponding edge displacement rate of Cu was increased by alloying with Mg and slowed down with Pd, Zr, or Sn additions.

9.8

Lifetime Distribution

9.8.1 Single-Damascene Line on W Via It is customary to plot the lifetime distribution in a log-normal function, which translates the lifetime from a linear to a log scale. The lognormal distribution, f, and log-normal cumulative probability, F, functions[22] can be described as follows: (lnt (ln t  lnt ln t ))  1  f
 e 2s2s 2ps  22

50 50

22

(13)

and




f

(lnt  llnt (ln t  nt )   1 2s 2s F
 e d ln t, 2ps  f 22

50 50

22

(14)

where t50 and s are the median lifetime and the deviation for the lognormal distribution function, respectively. A least-squares fitting method was used to obtain the best adjustable fitting parameter values of t50 and s by minimizing c2, which is defined by:[112] n (Fi  yi)2 c2
Σ  , e2 i
1

(15)

where the values of Fi and yi are the estimated function and data point at ln(t i), respectively; e i is the uncertainty of yi; and n is the sample size. The fitting procedure, which is dependent on the values of e i, is assumed to be 1 for the case of cumulative probability data. On the other hand, the frequency count in failure time distribution is graphed as a histogram showing the number of observations fi recorded for each ln(ti), and an error bar [113] for the frequency fi can be estimated as f i(1 f. in) Figure 9.16(a) shows the normalized line resistance vs. time at a sample temperature T
296°C with current density, j, of 32 mAmm2 in M1 and electron flow from W M0 to W CA to Cu M1 embedded in SiO2. The line resistance initially slowly decreased, then increased, followed by a

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Figure 9.16 Resistance change in single-damascene test lines vs. time.

period of rapid rise. The initial line resistance decrease was most likely due to reduction of contact resistance and/or purification of the Cu line. A sharp resistance increase occurred when a void grew completely across the line or line/via interface as the current had to pass through the high-resistance thin liner. With this rapid increase in line resistance, the lifetime difference between ∆R
R
1% and ∆R
R
20% in most of the samples tested was small. However, if the thickness of the liner were increased, there would have been a gradual line resistance increase during this time period, as shown in the case of Fig. 9.7. Here the lifetime defined by time at ∆R
Ro
1%, 20%, or 60% will differ significantly. Figure 9.17 is a graph of cumulative failure probability (%) vs. lifetime (t) on a log-normal scale from the data shown in Fig. 9.16. For comparison, the data for the electron flow from Cu V1 via to Cu M1 line at 296°C and from CA to M1 at a sample temperature at 255°C are also plotted. In this test structure, the V1 diameter is twice the linewidth. A linear behavior in a log-normal scale indicates that the failure distribution can be well represented by a single log-normal function. A rather tight distribution with deviation s of around 0.3 was obtained in all cases. The similar median lifetime between the samples tested with electron flow from W CA to M1 and Cu V1 to M1 suggests that the liners

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Figure 9.17 Cumulative percentage failure vs. t for single-damascene lines on a log-normal scale. Solid and dashed lines are the least-squares fits to the data.

at the bottom of V1 in these samples are good diffusion blocking boundaries. Figure 9.18 is a graph of the frequency count as a histogram in failure time distribution showing the number of observations fi recorded for each ln(ti) and their error bars. The plot shows a Gaussian peak, and the fluctuating character reflects the finite collection of data points. The solid curve represents a Gaussian distribution of the data to Eq. (13) by minimizing c2. The extracted value of c2v
(c2
n) of 0.9 from the least-squares fit indicates reasonably good fit, where n is the degree of freedom. The extracted values of s and median lifetime from either frequency count or cumulative failure probability methods are in good agreement.

9.8.2 Dual-Damascene Line on W Line 9.8.2.1 Multiple-Cu Vias Let us discuss a case of a 0.9-mm-wide Cu line embedded in SiO2 where the cathode end of the line has four Cu vias connected to a W

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Figure 9.18 Histogram of frequency, f, vs. measured log(t). The solid curve is the estimated Gaussian function.

underlying line.[45] This structure can be divided into a line section and a reservoir section (line/via overlapped region). The current distribution in multiple via test structures has been reported.[114] It was shown that the highest current density was at the line section. The material that drifted away along the Cu/SiNx interface in the line section was replenished by the material from the reservoir section and by the section without electrical current flow. Voids grown in the overlying Cu lines/Cu studs produce resistance changes over time. The observation of a stepwise resistance increase during the current stress was due to mass depletion within the four-via region of the reservoir section. Each resistance step probably corresponded to the formation of a void in one of the four studs at the Cu/underlying W interface. Once a void is formed across the entire intersection of the reservoir and line sections, the supply of material from the reservoir is cut off, marking the end of the resistance incubation period. A sharp increase in line resistance is observed after the small resistance change period, since the remaining overlying line is connected to W by a resistive Ta/TaN liner. Figure 9.19(a) through (c) shows optical micrographs

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Figure 9.19 Optical micrographs of 0.9-mm-wide Cu lines after electromigration stress with a current density of 2  106 A/cm2 at 350°C for (a) 181, (b) 382, and (c) 595 hours.

of Cu samples that were electromigration-stressed at 350°C and failed at 181, 382, and 595 hours, respectively. No formation of voids and hillocks was found anywhere in the line section, but voids did grow in the vicinity of the cathode studs (reservoir section). Figure 9.19(a), the earliest failed sample, shows no visible void anywhere in the Cu overlying line. The FIB

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Figure 9.20 FIB image of the electromigration-tested line from Fig. 9.19(a). The image was taken at an ion beam angle of 45 degrees.

cross-section image in Fig. 9.20 shows that voids formed at the four interfaces of the Cu studs/W underlying line, which resulted in failure but were not observed from top-down optical micrographs. A void size of several microns is evident in the reservoir section of the sample that failed at an intermediate time [Fig. 9.19(b)], apparently before the four Cu stud/W interfaces opened. Figure 9.19(c) shows a case of complete depletion of the entire 4.2-mm-long reservoir section, which appeared in the sample with the longest lifetime. Log-normal fitting of the lifetime cumulative data results in the large values for s of 0.4 to 1, in contrast to the small (0.1 to 0.3) values of s observed for two-level single-damascene structures. The large values of s are due to void formation in different locations from sample to sample. A more reasonable approach for failure analysis is to separate failure times into three different groups corresponding to three different individual failure modes, as shown in Fig. 9.19(a) through (c).[45] As an example, the s for the samples that showed 4.2-mm depletion mode is reduced to about 0.2.

9.8.2.2 Single-Cu Via Now let us consider a simpler case of a dual-damascene line with one Cu via on W instead of four vias as in the case discussed in Section 9.8.2.1. We attempt to correlate the lifetime failure distribution with the void growth and location and to present methods of data analyses in detail for the failure lifetime distributions. Even in this simple structure, the lifetime distribution has been reported and was also found to have multiple failure modes.[37] A wide spread in the lifetime distribution

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Figure 9.21 Histogram of frequency, f, vs. measured log(t). The solid curve is the estimated double-Gaussian function.

can be seen in Figs. 9.21 and 9.22, which are the graphs of ln(t) vs. frequency count and cumulative failure probability (%) vs. lifeime (t) on a log-normal scale, respectively. The data in Fig. 9.21 clearly show a two-Gaussian function. The solid curves respresent a double-Gaussian function fit. The extracted value of cv2
(c2 v) of 0.3 from the leastsquares analysis implies a reasonably good fit. The nonlinear behavior in Fig. 9.22 shows that the failure distribution is not a single lognormal function; however, it can be plotted as two individual lognormal functions. For samples tested at 296°C, the data points can also be separated into two individual groups as shown in Fig. 9.22. The vertical dotted line is the border between these two distributions; this point was determined by the transition of void growth from the bottom via to the line/via or to the line itself, which will be discussed later in this section. The cumulative failure distribution is analyzed either by separating the data points into two individual groups[115] or by using a bimodal distribution function. [34, 37–39] The bimodal distribution

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Figure 9.22 Cumulative percentage failure vs. t on a log-normal scale. Solid lines are the least-squares fits to the data. The dotted line is the border between two Gaussians. The solid circles are a replot of the data points (open circles) into two individual groups. Open circles with center dots correspond to the first-failure group.

function is given by:





0

(ln(t
t )) (t
t )) a1 (1  a1) (ln   e 2s d ln t   e 2s d ln t, F
 2ps 2 0 2ps 1 2

2 1

1

2

2 2

2

(16)

where t1, s1 and t2 , s2 are the median lifetimes and the deviations for the first and second log-normal distribution functions, respectively, and a1 is the fraction of the total population in the first log-normal function. A nonleast-squares fitting method was used to obtain the best adjustable fitting parameter values of t1, s1, t2 , s2, and a1. The voids in most of the electromigration-damaged lines were cross-sectioned and imaged using a FIB. For example, Fig. 9.23(a) to (d) shows cross-sectional-view FIB images of voids in the Cu dual-damascene lines tested at 296°C for lifetimes of 64, 216, 220, and 301 hours, respectively, taken at an ion beam

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Figure 9.23 FIB images of samples tested at 296°C, taken at an ion beam angle of 45 degrees, for lifetimes of (a) 64, (b) 216, (c) 220, and (d) 301 hours, respectively.

angle of 45°. The samples with a short lifetime of 64 hours, Fig. 9.23(a), and a medium lifetime of 216 hours, Fig. 9.23(b), showed that no voids grew in the line, although voids did grow at the bottom of the Cu via. However, the samples in the long-lifetime t2 group had mixed mode line/via voids [see Fig. 9.23(c) and (d)] or line-only voids. From the failure analysis sampling, voids were only seen at the bottom of the via for samples in the short-lifetime t1 group. From our analyses, the tested samples with lifetimes around 200 hours have a mixed mode, and with lifetimes greater than 216 hours no longer will have void growth at the via bottom only. Once the separation of the two groups was determined by FIB analysis, the data points could be simply plotted into two individual log-normal distribution functions (Two-Probit), as shown in Fig. 9.22. In this way, the strong correlation between five fitting-adjusted parameters in a bimodal function can be reduced. This method allows a more precise fit of the data to a log-normal function. The extracted values from the three methods

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Table 9.1. Fitting Parameter Values from the Various Methods

Bimodal (F)

Double Gaussian (f )

Two-Probit

Function Temp (°C)

350

296

350

296

350

296

t1(h) s1 t2(h) s2 a1

16 0.24 45 0.12 0.58

111 0.65 289 0.14 0.46

16 0.34 46 0.16 0.63

72 0.26 290 0.37 0.22

17 0.36 45 0.19 0.63

89 0.47 277 0.19 0.33

discussed above are listed in Table 9.1, which shows reasonably good agreement among the fitting methods. The table also shows that the median lifetime t1 in the first group is about one-third to one-fourth of those in the second group. The first group of voids grew at the bottom of the Cu via only and thus required less Cu to be removed compared to the second group of line/via voids. The extracted best fitting parameter values from a least-squares fit are shown in Fig. 9.24. The first group has a tighter distribution than the second group, and a1 is found to be weakly dependent on T. A similar electromigration activation energy of 1.0 eV was obtained for both t1 and t2 from the data in Fig. 9.25. This indicates a common diffusion mechanism for both t1 and t2, and diffusion along the Cu/SiNx interface, even though the void location varied for the two populations.

Figure 9.24 Values of fitting parameters to log-normal distributions vs. 1/T for (a) s and (b) a1 (see also Table 9.1).

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Figure 9.25 Plots of median lifetimes vs. 1/T (see also Table 9.1).

In summary, the investigation of electromigration in dual-damascene Cu lines on W has suggested that the dominant diffusion path was the Cu/SiNx interface, not the Cu/Ta interface, in the samples presented in this section. This was concluded from the similar void growth rate and activation energy, independent of void location. Furthermore, if the Cu/Ta interface diffusion was the dominant path, the lifetime distribution would have been more closely represented by a log-normal distribution, not a bimodal function, since the mass drifted away along the Cu/Ta interface and caused void formation at the via bottom to be the dominant failure mode.

9.8.2.3 Dual-Damascene Cu Line on Cu Line Figure 9.26 shows the plot of some normalized line resistance as a function of time for 0.27-mm-wide lines with a Fig. 9.3(b) test structure stressed with a current density of 22 mA
mm2 at a sample temperature of 350°C. The dotted curves represent the samples measured with the

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Figure 9.26 Line resistance vs. time for a Fig. 9.3(b) test structure at 350°C. Solid lines represent electron flow from W to Cu lines; dotted lines represent electron flow from the Cu V1 via to the Cu M1 line.

electron flow from dual-damascene Cu M2-V1 to Cu M1 embedded in SiO2. For comparison, the samples tested with electron flow from W CA to a single-damascene Cu M1 are also plotted (solid curves). There was a marked difference in failure lifetime between electron flow from CA to M1 and V1 to M1. However, for both cases, the line resistance initially decreased slowly, then increased, followed by a period of rapid rise. A rather tight failure time distribution (t50 of 26 hours and s of 0.3) was obtained in the case of CA to M1 electron flow. For the electron flow from V1 to M1, a wide range of failure times was observed. Only 50% of the samples in this case had a similar lifetime of around 26 hours, as in the CA to M1 case. The other 50% of the samples showed a much longer lifetime, some with no resistance increase even after 180 hours. The observations of the short- and long-lifetime groups were for all sample temperatures investigated from 250 to 350°C. The drift velocity or mass flow of Cu in a Cu M1 line should be identical regardless of the direction of electron flow. If either end of the M1 line was contacted to a completely blocking boundary, mass depletion at the cathode end and mass accumulation at the anode end of the line should occur. Thus for the case of

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electron flow from a blocking boundary W CA to M1, the void growth rate in M1 near the W CA should follow Eq. (1). On the other hand, if the liner at the M1
V1 interface were only a partial blocking boundary for the case of electron flow from V1 to M1, the migration of Cu M1 beneath the V1 liner could be replenished from V1
M2 Cu. Hence a very long lifetime would result. Figure 9.27(a) and (b) shows the SEM images of the two dual-damascene Cu samples tested at 294°C using a Fig. 9.3(b) test structure and lasting for 271 and 2200 hours, respectively. The electron flow is from V1 to M1 for both samples. A sharp line resistance increase was observed after testing for 271 hours in the former, but no line resistance increase was observed in the latter, even though the testing time was about 8 times longer. Figure 9.27(a) shows that a void grew (edge displacement) beneath the bottom V1 and beyond the V1 via
M1 line overlapping section for the sample tested for 271 hours. This void location resulted in a sharp resistance increase and caused the line failure, since the current had to pass through the thin liner to connect the remaining Cu line and via. The void growth rate obeys Eq. (1) and is consistent with the Cu electromigration drift velocity estimated from the samples with a completely blocking W boundary. The data suggest that a good diffusion barrier layer was formed in the sample shown in Fig. 9.27(a). However, two tiny voids on the surface of a single-crystal M1 near V1 and a void at the end of the M2 were seen for the sample shown Fig. 9.27(b). This observation suggests that Cu atoms migrating along the Cu M1 top interface were partly replenished from the M2
V1 Cu. To further explore the electromigration lifetime enhancement due to a thin liner at the M1
V1 interface, the tested samples with long lifetimes were examined by FIB imaging from cross sections prepared using FIB milling. Figure 9.28(a) and (b) shows FIB images of the two tested samples that used the Fig. 9.3(b) and (d) test structures, respectively. The direction of electron flow is from M2 to V1 to M1 in Fig. 9.28(a) and from M1 to V1 to M2 in 9.28(b). Both samples were run at 295°C and j
22 mA
mm2 for more than a thousand hours without an increase in line resistance. Here M2
V1 and M1 under V1 are the anode ends for Fig. 9.28(a) and (b), respectively. The Cu accumulation and compressive stress at the anode ends of the lines would occur if the bottom V1 liners were blocking boundaries. Surprisingly, voids, not hillocks, grew at the anode end of M2
V1 in Fig. 9.28(a). Moreover, an estimated void size on the order of 10 mm in M1 under V1 (cathode end at M1) in Fig. 9.28(a) and in M2 (cathode end at M2) in Fig. 9.28(b) should have been observed using Eq. (1) and the measured electromigration drift velocity. But no void was observed at the cathode end of the M1 fine test line in Fig. 9.28(a), and small voids in V1 and the corner of M2 in Fig. 9.28(b)

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Figure 9.27 SEM micrographs of the M2/V1/M1 using Fig. 9.3(b) test structures. The lines were tested for (a) 271 hours and (b) more than 2000 hours without failing. The arrows show the direction of electron flow.

