Quasicrystals

Quasicrystals HANDBOOK OF METAL PHYSICS SERIES EDITOR Prasanta Misra Department of Physics, University of Houston, Ho...

Author: Takeo Fujiwara | Yasushi Ishii

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Quasicrystals

HANDBOOK OF METAL PHYSICS SERIES EDITOR Prasanta Misra Department of Physics, University of Houston, Houston, Texas 77204, U.S.A.

Quasicrystals Volume Editors TAKEO FUJIWARA Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan

and YASUSHI ISHII Department of Physics, Chuo University, Kasuga, Tokyo 112-8551, Japan

AMSTERDAM . BOSTON . HEIDELBERG . LONDON . NEW YORK . OXFORD PARIS . SAN DIEGO . SAN FRANCISCO . SINGAPORE . SYDNEY . TOKYO

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands Linacre House, Jordan Hill, Oxford OX2 8DP, UK

First edition 2008 Copyright 2008 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-444-51418-9 ISSN: 1570-002X For information on all Elsevier publications visit our website at books.elsevier.com Printed and bound in Hungary 08 09 10 11 12 10 9 8 7 6 5 4 3 2 1

The Book Series ‘Handbook of Metal Physics’ is dedicated to my wife Swayamprava and to our children Debasis, Mimi and Sandeep

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Preface

Metal Physics is an interdisciplinary area covering Physics, Chemistry, Materials Science, and Engineering. Due to the variety of exciting topics and the wide range of technological applications, this field is growing very rapidly. It encompasses a variety of fundamental properties of metals such as Electronic Structure, Magnetism, Superconductivity, as well as the properties of Semimetals, Defects and Alloys, and Surface Physics of Metals. Metal Physics also includes the properties of exotic materials such as High-Tc Superconductors, Heavy-Fermion Systems, Quasicrystals, Metallic Nanoparticles, Metallic Multilayers, Metallic Wires/Chains of Metals, Novel Doped Semimetals, Photonic Crystals, Low-Dimensional Metals and Mesoscopic Systems. This is by no means an exhaustive list and more books in other areas will be published. I have taken a broader view and other topics, which are widely used to study the various properties of metals, will be included in the Book Series. During the past 25 years, there has been extensive theoretical and experimental research in each of the areas mentioned above. Each volume of this Book Series, which is self-contained and independent of the other volumes, is an attempt to highlight the significant work in that field. Therefore the order in which the different volumes will be published has no significance and depends only on the timeline in which the manuscripts are received. The Book Series ‘Handbook of Metal Physics’ is designed to facilitate the research of Ph.D. students, faculty and other researchers in a specific area in Metal Physics. The books will be published by Elsevier in hard cover copy. Each book will be either written by one or two authors who are experts and active researchers in that specific area covered by the book or by multiple authors with a volume editor who will co-ordinate the progress of the book and edit it before submission for final editing. This choice has been made according to the complexity of the topic covered in a volume as well as the time that the experts in the respective fields were willing to commit. Each volume is essentially a summary as well as a critical review of the theoretical and experimental work in the topics covered by the book. There are extensive references after the end of each chapter to facilitate researchers in this rapidly growing interdisciplinary field. Since research in various sub-fields in Metal Physics is a rapidly growing area, it is planned that each book will be updated periodically to include the results of the latest research. Even though these books are primarily designed as reference books, some of these books can be used as advance graduate level text books. The outstanding features of this Book Series are the extensive research references at the end of each chapter, comprehensive review of the significant theoretical work, a summary of all important experiments, illustrations wherever necessary, and discussion of possible technological applications. This would spare the active researcher in a field to do extensive search of the literature before she or he would start planning to work on a new research topic or in writing a research paper on a piece of work already completed. vii

viii

Preface

Since each volume will have an introductory chapter written either by the author(s) or the volume editor, it is not my intention to write an introduction for each topic (except for the book being written by me). In fact, they are much better experts than me to write such introductory remarks. Finally, I invite all students, faculty and other researchers, who would be reading the book(s) to communicate to me their comments. I would particularly welcome suggestions for improvement as well as any errors in references and printing.

Acknowledgements I am grateful to all the eminent scientists who have agreed to contribute to the Book Series. All of them are active researchers and obviously extremely busy in teaching, supervising graduate students, publishing research papers, writing grant proposals and serving on committees. It is indeed gratifying that they have accepted my request to be either an author or volume editor of a book in the Series. The success of this Series lies in their hands and I am confident that each one of them will do a great job. I have been greatly impressed by the quality of this volume on Quasicrystals edited by Professors Takeo Fujiwara of University of Tokyo and Professor Yasushi Ishii of Chuo University of Japan. The contributors to the volume are some the pioneers and best known experts in the field of ‘Quasicrystals’. The idea of editing a Book Series on Metal Physics was conceived during a meeting with Dr. Charon Duermeijer, publisher of Elsevier (she was Physics Editor at that time). After several rounds of discussions (via e-mail), the Book Series took shape in another meeting where she met some of the prospective authors/volume editors. She has been a constant source of encouragement, inspiration and great support while I was identifying and contacting various experts in the different areas covered by this extensive field of Metal Physics. It is indeed not easy to persuade active researchers (scattered around the globe) to write or even edit an advance research level book. She had enough patience to wait for me to finalize a list of authors and volume editors. I am indeed grateful to her for her confidence in me. I am also grateful to Drs. Anita Koch, Manager, Editorial Services, Books of Elsevier, who has helped me whenever I have requested her, i.e., in arranging to write new contracts, postponing submission deadlines, as well as making many helpful suggestions. She has been very gracious and prompt in her replies to my numerous questions. I have profited from conversations with my friends who have helped me in identifying potential authors as well as suitable topics in my endeavor to edit such an ambitious Book Series. I am particularly grateful to Professor Larry Pinsky (chair) and Professor Gemunu Gunaratne (Associate Chair) of the Department of Physics of University of Houston for their hospitality, encouragement and continuing help. Finally, I express my gratitude to my wife and children who have loved me all these years even though I have spent most of my time in the physics department(s) learning physics, doing research, supervising graduate students, publishing research papers and writing grant proposals. There is no way I can compensate for the lost time except to dedicate this Book Series to them. I am thankful to my

Preface

ix

daughter-in-law Roopa who has tried her best to make me computer literate and in the process has helped me a lot in my present endeavor. My fondest dream is that when my grandchildren Annika and Millan attend college in 2021 and Kishen and Nirvaan in 2024, this Book Series would have grown both in quantity and quality (obviously with a new Series Editor in place) and at least one of them would be attracted to study the subject after reading a few of these books. Prasanta Misra Department of Physics, University of Houston, Houston, TX, USA


Contents

Preface

vii

Chapter 1. Introduction to Quasicrystals Takeo Fujiwara and Yasushi Ishii 1.1. Discovery of non-crystalline rotational symmetry 1.2. Quasiperiodic lattice

1 2

1.3. Icosahedral quasilattice and its symmetry

4

1.4. Phonons and phasons 1.5. Electronic structure and related physical properties

6 8

Chapter 2. Recent Developments of Quasicrystallography Akiji Yamamoto and Hiroyuki Takakura 2.1. Introduction

11

2.2. nD description of quasicrystals 2.3. Decagonal, dodecagonal and icosahedral coordinate systems

14 15

2.4. Low-density elimination method

17

2.5. Modification of models in the structure refinement 2.6. Higher-dimensional cluster models of decagonal quasicrystals

19 23

2.7. Higher-dimensional cluster models of icosahedral quasicrystals 2.8. Quasicrystal models with fractal occupation domains

29 36

2.9. Symmetry breaking in clusters

39

2.10. Modulation functions for quasicrystals 2.11. Summary

43 43

Chapter 3. New Group of Icosahedral Quasicrystals Tsutomu Ishimasa 3.1. Introduction

49

3.2. Classification of icosahedral quasicrystals 3.3. Approximants including Tsai-type cluster

50 50

3.4. Preparation methods of Zn- and Cu-based quasicrystals

54

3.5. Zn–Mg–Sc quasicrystal as a prototype 3.6. Other Zn- and Cu-based quasicrystals

55 58

3.7. Single-quasicrystals and microvoids

63

xi

xii

Contents

3.8. Central structure of the Tsai-type cluster 3.9. Alloy chemistry and the Tsai-type quasicrystals 3.10. Conclusion

64 67 71

Chapter 4. Quasicrystals and Approximants in Cd-M Systems and Related Alloys An-Pang Tsai and Cesar Pay Gomez 4.1. Introduction

75

4.2. Approximants in the Cd-M systems

78

4.3. Stable quasicrystals of the Cd–Yb class 4.4. Hume-Rothery conditions for the stable quasicrystals

90 98

4.5. Concluding remarks

104

Chapter 5. Phason Modes in Aperiodic Crystals M. de Boissieu, R. Currat, and S. Francoual 5.1. Introduction: Hydrodynamic modes and quasiperiodic structures 5.2. Modulated crystals

107 116

5.3. Binary composites

126

5.4. Hydrodynamics of icosahedral phases 5.5. Phason modes in the icosahedral AlPdMn quasicrystal

129 142

5.6. Phason modes in other quasicrystals 5.7. Conclusion

155 162

Chapter 6. Electronic Structures and Stability Mechanism of Quasicrystals Yasushi Ishii and Takeo Fujiwara 6.1. Introduction

171

6.2. Stability mechanism – Hume-Rothery versus hybridization

173

6.3. Ab initio methods for calculating electronic structures of QCs 6.4. Electronic structure of QC-related compounds

176 179


203

Chapter 7. Quantum Transport in Quasicrystals and Complex Metallic Alloys Didier Mayou and Guy Trambly de Laissardie`re 7.1. Introduction 7.2. Quantum formalism for electronic transport

209 211

7.3. Anomalous quantum diffusion and conductivity in periodic and quasiperiodic systems

228

Contents

xiii

7.4. Evidence of anomalous quantum diffusion in quasicrystals and approximants

245

7.5. Conclusion

261

Chapter 8. Elastic and Plastic Properties of Quasicrystals S. Takeuchi and K. Edagawa 8.1. Introduction

267

8.2. Elastic properties

268

8.3. Dislocations and their motion 8.4. Mechanical properties

282 291

8.5. Deformation mechanisms

297

Chapter 9. Ab-Initio Studies of Quasicrystalline Surfaces M. Krajcˇı´ and J. Hafner 9.1. Introduction

313

9.2. Computational method

316

9.3. Fivefold surface of i-Al–Pd–Mn 9.4. Study of the tenfold surface of d-Al–Co–Ni

316 337

9.5. Summary 9.6. Outlook

349 351

Index

357


Chapter 1

Introduction to Quasicrystals Takeo Fujiwara1 and Yasushi Ishii2 1 Department of Applied Physics, University of Tokyo, Tokyo 113-8654, Japan and 2 Department of Physics, Chuo University, Tokyo 112-8551, Japan

1.1.

Discovery of non-crystalline rotational symmetry

Crystals hold the translational periodicity and allow 1-, 2-, 3-, 4-, and 6-fold rotational symmetries. Then, for many decades, people believed beyond a shadow of doubt that the fivefold rotational symmetry cannot exist in equilibrate condensed phases. Mineralogists have discovered many alloy materials with icosahedral units, either isolated or linked, but these materials are globally of crystalline rotational symmetry. In the meantime, our understanding about tessellation has been widened. An icosahedron is locally the most densely packed arrangement and has been seen in aperiodic materials such as liquids and amorphous. An icosahedral rotational symmetry contradicts the translational periodicity and it was thought that a long range order of icosahedral symmetry is unlikely to be found in realistic condensed matter, even though a two-dimensional pentagonal tiling, called Penrose tiling, was discovered by Penrose in 1974 [1]. After the invention of Penrose tiling, de Bruijn [2], Mackay [3], and Kramer [4] studied two- and three-dimensional tiling and their mathematics. Shechtman, Blech, Gratias, and Cahn discovered sharp diffraction patterns of icosahedral symmetry (Figure 1.1) in Al–Mn alloy rapidly quenched from the melt [5]. The icosahedral phase in rapidly solidified Al–Mn alloy is metastable and resistant to crystallization up to 350C. Independent of the discovery by Shechtman et al., Levine and Steinhardt generalized the notion of crystal to include a quasiperiodic translational order and named it ‘quasicrystal’ [6]. ‘Quasicrystal’ is specified by two fundamental characteristics, i.e., long-range quasiperiodic translational order and long-range crytallographically forbidden orientation symmetry. There are several different classes of quasicrystals [7]: Al-based transition metal alloys (TM alloys, e.g., AlMn, AlMnSi, AlCuFe, and AlPdMn), alloys without transition metal elements having similar composition to that of Frank-Kasper HANDBOOK OF METAL PHYSICS ISSN 1570-002X/DOI 10.1016/S1570-002X(06)02001-5 1

2008 ELSEVIER B.V. ALL RIGHTS RESERVED

2

Chapter 1. Introduction to Quasicrystals

79.2° ° 8.29

5 °

7 7.3

3

31.72 63.43

Figure 1.1. Diffraction pattern of fivefold symmetry (area proportional to the intensity) and stereogram of the icosahedral group.

phase with tetrahedrally closed-packed structures (FK alloys, e.g., Al–Cu–Li, Zn– Mg–RE (RE ¼ rare earth)), and stable binary alloys (Cd alloys, e.g., Cd5.7Yb and Cd17Ca3) [8]. In AlMn, AlFe, AlCuCo, and AlCoNi, one can observe a decagonal phase with a diffraction pattern of tenfold rotational symmetry, which has a onedimensional translational periodicity along the tenfold rotational axis [9]. We also have now two-dimensional octagonal quasicrystals CrNiSi and VNiSi [10], dodecagonal quasicrystals TaTe [11], and a polymer alloy [12].

1.2.

Quasiperiodic lattice

Structures of quasicrystals can be generated mathematically by inflation–deflation operation [2], utilization of matching rules [2], the grid method [2], strip projection method [4,13], cut projection method [14] or generalized dual method [15]. These methods are closely related to each other. Lattice points in the hypercubic lattice in the n-dimensional space En are projected, in the strip projection method, on vertices in the d-dimensional quasiperiodic lattice. Let us consider the n-dimensional space En decomposed into two subspaces, En ¼ Ejj þ E? ;

ð1:1Þ

where the subspace Ejj is the d-dimensional physical space and E? is orthogonal to Ejj , called ‘perpendicular-space’ (perp-space). Let W be the projection of a unit ‘hypercube’ onto E? , which we call the ‘window’. For any x0 in Ejj , we will write W0 for W+x0. When a lattice point in En is located inside W0, its projection onto Ejj is selected to be a vertex in the physical space Ejj . If the ‘slope’ of the hyperplane Ejj is rational to the hypercublic lattice in En, the projected lattice in Ejj is periodic. On the other hand, if the ‘slope’ of the hyperplane Ejj is irrational, the projected lattice becomes quasiperiodic. The n-dimensional hypercubic lattice, the directions of the two orthogonal subspaces E? and Ejj , and the window W must be invariant under operations of the non-crystalline group of the quasicrystal.

3

1.2. Quasiperiodic lattice

E⊥

E||

W Figure 1.2. One-dimensional quasiperiodic lattice (cross marks on the Ejj -axis) generated by projection.

We show, in Figure 1.2, a one-dimensional quasiperiodic lattice model by the projection from the two-dimensional square lattice. Then one-dimensional quasiperiodic lattice called the Fibonacci lattice is generated, when the Ejj -axis is oriented atpffiffi=arctan 1/ to the x-axis, where tan =1/ and is the golden mean ffi ð 5 þ 1Þ=2. A distribution function of lattice points ðxÞ is given by X ðxjj j cos l sin Þ ðx? þ j sin l cos Þ; ð1:2Þ ðxÞ ¼ðxjj ;x? Þ ¼ jl

and the distribution function of the projected lattice on Ejj , is Zþ1

jj

0 ðx Þ ¼

dx? ðxjj ; x? ÞWðx? Þ;

ð1:3Þ

1

where the window W is a function Wðx? Þ ¼

1; 0;

if 2 jx? j < cos þ sin : otherwise

ð1:4Þ

Then, the diffraction pattern of the one-dimensional Fibonacci lattice can be expressed by a structure factor Zþ1 fðkÞ ¼

dxjj expðikxjj Þ0 ðxjj Þ

1 þ1 X

sin k? w ðk 2ðm sin þ n cos ÞÞ; k? w m;n¼1

ð1:5Þ

where w ¼ 12 ðcos þ sin Þ and k? ¼ 2ðm cos n sin Þ. This procedure of projection and calculation of structure factor are generalized to higher dimensional cases and the pattern of the diffraction spots is generally a countable infinite number of densely distributed d-functions [6,16]. The pentagonal quasicrystal can be obtained by a projection of the five-dimensional hypercubic lattice to a two-dimensional space and the icosahedral quasicrystal by that of

4


the six-dimensional one to a three-dimensional space. The three-dimensional icosahedral quasicrystals are constructed by two rhombohedral units, prolate and oblate. For constructing the quasicrystals, the matching rule may be one of possible principle and, in the icosahedral quasicrystals, four types of unit cells appear with a uniquely defined matching rule: a triacontahedron, a rhombic icosahedron, a rhombic dodecahedron, and a prolate rhombohedron with volumes in the ratios 10:5:2:1 [17]. Actually the former three types of unit cells can be divided into several prolate and oblate rhombohedral without uniqueness.

1.3.

Icosahedral quasilattice and its symmetry

The mass density for the icosahedral case ðrÞ can be expanded in a Fourier series ðrÞ ¼

X

Gn exp iGn r ¼

X

Gn

jGn j exp iðGn r þ Gn Þ,

ð1:6Þ

Gn

where Gn is the reciprocal vector, corresponding to the positional vectors of vertices of an icosahedron shown in Figure 1.3; G ffi ð1; 0; Þ; G0 ¼ pffiffiffiffiffiffiffi 1þ 2 G ffi p ffiffiffiffiffiffiffi ð; 1; 0Þ, G3 ¼ 2 1þ

G G1 ¼ pffiffiffiffiffiffiffiffi ð0; ; 1Þ; 1þ 2 G ffi p ffiffiffiffiffiffiffi G4 ¼ ð0; ; 1Þ, 2

G G2 ¼ pffiffiffiffiffiffiffiffi ð; 1; 0Þ, 1þ 2 G ffi p ffiffiffiffiffiffiffi and G5 ¼ 1þ 2 ð1; 0; Þ.

1þ

Let us discuss the icosahedral quasicrystals whose reciprocal lattice space is spanned by six vectors fGn g. Correspondingly we consider the icosahedral group Y, the number of whose symmetry operations is 60. There are 60 elements, five classes in Y, i.e., the identity E, rotations by 2/5 about the axes through the vertices of the icosahedron C5 , rotations by 4/5 about the axes through the vertices of the icosahedron C25 , rotations by 2/3 about the axes through the faces of the icosahedron C3 , and rotations by about the edge centers of the icosahedron C2 . The character table of the icosahdral group Y is given in Table 1, which gives five irreducible representations 1 5 . Three translational vectors in the physical space transform under the representation 3 . The set of the complementary vectors 2

3

5 1

4

0 Figure 1.3. Reciprocal vectors Gn of icosahedral quasicrystals.

5

1.3. Icosahedral quasilattice and its symmetry Table 1.1. Character table of the icosahedral group I

1 3 30 4 5 D

E

12C5

12C25

20C3

15C2

1 3 3 4 5 6

1 1 1 0 1

1 1 1 0 1

1 0 0 1 1 0

1 1 1 0 1 2

G? 5 ¼ G5 ;

G? n ¼ G5 ;

n ¼ 0; . . . ; 4

ð1:7Þ

transform under the representation 30 , where h3ni5 ¼ 3n mod 5. The transformation by the symmetry operations are, for example, as following: C5 : G5 ! G5 ;

G0 ! G1 ! G2 ! G3 ! G4 ! G0 ;

C3 : G1 ! G2 ! G5 ! G1 ; C2 : G2 $ G3 ;

G0 $ G5 ;

G3 ! G0 ! G4 ! G3 ; G1 ! G1 ;

G4 ! G4 :

ð1:8Þ

The basis set of Gn is governed by the representation D in Table 1.1. The representation D is reducible into two irreducible representations D ¼ 3 30 , which represent the rotational symmetry of three-dimensional vectors in the physical and perpendicular spaces, respectively. General Bragg spots can be specified by reciprocal vectors G¼

5 X mn Gn ; a n¼0 G

ð1:9Þ

where mn are integers and a represents an edge length of the rhombohedral unit cell, which is called a quasilattice constant. The reciprocal lattice point Gn is associated with a unique partner in the perpendicular space G? ¼

5 X mn G? n : a n¼0 G

ð1:10Þ

These two associated vectors and appear already in an example of the one-dimensional Fibonacci lattice Eq. (1.5); i.e., k ¼ 2ðm sin þ n cos Þ ! G and k? ¼ 2ðm cos n sin Þ ! G? . Then the scattering intensity at the Bragg spots is calculated with an integration restricted within the window W of E? space as Z dx? expðiG? x? Þj2 : ð1:11Þ IðGÞ / j W

6


In icosahedral quasicrystals. the Bragg spots are indexed with a set of integers ðm5 ; m0 ; m1 ; m2 ; m3 ; m4 Þ [18].

1.4.

Phonons and phasons

The most generic property in quasiperiodic lattices is ‘phason’ degree of freedom. The phase Gn in Eq. (1.6) represents the effects of the displacement of the lattice points in physical and perpendicular spaces and, therefore, is written as Gn ¼ Gn u þ G? n w;

ð1:12Þ

where u and w are displacement vectors in the physical and the perpendicular spaces, Ejj and E? , respectively. When the displacement vectors u and w depend upon the position vector r, u(r) and w(r) represent the phonon and the ‘phason’ displacement vectors. The phonon displacement u(r) modulates distances between nearest lattice points and the corresponding lattice points in n-dimensional hypercubic lattice moves within the window. By the phason displacement w(r), some lattice points in the n-dimensional hypercubic lattice move out of the window and some others move into it. This movement rearranges lattice points or the edges of tiles, which can be clearly seen in the grid method [19,20]. The modulation of grids and the corresponding tiling are depicted in Figure 1.4, where the phason displacement generates a rearrangement of edges of tiles and introduces a mismatch defect. The phonon causes thermal diffuse scatterimg around Bragg peaks, whose intensity is reduced by the Debye-Waller factor. As for the phasons, if the phason strain is spatially uniform (linear phason stain wðrÞ ¼ Mr, where M is a matrix.), the Bragg spots systematically shift from G to G þ G? M. This is actually observed in several diffraction experiments. Quenched random phason strains contribute to finite widths of diffraction spots. When the quenched phason is a random Gaussian variable, the widths of the spots are proportional to a power of jG? j [21]. If the phason strain is appropriately chosen, one can generate a periodic structure, which is called approximant. Several complex crystals with large unit cells can be understood as lower symmetric approximant phases of icosahedaral quasicrystals where the icosahedral symmetry is broken by mode locking of the phason stain [22].

Figure 1.4. Modulation of grids and corresponding phason. Dots indicate mismatch of tiling.

7

1.4. Phonons and phasons

Other models of the icosahedral phase against the quasiperiodic lattice are the icosahedral glass model [23] and the random tiling model [24]. Both models show the long-range bond orientational order and a diffraction pattern of icosahedral symmetry. In the icosahedral glass model, icosahedral cluster units are randomly aggregated whereas the random tiling model is such that the entire space is filled with tiles without overlaps and gaps. These models can be constructed by the projection method from a distorted or fluctuating higher dimensional lattice and understood in terms of the phason degrees of freedom. In the random aggregate of icosahedral clusters, a mean square deviation of the phason variables grows as a power of spatial distance and hence the quasiperiodic positional correlation is lost at the long distance. The diffraction spots for the icosahedral glass model have finite widths as realized in the case of quenched random phasons. In the random tiling model, unlike the glass model, the quasiperiodic positional correlation is maintained at the long distance. The random tiling model is understood as an icosahedral quasiperiodic lattice with highly condensed short wavelength phasons and the diffraction spots for the threedimensional random tiling model are sharp as for the ideal quasiperiodic lattice but with diffuse tails [25,26]. The merit of these models is a gain of configurational entropy through the fluctuation of the phason variables and it is expected that the quasicrystalline phase is stabilized by entropy at higher temperatures [24]. In Table 1.2, we summarize the models of quasicrystals and characteristic spatial variation of the phason fields. Phenomenological theory of the phonons and phasons is derived from symmetry considerations [20]. The spatial derivatives of the phonon displacements represent rigid rotation and elastic deformation. In the icosahedral quasicrystals, the elastic strains are reduced to normal and shear strains, which transform under G1 and G5 representations, respectively and hence the elastic properties of the icosahedral quasicrystals are described with two independent elastic constants, which are expressed as the Lame coefficients l and m for isotropic elastic medium. The phason strains are classified into two modes transforming under the G4 and G5 representations, and thus we need two independent elastic constants for these phason modes. In addition, the G5 phason strain couples with the shear strain and we have another parameter describing the phonon–phason coupling. Stability of the icosahedral phase was discussed phenomenologically within the elasticity theory [26]. Table 1.2. Models of quasicrystals Model

Spatial variation of the phason fields

Diffraction spots

Quasiperiodic Linear phason strain (approximant) Icosahedral glass (quenched random phason)

No Linear ð~ wð~ r Þ ¼ M~ rÞ Random ~ð~ ~ð0Þj2 i / r Þ ðhjw rÞw Random ~ð~ ~ð0Þj2 i finiteÞ ðhjw rÞw

Sharp peak ~¼ G ~? MÞ Peak shift ðDQ

Periodic ð~ wð~ r Þ / sin ~ q ~ rÞ

Satellite scatterings ~ ¼ ~ ðDQ qÞ

Random tiling Phasonic modulation

~? j2= ) Finite width (/ jG Sharp peak with diffuse tails

8


1.5.

Electronic structure and related physical properties

Important point in electronic structures is that it seems to be correlated strongly to the structures of quasicrystals. Since stable icosahedral quasicrystals can be grown only in a narrow range of the electron-per-atom ratios (e/a ratio) with a very definite number of the ratios of the atomic diameter d to the quasilattice constant aR , quasicrystals may be stabilized primarily by the electronic mechanism [7]: e 1:6 1:8 ðTMÞ; a d 0:61 ðTMÞ; aR

e 2:1 2:25 ðFKÞ a

d 0:57 ðFKÞ: aR

Quasicrystals lack periodicity and the Bloch theorem can not be applied to the electronic structures. Then it seems that the electronic wavefunctions could not be extended over whole system. On the other hand, quasicrystals consist of metallic elements, especially the FK-type alloys contain only good metallic ones. Therefore, we can imagine that quasicrystals are good metals or presumably similar to crystals or amorphous metals. On the other hand, wave propagation properties in onedimensional quasiperiodic lattices were investigated throughout and the spectrum is very anomalous. When stable quasicrystals with almost perfect structural order were discovered, we realized that the physical properties are quite different from crystalline alloys as summarized below; 1. Anomalously large resistivity at low temperatures, e.g., 1:0 cm in AlPdRe at 4.2 K. 2. Decreasing resistivity with increasing temperature, e.g., (4K)/(300K) 190 in AlPdRe. 3. Lower conductivity in higher structurally ordered samples. 4. Temperature dependent Hall conductivity and thermoelectric power. 5. Complicated behavior of low-field magnetoconductivity at low temperatures and the importance of the quantum coherence. 6. No Drude peak in infrared spectra. In this volume, the state of the arts in several fields of quasicrystals will be discussed.

References [1] R. Penrose, Bull. Inst. Math. Appl. 10 (1974) 266; M. Gardner, Sci. Am. 236 (1977) 110. [2] N. G. de Bruijn, Ned. Akad. Weten. Proc. Ser. A 43 (1981) 27; 43 (1981) 29; 43 (1981) 53. [3] A. L. Mackay, Physica 114A (1982) 609. [4] P. Kramer, Acta Cryst. A38 (1982) 257. [5] D. Shechtman, I. Blech, D. Gratias, and J.W.Cahn, Phys. Rev. Lett. 53 (1984) 1951. [6] D. Levine and P. J. Steinhardt, Phys. Rev. Lett. 53 (1984) 2477.

References

9

[7] A. P. Tsai, Physical Properties of Quasicrystals, edited by Z. M. Stadnik (Springer, 1999), p.5. [8] A. P. Tsai, J. Q. Guo, E. Abe, H. Takakura, and T. J. Sato, Nature 408 (2000) 537. [9] L. Bendersky, Phys. Rev. Lett. 55 (1985) 1461. [10] N. Wang, H. Chen, and K. H. Kuo, Phys. Rev. Lett. 59 (1987) 1010. [11] M. Conrad, F. Krumeich, and B. Harbrecht, Angew. Chem. Int. Ed. 37 (1998) 1384. [12] X. Zeng, G. Ungar, Y. Liu, V. Percec, A. E. Ducley and J. K. Hobbs, Nature 428 (2004) 157; T. Dotera and T. Gemma, Philos. Mag. 86 (2006) 1085. [13] P. A. Kalugin, A. Yu. Kitaev, and L. S. Levitov, Pis’ma Zh. Eksp. Teor. Fiz. 41 (1985) 119. [14] P. Bak, Phys. Rev. Lett. 56 (1986) 861. [15] J. E. S. Socolar, P. J. Steinhardt, and D. Levine, Phys. Rev. B32 (1986) 5547. [16] D. Levine and P. J. Steinhardt, Phys. Rev. B34 (1986) 596. [17] J. E. S. Socolar and P. J. Steinhardt, B34 (1986) 617. [18] P. A. Bancel, P. A. Heiney, P. W. Stephens, A. I. Goldman, and P. M. Horn, Phys. Rev. Lett. 54 (1985) 2422; V. Elser, Phys. Rev. B32 (1985) 4892. [19] P. Bak, Phys. Rev. B32 (1985) 5764; J. E. S. Socolar, T. C. Lubensky, and P. J. Steinhardt, Phys. Rev. B34 (1986) 3345. [20] D. Levine, T. C. Lubensky, S. Ostlund, S. Ramaswamy, P. J. Steinhardt and J. Toner, Phys. Rev. Lett. 54 (1985) 1520. [21] T. C. Lubensky, J. E. S. Socolar, P. J. Steinhardt, P. A. Bancel, and P. A. Heiney, Phys. Rev. Lett. 57 (1986) 1440; P. M. Horn, W. Malzfeldt, D. P. DiVincenzo, J. Toner, and R. Gambino, Phys. Rev. Lett. 57 (1986) 1144. [22] V. Elser and C.L. Henley, Phys. Rev. Lett. 55 (1985) 2883; Y. Ishii, Phys. Rev. B39 (1989) 11862. [23] D. Shechtman and I. A. Blech, Metall. Trans. 16A (1985) 1005; P. W. Stephens and A. I. Goldman, Phys. Rev. Lett. 56 (1986) 1168. [24] C. L. Henley, J. Phys. A21 (1988) 1649; M. Widom, D. P. Deng and C. L. Henley, Phys. Rev. Lett. 63 (1989) 310. [25] M. V. Jaric and D. R. Nelson, Phys. Rev. B37 (1988) 4458. [26] M. Widom, Philos. Mag. Lett. 64 (1991) 297; Y. Ishii, Phys. Rev. B45 (1992) 5228.


