Physical Properties of Quasicrystals

449 Pages·1999·30.989 MB·English

by  Zbigniew M. Stadnik (auth.), Professor Dr. Zbigniew M. Stadnik (eds.)

Springer Series in
SOLID-STATE SCIENCES 126
Springer-Verlag Berlin Heidelberg GmbH
Springer Series in
SOLID-STATE SCIENCES
Series Editors:
M. Cardona P. Fulde K. von Klitzing R. Merlin H.-J. Queisser H. Stornier
126 Physical Properties of Quasicrystals
Editor: Z.M. Stadnik
Volumes 1-125 are listed at the end of the book.
Zbigniew M. Stadnik (Ed.)
Physical Properties
of Quasicrystals
With 221 Figures
Springer
Professor Dr. Zbigniew M. Stadnik
University of Ottawa, Department of Physics
150 Louis Pasteur, Ottawa
Ontario KiN 6N5, Canada
e-mail: [email protected]
Series Editors:
Professor Dr., Dres. h. c. Manuel Cardona
Professor Dr., Dres. h. c. Pete r Fulde*
Professor Dr., Dres. h. c. Klaus von Klitzing
Professor Dr., Dres. h. c. Hans-Joachim Queisser
Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
* Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38
D-01187 Dresden, Germany
Professor Dr. Roberto Merlin
Department of Physics, 5000 East University, University of Michigan
Ann Arbor, MI 48109-1120, USA
Professor Dr. Horst Stornier
Dept. Phys. and Dept. Appl. Physics, Columbia University, New York, N Y 10023 and
Bell Labs., Lucent Technologies, Murray Hill, NJ 07974, USA
ISSN 0171-1873
ISBN 978-3-642-63593-9 ISBN 978-3-642-58434-3 (eBook)
DOI 10.1007/978-3-642-58434-3
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Physical properties of quasicrystals I Zbigniew M. Stadnik (ed.). Berlin ; Heidelberg ; New York ; Barcelona ;
Hong Kong; London ; Milan; Paris; Singapore ; Tokyo : Springer 1999
(Springer series in solid-state sciences; 126)
ISBN 978-3-642-63593-9
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Preface
The discovery in 1984 of crystals with "forbidden" symmetry, materials that
were dubbed quasicrystals, opened a new branch of crystallography and solid
state physics. These structures possess long-range aperiodic order and crys
tallographically forbidden rotational symmetries, and are thus fundamentally
different from the two other known types of solid structure, crystalline and
amorphous. The literature devoted to the physical properties of quasicrystals
has been growing so rapidly that it is very difficult for a scientist to remain
well informed on more than a narrow speciality within this area. Some def
inite and intriguing new physical phenomena have already been discovered,
and other surprising results will probably follow. Our knowledge of the physi
cal properties of quasicrystals has now reached such a maturity that it seems
necessary to provide a comprehensive review. I think this is an opportune
time to produce a research-level monograph on the subject. As yet, no com
prehensive treatise exists at this level. It is very unlikely that a single author
could produce a monograph at the level and depth that is needed to do jus
tice to the physical properties of quasicrystals. I hope that this book comes
close to being such a monograph.
This book is intended for researchers in the field of the physics of qua
sicrystals, solid-state physicists, materials scientists, crystallographers, as well
as for graduate students working in the area of new materials. Written by
active researchers in their respective fields, it summarizes in a critical fash
ion much of our present knowledge and understanding of the complex, yet
exciting physical properties of quasicrystals. I hope that it conveys some of
the fascination I have found during my work in this field. We are still far
from a complete understanding of these materials; this book will have served
its purpose if it stimulates new research and helps to guide that research in
fruitful directions.
I wish to thank the authon, for their thorough and competent work. I
thank my departmental colleague, Dr. Brian A. Logan, for his assistance
with the English of the book. I am grateful to Dr. Claus Ascheron from
Springer-Verlag for his advice and friendly patience concerning some unex
pected delays. Finally, I express my thanks to Dr. Gerhard Stroink from
Dalhousie University who introduced me in 1986 to the fascinating research
area of quai'iicrystals.