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Figure 9.28 FIB images of M2/V1/M1 after electromigration stressing. Test structures in (a) and (b) are of the type shown in Fig. 9.3(b) and (d), respectively. The lines were tested for more than 1000 hours without failing. The arrows show the direction of electron flow. Bamboo-like Cu grains with twins in the via and line are shown.

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were seen. These results suggest that Cu atoms (or vacancies) actually moved through the bamboo-like V1 vias, resulting in long t  1000 hours. The voids, shown in the micrographs, were not the cause of the significant resistance change seen in the electromigration testing, since the metal line was still connected by high-conductivity Cu. According to Eq. (2), the drift velocity is mainly determined by the fastest diffusion path; in this case, Cu atoms drifted along the Cu
SiNx interface (the top Cu surface) with an electromigration activation energy of about 1 eV. The depletion of atoms creates voids; thus, voids occurring at the cathode end of the M1 in Fig. 9.28(b) would be expected. However, the observations of only small voids in Fig. 9.28(b) for the long test time and the existence of voids at the anode end of V1 and M2 in Fig. 9.28(a) were surprising. Here, some voids even grew in regions where there was little electric field (corners of the M2 lines). Similar experimental observations of void formation in a metal section outside of the region carrying current were previously reported.[104, 116–118] This is explained by the following argument. The mass transport of Cu along the top surfaces of M1 in Fig. 9.28(a) and M2 in Fig. 9.28(b) generates excess vacancies and tensile stress. The forces from the large vacancy concentration and stress gradients generate vacancy flux. The flux moves to vacancy sinks such as interfaces, grain boundaries, or the ends of the lines, and forms voids. These excess vacancies migrate easily through the Cu because of their low migration energy. The failure mechanism is then similar to electromigration in a single line with large reservoirs at both ends of the line. We can estimate the time required for vacancies to diffuse through the Cu using the vacancy diffusivity, D
Do exp(Qm
kT), where the pre-factor Do
0.16 cm2/sec and values of the vacancy migration energy Qm are reported to be 0.71 eV[101] or 0.78 eV.[84] The time required for vacancy diffusion at 295°C over a 2-mm diffusion length is estimated to be within one minute. The large Cu mass displacement in the long-lifetime samples can be clearly seen in Fig. 9.29(a). A huge Cu extrusion occurred at the anode end of the 0.27-mm-wide M1 line at the blocking W CA with test structure Fig. 9.3(b) and electron flow from M1 to W CA. The large compressive stress (mass accumulation) created by the electromigration driving force[50] cracked the SiO2 dielectric material and allowed Cu to extrude in the crack. However, Fig. 9.29(b) shows that no clear extrusion was observed at the anode end of the M2 line (electrons moving from M2 to V2) for the samples tested for more than 1000 hours with a Fig. 9.3(c) test structure. Here, the force from the electromigration-induced compressive stress appears to be sufficiently large that apparently even the 10-nm liner could not withstand the constant incoming Cu flux of j
22 mA
mm2 in a long M2 Cu line.

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Figure 9.29 FIB micrographs of the samples tested for more than 1000 hours showing the anode ends of (a) M1 and (b) M2. Extrusion caused by a blocking boundary was observed in (a) but not (b).

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Figure 9.30 Plot of cumulative percentage failure vs. log(t ) for the samples using a test structure shown in Fig. 9.3(c) with 30- and 3-nm-thick liners at the bottom of V1 with electron flow from V1 to M2. The dotted, dashed, and solid lines are the least-squares-fitted lines using single log-normal, bimodal, and triple-log-normal functions, respectively.

As these cases illustrate, the exact condition of the liner at the Cu via and Cu line interface can strongly influence the lifetime of the tested lines, even though the electromigration drift velocities are the same in all the cases. For example, Fig. 9.30 shows a plot of cumulative percent fails as a function of log (t) using the test structure of Fig. 9.3(c) from two wafers with two different V1 liner thicknesses: 3 and 30 nm. The electrons flow from V1 to M2 in both samples. Nonlinear behavior in a log-normal probability plot was observed. The failure time distributions were analyzed by log-normal, double-log-normal (bimodal), and triple-log-normal (trimodal) functions. The cumulative failure distribution for a trimodal, F, is given by





(ln (t
t )) (ln (t
t )) a2 a1   e 2s dln t   e 2s d ln t F
 2ps 2 2ps 1 2

2 1

1



(t
t )) (1  a1  a2) (ln  e 2s d ln t,   2ps 3 2

2 3

3

2

2 2

2

(17)

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where t and s are the median times to failure and the deviations, to be taken with subscripts 1 through 3, for the first, second, and third lognormal distribution function, respectively. A1 and A2 are the amplitude of the first and second lifetime group, respectively. As discussed in Sec. 9.8.2.2, the void growth is strongly influenced by partially and completely blocking boundaries. In addition, voiding in the interconnects could occur in several ways, such as by interface void, edge displacement type (vertical void), grain-thinning-type void, and void growth with reservoirs. Therefore, data analysis with more than one log-normal distribution is justified. A nonlinear least-squares method was used to obtain the optimum values of these parameters, by minimizing c2. As can be seen in Fig. 9.30, the data were best fitted using a tri-modal function. The reduced c2
n values dropped by a factor of 2 when fitted with a log-normal versus a tri-modal function, where n is the degrees of freedom. The values for the best fitting parameters are shown in Table 9.2. The uncertainties in the median lifetime and s are about 6 to 20% and 30 to 50%, respectively. The Cu lifetime was enhanced by an order of magnitude when the thinner liner was used. Table 9.2 shows that the majority of samples fell in the third lifetime group (longest). Apparently, the incoming Cu flux from the 5-mm-wide M1 even with j
1 mA
mm2 generated sufficient electromigration-induced compressive stress under the V1 liner to allow the Cu atoms to punch through the 3-nm-thick V1 liner to M2, but not the 30-nm-thick liner. A completely blocking boundary at the cathode end of the Cu line seems to be the case for the 30-nm-thick liner case, and a short lifetime would be expected when a void forms at the V1
M1 interface (bottom of V1). Only a small void at this interface is required to cause the line to fail [shown in Fig. 9.31(a)]. In the case of the 3-nm-thick liner, however, the vacancies in M2 flowing into the V1
M1 interface can be filled by Cu atoms from M1. Hence, Eq. (6) is no longer valid (Jb is not equal to 0), and Eq. (10) should be used. Continuous or partial flow at the boundaries will enable a very long lifetime and eliminate most of the via bottom voids. However, despite the variation in liner thickness, if vacancies pile up near the end of the M2 line to reduce stress and vacancy concentration gradients, the Table 9.2. Best Fitting Parameter Values for a Tri-Log-Normal Function Obtained from the Data in Fig. 9.30

LinerThickness (nm)

t1(h)

s1

30 3

66 339

0.13 0.06

a1

t2(h)

0.23 1237 0.1 780

s2

a2

t3(h)

s3

0.22 0.15

0.33 0.19

177 1679

0.88 0.55

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Figure 9.31 FIB image of a sample with a 30-nm-thick liner at the bottom of V1 showing an interface void.

lifetime for both cases should be within an order of magnitude. Therefore, lifetimes of around 180 to 340 hours were observed in the late-failure group with the 30-nm-thick liner and the first-failure group with the 3-nmthick liner, respectively. The Cu interconnections discussed above were embedded in SiO2 with near-vertical vias and thin liners on via side walls. When a void grew in the via bottom during electromigration testing, the thin liner at the via/line interface could not withstand the high current and caused a large resistance increase or line open. We will next discuss the case of Cu interconnects embedded in high-thermal-expansion-coefficient SiLK dielectric with shallow sloped via and thick liner structures. Cu interconnects in SiLK should generate higher tensile stresses in the Cu line/via region than in Cu
SiO2 structures. This could result in a less favorable site for vacancy flow to the via bottom, which would then reduce the fraction of the first mode failure population (see Sec. 9.11.2).

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In addition, a thick liner could provide a path for stable current passage where a marked sharply resistance increase would not occur when a void grew across the via or line. For a lifetime, for example, at ∆R
Ro
20% with a typical Cu line length of 400 mm, the test lines with via bottom voids would have sufficient time for the void to grow into a line/via void location. We expect that interconnects with a stable liner would have reduced the distribution in the first-failure lifetime group and resulted in a single log-normal lifetime distribution.

9.9

Current Density Dependence

Figure 9.32 shows a plot of mean lifetime t as a function of current density for 0.28-mm-wide single-damascene Cu lines on W [Fig. 9.3(b)] at 370°C. In this structure, the cathode end of the line is connected to a completely blocking boundary; the void growth rate is therefore equal to electromigration drift velocity. Following Eq. (7), the Cu lifetime is t
∆Lcr
vd or t
toj1. However, the equation of lifetime to current density is usually expressed by t
tojm, where to is a constant. The value of m will be determined by the experimental data. The values of m in Fig. 9.31 from the data of 0.28-mm-wide-line Cu
SiO2 structures were found to be 1.1 ± 0.2 for j  25 mA
mm2 and 1.8 for j from 25 to 140 mA
mm2. The value of m
1 is consistent with the growth of a void in the pure metal as predicted by Eq. (7), while the value of m
1.8 is believed to be dependent on the power used, the critical void size, the liner thickness and resistivity, the contact resistance, the surrounding material, and the ambient, which strongly reflects a local joule-heating effect from the liner, as discussed in Sec. 9.6. When extrapolating to the actual use condition from accelerated testing, this nonlinear dependence must be considered, especially in the case where accelerated test current density is above the 25-mA
mm2 level for the 0.28-mm-wide lines and the use current density is lower. For fat wires, 1-mm-wide and 1-mm-thick Cu damascene lines in polyimide dielectric, the value of m
2.2 was observed,[119] which deviates greatly from 1. This is probably due to the poor thermal conductivity of polyimide and the high power used in the fat lines, which then reflects a large jouleheating effect. The relationship between lifetime and current density was further explored by the measurements of the void growth rate as a function of current density in a Blech-type drift velocity test structure. This type of test structure closely resembles a two-level test line connected to W vias, completely blocking boundaries. When a void grows at the cathode end of the line in this interconnect structure, the liner under the void would be the

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Figure 9.32 Plot of log mean lifetime (t) as a function of log(j) for 0.28-mmwide lines.

only connection between the W via and the remaining Cu line. Joule heating is generated from conduction through the thin liner, which then causes increased drift velocity, especially in a higher current density case. However, in the Blech-type test structure of Fig. 9.3(a), the underlayer is rather thick compared to the liner in the interconnect structure, and the joule-heating effect is minimized. Therefore, the void growth rate in the Blech-type test structure would be a good test sample in which to examine the electromigration void growth rate theory. The typical resistance changes, ∆R, corresponding to void growth as a function of time for Ta/Cu/Ta line segments on W are plotted in Fig. 9.33. The linear behavior of these lines indicates that the resistance change rate (corresponding to drift velocity) of Cu is independent of time. The line intercept is a time close to zero, which suggests little incubation time in pure Cu lines. The resistance change rate related to drift velocity as a function of current density is plotted in Fig. 9.34. The current density exponent measured from the resistance change rate or drift velocity can be directly applied to

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Figure 9.33 Line resistance change vs. electromigration test time.

failure lifetime as t
tojm proportion to (∆R
∆t)m due to the constant electromigration drift velocity in the pure metallic films. The value of the measured exponent, m, of the current density for pure Cu was found to be 1.1 ± 0.2, which is in good agreement with the theoretically predicted value of 1 in a pure metal, as given in Eq. (1). Although m
3.6 has been reported in the literature from a single-stripe test structure (with pads as reservoirs),[88] it was obtained on a different test structure than that described here.

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Figure 9.34 Plot of resistance change rate (drift velocity) as a function of current density j for pure Cu.

For the dual-damascene line on W, the lifetime distribution is a multiple failure distribution function, and a large number of samples is required to obtain reasonably accurate data. A bimodal distribution function, Eq. (16), is used for fitting the cumulative failure probability. The sample temperature was set at 296°C in this case. The variation of lifetime distributions for the samples taken from the same wafer is small. However, the median lifetimes from one wafer to another wafer can vary by a factor of 2, which is influenced by a slight variation in the wafer fabrication processing, such as linewidth and thickness, line/via overlap area, cleaning steps, or Cu/SiNx interface property. Figure 9.35 shows cumulative lifetime distributions obtained from three wafers. The data shift among wafers is shown. Seven wafers were used to obtain the relationship between lifetime and current density. To overcome the statistical

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Figure 9.35 Cumulative percentage failure vs. t on a log-normal scale from three different wafers. The curves are the least-squares fits of the data to the bimodal function.

error from the wafer variation, the lifetime of every wafer was measured at a current of 2 mA (30 mA
mm2). The scaling factors between the wafers were obtained by normalizing the median lifetimes at 30 mA
mm2 to a chosen wafer. Then the median lifetimes from the other current densities were normalized according to these scaling factors. Figure 9.36 shows the median lifetime of t1 and t2 as a function of current. The average sample temperature increase from the Joule heating at the beginning of the current stress test was estimated from the test line resistance as a function of current and temperature. The relationship between R and T within the 25 to 350°C temperature range was found to be linear, with a temperature coefficient of resistance (a) measurement of 0.0032Ω
°C. Using test line resistance as a function of current and the value of a, the average sample temperature increase due to joule heating was found to be less than 3°C when 7 mA was applied. No attempt at sample temperature correction from joule heating was made, although the temperature

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Figure 9.36 Plots of median lifetimes vs. log (current). The solid and dotted lines are the least-squares fits of the data below and above 2.5 mA, respectively.

rise at the W
via bottom contact area could be higher. Values for the current density exponent for a current density 30 mA
mm2 was found to be m
1.3 for t1 and 1.1 for t2, which reflected the possible contribution of joule heating to the contact resistance at the W/TaN/Ta/Cu via bottom interface.

9.10

Lifetime vs. Linewidth

In the Cu interconnection structure of Fig. 9.3(b), a W stud is connected to a Cu damascene line that has the top surface covered by silicon nitride and the bottom and sides covered with a TaN
Ta liner. If it is assumed that diffusion at the top Cu/silicon nitride interface dominates in Cu, the electromigration flux is constrained to the top interface in an area of dsw, where ds is the effective thickness of the interface region and w is the linewidth. Then the relative amount of flux, at constant

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line current density, j, flowing through the interface region is proportional to the interface area/line area ratio, dsw
(wh) or ds
h, where h is the line thickness. The lifetime, t, obtained from Eqs. (1) and (8) can be written as follows: t
∆Lcr h kT
(dsDsZ*s erj),

(18)

where ∆Lcr is the critical void length and Ds is the Cu/SiNx interface diffusivity. To verify Eq. (18), chips with 0.69-, 0.26-, and 0.24-mmwide M2 lines were tested using a Fig. 9.3(d) test structure. The electrons flowed from V2 to M2, and the thickness of the via bottom liner was 10 nm. After testing, the samples were examined by FIB or SEM on cross sections prepared by FIB. The FIB or SEM images revealed that the typical degradation mode of void formation was at the cathode end of the M2 under the bottom of the V2 liner. The void growth was due to Cu drift away from the M2
V2 interface. The observation suggested that the 10-nm-thick diffusion barrier liner at the V2
M2 interface was thick enough to block the Cu migration from M3 to V2 to M2 when the M3 current density was only j
1 mA
mm2. The typical value of s in the log-normal failure distribution was about 0.4. The various void locations and shapes near the cathode end of the M2 lines were found to be the main reason for the spread of lifetime in the lifetime distribution. In general, if a void only grew from the top surface of the M2 (grain thinning[86, 103]) under the V2 and caused the Cu to disconnect between Cu M2 and V2, a rapid line resistance change and a short electromigration lifetime would result, as seen in Fig. 9.37(a). However, if a void grew by edge displacement (vertical voiding) to deplete all the M2 Cu under V2 [see Fig. 9.37(b) and 9.27(a)], or the grain surface thinned at a distance away from the V2 liner, a long lifetime could be obtained. A vertical void can be created when Cu migrates along the top surface of the Cu line and is then fed by Cu atoms from the bottom of the Cu line. A large volume of mass in a vertical void has to be removed before a marked line resistance change can take place. Similar behavior has been reported for electromigration in Al interconnections.[120] Figure 9.38 shows the median lifetime, t50, as a function of linewidth at constant thickness at sample temperatures of 244 and 309°C. The lifetimes are nearly independent or increase slightly as linewidth decreases, which is contradictory to the Cu/liner interface diffusion model. The model predicts that the lifetime would drop by a factor of 1.9 in tests where linewidth is changed from 0.69 to 0.24 mm [see Eq. (5)]. The data shown in Fig. 9.38 are mostly consistent with the proposed model that Cu

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Figure 9.37 FIB images of electromigration damage samples tested for (a) 18 hours and (b) 42 hours.

atoms migrate along the top Cu
SiNx interface and the lifetime is proportional to thickness only as in Eq. (5). According to Eq. (18), the electromigration Cu lifetime in a bamboo-like grain structure line is proportional to the Cu migration rate, the (metal line thickness)1 dependence, and the via/line overlap section that defines the critical ∆Lcr. All values for the 0.24- to 0.69-mm-wide lines were tested in this study. A slight

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Figure 9.38 Mean lifetime vs. metal linewidth. The electron flow was from V2 to M2 using a Fig. 9.3(c) test structure.

improvement in lifetime for narrow lines is partially due to the possibility of an additional fast path from grain boundary diffusion in wide lines and partially due to the small V2 via diameter of 0.3 mm on the wide 0.69-mmwide M2.