Chapter 2

Recent Development of Quasicrystallography Akiji Yamamoto1 and Hiroyuki Takakura2 1 National Institute for Materials Science, Tsukuba, 305-0044, Japan 2 Applied Physics, Graduate School of Eng. Hokkaido Univ, Sapporo, 060-8628, Japan

2.1.

Introduction

The icosahedral quasicrystal (QC) found in an Al–Mn alloy system in 1984 [1] opened a door for new crystallography. This has no periodicity along any direction in the three-dimensional (3D) physical space but shows Bragg spots in the diffraction patterns. Before 1984, it has been known that there exist two kinds of crystals without periodicity (at least along one direction) which show Bragg spots in the diffraction pattern. They are called modulated and composite crystals. A characteristic feature of the QC is in its point symmetry. It is non-crystallographic in contrast to the crystallographic point symmetry for the former two. This allows the existence of 5-fold, 8-fold, 10-fold, and 12-fold rotation axes, which never appear in the former two. Solids which have no periodicity but show Bragg spots are called aperiodic crystals. They can be classified into two classes. Let us call aperiodic crystals with a crystallographic point group semicrystals to distinguish from QCs. All aperiodic crystals found so far can be described as crystals in an n-dimensional (nD) space with n > 3 [2]. Therefore their symmetry can be specified by an nD space group. The 3D physical space is called the external and the complementary space normal to it, the internal space. Because of the non-crystallographic point symmetry, the structure analysis of QCs requires a new method which is largely different from that of semicrystals, although all of them are described as crystals (periodic structures) in nD space. Therefore we call crystallography necessary for QCs, quasicrystallography. Historically, a higher-dimensional crystallography was introduced for the analysis of modulated crystals [3]. This was extended so as to include composite crystals. After the discovery of QC, it is further extended to accommodate QCs. There are two classes in QCs, one of which has a periodicity along one direction, while the other has no period along any direction. The former is called dihedral HANDBOOK OF METAL PHYSICS ISSN 1570-002X/DOI 10.1016/S1570-002X(06)02002-7 11


12

Chapter 2. Recent Development of Quasicrystallography

(polygonal) QCs, while the latter is said as icosahedral QCs. In the dihedral cases, octagonal, decagonal, and dodecagonal QCs were found so far, which have 8-fold, 10-fold, and 12 fold axes, respectively. Icosahedral QCs have six 5-fold, ten 3-fold, and fifteen 2-fold axes, forming an icosahedral group. Soon after the discovery of quasicrystals, some kinds of atom clusters with high symmetry and same orientation were frequently observed in high-resolution transmission electron microscopy (HRTEM) images. In general, alloys are considered to have no structural units as molecules within which atoms are strongly bonded. Nevertheless, a quasicrystal model consisting of atom clusters facilitates the description of a whole quasicrystal structure. This is called a cluster model. In this model, the quasicrystal can be described by the clusters and glue atoms linking them. It is obvious that if more atoms are included in the cluster the description becomes simpler, because many atom positions are determined by one cluster center position. The higher-dimensional cluster model is a model paying attention to this point of view. In this model, aperiodically distributed clusters in the 3D space are described by a periodic structure in nD space. It is known that X-ray structure analysis is the most powerful method for the determination of global crystal structures. The same holds for QCs, although the HRTEM is also useful for observing their local structures. Since the QC is aperiodic in the external space, it is necessary to employ an nD crystallography for describing their structure with a finite number of parameters. In particular, the X-ray analysis became more important after the success in growing single crystals [4]. This enabled us to determine atomic structures as accurate as usual crystals in principle. The difficulties which are specific to QCs are the parametrization of an nD crystal and the phase determination of diffraction intensities. As mentioned above, it is clear that we need the nD crystallography. However, we can consider several models to describe an nD structure. Basic mathematical concepts for describing QCs in nD space have been shown before the discovery of QCs [5,6]. Soon after the discovery, the basis of the QC symmetry has been given [2]. There are several approaches to describe QC structures [7]. Among them, the section method is most important for the quasicrystallography [2,8], where the structure in the external space is given by an intersection of an nD periodic structure passing through a 3D hyper-plane parallel to the external space. In QCs, their structure is determined by the occupation domain (OD), which is extended in the internal space. Different from the modulation function in semicrystals, the OD is defined within a limited area (or volume).1 An atom position of each atom in the external space is obtained as the cross point between an OD and the external space. The point density of atoms in the external space is proportional to the size (area or volume) of the OD. Therefore, the determinations of the location of ODs and their shape and size are essentially important for the structure analysis of QCs. Most essential concepts necessary for the structure description and determination of QCs have been given before 1996. For these subjects, more detailed reviews are available [7,10]. On the other hand, detailed structure analyses of QCs using more than several hundreds independent reflections have been carried out after 1998. 1

Recently, a special kind of semicrystals was shown to have ODs [9].

2.1. Introduction

13

High quality single-domain QCs enabled us to analyze their detailed structure. The combination of high-quality samples and synchrotron radiation experiments gave more than 50000 observed intensities which include more than 1000 independent reflections. Such data include many weak diffraction intensities and in order to explain them, more sophisticated structure models developed recently are needed. Most detailed structure analyses of QCs made so far are based on higherdimensional cluster models, in which several kinds of atom clusters are included as important building units. The existence of the clusters in QCs is confirmed by HRTEM images in particular in decagonal QCs. (The term ‘cluster’ is used for ‘structure motif ’ or ‘structural unit’ in the quasicrystal community). The structure analysis based on the nD quasicrystallography was recently clarified to be efficient for any QCs [11]. This has, however, a limitation which can be complemented by other methods [12,13]. Recently, a methodologically important progress has been achieved in the direct method, which facilitates the building of an initial model for the structure refinement. The low-density elimination method (LDEM), which was developed for complicated crystals like protein, was extended so as to be applicable to icosahedral QCs [14]. It is well known that if we know the diffraction intensity of a QC and its phase, we can calculate its electron density distribution [7]. This can determine the rough shape and size of ODs from the electron density map in nD space which is obtained by the Fourier synthesis [15] as well as their location. This facilitated the initial model building of QCs, although the model building is still necessary and is the most difficult parts of the QC structure refinement. To obtain high resolution electron density maps, we need to include reflections with large diffraction vectors in both external (physical) and internal (complementary) spaces. In particular, reflections with a diffraction vector with large internal space component give a high resolution in the shape of ODs. Such a diffraction intensity, however, decreases rapidly with the increase of the internal space component. Therefore if we want to know more detailed shape of the OD, we need weaker diffraction intensities. To measure such weak intensities accurately, diffraction experiments using synchrotron radiation are known to be efficient. They may clarify more detailed initial models but at the same time, we need to consider that many weak reflections may be affected by the multiple scattering effect which is usually neglected in the structure analysis. A solution of such a problem in experiments will not be stated in this chapter. Another progress is seen in the model building. The ODs of Elser model for icosahedral Al–Pd–Mn (i-Al–Pd–Mn) quasicrystals were given by Kramer et al. [16]. This model includes many statistically occupied ODs, since the total volume of the ODs are much larger than a necessary volume giving observed density. The subdivision of ODs for another model for i-Al–Pd–Mn proposed earlier was given recently by Gratias et al. [17]. This clarified the different local environment of the so called Katz-Gratias model, which includes no statistical sites. The structure analyses based on these models have not yet been performed. Detailed structure analyses of quasicrystals have been performed in decagonal Al–Mn–Pd (d-Al–Mn–Pd) for the first time [18]. In this analysis, building units (clusters) were considered to maintain 10-fold symmetry everywhere as an approximation but it was clarified that this is not good enough to explain very weak reflections in d-Al–Ni–Co. The use of many weak reflections observed by synchrotron experiments gave us more detailed

14


information. This seems to require symmetry breaking in constituent clusters in general. This is quite natural because clusters in QCs are in different low-symmetric local environments if we consider their far neighbors, since the QC has no period in the (at least 2D) external space. A model with lower-symmetric clusters were introduced in the structure refinement of d-Al–Ni–Co [19,20]. A structure refinement of icosahedral quasicrystals are performed in i-Al–Pd–Mn for the first time, followed by similar analyses for i-Al–Cu–Fe, i-Al–Cu–Ru and i-Al–Pd–Re [21–23]. We will review such a development in model building. We do not, however, intend to give a comprehensive review in the recent development of quasicrystallography but mainly overview subjects which are important in the structure analysis of quasicrystals. A more comprehensive review for recent works in decagonal quasicrystals which are not mentioned here is available [13,24]. The arrangement of this chapter is as follows. In the next section, a description of QCs in nD space is briefly summarized. The coordinate systems used in this chapter is described in Section 2.3, since several different coordinate systems are used in the literature. Section 2.4 gives the low-density elimination method. In Section 2.5, several useful methods for checking the validity of models in the structure analysis are described. The following sections are devoted to the model building. In Section 2.6, a primitive 5D cluster model for decagonal Al–Cu–Co QCs is discussed. This provides the basis of more complicated and sophisticated model buildings. In Section 2.7, this is extended to icosahedral QCs, for which we need 6D cluster models. In dodecagonal Ta–Te (dd-Ta–Te) QCs, we encountered a fractal ODs. This is described in Section 2.8. In Section 2.9, we discuss the symmetry breaking in clusters for the basic d-Al–Ni–Co phase. In Section 2.10, we briefly discuss a modulation functions of QCs for future studies.

2.2.

nD description of quasicrystals

This section gives a short summary of quasicrystallography which is necessary for the understanding of recently developed methods or models discussed later. In order to describe QCs as crystals in nD space, it is necessary to consider periodic structures in 5D space for dihedral and 6D space for icosahedral QCs. The necessary space dimension is equal to the number of vectors which is needed for the indexing of diffraction patterns. For octagonal, decagonal, and dodecagonal QCs, five vectors are necessary, so that they are described as crystals in 5D space, while for icosahedral QCs, it is six. It is known that the diffraction intensity, I, is proportional to the square of the Fourier amplitude F of the electron density ðxÞ (for X-ray diffraction, nuclear density for neutron diffraction): I / jFj2. The (complex) Fourier amplitude is called the structure factor and it is, by the definition, given by Z e ðxe Þexpð2i he xe Þdxe Fðh Þ ¼ ð2:1Þ

where xe is the positional vector in the external space and the integration is taken over the entire volume of the crystal . (The superscript e means the external space

2.3. Decagonal, dodecagonal, and icosahedral coordinate systems

15

component of an nD vector, while its internal space component is shown by the superscript i). If the electron density ðxe Þ has a periodicity, the Fourier amplitudes has a non-zero value at the nD reciprocal lattice points in the external space, giving the Bragg spots (Bragg reflections) in the diffraction pattern. Then each Bragg reflection is indexable with three unit vectors a , b , c and three integers h; k; l: he ¼ ha þ kb þ lc . The vector he is called a diffraction vector, since it determines the direction of a diffracted X-ray beam. Aperiodic crystals show Bragg spots in the diffraction pattern but they are not e e indexable with three vectors and necessitatePn ðn > 3Þ vectors, d e 1 ; d2 ; ::; dn , and n n e e integers, h1 ; h2 ; ::; hn , for the indexing: h ¼ i¼1 hi di . This relation enables us to consider the following interpretation of the diffraction patterns. In the nD crystallography, an observed structure factor Fðhe Þ with the of the structure factor FðhÞ at an diffraction vector he is regarded as the projection P nD reciprocal lattice point specified by h ¼ ni¼1 hi d i onto the external space, where d i ði ¼ 1; 2; ::; nÞ are nD vectors with their external space components d e i . Because of the mathematical properties of the Fourier transformation (Fourier integral), the Fourier synthesis by the projected structure factors gives a cut of the nD electron density which is normal to the projection axis. The nD electron density is expressed in terms of the structure factor FðhÞ ¼ Fðhe Þ: ðxÞ ¼

1X FðhÞexpð2i h xÞ V h

ð2:2Þ

where V is the nD unit cell volume. In fact the cut of ðxÞ at a 3D hyperplane, ðxe Þ, is given by ðxe Þ ¼

1X 1X FðhÞexpð2i h xe Þ ¼ Fðhe Þexpð2i he xe Þ V h V he

ð2:3Þ

since h xe ¼ he xe . (Note that the last term in the above equation is the Fourier synthesis by the projected structure factors.) This property is independent of the space dimension. Therefore, the electron density in the external space is given by the intersection of the nD electron density at a 3D hyperplane which is normal to the ðn 3ÞD internal space. Thus an aperiodic crystal structure can be described as a periodic structure in nD space and its electron density in the external space is obtained from the corresponding nD electron density by cutting it at a 3D hyper-plane. From this fact, this method is called the section method. (Crystallographers frequently use its inverse relation to obtain a projected electron density by using reflections on a plane which is normal to the projection axis). It is shown that such hyper-planes are locally isomorphic. This means that each structure is the same except for the origin translation.

2.3.

Decagonal, dodecagonal, and icosahedral coordinate systems

In this section, we give only decagonal, dodecagonal and icosahedral coordinate systems since we discuss QCs with only these symmetries. As mentioned above, the

16


necessary dimension for describing QCs as a periodic structure in a higher-dimensional space agrees with the number of vectors needed for indexing of the diffraction pattern [2]. This is five for decagonal and dodecagonal and six for icosahedral cases. To describe a QC as a crystal in 5D or 6D space, we use the following coordinate systems. The unit vectors d j in the decagonal reciprocal lattice are given as follows. pffiffiffi d j ¼ ða = 5Þ½cðj1Þ a1 þ sðj1Þ a2

pffiffiffi ð2:4Þ þ ða 0 = 5Þ½c2ðj1Þ a4 þ s2ðj1Þ a5 ð j ¼ 1; 2; 3; 4Þ; d 5 ¼ c a3 ¼ c ; where cj ¼ cosð2j=5Þ, sj ¼ sinð2j=5Þ, a , and a 0 are the lattice constants of the 4D reciprocal lattice in the external and internal spaces. The vectors a1 , a2 , a3 represent the unit vectors of the external space and a4 , a5 , those of the internal space. They are mutually orthogonal: ai aj ¼ ij . Since the scaling in the internal space is physically meaningless, a 0 is often taken to be equal to a . We take a ¼ a 0 throughout this chapter. Then the unit vectors of the lattices reciprocal to d j are given by pffiffiffi dj ¼ ð2a= 5Þ½ðcj 1Þa1 þ sj a2 þ ðc2j 1Þa4 þ s2j a5 ð j ¼ 1; 2; 3; 4Þ; d5 ¼ ca3 ¼ c;

ð2:5Þ

where a and c are the lattice constants of the direct lattice which are given by a ¼ 1=a and c ¼ 1=c . Similarly, the dodecagonal unit vectors are defined by pffiffiffi d j ¼ ða = 2Þ½cðj1Þ a1 þ sðj1Þ a2 þ c5ðj1Þ a4 þ s5ðj1Þ a5 ð j ¼ 1; 2; 3; 4Þ; d 5 ¼ c a3 ¼ c

ð2:6Þ

and pffiffiffi djþ1 ¼ ð2a= 6Þ½cðj1Þ a1 þ sðj1Þ a2 þ c5ðj1Þ a4 þ s5ðj1Þ a5 ð j ¼ 0; 1Þ; pffiffiffi djþ1 ¼ ð2a= 6Þ½cðjþ1Þ a1 þ sðjþ1Þ a2 þ c5ðjþ1Þ a4 þ s5ðjþ1Þ a5 ð j ¼ 2; 3Þ; d5 ¼ ca3 ¼ c;

ð2:7Þ

where cj ¼ cosð2j=12Þ, sj ¼ sinð2j=12Þ. On the other hand, the unit vectors of the reciprocal lattice in icosahedral QCs are given by d 1 ¼ ða =2Þ½a3 þ a6 ; d i ¼ ða =2Þ½ðci a1 þ si a2 Þs þ ca3 þ ðc2i a4 þ s2i a5 Þs ca6 ði ¼ 2; 3; :::; 6Þ

ð2:8Þ

pffiffiffi where ci ¼ cosð2i=5Þ, si ¼ sinð2i=5Þ, c ¼ cos ¼ 1= 5, s ¼ sin . The vectors a1 , a2 , a3 represent the unit vectors of the external space and a4 , a5 , a6 , those of the internal space.

2.4. Low-density elimination method

17

The unit vectors di ði ¼ 1; 2; :::; 6Þ of the direct lattice are defined by d1 ¼ a½a3 þ a6 ; di ¼ a½ðci a1 þ si a2 Þs þ ca3 þ ðc2i a4 þ s2i a5 Þs ca6 ði ¼ 2; 3; :::; 6Þ

ð2:9Þ

where a ¼ 1=a . In diffraction experiments, we cannot obtain the phase of the structure factor F but the direct method mentioned in the next section can estimate a probable phase of each structure factor. Once we have phases of structure factors for all observed reflections, we can calculate the electron density by Eq. (2.2). The QCs show a characteristic electron density distribution in nD space, which is common to all QCs. The electron density peak continuously spreads within some limited region almost parallel to the internal space and outside of this region, it rapidly drops down to zero, suggesting the existence of ODs. The atom position of each atom in the external space is given by a cross point of the occupation domain and the external space as mentioned later. It is, therefore, essentially important to determine the location of the ODs in nD space and their shape and size for the structure analysis.

2.4.

Low-density elimination method

To obtain an initial model for the structure refinement, we can employ a direct method in nD space, which was first applied to an icosahedral QC by extending the low-density modification method for complicated crystals like proteins [25]. The method uses random phases as initial phases of independent reflections under an assumed space group symmetry. The initial electron density given by the Fourier synthesis (Fourier transformation) shows positive and negative peaks in general. Provided that the electron density of the Fourier map is almost flat near the zero level, the method suppresses small peak height by the density modification function. The prior density ðrÞ are modified by the following formula to obtain a new density 0 ðrÞ. (

"

#) 1 ðrÞ 2 for ðrÞ 0; ðrÞ ¼ ðrÞ 1 exp 2 0:2c 0

0

ðrÞ ¼ 0

ð2:10Þ

for ðrÞ < 0;

where c is an average peak height of the electron density distribution. The phases of the next cycle are calculated from the modified electron density by the inverse Fourier transformation. The electron density is calculated again by the Fourier synthesis. This cycle is repeated so far as the phases change. This method is based on the fact that the electron density is positive everywhere and it is almost flat except for the vicinity of atoms and peaks smaller than real atom peaks do not appear. This holds both in usual crystals and aperiodic crystals including modulated structures and QCs. This was first applied to an icosahedral QC [14] by modifying a program for usual crystals. Then the program has been extended for the use in 3D to 6D crystals [26]. This covers all known semicrystals and QCs.

18


The resulting electron density, however, depends on the initial phases. This means that we need to choose probable results from several sets among obtained electron densities. A true electron density map should be almost flat in the lowelectron density region and high-electron density appears only in the vicinity of atom position. In the Fourier synthesis, we can obtain the absolute value of the structure factor jFðhÞj from the corresponding diffraction intensity by IðhÞ but we do not know the scale factor . In addition, intensity at h ¼ 0 is not given by diffraction experiments. Therefore to obtain initial electron density used in LDEM, we use only observed reflections in Eq. (2.2) excluding Fð0Þ in the summation. The resulting electron density takes positive and negative values depending on the position. The method leads to different electron density depending on the initial phase as mentioned above. An important measure to select the most probable model is the R-factor, which is obtained from the observed and calculated structure factor. Note that the calculated structure factor is given by the calculated density 0 . Our experiences show that the probable model has the R-factor of about 0.3. This method enables us to determine the phases of observed reflections. From the Fourier synthesis after the phase determination, atom positions emerged in the nD space. We could easily distinguish light atom (like Al) position from transition metal atom position in the Fourier map. (Figure 2.1.)

(b)

(a)

b

b

c a

(c)

Z(min) = –0.333 Z(max) = 16.252

c

(d)

a

Z(min) = –0.043 Z(max) = 20.873

Figure 2.1. The isosurfaces of the electron density in the internal space around the origin and ð1=2; 0; 0; 0; 0; 0Þ of i-Al–Pd–Mn QCs (a) and (b) and their 2D cut planes (c) and (d) normal to the 2-fold axis (The 3D graphics program VEND is used [29]).

2.5. Modification of models in the structure refinement

2.5.

19

Modification of models in the structure refinement

Based on the initial electron density given by LDEM, we can construct an initial model in principle. The model building is, however, not straightforward in contrast to usual crystals. This is the subject of the following sections. Before discussing these, we shall mention other techniques which are efficient for the structure refinement of QCs in this section. In the course of the structure refinement, we need to check the validity of a model used as in usual crystals. The difference Fourier maps in nD space or the maximum entropy method (MEM) maps are available for it. The calculations of MEM maps use the Fourier transformation repeatedly as mentioned below. For the calculation of these maps the use of the fast Fourier transformation (FFT) is quite efficient. In the calculations of nD Fourier maps by this method, we need, however, large computer memory. As is shown in the following, this can be avoided by the use of the similarity transformation which is specific to QCs [27]. Once we obtain the phases of observed structure factors, we can calculate an electron density in nD space by the Fourier synthesis. This is, however, affected by the well-known truncation effect coming from the truncation of the Fourier series: in Eq. (2.2), we cannot observe all non-zero structure factors because of the limitation in an experimental method used. The observed reflections are limited by the wave length of X-ray used and we cannot observe the reflections with the diffraction vectors which are larger than the diameter of the Ewald sphere given by 2= . In addition, a stronger limitation is caused by the geometry of the diffractometer. In alloys, there exist many strong reflections out of such an observation limit. This may cause a strong truncations effect, although the use of high-energy synchrotron radiation can improve the situation very much. To reduce the truncations effect and to obtain more realistic electron density, MEM was applied to QCs. This is used to obtain detailed electron density of usual crystals but it is also applicable to QCs. Two ways have been so far proposed. One method calculates the electron density in the external space and after that it is lifted up to the nD space [28]. In the second method, the electron density of nD space is directly calculated. In the latter method, some sophisticated calculation technique is sometimes needed in order to overcome limited size of the computer memory [27].

2.5.1.

Maximum entropy method

For known structures, we know the phases of all observed reflections. In MEM, the electron density ðxÞ at x in nD space is calculated from the prior density ðxÞ repeatedly by ðxÞ ¼ ðxÞ expðDðxÞÞ

ð2:11Þ

20


with DðxÞ ¼

P

½Fobs ðhÞ Fcal ðhÞ expð2i h xÞ= 2 ðhÞ;

h

ð2:12Þ

where and ðhÞ are the Lagrange multiplier and the estimated error of reflections. The calculated structure factors are given by the Fourier transform of ðxÞ: Fcal ðhÞ ¼ V

P

ðxÞ expð2i h xÞ;

x

ð2:13Þ

where V is the unit cell volume. They are scaled so as to give the same average electron density which is equal to Fobs ð0Þ=V, where 0 is the zero vector. The rescaled structure factors are used in the DðxÞ calculation in the next cycle. To calculate all the necessary structure factors efficiently, we can use the fast Fourier transformation. An initial average electron density can be obtained by the structure refinement based on a model or the measured density and the chemical composition of a sample. The initial density used in MEM can be ðxÞ ¼ Fð0Þ=V, which is flat everywhere. The iteration of the density modification mentioned above converges an electron density which is positive everywhere as is ensured by Eq. (2.11). The inverse Fourier transformation of this density gives many non-zero structure factors which are not observed but they are necessary to obtain an electron density which is positive everywhere. In the iteration, the Lagrange multiplier can be set at some value. If it is too large, the iteration does not converge, while if it is too small, the convergence is slow. An appropriate value depends on the cycle or the prior density. To check the convergence, the R-factor is available. In Eq. (2.11) the electron density is calculated at each grid point in the nD unit cell. In Eq. (2.13), the summation with respect to x runs all the grid points. This requires large computer memory in particular in the icosahedral case as mentioned below. To calculate an accurate electron density function ðxÞ with the minimum grid points, we can use the self similarity of QC structures as shown in the following.

2.5.2.

Similarity transformations

A problem of MEM in higher-dimensional crystallography is the lack of a computer memory. In this method, the unit cell includes many grid points where the electron density is calculated. The method was first used in the analysis of a decagonal Al–Mn–Pd QC [18], which required (20)5 grid points in a unit cell. A similar method may need (20)6 points or more for a face-centered icosahedral QC which is described in 6D space. This will necessitate a computer with several GB memory. The situation can be improved if we note that the resolutions in the external (physical) and internal (complementary) spaces are different. As is well-known, a reflection having the diffraction vector with a large internal space component is weak [7] and usually cannot be observed. This suggests that the resolution in the internal space is limited

2.5. Modification of models in the structure refinement

21

in principle. In the following, we demonstrate an efficient use of the similarity transformation in face-centered icosahedral QCs, where the memory size problem is most serious. The similarity transformation matrix in QCs transforms a set of unit vectors in nD space into an equivalent one [7]. The transformed unit vectors should give the same unit cell volume, so that the determinant of S is one. P The diffraction vectors of a face-centered icosahedral lattice are given by h ¼ 6i¼1 hi d i with hi all even or all P odd. The corresponding lattice vectors in the direct space are given by l ¼ 6i¼1 li di combined with 32 centering translations [2]. In this case, the matrix S is given by 0

1 B1 B B 1 1B S¼ B 2B 1 B B @1 1

1 1

1 1

1 1 1 1

1 1

1 1

1 1 1 1

1

1

1

1

1 1 1C C C 1 C C: 1 C C C 1A

1

1

1

1

1

ð2:14Þ

P The unit cell volume defined by the vectors d0i ¼ 6j¼1 Sij dj is the same as the original one but the external components of d0i are multiplied by the factor p ¼ ð1 þ 5Þ=2 and internal ones, by its inverse. The diffraction vectors with P small internal space components can be given by h ¼ 6i¼1 hi d i ’ with P d i ’¼ 6j¼1 S~nij d j ðhmax =2 hi hmax =2Þ with small hmax , where the tilde means the transposition. The corresponding unit vectors in the direct space are P d0i ¼ 6j¼1 Sn ij dj . Since the resolutions of the electron density in the external and internal spaces are determined by the cut-off values of diffraction vectors in the external and internal spaces, the coordinate system appropriate for the generation of diffraction vectors with small internal components is also suitable for MEM calculations. Reasonable resolutions can be obtained by choosing appropriate n and hmax . P 0 If the transformed indices hi ¼ 6j¼1 hj S~nji for observed reflection are within the 0

range hmax =2 hi hmax =2, we can use all observed reflections in the Fourier transformation. Since the necessary memory size is proportional to ðhmax Þ6 , a slight increase of hmax leads to a large total size. Therefore for getting smallest hmax necessary for the Fourier map, we need to choose the best n. This means that if an appropriate coordinate system is chosen at the beginning, such a similarity transformation is not necessary. The coordinate systems used so far in the literature are however not necessarily suitable for MEM. In the case of i-Zn– Mg–Y, the best results were obtained for n=2 in the coordinate system with a ¼ 2a0 ¼ 10:4 A˚ [27]. In face-centered icosahedral QCs, the conventional unit cell volume is 64 ð26 Þ times larger than that of primitive icosahedral QCs because of the doubling of the lattice constant but its primitive cell has only doubled volume since 32 centering translations exist. Therefore the use of the primitive cell reduces the number of necessary grid points drastically.

22


The unit vectors of the primitive cell are given by dip ¼ 0

0 B0 B B 1 1B M¼ B 2B B0 B @0

1 0

0 1 1 0

0 0

0 0

0 0

0

0

1

0

0

0

1

P6 j¼1

1 0 0 0 1 C C C 1 0C C: 1 1C C C 1 0A 0

Mij dj ’ with

ð2:15Þ

1

Since the conventional coordinate system is convenient to describe the structure, we can use the coordinate system given by dip only for the calculation of MEM maps. In this method, the MEM maps for face-centered icosahedral QCs can be calculated at ð12Þ6 ð16Þ6 grid points in the primitive cell. This technique enabled us to use recent personal computer for the calculation of 6D Fourier or MEM maps [30]. From the electron density calculated at the grid points in nD unit cell, we can calculate the density at grid points in any 2D or 3D intersection plane by interpolation. This method is available for obtaining a 2D electron density map and isosurface in 3D space. As an example of isosurfaces, we show those of the icosahedral Al–Pd–Mn QCs in the internal space which are calculated from a 6D electron density by Fourier synthesis. Figures 2.1(a) and (b) show the isosurfaces of the electron density in the internal space passing through ð0; 0; 0; 0; 0; 0Þ and ð1=2; 0; 0; 0; 0; 0Þ. In i-Al–Pd–Mn, there are large ODs at these points and at ð3; 1; 1; 1; 1; 1Þ=4. The isosurfaces reflect the point symmetries of these positions and indicate the shapes of ODs located at these points. As is clear from the figures, the isosurfaces are not spherical but show a complicated shape maintaining icosahedral symmetry. (Note that their site symmetry is both icosahedral: m35). From the 2D maps shown in Figures 2.1(c) and (d), we can estimate the distribution of elements within these ODs, since the high-density parts correspond to heavy atoms.

2.5.3.

Application of LDEM to quasicrystals

The LDEM was first applied to an i-Zn–Mg–Ho quasicrystal [14,31]. This is a facecentered icosahedral quasicrystal. In this alloy system, a primitive icosahedral quasicrystal was also found and its 6D electron density was also obtained [32]. Another icosahedral quasicrystals Cd–Yb with different constituent clusters were also solved by LDEM [33].

2.5.4.

Application of nD maximum entropy method

A 6D MEM was first applied to i-Zn–Mg–Y quasicrystals for getting their initial models [27], while a 5D MEM was applied to Al70 Mn17 Pd13 for its structure refinement [34,18].

2.6. Higher-dimensional cluster models of decagonal quasicrystals

2.6.

23

Higher-dimensional cluster models of decagonal quasicrystals

Although LDEM gives an electron density of a probable model, the boundary of ODs are not sharp enough to determine the shape of ODs uniquely. Therefore to fix the model used in the structure refinement, we need additional information. In the case of decagonal QCs, the most powerful information was obtained from HRTEM images of decagonal QCs. In all cases found so far, many atom clusters are seen in the images. Their arrangement has no periodicity in at least 2D plane normal to the 10-fold axis but is quasiperiodic and not random. Therefore it is reasonable to construct a model consisting of such atom clusters. The atomic structure of the clusters are usually found in their crystal approximants. This is a crystal which has a chemical composition similar to that of a corresponding QC. Such crystal approximants show similar clusters found in HRTEM images but the clusters are arranged periodically. Therefore, we can imagine that QCs are composed of such clusters and the essential difference between the approximants and QCs exist in the arrangement of the clusters. The higher-dimensional cluster model shown below is derived from such an idea. In the cluster models, we construct the whole atoms in a QC as an arrangement of several (usually two or three) kinds of high symmetric clusters. Such a cluster can be derived by a higher-dimensional model in the following way. The model in the higher dimensional space producing a cluster in the 3D space is called a higherdimensional cluster model. In this model, we consider first the location of the cluster center and corresponding ODs producing such arrangements by a cut at a 3D hyperplane. For the arrangement of the clusters, we usually assume an ideal tiling with simple ODs. These are usually given by small ODs located at special positions in nD unit cell with high site symmetry. The site symmetry restricts the shape of the ODs since it must have the same or higher symmetry than that of its position. The positions of atoms included in the cluster can be obtained from the OD of the cluster center by shifting it in parallel to the external space. When the OD for the cluster center is located at x0 and an atom is located at xe from the center in the external space, the atom position is obtained by the same OD which is at x0 þ xe . This method generates all atoms around the cluster center [7] (Note that if we have n atoms around the cluster center, we have n shifted ODs at n xe ’s). To obtain the arrangement of clusters in decagonal QCs, HRTEM images projected along the 10-fold axis were proved to be efficient [35]. For icosahedral QCs, however, it is less efficient because the clusters are overlapped in the image. The atom cluster is not aligned along the projected axis even when the projection is parallel to 5-, 3-, or 2-fold axis. Furthermore, the point densities or chemical compositions for overlapped atoms are different in general [36]. Therefore the arrangement is assumed on the basis of a quasiperiodic tiling. As an example, we describe the model building taken in d-Al–Cu–Co [37].