Ottawa, September 1998 Zbigniew M. Stadnik
Contents
1. Introduction
Zbigniew M. Stadnik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References ..................................................... 3
2. Metallurgy of Quasicrystals
An Pang Tsai .................................................. 5
2.1 Introduction............................................... 5
2.2 Preparation of Quasicrystals ................................ . 6
2.2.1 Rapid Solidification .................................. . 6
2.2.2 Vapor Condensation .................................. . 7
2.2.3 Mechanical Alloying .................................. . 8
2.2.4 Crystallization of Melt-Quenched Amorphous Ribbons .... . 9
2.2.5 Conventional Solidification ............................ . 9
2.3 Structural Classification from Diffraction Patterns. . . . . . . . . . . . .. 10
2.3.1 Three-Dimensional Quasicrystals . . . . . . . . . . . . . . . . . . . . . . .. 11
2.3.2 Two-Dimensional Quasicrystals ........................ 13
2.3.3 One-Dimensional Quasicrystals ......................... 17
2.4 Quasicrystalline Alloy Systems
and the Formation of Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . .. 17
2.4.1 Metastable Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17
2.4.2 Stable Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20
2.4.3 Quasicrystals as Hume-Rothery Phases .................. 23
2.5 Phase Transformation
from Amorphous to Icosahedral Phase. . . . . . . . . . . . . . . . . . . . . . . .. 26
2.6 Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29
2.6.1 AI-Li-Cu............................................. 29
2.6.2 AI-Cu-Fe............................................. 31
2.6.3 Al-Pd-Mn............................................ 31
2.6.4 Zn-Mg-Y............................................. 33
2.6.5 Al-Ni-Co............................................. 33
2.7 Growth of Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35
2.7.1 Morphologies of Quasicrystals .......................... 35
2.7.2 Solidification of Quasicrystals. . . . . . . . . . . . . . . . . . . . . . . . . .. 39
2.7.3 Growing Large Single Grains. . . . . . . . . . . . . . . . . . . . . . . . . .. 43
VIII Contents
2.8 Summary.................................................. 44
References ..................................................... 45
3. Crystallography of Quasicrystals
Walter Steurer and Torsten Haibach. . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. 51
3.1 Introduction............................................... 51
3.2 N-Dimensional Description of Quasicrystals. . . . . . . . . . . . . . . . . . .. 52
3.2.1 Embedding of Direct and Reciprocal Space. . . . . . . .. . . . . .. 52
3.2.2 Structure Factor ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 57
3.3 One-Dimensional Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58
3.3.1 Indexing............................................. 59
3.3.2 Symmetry............................................ 60
3.3.3 Example of a One-Dimensional Quasicrystal:
Fibonacci Phase ...................................... 60
3.4 Decagonal Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65
3.4.1 Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65
3.4.2 Symmetry............................................ 66
3.4.3 Example of a Decagonal Phase: Layers of Penrose Tilings .. 68
3.5 Icosahedral Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76
3.5.1 Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76
3.5.2 Symmetry............................................ 79
3.5.3 Example of a Three-Dimensional Quasilattice:
Ammann Tiling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82
References ..................................................... 89
4. Experimental Determination
of the Structure of Quasicrystals
Michel Boudard and Marc de Boissieu . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 91
4.1 Introduction............................................... 91
4.2 X-ray and Neutron Diffraction ............................... 94
4.2.1 Patterson Analysis .................................... 95
4.2.2 Contrast Variation .................................... 96
4.3 Structure of the AI-Pd-Mn Icosahedral Phase. . . . . .. .. . . . . . . . .. 97