9.11

Lifetime Scaling Rule

9.11.1 Single-Damascene Line In Cu damascene structures, the top surface is covered by silicon nitride, and the bottom and sides are covered with a Ta-based liner. Diffusion at the top Cu/silicon nitride interface dominates in Cu for bamboo microstructure lines. The lifetime, t, from Eq. (18) can then be simplified to: t
C ∆Lcrh
Ds,

(19)

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Figure 9.39 Plot of t50 vs. 1/T with various line areas for Cu.

where C is a constant. In the present case, the via overlap region, ∆Lcr, was made to be about equal to the via diameter or linewidth. Test results from 1.3- to 0.24-mm-wide lines, shown in Fig. 9.39, validate the model. In Fig. 9.40, the relationship between the lifetime ratio of t50(w) to t50 at the 1.3-mm-wide and 0.9-mm-thick line is plotted as a function of ∆Lcr  h. The t50 clearly decreases with the decrease in the area ∆Lcrh. The activation energy for electromigration Q for these electroplated Cu lines appears to be constant at about 1 eV. The dependencies on interface transport and line area are consistent with reports of electromigration failure mechanisms in Cu lines deposited by PVD and chemical vapor deposition techniques.[26]

9.11.2 Dual-Damascene Line Let us consider the case of dual-damascene Cu lines and Cu mass transport occurring primarily at the top-surface interface and not at grain boundaries nor the Cu/liner interface. If diffusion at the top Cu/silicon nitride interface is dominant in Cu damascene lines, the lifetime t for a single Cu damascene line with a blocking boundary is given by Eq. (18).

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Figure 9.40 Normalized lifetimes: t50
t50(1.3  0.9 mm2) as a function of line area estimated from data in Fig. 9.39.

From Sec. 9.11.1, the Cu electromigration lifetime for a single-damascene line with a fixed line current density, jLine, was shown to be proportional to the via width (∆Lcr) times the metal line thickness (h). When ∆Lcr and h are scaled down in each new generation, the prediction of Eq. (18) for Cu electromigration lifetime will be reduced. The trend can be reversed, if we could modify parameters in Eq. (18), such as the via size (∆Lcr), metal thickness, and, especially, interface diffusivity (Ds). For Cu dual-damascene lines connected to W underlying lines, multiple failure modes are shown in Sec. 9.7.2. This behavior was found to correlate to the variations of void location where voids are formed in the line and/or the bottom of the via. Void growth was also attributed to Cu atom diffusion along the fast diffusion path, the Cu
SiNx interface in these samples. The drift of Cu atoms along the Cu
SiNx interface at the cathode end of the line by an electron current generates excess vacancies and concomitantly a high tensile stress. The forces from the large vacancy concentration and stress gradients generate the vacancy flux. These excess vacancies easily migrate through the Cu bulk, because of their low migration energy, and flow to relieve the stress gradient to vacancy sinks, such

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as via bottoms, interfaces, grain boundaries, or the ends of the lines, where voids may be formed. The short-lifetime group, for example, is found to result from void growth at the via bottom. The initial Cu line/via stress generated from the thermal expansion mismatch between Cu and its surrounding materials and the grain orientation near or at the cathode end of the line/via may play an important role in determining the direction and accumulation sites of the vacancy wind. The anisotropic variation in mechanical properties for Cu may have a large effect. The value of Young’s modulus for copper in the 111 direction is 2.9 times higher than that in the 100 direction;[121] thus a mixture of 111 and 100 orientated grains at the cathode end of the line/via will generate higher local stress gradients as well as higher overall stress than would be generated by all 100 orientated grains. The early failure mode determines the electromigration lifetime of the IC chips. Thus it is important to know the relationships among via size, linewidth, and thickness in the scaling rule of dual-damascene Cu interconnections. The via bottom void results from the collection of vacancies produced by electromigration at the top Cu
SiNx interface, as shown in Fig. 9.41. This TEM image shows the via bottom void at a Cu via and W interface. Since the electromigration flux is constrained to the top interface within an area of dsw, the electromigration lifetime of the via bottom void, the first lifetime group, can be written as: tvia
(Avia ∆Lvia) kT
(dswDs jLineeZ*r)

(20a)

tvia
(h ∆Lvia) kT
(dsDs jviaeZ*r).

(20b)

or

Here Avia and ∆Lvia are the area of the bottom via and the critical void height that causes failure, respectively; jvia is the current density at the via bottom; and jLine is the current density in the line. Equations (20a) and (20b) show that the scaling of lifetime for via bottom voids in a dualdamascene line on a Cu blocking boundary is controlled by either jvia or Avia
w for a constant line current density. They show that lifetime decreases as the linewidth increases for constant jLine and constant Avia. Figure 9.42(a) and (b) shows the lifetime cumulative probability plot for dual-damascene 0.27- and 0.90-mm-wide Cu lines stressed at a sample temperature of 296°C and currents of 2 and 3.2 mA, respectively, using the test structure shown in Fig. 9.3(c). In this case, both lines are connected to a single Cu via, 0.32 mm in diameter, and a W underlying line. The cumulative percentage failure distribution is fitted to a bimodal function. The median lifetimes of the first-failure group (t1)

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Figure 9.41 Cross-section TEM image of the first failed sample showing the via bottom voids. The dotted line shows the possible direction of excess vacancy flow. The solid line arrow shows the direction of biased mass motion and electron flow.

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Figure 9.42 Cumulative percentage failure of 0.27- and 0.90-mm-wide dualdamascene Cu lines on W. Both lines are connected to a single via, 0.32 mm in diameter. The dual-damascence lines were tested with (a) 2 and (b) 3.2 mA at 296°C. Solid and dotted curves are the best fits of the data to a bimodal function. The parameters t1, s1 and t 2, s 2 are the median time to failure and the deviation for the first and second log-normal distribution function, respectively; a1 is the amplitude of the first log-normal function.

corresponding to void formation at the via bottom (see Fig. 9.43) are 63 and 44 hours with 2 mA, and 41 and 45 hours with 3.2 mA, for 0.27- and 0.9-mm-wide lines, respectively. Since Avia and wjLine are the same for both linewidths, and both linewidths have similar t1, Eq. (19a) is verified. The second lifetime group (t2) was caused by voids formed in both the line and via or in the line only. The lifetimes of 486 and 311 hours were obtained from the 0.9-mm-wide lines for 2 and 3.2 mA, respectively, and 179 and 111 hours from the 0.27-mm-wide lines for 2 and 3.2 mA, respectively. The three times longer median lifetime in the second lifetime group for the 0.90-mm-wide lines compared to the 0.27-mmwide lines is due to a lower line current density (3.3 times lower) in the wide lines, as in Eq. (18), and the slight variation in line shape above a single Cu via. Figure 9.43(a) and (b) shows the FIB images from the tested lines in the first lifetime group for 0.27- and 0.9-mm-wide lines, respectively, taken at an ion beam angle of 45 degrees. The samples were sectioned using FIB along the line in (a) and across the lines in (b), respectively. Similar void sizes and shapes were observed for both linewidths. Although the current density in a 0.9-mm-wide line is about one-third of the value of a 0.27-mm-wide line, the fluxes at the top surface and void size at the via bottom are about the same, which resulted in similar lifetimes, as predicted by Eq. (19).

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Figure 9.43 FIB images of early failure mode samples tested at 298°C. (a) Metal linewidth of 0.24 mm; the arrow shows the direction of electron flow from right to left. (b) Metal linewidth of 0.9 mm; the circles show the directions of electron flow into the paper.

To further validate the proposed model, electromigration in Cu dualdamascene lines on W in SiLK dielectric with various linewidths and via bars was investigated. Figure 9.44(a) and (b) shows no large variation in the values of t1, t2, s1, and a1 for metal linewidths from 0.13 to 0.73 mm, even with constant jLine, when the via bottom areas are roughly scaled with metal linewidths. These results are in agreement with the prediction of Eq. (19b).

Figure 9.44 Plots of the dependence of linewidth on (a) median lifetime of the first, t 1, and second, t 2, and (b) s1 and a1 fitting parameters from failure time bimodal distribution functions of samples stressed with a via bar structure. A constant line current density was applied.

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The bottom via areas estimated from FIB images are 0.13  0.12, 0.18  0.18, 0.25  0.20, and 0.65  0.23 mm2 for 0.13-, 0.18-, 0.28-, and 0.73-mm-wide lines, respectively. The variations in the values of t1 are partly due to the variations of the bottom via current densities, statistical error, and error generated when measuring the via bottom Cu areas. Also, in this sample set, error could have been generated by the lithography resolution when attempting to pattern sub-0.18-mm-wide lines. The current density jLine in the lines was 30 mA
mm2, while via bottom current densities jvia were estimated to be 77, 62, 60, and 50 mA
mm2 for 0.13-, 0.18-, 0.28-, and 0.73-mm-wide lines, respectively. The dotted lines are the mean values of t1 and t2 together with their uncertainties. Comparing the data obtained from Fig. 9.42 and Fig. 9.44(b), a slightly lower value of a1 in Cu
SilK (0.2) than in Cu
SiO2 (0.5) was obtained, which may be attributed to the different thermal expansion mismatch between Cu
SiLK and Cu
SiO2. Figure 9.45(a) is a FIB image of a tested sample in the first-failure group with a 0.13-mm-wide line. Figure 9.45(b) and (c) shows serial images of the t1 group from a 0.73-mmwide line taken 0.3 mm apart. In all three images, voids are observed at the bottom of the Cu via, although via bottom voids are reduced in the Cu
SilK dielectric compared to those in Cu
SiO2 structures. The electromigration lifetime of a dual-damascene line connected to a completely blocking boundary W line with various Cu via sizes and linewidths can be reasonably represented by a bimodal function. The dominant Cu mass motion is along the Cu/dielectric interface. The firstfailure group had void formation at the via bottom and was found to be a function of metal line thickness divided by the current density at the bottom of the via or bottom via area divided by linewidth. The results from stressing 0.13- to 0.90-mm-wide dual-damascene Cu lines are consistent with our proposed model and suggest the importance of the via size used in Cu interconnections, especially in wide lines.

9.12

Short-Length Effect

A two-level Cu interconnect structure, M1
V1
M2, was used for this study.[122] Figure 9.46(a) and (b) shows the schematic diagrams of the test structure. The thicknesses of M1 and M2 are 0.31 and 0.35 mm, respectively. M1 is 3 mm wide and 12 mm long. M2 consists of one 0.21-mmwide fine test line and two 3-mm-wide lines on either side of the test line. The fine M2 is 375 mm long. Each end of the 0.21-mm-wide M2 connects to one end of the M1 through a V1 via. The top and bottom diameters of the V1 via are 0.35 and 0.2 mm, respectively. The other end of the M1 is connected to the 3-mm-wide M2 using three 3-mm-long V1 slots. The widths of the top and bottom V1 slots are 0.46 and 0.2 mm, respectively. The Cu M2 was passivated with SiNx
SiO2. A dc current of 1.6 mA was

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Figure 9.45 FIB images of the failed samples tested at 298°C, taken at an ion beam angle of 45 degrees, for metal linewidths of (a) 0.13 mm; (b) and (c) 0.73 mm. Voids located at the bottom of vias are shown.

applied in the test lines, which resulted in current densities, j, in the 0.21-mmwide fine M2, 3-mm-wide M2 and M1 of 22, 2.3, and 2.6 mA
mm2, respectively. The samples were run at a temperature range from 253 to 365°C. All the electromigration tests were run in a vacuum furnace with

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Figure 9.46 Schematic diagrams of the test structure. (a) Top view; (b) crosssection view of one side of the test structure.

an atmosphere of 10 to 30 torr of forming gas (N2-5% H2). The average grain sizes were found to be roughly about 0.7 and 1.4 mm for the 0.21 and 3-mm-wide lines, respectively. The liner thicknesses at the M1
V1 via and M1
V1 slot interfaces were measured from TEM micrographs to be 8 and 19 nm, respectively. The variation in the liner thickness is due to the variation of the PVD step coverage in the various structures. If the liner at the via/line interface provided complete blocking boundaries for the Cu fluxes, the void growth rate would be the same as the drift velocity, since the Cu atoms would drift away from the liner at the contact interface (the cathode end of the line), creating a void. The Cu would then drift to the liner at the anode end of the line, generating an t extrusion. The lifetime t is obtained by ∆Lf
 nd dt, where t is the 0 amount of time required to grow a critical void size ∆Lcr necessary for ∆R
Ro
2%. In the case of partial blocking boundaries, voids and extrusions in metal lines occurred when there was an imbalance of Cu fluxes at a particular location, x, given by Eq. (10). Figure 9.47 presents data for cumulative percent failure vs. lifetime. It shows two distinguishable groups of failure times (short- and long-lifetime groups) that cannot be fit by a

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Figure 9.47 Plot of cumulative percentage failure vs. lifetime t. Solid lines are the least-squares fits to the bimodal functions.

single log-normal distribution. This is similar to the case of Cu
SiO2. A bimodal distribution function was used. The median lifetime of the shortlifetime group was about an order of magnitude less than that of the longlifetime group. The samples were examined in a FIB microscope to understand the correlation between void growth, open failure, and lifetime.

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Figure 9.48 FIB image of a failed sample from the short-lifetime group, showing an open failure at the M2/V1.