2.6.1.

Arrangement of cluster centers

There are two models with different cluster arrangements in d-Al–Cu–Co, which are proposed by Burkov and the present author [38,37]. The position of the cluster

24


a4

1 a3 2

Figure 2.2. The occupation domain for generating the cluster centers in the present model (innermost decagon) and that of the Burkov model (middle decagon). The arrows 1 and 2 represent 2 ei1 and 3 ðei1 ei3 Þ. The innermost and outermost decagons generate the Penrose pentagon tiling with the edge length of 20 A˚ and 12 A˚, respectively.

center in the Burkov model is obtained from a decagonal occupation domain placed at the origin of the 5D decagonal lattice. This is a pattern consisting of regular decagons and regular pentagons and their fragments with an edge-length of about 12 A˚ (Figure 2.3(b)). It is obtained from the decagonal occupation domain with the radius of 2 eij (j ¼ 0,1, ... ,4) located at the lattice points of the 5D decagonal lattice, where eij is the internal space component of the 5D vector pffiffiffi ej ¼ ð2a= 5Þðcj a1 þ sj a2 þ c2j a4 þ s2j a5 Þ. (The middle decagon in Figure 2.2). It is known that in the decagonal lattice, the lattice constant with n a with any integer n corresponds to a different setting of the same lattice. We use the coordinate system pffiffiffi with 2a= 5 ’ 2:4 A˚. (It should be noted that if we use a different setting, the same point in 5D space has different coordinates). This is a subset of the occupation domain for the Penrose pentagon tiling (PPT) with the same edge length, which is obtained from the decagon with the radius of 2 ðei1 ei3 Þ etc. [39] (the outermost decagon in Figure 2.2). On the other hand, the PPT with the edge length of 20 A˚ (Figure 2.3(a)) is obtained from the decagon which is smaller by 1 than this because of the self-similarity of the PPT (The innermost decagon). The comparison of Figures 2.3(a) and (b) and HRTEM images (Figure 2.4(a) in Hiraga et al. [40] and in particular, Figure 2.4(b) in Hiraga et al. [41]) clearly supports the former in d-Al–Cu–Co or d-Al–Ni–Co. The positions of all atoms are derived from the arrangement of cluster centers and the structure of constituent atom clusters as shown in the following. On the other hand, the cluster arrangement of d-Al–Mn–Pd is similar to Figure 2.3(b) but with an edge length of 20 A˚. Such a figure is obtained from the OD similar to the middle in Figure 2.2 but its size (edge length) is smaller by 1 . The structure analysis of d-Al–Mn–Pd based on such a model has been performed [18].

2.6.2.

Two-dimensional structure of d-Al–Cu–Co

As examples of the model construction discussed in the previous subsection, we consider two models of d-Al–Cu–Co and d-Al–Ni–Co [38,42].

25


(a)

(b)

a2

a2

a1

a1

Figure 2.3. The arrangements of cluster centers in the Yamamoto model (a) and the Burkov model (b). In (a) and (b), the edge-lengths of the pentagon are 20 and 12 A˚, respectively. In (b), the cluster is located at the center of each decagon in addition to every vertex. The pattern (a) is called the Penrose pentagon tiling.

4

B

A

E

1

7

4

8

5

2 6

A 2

(d)

8

6 4 3

1

2

(c)

B 3

1

1

7

9

(b)

(a)

E

5

3

3 2

4

2 1

2 3

(e)

1

(f)

Figure 2.4. The occupation domains of the Yamamoto model (a)–(c), and the Burkov model (d)–(f) of d-Al–Cu–Co. The domains E, A and B are located at ð0; 0; 0; 0; 0Þ, ð1; 1; 1; 1; 5 0:25Þ=5 and ð2; 2; 2; 2; 5 0:25Þ=5. The domain A is occupied by transition metals and B by Al, while the domain E is not occupied by any atoms but gives the vertices of the framework. In (b), (c), (e), and (f), the domains obtained from the X-ray analysis are denoted by dashed lines. The small domains in E, A, and B with the number i ði ¼ 1; 2; :::Þ and those related with them by the point symmetry 5m are referred to as Ei , Ai , and Bi in the text. The ODs E1 in (a), and (d) agree with the innermost and middle decagon in Figure 2.2, so that they create the vertices in Figures 2.3(a) and (b).

Both models are composed of similar small and large clusters. It is convenient to consider their atom distributions in the 2D external space normal to the 10-fold axis (projected structure) in the following two steps (a) and (b) shown below.

26


(a)

(b)

a2

C

a2

D C

A

E

a1

A

B

a1

B

Figure 2.5. The projected structures of the Yamamoto model (a) and the Burkov model (b) along the ten-fold axis. The black and gray circles represent transition metal and Al atoms. The inter-cluster distance AB is about 20 A, and that of CD is about 12 A in (a), while the corresponding distances in (b) are BC and AB. The clusters in the decagonal and pentagonal cages in (b) are the same as those in (a).

(a) Framework. We consider first the framework of decagons (cage) in Figure 2.5(a) in which the smaller atom clusters are accommodated. (The solid lines in Figure 2.5(a)). For simplicity, the domains with the number i in Figure 2.4 and equivalent to it under the point group symmetry are hereafter simply denoted by Ei , Ai , Bi , etc. The framework is obtained from the occupation domain E at the origin of the decagonal lattice (Figure 2.4(a)). This consists of decagons and pentagons. The central and ten overlapped decagons are same in size as the innermost decagon in Figure 2.2. The central decagon E1 in Figure 2.4(a) generates the centers (A, B, etc.), while the outer ten decagons E3 etc. at ðe1 þ ei2 ei4 ei5 Þ from the center generate ten vertices of the decagon (in the framework) with the radius of 7.6 A˚ around each cluster center (Figure 2.5(a)). A derivation of such points around some points in the external space is explained in detail in (b) below. The other vertices of the framework are obtained from the small pentagons with the radius of 4 ei1 etc. (E2 and E4 in Figure 2.4(a)). All the vertices of the framework are not occupied by atoms. Therefore the framework is only for the guide to the eyes but it is convenient to understand the structure. Similarly, the framework in the Burkov model (Figure 2.5(b)) is obtained from Figure 2.4(d), where the central decagon, E1 , is the same as the middle decagon in Figure 2.2. In Figure 2.5(b), most decagonal cages are overlapped in contrast to Figure 2.5(a). The centers of the decagonal cages correspond to the vertices in Figure 2.3, since they are obtained from the same occupation domains located at the origin. The centers of the pentagons and stars are however not obtained from the occupation domain at the origin but from those at ð2; 2; 2; 2; 5zÞ=5 and ð1; 1; 1; 1; 5zÞ=5, where in ðx1 ; x2 ; x3 ; x4 ; x5 Þ, xj ð j ¼ 1; 2; ::: ; 5Þ are the coordinates with respect to dj . The fifth coordinate z is irrelevant for the projected structure and discussed later. The points ð1; 1; 1; 1; 5zÞ=5 and ð2; 2; 2; 2; 5zÞ=5 are the special positions which are invariant under the point group 5m [2].

27


(a)

a2

–f4 –f3 –f5 –f –f1 2

(b)

a1

a2

f2 f1 f f3 f 5 4

a1

Figure 2.6. The two layers of d-Al65 Cu20 Co15 at z ¼ 1=4 (a) and 3/4 (b). The label fj ðj ¼ 1; 2; ::; 5Þ denote eej . (See text.)

The centers of the pentagons in Figures 2.5(a) and (b) are obtained from B1 +B2 and B1 in Figures 2.4(c) and (f), while the centers of the stars (e.g., E in Figure 2.5(a)) are from A1 in Figure 2.4(b). (b) Atom configuration. In both models, the common atom clusters are accommodated in each decagonal pffiffiffi and pentagonal cages. (See Figures 2.5(a) and (b).) The atoms at eej ¼ ð2a= 5Þðcj a1 þ sj a2 Þ ( j=0,1, ... ,4) in the external space from the cluster center are generated from the same occupation domains as that of the cluster center at the origin but located at eej . Then if the external space cut the OD for the center, it always cut these ODs, since the shift eej is parallel to the external space and the ODs are theP same as the OD for the center. From the relations ej ¼ eej þ eij ¼ dj 4i¼1 di =5 (see Eq. 2.5), the first nearest neighbor P atoms at

fj ¼ eej (Figure 2.6) are derived from the domains at ½dj 4i¼1 di =5 eij . the atoms at eej is located at Since eij is parallel to the internal space, the OD Pfor 4 the 2D internal space passing through dj þ i¼1 di =5. The coordinates of e1 , ð4; 1; 1; 1; 0Þ=5, are equivalent to ð1; 1; 1; 1; 0Þ=5 under the lattice translations. Thus the first nearest neighbor atoms at ee1 from the cluster center are obtained from the domain A4 at ð1; 1; 1; 1; 0Þ=5 in Figure 2.4(b) and corresponding domains D4 at ð1; 1; 1; 1; 0Þ=5. The occupation domains D and C located at ð1; 1; 1; 1; 0Þ=5 and ð2; 2; 2; 2; 0Þ=5 are the inversion of the domains A and B in Figure 2.4. The same consideration leads to the conclusion that atoms at eej are generated by the ODs which is obtained from A4 by the ð72 jÞ degree rotation around the center of OD A. A similar consideration concludes that the second nearest neighbor atoms at ðeej þ eejþ1 Þ are given by the occupation domains B4 , C4 located at ð2; 2; 2; 2; 0Þ=5. The position of all vertices mentioned in (a) are derived by the same consideration. Thus in this method, all atom positions (or vertices) around several kinds of cluster centers are derived from ODs which are obtained from those cluster centers by shifting them in parallel to the external space by an appropriate distance. The shifted ODs are frequently located around some special positions in the lattice. In the present case, all ODs are in the internal space passing through ði; i; i; i; 5zÞ=5

28


ði ¼ 0; 1; ::: ; 4Þ. This is the reason why the large ODs in Figure 2.4 are composed of a small number of building units (several small ODs). The same method is applicable to obtain all the atom positions in the cluster proposed by Hiraga et al. [41]. The resulting occupation domains are shown in Figures 2.4(b) and (c), where we assumed the occupation domains A (Figures 2.4(b)) and D are occupied by the transition metal and B and C are occupied by Al. These generate the structure in Figure 2.5(a). For comparison, the occupation domains of the Burkov model (Figure 2.5(b)) are also shown in Figures 2.4(e) and (f) [38].

2.6.3.

Three-dimensional structure of d-Al–Cu–Co

The three-dimensional structure can easily be obtained from the above model and the superspace group P105 =mmc, which was determined by diffraction experiments. If the domain A is at ð1; 1; 1; 1; 5zÞ=5, the domain D is located at ð1; 1; 1; 1; 5zÞ=5 because of the inversion symmetry. This is also obtained by the hyper-screw axis 105 with non-primitive translation parallel to the fifth axis. Then D is at ð1; 1; 1; 1; 5ðz þ 0:5ÞÞ=5. Similarly the domain B at ð2; 2; 2; 2; 5z0 Þ=5 leads to the domain C at ð2; 2; 2; 2; 5z0 Þ=5 and ð2; 2; 2; 2; 5ðz0 þ 0:5ÞÞ=5. Since the period along the 10-fold axis is about 4 A˚, the structure is considered to consist of two layers and interlayer distance is about 2 A˚ [8]. This implies that z ðz0 Þ is equivalent to z þ 1=2 ðz0 þ 1=2Þ and limits z ðz0 Þ to 0:25 where the mirror planes perpendicular to the ten-fold axis are located. Furthermore, the appearance of crystallochemically too short interatomic distances has to be avoided. This fixes the z coordinate of each domain: the domain A and B are at z ¼ 1=4 while C and D are at z ¼ 1=4. The two layers of the present model are shown in Figure 2.6. A large cluster consists of atoms in the decagon and ten pentagons around it in Figure 2.5(a) or Figure 2.6. Two clusters with the intercluster distance of 20 A˚ share atoms in the two pentagons. (The cluster pair AB in Figure 2.5(a).) This linkage corresponds to the 20 A˚ edge in Figure 2.3(a). When the cluster centers are on the vertices with the distance of 12 A˚ in Figure 2.3(a), e.g. those on the short diagonal of an acute rhombus, the clusters are merged by removing two pentagons. (CD in Figure 2.5(a).) The cluster is comparable with that in Figure 2.6 of Hiraga et al. [41]. It is clear that these two are quite similar except for the difference in the distribution of Al and transition metal atoms but the occupied and unoccupied centers of the pentagonal cages in Figure 2.6 are interchanged in Figure 2.6 of Hiraga et al. The determination of these points requires a careful simulation of the TEM image or a structure refinement based on the X-ray diffraction data.

2.6.4. Application of higher-dimensional cluster models in decagonal quasicrystals The higher-dimensional cluster model discussed above is first used in the structure refinement of d-Al–Mn–Pd [18]. This has a 12 A˚ period along the 10-fold axis, so that the model described above is not applicable. The occupation domain of the

2.7. Higher-dimensional cluster models of icosahedral quasicrystals

29

cluster center was assumed to be the same tiling as that in Figure 2.3(b) but with an edge length 20 A˚. Therefore it is generated by the OD similar to the middle decagon in Figure 2.2 but scaled down by 1 . The similar consideration has been made for the model building of the S1 superstructure phase in the Al–Ni–Co alloy system. The phase transition between the basic-Ni phase and the S1-phase was discussed [43], and the possibility of the phase transition due to the inter-layer phason flip was suggested from the comparison of the structures of these two phases.

2.7.

Higher-dimensional cluster models of icosahedral quasicrystals

The model-building method mentioned in the previous section is equally applicable to icosahedral QCs. In this case, they are described as a crystal in 6D space, so that the ODs are extended to some extent in 3D internal space. A building principle similar to that used in d-Al–Cu–Co was applied to i-Al–Pd–Mn for the first time and its structure was refined successfully [21,44]. Since their ODs are 3D object defined in the internal space, they have no period in any direction in the external space. These facts make the model building difficult. However, the method of the model building is same as that applied to dihedral quasicrystals mentioned in the previous section. This is exemplified in i-Al–Pd–Mn quasicrystals. The 6D model employed in the structure refinement of i-Al–Pd–Mn is a cluster model based on the 3D Penrose pattern (three-dimensional generalization of the Penrose pattern, 3DPP), where large icosahedral clusters are located at the 12-fold vertices of the 3DPP with an edge length of about 20 A˚. The OD of 3DPP is obtained from the projection of a unit cell in the 6D icosahedral lattice onto the 3D internal space [45,46]. This has the 6D icosahedral space group Fm35 with a doubled lattice constant a compared with that with Pm35 as in i-Cd–Yb, or i-Zn–Mg–Ho found recently. In addition, reflections with all even indices are systematically stronger than those with all odd indices. Therefore it is regarded as a superstructure of the latter. We write the lattice constants of the average structure in the external and internal spaces as a0 ¼ a=2 and consider the 3DPP with an edge length of a0 . In order to consider an initial model for the structure determination, we can use LDEM mentioned in section 4. This gave large ODs at the origin and ð1; 0; 0; 0; 0; 0Þ=2 and a small one at ð3; 1; 1; 1; 1; 1Þ=4. The characteristic feature of this structure is that there is no OD at ð1; 1; 1; 1; 1; 1Þ=4. Another important information was obtained from the 2/ 1 approximant, from which the structure of the cluster was known [47]. The guiding principle of the model building is therefore to include many such clusters and to get ODs of the model which agrees with the observed ones. Furthermore, we need to have no empty holes in the structure since it is an alloy and an alloy is characterized by dense packing of atoms in general. In order to get such a dense structure we consider cages which is specific to the 3DPP and use its self-similarity. It is known that the 3DPP has the 6D space group Pm35 and the primitive icosahedral lattice has a similarity ratio of 3 . The above considerations enable us to use the following process for the model building of i-Al–Pd–Mn. First we consider the 3DPP. Its OD is the rhombic triacontahedron with an edge length a0 and each edge of the triacontahedron is parallel to one of dij ðj 6Þ. The 3DPP consists of two kinds of rhombohedra with an edge length a0 . An acute

30


rhombohedron has edges parallel to de1 , de2 and de3 , while, an obtuse one to de4 , de5 and de6 . (See Figures 2.8(c) and (f).) From the self-similarity of the (primitive) icosahedral lattice, the 3DPP with 3 a0 (’ 20 A˚) edges can be obtained from an OD with rhombic triacontahedral shape with an edge length of 3 a0 , The real i-Al–Pd–Mn QC is face-centered and has a doubled lattice constant a ¼ 2a0 . We distinguish the coordinates with respect to a and a0 by ðx1 ; x2 ; x3 ; x4 ; x5 ; x6 Þ and ðx1 ; x2 ; x3 ; x4 ; x5 ; x6 Þ0 . Then the same 3DPP is obtained from the rhombic triacontahedra at ð0; 0; 0; 0; 0; 0Þ and ð1; 0; 0; 0; 0; 0Þ=2 and positions equivalent to them under the centering translations. The 32 centering translations of the face-centered icosahedral lattice are given by ð0; 0; 0; 0; 0; 0Þ, ð1; 1; 0; 0; 0; 0Þ=2, ð1; 0; 1; 0; 0; 0Þ=2, ..., ð1; 1; 1; 1; 1; 1Þ=2. The even–parity vertices of the 3DPP are generated by the ODs at ð0; 0; 0; 0; 0; 0Þ and other 31 equivalent positions, while the odd-parity ones by the ODs at ð1; 0; 0; 0; 0; 0Þ=2 and other 31 equivalent positions. We call the 3DPP with the edge length of 3 a0 an inflated 3DPP. The inflated 3DPP can be decomposed into dodecahedral stars (DSs), rhombic icosahedra (RIs), rhombic triacontahedra (RTs) and obtuse rhombohedra (ORs) with an edge length of a0 (Figures 2.7 and 2.8). It is known that the DS is located at each vertex of the inflated 3DPP, the RI is at the edge center of each edge and the RT is at two positions on the body diagonal of each acute rhombohedron of the inflated 3DPP (Figure 2.8) [48]. Therefore we can consider three kinds of clusters accommodated in these three polyhedra, which are called cages hereafter. The OR appears at a gap of them in the inflated OR. [Figure 2.8(f)]. These three building units of the inflated 3DPP can be decorated based on the polyhedral occupation domains with the same shape as the building units in Figure 2.7 but the length of each edge is 3 a0 . (Note that building units in Figure 2.7 are the polyhedra in the external space, while ODs discussed here are those in the internal space.) For convenience, polyhedral ODs with edge lengths of 3 a0 and a0 are hereafter called small and large ODs. If we decorate DS, RI and RT constituting the inflated 3DPP in a similar manner, we can obtain a definite cluster model, where the same atom clusters appear at each vertex, edge center or body diagonal two positions of the acute rhombohedron, although the face-centered lattice requires different clusters for the even- and odd-parity vertices. Such a model can be given on the basis of the small RT placed at ð1; 1; 1; 1; 1; 1Þ=4 and ð3; 1; 1; 1; 1; 1Þ=4 of the

(a)

(b)

(c)

Figure 2.7. The building units of the inflated 3D Penrose pattern. (a) the dodecahedral star (DS), (b) the rhombic icosahedron (RI) and (c) the rhombic triacontahedron (RT) with the edge length of a0 . They appear also as the building units of occupation domains in a higher-dimensional cluster model. In this case, the edge length is 3 a0 .


(a)

(d)

(b)

(e)

31

(c)

(f)

Figure 2.8. The inflated acute and obtuse rhombohedra composed of building units shown in Figure 2.7. (a) the dodecahedral stars located at vertices of the inflated acute rhombohedron. (b) the rhombic icosahedra at edge centers. (c) two rhombic triacontahedra at the body diagonal. (d) the dodecahedral stars at vertices of the inflated obtuse rhombohedron. (e) the rhombic icosahedra at the edge centers. (f) the six obtuse rhombohedra. In (c) the two rhombic triacontahedra share one obtuse rhombohedron and in (d), one acute rhombohedron is shared by two dodecahedral stars on the short diagonal of the inflated obtuse rhombohedron.

face-centered icosahedral lattice as shown below. (Note that there are 64 ODs related to them in the unit cell, because of the 32 centering translations. For simplicity, we mention about only independent ODs in the following.) From the self-similarity of the 3DPP, the small RTs centered at these two positions give the inflated 3DPP. The edge center of the inflated 3DPP is derived from the small RI at ð1; 0; 0; 0; 0; 0Þ=4, while the body-diagonal two positions are obtained from the small DS at the origin and ð1; 0; 0; 0; 0; 0Þ=2 as discussed in the next section.

2.7.1.

Atom positions in six-dimensional space

As mentioned before, the 3DPP with the edge length of a0 is generated by the RT with the edge length of a0 placed at the origin of the primitive icosahedral lattice [45,46]. Similarly, the edge center of the 3DPP is given by the RI at ð1; 0; 0; 0; 0; 0Þ0 =2 and the body-diagonal two positions of the acute rhombohedron, the DS at ð1; 1; 1; 1; 1; 1Þ0 =2 [49]. The RI with its 5-fold axis parallel to d1 is given by the

32


intersection of the 2 RTs separated by d1 , while the acute rhombohedron (AR) with the 3-fold axis parallel to d1 þ d2 þ d3 , by the intersection of 4 RTs at the origin and d1 , d2 and d3 . The DS is composed of 20 such ARs. For the corresponding sites in the inflated 3DPP is easily obtainable by the similarity transformation for the icosahedral lattice. The similarity transformation matrix S for the face-centered icosahedral lattice is given by Eq. (2.14). On the other hand, that of the primitive icosahedral lattice is given by S3 [50]. This inflates the external component of the unit vector di by 3 while deflates the internal component by the factor of 3 . The three kinds of cluster centers in the inflated 3DPP mentioned above are therefore obtained from small RT, small RI and small DS placed at ð0; 0; 0; 0; 0; 0Þ0 , S3 ð1; 0; 0; 0; 0; 0Þ0 =2 and S3 ð1; 1; 1; 1; 1; 1Þ0 =2, respectively. In the face-centered icosahedral lattice, the first one, ð0; 0; 0; 0; 0; 0Þ, is equivalent to ð0; 0; 0; 0; 0; 0Þ and ð1; 0; 0; 0; 0; 0Þ/2, the second one, to ð0; 1; 1; 1; 1; 1Þ=4 and ð2; 1; 1; 1; 1; 1Þ=4, and third one, to ð1; 1; 1; 1; 1; 1Þ=4 and ð3; 1; 1; 1; 1; 1Þ=4. The two positions for RI are however equivalent under the inversion. In the structure analysis, the origin was taken at the center of small DS. In this coordinate system, the coordinates of small RT and small DS are interchanged and those of RI are ð1; 0; 0; 0; 0; 0Þ=4. The latter is equivalent to ð1; 0; 0; 0; 0; 0Þ=4 as mentioned above. We use this setting in the following. We consider real atom positions in the cages. In the higher-dimensional cluster model, the ODs creating atoms around the cluster centers are obtained from the ODs for the centers by shifting them along the external space as mentioned in the previous section [51]. In a realistic model, the atom shifts from the ideal positions given by the location of the centers of ODs are strongly constrained. From the electron density given by experiments, large ODs are observed only at ð0; 0; 0; 0; 0; 0Þ, ð1; 0; 0; 0; 0; 0Þ=2 and ð3; 1; 1; 1; 1; 1Þ/4. This means that the shifted small ODs have to be a part of the large ODs situated at these points. (Otherwise, many small ODs located separately are expected.) Such a subdivision of the large ODs into small ones is achieved inversely by reconstructing the experimentally obtained ODs with shifted small ODs. This reconstruction is practical because only rough size and shape of each large OD can be obtained from the MEM or Fourier maps obtained by LDEM. The ODs at ð0; 0; 0; 0; 0; 0Þ and ð1; 0; 0; 0; 0; 0Þ=2 (called the ODs A and B hereafter) can be constructed mainly by the three kinds of small occupation domains shown in Figure 2.7(a–c). As mentioned above, the small DSs are situated at the centers of ODs A and B. For the reconstruction of the ODs A and B, we can use the connection of small ODs which is similar to the cage connections in the external space. The positions of the independent small occupation domains are shown in Figures 2.9(a,b). For OD C (Figure 2.9(c)), we need to consider new subdivision of the large ODs. This is formed by the small RTs centered at ð3; 1; 1; 1; 1; 1Þ=4 and at ð1; 1; 1; 0; 0; 0Þi0 from it. The latter can be subdivided into two small rhombic dodecahedra (RDs) along the 2-fold axis and a small AR on the 3-fold axis and 3 ARs around it. [In fact a small part of the RD in OD 36 in Figure 2.9(c) has to be removed but this is neglected in this chapter]. The symmetry operations create 20 inter-penetrated small RTs sharing a small RD [33 in Figure 2.9(c)] and a part of another RD (36). Each of these 20 small RTs share a small obtuse rhombohedron with the central small RT (32). This leads to Figure 2.10(c).

33


(a)

(b)

3f

(c)

3f 2f

2f

29

3f 2f 35

13 34 26

15

9

5f

30 24

33

11

25 23

21

8 37

(d)

5

19 17

5f

14

7

6

22

32

5f

12

10

27 28

20

40

4 3

16

5f

36

38 41 42

Figure 2.9. The independent parts of large occupation domains at ð0; 0; 0; 0; 0; 0Þ (a), at ð1; 0; 0; 0; 0; 0Þ=2 (b), at ð3; 1; 1; 1; 1; 1Þ=4 (c), and at ð1; 0; 0; 0; 0; 0Þ=4 (d). For visibility, a small gap is introduced between small occupation domains. The edge length of each polyhedron is 3 a0 . Domains without shading are not occupied by any atoms in i-Al–Pd–Mn but added for discussions. (See text.) Green, red, yellow and blue ODs are assumed to be statistically occupied by Al/Mn, Pd/Mn, Mn/Al and Mn/Pd. In each pair, the first atomic species was expected to be predominant in the initial model of the refinement. Arrows show the five-fold (5f), three-fold (3f) and two-fold (2f) axes passing through the center of the large occupation domains. [Note that the OD 38 is at ð1; 0; 0; 0; 0; 0Þ=4 in (d)].

(a)

(b)

(c)

(d)

Figure 2.10. The total occupation domains used in the refinement of i-Al–Pd–Mn. (a) the occupation domain A at ð0; 0; 0; 0; 0; 0Þ, (b) B at ð1; 0; 0; 0; 0; 0Þ=2 , (c) C at ð3; 1; 1; 1; 1; 1Þ=4 and (d) D at ð1; 0; 0; 0; 0; 0Þ=4. The unoccupied ODs shown as unshaded polyhedra in Figure 2.9 are not written.

34


As a final model for the structure refinement, additional small ODs around ð1; 0; 0; 0; 0; 0Þ=4 shown in Figure 2.9(d) had to be introduced. In the real i-Al–Pd–Mn, non-shaded small ODs are not occupied and the OD 40 is statistically occupied with a small occupation probability. It is noted that the symmetry of the occupation domain around ð1; 0; 0; 0; 0; 0Þ=4 is 5m in contrast to other three large ODs. This is because of the site symmetry of the OD centers: all the ODs in Figure 2.9 are on a 5-fold axis, while the other three ODs (Figures 2.10(a)–(c)) are situated at the special position with the site symmetry of m35. The whole ODs are obtained from their independent parts shown in Figure 2.9 by applying symmetry operations. The large ODs A, B, C and D occupied by atoms are shown in Figure 2.10. Among ODs shown in Figure 2.9(d), the OD 38 is at ð1; 0; 0; 0; 0; 0Þ=4 and the ODs having the same shape but different orientations are located at other 11 equivalent positions in the unit cell. Similarly, ODs 37 and 40 have 11 equivalent ones. The ODs around ð1; 0; 0; 0; 0; 0Þ=4 are isolated from their equivalent ones and from ODs A, B, and C. The ODs shown in Figure 2.10 lead to the decoration of the cage shown in Figures 2.11–2.13. (a)

(b)

#8 (d)

(c)

#23 (e)

#26

#37 (f)

#6

#27

(g)

#34 Figure 2.11. The decoration of the dodecahedral star in Figure 2.8 located at the even-parity vertices of the inflated 3DPP (Green: Al/Mn, red: Pd/Mn, blue: Mn/Pd, yellow: Mn/Al). The number in each figure means the atoms are generated by the OD in Figure 2.9 with the same number.


(a)

(b)

(c)

#16 (blue) #40 (green)

(d)

35

#5

#24

(e)

#15

#35 (red) #36 (green)

Figure 2.12. The decoration of the rhombic triacontahedron in Figure 2.8 located on the body-diagonal of the inflated 3DPP (Green: Al/Mn, red: Pd/Mn, blue: Mn/Pd).

(a)

(b)

#38

(d)

(c)

#11, 25 (green) #4 (blue) #17 (green)

#7 (red) #21 (green)

(e)

#15, 26 #37

#30

Figure 2.13. The decoration of the rhombic icosahedron in Figure 2.8 located at the edge center of the inflated 3DPP. (Green: Al/Mn, red: Pd/Mn, blue: Mn/Pd) In (b) and (d), the ODs 4, 17, and 37 are on the 5-fold axis of the RI.

2.7.2. Application of higher-dimensional cluster models in icosahedral quasicrystals Icosahedral quasicrystals i-Al–Pd–Re, i-Al–Cu–Fe and i-Al–Cu–Ru have been considered to be isostructural to i-Al–Pd–Mn from the intensity distributions. Therefore the

36


same model is applied to their structure refinements [22,23]. They have the facecentered icosahedral space group Fm35 in 6D space. Another model has been recently proposed in an i-Yb–Cd QC with the space group Pm35[52] and successfully applied to its structure refinement. The resulting structure consists of partially interpenetrated clusters with 146 atoms, which covers about 95% of the total atoms. In this model, we could reduce number of the glue atoms which are located between the clusters. Furthermore, this model simplifies the description of quasicrystal structures. Since we can know the cluster structures from the analysis of the corresponding crystalline approximants, if we determine the distribution of the clusters, we can fix most atom positions. The application of this model to other quasicrystals will be a good subject of future studies.

2.8.