4.3.1 Space Group Determination. . . . . . . . . . . . .. . . . . . . . . . . . . .. 98
4.3.2 Patterson Analysis .................................... 101
4.3.3 First-Order Model ..................................... 105
4.3.4 About the Resulting Atomic Structure ................... 106
4.3.5 Limitations of the Direct Approach ...................... 110
4.3.6 Modeling ............................................. 110
4.4 Structure of the AI-Ni-Co Decagonal Quasicrystal .............. 119
4.5 Conclusions ................................................ 124
References ..................................................... 124
Contents IX
5. Electronic Transport Properties
of Quasicrystals - Experimental Results
Osten Rapp .................................................... 127
5.1 Introduction ............................................... 127
5.1.1 Background .......................................... 127
5.1.2 Resistance Anomalies .................................. 128
5.1.3 Outline .............................................. 130
5.2 Experimental Results ....................................... 130
5.2.1 Overview ............................................ 130
5.2.2 High-Temperature Electrical Resistivity .................. 134
5.2.3 Hall Effect and Thermoelectric Power .................... 135
5.2.4 Icosahedral Approximants .............................. 137
5.2.5 Decagonal Quasicrystals ............................... 138
5.3 Towards Understanding Transport Properties .................. 140
5.3.1 Strong Sensitivity to Electron Concentration .............. 141
5.3.2 Magnitude of the Electrical Resistivity ................... 144
5.3.3 The Magnetoresistivity ................................ 147
5.3.4 p(T) at Low and Intermediate Temperatures .............. 154
5.3.5 Is There a Metal-Insulator Transition
in Icosahedral AI-Pd-Re? ............................... 156
5.4 Concluding Remarks ........................................ 159
References ..................................................... 162
6. Theory of Electronic Structure
in Quasicrystals
Takeo Fujiwara ................................................. 169
6.1 Introduction ............................................... 169
6.2 Electronic Structure
in One- and Two-Dimensional Quasilattices . . . . . . . . . . . . . . . . . . .. 172
6.2.1 One-Dimensional Quasilattice: Fibonacci Lattice .......... 172
6.2.2 Two-Dimensional Quasilattice: Penrose Lattice ............ 175
6.3 Electronic Structure in Quasicrystals .......................... 179
6.3.1 Method of Calculations:
Tight-Binding LMTO and Related Methods . . . . . . . . . . . . . . 179
6.3.2 Quasi-Brillouin Zone
and Modification of DOS of Model Icosahedral Al ......... 180
6.3.3 Electronic Structure in MI-Type Icosahedral Quasicrystals . 181
6.3.4 Electronic Structure in TC-Type Icosahedral Quasicrystals . 185
6.3.5 Electronic Structure in Decagonal Quasicrystals ........... 185
6.3.6 General Characteristics of DOS and Wave Functions ....... 188
6.3.7 Experimental Study of Electronic Structures .............. 194
6.4 Transport Properties in Quasicrystals ......................... 194
6.4.1 Scenario of Transport in Random Systems ................ 194
6.4.2 Experimental Observations ............................. 196
X Contents
6.4.3 Effects of Randomness ................................. 196
6.4.4 Boltzmann Theory and Relaxation-Time Approximation ... 197
6.4.5 Anomalous Diffusion ................................... 200
6.4.6 Scaling Behavior ...................................... 200
6.4.7 Scenario of Transport in Quasicrystals ................... 202
6.5 Summary .................................................. 203
References ..................................................... 204
7. Elementary Excitations
and Physical Properties
Jurgen Hafner and Marian KrajCf ................................. 209
7.1 Introduction ............................................... 209
7.1.1 Quasiperiodic Structure ................................ 209
7.1.2 Physical Properties .................................... 210
7.1.3 Spectral Properties of Quasiperiodic Hamiltonians ......... 211
7.2 Quasiperiodicity, Symmetry,
and Elementary Excitations .................................. 214
7.3 Modelling Quasicrystalline Structures