Figure 9.48 is a FIB image of the 0.21-mm-wide M2 line from the shortlifetime group; it shows a void at V1
M2 (open fail). The failure time of 64 hours at 298°C in the short-lifetime group is consistent with the time to grow a 0.3-mm-long void at the cathode end of the 0.21-mm-wide line with a blocking boundary as measured in Cu
SiO2 structures. However, in the case of the long-lifetime group, the voids were not observed at either end of the 0.21-mm-wide M2 in most of the samples. No visible voids at the V1 via
M1 interface are seen, as shown in Fig. 9.49(a), even though the line was stressed for 585 hours. However, a void in the 3-mm-wide M1 short line is shown in Fig. 9.49(b). Voids in the 3-mm-wide M2 lines are also often observed. For an electromigration drift velocity (Cu mass migration rate) at 298°C of 0.006 mm
h for the 0.21-mm-wide line, after a lifetime of 600 hours, we would expect to see a 3.5-mm-void in the fine M2 test line of Fig. 9.49(a) if the liner at the V1
M1 interface were a blocking boundary. In fact, no voids were observed near the M1
V1 via
M2 interfaces in most of the tested samples. This result suggests that the migration of Cu in the fine M2 line is replenished by the wide M1 line through the V1 via. However, voids were found in the 3-mm-wide M1 and M2, and some were observed at the end of the M1 section where there was little or no electrical field, as shown in Fig. 9.49(b). Similar behavior has been reported in Al
SiO2[104, 117, 118] and Cu
SiO2[40, 45] During electromigration, the Cu atoms will drift along the Cu
SiNx interface (the top Cu surface), and the depletion of atoms is responsible for creating the voids. Thus, voids would be expected to occur at the cathode ends of the fine M2 lines. However, a large portion of the tested samples showed no voids at the fine M2
V1 via interface in the long-lifetime group. The mass transport of Cu toward the anode along the top M2 surface generates excess

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Figure 9.49 FIB images of a line tested at T = 298°C for 585 hours. (a) M1/V1/M2 fine test line interface; (b) wide M2 and M1 lines.

vacancies and tensile stress near the cathode. The vacancy concentration and stress gradients generate a vacancy flux that moves to vacancy sinks such as interfaces, grain boundaries, or the ends of the M1 and M2 lines to form voids. Some vacancies pile up at the end of the M2 and some migrate through to M1, resulting in short and long lifetimes, respectively. This may be rationalized by the following discussion. If the liners at the V1
M1 interfaces were good blocking boundaries, mass accumulation at the anode end of the M1 lines should occur. This electromigration-induced mass accumulation would generate a stress gradient in a line that can cancel out the electromigration driving force, Z*erj
Ω(∆s
Lc), where Ω, ∆s, and Lc are the Cu atomic volume, electromigration-induced stress, and critical length, respectively. No damage should occur in a metal line length below Lc. The estimated value of Lc is 20 mm for an unpassivated Cu line with j at 2.6 mA
mm2 (jLc of 530 A
cm).[91] However, the observation of voids in 12-mm-long M1 lines with 2.6 mA
mm2 supports the notion that thin liners at M1
V1 interfaces

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used in the present test structure cannot withstand the incoming Cu electromigration flux and Cu migration becomes continuous through M1 to V1. Then the failure mechanisms in this structure are similar to that of testing a fine test line connecting two large reservoirs at both ends of the line; that is, continuous or partial flow will enable a very long lifetime. However, with the anode end of the line connected to a blocking boundary, such as a thicker liner or W via, high threshold values of jLc, 8000 A
cm,[77] 4000 A
cm,[123] 3000 A
cm,[124] and 7000 A
cm[125] for lines with electron flow to a blocking boundary have been obtained. Electromigration in two-level Cu dual-damascene lines in a lowdielectric-constant material (SiLK dielectric) has been discussed. The failure distribution cannot be fitted to a single log-normal distribution. At least two distinguishable lifetime groups were observed. The shortlifetime group is consistent with the time required to grow a line void at the end of the 0.21-mm-wide test line. The results are also in a good agreement with the lifetimes obtained from the Cu
SiO2 system. The samples with lifetimes in the long-lifetime group suggest that the migration of Cu in the M2 is replenished by Cu from the M1 through the V1
M1 interface. These observed results suggest that the fundamental Cu
SiLK dielectric electromigration failure modes are similar to Cu
SiO2 and that the observed bimodal distribution is not the result of using a low-dielectricconstant material. A thin liner at the Cu via
Cu line interface that cannot withstand the incoming Cu fluxes can also eliminate the short-length effect in a Cu line.

9.13

Reduced Cu Interface Diffusion

As discussed in Secs. 9.10 through 9.12, the electromigration Cu lifetime is most dependent on atomic transport at the Cu/dielectric interface. Some investigators have, however, reported that the Cu/liner interface is the dominant fast path. There may be significant differences in the integration processes and materials, leading to differences in the interfaces. However, in both cases, the fraction of total Cu atoms situated on interfaces increases as the dimension of the interconnection is scaled down. Accordingly, the Cu electromigration lifetime and the allowed current density specification in Cu conductors may be reduced in every new generation of interconnect. The ability to significantly extend Cu conductor technology to smaller dimensions can be realized by altering or modifying the Cu/dielectric interface to reduce the Cu transport properties; for example, 50-nm-thick CoP[126] or 100 to 200-nm-thick CoWP[127, 128] selectively deposited electroless films can serve as diffusion barrier layers. Recently, improved electromigration Cu lifetime was reported from

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Figure 9.50 TEM cross-section image of a Cu interconnect coated with CoWP using a Fig. 9.3(d) test structure.

testing Cu lines with a thin (10 to 20 nm) surface layer of electroless CoWP, CoSnP, or Pd.[129] Figure 9.50 shows a TEM image of a 10-nmthick CoWP film on top of a Cu M1 line. The amorphous a-SiCxHy capping layer and a low-dielectric-constant (k
2.6) SiLK dielectric are also shown. A smooth, uniform layer of CoWP and large Cu grains were observed. From this TEM sample, an x-ray energy dispersive spectroscopy (EDS) line scan was taken across the top Cu M1 surface. Figure 9.51 shows a plot of x-ray intensity as a function of element and probe position for the line scan. The scan started in the Cu line near the top surface, proceeded across the CoWP and amorphous SiCxHy layers, and ended in the SiLK dielectric. In Fig. 9.51, the boundaries of these layers are labeled. The EDS line scan shows that the Cu signal extends somewhat into the CoWP layer, suggesting that Cu is mixed into the cap. The Cu signal remains above background noise counts even in the amorphous SiCxHy and SiLK dielectric layers because the samples were mounted on a membrane on a Cu grid. Spurious X-rays hitting the Cu grid provided a background Cu signal. The CoWP grain size was measured

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Figure 9.51 Intensities of elements from EDS measurements. The electron probe moved from the top surface of a Cu damascene line, through the CoWP and amorphous SiCxH y coating layers, and ended in the SiLK dielectric. The vertical dotted lines indicate the boundaries between the layers.

to be 15 to 20 nm from a series of plan-view TEM images. The RBS analyses determined the composition of the CoWP metal cap to be Co(3% W, 6% P). Electromigration was performed on test structures of either single-damascene Cu lines in SiO2 on W or dual-damascene Cu lines in SiLK on W. The metal linewidths ranged from 0.1 to 2 mm. The current densities in the via bottom and line were 120 and 35 mA/mm2, respectively. Figure 9.52 shows typical dual-damascene Cu test line resistance as a function of time for 2-mm-wide polycrystalline lines tested with CoWP
a-SiCxHy and a-SiCxHy-only thin cap layers in SiLK dielectric at a sample temperature of 350°C. The data clearly show a remarkable improvement in lifetime for samples with CoWP coatings. Voids were observed in the via bottom and/or the line in dual-damascene Cu lines with an a-SiCxHy-only cap. However, only via bottom voids were observed in the case of a CoWP cap. Figure 9.53(b) shows a via bottom void for a Cu dual-damascene line with a CoWP cap.

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Figure 9.52 The resistance of a damascene Cu conductor, with and without a thin metal film on the top surface, versus time.

Figure 9.54 shows an Arrehenius plot of the median lifetimes of Cu damascene lines with various capping layers. For comparison, the data points from the samples of 0.27-mm-wide single-damascene lines capped with SiNx are also included in Fig. 9.54. The solid lines are the leastsquares fits. The value of electromigration activation energy for 2-mmwide dual-damascene CoWP capped polycrystalline lines was found to be 1.0 ± 0.1 eV. In the case of lines with bamboo-like grains, the activation energies of 0.90 ± 0.05 eV, 1.0 ± 0.1 eV, and 1.9 ± 0.2 eV were obtained

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Figure 9.53 FIB images of Cu lines electromigration stressed with a bottom via current density of 12  106 A/cm2 and at a sample temperature of 280°C. (a) 2.8 hours with SiCxHy; (b) 1100 hours with CoWP coating.

for a 0.18-mm-wide a-SiCxHy capped dual-damascene line, 0.27-mm-wide SiNx capped single-damascene line, and a 0.7-mm-wide CoWP capped dual-damascene line, respectively. Significant lifetime improvement for the samples with CoWP capping compared to SiNx or a-SiCxHy capping suggests that CoWP coating significantly reduces Cu diffusion along the

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Figure 9.54 Plot of median lifetime t50 vs. 1/T for Cu lines with various capping layers.

Cu/cap interface. The similar lifetimes obtained for 0.18-mm-wide bamboo-like and 2-mm-wide polycrystalline lines with a SiCxHy cap indicated that the dominant diffusion path in these lines is along the Cu
SiCxHy interface, not grain boundaries. However, in the case of CoWP capped samples, the measured activation energy of 1.0 eV for the 2-mmwide lines is in good agreement with the activation energy of Cu grain boundary diffusion.[81–83] Furthermore, the data in Fig. 9.54 show a drastic increase in Cu lifetime from the 2-mm-wide line to the 0.7-mm-wide bamboo-like line. These results suggest CoWP capping caused a great reduction in Cu interface diffusion and the Cu mass motion in CoWP capped 2-mm-wide polycrystalline lines was primarily influenced by Cu grain boundary diffusion. The 2-mm-wide CoWP line had an average grain size d of ∼1.2 mm, while the 0.28-mm-wide silicon nitride capped lines had a metal line thickness h of 0.25 mm. We would expect a factor of 5 reduction in void growth rate from d
h in Eq. (4) if the dominant migration path were from the Cu/silicon nitride interface for the 0.28-mmwide uncapped line and from grain boundaries for the 2-mm-wide CoWP line, even though similar activation energies were obtained in both cases.

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In addition, the existence of single-crystal grains capped with CoWP in the 2-mm-wide line would create a blocking path for grain boundary diffusion, which would further increase the lifetime for the samples with a CoWP cap. Therefore, we would expect more than the factor of 5 in the 2-mm-wide CoWP capped line samples as compared with the 0.28-mmwide Cu/SiNx capped bamboo line. The observed activation energy of 1.9 eV for the bamboo-like lines capped with CoWP is close to the value of 2.07 to 2.2 eV for Cu bulk diffusion.[72, 84] For these data, it is not known if the diffusion path in a bamboo-like grain line structure capped with CoWP is along the Cu
Ta, Cu
CoWP interfaces or in the bulk Cu. The existence of voids at the bottom of the via could suggest that diffusion in the Cu-Ta interface or in Cu bulk was dominant, but this voiding could have been caused by top-interface Cu diffusion and vacancy migration to the via bottom (vacancy sink).[11] The mechanism of reduction of the interface diffusion by substitution of Cu/metal for Cu
SiNx or Cu/amorphous SiCxHy interfaces is not totally understood, although it is tempting to speculate that the Cu migration is affected by the number of interfacial defects, the interface bond strength, and/or the surface migration energy of Cu atoms directly in contact with the cap material. Increased improvement in electromigration resistance is expected to translate to exceptional flexibility for the circuit designers, effectively removing electromigration as the limiting factor for use of high currents. The results of the tests further support the hypothesis that the uncoated surfaces or interfaces of Cu with the dielectric are the major sources of electromigration and thus reliability degradation. In summary, an investigation of Cu electromigration in Cu damascene interconnections with and without thin CoWP coatings on the top surface of the Cu line showed that electromigration failure lifetimes can be drastically improved. The diffusion of Cu at the top surface of a Cu damascene line was greatly reduced in the samples with CoWP cap so that the Cu electromigration lifetime was markedly improved. The activation energy for electromigration in Cu damascene lines capped with CoWP was found to be 1.9 and 1.0 eV for bamboo-like and polycrystalline grain structures, respectively, and 0.90 eV for SiCxHy cap-only samples.

9.14

Conclusion

This chapter has discussed electromigration in three-level and twolevel Cu single- and/or dual-damascene lines. The mass transport in Cu interconnects occurs mainly by interface diffusion. Fast diffusion along either the Cu/metal liner or Cu/dielectric interface has been reported. These results suggest that the fast diffusion paths in Cu interconnects

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are very sensitive to the nature of the interface, which is dependent on the selected fabrication materials and processes. The main observations follow: 1. Similar electromigration activation energy and void growth rate were found for the via bottom and the line voids. 2. Line void growth rate is related to 1/(metal line thickness). 3. Bimodal failure distributions were found for Cu dualdamascene lines on W. 4. Via bottom void growth rate is related to the via current density. 5. A drastic reduction of the void growth rate found for a thin metal (CoWP, CoSnP, Pd) layer capped on Cu line surfaces with a bulk-like activation energy for a bamboo-like Cu line suggests that the fast diffusion in Cu lines is along the Cu/ dielectric interface rather than the Cu/Ta interface. Primarily, two types of void growth in Cu lines are observed, grain thinning and edge displacement, even though mass transport is along the top surface of the line. The grain thinning phenomenon is void growth by thinning a single-crystal Cu grain from the top surface down, one grain at a time, thereby removing the Cu grains in layers. Edge displacement voiding pertains to a vertical void growth where Cu atoms from the bottom of the line/via edge feed Cu atoms drifted away from the top interface. Thus, the Cu lifetime distribution for via-to-line current flow will differ for the cases of via above and via below the Cu line. These differences can be seen particularly for the case of a via fully landed inside a line with electron flow from via to line and a blocking boundary at the bottom of the via. Grain thinning void growth can quickly separate the via from the line, which will then cause a sharp resistance change at early failure lifetimes. Edge-displacement voids will take more time to grow across the line under the via to cause failure. Therefore, in this case, edge-displacementvoid growth will have a longer lifetime than grain-thinning-void growth. On the other hand, for a Fig. 9.3(b) test structure, grain-thinning-void growth would have a longer lifetime if the grain in the line on top of the W via were larger than the W via diameter. The Cu lifetime distribution is further complicated by partial blocking boundaries at the line/via interface, such as the quality and thickness of the liner. Observations of very long electromigration lifetime have suggested that a thin liner at the via/line interface allowed Cu atoms to diffuse from one Cu line level to the other level, despite a Cu via bamboo microstructure. The fraction of the total Cu atoms present at interfaces increases as the dimensions of the Cu interconnections are scaled down, which suggests

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that the void growth rate will increase for future generation interconnects. In addition, the volume of the via and line will also be scaled down. Therefore, the combination of a faster void growth rate and a smaller void size required to cause failure will reduce chip lifetime in every new generation. The methods of enhancing Cu electromigration lifetime should focus on improving the Cu/dielectric interface, for example, by metal capping or by impurity segregation on the top of the Cu line surface. Drastic improvement in electromigration lifetimes for chips with metal caps or impurity-rich surface layers will give exceptional flexibility to circuit designers and may rule out electromigration as the limiting factor for use of high currents.

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10 Diffusion in Some Perovskites: HTSC Cuprates and a Piezoelectric Ceramic Devendra Gupta IBM T. J. Watson Research Center, Yorktown Heights, New York

10.1

Introduction

Understanding of diffusion processes in perovskites, represented by the high-temperature superconducting (HTSC) cuprates and a piezoelectric ceramic, of all the constituent elements as well as of some foreign atomic species, is important for scientific as well as technological reasons. Self-diffusion of various cations and anion species in these compounds is a basic material property; it has an impact on the superconducting properties of the former and the physical response of the latter to electrical, mechanical, and thermal fields. The interplay of diffusion with the microstructures ultimately controls the reliability of the devices in actual applications. One good example is the control of twin-density and grain refinements accomplished recently through cation doping, which in some cases alleviates the flux-pinning problem in the matrix.[1] There are many situations in which cation diffusion manifests itself in the fabrication of HTSC elements. In the bulk production of tapes and wires produced by the oxide-powder-in-tube (OPIT) method, for example, silver sheath is typically used and the composite is subjected to severe thermomechanical deformation.[2] In such a fabrication process, silver sheath may partially dissolve in the fabrication process and reach the oxide core by diffusion, thereby reducing the current-carrying capacity of the connectors. Similarly, contacts to the HTSC and piezoelectric thin-film microelectronic devices involve metallic electrodes that typically consist of AgPd alloys. A controlled amount of diffusion would promote adhesion and strength between the film and the substrate in the fabrication processes. However, large and uncontrolled diffusion may degrade the critical superconducting temperature (Tc), the critical current (Jc), and other physical characteristics, as has been shown in the substitution studies of several metals,[3] and may also lead to device degradation due to material reactions. Similarly, degradation of the dielectric constants and Curie temperature may be expected in the piezoelectric

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ceramics, which consist of highly reactive base metals such as Pb, Mg, Nb, Ni, Ti, Sr, and Zr. Diffusion of the anion (oxygen) species is also important. Ordering and stoichiometry of oxygen are directly related to the Tc and Jc through the vacancy structure in the basal planes.[4] In this chapter it will be shown that diffusion and stoichiometry of oxygen are themselves related to the Cu cation self-diffusion kinetics in HTSC. We have use x to denote deviation from the stoichiometry in the YBa2Cu3O7
x superconductor to avoid confusion of the usage of d for grain boundary width. This chapter is largely based on a recent review of diffusion in HTSC cuprates.[5] We first discuss the self-diffusion of all constituent cation atomic species in the polycrystalline YBa2Cu3O7
x (YBCO) compound and its epitaxial thin films grown on (100) SrTiO3 from the studies conducted in several laboratories.[6–9] In all studies, well-characterized YBCO specimens showing Tc in the high 88 to 90 K range have been used, and radioactive-tracer diffusion studies have typically been conducted under an oxygen pressure of 105 Pa (1 bar). From these investigations, a comprehensive picture of cation diffusion in YBCO has emerged. Initially, in Sec. 10.2.1, we discuss the general characteristics of the YBCO bulk and thin-film specimens. Because thin films are vital for actual applications, diffusion and interaction between the YBCO thin films and the substrate are also discussed. This is followed by a description of the atomic mechanisms of self-diffusion for the three kinds of constituent cations in YBCO, which show a marked variation due to their site preference. To a limited extent, we describe the anisotropy of diffusion in YBCO single crystals and the perturbations, which are caused by charge variation during diffusion of the impurity cations. A comprehensive discussion of anion (O) diffusion kinetics in a large number of HTSC cuprates is provided, as well as information on defect equilibria and the underlying defect mechanisms. While the anion diffusion kinetics seem to vary widely in the various HTSC cuprates, their overall behavior can be unified when stoichiometry and ordering of oxygen are taken into account. From the cation Cu self-diffusion data in the YBCO, it is possible to obtain lower and upper bounds for the diffusion kinetics for the anion (O) species as well in the large number of HTSC cuprates examined; the procedures to obtain them are described. Diffusion of several cations in the same piezoelectric group of perovskites, such as PbNbNiO3 (PNN), PbTiO3 (PT), PbZrO3 (PZ), PbMgNbO3 (PMN), and SrTiO3, and their solid solutions, are discussed. Diffusion in the lattice and grain boundaries has also been studied recently in some of these materials[10] and compared with the HSTC cuprates. Experimental techniques commonly used for diffusion measurements are described by Rothman[11] and are not included here.