Quasicrystal models with fractal occupation domains

Most quasicrystals known so far are analyzed based on models with polygonal or polyhedral ODs. Such models are not required by the symmetry of quasicrystals. As another possibility, we can consider models with fractal ODs fulfilling the symmetry. As an example of such QC models, we show a model of dodecagonal (DD) Ta–Te QCs. Although most quasicrystals found so far are ternary alloys, this is a calcogenide QC [53]. As mentioned in Section 2.3, this dodecagonal quasicrystal is described as a periodic structure in 5D space. The model is based on the modified Stampfli pattern (MSP) as mentioned below. The framework (thin lines) with a few distorted hexagons seen in Figure 2.14 can be generated by the large complicated OD shown in Figure 2.15(c), which includes the ‘pinwheel’ (Figures 2.15(a) and (b)) [54]. We call this pattern the modified Stampfli pattern

(a)

(b)

Figure 2.14. Model structure of the Ta-Te quasicrystal. (a) three Ta layers projected along the d5 (periodic) axis. (b) its lower two layers. Solid lines joining Ta atoms at u ¼ 0 stand for the framework of the quasicrystal. Te atoms are at the center of each square or triangle similar to those of the approximant. Two hexgonal holes are seen in this region. For removing the hexagonal holes, see the text. Note that the two hexagonal holes are seen in this region.

2.8. Quasicrystal models with fractal occupation domains

37

(c) (e) P1 (a) A

C

(d) (f)

P2 (b) B

D

Figure 2.15. OD’s of the 5D model of the Ta-Te QC. (a),(b) two pinwheel patterns appearing in the model, (c) the OD at the origin, (d) at (0,1/2,0,0,u), (e) at (0,2/3,0,1/3,u) and (f) at (0,1/2,1/2,0,u) and (0,1/2,z,0,u). The coordinates show the center of the occupation domain except for the last, while (0,1/2,z,0,u) show a point on the vertical line through the center in (f). A corner vector of the central dodecagon of (c) is 2 ð1; 0; 0; 0; 0Þ. These polygonal OD’s are of the first generation of the model. They become truly fractal as the generation N becomes infinity.

(MSP) of the first generation. The MSP of the zero-th generation is givenpby pffiffiffi ffiffiffi the central dodecagon of Figure 2.15(c) [55], scaled up by 2 , where ¼ ð1 þ 3Þ= 2 is the similarity ratio of the dodecagonal lattice. This consists of the same triangles, squares and hexagons but include more hexagonal holes compared with the first generation. To obtain a 5D-crystal model for the real DD phase, we consider the structures of its crystal approximants [56,57]. We remark that Ta atoms decorating the framework of the approximant are located at the center of each edge, the face center of each triangle and four positions in each square. Furthermore, the edges are classified into two groups, one of which is parallel to the six directions related by the 6-fold rotations, and the other which makes an angle of 30 from them. The edge center atoms of the first group are above the frame work, while those of the second are below it. Accordingly, the triangles are classified into two groups according to their orientations. The edge-center atoms of the first and second groups are situated above and below the framework, respectively. Similarly, the squares form three groups are related by 30 rotations. Each square has two pairs of atoms in its interior. A given pair is located above (below) the framework if it is parallel to an edge belonging to the first (second) edge-group. The same decoration of the MSP gives a model of the DD phase.2 The decoration breaks a mirror plane normal to the

2

A similar decoration for a different dodecagonal tiling has been given [58].

38


Table 2.1. Positions of occupation domains of the 5-dimensional model of dodecagonal Ta-Te QCs with the space group P12m2ð125 mmÞ. The first and third columns show the Wyckoff symbol and a representative of several equivalent positions. The number in the Wyckoff symbol means the order of the special position. The atom and the OD in Figure 2.15 are listed in the fourth and fifth columns. W.S. 1a 2a 4a 6a 6b 12a

Site symmetry 5

12m2ð12 mmÞ 6mmð65 mmÞ 3mð32 mÞ mm2ðmm1Þ mm2ðmm1Þ m11ðm11Þ

rep. position

Atom

O.D.

0; 0; 0; 0; 0 0; 0; 0; 0; u 0; 2=3; 0; 1=3; u 0; 1=2; 0; 0; u 0; 1=2; 1=2; 0; u 0; 1=2; z; 0; u

Ta Te Ta,Te Ta Te Ta

(c) (c) (e) (d) (f) (f)

12-fold axis, leading to the space group P12m2ð125 mmÞ. All the atom positions of the DD phase in the 3D external space are obtained from the OD’s at the special positions listed in Table 2.1. The method of obtaining such decorations is the same as that for simple polyhedral ODs mentioned in Sections 2.6 and 2.7 [51,7]. The vertices of the framework come from the OD at the origin (1a in Table 2.1). There are six sites for the edge center atoms. The edge center positions are obtained from the OD’s at (0,1/2,0,0,u) and positions equivalent to them (6a). Similarly, the face center positions are obtained from the OD’s at 4 sites (4a) and the four sites in the square come from the occupation domains at 12 positions (12a). All these sites are occupied by Ta atoms, while Te atoms are above and below the vertices of the framework (2a) and the face center of each square and triangle (4a; 6b). The shape of the OD’s for these atoms can be obtained by considering the intersections of the OD’s. For example, the OD for the edge center atoms at (0,1/2,0,0,u) (Figure 2.15(d)) are the intersection of the OD’s for the framework at (0,0,0,0,u)i and (0,1,0,0,u)i .3 This procedure gives rise to a 5D model which provides a 3D structure consistent with the approximants. All independent OD’s are shown in Figures 2.15(c)–(f). The hexagonal holes in the MSP of the first generation (Figure 2.14) can be removed by adding one point in each hexagon which divides the hexagon into a square, two triangles and a skinny rhombus. This point is obtained from a small (open) OD at the position shown by the arrow in Figure 2.15(c) and 11 ODs equivalent to it under the DD symmetry. The same decorations for the new squares and triangles also need to add corresponding small domains in Figure 2.15(d)–(f). The MSP of the first generation includes many DD clusters (but with hexagonal symmetry) in two orientations, which are related by 30 rotation to each other, as in the crystal approximant. In order to reduce the number of the hexagonal holes, we can use the self-similarity of the dodecagonal lattice. We consider the second generation ODs. We note that the central parts of ODs A, B, C, and D are composed of a dodecagon in A, two kinds of (concave and convex) octagons (in B and D), a hexagon (in C) and small triangles (around a central dodecagon in A). Remove the triangles and replace the small dodecagon at the center of A by total A in Figure 2.15(c) 3

In [7], ‘union’ in Section 2.15 should be read as ‘intersection’.


39

after scaling down by 2 . Similarly replace convex and concave small octagons in A, B, C, D by the scaled B, D with the same orientation. Finally replace the small hexagon at the center of C with scaled C after the inversion. Central parts of new ODs are dense except for small holes in the removed triangles, since the pinwheels P1 and P2 with two orientation in Figure 2.15 form a dense dodecagon without hole. Finally the pinwheels are replaced by the new pinwheels, which are obtained from A, by removing six thin regions with an angle of 30 as in (a) or (b) and scaling down by 2 . In the next generation, new A, B etc. are used for A, B etc. This procedure is repeated. An infinite number of repetitions leads to ODs with fractal shape. The procedure reduces the number of hexagons seen in Figure 2.14 and finally, they disappears in the fractal ODs. As a result, the resulting pattern consists of only squares and triangles. (This is similar to the Stampfli pattern, which are obtained by the inflation-deflation method [59]). It should be noted that from the fractal OD for the framework, the ODs for the edge-center sites, face-center sites and sites in the square are obtained by taking the intersection same as in the polygonal ODs. This indicates that the model building for a higher-dimensional cluster model is possible regardless of the fractalness of ODs. The distribution of the cluster centers in higher-dimensional cluster models is usually assumed to be obtained from a simple polygonal or polyhedral OD. This assumption is however not always necessary. At least, the fractal ODs are more probable in dodecagonal quasicrystals like dd-Ta–Te than polygonal ODs.

2.9.

Symmetry breaking in clusters

It is known that the Al–Ni–Co alloy system has several different decagonal quasicrystals with different diffraction patterns [60]. Some of them are considered to be a superstructure of the other. Among them, Al72Ni20Co8, has a simplest structure and is called a basic-Ni phase. This is considered to be a high-temperature phase and at lower temperatures, another phase appears. This structure is understandable based on the 2D Penrose tiling (2DPT) consisting of fat and skinny rhombi with an edge-length of 2.43 A˚. A 5D model of the basic Ni phase can also be constructed theoretically on the basis of the higherdimensional cluster model as shown in Section 2.6. In this section, we show another approach to the determination of its ODs, which is more experimental although it is still based on the ODs of the 2DPT. In an initial model, we consider large ODs, which generate all probable atom positions, a part of which are really occupied by atoms (Figure 2.16). The ODs of the initial model are similar to those proposed by Yamamoto et al. [42] for Al70Ni10Co20 and by Saito et al. [61] for Al72Ni20Co8. It is also similar to a model for Al–Cu–Co by [62] except for the central part of OD B. The structure analysis of the basic Ni phase suggests that the constituent clusters are lower-symmetric. We do not, therefore, employ any symmetric cluster model. This is natural since the cluster size considered is small (at most 6.36 A˚) as shown below and the clusters are situated at lower symmetric local environments. We discuss how to break the symmetry of clusters within the higher-dimensional cluster model.

40


m

m m

A'

m B

A' m

A'

m

A

b

A' A' m

m

m α

m

β

Figure 2.16. The occupation domains of an initial model for Al72Ni20Co8 constructed on the basis of the 2D Penrose tiling with edge length of 2.43 A˚.

2.9.1.

Model building of lower symmetric cluster models

To realize such low-symmetric clusters, new partitioning of ODs are necessary. In higher dimensional models used so far, we assume same displacement everywhere in the small OD. In this case, constraint conditions for the displacement are determined by the highest site symmetry among all points within the OD. Therefore if we choose to subdivide an OD into smaller one so as to avoid to include a high symmetric point within the OD but to locate it at the boundary, the constraint of the resulting OD can weaken. In particular, if the small OD does not include any special position in it, the OD can be shifted along any direction. This means that atoms generated by the OD can shift along any direction and can forms low symmetric clusters. There are, however, several ways for such a partitioning. After various modifications based on the structure refinement, the final partitioning of the ODs shown in Figure 2.17 were employed in the refinement of the basic-Ni phase [19]. It should be noted that most small ODs are triangles with several different shapes and sizes, which are located at general positions. However, several triangles are on the mirror, the location of which is indicated by m in Figure 2.16. Such ODs at lower-symmetric positions enable us to move them so as to break a pentagonal symmetry in the atom arrangement within the clusters. In the final ODs, there are 23 independent small ODs. Among them, the small domains A9, A10, B11, B15 and B17 were treated as partially occupied ones.

2.9.2.

Projected structure

The ODs in Figure 2.17 lead to the structure projected along the c axis shown in Figure 2.18. The structure obtained after the structure refinement is shown in Figure 2.19. As seen in the figure, most atoms show only small shifts from their initial


9 8 7 4 5 2 1 6 3 10

41

23 19 20 18

16 14 22 21 13 12 11 17 15

Figure 2.17. The final occupation domains as a result of structure refinement of Al72Ni20Co8 constructed on the basis of the 2D Penrose tiling with edge length of 2.43 A˚. The darkness of the gray tone indicate the concentration of transition metals in the assigned small domains.

Figure 2.18. The projection of the atom positions (41 41A˚2) created from the occupation domains in Figure 2.16 along the c axis. The atoms at z=0.25 or –0.25 are indicated by open or filled circles, respectively. The black lines show the 2D Penrose tiling with an edge length of 2.43 A˚. The gray lines indicate the hexagon, boat and star (HBS) tiling with an edge length of 6.36 A˚.

42

Chapter 2. Recent Development of Quasicrystallography 17 6

4

16 4

20

13

22

9 14 3 22

5 16

5 19 23

8 13

13

18 4

9

4 22 15

7

17 7

5

16 8 14

22

8

5

13 8

4 13

1

18

8 20 7 10 21 18 5 5 8 12 14 1 1 4 16 22 11 21 4 13 16 10 6 19 23 14 15 21 7 18 4 7 8 22 16 22 7 4 18 15 7 7 21 14 3 23 10 16 19 8 16 4 8 5 13 4 7 21 13 2 14 12 12 22 1 20 4 18 10 23 9 21 10 9 9 18 22 22 22 8 8 19 3 18 13 7 15 5 13 5 5 16 13 8 8 5 20 20 0 9 4 4 5 14 16 8 15 14 9 4 22 4 18 13 6 17 8 21 22 8 7 18 20 17 22 18 8 7 7 5 19 23 1 3 6 3 15 10 1 15 17 12 11 12 13 21 16 5 7 7 8 22 20 17 18 7 8 21 22 8 16 23 6 17 22 4 4 14 9 8 15 5 14 14 16 19 4 4 9 20 0 20 8 5 13 8 16 5 5 13 5 7 14 18 13 3 19 8 8 22 22 21 18 9 9 10 10 23 21 9 20 12 1 12 22 4 18 14 13 2 21 4 7 13 8 5 16 4 8 16 17 19 10 23 3 14 21 7 7 15 18 4 7 16 8 22 7 4 18 21 7 15 1 5 14 23 19 6 16 10 13 21 22 11 4 16 12 14 1 1 8 14 5 8 18 5 7 4 9 21 10 23 21 20 7 8 22 22 8 19 17 9 3 22 13 18 5 8 13 8 14 5 15 7 16 13 4 4 20 21

8 14

9

5

1

16

4

17 6

9 14

5

8

13

18

4 22

5

16

13

4

1

Figure 2.19. The projection of the refined structure (45 30A˚ ) of Al72Ni20Co8 along the c axis. The white lines indicate the HBS tiling with an edge length of 6.36 A˚. The number indicates that atoms come from the small occupation domain with the same number in Figure 2.17. The darkness of the gray tone indicate the concentration of transition metals in the assigned small domains. 2

positions except for the atoms from subdomain A9 . At the same time, small pentagonal cluster seen in Figure 2.18 at each vertex of the HBS tiling are slightly distorted in Figure 2.19. As is clear from Figure 2.19, the arrangement of atoms and their assignment of atomic species are quite different from the 20 A˚ tenfold symmetric cluster which is discussed in Section 2.6. It can be considered as atomic decoration of the hexagon, boat, and star (HBS) tiling [63] with edge length of 6.36 A˚. This tiling is given by the white lines in Figure 2.18. (In the figure, atoms are labeled by the number in ODs which generates them). This is generated by a central part of OD B in Figure 2.16 (and corresponding part of OD C), which is a pentagon obtained from OD A by scaling down by 1 (OD b in the figure). The 20 A˚ cluster region in d-Al–Cu–Co corresponds to the (white) decagon near the upper left or lower right corner in Figure 2.18. As seen in the figure it is decomposed into two hexagons and one boat. As a result, this decagon has only a mirror symmetry. Therefore there is no 20 A˚ cluster with 10-fold or 5-fold symmetry in the basic Ni phase. In addition, most 6.36 A˚ clusters located at the vertices of the HBS tiling are low-symmetric in the positions of the first shell atoms and in their TM occupation probabilities denoted by the gray tone. The vertices of the tiles are mainly occupied by Al atoms, and edges of them are occupied by a pair of TM and TM/Al atoms. The latter is a simple realization of the matching rules of the 2DPT. The partitioning of the ODs in the present model is empirically determined and therefore does not rely on the higher-dimensional cluster model. In the model based on the higher-dimensional cluster model, whole ODs have to consist of small overlapped pentagons for cluster centers. If we can remove the OD 23 in Figure 2.17, we can construct such a model. Then two large ODs can be decomposed into overlapped two pentagons, one of which is given by OD b in Figure 2.16 and the other is the small pentagon consisting of ODs 1 and 2 in Figure 2.17. However, the structure refinement confirmed that the ODs in Figure 2.17 gives acceptable R factor. In this

2.11. Summary

43

partitioning, many ODs except for ODs 1,2,3,5,6,10,12,15,17,19,23 are located not at special positions but at general positions. These ODs are responsible for the lowsymmetric clusters. The structure refinement has shown that the introduction of low-symmetric clusters is essentially important in this phase [19]. In general, the clusters are situated at lower-symmetric local environments, so that the symmetry of the clusters is affected by such a site symmetry. If the clusters have approximately high symmetry and their size is large, the perturbation for atoms near the center will be small because the low-symmetric atoms around the center is far from atoms near the center. In such a case, the atoms near the center can be approximated by the ideal high-symmetric arrangement. In contrast, the basic-Ni phase in d-Al-Ni-Co consists of small clusters. Therefore, it is easy to consider that this approximation will be not be good enough.

2.9.3.

Application of low-symmetric cluster models

The structure refinement based on a model which allows low-symmetric clusters were made for the first time in the basic-Ni phase in d-Al72Ni20Co8 [19]. The space group used is the centrosymmetric P105 =mmc. A similar alloy but with a slightly different chemical composition d-Al70.6Ni22.7Co6.7 shows weak satellite reflections corresponding to 8A˚ modulation along the c axis. Its average structure with a 4 A˚ period has been analyzed based on a low-symmetric space group P10 by using a slightly different model [20]. In the analysis, initial ODs which are much larger than those used in the analysis of d-Al72Ni20Co8 were used. This is an analysis which includes more than 700 parameters. It is possible to introduce similar models with non-icosahedral clusters for icosahedral quasicrystals. An analysis using a model, which allows a low-symmetric cluster, was performed recently in i-Cd–Yb [52].

2.10.

Modulation functions for quasicrystals

In the structure analysis performed so far, large ODs are subdivided into smaller ones and it was assumed that shift of small ODs from their ideal positions is same within everywhere in the small ODs. This is an approximation to simplify the description of quasicrystals. It is in general not required by the constraint caused by the symmetry. We can consider the situation in which the shift is different everywhere in the small ODs. In order to consider such a case, it is needed to introduce modulation functions for the shift and the occupation probability. A possible modulation function for the scalar properties like the occupation probability and isotropic temperature factor has been discussed recently [64].

2.11.

Summary

An overview of the recent development of quasicrystallography has been given. In the latest decade, detailed structure refinements of quasicrystals were performed owing to the development of the direct method, model building and computer

44


software. The development of the direct method for quasicrystals enables us to obtain electron density maps in nD space almost automatically. One of the direct methods, the low density elimination method for quasicrystal, was described. This facilitates largely the model-building of an initial model of quasicrystals. Most structure analysis performed so far are based on the higher-dimensional cluster models. The model building based on this method was demonstrated for d-Al–Cu–Co and i-Al–Pd–Mn. Their models have polygonal or polyhedral occupation domains (ODs). Such ODs may not necessarily be needed in general. As an example, a model with fractal ODs was shown in dodecagonal Ta-Te quasicrystals which showed that the higher-dimensional cluster model is applicable to such cases. In most structure analyses, the clusters are assumed to maintain high symmetry everywhere in quasicrystals independent of their local environments. In more detailed structure analysis, we may need to consider lower-symmetric clusters since in general the local environments of clusters is low-symmetric. In particular when the cluster size is small, the use of lower-symmetric clusters will be important. Such an analysis was shown in the basic-Ni phase in decagonal Al–Ni–Co quasicrystals.

References [1] D. Shechtman, I. Blech, D. Gratias, J. W. Cahn, Metallic Phase with LongRange Orientational Order and No Translational Symmetry, Phys. Rev. Lett. 53 (1984) 1951. [2] T. Janssen, Crystallograhy of Quasi-Crystals, Acta Crystallogr. A 42 (1986) 261. [3] P. M. D. Wolff, The Pseudo-Symmetry of Modulated Crystal Structures, Acta Crystallogr. A 38 (1974) 777. [4] A. P. Tsai, A. Inoue, T. Masumoto, Stable Al-Cu-Fe, Jpn. J. Appl. Phys., Part 2 26 (1987) L1505. [5] N. G. D. Bruijn, Algebraic theory of Penrose’s non-periodic tilings of the plane. I, II, Proc. Nederl. Acad. Wetensch. Proc. Ser. A (1981) 39. [6] P. Kramer, R. Neri, On periodic and non-periodic space filling of Em obtained by projection, Acta Crystallogr. A 40 (1984) 580. [7] A. Yamamoto, Crystallography of Quasiperiodic Crystals, Acta Crystallogr. A 52 (1996) 509. [8] A. Yamamoto, K. N. Ishihara, Penrose patterns and related structures II. Decagonal quasicrystals, Acta Crystallogr. A 44 (1988) 707. [9] U. Michiue, A. Yamamoto, M. Onoda, A. Sato, T. Akashi, H. Yamane, T. Goto, Incommensurate crystallographic-shear structures of Bax Bi22x Ti4x O114x (x=0.275), Acta Crystallogr. B 61 (2005) 145. [10] D. P. DiVincenzo, P. J. Steinhardt, in: D. P. DiVincenzo, P. J. Steinhardt (Eds.), Quasicrystals, the state of the art, World Scientific, Singapore, 1998. [11] A. Yamamoto, H. Takakura, Structure Refinement of Quasicrystals, Ferroelectrics 305 (2004) 223. [12] C. L. Henley, M. Mihalkovic, M. annd Widom, Total-energy-based structure prediction for d(AlNiCo), J. Alloys and Comp. 342 (2002) 221.

References

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[13] W. Steurer, Twenty years of structure research on quasicrystals. Part I, Z. Krist. 219 (2004) 391. [14] H. Takakura, M. Shiono, T. J. Sato, A. Yamamoto, A. P. Tsai, Ab Initio Structure Determination of Icosahedral Zn-Mg-Ho Quasicrystals by Density Modification Method, Phys. Rev. Lett. 86 (2001) 236. [15] J. W. Cahn, D. Gratias, B. Mozer, Patterson Fourier analysis of the icosahedral (Al,Si)-Mn alloy, Phys. Rev. B 38 (1988) 1638. [16] P. Kramer, A. Papadopolos, W. Liebermeister, Atomic Positions in Icosahedral Quasicrystals, in: S. Takeuchi, T. Fujiwara (Eds.), Proc. 6-th International Conference on Quasicrystals, World Scientific, Singapore, 1998, p. 71. [17] D. Gratias, F. Puyraimond, M. Quiquandon, A. Katz, Atomic clusters in icosahedral F-type quasicrystals, Phys. Rev. B 63 (2000) 024202. [18] S. Weber, A. Yamamoto, Non-centrosymmetric structure of a decagonal Al70Mn17Pd13 quasicrystal, Acta Crystallogr. A 54 (1998) 997. [19] H. Takakura, A. Yamamoto, A. P. Tsai, The structure of a decagonal Al72Ni20Co8 quasicrystal, Acta Crystallogr. A 57 (2001) 576. [20] A. Cervellino, T. Haibach, W. Steurer, Structure solution of the basic decagonal Al-Co-Ni phase by the atomic surfaces modelling method, Acta Crystallogr. B 58 (2001) 8. [21] A. Yamamoto, H. Takakura, A. P. Tsai, Structure refinement of i-Al-Pd-Mn quasicrystals by IP-Weissenberg camera data, J. Alloys Comp. 342 (2002) 159. [22] A. Yamamoto, H. Takakura, A. P. Tsai, Refinement of i-Al-Cu-Fe and i-Al-Cy-Ru Quasicrystal Structures, Ferroelectrics 305 (2004) 279. [23] A. Yamamoto, H. Takakura, T. Ozeki, A. P. Tsai, Y. Ohashi, Structure refinement of i-Al-Pd-Re by synchrotron radiation data., J. Noncryst. Solids 334&335 (2004) 151. [24] K. Hiraga, The Structure of Quasicrystals Studied by Atomic-Scale Observations of Transmission Electron Micorscopy, Advances in Imaging and Electron Physics 122 (2002) 2. [25] M. Shiono, M. M. Woolfson, no title, Acta Crystallogr. A 48 (1992) 451. [26] A. Yamamoto, Quasicrystal structure analysis. The state of the art, Acta Crystallogr. A 61 (2005) C2. [27] A. Yamamoto, S. Weber, A. Sato, K. Kato, K. Ohshima, A. P. Tsai, A. Niikura, K. Hiraga, A. Inoue, T. Masumoto, Electron density of Icosahedral Zn-Mg-Y quasicrystals determined by a six-dimensional maximum entropy method, Phil. Mag. Lett. 73 (1996) 247. [28] W. Steurer, The N-dim maximum-entropy method, in: j. M. Pe´rez-Mato, F. J. Z5Diga, G. Madariaga (Eds.), Methods of Structural Analysis of Modulated Structures and Quasicrystals, World Scientific, Singapore, 1991, p. 344. [29] F. Izumi, A. Dilanian, Commission on Powder Diffraction, IUCr. Newslett. 32 (2005) 59. [30] A. Yamamoto, QUASI06, http://www.nims.go.jp/aperiodic/yamamoto/. [31] H. Takakura, M. Shiono, T. J. Sato, A. Yamamoto, A. P. Tsai, Structure of icosahedral Zn-Mg-Ho quasicrystals determined by a density modification method, Ferroelectrics 250 (2001) 301.

46


[32] H. Takakura, A. Yamamoto, T. J. Sato, A. P. Tsai, Y. Ozawa, N. Yasuda, K. Toriumi, Ab initio reconstruction of p-type icosahedral An-Mg-Ho quasicrystal structures, Phil. Mag. 86 (2006) 621. [33] H. Takakura, A. Yamamoto, M. D. Boissieu, A. P. Tsai, Ab Initio Structure Solution of Icosahedral Cd-Yb Quasicrystals by a Density Modification Method, Ferroelectrics 305 (2004) 209. [34] S. Weber, A. Yamamoto, Application of the five-dimensional maximum entropy method to the structure refinement of decagonal , Phil. Mag. A 76 (1997) 85. [35] K. Hiraga, in: P. W. Hawks (Ed.), Advances in Imaging and Electron Physics, Academic Press, London, 1998, pp. 37. [36] A. Yamamoto, Seciton method for projected structures of icosahedral quasicrystals, Phys. Rev. Lett. 93 (2004) 195505. [37] A. Yamamoto, Structure of Decagonal Quasicrystals, Sci. Rep. RITU A 42 (1996) 207. [38] S. E. Burkov, Structure model of the Al-Cu-Co decagonal quasicrystal, Phys. Rev. Lett. 67 (1991) 614. [39] K. Niizeki, A classification of the space groups of approximant lattices to a decagonal quasilattice, J. Phys. A Math. Gen. 24 (1991) 3641. [40] K. Hiraga, W. Sun, F. J. Lincoln, Structural change of Al-Cu-Co decagonal quasicrystal studied by high-resolution electron microscopy, Jpn. J. Appl. Phys. 30 (2B) (1991) L302. [41] K. Hiraga, F. J. Lincoln, W. Sun, Structure and structural change of Al-Ni-Co decagonal quasicrystal by high-resolution electron microscopy, Mater. Trans., JIM 32 (4) (1991) 308. [42] A. Yamamoto, K. Kato, T. Shibuya, S. Takeuchi, Atomic structure of a decagonal Al-Co-Ni quasicrystal, Phys. Rev. Lett. 65 (1990) 1603. [43] A. Yamamoto, H. Takakura, E. Abe, Five-dimensional model of S1-superstructure phase in Al-Ni-Co quasicrystals, Phys. Rev. B 72 (2005) 144202. 144202–6. [44] A. Yamamoto, H. Takakura, A. P. Tsai, Six-dimensional model of icosahedral Al-Pd-Mn quasicrystals, Phys. Rev. B 68 (2003) 094201. [45] M. Duneau, A. Katz, Quasiperiodic Patterns, Phys. Rev. Lett. 54 (1985) 2688. [46] A. Katz, M. Duneau, Quasiperiodic patterns and icosahedral symmetry, J. Physique 47 (1986) 181. [47] K. Sugiyama, N. Kaji, K. Hiraga, T. Ishimasa, -AlPdMnSi (2/1), Z. Krist. 213 (1998) 90. [48] T. Ogawa, On the structure of a quasicrystal -Three dimensional Penrose transformation, J. Phys. Soc. Jpn. 54 (1985) 3205. [49] A. Yamamoto, Ideal structure of icosahedral Al-Cu-Li quasicrystals, Phys. Rev. B 45 (1992) 5217. [50] V. Elser, Indexing problems in quasicrystal diffraction, Phys. Rev. B 32 (1985) 4892. [51] A. Yamamoto, K. Hiraga, Structure of an icosahedral Al-Mn quasicrystals, Phys. Rev. B 37 (1988) 6207. [52] H. Takakura, C. P. Gcmez, A. Yamamoto, M. de Boissieu, A. P. Tsai, Atomic structure of the binary icosahedral Yb-Cd quasicrystal, Nature Materials 6 (2007) 58.

References

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[53] A. Yamamoto, Dodecagonal Ta-Te quasicrystal and a 5-dimensional model with fractal occupation domains , Acta Crystallogr. A 60 (2004) 142. [54] A. P. Smith, The sphere packing problem in quasicrystals, J. Non-Crys. Solids 153&154 (1993) 258. [55] K. Niizeki, H. Mitani, Two-dimensional dodecagonal quasilattices, J. Phys. A 20 (1987) L405. [56] M. Conrad, B. Harbrecht, Structure properties of Ta97 Te60 and Ta181 Te112 , two approximants of a dodecagonal tantalum telluride, in: M. Boissieu, J. L. Verger-Gaugry, R. Currat (Eds.), Aperiodic ’97, World Scientific, Singapore, 1998, p. 205. [57] M. Conrad, B. Harbrecht, F. Kurumeich, Structure Properties of a Dodecagonal Titanium Telluride quasicrystal, in: Aperiodic ’97, World Scientific, Singapore, 1998, p. 199. [58] F. Ga¨hler, dodecagonal model, in: C. Janot, J. M. Dubois (Eds.), Quasicrystalline Materials , World Scientific, Singapore, 1988, p. 272. [59] P. Stampfli, A dodecagonal quasiperiodic lattice in two dimensions, Helv. Phys. Acta 59 (1986) 1260. [60] S. Ritsch, C. Beeli, H. U. Nissen, T. Godecke, M. Scheffer, R. Lu¨ck, The existence regions of structural moddifications in decagonal Al-Co-Ni, Phil. Mag. Lett. 78 (1998) 67. [61] K. Saitoh, K. Tsuda, M. Tanaka, New Structural Model of an Al72Ni20Co8 Decagonal Quasicrystal, J. Phys. Soc. Jpn. 67 (1998) 2878. [62] E. Cockayne, M. Widom, Ternary model of an Al-Cu-Co decagonal quasicrystal, Phys. Rev. Lett. 81 (1998) 598. [63] X. Z. Li, Structure of Al-Mn Decagonal Quasicrystal. I. A Unit-Cell Approach, Acta Crystallogr. B 51 (1995) 265. [64] A. Yamamoto, Modulation Functions for Quasicrystals, to be published in Phil. Mag.


Chapter 3

New Group of Icosahedral Quasicrystals Tsutomu Ishimasa Division of Applied Physics, Graduate School of Engineering, Hokkaido University, Kita-ku, Sapporo 060-8628, Japan

3.1.