and Approximant Phases .................................... 216
7.3.1 Icosahedral Quasicrystals ............................... 216
7.3.2 Decagonal Quasicrystals ............................... 218
7.3.3 Approximant Structures ............................... 219
7.4 Numerical Characterization
of Elementary Excitations ................................... 222
7.4.1 Direct Diagonalization ................................. 222
7.4.2 Real-Space Recursion .................................. 223
7.4.3 Comparison With Experiment .......................... 224
7.5 Phonons in Quasicrystals .................................... 224
7.5.1 Interactomic Force Law
and Quasiperiodicity - Modulated Quasicrystals .......... 224
7.5.2 Phonons in Icosahedral Quasicrystals .................... 226
7.5.3 Phonons in Decagonal Quasicrystals ..................... 231
7.5.4 Phonons - Summary ................................... 234
7.6 Electrons in Quasicrystals ................................... 234
7.6.1 s,p-Bonded Icosahedral Alloys as Hume-Rothery Phases ... 235
7.6.2 Icosahedral
and Decagonal Aluminum-Transition Metal Alloys ......... 238
7.6.3 Titanium-Based Quasicrystals .......................... 242
7.6.4 Fine Structure of the Electronic Spectrum,
Pseudogaps, and Real Gaps ............................ 242
7.6.5 Band-Structure Effects in Electronic Transport ........... 246
7.6.6 Magnetic Properties of Quasicrystals .................... 248
7.6.7 Electrons - A Summary ................................ 249
7.7 Final Remarks ............................................. 251
References ..................................................... 251
Contents XI
8. Spectroscopic Studies
of the Electronic Structure
Zbigniew M. Stadnik ............................................ 257
8.1 Introduction ............................................... 257
8.2 Theoretical Predictions ...................................... 258
8.2.1 Pseudogap in the Density of States ...................... 258
8.2.2 Fine Strucure of the Density of States ................... 259
8.3 Experimental Results ....................................... 260
8.3.1 s,p-Bonded Icosahedral Alloys .......................... 260
8.3.2 AI-Cu-Transition Metal Icosahedral Alloys ................ 263
8.3.3 AI-Pd-Mn Icosahedral Alloys ........................... 271
8.3.4 AI-Pd-Re Icosahedral Alloys ............................ 275
8.3.5 AI-Co-Cu Decagonal Alloys ............................. 277
8.3.6 AI-Ni-Co and AI-Ni-Rh Decagonal Alloys ................. 278
8.3.7 Fine Structure of the Density of States ................... 280
8.4 Uniqueness of the Electronic Structure
of Quasicrystals ............................................ 284
8.5 Quasiperiodicity and Unusual Physical Properties ............... 287
8.6 Conclusions and Outlook .................................... 288
References ......................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
9. Magnetic Properties of Quasicrystals
Kazuaki Fukamichi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
9.1 Introduction ............................................... 295
9.2 AI-Based Quasicrystals ...................................... 296
9.2.1 Paramagnetism, Effective Magnetic Moment
and Saturation Magnetization .......................... 296
9.2.2 Fraction of Magnetic Mn Atoms
and Giant Magnetic Moment ........................... 299
9.2.3 Difference Between Magnetic Moments in Icosahedral
and Decagonal Phases .......... . . . . . . . . . . . . . . . . . . . . . . . 301
9.2.4 Spin-Glass Behavior ................................... 303
9.2.5 Low-Temperature Specific Heat ......................... 306
9.2.6 Model for Magnetism and Pauling Valence ............... 308
9.2.7 Phasons, Diamagnetism, and Pauli Paramagnetism ........ 309
9.2.8 Ferromagnetism ....................................... 312
9.3 Mg-RE-Zn Quasicrystals ..................................... 315
9.3.1 Susceptibility and Spin-Glass Behavior ................... 316
9.3.2 Low-Temperature Specific Heat ......................... 318
9.3.3 Antiferromagnetism ................................... 321
9.4 Summary .................................................. 323
References ..................................................... 324