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491

Cation Diffusion

10.2.1 Characteristics of YBCO Bulk and Thin-Film Specimens In general, diffusion in materials is very sensitive to their microstructure, as discussed in Chapter 1. Defects such as dislocations and grain boundaries enhance diffusion by many orders of magnitude. Oxides and ceramic materials are no exceptions in this respect. In fact, interconnected sintering porosity, commonly present in this class of sintered materials, becomes an additional source for acceleration of diffusion. Microstructural defects manifest themselves in several ways: They increase the depth of diffusion, lower the activation energy, and distort the diffusion profiles, causing nonlinearity in the log of the specific activity vs. the penetrationdistance-squared Gaussian plots. Generation of non-Gaussian profiles is the first indication that the short-circuiting paths in the microstructure

Figure 10.1 Polarized light micrograph of the bulk YBa2Cu3O7
x showing singlephase microstructure with more than 96% density in the area imaged. The heat treatment in O2 produced univariant twinning in each grain.[7]

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have become active. In this section, we show the role of microstructure on cation diffusion in polycrystalline bulk YBCO, the epitaxial thin films, which are free from grain boundaries but may still contain dislocations, and the truly single-crystal specimens. Figure 10.1 (pg. 491) shows the microstructure of the bulk polycrystalline YBCO specimens sintered at 960°C in 1 bar pressure of pure O2 where a density fraction greater than 95% was obtained with a stoichiometry of 6.9, that is, x  0.1. Large grains with plate shapes of 5 to 10 mm diameter are seen,[7] and there is no evidence of any porosity left due to any incomplete sintering. The susceptibility versus temperature behavior of these bulk specimens is shown in Fig. 10.2(a), where a Tc of 88 K is observed. Cation diffusion has also been measured in the epitaxial YBCO thin films grown on (100) SrTiO3 substrate by the ex situ metal oxidation technique.[12] In Fig. 10.2(b), superconducting transformation in such a film is shown at a Tc of 90 K.[6] Figure 10.3, taken from LeGoues,[13] shows a transmission electron microscopy (TEM) cross section of the epitaxial YBCO thin film deposited on (100) SrTiO3 substrate. No grain boundaries are observed in these films; however, the structure did contain misfit dislocations, which

Figure 10.2(a) Susceptibility (c) vs. temperature curves for a thin-plate-shaped YBa2Cu3O7
x specimen (demagnification factor  0). The zero-field-cooled (ZFC) curve shows close to 100% shielding, confirming the single-phase nature of the specimen, and the field-cooled (FC) curve shows about 30% Meissner effect.[7]

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Figure 10.2(b) Temperature vs. resistance curve for the as-prepared YBa2Cu3O7
x superconducting epitaxial film on (100) SrTiO3 substrate.[6]

were fully characterized. An additional variable to consider is the possible diffusion and reaction of the HTSC film with the substrate material, which is likely to affect its diffusion. All these issues are considered in Sec. 10.2.2.

10.2.2 Diffusion and Interactions Between YBCO Thin Films and Substrates As mentioned above, there is always a concern about the extent of dissolution of the substrate materials into the superconducting films during deposition, which involves physical vapor deposition of the constituent metals followed by oxidation at high temperatures or alternative techniques such as sputtering and laser ablation. All these processes may involve high-temperature annealing. This problem has received wide attention because, in general, degradation of Tc and the critical current Jc may be expected. We mention a few important findings here.

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Figure 10.3 Cross-sectional TEM lattice image of the interface between (100) SrTiO3 substrate and YBa2Cu3O7
x epitaxial thin film. Note the atomic registry at the interface.[13]

Interdiffusion and interfacial reactions between the sputter-deposited YBa2Cu3O7–x thin films on MgO, sapphire, quartz, and Si substrates has been studied by Nakajima et al.[14] using Rutherford backscattering spectroscopy (RBS) at 876 to 1226 K. All the constituent metals of YBCO were found to diffuse and react with the substrates, although at variable rates. Cu was reported to diffuse into the substrates at the fastest rate, and the MgO substrate was found to be the most stable thermally. Madakson et al.[15] studied diffusion in La1.8Sr0.2CuO4 and YBa2Cu3O7 superconducting thin films deposited by dual-ion-beam sputtering on Nd-YAP, MgO, SrF2, Si, CaF2, ZrO2-9%Y2O3, BaF2, Al2O3, and SrTiO3 substrates. The films were characterized by RBS, resistance measurements, TEM, x-ray diffraction (XRD), and secondary ion mass spectroscopy (SIMS) techniques. Significant substrate/thin-film interactions were observed in all cases, accompanied by diffusion of the metal species from the substrate into the films and, in some cases, diffusion of the thin-film components into the substrate. The most interesting observation was in the YBa2Cu3O7–x films deposited on the SrTiO3 substrate, onto which epitaxial films can be readily grown.[12] Both Sr and Ti migrated into the film; in addition, Ba from the film replaced Sr in the substrate. However, the

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epitaxial character and the critical superconducting temperature (Tc) were not significantly affected during the replacement process until about 40% Sr was incorporated into the film. Migration of Sr atoms from the (100) SrTiO3 substrate to YBa2Cu3O7–x superconducting epitaxial films was studied by Gupta et al.[16] as a function of temperature by using the radioactive 85Sr marker at the interface. Initially, 85Sr was diffused into the (100) SrTiO3 substrates in the 700 to 900°C range. Diffusion of 85Sr into (100) SrTiO3 substrate is described by 1.5  10–7exp(
212 kJ/mol/kT) m2/sec (see Table 10.1). Four YBa2Cu3O7
x films of 0.25 mm thickness were grown in the 700 to 950°C range by metal evaporation and ex situ O2 annealing for 8 min. onto the (100) SrTiO3 substrates, in which 85Sr was previously diffused at 700°C for Table 10.1. Cation Diffusion in YBa2Cu3O7
x, Copper Metal, and PMN-PT Piezoelectric Ceramic

No. Host

Tracer (Technique)

Stoichiometry* Do (cm2/s) or (x) D(T) 0.1 0.1

Do  4.0 Do  1.3

255 260

0.1

Do  1.0

241

P [7, 8] Epitaxial film [6] P [7]

280 247

C [8] C [8] C [8] P [8] P [8]

1 2

YBa2Cu3O7
x YBa2Cu3O7
x

67

3 4

YBa2Cu3O7
x YBa2Cu3O7
x T  700°C T  600°C T  600°C YBa2Cu3O7
x YBa2Cu3O7
x YBa2Cu3O7
x T  798°C T  850°C T  900°C YBa2Cu3O7
x YBa2Cu3O7
x YBa2Cu3O7
x Cu selfdiffusion PMN-PT SrTiO3

63

5 6 7

8 9 10 11 12 13

Cu, 64Cu Ni

63

Ni

Ni (SIMS)

0.1

Co (SIMS) 0.1 Zn Ni (SIMS) 0.1

65

Zn

110

Ag Ba (SIMS) 88 Y (SIMS) 67 Cu 133

110

Ag Sr

85

0.1 0.1 0.1 0.1 Metal

D(T)  1.9  10
15 D(T)  1.2  10
16 D(T)  1.4  10
16 Do  14 Do  2.0 D(T)  2.7  10
12 D(T)  8.5  10
12 D(T)  3.5  10
11 Do  10

Q Comment**/ (kJ/mol) Reference

P [8]

Do  0.78

227 890 1000 211

P [9] P [9] P [9] C [20]

Do  0.0034 Do  0.0015

277 212

C [10] C [16]

The stoichiometry of 6.9 with x  0.1 was obtained by diffusion annealing under oxygen pressure of 105 Pa. ** P: polycrystalline; C: c axis or AB plane; Epitaxial film: epitaxial film grown on (100) SrTiO3 substrate *

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128 min. The superconducting temperatures (Tc) were also measured at each growth temperature. The spread of the 85Sr tracer and Sr from the substrate to the films was measured by sputtering with a neutral Ar atom ion-beam generated in a Kaufman source at 500 eV and current density of 2 mA/cm2. The sputtered-off material was quantitatively collected. The details of the techniques can be found elsewhere.[11, 17, 18] The 85Sr tracer profiles in the epitaxial YBa2Cu3O7–x films and the substrate are shown in Fig. 10.4, which follows their growth at 700, 800, 900, and 950°C. Temperature-versus-resistance curves for the various films are shown in Fig. 10.5(a) and (b). In the film grown at 700°C, no superconducting behavior was observed [Fig. 10.5(a)]. The films grown in the 800 to

Figure 10.4 85Sr radiotracer profiles in the epitaxial YBa2Cu3O7
x films and the (100) SrTiO3 substrate interface at various O2 annealing and oxidation conditions. Note the progressive migration of the 85Sr radiotracer from the substrate into the films with increasing temperatures.[16]

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Figure 10.5 Temperature vs. resistance curves for the YBa2Cu3O7
x thin films deposited on (100) SrTiO3 substrate. Prior to deposition, 85Sr radiotracer was used at the interface as a marker. The ex situ O2 annealing was conducted for 8 min. in the 700 to 950°C range. (a) At 700°C, the film did not develop superconducting properties. At (b) 800°C, (c) 900°C, and (d) 950°C, the films became superconducting. Note that the O2 annealing near 900°C produces the highest Tc accompanied by the least resistance ratio during the transition.[16]

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950°C range showed superconducting transitions [Fig. 10.5(b)]. The best results were obtained in the film grown at 900°C, which showed the least electrical resistance, sharp transition, and the highest Tc of 88 K. The Tc in the epitaxial films is actually almost the same as observed in the bulk polycrystalline YBCO specimens mentioned in Sec. 10.2.1. The accumulation of the Sr is seen to peak at about half the film thickness at 900°C following an 8-minute annealing period. The diffusion distance traced at the peak is about the same as may be estimated by a diffusion coefficient of 4  10–17 m2/sec within a factor of 2. A larger spread may be due to the Sr migration in the metal evaporation phase itself. The exchange of Ba with Sr in the YBCO epitaxial films in small concentration is believed to be not only benign but beneficial as well because the superconducting transition is narrower and the film resistance lower. Furthermore, the films were found to have better shelf lives against degradation by atmospheric moisture.

10.2.3 Self-Diffusion of the Constituent Cations (Y, Ba, and Cu) of YBCO In Table 10.1, the activation energy Q and the pre-exponential factor Do are listed for constituent cations (Y, Ba, and Cu) species from the studies of Gupta et al.,[7] Routbort et al.,[8] and Chen et al.[9] Diffusion profiles for 67Cu tracer in bulk YBCO are shown in Fig. 10.6. Diffusion coefficients were computed from the linear segments of the profiles according to the Gaussian solution [see Chapter 1, Eq. (10)]. Two 67Cu penetration profiles show excellent linear (Gaussian) behavior at 710 and 730°C. However, at lower temperatures, they are significantly curved due to grain boundary contributions in these polycrystalline bulk specimens, which are discussed in Sec. 10.4. Diffusion coefficients for 67Cu at 618 and 585°C could be obtained from the near-surface data points after extrapolating and deducting grain boundary contributions. Figure 10.7 shows the Arrhenius dependence for Cu, Ba, and Y tracer diffusion. The Cu diffusion data from the two investigations[7, 8] are in excellent agreement. Among the three constituent cations in YBCO, the magnitude of diffusion could be different due to their locations in the lattice. Cu is an important species, however, since it is the most abundant and it is the principal current carrier in the superconducting state. It also determines many physical properties such as the critical temperature (Tc) and current (Jc). As mentioned earlier, it is a key for understanding diffusion processes in YBCO and is discussed in detail in Sec. 10.2.3.1.

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Figure 10.6 67Cu radiotracer diffusion penetration profiles in bulk polycrystalline YBa2Cu3O7
x specimens.[6]

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Figure 10.7 Arrhenius plots Diffusion of 67Cu, Ba, and Y tracers in bulk polycrystalline YBa2Cu3O7
x specimens in 1 bar O2 pressure.[9]

10.2.3.1 Self-Diffusion of Cu in YBCO A high value of activation energy for Cu cation self-diffusion of 255 kJ/mol seen in Table 10.1 in the YBCO with x  0.1 is indicative of the absence of contributions from the oxygen vacancies present in the basal plane. We will examine the sites in the YBCO lattice, which are likely to be involved in the Cu cation diffusion process. In Fig. 10.8, the various sites in the YBCO lattice are shown from the neutron diffraction studies reported by Jorgenson et al.[19] It was

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Figure 10.8 Structure of YBa2Cu3O7
x in the orthorhombic phase. Possible diffusion-jump directions of the 110 and 301 types for Cu and homovalentcation impurities replacing Cu are shown by dashed lines; the other crystallographic multiples are not shown. After Jorgensen et al.[19]

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observed that long-range Cu diffusion is a difficult process in both the Cu (1)-O (1) chains and the Cu (2)-O (2) planes. If a certain amount of thermal vacancies are assumed to exist on the Cu sites, similar to those commonly observed in the metal and transition-metal oxides,[20, 21] Cu diffusion is possible without the mediation of oxygen vacancies or excursions on the oxygen sublattice. Cu atom migration could then be thought to take place in all combinations of atomic jumps of the types 110, 301, and 031, as shown in Fig. 10.8. For example, Cu atoms would diffuse along the Cu (1)-Cu (1) diagonal in the basal plane, along Cu (2)-Cu (2) diagonals in the Cu-O planes and (001) planes, and so forth. Note that communication between the chains and planes is permissible only by the diagonal jumps between Cu (1)-Cu (2) sites. The alternative Cu (2)-Cu (2) jumps along 001 are somewhat shorter, but their contribution to diffusion may not be substantial. Since there would be fewer thermal vacancies in the Cu sites than in the oxygen O (1) sites, and they would be accompanied by longer jumps, cation diffusion may be expected to be very slow. Therefore, both Do and Q would be larger for cation diffusion compared to anion diffusion in the YBCO lattice and they may resemble the corresponding values in closely packed lattices of pure metals, such as Cu.[22] The analysis can be extended to show that cation diffusion in the YBCO lattice resembles self-diffusion in the Cu lattice. From the lattice parameters of the neutron diffraction studies,[19] an average length of
5.5Å for the Cu (1)-Cu (1), Cu (1)-Cu (2), and the Cu (2)-Cu (2) diagonal jumps (l) may be estimated. To the first approximation, the pre-exponential term Do (cm2/sec) may be written as:

Do (cm2/sec)
l2 n e (∆S/k ),

(1)

where n is the lattice frequency
7  1012/sec, ∆S is the entropy for diffusion, and k is the Boltzmann constant. Hence the measured value of Do (Table 10.1) for Cu in YBCO leads to the entropy ∆S
5k. The corresponding value for self-diffusion in Cu is ∆S
3k.[22] Considering the dissimilarities of the lattice and the nature of the nearest neighbors, entropy as well as the enthalpy for Cu cation diffusion in the YBCO have values similar to those for Cu self-diffusion in the Cu metal. Note that the orthorhombic to tetragonal transformation has no effect on cation diffusion kinetics. The Arrhenius plot, shown in Fig. 10.7, remains straight through the transformation that occurs at 700°C. This is consistent with the fact that the YBCO cell volume is preserved during this second-order transformation.[19] The cation diffusion processes are, however, expected to be anisotropic, but the available data are sparse. Considering the diffusion

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mechanism proposed above for Cu, we do not expect any significant anisotropy for Cu diffusion, although it is not ruled out for the Ba and Y species.