Introduction

Quasicrystal is a solid exhibiting Bragg reflections, but with no periodicity in its atomic arrangement. In this definition there is no restriction on the symmetry. For example, a quasicrystal with cubic diffraction symmetry is possible. However, usually one uses the word ‘quasicrystal’ as a solid exhibiting non-crystallographic diffraction symmetry: for example, 5-fold rotational symmetry. The fact that the Bragg reflections are generated from a quasicrystal indicates the presence of a special type of longrange order. That is called quasiperiodicity. The Ammann tiling, or 3-dimensional Penrose tiling, is a famous example of an icosahedral quasiperiodic lattice consisting of two kinds of Penrose rhombohedra. After the discovery of Al–Mn quasicrystal by D. Shechtman et al. [1], this non-periodic ordered state has attracted many attentions of physicists, chemists, and metallurgists. The physical properties, namely electronic, magnetic, and mechanical properties, of a quasicrystal have been investigated, and the following have been discovered: the presence of pseudo-gap at Fermi level [2], large electrical resistivity with negative temperature coefficient [3,4], large thermoelectric power [5], positive temperature dependence of Pauli paramagnetism [6], short-range magnetic ordering satisfying the icosahedral symmetry [7], excess specific heat at high temperature beyond the Dulong-Petit law [8], phason hopping [9], and low friction coefficient [10]. This progress is based on the discoveries of stable quasicrystals with nice structural quality such as Al–Pd–Mn [11] and Zn–Mg–R (R ¼ Y, and lanthanoids) [12] by A.P. Tsai’s group. These quasicrystals form as equilibrium phases at certain composition and temperature regions, and then the growth of single-grain samples has become possible. Such single quasicrystals have stimulated the studies on atomic structure and the physical properties of the quasicrystals. In the process of these studies, interesting question has arisen; Is there any other hidden quasicrystal alloy? This article treats of a new group of icosahedral quasicrystals discovered from this simple question. HANDBOOK OF METAL PHYSICS ISSN 1570-002X/DOI 10.1016/S1570-002X(06)02003-9 49


50

3.2.

Chapter 3.

New Group of Icosahedral Quasicrystals

Classification of icosahedral quasicrystals

The icosahedral quasicrystals described here is new not only as an alloy but also as a structure type. Before describing this new type, a brief summary on the classification of the icosahedral quasicrystals is given here. Main characteristics of a quasicrystal structure are understood as a combination of a quasilattice (quasiperiodic lattice) and a cluster decorating it. Three types of icosahedral quasilattices, P-, F-, and I-types, are known theoretically [13–15]. Among them the P- and F-types have been observed experimentally, and they are distinguished by reflection conditions. The reflection vector g of an icosahedral quasicrystal is expressed as follows: g¼

6 1 X mi ei== ; a6D i¼1

ð3:1Þ

where the six ei== vectors with length 2–1 are parallel to the lines connecting the center and vertices of an icosahedron, and a6D denotes the lattice parameter of the 6-dimensional hypercubic lattice in the frame work of the section (or projection) method [16–18]. While there is no restriction for the indices in the case of P-type, reflections with all odd or all even indices only appear in F-type [19]. Furthermore, the difference in reflection conditions causes that in scaling rules with respect to the positions of the reflections. The F-type exhibits -scaling rule (: golden mean), while the P-type does 3-scaling rule. In the real space, a P-type quasilattice can be decomposed into two F-type sub-lattices. Based on this relation, an F-type quasicrystal has been interpreted as an ordered phase, in which two kinds of atomic clusters with different atomic decoration are arranged regularly. The high degree of structural perfection observed in the F-type quasicrystals, for example Al–Pd–Mn, has been considered due to this type of ordering. The clusters decorating the quasilattices consist of several tens of atoms that are arranged with approximate icosahedral symmetry m35 [20,21]. In the case of Al-transition metal quasicrystals, Mackay-type cluster is considered as a basic structural unit. The 54 atoms form the triple shells as presented in Figure 3.1(a). The first and the second shells are an icosahedron and an icosidodecahedron, respectively. They are composed of Al. The third shell is larger icosahedron of the transition metal. In the case of ternary alloys, the following five alloys have been reported as a stable Mackay-type: Al–Cu–(Fe, Ru, Os) [22,23] and Al–Pd–(Mn, Re) [11]. In the case of Zn-based quasicrystals, Bergman cluster presented in Figure 3.1(b) has been proposed, which includes 104 atoms [21]. This cluster has four concentric shells. An icosahedron, a dodecahedron, a larger icosahedron, and a truncated icosahedron (soccer ball) are arranged in this order. The following ten ternary alloys are known to be Bergman-type: Al–Li–Cu, Zn–Mg–Ga, Ti–Zr–Ni, Mg–Al–Pd, and Zn–Mg– ( Y, Dy, Gd, Ho, Tb, and Er) [12].

3.3.

Approximants including Tsai-type cluster

The structure model of Zn17Sc3 crystal is presented in Figure 3.2, which was proposed by Andrusyak et al. [24]. Depending on details in structural model, this

51

3.3. Approximants including Tsai-type cluster

(a)

2.45 Å 4.70 Å

4.83 Å

(b)

2.48 Å 4.50 Å

4.93 Å 6.78 Å

Figure 3.1. Icosahedral atomic clusters. (a) Mackay-type in Al–Mn–Si approximant and (b) Bergmantype in Zn–Mg–Al approximant. The radius of each shell is inserted.

13.843 Å

Figure 3.2. Structure model of Zn17Sc3 approximant after Andrusyak et al. [24]. Large and small balls correspond to Sc and Zn atoms, respectively.

phase is sometimes referred to as Zn6Sc [25]. The crystal has body-centered cubic (BCC) structure with a lattice parameter a ¼ 13.843 A˚. This structure is interpreted as combination of a cage network and embedded cluster. The cage network has holes arranged with the translational symmetry of BCC as presented in Figure 3.2. In each hole, the cluster shown in Figure 3.3 is embedded. This cluster named Tsaitype consists of triple shells. They are the smallest dodecahedron, the second icosahedron, and the third icosidodecahedron. The first and the third shells are composed of Zn, and the second is of larger Sc. This atomic configuration of Tsaitype is different from Mackay- and Bergman-types. From the geometrical view point, it is possible to construct this triple shell structure as close-packing of two kinds of hard spheres [26]. The close-packing condition is satisfied, if EC ¼ ED and

52

Chapter 3.


F E C

D

Figure 3.3. Hard sphere model of Tsai-type cluster. The ideal ratio of two spheres is determined by the close-packing conditions EC ¼ ED and EF ¼ CF.

EF ¼ CF in Figure 3.3. In this case, the radii of the larger sphere A and the smaller B need to satisfy the following relationship: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ffi pffiffiffi rA =rB ¼ ð10 þ 5 þ 2 3Þ=3 1 1:288:

ð3:2Þ

However, there remains small cavity inside the cluster, which can include approximately four Zn atoms. In a real structure this cavity must be filled in some fashion, but there seems no simple solution due to mismatch between the cavity size and the icosahedral symmetry of the outer shells. This interesting problem is related to the low-temperature transitions in the Cd- and Zn-based approximants recently reported [27–29], and will be discussed later. In general the word ‘approximant’ denotes a periodic structure that approximates local structural properties of a quasicrystal, and includes an icosahedral cluster as a local structural unit as in the case of Zn17Sc3. Such approximant experimentally tends to form at the composition near to that of corresponding quasicrystal. Furthermore, in the theoretical treatment of the structure analysis, both approximant and quasicrystal originate from a unique structure in the high dimensional space [18,20]. The structure of the approximant can be generated from quasicrystal by introducing a linear phason strain. In the case of a cubic approximant of an icosaheral quasicrystal, the following relationship is satisfied between the lattice parameter a and the 6-dimensional lattice parameter a6D: a¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=2 þ ðFn þ Fnþ1 Þa6D ;

ð3:3Þ

where Fn and Fn+1 are successive Fibonacci numbers [20,30]. The lattice parameter a of the Zn17Sc3 corresponds to the case of Fn ¼ 1 and Fn+1 ¼ 1, and then this is called 1/1 approximant. The crystalline alloys isomorphous, or almost isomorphous, to the Zn17Sc3 approximant are listed in Table 3.1. They are Be, Ga, Cu,

53

3.3. Approximants including Tsai-type cluster

Table 3.1. 1/1 approximants including Tsai-type cluster. R denotes lanthanoids Pr, Nd, Sm, Eu, Gd, Dy, or Yb [37]. a (A˚)

Alloy Be17Ru3 Ga3.85Ni2.15Hf Ga3.22Ni2.78Zr Ga3.64Ni2.36Sc Cu3.7Ga2.3Sc Cu3.4Ga2.6Lu Zn17Sc3 Zn17Yb3 Ag47Ga38Yb15 In4Ag2Yb In4Ag2Ca Cd6Y Cd6R Cd6Ca Cd6Sr

References

11.337 13.319 13.374 13.440 13.472 13.745 13.843 14.291 14.701 15.362 15.454 15.482 15.441 15.661 15.702 16.044

[31] [32] [32] [32] [32] [33] [24] [34] [35] [35] [36,37] [37,38] [37,39] [39]

Zn, Ag, In, and Cd-based alloys with lattice parameters ranging rather widely from 11 to 16 A˚. They belong to the space group either Im3 or I23. Electron diffraction patterns of the Zn17Sc3 crystal are presented in Figure 3.4. These patterns have similarity to the corresponding patterns of the icosahedral quasicrystal. Namely Figures 3.4(a), (b), and (c) approximate 2-, 3- and 5-fold diffraction patterns of the quasicrystal. In these patterns reflections with Miller indices consisting of Fibonacci numbers tend to have strong intensity, e.g., 235 reflection. Using the approximants listed in Table 3.1 as a starting material, a new quasicrystal alloy has been searched by changing alloy composition or adding the third or fourth elements, and finally the new group of quasicrystals has been discovered. The new quasicrystals are summarized in Table 3.2. Cd- and Ag-In based alloys in this table were found by Guo et al. [48–50]. In this article, the experimental results on the Zn- and Cu-based quasicrystals will be mainly presented.

B D A

A C

(a)

(b)

(c)

Figure 3.4. Electron diffraction patterns of 1/1 approximant Zn17Sc3. (a) [100], (b) [111] and (c) [ 0 1]. The indices of reflections are as follows; A: 060, B: 03 5, C: 23 5, and D: 25 3.

54

Chapter 3.


Table 3.2. Tsai-type icosahedral quasicrystals discovered in the recent six years. For the alloy composition, analyzed composition is used, if available. Alloys

Stability

Cu46Al38Sc16 Cu48Ga34Mg3Sc15 Zn81Mg4Sc15 Zn74Mn10Sc16 Zn77Fe7Sc16 Zn78Co6Sc16 Zn74Ni10Sc16 Zn75Pd9Sc16 Zn74Pt10Sc16 Zn72Cu12Sc16 Zn74Ag10Sc16 Zn74Au10Sc16 Zn77Fe7Sc8Ho8 Zn77Fe7Sc8Er8 Zn77Fe7Sc7Tm9 Zn76Mg10Yb14 Ag42In42Ca16 Ag42In42Yb16 Cd65Mg20Ca15 Cd65Mg20Y15 Cd65Mg20Gd15 Cd65Mg20Tb15 Cd65Mg20Dy15 Cd65Mg20Ho15 Cd65Mg20Er15 Cd65Mg20Tm15 Cd65Mg20Yb15 Cd65Mg20Lu15 Cd85Ca15 Cd84Yb16

– Stable Stable Meta Stable Stable Stable Stable – Stable Stable – Stable Stable Stable Meta Stable Stable Stable Stable Stable Stable Stable Stable Stable Stable Stable Stable Stable Stable

3.4.

a6D(A˚)

a6D/r

rA/rB

e/a

References

6.959 6.938 7.115 7.107 7.083 7.076 7.051 7.117 7.112 7.065 7.149 7.136 7.165 7.170 7.166 7.371 7.928 7.905 8.105 7.928 7.987 7.959 7.959 7.955 7.951 7.922 8.099 7.879 8.105 8.057

4.99 5.00 4.94 5.00 4.97 4.97 4.96 4.97 4.96 4.98 4.97 4.96 4.99 5.00 4.99 4.94 4.89 4.89 4.95 4.92 4.96 4.95 4.96 4.96 4.96 4.95 4.97 4.92 4.98 4.95

1.22 1.22 1.17 1.19 1.19 1.19 1.19 1.18 1.18 1.19 1.17 1.17 1.23 1.23 1.23 1.37 1.27 1.25 1.25 1.14 1.14 1.13 1.13 1.12 1.12 1.11 1.23 1.10 1.26 1.24

2.08 2.01 2.15 2.15 2.09 2.09 2.02 2.07 2.06 2.04 2.06 2.06 2.09 2.09 2.09 2.00 2.00 2.00 2.00 2.15 2.15 2.15 2.15 2.15 2.15 2.15 2.00 2.15 2.00 2.00

[40] [41] [42] [43] [43] [43] [43] [44] [44] [45] [44] [44] [46] [46] [46] [26] [47] [47] [48,49] [48] [48] [48] [48] [48] [48] [48] [48,49] [48] [50] [50]

Preparation methods of Zn- and Cu-based quasicrystals

Because of both the high vapor pressure of Zn and the large difference in melting temperatures between Zn and transition metals, one needs to apply special technique to synthesize Zn-transition metal alloys. Two methods have been employed for this purpose. One is packaging with Mo thin foil [42] or arc-welded Nb container [45] and the other is melting under the condition of temperature gradient. The detail of the latter method was described elsewhere [51]. In both cases weight loss during the melting and annealing processes was controlled usually within 0.1 and 1 wt.% at worst. The Cu-based alloys were prepared by either induction or arc-melting in an Ar atmosphere [40,41]. The homogeneity and the alloy composition have been checked by means of electron probe microanalysis. In the case of single-phase specimen, the difference between the nominal and the analyzed compositions was controlled within 1 at.% for the Zn- and Cu-based alloys listed in Table 3.2.

55

3.5. Zn–Mg–Sc quasicrystal as a prototype

3.5.

Zn–Mg–Sc quasicrystal as a prototype

The Zn–Mg–Sc quasicrystal is thermodynamically stable at approximately 650 C near the alloy composition Zn81Mg4Sc15, and forms with the Zn17Sc3 approximant and the Zn2Mg-type Laves phase [42]. The Zn17Sc3 approximant can contain 2 at.% Mg, and then the composition difference between this approximant and the quasicrystal is less than 3 at.% Mg. The lattice parameter of the Zn17Sc3 is a ¼ 13.843 A˚, and can be related to the 6-dimensional lattice parameter of the quasicrystal a6D ¼ 7.111 A˚ as a 1/1 approximant using the equation (3) as described above. In addition to these phases, Lin and Corbett have reported the presence of a 2/1 cubic approximant with a lattice parameter a ¼ 22.412 A˚ at the alloy composition Zn84Mg3Sc13 [52]. According to their results, this approximant has wider phase region than the quasicrystal at 400 C (see Figure 3.2 in [52]). Figure 3.5 presents diffraction patterns of the Zn–Mg–Sc quasicrtystal with the beam incidence along the 2-, 3-, and 5-fold axes, which reveal the icosahedral symmetry m 3 5 in the reciprocal space. In Figure 3.5 weak reflections have point like shape and stronger ones have circular shape. They are located at almost ideal positions. In the case of low-quality quasicrystal including phason strains, zigzag arrangement of diffraction spots has been reported [53,54]. However in this quasicrystal, the deviation Dg ¼ Dq/2 of each spot is very small, and was measured smaller than 0.0007 A˚–1. Intensity distribution in Figure 3.5(c) satisfies almost perfectly the 10-fold symmetry. The X-ray Laue patterns of the Zn–Mg–Sc quasicrystal are presented in Figure 3.6. High degree of structure perfection of the Zn–Mg–Sc quasicrystal can be noticed by the number of reflections, their sharpness, and small deviations from the ideal positions. Furthermore recent experiment with synchrotron radiation has revealed higher quality of this quasicrystal than that of Al–Pd–Mn with respect to the number of many weak reflections and weaker diffuse scattering around the strong reflections [55]. The 200121 reflections of the two quasicrystals are compared in Figure 3.7, which are also referred to as 20/32 using Cahn’s indexing scheme [56]. In the case of Zn–Mg–Sc, even very weak reflections with intensity 10–6 times smaller than the strong Bragg reflection 200121 were recognized as sharp peaks, while those

(a)

(b)

(c)

Figure 3.5. Electron diffraction patterns of the Zn–Mg–Sc icosahedral quasicrystal. (a) 2-fold, (b) 3-fold, and (c) 5-fold patterns.

56

Chapter 3.

(a)


(b)

(c)

Figure 3.6. Laue patterns of the Zn–Mg–Sc icosahedral quasicrystals. (a) 2-fold, (b) 3-fold, and (c) 5-fold patterns (Courtesy of H. Takakura).

108 107

I (e.u./Å3)

20/32

i-AlPdMn i-ZnMgSc

106 105 104 1000 100 10 1 2.5

3

3.5

Qpar (2π/a6D unit) Figure 3.7. Comparison between Zn–Mg–Sc and Al–Pd–Mn quasicrystals from [55]. Intensity distribution was measured along a 2-fold axis.

are smeared out by diffuse scattering in the Al–Pd–Mn. Preliminary results of single-crystal X-ray diffraction on the series of Zn–M–Sc for M ¼ Mg, Co, Pd, and Ag have suggested that the best quality is realized for Mg, and the second for Ag [57]. A high-resolution image of the Zn–Mg–Sc quasicrystal is presented in Figure 3.8, which was observed along the 5-fold axis. This image reveals non-periodic but ordered feature of a quasicrystal. By looking at grazing angle, parallel lines of bright contrast can be observed in the five directions as indicated by arrows in Figure 3.8. The lines are arranged non-periodically with two kinds of intervals, 0.67 and 0.42 nm. The ratio of two intervals is approximately equal to the golden mean . Five such lattice lines are oriented with a rotation angle /5. This fact indicates the existence of a quasilattice and thus the long-range orientational ordering in the real space. The properties of the Zn–Mg–Sc quasicrystal described here are commonly observed in this series of icosahedral quasicrystals, Zn–M–Sc with M ¼ Mg, Fe,

3.5. Zn–Mg–Sc quasicrystal as a prototype

57

Figure 3.8. High resolution electron micrograph of the Zn–Mg–Sc quasicrystal observed along the 5-fold axis.

Co, Ni, Cu, Pd, and Ag, Zn–Fe–(Sc, R) with R ¼ Ho, Er, and Tm, and Cu–Ga–Mg– Sc. Formation in narrow composition range as an equilibrium phase, ternary or pseudo-ternary alloy containing small amount of the third element M, structural relationship to the Zn17Sc3-type approximant, and a P-type icosahedral quasicrystal with high degree of structural perfection are the common features of these quasicrystals. The last property, P-type with high quality is in contrast to our old consensus that the high structural quality is limited only for the F-type quasicrystals

58

Chapter 3.


such as Al–Pd–Mn. Among the Zn- and Cu-based Tsai-type quasicrystals, the Zn–Mg–Sc has the best structural quality, and then this can be regarded as a prototype of this series.

3.6.

Other Zn- and Cu-based quasicrystals Zn–M–Sc quasicrystals with M ¼ Cu, Ag, Au, Pd, or Pt

3.6.1.

In Figure 3.9(b), powder X-ray diffraction pattern of the Zn74.5Ag9.5Sc16 alloy is presented, which was annealed at 600 C. All peaks are indexed as a P-type icosahedral quasicrystal with a6D ¼ 7.146 A˚ by using the indexing scheme (1). In this pattern the peaks are very sharp. For example, 02 23 03 reflection located at 2 ¼ 66.60 in Figure 3.9(c) has a full width 0.12 at the half maximum. This width corresponds to Dg ¼ Dq/2 ¼ 0.0012 A˚–1, which is comparable with the peak width 0.0010 A˚–1 of Si measured under the same condition. In this alloy system, the Zn17Sc3 type approximant also forms. In Figure 3.9(a), the X-ray diffraction pattern of the Zn74.5Ag11Sc14.5 alloy is presented. All the peaks are indexed as a 1/1 approximant with a ¼ 13.914 A˚. This lattice parameter again

400

011202

(b)

1031,952,765

1033,961

941,853,770

831,750,743

620 541

330,411 420 333 422 510,431 521

200

220 310 222 321

(c)

300

2

100

66.5

67.0

67.5

68.0

68.5

222224, 131304

66.0

131303 122214, 131313 022304, 222214

0 65.5

022303

111212,011103 111212,020202

111111,010102 021101,010112 011102

011101

010101

000001

1

021202 121202,111113

200

111112

Intensity (cps)

3

110

600,442

4 × 103

631 710,550,543 640 721,552 642

530,433

611,532

(a)

0 10

20

30

40

50

60

70

80

2 θ (degree) Figure 3.9. Powder X-ray diffraction patterns of the Zn–Ag–Sc alloys measured by Cu Ka radiation. (a) 1/1 approximant in Zn74.5Ag11.0Sc14.5 alloy. (b) Quasicrystal in Zn74.5Ag9.5Sc16 alloy. Notice overall similarity to (a) in intensity distribution. (c) Magnified part of 02 23 03 reflection in (b).

59

3.6. Other Zn- and Cu-based quasicrystals

coincides with the value 13.910 A˚ estimated from the a6D using the equation (3.3). Although one is a quasicrystal and the other crystal, overall intensity distribution is similar in these two patterns. These observations as well as the similarity in compositions suggest that the Zn–Ag–Sc quasicrystal also includes the Tsai-type cluster as a local structural unit. Very recently Takakura et al. [58] have carried out structure analysis of the binary Cd–Yb quasicrystal, and obtained a direct evidence of the presence of the Tsai-type cluster. Their result may support the presence of Tsai-type cluster in the Zn–Sc-based quasicrystals, because the Zn17Sc3 and the Cd6Yb approximants are isomorphous. The structure analysis of the ternary Zn–Sc-based quasicrystals is a subject in the next step. In Figure 3.10, the formation conditions of the quasicrystal and the approximant in the Zn–Ag–Sc system are presented. The approximant forms in the elongated region with constant Sc composition of 14.5 at.%. At 600 C the solubility limit of Ag in the 1/1 approximant is 13 at.%. By the addition of Ag the lattice parameter increases almost linearly from 13.843 to 13.940 A˚. The composition difference between the approximant and the quasicrystal is very small in this alloy system. Interestingly simple addition of Ag in the Zn17Sc3 approximant does not cause formation of the quasicrystal, but the small increase of Sc is necessary. The corresponding result has been reported in the Zn–Cu–Sc system [45]. Similarly to Cu and Ag, an addition of other noble metals, Au, Pd or Pt cause formation of P-type icosahedral quasicrystals [44]. Their X-ray diffraction patterns are presented in Figure 3.11. Among them it was confirmed that the Zn–Pd–Sc quasicrystal is stable. However, there is no detailed study on the formation conditions of these three quasicrystals.

Zn–T–Sc quasicrystals with T ¼ Mn, Fe, Co, or Ni

3.6.2.

In Figures 3.12(a–d), powder X-ray diffraction patterns of as-cast Zn75T10Sc15 alloys are presented, where T denotes Mn, Fe, Co, or Ni. In these alloys, icosahedral quasicrystals form as a main phase. In Mn, Fe, and Ni alloys, the quasicrystals form almost exclusively in the as-cast alloys. Diffraction patterns of the

82

83

Zn

81

80

79

78

77

76

75

74

73 17

84

16

Sc

85 15 86 0

14 1

2

3

4

5

6

7

8

9

10

11

12

13

Ag Figure 3.10. Formation conditions of icosahedral quasicrystal and 1/1 approximant in the Zn–Ag–Sc system at 600 C. All specimens were annealed at 600 C for 20–50 h, except for the Zn79.2Ag5.2Sc15.6 alloy annealed at 740 C. The square, star, and hexagon marks denote the 1/1 approximant, the quasicrystal and the hexagonal Zn53Sc17 phase, respectively.

60

Chapter 3.

4000


(a)

Zn75Pd9Sc16 a 6D = 7.117 Å 761 °C for 50 h

3000 2000 1000 0

Intensity (cps)

(b) Zn74Pt10Sc16 a 6D = 7.112 Å as cast

2000

1000

0 3000

(c) Zn74Au11Sc15 a 6D = 7.136 Å as cast

2000

1000

0 10

20

30

40

50

60

70

80

2θ (deg)

Figure 3.11. X-ray diffraction patterns of (a) Zn75Pd9Sc16, (b) Zn74Pt10Sc16, and (c) Zn74Au11Sc15 alloys measured by Cu Ka radiation. The Zn75Pd9Sc16 quasicrystal forms in the alloy annealed at 761 C, and is stable.

annealed alloys are presented in Figures 3.12(e–h). The annealing experiments have revealed thermodynamical stability of three quasicrystals containing Fe, Co, and Ni. The stoichiometric compositions of the stable phases are Zn77Fe7Sc16, Zn78Co6Sc16, and Zn74Ni10Sc16. They have very narrow single-phase region, and approximately 1 at.% difference in transition metal composition causes appearance of impurity phase. In the case of Mn alloy, the quasicrystal transformed to mainly the 1/1 approximant by the annealing at 700 C. Then Zn–Mn–Sc quasicrystal is metastable. This series is interesting from a view point of magnetism. The temperature dependence of magnetic susceptibility is presented in Figure 3.13. These curves obey Curie-Weiss low in the wide temperature regions. Kashimoto et al. [59] have estimated relatively large magnetic moment 3.7 mB for Fe, and 0.07–0.31 mB for Mn, Co, and Ni from the Curie constants under the assumption that only the transition elements, Fe, Mn, Co, or Ni have magnetic moments. This is the first report of large magnetic moment of the 3d transition element, Fe, in a stable icosahedral quasicrystal. They also revealed that the cusps observed in the Mn and Fe alloys are due to the appearance of spin-glass state with freezing temperatures 14 and 7 K, respectively. The magnetic properties are summarized in Table 3.3. Very recently it has been shown that the rare-earth metals, Ho, Er, and Tm can be partially

61

3.6. Other Zn- and Cu-based quasicrystals

3000

3000

(a) Zn75Mn10Sc15

(e) Zn75Mn10Sc15

2000

2000

1000

1000

0 3000

3000

(b) Zn75Fe10Sc15

(f) Zn77Fe7Sc16 2000

2000

1000

1000 0 1500

0 3000

(c) Zn75Co10Sc15

1000

2000

500

1000 0 3000

0

(d) Zn75Ni10Sc15 2000

(g) Zn78Co6Sc16

(h) Zn75Ni10Sc15

2000

1000

1000

0

0 10

20

30

40

50

60

70

80

10

20

30

40

50

60

70

80

Figure 3.12. Stability and metastability of Zn–T–Sc icosahedral quasicrystals with T ¼ Mn, Fe, Co, or Ni. The specimens presented in (a)–(d) are as-cast ones, and in (e)–(h) are annealed ones at 647–703 C. All diffraction patterns were measured by Cu-Ka radiation. Notice the appearance of 1/1 approximant in the annealed Zn75Mn10Sc15 alloy. The composition of the quasicrystal in Zn75Ni10Sc15 alloy was analyzed to be Zn74Ni10Sc16 by means of electron probe microanalysis.

substituted for Sc in the Zn–Fe–Sc quasicrystal [46]. This new series may provide an opportunity to study magnetic properties in those including two kinds of magnetic elements located at different crystallographic sites.

3.6.3.

Cu-based quasicrystals

Because of the presence of the Cu3.7Ga2.3Sc crystal [32] isomorphous to the Zn17Sc3, Cu-based quasicrystals have been searched and actually found by the following methods: rapid cooling using melt spun technique [40], adding small amount of Mg [41], or replacing Ga with Al [40]. By the rapid cooling, a quasicrystal is formed near the alloy composition Cu48Ga34Sc18. However, this quasicrystal is not stable, but the 1/1 approximant at 700 C. At the alloy composition Cu48Ga34Mg3Sc15, an icosahedral quasicrystal forms exclusively, and this phase is stable at 700 C [41,60]. Systematic study on the series of alloys (Cu0.59Ga0.41)85-xMgxSc15 has revealed that the addition of Mg is necessary for the formation of the quasicrystal. Electron diffraction patterns of the Cu–Ga–Mg–Sc quasicrystal are presented in Figure 3.14. These patterns indicate that this quasicrystal belongs to P-type and isomorphous to the Zn–Mg–Sc quasicrystal. By careful inspection, displacements of

62

Chapter 3.


10–4

χ /cgsemu g–1

Zn77Fe7Sc16 10–5

Zn74Mn10Sc16

10–6

Zn78Co6Sc16

Zn74Ni10Sc16 0

50

100

150

200

250

300

T /K Figure 3.13. Magnetic susceptibilities of Zn-T-Sc quasicrystals with T ¼ Mn, Fe, Co, and Ni. Thin lines indicate results of Curie-Weiss fitting. Notice cusps of Mn and Fe alloys at low temerature.

Table 3.3. Magnetic properties of Zn–T–Sc quasicrystals with T = Mn, Fe, Co or Ni. Curie constants C were estimated by Curie-Wiess fitting in high temerature regions. See [59] for details. Quasicrystal Zn74Mn10Sc16 Zn77Fe7Sc16 Zn78Co6Sc16 Zn74Ni10Sc16

C (cgsemu Kg–1)

(K)

p (mB)

TF (K)

1.17 10–5 1.95 10–3 8.92 10–6 5.47 10–7

–9.3 4.2 –4.7 –0.8

0.31 3.7 0.27 0.07

14.0 7.0 – –

Figure 3.14. Electron diffraction patterns of Cu48Ga34Mg3Sc15 quasicrystal. (a) 2-, (b) 3-, and (c) 5-fold patterns. (d) Magnified image of the rectangular region in (a). Displacements due to a linear phason strain are indicated by arrows.

3.7. Single-quasicrystals and microvoids

63

diffraction spots from the ideal position can be detected, which are indicated by arrows in the magnified image, Figure 3.4(d). This observation indicates that the magnitude of the linear phason strain [53,54] is larger than that in the Zn–Mg–Sc quasicrystal. By replacing Ga with Al, a ternary Cu46Al38Sc16 quasicrystal has been discovered very recently [40]. This quasicrystal is formed in the as-cast alloy and remains after the aging at 600 C for 61 h.

3.7.

Single-quasicrystals and microvoids

In the case of stable quasicrystals, slow cooling from liquid state sometimes causes growth of a large single domain reaching to millimeter size. Two types of external shapes have been observed in the Tsai-type quasicrystals. One is a rhombic triacontahedron presented in Figure 3.15(a), which was grown in the Zn81Mg4Sc15 alloy. This shape is surrounded by planes normal to the 2-fold axes. The triacontahedral shape is rather rare, and other example can be found in Bergman-type Al–Li–Cu [61]. The other is a dodecahedron of the Zn77Fe7Sc8Er8 quasicrystal presented in Figure 3.15(b). This is a typical shape of an icosahedral quasicrystal, and can be found in many alloys, e.g., Mackay-type Al–Cu–Fe [22] and Bergman-type Zn–Mg–Ho [62]. In both cases the external shapes certainly reflect the diffraction symmetry m35, and indicate the presence of densely packed atomic planes perpendicular to the 2- and 5-fold axes. From the theoretical view point, the growth of a quasicrystal is an interesting problem, because the ideal quasiperiodicity is global and strict property not realized simply from a local rule. Theoretical studies have predicted a slow growth velocity of a quasicrystal comparing with a conventional crystal [63]. However, in the experiments it is not so difficult to synthesize a large single quasicrystal. In a millimeter sized single quasicrystal of Zn–Fe–Sc, facetted microvoids were frequently observed as presented in Figure 3.16 [64]. They are ‘negative crystals’. The diameters of these voids range from 1 to 50 mm, and were controlled by cooling speed from the melting point (approximately 823 C) as well as annealing time at 700 C. The size is smaller in faster cooled specimen to 700 C from the melting point. This tendency is natural, if the growth of microvoids is governed by condensation of excess vacancies as pointed out by C. Beeli et al. on the Al–Pd–Mn quasicrystal [65].

a

b

1 mm 0.1 mm Figure 3.15. Single quasicrystals. (a) Scanning electron micrograph of the Zn–Mg–Sc quasicrystal. (b) Optical micrograph of the Zn–Fe–Sc–Er quasicrystal.

