Gupta D. (Ed.). 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.
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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
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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
1.1 Introduction ................................................................................ 1
1.2 Diffusion in Single Crystals ...................................................... 2
1.2.1 Mathematical Basis .......................................................... 2
1.2.2 Atomistic Nature of Diffusion ......................................... 8
1.2.3 Pressure and Mass Dependence of Diffusion .................. 15
1.2.4 Linear Chemical Diffusion Regime:
Finite Driving Force F on Individual Atoms ................... 20
1.2.5 Nonlinear Chemical Diffusion Regime ........................... 23
1.3 Structurally Inhomogeneous Samples ....................................... 27
1.4 Some Illustrative Experimental Data ......................................... 33
1.4.1 Diffusion Profiles in Au Having Variable
Microstructure .................................................................. 34
1.4.2 Self-Diffusion Data in the Au Lattice .............................. 38
1.4.3 Self-Diffusion in the Au and Au-1.2 at.%
Ta Alloy Grain Boundaries .............................................. 41
1.5 General Characteristics of Grain Boundary Diffusion .............. 44
1.5.1 Anisotropy of Diffusion in Grain Boundaries ................. 44
1.5.2 Diffusion Mechanisms in Grain Boundaries ................... 46
1.5.3 Interrelationship Among Grain Boundary,
Lattice Diffusion, and Energy .......................................... 48
1.5.4 Grain Boundary Solute Segregation Effects .................... 52
1.6 Diffusion in Quasicrystalline and Amorphous Alloys ............... 56
1.6.1 Diffusion in Quasicrystalline Alloys ............................... 57
1.6.2 Diffusion in Amorphous Alloys: Metallic Glasses .......... 60
1.7 Summary .................................................................................... 63
Acknowledgment .............................................................................. 64
References ......................................................................................... 64
2 Solid State Diffusion and Bulk Properties ................................... 69
2.1 Introduction ................................................................................ 69
2.2 Correlations with Bulk Properties ............................................. 71
2.2.1 The Melting Parameters ................................................... 71
2.2.2 Elastic Constants .............................................................. 82
2.2.3 Bulk Modulus .................................................................. 85
2.2.4 The Debye Temperature ................................................... 87
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viii CONTENTS
2.2.5 Valence Bond Parameter ................................................ 88
2.2.6 Electron-to-Atom Ratio ................................................. 91
2.2.7 Summary of Empirical Correlations .............................. 92
2.3 Equilibrium Thermodynamic Calculation of
Diffusion Parameters ............................................................... 93
2.3.1 The Activation Volume .................................................. 94
2.3.2 Activation Entropy and Diffusion Frequency ............... 97
2.3.3 Specific Heat of the Activated Complex ....................... 99
2.3.4 Magnitude of Estimated Values ..................................... 99
2.3.5 Reliability of Estimated Parameters .............................. 104
2.4 Summary .................................................................................. 105
Acknowledgment ............................................................................ 105
Appendix 2A. Taylor Series Expansion of ∆G
*
............................ 105
Appendix 2B. Evaluation of Errors in Estimated Parameters ....... 107
2B.1 ∆S
*
.................................................................................. 108
2B.2 u ...................................................................................... 108
2B.3 ∆C
*
V
................................................................................. 108
2B.4 ∆C
*
P
................................................................................. 109
References ....................................................................................... 109
3 Atomistic Computer Simulation of Diffusion ............................ 113
3.1 Introduction ............................................................................... 113
3.2 Atomic Interaction Models ....................................................... 114
3.2.1 Embedded-Atom Method ............................................... 114
3.2.2 Angular-Dependent Potentials ....................................... 116
3.2.3 More Accurate Methods ................................................. 118
3.3 Molecular Statics ..................................................................... 119
3.3.1 Simulation Block and Boundary Conditions ................. 119
3.3.2 Point-Defect Formation Energy ..................................... 120
3.4 Harmonic Approximation ........................................................ 122
3.4.1 Harmonic Entropy of Point Defects .............................. 122
3.4.2 Effect of Boundary Conditions on
Point-Defect Entropy ..................................................... 123
3.4.3 Embedded Cluster Method ............................................ 125
3.4.4 Local Harmonic Approximation .................................... 127
3.4.5 Quasi-Harmonic Approximation .................................... 128
3.5 Equilibrium Defect Concentrations .......................................... 129
3.5.1 Elemental Solids ............................................................ 129
3.5.2 Non-Stoichiometric Compounds ................................... 130
3.5.3 Effect of Pressure ........................................................... 137
3.6 Transition Rate Calculations .................................................... 138
3.6.1 Transition State Theory .................................................. 138
3.6.2 Finding the Saddle Point ................................................ 141
3.7 Kinetic Monte Carlo Simulations ............................................. 143
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CONTENTS ix
3.7.1 Basic Idea of the Method ............................................... 143
3.7.2 Grain Boundary Diffusion ............................................. 145
3.7.3 Diffusion in Alloys and Compounds ............................. 149
3.7.4 On-the-Fly Monte Carlo Simulations ............................ 151
3.8 Molecular Dynamics ................................................................ 153
3.8.1 Calculation of Diffusion Coefficients ............................ 153
3.8.2 Diffusion Mechanisms in Grain Boundaries ................. 155
3.8.3 Diffusion Mechanisms in Intermetallic Compounds ..... 157
3.8.4 Accelerated Molecular Dynamics .................................. 162
3.9 Conclusions .............................................................................. 165
Acknowledgment ............................................................................ 166
References ....................................................................................... 166
4 Bulk and Grain Boundary Diffusion in
Intermetallic Compounds ............................................................ 173
4.1 Introduction .............................................................................. 173
4.2 Crystal Structures and Point Defects in Ni, Ti,
and Fe Aluminides ................................................................... 174
4.3 Diffusion Mechanisms in Intermetallics ................................. 182
4.4 Experimental Results on Bulk Diffusion
in Ordered Aluminides ............................................................ 188
4.4.1 Ni
3
Al .............................................................................. 189
4.4.2 Ti
3
Al ............................................................................... 196
4.4.3 TiAl ................................................................................ 199
4.4.4 NiAl ............................................................................... 205
4.4.5 Fe-Al System ................................................................. 213
4.5 Discussion of Lattice Diffusion in Intermetallics ................... 221
4.6 Grain Boundary Diffusion ....................................................... 224
4.6.1 Ni
3
Al .............................................................................. 225
4.6.2 Ti
3
Al ............................................................................... 227
4.6.3 TiAl ................................................................................ 229
4.6.4 NiAl ............................................................................... 229
4.6.5 Fe
3
Al .............................................................................. 232
4.7 Summary .................................................................................. 232
Acknowledgments ................................................................... 234
References ................................................................................ 234
5 Diffusion Barriers in Semiconductor Devices/Circuits ............. 239
5.1 Introduction .............................................................................. 239
5.2 Diffusion Barriers from the 1960s Through the 1990s ........... 241
5.3 Brief Review of Diffusion and the
Influencing Material Factors ................................................... 244
5.3.1 Diffusion in the Lattice and Grain Boundaries ............. 244
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