10.2.3.2 Self-Diffusion of Ba and Y in YBCO As Fig. 10.8 shows, both Ba and Y atoms are deeply embedded in the YBCO lattice, and their atomic movement is expected to be even more difficult than Cu. Diffusion of Ba and Y atoms would require excursions to the Cu sites, which would disturb the thermodynamic equilibrium of the occupancy of sites. In addition, the chemical environments for Ba and Y cations are different than that of Cu. Ba resides in Ba-O insulating layers, while Y is surrounded by Cu (2)-O layers, which have partially free electrons. Thus the state of defects, the energetics, and the diffusion paths for Ba and Y may be expected to be vastly different. A long-range diffusive motion may in fact involve cooperative atomic jumps between the Cu and Ba or Y sites. The activation energies for Ba and Y diffusion are expected to be much larger than that for Cu, and diffusion is likely to be highly anisotropic. Table 10.1 shows that the activation energies measured by Chen et al.[9] for Ba and Y species are indeed very large, of the order of 900 to 1000 kJ/mol, and diffusion of Ba shows an anisotropy of at least 1000. These activation energies are similar to those observed for steadystate creep in transition metal oxides.

10.2.4 Diffusion of Cation Impurities in YBCO Bulk and thin-film YBCO specimens described in Sec. 10.2.1 were used to study diffusion of the 63Ni cation[6, 7] under an oxygen pressure of 105 Pa. The corresponding diffusion profiles, plots of the log-specific activity versus the square of the cumulative penetration distance, are shown in Fig. 10.9(a) and (b). All 63Ni diffusion profiles displayed in Fig. 10.9, in bulk as well as the epitaxial thin-film specimens, showed initially high data points and subsequently became linear. Unlike 67Cu tracer (see Fig. 10.6), which should not have had any problems of solubility in the host YBCO specimens, the 63Ni tracer may have exceeded the solubility limit because 63 Ni has a half-life of 85 years and a large number of Ni atoms are required to obtain detectable radioactivity. Therefore, we relied on the deeper linear (Gaussian) segments of the 63Ni profiles to extract the diffusion coefficients. Thus, 63Ni diffusion coefficients have been obtained by fitting the Gaussian solution of the Fick’s law for an instantaneous

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Figure 10.9(a) 63Ni radiotracer diffusion penetration profiles in bulk polycrystalline YBa2Cu3O7
x specimens.[7]

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Figure 10.9(b) 63Ni radiotracer diffusion profiles in YBa2Cu3O7
x epitaxial thin films on (100) SrTiO3 substrate. Note the scale factor (SF) for the abscissa marked on each profile.[6]

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source and semi-infinite boundary conditions to the linear segments in the profiles shown in Fig. 10.9(a) and (b). This section compares and contrasts the impurity cation (63Ni) diffusion parameters obtained in polycrystalline bulk and epitaxial films with similar data in single crystals obtained by Routbort et al.[8] Then we examine diffusion of Co, Cu, Ag, and Zn impurity cations in the polycrystalline YBa2Cu3O7– d specimens listed in Table 10.1. We first discuss the Ni tracer diffusion, which behaves more or less similarly to the Cu tracer and is known to substitute for Cu in the YBCO lattice.

10.2.4.1 Ni Tracer Cation Diffusion in YBa2Cu3O7
d In Fig. 10.10, the Ni diffusion data are plotted for epitaxial,[6] polycrystalline,[7] and single-crystal specimens[8] to show the effect of microstructure and crystal anisotropy. As Table 10.1 shows, the activation

Figure 10.10 Arrhenius plots of Ni diffusion in YBa2Cu3O7
x polycrystalline bulk,[7] epitaxial thin films on (100) SrTiO3 substrate,[6] and single-crystal specimens.[8]

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energy for Ni diffusion in the YBCO specimens (epitaxial thin-film, polycrystalline, or single-crystal specimens) is very similar to that for the Cu tracer between 240 to 260 kJ/mol. This is understandable since both occupy the same sites and have similar electrical charge. Figure 10.10 shows that the 63Ni radioactive tracer data in the (100) epitaxial films agree reasonably well with the secondary ion mass spectroscopy (SIMS) diffusion data of Ni in the c direction of single crystals of YBa2Cu3O7
x. The principal difference is in Do for the 63Ni data in polycrystalline bulk specimens due to contributions of grain boundaries present in the specimens (see Fig. 10.1). The small difference between the Ni diffusion coefficients in the epitaxial films and the single crystals seen in Fig. 10.10 may be attributed to the presence of a higher density of dislocations in the former rather than anisotropy of the diffusion coefficients, because both were diffused in the c direction.

10.2.4.2 Cation Impurity Diffusion and Effect of Charge Imbalance In Fig. 10.11, impurity cation diffusion for Zn, Ni, Ag, and Co[6–8] is compared with the Cu self-diffusion data.[7, 8] Note that charge may be an important factor in determining the sites occupied by impurity cations; consequently, the diffusion kinetics will be affected. It is well recognized that Zn and Ni in low concentrations, of the order of 0.05%, substitute Cu in Cu (1) sites,[23] but in higher concentrations (≈0.30) they occupy Cu (1) and Cu (2) sites randomly.[24] Ag also replaces Cu on Cu (1) and Cu (2) sites,[25] while Co occupies exclusively the Cu (1) sites.[26] For radiotracer studies that involve very small concentrations, we may consider that Zn, Ni, and Co all go into Cu (1) sites. Because of their close proximity to oxygen-ion vacancies, diffusion along Cu (1) sites is much faster than along Cu (2) sites. Furthermore, the valent states of (a) Cu, Zn, (b) Co, Ni, and (c) Ag are 2, 2 or 3, and Ag 1, respectively. As Fig. 10.11 shows, that diffusion of Zn in bulk YBCO specimens falls on the Cu selfdiffusion Arrhenius plot. This is not surprising since both Cu and Zn have 2 states and Zn occupies Cu (1) sites. Ni diffusion in the epitaxial films also come very close to the Cu self-diffusion line, indicating that it is likely to diffuse in the Cu (1) sites in the Ni2 state. Ag, on the other hand, diffuses substantially faster than Cu or Zn. Routbort et al.[8] have advanced an explanation to account for this difference: Ag1, being negatively charged with respect to the Cu2 ion it replaces, may attract oxygenion vacancies for charge compensation, thereby lowering the activation energy for motion at the saddle point. As a converse corollary, if Co

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Figure 10.11 Comparison of diffusion of cation impurities (Ag, Ni, Co, and Zn) with Cu self-diffusion in YBa2Cu3O7
x. (Table 10.1 gives the source of the data.) The arrow indicates the orthorhombic-to-tetragonal transformation.[8]

occurred in the 3 state, it should repel oxygen-ion vacancies, increase the saddle-point energy, and slow down its diffusion. This is, in fact, observed, as shown in Fig. 10.11. It is therefore seen that diffusion of the cation impurities, which lie close to Cu in the Periodic Table, is very similar to Cu self-diffusion, with some perturbations caused by their charge dissimilarities, if any.

10.2.4.3 Cation Impurity Diffusion in Piezoelectric Perovskites Besides cation diffusion in YBCO, similar data are also available in several other perovskites, notably the single crystals of the PbMnNbO3PbTiO3 (PMN-PT) system.[10] Figure 10.12 compares Ag radiotracer diffusion in the PMN-PT piezoelectric ceramic and the YBCO. It also shows the Cu self-diffusion in the YBCO. Ag diffusion in the PMN-PT single crystals compares well with the self-diffusion of Cu in the HTSC YBCO.

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Figure 10.12 Arrhenius plots of diffusion of 110Ag in PMN-PT single crystals and polycrystalline superconducting YBa2Cu3O7
x. 67Cu self-diffusion in the latter is also shown for comparison.[10] (See Table 10.1)

The Ag diffusion data in the polycrystalline YBCO, however, are significantly higher due to the contribution of the grain boundaries. Nevertheless, the diffusion process in two perovskites, YBCO and PMN-PT, appears to be similar and to involve primarily thermal vacancies on Cu or the Pb sites, respectively.

10.3

Anion Diffusion in Several HTSC Cuprates

Soon after the discovery of the HTSC cuprates, it was realized that they were essentially nonstoichiometric compounds and that the deviation from the stoichiometry (x) is controlled largely by the anion (oxygen) defect equilibria. The electronic defects, namely the electrons and the holes, are also related to the defect equilibria. In fact, oxygen stoichiometry turned out to be a critical parameter in determining the superconducting transition temperature, Tc , and the critical current, Jc.[27] Diffusion of anions is also expected to be closely related to the concentration and

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mobility of defects and hence to the stoichiometry of the compounds. Therefore, it is not surprising that the early investigators noted large changes in the physical properties of the compounds upon heating and cooling, notably color, electrical resistance, and specific gravity. Some of these effects became bases of measurements of fast in-and-out diffusion of oxygen in these compounds, as discussed later in this section. Fast diffusion kinetics of oxygen in a large number of superconducting cuprates and their defect equilibria have been comprehensively reviewed by Routbort and Rothman.[28] We will discuss here only the salient features of oxygen diffusion in the HTSC cuprates and the manner in which they may be related to self-diffusion of the Cu cation, in YBCO in particular. Bakker et al.[29] provide details of the defect equilibria in YBCO in their excellent paper.

10.3.1 Oxygen Diffusion Data in HTSC Cuprates In Table 10.2, we have compiled critical data on oxygen diffusion in a number of HTSC cuprates measured by the SIMS technique using 18O stable isotope. The data are also displayed in Fig. 10.13. The scatter among data from material to material is rather large, varying 6 to 10 orders of magnitude. The anion diffusion coefficients are also many orders of magnitude larger than those for cation species (compare data in Tables 10.1 and 10.2), although in some cases diffusion kinetics do come close to each other (for example, in YBa2Cu4O7
d and Bi2Sr2Cu2Ox compounds). For rigorous comparison of diffusion data in dissimilar compounds, a homologous temperature scale should be used; however, this is not possible because these compounds decompose and sublime before melting takes place. Figure 10.8 clearly shows that diffusion of oxygen would be quite complex because it occupies five different kinds of sites due to differing environments, state of occupancy, and, in particular, ordering of oxygen atoms and vacant sites in the basal planes. Departure from stoichiometry, as measured by x, is the controlling parameter, but it is not a fixed quantity. It increases at higher temperatures and decreasing partial pressure of O2. The deviation of the stoichiometry decreases as the occupancy in O (1) and O (4) sites is reduced and an increase in O (5) sites takes place. The reductions or increments of occupancy may be neither proportional nor compensatory. The magnitude of x and its thermodynamic kinetics vary from material to material. Diffusion of oxygen may also be expected to be anisotropic by several orders of magnitudes.[30] Hence these numerous variables result in a very large scatter in the diffusion data for oxygen, as seen in Fig. 10.13. However, meaningful inferences have been drawn from

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Table 10.2. Oxygen Diffusion in Some HTSC Cuprates

No.

Host

Technique

Stoichiometry (x)

Do (cm2s)

Q Comment*/ (kJ/mol) Reference

1.6  10
6 0.55  10
6 0.2  10
6 5.9  10
6 12 1.36  10
4

80.0 62.6 77.0 126.2 234.1 93.5

1 2 3 4 5 6

La2
xSrxCuO4 La2
xSrxCuO4 La2
xSrxCuO4 La2
xSrxCuO4 La2
xSrxCuO4 YBa2Cu3O7
x

SIMS(18O)

0 0.10 0.10 0.10 0.24 0

S [33] P [33] P [34] C [33] P [33] P [31]

7

YBa2Cu3O7
x

Resistance

0.7

3.5  10
2

125.3

P [32]

8

YBa2Cu3O7
x

Calculation

0.1

0.04

125.3

P [35]

0 0 8 8 6

0.08 75 1.7  10
5 0.6 0.06

200.6 296 89.3 212.3 203.6

P [36] C [36] P [37] C [37] AB plane [38]

9 10 11 12 13

YBa2Cu4O7
x YBa2Cu4O7
x Bi2Sr2Ca1Cu2Ox Bi2Sr2Ca1Cu2Ox Bi2Sr2Cu1Ox

14

YBa2Cu3O7
x

Radiotracer 67 Cu

0.1

4.0

255.3

P [6, 7] Cation

15

YBa2Cu3O7
x

Molecular dynamics

0.09

1.4  10
4

94.4

Theory [39]

SIMS (18O)

*

P: polycrystalline; C: c axis or AB plane; S: single crystal; NA: not available

these data; these have been summarized by Routbort and Rothman for YBCO,[28] as follows: 1. The measured oxygen tracer diffusion coefficients (D) are relatively insensitive to the oxygen partial pressure (PO2). 2. The Arrhenius plot at the fixed partial pressure of 105 Pa is straight over a large temperature range and remains unbroken through orthorhombic-tetragonal transformation, similar to the Cu cation discussed in Sec. 10.2. 3. In single crystals, Db W Da at small values of x obtained at 300°C and PO2  105 Pa, but Db ≈ Da at higher values of x obtained at 600°C and PO2  2  104 Pa, where the subscripts a through c refer to a and b directions in the AB plane and c is the direction normal to it (see Fig. 10.8). 4. Dc V Dab at low temperatures, and both Q and Do are much larger in the c direction, indicating that oxygen diffusion is occurring by individual lattice vacancies far away from the basal planes. 5. In polycrystalline material, Dpoly ≈ Dab.

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Figure 10.13 Anion (O) diffusion in some HTSC cuprates with upper and lower limits shown by heavy lines. The upper limit is computed from the estimated activation energy for vacancy motion only in the YBa2Cu3O7
x lattice. The lower limit is provided by the Cu cation self-diffusion in nearly stoichiometric HTSC compounds. The numbers in parentheses refer to Table 10.2, which gives the sources of the data.

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Many of these effects can be discerned from the data listed in Table 10.2 and displayed in Fig. 10.13.

10.3.2 Comparison Between Cation and Anion Diffusion The basic premises of the oxygen diffusion mechanism have been laid down by Routbort and Rothman.[28] In view of the multiplicity of sites, fast diffusion of oxygen occurs only in the basal planes composed of Cu (1)-O (1) chains and largely unoccupied O (5) sites. Routbort and Rothman have suggested an oxygen diffusion mechanism in the AB plane of YBa2Cu3O7
x in which oxygen ions break loose from the chain ends, diffuse parallel to the chains, and pop into a vacant chain-end site. In this mechanism, some atomic vacancy interchange is responsible for diffusion in the orthogonal and tetragonal phases, and no dependence on PO2 is expected. Furthermore, due to the preponderance of vacancies in O (5) sites, only an activation energy for migration will be required. Diffusion in other oxygen sites, O (2), O (3), and O (4), is many orders of magnitudes slower since no chains form and only isolated oxygen vacancy interchanges take place. Indeed, the activation energy for anion diffusion lies in the range of 100 to 125 kJ/mol, according to the studies of Rothman et al.[31] and Tu et al.[32] It is possible to estimate the migration energy for the atomic-jump mechanism suggested by Routbort and Rothman[28] for oxygen diffusion in the basal planes of YBa2Cu3O7
x from the somewhat empirical approach given below. In most closely packed metals, the activation energy for vacancy motion is about one-half that for self-diffusion.[22] If this rule is applied a priori to YBa2Cu3O7
x, the activation energy for vacancy motion should be one-half of the activation energy for Cu diffusion (255 kJ/mol), that is, ≈125 kJ/mol. As mentioned in Sec. 10.3.1, there is a large vacancy concentration next to the oxygen atoms in the basal plane, hence the formation of vacancies is not required for diffusion to occur. This is indeed the case, as shown in Table 10.2, items 6 and 7. The pre-exponential factor for anion (O) diffusion in YBa2Cu3O7
x phase can also be estimated using the same scaling factor for the entropy of vacancies. It should contain only the entropy of motion of vacancies as one-half that for Cu self-diffusion [Eq. (1)] and equal to ≈2.5 k. In addition, the atomic-jump length between the O (1) sites and the ordered vacancies on O (5) sites is about one-half that for the Cu jumps (l), as shown in Fig. 10.8, and ≈2.25 Å. For these two reasons, the pre-exponential factor for anion diffusion should be smaller by a factor of 0.01 compared with that for Cu cation diffusion. Thus, a value for Do ≈ 0.04 cm2/sec may be estimated for anion diffusion.