64

Chapter 3.


5 μm a

b

20 KV

x750

10 μm

032603

Figure 3.16. Scanning electron microscope image of the cleavage surface of the Zn–Fe–Sc quasicrystal. (a) Dodecahedral microvoids in the specimen quenched from 600 C. (b) Highly faceted microvoid in the specimen quenched from 823 C.

When the quenching temperature is higher than 800 C, the complicated facetted voids, as presented in Figure 3.16(b), were mainly observed. The external shape is composed of the 2-, 3-, and 5-fold planes and narrow additional planes. The number of dodecahedral voids is increased in the specimens quenched from lower temperature, e.g., 600 C as presented in Figure 3.16(a). In this image, the parallel orientation of these voids suggests the presence of a millimeter sized single quasicrystal, which was also confirmed by means of Laue method.

3.8.

Central structure of the Tsai-type cluster

The most mysterious point in the Tsai-type cluster is its central structure, namely the inside structure of the first dodecahedral shell. As described already in the Section 3.3, the structure of the triple shells of the Tsai-type cluster is basically understood by the close-packing rule. However, this simple rule is not applicable for the central region, even though that is occupied by the same type of atoms as in the first and the third shells. There are two possible models for the central structure. They are presented in Figure 3.17. In Figure 3.17(a), a regular tetrahedron of Zn is surrounded by 20 Zn atoms forming a dodecahedron. This model corresponds to 100% occupied version of Cd6Yb proposed by Palenzona [38]. Four 3-fold axes of the tetrahedron are parallel to those of the dodecahedron, and a short interatomic distance, 2.22 A˚ in the case of Zn–Sc, appears along the four 3-fold axes. The other model is presented in Figure 3.17b, in which an irregular tetrahedron with the point symmetry 2mm is situated in a dodecahedron with two common mirror planes. In this case the 2-fold axis of the tetrahedron is parallel to one of the dodecahedron. A short interatomic distance 2.07 A˚ appears at the edge CD in Figure 3.17(b) of the tetrahedron in the case of Zn–Sc. This is fully ordered version of the Zn6Sc model proposed by Lin and Corbett [25], and corresponds to the Cd6Y model by Larson and Cromer [36]. In both models in Figure 3.17, unrealistic short atomic distances

65

3.8. Central structure of the Tsai-type cluster

(a)

(b)

B A C

D

Figure 3.17. Two models of the central structure of the Tsai-type cluster. (a) Regular tetrahedron with its four 3-fold axes parallel to those of the outer dodecahedron. The distance AB is 2.22 A˚. (b) Irregular tetrahedron satisfying 2mm in the outer dodecahedron with two common mirror planes. The distance CD is 2.07 A˚.

appear, and the close-packing rule is not applicable for understanding the central structure. Up to the present all structure analyses of Cd- and Zn-based 1/1 approximant have revealed a fractional occupancy for the inclusion site. The fractional occupancy has been interpreted by statistical averaging of various orientations of the tetrahedra. This situation was beautifully demonstrated by Go´mez and Lidin [37]. The structure model of a quasicrystal has essentially the same problem, and actually no information has been obtained for the inclusion site in the analysis of the Cd–Yb quasicrystal [58]. Recent discovery of low-temperature transitions in the Cd- and Zn-based approximants [27–29] has opened the way to solving this interesting problem. Tamura et al. suggested that this ordering is related to a periodic arrangement of the included tetrahedra. The structural analysis of the Zn–Sc low-temperature phase at 92 K has been carried out by powder X-ray Reitveld analysis using synchrotron radiation [66]. The low-temperature phase is monoclinic belonging to the space group C2/c and has lattice parameter a ¼ 19.47 A˚, b ¼ 13.79 A˚, c ¼ 19.55 A˚, and b ¼ 89.931 . This unit cell includes four Tsai-type clusters whose centers are situated at [0 -1/4 -1/4], [0 1/4 1/4], [1/2 1/4 -1/4], and [1/2 -1/4 1/4]. Then the 2-fold axis penetrates the center of the cluster along the b-axis. Because these four clusters are related by the symmetry operations of the C-center and the c-glide, they have unique structure. In other words, there is only one type of cluster in the Zn-Sc lowtemperature phase. The result of the Reitveld analysis is presented in Figures 3.18 and 3.19. The cluster consists of almost regular tetrahedron and triple shells. The edge length of the tetrahedron ranges from 2.48 to 2.62 A˚, and then the tetrahedron is regarded as almost regular. The first dodecahedral shell is highly distorted as illustrated in Figure 3.18. For example the lengths EF and GH in Figure 3.18a are 7.31 and 6.68 A˚ respectively, which should be the same in a regular dodecahedron. The tetrahedron is rotated against the dodecahedron around the common 2-fold axis by approximately 13 . This rotation can be noticed in Figures 3.18(a) and (b). This model is similar to that in Figure 3.17(b), but different with respect to the following three points; the presence of almost regular tetrahedron, the distortion of

66

Chapter 3.


(a)

(b) E

y

G

C

D

F

F

D

A y

x A

B

H

(d)

C

E z

y

z

(c)

z x

(e)

(f)

B

x

z

y

y

D

y

C

z x

z

x

Figure 3.18. Structure model of the central region of the Tsai-type cluster at [0 1/4 1/4] determined by Rietveld analysis. (a) [–1 0 1], (b) [0 1 0], (c) [–0.55, –0.63, 0.55], (d) [0.55, –0.63, –0.55], (e) [–0.32, 0.90, –0.31], (f) [0.29, 0.92, 0.25] projections. All atoms are Zn. The line CD is rotated from the line EF by 13 as seen in (b). The atoms A – D in (c)–(f) correspond to those in (a) and (b).

(a)

(b)

(c) I F

y z

5.20 Å Figure 3.19. Structure model of Tsai-type cluster in the low-temperature phase. (a) Inside tetrahedron surrounded by the first and the second shells. (b) The first and the third shells. (c) The second and the third shells. Dark and bright spheres denote Sc and Zn atoms, respectively.

the surrounding dodecahedron and the small rotation. With the help of these modifications, the problem of the unrealistic short interatomic distance is now resolved. Figures 3.18(c)–(f ) illustrate the geometrical relationship between the Zn atoms belonging to the tetrahedron and the nearby pentagons. The Zn atom, AD, is seen near the center of the distorted pentagon that covers the Zn atom in each figure. Actually the shortest distance between them is 2.46 A˚. The tetrahedron at the cluster center also affects the geometry of the outer shells, but no unrealistic situation occurs. The model of the outer shells is presented in Figure 3.19. The icosahedral shell of Sc has also a small deviation from the regular one, of which edge lengths range from 4.92 to 5.39 A˚. In the third shell of the icosidodecahedron, the edge lengths range from 3.06 to 3.95 A˚. In spite of these distortions, no serious packing problem occurs in the triple shells as illustrated in Figures 3.19(b) and (c). Actually the shortest interatomic distance in the hole model is FI ¼ 2.38 A˚ in Figure 3.19(b), which is acceptable for Zn–Zn distance.

3.9. Alloy chemistry and the Tsai-type quasicrystals

67

The combination of the rotated regular tetrahedron and the distorted dodecahedron is the characteristic of this model. Corresponding features have been clarified also by the first principles calculations [67,68]. It is very natural to imagine that the Tsai-type cluster in a quasicrystal has similar structural property. Other details of the structure model is described elsewhere [66].

3.9. 3.9.1.

Alloy chemistry and the Tsai-type quasicrystals Linear relationship between a6D and average atomic radius

The quasicrystals listed in Table 3.2 have common properties with respect to atomic structure as well as alloy chemistry. It is probable that all the quasicrystals listed in Table 3.2 belong to one unique structural type named Tsai-type as described in Sections 3.5 and 3.6.1, while there is a few direct evidence indicating the presence of the Tsai-type cluster at present [58]. The additional indirect evidence is the relationship between the lattice parameter a6D and the average atomic radius r of the constituent elements [48,49]. Here, the lattice parameter p a6D is related to the edge length aR of the 3D Penrose rhombohedra by a6D ¼ 2 aR [16,17]. The atomic decoration scheme of the rhombohedra depends on the type of an atomic cluster, and then their edge length depends on the decoration scheme. Therefore the ratio a6D/r is expected to reflect the cluster type. Actually it is known that for the Mackay-type this ratio is approximately 4.58, and for the Bergman-type approximately 4.89. The ratios of the new quasicrystals are summarized in the fourth column in Table 3.2, which range from 4.89 to 5.00. For the calculation of the average radius r, the atomic radii listed in [69] were used, where the coordination number 12 was assumed for all the atoms. As a whole there is a linear relationship, a6D ¼ 4.96 r, in the relatively wide range of r from 1.39 to 1.64 A˚ as presented in Figure 3.20. In Figure 3.20 quasicrystals with the same base metal form a group, namely Cu-based, Zn-based, and Cd- and Ag–In-based quasicrystals, while the latter two are not separated. The Zn–Mg–Yb quasicrystal is also located on the line, but isolated from other groups. The average ratio 4.96 is clearly different from that of Mackay-type, but only slightly different from that of Bergman-type. Actually it is difficult to distinguish them by only this criterion. For example in the former report [51], the following quasicrystals were treated as Tsai-type or related type: Zn84Mg8Ti8 [70], Zn84Mg9Zr7, Zn84Mg9Hf7 [71], and Zn76Mg14Ho10 [72]. They have the ratios a6D/r ranging from 4.96 to 4.98. However, recent structure analyses of the corresponding approximant [73] or the quasicrystal itself [74] have indicated that these quasicrystals may belong to Bergman-type. Certainly it is important to carry out structure analysis of each quasicrystal or at least typical quasicrystals in Table 3.2.

3.9.2.

Substitution rules

There are interesting substitution rules in the alloys listed in Table 3.2. These alloys consist of three components: the base metal, the second and the third components.

68

Chapter 3.


8.2

8.0

Cd- and Ag-In based alloys

7.8

a 6D (Å)

7.6

7.4

Zn-Mg-Yb

7.2 Zn-based alloys 7.0 Cu-based alloys 6.8 1.40

1.45

1.50

1.55

1.60

1.65

r (Å) Figure 3.20. Linear relationship between the lattice parameter a6D and the average atomic radius r of the constituent elements.

The base metals are non-transition elements such as Zn, Cd, Cu–Al, Cu–Ga and Ag–In. They are divalent or effectively divalent on average. As a second component, they typically contain 16 at.% trivalent transition element such as Sc, Y, or lanthanoids. Non-magnetic character of Yb is known in the Cd–Yb and Zn–Mg–Yb quasicrystals, and then Yb is regarded as divalent. The divalent Ca or Yb as the second component seem apparent exceptions. The combination of the base metal and the second component is important for the formation of the approximants listed in Table 3.1. Except for the binary Cd–Ca and Cd–Yb quasicrystals, an addition of several percents of the third component plays the decisive role to stabilize a quasicrystal instead of an approximant. In the case of Zn–Sc alloys, there is a variety of the third components. They are Mg, Mn, Fe, Co, Ni, Cu, Pd, Ag, Pt, and Au. The variety of the Tsai-type quasicrystals listed in Table 3.2 originates from the flexibility in the base metal as well as the second and the third component. Such systematic feature has not been observed in the Mackay- and Bergman-type quasicrystals, and allows us to study the formation conditions of the Tsai-type quasicrystals.

3.9.3.

Hume-Rothery rules 1: Near equality in e/a

By examining the substitution rules, one can find the important role of the electronic contribution in the total energy in stabilizing the quasicrystal structures. The value

69

3.9. Alloy chemistry and the Tsai-type quasicrystals

of an average electron concentration, e/a, is summarized in Table 3.2. In this calculation, the following valences were used: 1 for Cu, Pd, Pt, Ag, and Au, 2 for Mg, Ca, Zn, Cd, and Yb, and 3 for Sc, Y, and lathanoids. The valences of the 3d transition metals cannot be defined uniquely, because the 3d and 4s levels are energetically adjacent to each other, and in delicate situation near the Fermi level. It is well known that their valences depend on alloying component. In the 1950s, many experimental works were carried out to obtain a reasonable set of valences of the transition elements which can systematically explain the alloy formation. There is a nice summary by Elliott and Rostker (see Table 3.5 in [75]). In the case of Al-transition metal alloys, negative valences of transition metals were proposed [76], and they have been successfully used for the discussion of the Al-based quasicrystals [77]. In the present report, we use the set by Haworth and Hume-Rothery: Mn: 1.9, Fe: 1.0, Co: 0.8, and Ni: 0.6. This set was proposed in order to explain the formation of a and b brasses in Cu–Zn-based alloys [78]. By applying these valences, almost constant magnitude of e/a ranging from 2.00 to 2.15 is calculated for all the quasicrystals listed in Table 3.2. This near equality indicates validity of Hume-Rothery rules in these Tsai-type quasicrystals. The physical meaning of the near equality can be seen by comparing the size of the Fermi surface and the Brillouin zone. In the strict sense, there is no Brillouin zone in a quasicrystal because of the non-periodicity. However, a quasicrystal generates strong Bragg reflections as in Figure 3.5, and they form energy gaps in dispersion curves as in a crystal [6,79,80]. Accordingly Brillouin zone formed by the Bragg reflections may have meaning. The size of Fermi sphere can be estimated using the following formula by assuming a free electron model. !1=3 2 3 N e=a A : ð3:4Þ kF ¼ 1=2 A Here , NA, A denote density of the quasicrystal, the Avogadro’s constant and the average atomic weight, respectively. The densities of some Tsai-type quasicrystals were measured by Archimedes’s method, and are listed in Table 3.4. The calculated Fermi wave vector kF agrees well with a half of the reflection vector g of the 02 1202 reflection. This reflection has large structure factor as seen in Figure 3.9(b). The multiplicity of the reflection is sixty. The zone formed by sixty 02 12 02 reflections is further truncated by twelve 11 1 1 13 reflections aligned along 5-fold axes. The final zone is presented in Figure 3.21. The large multiplicities of reflections in an icosahedral quasicrystal tend to make the Brillouin zone like a sphere. Such highly Table 3.4. Matching between kF and g/2 of the 02 12 02 reflection for some Tsai-type quasicrystals. Quasicrystal Zn81Mg4Sc15 Zn77Fe7Sc16 Zn74Ni10Sc16 Zn74Ag10Sc16 Cu48Ga34Mg3Sc15

(g/cm3)

e/a

kF (A˚–1)

g/2 (A˚–1)

– – – – –

2.15 2.09 2.02 2.06 2.01

0.251 0.251 0.251 0.248 0.251

0.2512 0.2522 0.2533 0.2498 0.2574

6.17 6.50 6.67 6.81 6.77

0.03 0.01 0.02 0.01 0.02

70

Chapter 3.


Figure 3.21. Brillouin zone formed by sixty 02 1 2 02 reflections with truncation by 12 11 11 13 reflections. Small spheres indicate the positions of these reflections in the reciprocal space. (Courtesy of H. Nakano).

faceted zone is responsible to the interaction with the Fermi surface. As the result the Fermi edge is located at the pseudogap in the curve of the density of states, and this matching causes the reduction of the electronic contribution in the total energy. This is called Hume-Rothery mechanism. Other mechanism has been proposed by Ishii and Fujiwara [81]. They calculated the band structures of the Cd-based and Zn-Sc approximants, and revealed that the sp-d hybridization is responsible for the formation of the pseudogap rather than the Hume-Rothery mechanism. Interestingly, the sp-d hybridization occurs also in the Cd–Ca approximant, and then Ca is regarded as a transition element with zero d-electron. With respect to the mechanism of the formation of the pseudogap at the Fermi level, Mizutani et al. have proposed different interpretation for the Al–Cu– Ru–Si approximant [82]. They emphasized that particular Fermi surface-Brillouin zone interactions are strongly coupled with the sp-d hybridization. These two theoretical studies have explained the reason why Tsai-type quasicrystals contain transition elements as the second components.

3.9.4.

Hume-Rothery rules 2: Near equality in ratio of atomic radii

The grouping in Figure 3.20 indicates an important role of the ratio of atomic radii rA/rB in the stabilization of the quasicrystals, where rA and rB denote the radii of the second component A and the base metal B, respectively. In the cases of Zn and Cu alloys the second component is Sc, and in Cd and Ag alloys Ca or lanthanoids. In other words there is a restriction with respect to the ratio rA/rB, which is listed in the fifth column in Table 3.2. For this calculation, the constituent elements were tentatively classified into two groups, namely the base or the second component in the cases of ternary and quaternary alloys. For example in Cu–Ga–Mg–Sc, Cu–Ga–Mg is regarded as a base. In Zn–Fe–Sc–Tm, Zn–Fe, and Sc–Tm are regarded as the base and the second component, respectively. The ratios rA/rB are distributed from 1.10

3.10. Conclusion

71

to 1.27 for the stable quasicrystals in Table 3.2. Then the atomic radius restriction is not so strict, but has some acceptable range. The ratios are slightly smaller than the geometrical one, 1.288, in the equation (3.2). This difference may be related to the fact that the real structure of the Tsai-type cluster is much more complicated than the close-packed model due to the center tetrahedron described in Section 3.8. However, the geometrical ratio may be applicable as a rough criterion, and further be regarded as the upper limit in Table 3.2. For example, the metastability of the Zn–Mg–Yb quasicrystal is related to the presence of this upper limit, even if the divalent atomic radii of Yb, 1.940 A˚, is overestimated. Another example of this restriction on rA/rB is the solubility limit of lanthanoids, L ¼ Ho, Er, and Tm, in the Zn–Fe–Sc–L quasicrystals. The limits are approximately 8 at.% for Ho and Er, 9 at.% for Tm [46]. These limits are interpreted by the larger atomic radii of the lanthanoids than Sc. In summary the near equality in rA/rB and in e/a composes the Hume-Rothery rules in Tsai-type quasicrystal. Recent discovery of Cu–Al–Sc quasicrystal is an example of the application of the Hume-Rothery rules [40].

3.10.

Conclusion

Experimental studies in the past six years have revealed the existence of new type of icosahedral quasicrystals. This discovery is forcing us to change the old common understanding of a quasicrystal in various aspects, namely in metallurgy, chemistry, and physics. Most surprising feature is its variety as an alloy. Now we know Cu, Ag, and Cd-based quasicrystals besides Al- and Zn-based quasicrystals. This is the largest group of quasicrystal alloys. The flexibility in the formation conditions allows us ‘tuning’ on structural quality as well as physical properties. It is certainly a nice starting point to the new field of quasicrystal research.

Acknowledgements The author is grateful to his colleagues, in particular Y. Kaneko, R. Maezawa, J. Matsunagi, C. Masuda, S. Kashimoto, S. Motomura, and S. Matsuo for their collaboration. This work has been partly supported by a Grant-in-Aid for the Scientific Research of Priority Areas ‘Panoscopic Assembling and High Ordered Functions for Rare Earth Materials’ from the Ministry of Education, Culture, Sport, Science and Technology of Japan.

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Chapter 4

Quasicrystals and Approximants in Cd-M Systems and Related Alloys An-Pang Tsai and Cesar Pay Gomez Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan

4.1.

Introduction

A diffraction pattern with icosahedral symmetry, long considered impossible in classiccrystallography, in the Al–Mn alloy prepared by a rapid cooling technique from the melt, had inspired the world of solid state science [1]. Meanwhile, together with icosahedral symmetry, the alloy phase in question generates a non-periodic array of diffraction peaks in the diffraction pattern, indicating the aperiodic arrangement of atoms. Traditionally, a crystalline structure is defined by a periodic repetition of a unit cell, and the periodicity is another definition for translational symmetry, the strictest form of long-range order. Thus, ‘Order in crystals is periodicity’ is the knowledge of long-standing in crystallography [2]. On the other hand, an icosahedron is locally the most densely packed arrangement but an icosahedral symmetry is incompatible with the translational symmetry because the icosahedral motif alone cannot fully fill space in three dimensions. Therefore, it was believed to be impossible that a long-range order with icosahedral symmetry would be formed in realistic matter. Nowadays, more than one hundred alloys have been verified to be stable quasicrystalline phases or/and quasicrystalline phases exist in the equilibrium phase diagrams. Undoubtedly, the quasicrystal is no longer a unique form of solid, it is universally verified in many metallic alloys, and possibly will be among polymers. It has been known for a long time prior to the discovery of quasicrystals that although five-fold and icosahedral symmetry is forbidden in the presence of two- and three-dimensional translation groups, local icosahedral arrangements of atoms in crystals are possible and in fact, for complex metallic alloys fairly common [3]. Many such alloys are now called ‘crystalline approximants’ (approximants) [4,5]. An approximant is a compound whose composition and structural units are very similar to those of a quasicrystal, but nevertheless it is a crystal. Another requirement for an approximant is that its space group should be a subgroup of its corresponding HANDBOOK OF METAL PHYSICS ISSN 1570-002X/DOI 10.1016/S1570-002X(06)02004-0 75


76

Chapter 4. Quasicrystals and Approximants in Cd-M Systems and Related Alloys

quasicrystal. Consequently, the approximant can be generated by a rational cut through high-dimensional space, and hence an approximant and a quasicrystal are related in structure through the term of ‘linear phason strain’ [4]. For example, in the case of an icosahedral quasicrystal, it is described that an identical icosahedral cluster exists in both the quasicrystal and the approximant but it arrays in a quasi-periodic arrangement for the former and arrays in a periodic way for the latter. Approximants have played a crucial role in understanding the structures of quasicrystals since the structures of approximants have been well defined and described as local packings of icosahedral clusters. Actually, icosahedral quasicrystals (i-QCs) can be classified into three classes according to hierarchic structures of icosahedral clusters derived from their corresponding approximants; the Al–Mn–Si [4] class, the Mg–Al–Zn [5] class and the Cd–Yb [6] class. The structures of the atomic shells for the three classes are shown in Figure 4.1. It has been recognized by

(a)

Vacant

12 Al/Si

12 Mn

30 Al/Si

(b)

Vacant

12 Al/Cu

20 Li

12 Al/Cu

(c)

4 Cd

20 Cd

12Yb

30 Cd

Figure 4.1. Shell structures of three types of icosahedral clusters derived from three 1/1 approximants of quasicrystals. (a) The Al–Mn–Si class or Mackay icosahedral cluster: the center is a vacant, 1st shell is a Al/Si icosahedron, 2nd shell is a Mn icosahedrons, and the 3rd shell is a Al/Si icosidodecahedron. (b) The Zn–Mg–Al class or Bergman cluster and the example is the R-AlLiCu: the center is a vacant, the 1st shell is a Al/Cu icosahedron, the 2nd shell is a Li dodecahedron, 3rd shell is a larger Al/Cu icosahedron. (c) The Cd–Yb class: the center is a Cd tetrahedron, the 1st shell is a Cd dodecahedron, 2nd shell is a Yb icosahedrons, and a 3rd shell is a Cd icosidodecahedron.

77

4.1. Introduction

Henley and Elser that two complex compounds known for a long time, -Mn12(Al,Si)57 [7] and Mg32(Al,Zn)49 [8] were indeed approximants of the i-QCs in the Al–Mn class and the Mg–Al–Zn class, respectively. The former is also quoted as the Mackay cluster and the latter quoted as the Bergman cluster. Each of these two structures is a bcc packing of clusters consisting of three concentric atomic shells with icosahedral symmetry as shown in Figure 4.1(a) and (b). The i-QCs in these two alloy systems were only obtained by conducting rapid solidification from the approximants, the i-QCs were realized to be energetically non-equilibrium phases or metastable phases with respect to the corresponding approximants. Lately, the stable i-QCs found in Al–Cu–Fe[9] and Al–Pd–Mn[10] alloys were grouped into the Al–Mn class and their corresponding approximants were found in the same alloys systems but at compositions deviated from those of i-QCs [11,12]. On the other hand, stable i-QCs in the Zn–Mg–RE (RE ¼ rare earth) systems were grouped into the Mg–Al–Zn class and no corresponding approximants were found in these systems. The third class of Cd–Yb is related to the Cd6Yb approximant [13] which has a bcc packing of identical icosahedral clusters. In the core of each cluster there is a tetrahedron created by four positionally disordered Cd atoms, as shown in Figure 4.1(c), which breaks the icosahedral symmetry. i-QCs and approximants of the Cd–Yb class have been found in a large number of alloys which thus constitutes the largest one of the three classes of i-QCs. Another simple way to distinguish the three classes of clusters is by using the Elser-Henley description [4,5] as shown in Figure 4.2. Elser and Henley used a threedimensional analogy inspired by the two rhombohedral Penrose cells; a prolate

(a)

Mn

(b)

AI/Si

Mg

(c)

Al/Zn

Yb

Cd

Figure 4.2. Elser-Henley description of prolate rhombohedron (upper) and rhombic dodecahedron (lower), showing their decoration by constituent atoms. Each edge has length aR and lies along one of the six icosahedral five-fold directions.

78


rhombohedron (PR) and an oblate rhombohedron (OR). The unit cell of an approximant was used to specify their atomic decorations. The description of the decoration is to group two PR and two OR that pack around a single vertex into a new packing unit; the rhombic dodecahedron (RD); see lower part of Figure 4.2. The tips of the PRs are identical to those of the RDs and correspond to the centers of the icosahedral clusters. The icosahedral clusters are assembled by packing six RD and eight PR around an identical tip. By using this description, the three classes of icosahedral clusters can be decomposed into PR and RD units with different atomic decorations [4,5,14] as shown in Figure 4.2. A main difference in the decoration among the three classes of clusters is the occupation of atoms at the midpoints of the edges. These are partially occupied in the Mackay class and fully occupied in the Bergman class, whereas there are no atoms at these positions in the Cd–Yb class. With this description one may immediately distinguish the differences in atomic decorations at a glance. Since the first two classes of structures have been reviewed and discussed in many books and review papers, in this article we will only focus on the approximants and i-QCs of the Cd–Yb class and the related systems.

4.2. 4.2.1.

Approximants in the Cd-M systems History

The finding of the binary Cd–Yb i-QC and its series of approximants has spawned a whole family of related quasicrystals and approximants in other, mainly ternary systems [6,15]. The importance of the Cd–Yb phase however, goes far beyond the role of simply being a prototype structure; the binary Cd–Yb system offers several advantages over other ternary families of quasicrystals in terms of well-defined chemical order and the availability of several high-quality approximant phases with compositions in the immediate vicinity of the quasicrystals [13,16–18]. Although binary i-QCs only exist in the Yb–Cd and Ca–Cd systems [6,19], the 1/1 approximant structures form with all the Lanthanides in the Cd–RE (RE ¼ Rare Earth) systems and several other binary and ternary systems [16,17,20,21]. Before discussing more recent studies, we will review the early works on the type structures of the 1/1 approximants in the Cd–RE (RE ¼ Rare Earth) family. Before anyone knew that they were quasicrystal approximants, the Cd6RE-phases were classified into one of the following three prototype structures: Cd6Yb[13], Cd6Y [22], and Be17Ru3 [23]. The original prototype structures were reported to be essentially isostructural, differing only in the description of the innermost core unit of the main cluster found in these phases. The structure of Cd6Yb was studied by Palenzona, who reported that the core unit of the cluster was composed of four Cd atoms distributed over the eight vertices of a cube. Larson and Cromer reported that the core unit of Cd6Y was composed of four Cd atoms distributed over the 12 vertices of an icosahedron. Finally the structure of Be17Ru3 was described with an empty cavity at the center of the cluster by Sands et al. Recent structural studies however, reveal an amazing array of structural variation within the group of Cd6RE approximants, clearly revealing the weaknesses of the previously assigned type


79

structures [16,17]. Though most of the Cd6RE phases exhibit disorder of the core unit at ambient conditions, this disorder varies between the different phases. Several of these phases slightly deviate from the Cd6RE composition and in some cases superstructures form at both ambient and low-temperature/high-pressure conditions [16,24–28]. Furthermore, the suspiciously large void in the clusters of the phases described by the Be17Ru3 prototype was recently shown by Corbett and Lin to be occupied by four Zn atoms in the structure of ScZn6 [20]. This casts serious doubt on the very existence of a prototype structure with an empty core.

4.2.2.

The disordered 1/1 approximants

Though the term of approximant is associated with a strict mathematical definition, it is sometimes used loosely to also encompass other structurally related phases [29,30]. However, when we here refer to the cubic 1/1 Cd6RE approximants we have defined the slope of the cut-space in relation to the 6D lattice of the parent i-QC. Most of the Cd6RE approximants crystallize in the space group Im-3 at ambient conditions. This is the space group that we usually expect for a 1/1 approximant of a P-type i-QC. However, among the Cd6RE phases there are also several examples where the space group is different due to superstructure formation [24,25]. The formation of superstructures has largely been attributed to the ordering of the four atoms in the core of the basic cluster that constitutes the structure of the Cd6RE phases. Before discussing the superstructures, it is necessary to understand the basic structures of the Cd6RE phases with disordered core units. The basic structure of the Cd6RE phases can be described as a bcc packing of identical clusters. Each cluster unit is composed of successive polyhedral shells with nearly perfect icosahedral symmetry (Figure 4.1). The four positionally disordered Cd atoms are located within the dodecahedral shell, near the center of the cluster. The electron density distributions corresponding to the innermost core units in the clusters of different Cd6RE and Cd6M (M ¼ Ca, Y) phases and are shown in Figure 4.3. It can be shown that these different electron densities can be reproduced by properly applying different disorder modes to a Cd4 tetrahedron [16], which sometimes is required to be distorted. The ‘type-1 disorder’ is a 90 rotational disorder of the Cd4 tetrahedron along its inherent two-fold axis, as shown in Figure 4.4(a). The ‘type-2 disorder’ is a triple splitting of the tetrahedral vertices, which arises when the three-fold axis of the tetrahedron fails to coincide with that of the cubic unit cell, as shown in Figure 4.4(b). The amount of splitting; or distance between a tetrahedral vertex atom and the nearest three-fold axis, is often variable even within the same structure. When examining electron densities generated from single crystal X-ray data, this is seen as continuous, smeared out densities. At the maximum amount of split, the electron density will concentrate at the vertices of a cube octahedron, Figure 4.4(c). This cube octahedron is often distorted as to approach the shape of an icosahedron (it is known that these two polyhedra can be interconverted by a simple distortion), and in either case it means that the disordered tetrahedron would have to be distorted. In the space group Im-3 it is impossible to have a perfectly ordered tetrahedron centered on the origin of the unit cell; this space group is thus a signature for disorder of the central core unit in the Cd6RE phases. The symmetry is merely a

80


M = Ce

M = Pr

M = Nd

M = Sm

M = Eu

M = Eu

M = Gd

M = Dy

M = Ho

M = Er

M = Tm

M = Lu

M = Yb

M = Ca

1)

2)

M=Y

Figure 4.3. Fobs electron density iso-surfaces at the 8.5 eA˚–3 level, corresponding to the disordered Cd4 tetrahedra in the different 1/1 Cd-M approximants. The refinements were performed on single crystal X-ray diffraction data. Note that the superstructure of Cd25Eu4 contains two symmetry independent Cd4 tetrahedra, labeled (1) and (2).

product of the average structure, and locally (at cluster level) the 2/m3 point symmetry must be broken if indeed only four atoms exist in the core of each cluster.

4.2.3.

The ordered 1/1 approximants

Superstructure formation in a Cd6RE-related phase was first observed in the compound formerly known as Cd6Ce by Armbru¨ster and Lidin in 2000 [24]. Single


81

(a)

(b)

(c)

Figure 4.4. Tetrahedral disorder modes found in the 1/1 Cd-M approximants.

crystal X-ray diffraction data clearly showed that the unit cell was primitive instead of body centered and systematic extinctions indicated the presence of an n-glide. The structure could successfully be refined in the space group Pn-3, and this was the first clear evidence that the previously assigned prototype structure Cd6Y did not describe the structure of this compound correctly. The superstructure was found to be stable at ambient and low temperature (measured down to 150 K), but two reversible phase transitions were noted at high temperatures (868 K and 1003 K). At the time, the existence of the Cd5.7Yb quasicrystals had not yet been reported, and the fact that the Cd6RE phases were quasicrystal approximants was not yet known to the general public. Later that same year, Tsai et al. reported the existence of the binary quasicrystals [6], and the Cd–RE systems gained renewed interest. Two years after the report of Armbru¨ster and Lidin, another superstructure was reported by Tamura et al. [26]. In the paper it was noted that Cd6Yb undergoes a reversible phase transition at low temperature (110 K), followed by the appearance of weak superlattice reflections in the electron diffraction pattern. A similar observation was also reported for Cd6Ca by the same authors [27], and the second superstructure known to be stable at ambient conditions was solved and reported by Go´mez and Lidin [25], this was the phase formerly known as Cd6Eu, also this phase showed a phase transition at high temperature (782 K) and it was later confirmed by electron diffraction that the superlattice reflections disappear at high temperature, and that the pattern can be indexed with a simple bcc lattice [31]. Finally, Watanuki et al. reported five phases in the temperature-pressure regime ranging up to 5.2 GPa and

82


(a) 300

bcc (disordered phase)

Temperature (K)

250

IV′

200

II IV

150

III′ 100

50

I

III

V

[110] Noncubic

cubic type

[110] Non-cubic

0 0

1

2

3

4

5

6

Pressure (GPa) (b)

(c)

Phase II, III

Phase I, IV, V

Figure 4.5. (a), Partial temperature-pressure phase diagram for the Cd6Yb phase. Five main phase regimes were observed in the investigated P-T area (I-V), slight modifications were also detected (III0 , IV0 ). (b) and (c), schematically show how different orderings of tetrahedra (light and dark) could give rise to the propagation of the observed superstructures along the and directions as indicated by arrows and planes.

down to 10 K for Cd6Yb very recently [28]. Figure 4.5 shows a temperature-pressure phase diagram for the Cd6Yb approximant. The single crystal refinements on the Ce and Eu containing superstructures revealed that these phases were stabilized by oriented Cd4 tetrahedra and interstitial Cd atoms located within cube-shaped Cd8 cavities in the structure. In all the Cd6RE phases, each cluster unit is associated with eight Cd8 cavities, which serve as direct links between adjacent clusters along threefold directions, see Figure 4.6. Though such cavities exist in all the Cd6RE phases, they are only found to contain additional Cd atoms for phases with RE atoms equal or larger in size than Pr, as seen in Figure 4.7. As a consequence of having additional Cd atoms in the Cd8 cavities, these phases deviate from the ideal Cd6RE stoichiometries; the refinements on the Ce and Eu containing phases give the stoichiometries Cd37Ce6 and Cd25Eu4. The superstructures in these two phases propagate along the directions, this propagation can be understood by looking at the Cd4 tetrahedra and Cd8 cavities found along the three-fold axes of the cubic unit cells,


83

Figure 4.6. Arrangement of Cd8 cubes around the dodecahedral cage in a typical 1/1 Cd–M approximant.

as seen in Figure 4.8. One can see that vacant Cd8 cavities are always surrounded by Cd4 tetrahedra with vertex atoms pointing towards the cavity. Other Cd8 cavities being surrounded by one or two Cd4 tetrahedra pointing away from them will host an additional Cd atom in the structures of Cd37Ce6 and Cd25Eu4. To date, most of the superstructures of the Cd6RE phases which form at non-ambient conditions remain unresolved. Ishimasa et al. recently studied the superstructure of the related Zn6Sc phase previously reported by Tamura et al., which also forms at low temperature [32,33]. The refinement indicates that the ordering of Zn4 tetrahedra is mainly responsible for the superstructure formation also in this phase. It is believed that the ordering of the tetrahedra plays a central role in the formation of superstructures of the Cd6RE-related phases. The different factors that stabilize the superstructures in all these phases; low temperature, high pressure, and additional atoms in the Cd8 cavities, are merely different ways of increasing the density of the compound. Looking only at the structures of Cd37Ce6 and Cd25Eu4, we can conclude that this density increase causes a distortion of the dodecahedral cage that contains the Cd4 tetrahedron. This distortion brings some of the vertex atoms of the dodecahedron closer to those of the Cd4 tetrahedron, thus enforcing a sterical restriction causing one or several disorder modes of the tetrahedron to disappear and locking it into specific orientations. The new long range order takes the local breaking of the 2/m3 point symmetry to the global scale, bringing on a consequent change of space group. So far, the superstructures that are stable at non-ambient

84


(a)

Atomic Radii (Å) 2.050 Eu

2.000

Ca Yb

1.950 1.900 1.850

Ce

Pr

Nd Sm Gd

1.800

Y Tb

Dy

1.750

Ho

Er

Tm

Lu

1.700

(b) 2)

1)

M = Ce

M = Pr

M = Eu

M = Eu

Figure 4.7. The diagram in (a) shows the atomic radii of different M-metals that form 1/1 Cd–M approximants. For all the large M atoms except Ca and Yb above the dashed line, the corresponding 1/1 Cd–M approximants contain additional atoms within the Cd8 cubes. In (b), electron density iso-surfaces at the 8.5 eA˚–3 level are shown for the Cd8 cubes of these phases. For Cd–M phases containing small M-metals (below the dashed line), the Cd8 cubes are vacant. Note that the superstructure of Cd25Eu4 contains three symmetry independent Cd8 cubes, of these only two are occupied and are labeled (1) and (2).

conditions have all been observed by electron diffraction, synchrotron X-ray diffraction or other techniques which are highly sensitive for detecting phase transitions or superstructural order. Attempts to detect the superstructure of the low-temperature Cd6Yb and Cd6Ca phases by single crystal X-ray diffraction using a standard laboratory X-ray source have failed, though the superstructures of the Cd37Ce6 and Cd25Eu4 phases are easily detectable using the same equipment. This could indicate that possibly the superstructural order of some of the phases that are stable at non-ambient conditions only acts at short or medium range.

4.2.4.

The 2/1 approximants

The first Cd-RE-related 2/1 approximant to be structurally determined was Cd76Ca13 [34]. Though the collected single crystal data was not of optimal quality, the refinement correctly showed the main structural features of the phase, such as the new cluster arrangement and structural building blocks. A refinement of higher quality was made on the Yb13Cd76 2/1 approximant [18], the results revealed several

85


(a)

Repeating sequence in Ce6Cd37

(b)

Repeating sequence in Eu4Cd25

Cd1c

Cd1t

Cd2c

Cd2t

Cd2t

Cd2c

Cd1t

Cd1c

Figure 4.8. Repeating sequences of Cd4 tetrahedra and Cd8 cubes along the three-fold axes of (a) Cd37Ce6 and (b) Cd25Eu4.

new details in the structure, which eventually led to a redefinition of the previously considered Cd-Yb-type polyhedral cluster of the Cd-RE approximants and quasicrystals. As previously mentioned, the structure of the cubic 1/1 Cd-RE approximants can be considered as a bcc packing of clusters. The conventional description of the cluster includes the atoms of all atomic shells up to the level Cd30 icosidodecahedron. In this description, atoms outside this shell are considered as ‘glue atoms’. It is important to understand that in principle any atomic shell and its subshells can be considered as a cluster, and that this definition is highly dependent on one’s purpose and point of view. The following section will be focused on structural understanding, and our definition of cluster will be that which best serves our purpose. With the structures of both 1/1 and 2/1 approximants we can find a different cluster description parting from the idea that the cluster should be the largest common building block in both approximants. A prerequisite for this definition is to have structural knowledge of more than one approximant of the same quasicrystal since the cluster is identified only by comparison of two or more structures. The conventional cluster definition of the Cd-RE phases was postulated before the 2/1 approximants in these systems were known, consequently it was based solely on the structures of the 1/1 approximants and a largest common building block could not be identified. With the structure solution of the 2/1 approximants a new cluster definition can be considered, that allows for the expansion of the conventional cluster description with yet another atomic shell, effectively eliminating the ‘glue atoms’ in the process. The outer atomic shell in the new cluster description of the Cd-RE approximants becomes a rhombic triacontahedron, with Cd atoms on all vertex and edge-center positions [16,18,34]. In the 2/1 approximants, some of the Cd atoms on three-fold vertices of this shell have partial occupancies as a consequence of positional disorder along the three-fold axes.

86


b-bond

(a)

c-bond (c)

(b)

Figure 4.9. The structures of (a), 1/1 Cd-M and (b), 2/1 Cd-M approximants shown as packings of triacontahedral clusters. As seen in (c), these clusters connect by sharing faces along two-fold directions (b-bond) and interpenetration along three-fold directions (c-bond).

In the 1/1 approximants however, these partially occupied positions are entirely missing; or rather these atoms are not positionally disordered and thus always form part of the Cd20 dodecahedron instead. The structures of both 1/1 and 2/1 approximants can now be visualized as simple packings of triacontahedral clusters that interpenetrate along three-fold directions and share rhombic faces along two-fold directions as shown in Figure 4.9(a)–(c). The rhombic triacontahedral shell contains all the previously mentioned atomic shells of the Cd-Yb-type cluster (Figure 4.1(c)), including the disordered Cd4 tetrahedron. However the disorder of the Cd4 tetrahedron is very different in the 2/1 approximants as compared to the 1/1 approximants, and it can no longer be described with the disorder modes shown in Figure 4.4(a)–(c). Electron density maps showing the disorder of the Cd4 tetrahedra in the 2/1 approximants of the Cd–Yb and Cd–Ca systems are shown in Figure 4.10(a) and (b). Though most of the electron density distribution is difficult to interpret, a prominent feature in both 2/1 approximants is the shape of a trigonal bipyramid.

+

(a)

(b)

(c)

Figure 4.10. Electron density iso-surfaces at the 11.3 eA˚–3 level, corresponding to the disordered Cd4 tetrahedra in the (a), 2/1 Cd76Yb13 and (b), 2/1 Cd76Ca13 approximants. In (c), a possible interpretation of the trigonal bipyramidal electron density distribution is shown.


87

This could be interpreted as two possible orientations of an off-center displaced Cd4 tetrahedron, seen in Figure 4.10(c). By off-center displaced we mean that the geometrical center of the Cd4 tetrahedron no longer coincides with that of the cluster, it is somewhat shifted along a threefold-axis. This shift places three atoms of the Cd4 tetrahedron at ideal capping positions for three of the surrounding pentagonal faces of the Cd20 dodecahedron. The fourth Cd atom of the tetrahedron is on the three-fold axis, but severely displaced along it, furthermore there is arbitrariness as to where this atom is pointed. In the binary 2/1 approximants, one can see a correlation between the pointing direction of this atom and the displacement of a vertex atom of the Cd20 dodecahedron into the ideal vertex position of the rhombic triacontahedron, as seen in Figure 4.11. A difference can be noted between the two 2/1 approximants in the Cd–Yb and Cd–Ca systems: The 2/1 approximant in the Cd–Yb system shows a low but significant electron density at the geometrical center of the cluster unit, this is not observed for the Ca-containing 2/1 approximant. Considering interatomic distances, neither Yb nor Cd can occupy this additional site without having unreasonably short distances to neighboring atoms, thus it is likely that when this site is occupied, the Cd4 tetrahedron is entirely absent. A similar observation has been made for the Zn–Yb 1/1 approximant [35], where a singular Yb atom is found at the center of the cluster and no Zn4 tetrahedron is observed. The different packing of clusters in the structures of the 2/1 approximants Cd76Yb13 and Cd76Ca13 requires yet another building unit apart from the triacontahedral cluster to fill space completely; this is the acute rhombohedron shown in Figure 12(a).

Figure 4.11. Schematic representation of the local distortion of the dodecahedral cage in the 2/1 approximants. The shifted tetrahedron pushes a vertex of the dodecahedron into the position marked by a circle, this is the perfect position of a triacontahedral vertex. All the shifted atoms in the figure are colored dark.

88


(a)

(b)

(c)

Figure 4.12. The three fundamental building blocks with unique atomic decorations that can be used to describe the structures of Cd-M approximants and quasicrystals. In (a), the acute rhombohedron, in (b), the obtuse rhombohedron and in (c), the triacontahedron.

This structural unit is completely defined by a double Friauf polyhedron (containing a Yb–Yb or Ca–Ca pair of atoms) and two capping Cd atoms, and is different in decoration from the acute rhombohedron in the Elser-Henley description as shown in Figure 4.2(c). It has been shown that the acute rhombohedron is one of the three fundamental building blocks with unique atomic decorations that are necessary to describe the structure of the i-CdYb5.7 quasicrystal [36]. The fact that this unit is first observed in the 2/1 cubic approximants emphasizes the importance of these phases for understanding the structures of quasicrystals. The other two fundamental building blocks are the rhombic triacontahedral cluster and the obtuse rhombohedron shown in Figure 4.12(b) and (c).

4.2.5.

From approximants to quasicrystals

Though approximant crystals can be of great interest for many reasons, one of those reasons is certainly structural understanding of local atomic order in the parent quasicrystals. The Cd–Yb system is unique in many ways; it contains binary icosahedral quasicrystals that can be grown to large crystals in equilibrium with the melt. Furthermore several approximant crystals have been synthesized and their structures studied by single crystal X-ray diffraction, all these phases existing within a compositional range of less than 2 at.%. Recent achievements in the structural analysis of these phases show that the Cd-RE family of quasicrystals and approximants is an excellent example of how hierarchic order can transcend between a quasicrystal and its approximants. Based on the structure analyses of the 1/1 and 2/1 approximant phases, we can identify three fundamental building units; the triacontahedral cluster, the acute rhombohedron and the obtuse rhombohedron. For a complete analogy with the 3D-generalization of the Penrose tiling, only two rhombohedra are needed. However, if the triacontahedral cluster is subdivided into acute and obtuse rhombohedra, the building units will no longer have unique atomic decorations, which is a serious drawback. Thus by using three building units, the problem of non-unique atomic

89


(a)

(b)

(c)

b-bond

c-bond

Figure 4.13. The figure shows how the fundamental building blocks and their linkage rules are identified from the 1/1 and 2/1 Cd-M approximants. The 1/1 approximant in (a), is composed of triacontahedra, the 2/1 approximant in (b), is composed of triacontahedra and acute rhombohedra. The obtuse rhombohedra are created at the intersection between two c-bonded triacontahedra as in (c), and thus exist in both 1/1 and 2/1 approximants.

decorations is solved. Apart from the building units, the approximants also provide us with a set of linkage rules that can be used as guidelines for constructing higher order approximants and even quasicrystal structures. A scheme showing the building units, linkages and where they are found in the approximant structures is shown in Figure 4.13(a)–(c). Single crystal X-ray diffraction data collected on a single grain of Cd–Yb quasicrystal has recently been successfully used to analyze the atomic structure of the Cd–Yb quasicrystal [36]. From the reconstructed electron densities, it is possible to analyze the cluster packing order in 6D space. This information can be used to generate the distribution of cluster centers in a theoretical approximant of any order. Combined with the knowledge of linkage rules and building blocks obtained from the 1/1 and 2/1 approximants, a complete theoretical model of a 3/2 approximant can be constructed [37]. By knowing the locations of cluster centers, triacontahedral clusters were placed at each of these centers and linked along two- and three-fold directions according to the linkage rules in Figures 4.9(c) and 4.13(c). The atomic decorations inside the triacontahedra are known, and the only thing remaining for the completion of the model is to fill the voids in between these clusters with the proper atomic arrangements. For this, we once again turn to the 2/1 approximants and note that in these structures the voids are filled with double Friauf polyhedra (acute rhombohedra), by applying the same approach to the model of the 3/2 approximant we see that the model is completed by assuming an arrangement of acute rhombohedra which locally is similar to that seen in the hexagonal Laves phase Cd2Yb [13] The schematic construction of the 3/2 approximant model is seen in Figure 14(a)–(c). This model showed the compatibility of the construction rules between approximants and quasicrystal, and furthermore gave rise to ideas of local atomic order in

90


(a)

(b)

(c)

Figure 4.14. Model of a theoretical 3/2 approximant. In (a), the distribution of cluster centers as obtained by high-dimensional projection is seen as light spheres. The dark spheres represent ‘glue atom’ cavities which need to be properly decorated by atoms to obtain a complete structure model. In (b), triacontahedra have been placed at the centers represented by the light spheres in (a), these triacontahedra are linked along two- and three-fold directions as seen in Figures 4.9(c) and 4.13(c). Now the shape of the ‘glue atom’ cavities can be seen in the model. In (c), the arrangement of acute rhombohedra which completely fills the ‘glue atom’ cavities is shown; thus the model is complete.

the quasicrystal that eventually were verified by refinement and resulted in a detailed structure solution for the icosahedral Cd–Yb phase [36].

4.3. 4.3.1.

Stable quasicrystals of the Cd–Yb class Binary stable i-QCs in the Cd–Yb and Cd–Ca systems

In the binary Cd–Yb phase diagram, a compound observed by Palenzona just adjacent to the Cd6Yb phase with the composition Cd5.7Yb [13], was reported as an unknown phase with very large lattice constant. On the other hand, the structure of Cd6Yb determined by single crystal X-ray diffraction is a bcc phase with space group Im-3. The crystal consists of 168 atoms and has a lattice parameter of 1.564 nm. As described previously, the Cd6Yb phase is a bcc packing of icosahedral clusters, which consist of four successive shells as shown in Figure 4.1(c). As shown in Figure 4.2, similar to -Mn12(Al,Si)57 and Mg32(Al,Zn)49, the structure of Cd6Yb can be described by a tiling of two decorated building blocks; a prolate rhombohedron and a rhombic dodecahedron with a long body diagonal identical to the lattice parameter. The rhombic dodecahedron is assembled by two acute and two oblate rhombohedra with identicalpedge length aR. Geometrically, the aR is ffiffiffi related to the equation of a ¼ aR(4+8/ 5)1/2 where a is the cell parameter of the 1/1 approximant and two rhombohedra in the three-dimensional generalization of the Penrose tiling and referred to as the icosahedral lattice constant of the corresponding i-QC. Thus, if the Cd6Yb is an approximants to the i-QC they should share the same aR. The aR derived from Cd6Yb and that estimated for the i-Cd5.7Yb phase by X-ray diffraction experiments agree very well, as shown in Figure 4.15. This figure shows the powder X-ray diffraction patterns for the 1/1 Cd6Yb and 2/1 Cd76Yb13 approximants, as well as the i-Cd5.7Yb phase. The 2/1 Cd76Yb13 approximant was identified by a single crystal X-ray diffraction experiments [34] after identification of the i-QC. Figure 4.16 shows X-ray Laue diffraction patterns

91

(333101) (422211)

(332002)

(222100) (311111)

(221001) (860),(1000)

(853),(941),(770)

Cd5.7Yb

Cd76Yb13

(532),(611)

(850),(843),(762) (530)

(600)

Intensity (arb.units)

(211111)

4.3. Stable quasicrystals of the Cd–Yb class

Cd6Yb 30

40

50

60

70

80

2θ (deg.) Figure 4.15. Powder X-ray diffraction patterns for the i-Cd5.7Yb, the 2/1 approximant of Cd76Yb13 and the 1/1 approximant of Cd6Yb.

(100)

(350) 6

6

(a)

4

4

2

2

0

0

–2

–2

–4

–4

–6

–6 –6 –4 –2

6

(b)

0

2

4

6

–6 –4

6

(c)

4

4

2

2

0

0

–2

–2

–4

–4

–6

–6 –6 –4 –2

0

2

4

6

–2

0

2

4

6

–6 –4 –2

0

2

4

6

(d)

Figure 4.16. Calculated diffraction patterns with incidences along [530] and [100] axes (a,b) for the Cd6Yb 1/1 approximant and transmitted X-ray diffraction patterns taken with incidences along five-fold (c) and two-fold (d) axes obtained from a single grained i-Cd5.7Yb.

92


2/1 Approximant

Temperature (K)

1/1 Approximant 900

Cd76Yb13

Quasicrystal Cd5.7Yb1

Cd6Yb1

800

700 0

5

10

15

20

at. % Yb Figure 4.17. A temperature-composition phase diagram containing the i-QC, the 2/1 and the 1/1 approximants for the Cd-Yb system. The Cd–Ca also has the same diagram.

taken along five- and two-fold axes of the Cd5.7Yb phase and calculated patterns along the [530] axis (pseudo five-fold) and [100] for Cd6Yb. The finding was the first evidence of a thermodynamically stable i-QC in a binary system. Shortly thereafter, it was noted that a similar phase diagram with an unknown phase existed for the Cd–Ca system [38]. The unknown phase, Cd17Ca3 (Cd5.7Ca), whose composition is very close to a Cd6Ca compound isostructural to Cd6Yb, was identified as an iQC [39]. Therefore, the correct phase diagram for the Cd–Yb (or Cd-Ca) system should be the one shown in Figure 4.17. Very recently, Kramer et al., studied the structure and thermal properties of solid and liquid Cd84Yb16 from 300 to 1050 K by means of high-energy synchrotron X-rays [40]. They verified that the i-Cd84Yb16 is stable up to its melting temperature of 914 K and melts congruently over temperature range of less than 1K. It should be stressed that the Cd17Ca3 phase contains no transition metals or rare earth metals, being the second example of a stable i-QC following the AlLiCu phase [41]. The identification of the first stable binary i-QC destroyed two notions that had arisen over the years; the idea that the i-QCs are found only among the metals on the right hand side of the Periodic Table, and that the i-QCs require at least three different elements. The binary phase offered a clear advantage in chemical simplicity for structural analysis.

4.3.2.

Cd–Mg–RE system

Stable i-QCs have been found in the Zn–Mg–RE systems only for REs with small atomic radii, i.e., Y, Gd, Er, Tb, Ho, and Dy [42]. An idea to search for new quasicrystal was based on the replacement of Zn by Cd in the Zn–Mg–RE system. Eventually, stable i-QCs were identified in seven Cd–Mg–RE alloys for the RE elements Y, Ho, Dy, Lu, Yb, Tm, and Dy [43]. Again, this is a very strong indication that the stability of the i-QCs is governed by the chemistry of the

93

4.3. Stable quasicrystals of the Cd–Yb class 2.1

(Atomic radii × 10)/nm

2.05

Eu

2 Ca

Cd84RE1

1.95

Yb 6

1.9

Cd65Mg20RE15

1.85 Nd Ce

1.75

Pr

Gd Sm Y

1.8 Tm

Er

Ho Dy

Tb

Zn60Mg30RE1 0

1.7

Figure 4.18. Formation of stable quasicrystals versus atomic radii of RE elements for the Zn–Mg–RE (RE: rare earth metals) and the Cd–Mg–RE systems.

constituent elements. Figure 4.18 shows the phase formation of stable i-QCs in the Zn–Mg–RE and the Cd–Mg–RE systems for REs with various atomic radii. In view of the combination of constituent elements, it seems that the stable i-QCs in these two systems belong to the same class. However, they are different according to the following criteria: First, in contrast to i-QCs in the Zn–Mg–RE systems which have face centered cubic lattices in six dimensions [42]; the i-QCs in the Cd–Mg–RE systems possess primitive lattices in six dimensions. Second, the compositions of the i-QCs in the Cd–Mg–RE systems are close to Cd65Mg20RE15 whereas those in the Zn-Mg-RE systems are close to Zn60Mg30RE10. Third, the 1/1 Cd6RE approximants form for most of the REs in the binary Cd–RE systems whereas no 1/1 approximant has been observed in the Zn–RE binary systems, with the exception of Zn–Yb [35]. Fourth, the arrangements of RE atoms in the i-QCs of the two systems are different; the RE atoms form dodecahedra with radii of about 0.75 nm in the Zn–Mg–RE [44] systems whereas they create icosahedra with radii of 0.56 nm in the Cd–Mg–RE systems. Clearly, the different arrangements of REs atoms give rise to the difference in RE concentration between the i-QCs in these two systems. The addition of Mg undoubtedly stabilizes the i-QCs in both systems, but the role of Mg in the stabilization mechanisms is not the same. On the other hand, the i-QCs show very high solubility of Mg in the Cd–Yb and the Cd–Ca systems; the compositional range of the i-QCs spans up to Cd25Mg60Yb15 and Cd30Mg55Ca15, respectively[45]. Replacement of Cd with Mg thus expands the compositional range of the i-QCs in the ternary Cd–Mg–Yb and Cd–Mg–Ca systems. Figure 4.19 shows a part of the powder X-ray diffraction patterns for the Mg60Cd24Yb16 and Mg60Cd26Yb14 alloys in comparison with those for the i-QC and the Cd6Yb 1/1 approximant. Note that the diffraction peaks significantly shift to the lower angle region, indicating that the replacement by Mg expands the lattice spacing of the i-QC. As shown in Figure 4.20, the replacement of Cd by Mg leads to a linear increase in lattice parameter with increasing Mg content. The atomic

(600)

(221001)

Mg60Cd24Yb16

(221001)

(211111) (211111)

32

33

Cd86Yb14

Cd84Yb16

(221001)

(211111)

Intensity (arb. unit)

(532) (611)


(530)

94

34

Mg60Cd26Yb14 36

35

37

38

2θ (deg.)

Figure 4.19. Powder X-ray diffraction patterns revealing the main diffraction peaks for Cd–Yb alloys and Mg–Cd–Yb alloys. Note that the addition of Mg promotes the formation of the i-QC rather than the 1/1 approximant.

0.585

Cd-Mg-Yb Cd-Mg-Ca

aR (nm)

0.58

0.575

0.57

0.565

0

10

20

30

40

50

60

70

Mg (at.%) Figure 4.20. Relationship the quasilattice constant and the Mg contents for the i-QC in the Cd–Mg–Yb and the Cd–Mg–Ca systems.

radii of Cd, Mg, and Yb are 0.158, 0.16, and 0.192 nm, respectively. Taking the atomic sizes of the constituent elements into account, it is reasonable to speculate that the linear increase of lattice parameter suggests that the structure remains the same throughout the Mg/Cd replacement and Mg is likely to substitute the atomic


95

site of Cd in the i-QC structure. One of the main reasons for the high solubility of Mg, as will be described in the next section, is the formation of binary stable i-QC in the Cd–Yb and Cd–Ca systems. Furthermore, Mg has almost the same atomic size and valence as Cd and replacement of Cd with Mg would thus not cause much strain or change in electron concentration. This indicates that both the size and valence are important factors in the formation of the stable i-QCs.

4.3.3.

i-QC and approximants in the In–Ag–RE systems

Figure 4.21 shows a part of the Periodic Table containing In, Cd, and Ag together with the corresponding atomic radii. Generally, stable i-QCs are described as Hume-Rothery compounds, where the structures are stabilized by the valence electron concentration, i.e., e/a ratio. This rule has been found to work well in most of the i-QCs identified so far. Replacement of Cd by half In and half Ag for Cd84Yb16 and Cd84Ca16 led to the formation of the stable In42Ag42Yb16 and In42Ag42Ca16 i-Qcs [46]. The Yb atoms in the approximants and i-QCs has been found to be divalent in all the Cd–Yb binary compounds and therefore the e/a ratio for the Cd84Yb16 and Cd84Ca16 stable i-QCs is 2.0. As shown in Figure 4.21, In located to the right of Cd is trivalent and Ag located to the left of Cd is monovalent, thus the e/a ratios of the two In–Ag stable i-QCs are kept at 2.0. This is another indication confirming that the stable i-QCs are stabilized with the criterion of the e/a ratio. Nevertheless, in this system the e/a ratio and the relative atomic size equally contribute to the stability of the i-QCs. For example, Cd, Ca, and Yb are all divalent in the quasicrystals and their atomic radii are 0.157, 0.198, and 0.194 nm, respectively. The atomic size ratios are close to rCd/rCa,Yb 0.8. This means that the valence electron concentration already determines the atomic size ratio. Similarly, the replacement of Cd with Ag and In does not cause any change in atomic size ratio, since the atomic radii for Ag and In are 0.145 and 0.166, respectively, and give an average atomic radius of 0.156 nm close to that of Cd. Interestingly, together with the i-QCs, a 2/1 and a 1/1 approximant also form in the In–Ag–Yb and the In–Ag–Ca systems [47]. Figure 4.22 displays selected area

Cd84Yb16 Cd84Ca16 e/a = 2.0 In42Ag42Yb16 In42Ag42Ca16 Cu Zn Ga Ag Cd In Au Hg Tl Valence 1 2 3 Figure 4.21. Formation of stable quasicrystal by replacement of Cd with In and Ag for Cd–Yb and Cd–Ca systems.

96


(a) [5-fold]

(b) [2-fold]

(c) [3-fold]

5-fold

(d) [110]

(g) [110]

(e) [100]

(h) [100]

(f) [111]

(i) [111]

Figure 4.22. Selected area diffraction patterns taken with incidences along the main axes for the i-QC (a, b, c), the 2/1 (d, e, f) and the 1/1 (g, h, i) approximants.

electron diffraction patterns taken along the main axes for the i-QC, the 2/1 and the 1/ 1 approximant in the In-Ag-Yb system [48]. Figure 4.23 shows the isothermal (a) and temperature-composition (b) phase diagrams for the In–Ag–Yb system. It seems that the approximants and i-QC have more flexibility in composition for the In–Ag–Yb system as compared with the Cd–Yb system. Compositional analyses by EPMA for the three phases revealed the composition is In46Ag40Yb14 for the 1/1 approximant, In44Ag41Yb15 for the 2/1 approximant, and In42Ag42Yb16 for the i-QC. It is noted that the concentration ratios of In and Ag are slightly different for the three phases indicating that their e/a ratios are different in the In–Ag–Yb system. The phase diagram shown in Figure 4.23(b) could be regarded as a pseudo-binary system with respect to the Cd–Yb system. Like the compounds in the Cd–Yb system, the In–Ag–Yb i-QC melts congruently and the approximants form by peritectic reactions. Since the i-QC is a congruent compound, a single grain of 10 mm in diameter is easily grown by using the Bridgman method. Figure 4.24 shows a single grain of

97


(a)

(b) 10

1/1 AP 2/1 AP i-QC

900

60

Liquid

at. %

875

Temperature/K

20 50

Yb

In 30

i-QC Ag42In42Yb16

850

825 2/1 AP Ag41In44Yb15

40 800

1/1 AP Ag40In46Yb14

40 30 30

40

50

775 5

60

10

15

20

Yb/at. %

Ag

Figure 4.23. Partial isothermal phase diagram at 823K (a) and the temperature-composition section containing the i-QC, the 2/1 and the 1/1 approximants (b) for the Ag–In–Yb system. Note that the Figure 4.23(b) is almost identical to that of Figure 4.17.