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This value of Do, together with the activation energy of 125 kJ/mol, results in anion diffusion coefficients that agree within an order of magnitude with the measurements of Rothman et al.[31] and Tu et al.,[32] as shown in Fig. 10.13. Figure 10.13 and Table 10.2 also show anion diffusion data in several other HTSC cuprates that display large variation of the deviation from stoichiometry x.[33–38] In the passing it may be mentioned that there is also good agreement with the results of a molecular-dynamics study of oxygen diffusion in YBa2Cu3O7
x by Zhang and Catlow.[39] Furthermore, this estimate provides an upper limit for oxygen diffusion coefficients in nonstoichiometric HTSC compounds in general since they all belong to the same perovskite family with the same crystal space group. The lower limit for oxygen diffusion should be the Cu cation diffusion itself, in which vacancy formation energy and entropy are included and little deviation from stoichiometry is implied. Both these limits are displayed in Fig. 10.13 and are seen to hold reasonably well.

10.4

Grain Boundary Diffusion and Solute Segregation Effects in Perovskites

Grain boundaries are important in all engineering materials. They assume even greater importance in the high-temperature superconductors and in perovskites, in general, because we must rely on polycrystalline materials for actual applications. In the YBCO, grain boundaries have been recognized as weak links, and they are deficient in electron carriers, thus impairing both Tc and Jc. In some situations, the superconducting weak links behave like Josephson junctions and have been beneficially exploited to fabricate the superconducting quantum intereference devices, the dc-SQUIDs,[40] and the Josephson field-effect transistor, JoFET,[41] operating at temperatures as high as 110 K. However, for power transmission, which relies on high Tc and Jc, grain boundary weak links are detrimental because they restrict the current-carrying capacity. An obvious solution is to introduce carriers in the grain boundaries by solid-state grain boundary diffusion. This approach has indeed been successful, as demonstrated by Hammerl et al.,[42] by replacing the Y3 ions by Ca2 in YBCO, thereby enhancing Jc by a factor of 6 at a Tc of 77 K in high-angle grain boundaries (24 degrees). Doping by Ag has also been reported to enhance Jc.[43] An important approach to achieve highly localized doping is through the use of hetero-structures, as discussed by Hilgenkamp and Mannhart.[44] The dopant is diffused through an overlying hetero-film, which can be removed after diffusion by techniques such as ion-etching or chemical-mechanical polishing, leaving behind the highly localized dopant only in the grain boundaries of the superconducting film.

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However, understanding of grain boundary diffusion and the associated solute segregation effects in HTSC and perovskites has been lacking, despite their technological importance. Just two measurements have been reported in the literature. The first consists of chemical diffusion measurements of Ca in the grain boundaries of the superconducting YBCO using SIMS.[45] The second measurement was on the 110 Ag radiotracer in HTSC and PZT,[46] using sintered specimens with a rather low density fraction (66%). The tracer penetration profiles were analyzed by error function instead of the more precise grain boundary diffusion analyses available from Suzuoka[47] or Whipple.[48] Thus both investigations, while potentially useful in actual applications, provide little insight into the basic grain boundary diffusion process itself. Grain boundary diffusion measurements of the constituent atomic species of the YBCO, the majority Cu species, for example, should be very valuable in this regard. Some Cu self-diffusion measurements in the polycrystalline YBCO were reported earlier;[7] the corresponding penetration profiles are shown in Fig. 10.6. The figure shows two profiles at 618 and 585°C to be nonlinear in the penetration distance squared vs. the log of the specific activity plots, indicating grain boundary contributions. These profiles are re-examined here to compute the grain boundary diffusion coefficients according to the Suzuoka-Whipple analysis. This discussion divides the grain boundary diffusion in HTSC and a perovskite into three groups: 1. Diffusion of nonsegregating cations: the Ca, Ni and Cu selfdiffusion in the YBa2Cu3O7
x 2. Diffusion of the segregating cations, primarily the Ag tracer in YBa2Cu3O7
x and PNN-PT-PZ 3. The grain boundary diffusion of the anion oxygen species available in the literature. The combined grain boundary diffusion coefficients, sdDb, thus extracted are listed in Table 10.3, where s is the solute segregation, d the width, and Db the diffusion coefficient for grain boundaries.

10.4.1 Grain Boundary Self-Diffusion of Nonsegregating Cations in the YBa2Cu3O7
x Superconductor In Fig. 10.14(a), two 67Cu tracer diffusion profiles at 618 and 585°C from the earlier investigation[7] are replotted according to the asymptotic

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Table 10.3. Cation Grain Boundary Diffusion in YBa2Cu3O7
x Superconductor and PNN-PT-PZ

Diffusant/ Sample Type

Temp (°C)

sdDb (T) (m3/s)

D
(T) (D
m2/s)

sdDbo (m3/s)

Qb Comment/ (kJ/mol) Reference

1.

Cu/YBCO Bulk

618 585

5.51  10
25 4.24  10
19 4.67  10
17 135 2.73  10
25 1.13  10
19

95% density [7]

2.

63

Ni/YBCO Film/Al2O3

700 618

1.49  10
24 1.36  10
18 1.54  10
16 149 2.74  10
25 7.00  10
20

[7]

3.

40 Ca/YBCO Film/(100)SrTiO3

870 798

4.20  10
24 6.20  10
18 8.90  10
15 113 3.10  10
24 2.70  10
18

SIMS [45]

4.

110

Ag/YBCO

800 600 500

2.90  10
19 8.50  10
15 2.20  10
21 2.48  10
17 4.00  10
10 188 8.00  10
23 4.30  10
19

80% density [46]

5.

110

Ag/PNN-PT-PZ 575–1038

3.70  10
9

6.

18

O/YBCO

5.50  10
12 135

67

400–700

168

99% density [10] SIMS [52]

solution given by Suzuoka[47] and Whipple:[48]



∂ln(c–) sdDb  0.661  ∂y65

 4Dt
53

l 12

,

(2)

where the combined grain boundary parameter sdDb has been defined above, y is the penetration distance, c– is the grain boundary concentration in the diffused specimen at y, D
is the diffusion coefficient in the adjoining lattice, and t is the time for diffusion. The last 10 to 12 points agree well with Eq. (2). Figure 10.14(b) shows two unpublished grain boundary diffusion profiles of the 63Ni tracer in the polycrystalline YBCO films grown on sapphire (Al2O3).[49] Figure 10.15 shows the Arrhenius behavior of the combined grain boundary diffusion coefficients, sdDb, obtained from the profiles shown in Fig. 10.14(a) and (b) and those obtained by Berenov et al.[45] Table 10.3 shows that the activation energies Qb for grain boundary diffusion of the Cu and Ni tracers are within the limits of 40(±20)% discussed in Chapter 1. Similarly, the pre-exponential factors, sdDbo, have the magnitude of ∼10
15(m3/sec) within a factor of 10 of what is commonly observed in pure metal grain boundaries (see Chapter 1, Table 1.4). Thus, these two tracers appear to have nonsegregating characteristics

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Figure 10.14(a) The curved segments of the 67Cu diffusion profiles at 585 and 618oC YBa2Cu3O7
x shown in Fig. 10.6 are re-evaluated for grain boundary diffusion.

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Figure 10.14(b) 63Ni radiotracer diffusion profiles at 618 and 700oC in polycrystalline YBa2Cu3O7
x films grown on sapphire, showing their grain boundary characteristics.

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Figure 10.15 Arrhenius plots of the combined grain boundary diffusion coefficients, sdDb, obtained from the profiles shown in Figs. 10.14(a) and (b) for 67Cu and 63Ni radiotracers and the 40Ca stable isotope. References are shown in the parenthe-

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with a value of s ∼1. Note that Ni diffusion in the YBCO films grown on sapphire (Al2O3) is significantly lower than the self-diffusion of Cu. Similar films grown on sapphire have been found to contain large amounts of Al migrating from the substrate to the superconducting film during the deposition process.[15] The presence of Al may be responsible for the observation of lower grain boundary diffusion coefficients, yet display diffusion parameters within the limits described above for a nonsegregating cation. However, a similar conclusion cannot be drawn with certainty about the Ca diffusion in the YBCO grain boundaries because of the chemical grain boundary diffusion process involved. The observation of significantly lower activation energy appears to be related to the Darken’s factor (see Chapter 1), which generally results in faster diffusion.

10.4.2 Grain Boundary Diffusion of a Segregating Cation in the YBa2Cu3O7
x Superconductor and a Piezoelectric Ceramic As mentioned in Sec. 10.4, there are diffusion measurements of 110Ag tracer in the YBCO[46] that may be related to grain boundary diffusion after re-evaluation of the penetration profiles. Accordingly, we have re-evaluated the profiles in the penetration distance to the power of 65th and computed the combined grain boundary diffusion parameters using Eq. (2). The needed lattice diffusion coefficients for 110Ag tracer, computed from the data of Chen et al.,[9] are listed in Table 10.1. The combined diffusion parameters are listed in Table 10.3 and displayed in Fig. 10.16. The 110Ag grain boundary diffusion data of Lewis et al.[10] obtained in the PNN-PT-PZ piezoelectric ceramic are also shown. A good agreement is seen among the various grain boundary diffusion data in these two perovskites. Although the density fraction of the sintered HTSC specimens used by Kulikov et al.[46] was only 66%, the grain boundary diffusion process appears to be dominant and the porosity was, perhaps, disconnected and isolated. Note that the grain boundary diffusion coefficients shown in Fig. 10.16 for the segregating Ag cation are much larger than those shown in Fig. 10.15 for the nonsegregating cations. Both sdDbo and Qb are affected by Ag segregation at the grain boundaries, resulting in s W 1 and enhanced activation energy Qb. The Ag segregation effect has been discussed in detail by Lewis et al.,[10] and it should be valid for the segregating cation grain boundary diffusion in perovskites in general.

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Figure 10.16 Arrhenius plots of the combined grain boundary coefficients, sdDb, for 110Ag tracer in PNN-PT-PZ piezoelectric ceramic and superconducting YBa2Cu3O7
x. References are shown in the brackets.

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10.4.3 Grain Boundary Diffusion of Oxygen in the YBa2Cu3O7
x Superconductor As mentioned in Sec. 10.3, oxygen (18O) diffusion in YBCO lattice is much faster than cation diffusion due to a preponderance of vacancies in the basal plane; hence only vacancy migration energy is required. Hence enhanced grain boundary diffusion of oxygen is generally not expected. However, enhanced grain boundary diffusion of oxygen in polycrystalline YBCO by 3 to 4 orders of magnitude has been reported.[50–52] Extensive SIMS studies of oxygen diffusion and surface exchange kinetics in single-crystal and polycrystalline YBa2Cu3O7
x superconductor have been carried out by Claus et al.,[52] who also discuss the results of earlier investigations.[50–51] Figure 10.17 shows the results of their grain boundary diffusion of oxygen in polycrystalline YBa2Cu3O7–x and compares them with those in its lattice. For easy comparison, d  1 nm has been taken out from the measured combined grain boundary diffusion parameters, sdDb, obtained by Claus et al. The oxygen diffusion data in the lattice and the grain boundaries are listed in Tables 10.2 and 10.3, respectively. Enhancement of grain boundary diffusion of oxygen is clearly seen in Fig. 10.17, accompanied by larger values of sdDbo and Qb of 4  10
12 m3/s and 135 kJ/mol, respectively. Figure 10.16 compares grain boundary diffusion of 18O by Claus et al.[52] with the 110Ag tracer in YBCO and PNN-PT-PZ perovskites. Oxygen grain boundary diffusion data in the YBCO show a similar trend, implying a segregation effect similar to that discussed for Ag cation grain boundary diffusion in YBCO and PNN-PT-PZ. Larger values of sdDbo and Qb for 18O diffusion may be attributed to s W1 and its accompanying temperature dependence, respectively. However, unlike Ag, oxygen is not an extrinsic impurity. It is indeed an integral part of the superconducting YBCO. Thus a combined pre-exponential factor sdDbo of the order of 10
15 m3/s and Qb
100 kJ/mol would have been expected. The oxygen segregation effect may actually stem from its interaction with the likely presence of second-phase domains and amorphous material at the grain boundaries, as proposed by Sabras et al.[50, 51] The presence of second-phase particles of BaCO3, CuO, and carbides has been confirmed by AES and SIMS studies by Claus et al.[52] The YBCO is known to be prone to uptake of CO2 and moisture during its exposure to the ambient and in the handling process, which could account for its presence in the grain boundaries. In any event, faster-than-lattice grain boundary diffusion of oxygen in polycrystalline YBCO has been observed.

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Figure 10.17 Comparison of 18O diffusion in the YBa2Cu3O7
x lattice and grain boundaries. For an adequate comparison, d  1 nm is removed from the measured combined coefficients sdDb shown in Fig. 10.16. References are shown in the brackets.

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Summary

A large body of data now exists in the literature on diffusion of cations and anions in a variety of high-temperature superconducting cuprates and some perovskites, which are reviewed and critically discussed in this chapter. Diffusion data are available on the YBa2Cu3O7
x superconductor of the constituent cations, Cu, Ba, and Y, and also of the impurity Ni, Co, Ag, and Zn cations. Self-diffusion of Cu takes place largely on the Cu sites through thermal vacancies with an activation energy of 255 kJ/mol. Self-diffusion of Ba and Y atoms also takes place on their respective sites and is characterized by much larger activation energies, in the 900 to 1000 kJ/mol range, because they are deeply embedded in the lattice. The large values of activation energies for Ba and Y reflect the difficulty of the diffusion process due to long atomic-jump distances and the lack of vacancies in which jumps can occur. The Ni, Co, Ag, and Zn impurity cations have diffusion parameters similar to the Cu self-diffusion in YBa 2Cu3O7
x because they all replace Cu. Some effect of ion charge is seen in the case of heterovalent cations. Extensive diffusion data on the anion species (O) also exist in about a dozen high-temperature superconducting cuprates, which are compiled and compared. The oxygen diffusion data show scatter of many orders of magnitudes. This is attributed to several factors, notably, the deviation from stoichiometry and the atomic arrangements in the various Cu-O planes, which differ from material to material. These factors, in fact, determine the upper and lower limits of the diffusion data. A comparison is also made between the diffusion kinetics of the cation and anion species. A semi-empirical relationship between cation and anion diffusion processes is shown to hold through the activation energy and entropy of motion for vacancies responsible for diffusion of the various species. Effects of diffusion and reaction between the YBa 2Cu3O7
x films and the substrates are discussed. Finally, grain boundary diffusion of several cations and the oxygen anion is briefly reviewed.

Acknowledgments The author is indebted to Drs. S. J. Rothman and J. L. Routbort for fruitful discussions and exchange of views on the subject matter discussed in this chapter over a period of a decade. They have pioneered much of the data and the concepts discussed, and permission to use them here is gratefully acknowledged. Much of the material presented in this chapter is reproduced from an earlier article by the author published in Metals, Materials and Processes, 11:233–246 (1999), with permission of Meshap Science Publishers, Mumbai, India.

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References 1. S. L. Shinde and T. M. Shaw, in Superconductivity and Ceramic Superconductors (K. M. Nair and E. A. Giess, eds.), Ceramics Trans., 3:579 (1990) 2. K. H. Sandhage, G. N. Riley, Jr., and W. L. Carter, J. Metals, 43(3):21 (1991) 3. G. Xiao, M. Z. Cieplak, A. Gavrin, F. H. Streitz, A. Bakhshai, and C. L. Chien, Phys. Rev. Lett., 60:1446 (1988) 4. D. Jorgensen, H. Shaked, D. G. Hinks, B. Dabrowski, B. N. Veal, A. P. Paulikas, L. J. Nowicki, G. W. Crabtree, W. L. Kwok, L. H. Nunez, and H. Claus, Phys. C, 153–155:578 (1988) 5. D. Gupta, Metals Mater. Proc., 11:233 (1999) 6. D. Gupta, R. B. Laibowitz, and J. A. Lacey, Phys. Rev. Lett., 64:2675 (1990) 7. D. Gupta, S. L. Shinde, and R. B. Laibowitz, in High Temperature Superconducting Compounds, vol. II (S. H. Wang, A. Dasgupta, and R. Laibowitz, eds.), Minerals, Metals and Materials Society, Warrendale, PA (1990), p. 377 8. J. L. Routbort, S. J. Rothman, N. Chen, and J. N. Mundy, Phys. Rev., B43:5489 (1991) 9. N. Chen, S. J. Rothman, and J. L. Routbort, J. Mater. Res. 7:1 (1992) 10. D. J. Lewis, D. Gupta, M. R. Notis, and Y. Imanaka, J. Am. Ceram Soc., 84(8):1777 (2001) 11. S. J. Rothman, in Diffusion in Crystalline Solids (G. E. Murch and A. S. Nowick, eds.), Academic Press (1984) 12. P. Chaudhari, R. H. Koch, R. B. Laibowitz, T. R. McGuire, and R. J. Gambino, Phys. Rev. Lett., 58:2684 (1987) 13. F. LeGoues, Philos. Mag., 57:167 (1988) 14. H. Nakajima, S. Yamaguchi, K. Iwasaki, H. Morita, and H. Fujimori, Appl. Phys. Lett., 1437 (1988) 15. P. Madakson, J. J. Cuomo, D. S. Yee, R. A. Roy, and G. Scilla, J. Appl. Phys., 63:2046 (1988) 16. D. Gupta, J. A. Lacey, and R. B. Laibowitz, Defect Diffusion Forum, 75:79 (1991) 17. F. Wenwer, A. Gude, G. Rummel, M. Eggermann, T. Zumskley, N. A. Stolwijk, and H. Mehrer, Meas. Sci. Technol., 7:632 (1996) 18. D. Gupta, Thin Solid Films, 25:231 (1975) 19. J. D. Jorgenson, M. A. Beno, D. G. Honks, L. Soderham, K. L. Volin, R. L. Hitterman, J. D. Grace, I. K. Schuller, C. U. Segre, K. Zhang, and M. S. Kleefisch, Phys. Rev., B36:3608 (1987) 20. S. J. Rothman and N. L. Peterson, Phys. Status Solidi, 35:305 (1969) 21. N. L. Peterson, Mater. Sci. Forum, 1:85 (1984) 22. P. Shewmon, Diffusion in Solids, 2nd ed., Minerals, Metals and Materials Society, Warrendale, PA (1989), p. 79 23. H. Shaked, J. Faber, Jr., B. W. Veal, R. L. Hitterman, and P. Paulikas, Solid State Comm., 75:445 (1990)

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24. R. S. Howland, T. H. Giballe, S. S. Laderman, A. F. Corbie, M. Scott, J. M. Tarascon, and P. Barboux, Phys. Rev., B39:9017 (1989) 25. Y. H. Kao, Y. D. Yao, L. Y. Jang, F. Xu, A. Krol, L. W. Song, C. J. Sher, A. Darowsky, J. C. Phillips, J. J. Simmins, and R. L. Snyder, J. Appl. Phys., 67:353 (1990) 26. R. Beyers and T. M. Shaw, Solid State Phys., 42:135 (1989) 27. A. L. Robinson, Science, 236:1063 (1987) 28. J. L. Routbort and S. J. Rothman, J. Appl. Phys., 76:5615 (1994) 29. H. Bakker, D. M. R. LoCasio, J. P. A. Westerveld, A. T. Gomperets, and M. T. van Wees, Defect Diffusion Forum, 75:19 (1991) 30. J. L. Routbort, N. Chen, K. C. Goretta, and S. J. Rothman, High Temperature Superconductors: Materials Aspects (H. C. Freyhardt, R. Flükiger, and M. Peuckert, eds.), Deutche Gesellschaftfür Materialkunde (1991), pp. 569–580 31. S. J. Rothman, J. L. Routbort, and J. E. Baker, Phys. Rev., B40:8852 (1989) 32. K. N. Tu, N. C. Yeh, S. I. Park, and C. C. Tsuei, Phys. Rev., B39:304 (1989) 33. E. J. Opila, H. L. Tuller, B. J. Wuensch, and J. J. Mauer, J. Amer. Ceram. Soc., 76:2363 (1993) 34. J. L. Routbort, S. J. Rothman, B. K. Flandermeyer, L. J. Mowicki, and J. E. Baker, J. Mater. Soc., 3:16 (1988) 35. D. Gupta, in Proc. DIMAT-92 Symposium (Kyoto, Japan) (M. Koiwa, ed.), Defect Diffusion Forum, 95–98:1111 (1993) 36. J. L. Routbort, S. J. Rothman, J. N. Mundy, J. E. Baker, B. Dabrawski, and R. K. Williams, Phys. Rev., B48:7505 (1993) 37. M. Runde, J. L. Routbort, S. J. Rothman, K. C. Coretta, J. N. Mundy, X. Xu, and J. E. Baker, Phys. Rev., B45:7375 (1992) 38. M. Runde, J. L. Routbort, J. N. Mundy, S. J. Rothman, C. L. Wiley, and X. Xu, Phys. Rev., B46:3142 (1992) 39. X. Zhang and C. R. A. Catlow, Phys. Rev., B46:457 (1992) 40. P. Chaudhari, J. Mannart, D. Dimos, C. C. Tsui, J. Chi, M. Oprysko, and M. Scheuermann, Phys. Rev. Lett., 60:1653(1988) 41. P. Chaudhari, C. A. Mueller, and H. P. Wolf, U.S. Patent No. 5401714, March 28, 1995 42. G. Hammerl, A. Schmehl, R. R. Schultz, B. Goetz, H. Bichefeldt, C. W. Schneider, H. Hilgenkamp, and J. Mannart, Nature, 407:162 (2000) 43. C. H. Cheng and Y. Zhao, J. Appl. Phys., 93:2292 (2003) 44. H. Hilgenkamp and J. Mannhart, Rev. Mod. Phys., 74:485 (2002) 45. A. V. Berenov, R. Marriot, S. R. Foltyn, and J. L. MacManus-Driscoll, IEEE Trans. Appl. Superconductivity, 11:1051 (2001) 46. G. S. Kulikov, R. S. Malkovich, E. A. Skoryatina, V. P. Usacheva, T. A. Shaplytina, S. F. Gafarov, and T. D. Dzhafarov, Ferroelectrics, 144:61 (1993) 47. T. Suzuoka, Jpn Inst. Metall., 2:25 (1961) 48. R. T. P. Whipple, Philos. Mag., 45:1225 (1954) 49. D. Gupta, “Grain boundary diffusion in YBa2Cu3O7
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50. J. Sabras, G. Perudeau, R. Berjoan, and C. Monty, J. Less-Common Met., 164/165:239 (1990) 51. J. Sabra, C. Dolin, J. Ayasche, C. Monty, R. Maury, and A. Fert, Colloq. Phys. C1, suppl. No.1, tome 51, 1035 (1990) 52. J. Claus, G. Borchardt, S. Weber, and S. Scherrer Z. Phys. Chem., 206:49 (1998)

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Index A accelerated dynamics: 162, 164–165 Arrhenius dependence: lattice diffusion 12 grain boundary diffusion 32 solute segregation 52 atomic vibrations: collective mechanisms 138, 140 entropy of 132 free energy of 122 frequency 123 harmonic approximation 122–125 local harmonic approximation 124–125,127 quasiharmonic approximation 128 B back stress 386 Boltzmann–Matano Analysis 6 Borisov’s conjecture 48 bulk Properties 71 bulk modulus 87 compressibility 95 vacancy formation energy 70–73 melting point 70–71 volume thermal expansion coefficient 94–96 C cap layer 379 chemical interactions 252, 330, 335–336, 353 condensation coefficient 6 cohesive energy 71,88 Coble creep 379 constrained diffusional creep 394 cracking of films 399 creep 379, 393–396 critical nucleus 330, 352 critical shear stress 383 D damascence process: single 461 dual 435, 462 Debye temperature 87 decagonal quasicrystals 57–60, 68 delamination of films 399

density–of–state effect 321 diffusion: activation volume 17, 47, 79, 94 alloys 149 amorphous metallic alloys 60–63 anisotropy in grain boundary 44 atomistic simulation 113 bulk metallic glasses 63 coefficients in metals (Table1.1) 13 computer modeling 113 correlation factor (f) 11–12, 14–17 enthalpy 69 epitaxial thin films 37 free energy 69 frequency 97 Gaussian solution 4, 34, 498, 503 grain boundary 21–26, 145, 155 impurity 14, 87 Instantaneous source condition 4 interdiffusion coefficient 6–8 interstitials 9,47 isotope effect (DE) 18–20 kinetic factors (DK) 19 linear chemical 20 mass dependence 19 mechanism 8, 44, 145,155,157 modulated thin films 26 nonlinear chemical 23–25 polycrystalline bulk 32–34 polycrystalline thin films 37 pressure dependence 17 single crystals 8 thick–layer Geometry 5 under chemical gradient 6 diffusion barriers: alloying of Cu 249, 269 APDB 235–239, 241–245, 271 borides 243, 260 carbides 243, 260 controlled microstructure 248 characteristics 240 electric field 259 free energy 258 heat of formation 258 high melting 242–243 ILD 240, 262, 275 impurity segregation 249–250 in Al technology 242, 248–250, 260, 272 in Cu technology 240, 248, 262 interdiffusion 246–247

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530

low-k 275 multilayer barriers 242, 261–262 nitrides 243, 260 phase diagram 250, 256 SAM 270 solid solubility 250–252 temperature effect 259 thin–film reactions 255 zero–Flux planes 272 dislocation in thin films: core spreading 391–392 interfacial 383 parallel glide 396 pileup 381 thermally activated glide 385 channeling/theading 382

INDEX

reservoir effect 406 short line length 468 stress gradient 418 surface diffusion 423 test structure 409 texture 411 void growth 419, 422, 454, 463 electron microscopy 49, 332, 352, 404, 416, 486 electron–to–atom ratio 91 embedded–atom method 114 embedded–cluster method 125–126, 152 embedded–defect method 115 enthalpy of formation 279 entropy of mixing effects 306 F

E elastic anisotropy in films 367–368 elastic constants 82 elastic strain energy 54 electric field 243, 253, 265–266, 270 electromigration in copper films: activation energy 425, 426 alloying effect 428 ambient effect 427 atomic flux 417 backflow 418 bamboo grain structure 412, 422,460 blocking boundaries 419 470 bulk diffusion 405 current density dependence 453 diffusion barriers 406 dual-damascene 474 drift velocity 414–415, 417 effective charge 415, 417, 426 effective diffusivity 416 electron wind 415 failure physics 406 fast diffusion paths 406 422 grain boundary diffusion 405, 415–416 grain thinning 459 interface diffusion 405, 474 Joule heating 453, 458 life time scaling rules 461 life time statistical distribution 433, 440–450 linewidth dependence 456 log-normal life time 411, 433 metal liner 405 microstructure 411, 412, 422, 460 multiple log-normal 419, 422, 454, 456, 463

fatigue 399 Fermi energy 91 Fick’s first law 2 Fick's second law 3,339, 347–348 Fick’s law, difficulties 286 first–principles calculations 118 film passivation 379 film/substrate interface 382–383 fluoropolymers 349, 353 Frenkel defect 12 G germanides: Co 320 Ti 316 grain boundary: diffusion 21–26, 28, energy 49–50, Grüneisen relation 99–103 H Harrison's kinetic regime 28–29 hyperdynamics 162–163 high temperature super conductors: anion (o) diffusion 503 anion-grain boundary diffusion 516 anion- grain boundary segregation 516–517 Ba self-diffusion 494 cation/anion diffusion comparison 507 Cu self-diffusion 492 film/substrate interaction 487 grain boundary diffusion-anion 505 grain boundary diffusion-cation 509–515

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grain boundary doping 509–510 grain boundary segregation 514 impurity cation diffusion 497 impurity cation diffusion-charge effect 501 lattice structure 595 Ni diffusion in single crystals 501 Oxygen diffusion parameters 505 Y self-diffusion 492 I icosahedral solids 57–60 immobilization 328–329, 341, 345, 347–351, 353 ions 241, 336–337, 351–352, 507–508 interconnection integration 407 intermetallic compounds: Al-Co 285 NiAl: diffusion mechanism 182–184, 210, 213 grain boundary diffusion 229 interdiffusion 210 self-diffusion 205 solute diffusion 212 state of defects 180 Ni3Al: diffusion mechanism 184, 189–193,195 grain boundary diffusion 225 interdiffusion 193–195 self-diffusion 189 solute diffusion 195 TiAl: diffusion mechanism 187–188 grain boundary diffusion 229 interdiffusion 203–204 self-diffusion 199 solute diffusion 204–205 Fe-Al: diffusion mechanism 216–218 grain boundary diffusion 232 interdiffusion 219 self-diffusion 213–218 solute diffusion 212 state of defects 180 M melting : entropy of fusion 75 latent heat 71–73 melting point 75 metallic glasses 60–63

531

micro–tensile testing 380 minimum energy path 139 modified embedded atom method 118 molecular dynamics: 153 accelerated dynamics 162 hyperdynamics 162 parallel replica dynamics 163 temperature–accelerated dynamics 164 Monte Carlo method: 143 kinetic 143 on–the–fly 150–151 Monte Carlo simulations 113–114, 118, 143, 150–151, 190, N Nernst–Einstein equation 75, 285, 415 polymer-metal nucleation 336 nudged elastic band method 141 O ordered Cu3Au rule 47, 294, 301 P parallel glide 389–391 passivation 379, 404, 412 phase growth: amorphous phases 315 Au-Cu 309 Co-Au 297 Cu6Sn5 300 first phase formed 294 formation energies 297, 301 grain boundary diffusion in 293, 307 impurity effect 307 lattice diffusion in 293 linear-parabolic kinetics 290 metastable phases Ni-Al 304 metastable phases NiSi 309 nucleation controlled NiSi2 301 nucleation controlled TiSi2 315 parabolic 285 reaction constants 292–293 sequential 292, 301 single 284 two 287 photoelectron spectroscopy 337, 342 piezoelectric ceramics: cation-ion diffusion 502 grain boundary diffusion 509 grain boundary segregation 41, 52, 54, 514 point defects: 9–12,47

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collective jumps 154, 158 composition–conserving complex 131 constitutional 131 delocalized 146 equilibrium concentration 129, 249 formation energy 115, 120–121 formation entropy 102, 122–126, formation volume 137 jump rate 138 migration energy 141, 507, 516 unstable 146, 154 polycrystalline films 37, 382 polymers: activation energy surface diffusion 339–340 adsorption energy 339–340 BPA-PC (Ag, Au) 350 chemical interaction 341 Cr 342, 352 Cs 342, 352 electron microscopy 344 FLARETM 355 HOSP (Cu, Al, Ta, Pt) 341, 357 K 342 Kapton® (O2, CO2, H2O) 349 low k 333 Monte Carlo simulation 346, 353 oxygen 348, 349 PMDA-ODA(Ag, Al, Au, Cu) 337, 348, 358 polyimide 337, 348 PTCDA 343, 348 SiLK® (Cu) 337, 348 Teflon® AF 337, 355 Ti 337, 342 TMC® (Ag, Au) 343, 351 Q quasicrystals: 57–60 Al-Pd 306 Co-Al 306 R reactive phase formation 284 S self–diffusion: activation entropy 69–71, 95 cohesive energy 71,88 diffusion frequency 97, 101 free energy of activation 69, 76, 94

INDEX

heat of activated complex 71, 95, 99–100 spinodal decomposition 23–25 sub–boundaries 37, 41, 43 solute effect: lattice 14 grain boundaries 52–56 parameters (Table 1.5) 55 silicides: CoSi2 320, 322 Fe 306, 322 Hf 310 Ir 302 Mo 300, 316 Ni, NiSi2 301–304, 317, 323–324 Rh 302 SiC 306 Si-Ge 306 SiH4 300 technology problems 309 Ti, TiSi2 (c49, C54) 309–310, 315 V 309 W 310 Zr 310, 316 sticking coefficient (metal/polymer) 265, 331–332, 335 strain hardening 379–381, 383, 396 stress relaxation 298, 366–367, 378 substrate curvature 368, 370 surface roughness, nucleation 317, 319 T texture 376, 399, 405 thermal expansion 94–95, 367, 370, 458, 463 thermally activated glide 378 thin Films 30–37 tight-binding method 118 transition state theory 138 V vacancy: enthalpy of formation and motion 12 entropy of formation and motion 12 valence bond parameter 88–90 voiding 399, 445, 474– 475 stress–induced 394–395, 397, 478, 481 X x–ray: stress measurement 373–376 XPS 337, 342, 354