(b)

(a)

Crystal growth direction

10 mm

Figure 4.24. A single grain of the i-QC of the Ag–In–Yb system grown by the Bridgman method (a) and its back Laue diffraction pattern taken with incidence along a five-fold axis (b).

In–Ag–Yb i-QC and a Laue back diffraction pattern taken from a single grain. So far, the same size of single crystal i-QCs with P-type structure have only been grown in the Al–Li–Cu and Cd–Yb systems, but they quickly deteriorate in air [49]. Therefore, studies on surface structures and bulk properties were only limited to the F-type structures of Al-based alloys. The i-QC in the In–Ag–Yb system is the first stable P-type single quasicrystal grown to this scale and it is stable in air. Therefore it is expected to enable the study of surface structure of a non-Al based quasicrystal for the first time. The partial phase diagram of the In–Ag–Ca system is almost the same as that of the In–Ag–Yb system. Similarly, the large single quasicrystals in the In–Ag–Ca system were also grown by slow cooling as well as by the Bridgman method.

98


Upon replacement of Cd with In and Ag in the Cd-RE systems, a similar tendency of phase formation has been confirmed. That is, stable i-QCs form only for alloys containing Yb and Ca, and a large number of approximants were newly identified in In–Ag–RE alloys [50]. Single crystal X-ray diffraction studies have verified that the structures of the 1/1 and the 2/1 approximants of the In–Ag–Yb system are isostructural to those in the Cd–Yb system [51]. In the 1/1 approximant of the In–Ag–RE alloys, tetrahedra at the center of the icosahedral clusters were also commonly observed. Apart from the phases of the i-QC and approximants, a monoclinic crystalline phase related to the quasicrystal in structure was also observed at the composition Ag40.6In36.7Yb22.7.

4.4.

Hume-Rothery conditions for the stable quasicrystals

The Hume-Rothery rules [52] for stable i-QCs have been intensively discussed in the Al–Mn–Si class. Since a large number of stable i-QCs and approximants have been observed in the Cd–RE systems, it is of benefit to discuss the stability in terms of Hume-Rothery rules also for these phases. Here, we consider two Hume-Rothery conditions, i.e., the valence electron concentration and the atomic size factor for comparison between the i-QCs and approximants of the Cd–Yb class with those of the Al–Mn–Si and Zn–Mg–Al classes.

4.4.1.

Valence electron concentration

Apart from the Al–Mn–Si class in which the stable i-QCs reveal a strict e/a of 1.75, those in the Zn–Mg–Al and the Cd–RE classes show a more flexible e/a around 2.0 2.15. Since all the stable i-QCs have the corresponding value of e/a, it is clear that the e/a is the most prominent factor to determine the stability of quasicrystalline structures. The e/a criterion and knowledge of the Cd6RE structures have been combined to discover new quasicrystals. Consequently, a replacement of 5 at.% of Zn by Mg in Zn6Sc led to the formation of the stable Zn80Mg5Sc15 quasicrystal, where the Zn6Sc phase is isostructural to the Cd6Yb approximant [53]. Further replacement of Zn or Mg by other elements has resulted in further discoveries of a large number of quasicrystals. This will further be described by Ishimasa in this book. The formation of a Hume-Rothery phase with a definite value of e/a was explained in terms of band theory by Jones, i.e., the interaction of the Fermi sphere and the Brillouin zone. Jones describes a density of states for the valence electrons as a function of their energy, similar to that of metals [54]. This is a critical assumption. For monovalent metals like Cu, Ag, and Au, the first Brilliouin zone has only half of its volume occupied by electrons. Addition of further electrons by alloy formation can proceed to fill the zones. When the Brillouin zone boundary touches the Fermi sphere, the structure corresponding to the zone will be stabilized. As a result, a pseudo-gap of density of states around the Fermi level will arise. This is the HumeRothery condition i.e., KP ¼ 2kF where KP is the diameter of the Brillouin zone and kF is the radius of the Fermi sphere. Although a strict definition of a Brillouin zone is

99

4.4. Hume-Rothery conditions for the stable quasicrystals 3.2

Al-TM group

Kp (A–1)

3

2.8

2.6 2k F

=K

Zn-based

p

2.4

Cd-based 2.4

2.6

2.8

3

3.2

2kF (A–1) Figure 4.25. Relationship of KP and 2kF for stable quasicrystal forming alloys. The Zn–Mg–RE and Cd-based alloys are deviated from the line of KP ¼ 2kF.

not available for a quasiperiodic structure, a few strong reflections ( and ) appear in powder X-ray patterns and electron diffraction patterns as shown in Figure 4.16 forming a polyhedral zone (decagon) which could be regarded as a pseudo-Brillouin zone. Thus, a KP value can be estimated from the diffraction pattern and kF can be deduced from the free electron theory, kF ¼ (32(N/V))1/3, where N/V is the number of electrons per unit volume which is estimated from the e/a and atomic volume obtained by measurements of density. Figure 4.25 shows the relationship of KP and 2kF for three classes of stable quasicrystals. The condition is effective for the Al–Mn–Si class but not for the Zn–Mg–Al and Cd–Yb classes. Furthermore, the relationship between KP and 2kF is estimated on the basis of the single crystal data, e.g., atomic density and peak positions for the Cd5.7Yb i-QC and the series of 1/1 and 2/1 approximants in the Cd-RE and Cd-Ca binary systems are given in Figure 4.26. Again, the relationship deviates from the line KP ¼ 2kF. However, a good linear relationship of KP and 2kF is observed for most Cd6RE approximants. Note that the points that deviate from the straight line are those of the 1/1 approximants having superstructures or containing larger RE atoms such as Ca, Yb, and Eu. The reason for the discrepancy among the classes is not clear but the use of the apparent negative valence for TM elements is probably a key factor. Regarding this, Friedel suggested the following [55]: In normal metals, the stability is associated with the Fermi sphere just touching a set of Brillouin zones limits with strong intensity. While for Al–TM alloys with a sufficient concentration of transition metal solute elements, the maximum stability is obtained when the Brillouin zone is just inscribed in the Fermi sphere. This would explain the apparent negative valence of these elements when the usual Hume-Rothery-Jones mechanism is applied. Recently, the electronic structure of the Cd6Yb approximant was studied

100

Chapter 4.

Quasicrystals and Approximants in Cd-M Systems and Related Alloys 2.5

Kp (A–1)

2.45

Cd-Yb(Ca)

2.4

Cd37Ce6 2.35

2.3 2.7

Cd25Eu4

2.75

2.8

2.85

2.9

2kF (A–1) Figure 4.26. Relationship of KP and 2kF for a series of Cd6M 1/1 approximants. The KP was estimated from the main diffraction peaks of [530], [532], and [611]. The 2kF was estimated from the atomic density of the structural model. The structural model was obtained by single-grained X-ray diffraction method in Refs. [29] and [30].

where an sp-d hybridization was reported [56]. The hybridization is an implication that the apparent valence is possibly smaller than 2.0 in the Cd–Yb class. However, it is still not clear how to treat the approximants within this criterion in the Zn–Mg–Al and the Al–Mn–Si classes. This is due to the fact that there are no appropriate corresponding approximants with similar composition to the stable i-QCs. For the Cd–Yb class, all Cd–RE binary systems form Cd6RE approximants and most of them form stable i-QCs when 20 at.% of Cd is replaced by Mg. For example, Cd65Mg20Y15 i-QC and Cd6Y both have an e/a of 2.15. Furthermore, e/a is 2.0 for both the Cd5.7Yb i-QC and the Cd6Yb approximant. Therefore it is reasonable to treat the approximants and the i-QCs as the same group within the e/a criterion.

4.4.2.

Atomic size factor

The atomic size factor of an alloy containing A and B elements is defined as = ( rA rB)/rA, where rA and rB are the atomic radii of atoms A and B, respectively. For the Al–Mn–Si class, i-QCs are formed at a very narrow compositional range with e/a limited between 1.7 and 1.8. For example, the stoichiometry of the i-QC in the Al–Cu–Fe system is around Al63Cu25Fe12, but alloys made at the composition Al62Cu25Fe13 form both i-QC and a b-phase, and alloys made at the composition Al65Cu24Fe11 form i-QC and w-phase [57]. This indicates that the condition of e/a is

4.4. Hume-Rothery conditions for the stable quasicrystals

101

absolutely strict. In other words, e/a is absolutely dominant for the formation of the i-QC in this class. For the Zn–Mg–Al class the stable i-QC shows a solubility range. The replacement between Mg and Zn reaches 5 at.% in this i-QC. The replacement of Zn with Mg induces an expansion in the lattice as observed in the X-ray diffraction patterns, confirming the dissolution of Mg into the i-QC structure. The value of for Zn/Mg is about 15% and the value corresponds to the limits of the solubility. The importance of the atomic size of the RE elements for the phase formation in this class is brought out from the observation that the stable i-QCs are formed in alloys containing RE elements with atomic radii in the range 0.173–0.179 nm, as shown in Figure 4.18. For the Cd–Yb class, the i-QC has a high solubility of Mg. It has been reported that the formation range of the i-QC expands towards the Mg-rich side as shown in Figure 4.19 [45]. The i-QC persists even when Cd is replaced with Mg by 60 at.%. The for Cd/Mg is 2 at.% and this allows for a large replacement of Cd by Mg. Such a high solubility rarely occurs in an intermetallic compound and is observed in a stable i-QC for the first time.

4.4.3.

Comparison for the three classes

The Hume-Rothery rules have been tested in three classes of quasicrystals. The stability and range of solubility of the quasicrystalline phases could be figured out within the frame work of the Hume-Rothery rules. However, the criteria are slightly different among the three groups e.g., the range of solubility of a stable i-QC is strongly dependent on the alloy system. The first priority of the criterion for the formation of the quasicrystalline phase is e/a. This is absolutely rigid and even stricter than the electron phases that were studied by Hume-Rothery and coworkers, thus it turns out to be a characteristic of the stable i-QCs. The fact that the formation of stable i-QCs from Cd–Yb and Cd–Ca alloys respectively extends to In–Ag–Yb and In–Ag–Ca ternary alloys, is solid evidence of the importance of the e/a. Recently, a comparative study on atomic structure and electronic structure for a number of approximants indicates that phase stability and the high resistivity in Al-TM approximants is due to sp-d hybridization at the atomic cluster level. This in turn is responsible for the shortrange order and for the Fermi-surface-Brillouin zone interaction, which further accounts for the long-range order [58]. This rule seems to work well for the Al–Mn–Si class. On the other hand, the Fermi-surface-Brillouin zone interaction is the only Hume-Rothery mechanism responsible for the stability of the approximants in the Al–Mg–Zn system. However, the new stable quasicrystals containing RE elements may not fully fit the case of Al–Mg–Zn, since the hybridization due to the involvement of the RE element cannot be ignored. This is the same for the Cd–Yb class. The small solubility of the stable i-QCs of the Al–Mn–Si class is due to a significant difference in valence among the constituent elements. For example, replacement of an Al atom by one Fe atom will induce a decrease of 5.66

102

Chapter 4.

Quasicrystals and Approximants in Cd-M Systems and Related Alloys

electrons. A slight change in composition ends up in a large deviation of e/a. This explains the small solubility and the rigid composition of the quasicrystals in the Al–Mn–Si class. Since both Zn and Mg are divalent, an exchange between Zn and Mg does not change the e/a for the Zn–Mg–Al class, and thus a certain amount of replacement is allowed. However, when the e/a is favorable the atomic size factor will become dominant and the large forces the maximum solubility of Mg to be limited to less than 5 at.%. A detailed study on phase formation indicated that 2 3 at.% replacement between Mg/Y is acceptable and the composition constraint is lower than in the Al–Mn–Si class. This can be attributed to the slight change in the e/a due to this replacement. For the Cd–Yb class as seen in Figure 4.19, the range of solubility (Cd/Mg replacement) in the i-QC expands to 60 at.%. This can be simply explained by the similarity in both valence and atomic size factor for Cd and Mg. The replacement of Cd/Mg first preserves the e/a criterion and only leads to a small of 2%. The product of and solubility is a measure of the distortion due to atom replacement. For smaller , a larger solubility is expected. A small amount of replacement of Yb is even allowed in this system.

4.4.4.

Phase selection between the i-QC and Approximants

Basically, the Cd6RE approximants and the Cd–RE i-QCs belong to the same class within the e/a criterion. However, the atomic size of RE elements seems to be important in phase selection between the approximants and the i-QC. Figure 4.27 summarizes the relationship between atomic radii of RE elements and

16

Cd25Eu4 Cell parameter or Cd6 M (nm)

15.9 15.8

Cd37Ce6 Cd6Ca

15.7

Cd6Yb

15.6 15.5 15.4 15.3 15.2 1.6

1.7

1.8

1.9

2

2.1

Atomic radius of M (nm) Figure 4.27. Relationship of cell lattice parameter for the Cd6M and atomic radii of the M.

4.4. Hume-Rothery conditions for the stable quasicrystals

103

the lattice parameters for all Cd6RE approximants. The stable i-QCs form in the alloys containing RE elements with atomic radii in two discontinuous regions; rRE < 0.182 nm and rRE 0.195. It can also be said that the size of the icosahedral shell created by twelve RE atoms determines the phase formation. For the RE at the region of rRE < 0.182 nm, the i-QCs only form in the ternary Cd–Mg–RE systems where the addition of Mg is necessary. Since the role of Mg in the formation of the i-QCs is not clear, we will not discuss this in detail. Here, we focus on the region of the Cd–(Yb, Ca, Eu) systems where the i-QCs and 2/1 approximants are obtained. In the binary Cd–Yb(or Cd–Ca) system, the Cd6Yb (or Cd6Ca) approximant is adjacent to the i-QC. Since the e/a criterion is analogous for the approximant and for the i-QC, both phases should be treated as the same class of Hume-Rothery compounds. In the mean time, a question of ‘what is the factor that determines the structure selection between the approximants and the i-QC?’ comes up. In a systematic investigation, it has been verified that higher order approximants (2/1) and the i-QCs are only obtained in the alloys containing Ca and REs with larger atomic radii such as Yb and Eu in several alloy systems, e.g., Cd–M, In–Ag–M, Au–In–M and Au–Sn–M systems where M ¼ REs or Ca. The In–Ag–M, Au–In–M and Au–Sn–M can be treated as pseudo-binary systems with respect to the Cd–M system. The formation tendency in these systems suggests that the larger REs or Ca are favorites for the formation of 2/1 approximants and i-QCs. Apparently, the ratios of atomic size between the M and the rest elements are important. In order to get insights into the role of this factor, we introduce a parameter, namely effective atomic size ratio, Rr,e, which is defined to be the atomic size ratio by taking concentration into account. For example, the Rr,e for the Cd86Yb14 is (rYb Cyb) /(rCd Ccd) ¼ (1.94 0.14)/(1.57 0.86) ¼ 0.201 and for the Cd84Yb16 i-Qc is 0.235, where the r and C is the atomic radius and concentration of each constituent element, respectively. In between the 1/1 approximant and the i-QC the Rr,e for the 2/1 approximant is 0.211. This parameter is more valid to distinguish the three phases in the In–Ag–Yb system, where the Rr,e for the i-QC, the 2/1 and 1/1 approximants is 0.240, 0.220, and 0.200, respectively. Confirmation of the universal validity of the Rr,e for the three phases in the Cd-M family is highly desired.

4.4.5.

Summary

It is clear that the three classes of i-QCs are stabilized by similar factors but with markedly different degrees of constrains. This can be interpreted by the criteria of structural stability and the range of solubility for Cu, Ag and Au binary alloys, as discussed by Hume-Rothery and coworkers, and for binary Al alloys as discussed by Raynor and coworkers. i-QCs are the group of solids that most strictly follow the Hume-Rothery rules among all intermetallic compounds. Although the estimation of the size of the Fermi sphere is speculative, a comparison between various groups would be helpful to get an insight into the stabilization mechanisms for the quasiperiodic structures. However, to reach a definite conclusion, further investigation on physical properties is needed.

104

4.5.

Chapter 4.


Concluding remarks

Twenty-five years after its discovery, the quasicrystalline phase has been identified to be an equilibrium state in a large number of alloys. The discoveries of the i-QCs and their series of approximants in the Cd–M systems have greatly contributed to the progress in understanding quasicrystals. They offered an opportunity to obtain the first complete atomic structure solution for the Cd–Yb i-QC[36], in which the structural information of the approximants played a key role. The knowledge of atomic structure can facilitate the manipulation of physical properties and designing of new alloys. The discovery of the alloys in the Cd–M systems has inspired the identification of more i-QCs and approximants in several alloys systems, e.g., Cu-, Ag-, Zn-, and Au-based alloys containing REs and transition metals. These alloys will be helpful in order to understand the effects of quasiperiodicity, complexity in physical properties and stabilization mechanisms. They also give rise to several interesting topics for further study from points of view of solid state chemistry, solid state physics and metallurgy.

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[21] J.F. Ruan, K.H. Kuo, J.Q. Guo, A.P. Tsai, J. Alloys Compd. 370 (2004) L23. [22] A.C. Larson, D.T. Cromer, Acta Cryst. B 27 (1971) 1875. [23] D.E. Sands, Q.C. Johnson, O.H. Krikorian, K.L. Kromholtz, Acta Cryst. 15 (1962) 1191. [24] M. Armbru¨ster, S. Lidin, J. Alloys Compd. 307 (2000) 141. [25] C.P. Go´mez, S. Lidin, Chem. Eur. J. 10 (2004) 3279. [26] R. Tamura, Y. Murao, S. Takeuchi, M. Ichihara, M. Isobe, Y. Ueda, Jpn. J. Appl. Phys. 41 (2002) L524. [27] R. Tamura, K. Edagawa, Y. Murao, S. Takeuchi, K. Suzuki, M. Ichihara, M. Isobe, Y. Ueda, J. Non-Crys. Solids 334&335 (2004) 173. [28] T. Watanuki, A. Machida, T. Ikeda, K. Aoki, H. Kaneko, T. Shobu, T.J. Sato, A.P. Tsai, Phys. Rev. Lett. 96 (2006) 105702. [29] C.P. Go´mez, S. Lidin, Solid State Sci. 4 (2002) 901. [30] S. Piao, C.P. Go´mez, S. Lidin, Z. Kristallogr. 221 (2006) 391. [31] R. Tamura, K. Nishimoto et al., in JPS meeting, Kashiwa, Tokyo University, of Science, Japan, 2006. [32] T. Ishimasa, Y. Kasano, A. Tachibana, S. Kashimoto, K. Osaka, in Aperiodic 2006, Philos. Mag., Zao, 2006, p. 1. [33] R. Tamura, K. Nishimoto, S. Takeuchi, K. Edagawa, M. Isobe, Y. Ueda, Phys. Rev. B71 (2005) 092203. [34] C.P. Go´mez, S. Lidin, Angew. Chem., Int. Ed. Engl. 40 (2001) 4037. [35] H. Takakura, Unpublished results, Division of Applied Physics, Graduate School of Engineering, Hokkaido University. [36] H. Takakura, C.P. Go´mez, A. Yamamoto, M.d. Boissieu, A.P. Tsai, Nat. Mater. 6 (2007) 58. [37] C.P. Go´mez, H. Takakura, A. Yamamoto, M.d. Boissieu, A.P. Tsai, in ICQ9, Ames, Iowa State University, USA, 2005. [38] G. Bruzzone. Gazz.Chim.Italy 102 (1972) 234. [39] J.Q. Guo, E. Abe and A.P. Tsai, Phys. Rev. B62 (2000) R14605. [40] M.J. Kramer, T.A. Lograsso and D.J. Sordelet, Philo. Mag. Lett. 85 (2005) 151. [41] M.D. Ball and D.J. Lloyd, Script.Met. 19 (1985) 1065. [42] A.P. Tsai, A. Niikura, A. Inoue, T. Masumoto, Y. Nishita, K. Tsuda, M. Tanaka, Phil. Mag. Lett. 70 (1994) 169. [43] J.Q. Guo, E. Abe, A.P. Tsai, Jpn.J.Appl.Phys. 39 (2000) L770. [44] H. Takakura, M. Shiono, T.J. Sato, A. Yamamoto, A.P. Tsai, Phys. Rev. Lett. 86 (2001) 236. [45] J.Q. Guo, E. Abe and A.P. Tsai, Phil. Mag. Lett. 82 (2002) 27. [46] J.Q. Guo and A.P. Tsai, Philo.Mag.Lett. 82 (2002) 349. [47] B.B. Deng and K.H. Kuo, J. Alloys and Compounds 366 (2004) L1. [48] S. Ohhashi, J. Hasegawa, S. Takeuchi and A.P. Tsai, Aperiodic 2006, Zao. Philo. Mag.B 87 (2007) 3089. [49] G.Krauss, S. Deloudi, A. Steiner, W. Steurer, A.R. Ross and T.A. Grasso, Mat.Res.Soc. Symp.Proc. 805 (2004) p. 41. [50] J.F. Ruan, K.H. Kuo, J.Q. Guo and A.P. Tsai, J. Alloys and Compounds 370 (2004) L23. [51] C.P. Go´mez et. al., in preparation.

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W. Hume-Rothery, J. Inst. Metals 35 (1926) 295. H. Kaneko, Y. Arichika, and T. Ishimasa, Philo. Mag. Lett. 81 (2001) 777. H. Jones, Proc. Phys. Soc. 49 (1937) 250. J. Friedel, Helv. Phys. Acta 61 (1988) 538. Y. Ishii and T. Fujiwara, Phys. Rev. Lett. 87 (2001) 206408. P. Bancel, Quasicrystals, The State of the Art, ed D.P. Divincenzo and P.J. Steinhardt, (World Scientific, Singapore 1991) p.17. [58] U. Mizutani, T. Takeuchi and H. Sato, J. Phys.-Condens Mat. 14 (2002) R767.

Chapter 5

Phason Modes in Aperiodic Crystals M. de Boissieu1, R. Currat2, and S. Francoual1,2 1 Laboratoire de Thermodynamique et Physico Chimie Me´tallurgiques, UMR CNRS 5614, INPG, ENSEEG, BP 75 38402 St Martin d’He`res Cedex, France 2 Institut Laue-Langevin, BP 156, 38042, Grenoble Cedex 9, France

5.1. 5.1.1.

Introduction: hydrodynamic modes and quasiperiodic structures Outline

This review deals with the subject of phasons, a concept which comes up very frequently in the field of quasiperiodic structures, with different meanings in different contexts. In displacively modulated crystals, phasons are thermally excited fluctuations, which may be constructed from a coherent superposition of phonons of the unmodulated parent structure. In quasicrystals on the other hand, phasons are viewed more as quenched or slowly relaxing structural defects. The link between the different aspects of the concept originates in the hyperspace description of quasiperiodic structures. The hyperspace embedding technique identifies phasons as the soft, hydrodynamic degrees of freedom specific of the quasiperiodic state and it gives phasons a very general and simple geometrical interpretation in terms of translational degrees of freedom in internal (or perpendicular or complementary) space. These new hydrodynamic variables are distinct from the ordinary acoustic variables, which reflect the continuous degeneracy of the system with respect to arbitrary translations in physical or parallel space. This review will emphasize dynamical aspects, i.e., how phasons fluctuate in time, and with the help of general hydrodynamic mode-counting arguments we shall make qualitative predictions on the dynamical characteristics of these modes. Hence, we shall spend some time presenting the ingredients which enter the hydrodynamic picture, such as the concepts of continuous broken symmetries and of Goldstone modes. We shall then review how these concepts can be applied to several classes of systems, viz., simple fluids, ordinary crystals and quasiperiodic systems, in order of increasing complexity, and what predictions can be made concerning the hydrodynamic-mode structure of these systems. HANDBOOK OF METAL PHYSICS ISSN 1570-002X/DOI 10.1016/S1570-002X(06)02005-2 107


108

Chapter 5. Phason Modes in Aperiodic Crystals

In Section 5.2, we specialize to the case of modulated crystals, which, typically, develop as a result of a continuous phase transition, starting from an ordinary crystalline phase. For weakly modulated systems, Landau theory allows one to make specific predictions on the form of the order-parameter fluctuation spectrum in the modulated phase. In particular, the decoupling of the spectrum into phase and amplitude components – the phason and amplitudon excitation branches – will be discussed and the dynamical characteristics of both types of excitations, derived. These predictions will be compared to available results on real modulated systems, of the displacive- as well as order/disorder-types, with, respectively, oscillatory (phonon-like) and relaxational (jump-like) dynamics. In Section 5.3, we briefly review the case of binary composites where one expects some degree of mixing between phason and acoustic degrees of freedom. Unfortunately, the experimental situation in that field is still rather unclear due to the lack of suitable model systems and suitable single-crystal specimens. We then switch to the subject of quasicrystals (Section 5.4) where, as for the order/ disorder-type modulated crystals, the phason variables are purely relaxational in nature. We emphasize the distinction between single-particle events consisting in an atomic jump or a local rearrangement of a few tiles and much slower collective modes involving a correlated sequence of such rearrangements. We then review a few experimental results on the AlPdMn icosahedral phase to illustrate the concepts of quenched and thermal phasons, of phason strains, spatially homogeneous or not, of phason elasticity and finally of phason-driven structural phase transitions (Section 5.5). In Section 5.6, experimental results on other quasicrystalline systems are reviewed. In Section 5.7, we give a brief outlook on future developments and remaining challenges in this exciting field.

5.1.2.

Quasiperiodic structures

Loosely speaking, quasiperiodic systems may be defined as having too many spatial periodicities – more periodicities than space dimensions. For a 3-dimensional system, this means that the indexation of the diffraction pattern requires more than three integer Miller indices [1]. Strictly speaking, too many periodicities is like no periodicity at all. However, as long as the system is long-range ordered, one can define a r-dimensional image of the system (r > 3) which has periodicity and space group symmetry. The original 3d physical system is recovered as the intersection of the higher-dimensional periodic structure with the 3d physical hyperplane. Conversely, in Fourier space, the diffraction pattern of the 3d physical system is obtained as the projection of the nodes of the r-d reciprocal lattice, onto 3d physical Fourier space. Figure 5.1 illustrates the principles of hyperspace embedding with a few elementary diagrams, drawn for a system of physical dimensionality 1. Direct space is shown on the left, Fourier space, on the right. The top row refers to a 1d modulated crystal and the bottom row shows a 1d quasicrystal. The physical dimension is labeled external (RE) or parallel (Rpar) and the extra dimension, internal (RI) or perpendicular (Rper).

109

5.1. Introduction: hydrodynamic modes and quasiperiodic structures

1D modulated crystal

RI

QI

QE RE Direct space RI

Fourier space RE

QE

QI

1D quasicrystal Figure 5.1. Relationship between hyperspace periodicity and physical space quasiperiodicity: (a) displacively-modulated 1d crystal as intersection of a 2d periodic structure (wavy lines) with direct physical space (RE axis); (b) same in Fourier space: the position and intensity of the diffraction spots are obtained by projection of the nodes of the 2d reciprocal lattice, onto physical Fourier space (QE axis); (c) 1d quasicrystal as intersection of a 2d array of linear segments, with physical space; (d) same in Fourier space.

One obvious difference between the two classes of systems is that some of the Bragg reflections of the modulated crystal are directly located in physical Fourier space and hence need not be projected. These are the fundamental reflections which define the average periodic structure of the modulated crystal. In contrast, there is no average periodic structure in the case of quasicrystals. In direct space, the origin of the difference can be traced back to the fact that one of the axes of the 2d unit supercell is normal to physical space for the modulated crystal and not so for the quasicrystal. What is the physical meaning of the direct-space diagrams in each case? For the modulated crystal the wavy line along Rper corresponds to the modulation function (Figures 5.1 and 5.2) which leads in Rpar to atomic displacements away from the average periodic positions. We see from Figure 5.2 that the perpendicular dimension is related to the phase of the modulation and that a translation of the physical space Rpar along Rper corresponds to a phase shift of the modulation with respect to the average periodic chain. As one sweeps the phase continuously between 0 and 2, each atom in the chain describes one of the wavy vertical lines which are the graphical representation of the shape of the modulation function f: Rl,par ¼ l þ f ðqsl þ Þ;

Rl,per ¼ ;

ð5:1Þ

where l denotes the atomic positions in the average periodic chain, f is a periodic function with period 2, qs is the modulation wavevector and is an arbitrary phase.

110

Rperp 2π

Phase shift


[l + f (qsl + φ ), φ ]

Modulation function

φ l + f (qsl ) Rpar

0 Sites of the regular chain

–qsl

‘Displaced’ atom positions

Figure 5.2. The modulated atomic chain in hyperspace: physical interpretation of Rper as the representation of the phase-shift degree of freedom of the modulated atomic displacements.

So, in that case, hyperspace embedding can be viewed simply as an operation by which one stacks an infinite number of physical systems ‘one on top of the other’ along the perpendicular dimension, all systems being identical except for the variable phase shift . Hyperspace embedding is somewhat less intuitive in the case of quasicrystals, but the important notion which Figure 5.1(c) illustrates, is that translations along Rper generate different sequences of short and long segments or tiles and as long as the slope of the cut is irrational, all these sequences are energetically equivalent. Furthermore a single hyperspace unit cell contains all the information necessary to generate the infinite quasiperiodic sequence of long and short tiles. An important aspect of the hyperspace embedding technique concerns the continuity of the atomic trajectories – the wavy lines in Figure 5.1(a) and the line segments parallel to RI in Figure 5.1(c) – which become atomic surfaces or atomic volumes for real systems, with physical dimensionality d = 3. For d = 3 quasicrystals, these atomic domains have been found piecewise discontinuous (see Yamamoto, this volume), whereas for modulated and composite systems, atomic domains can be either continuous, as in Figure 5.1(a), or discontinuous, as in commensurate, quasi-commensurate or, more generally, strongly modulated systems. The transition between the two regimes corresponds to a transition from analytic to non-analytic behavior for the modulation function f in Eq. (5.1) – the so-called transition by breaking of analyticity of Aubry [2]. A related question concerns the nature of the degeneracy of quasiperiodic systems with respect to translations of the cut along the Rper axis. Is this degeneracy continuous in a mathematical sense? Even when the initial and translated configurations are energetically equivalent, there may be collective energy barriers, i.e., friction, involved in the transformation. The existence or not, of such barriers, is closely connected with the continuous or discontinuous character of the atomic domains, at least for modulated and composite systems. From a dynamical


111

standpoint, the analytic regime of the modulation function is clearly associated with the existence of a hydrodynamic Goldstone phason branch and the breaking of analyticity is thus expected to signal the opening of a long-wavelength phason gap. In the following sections, we shall examine in more detail how this general scheme applies in each class of quasiperiodic structures and to which extent theoretical predictions are borne out by available experimental results.

5.1.3.

Hydrodynamic modes

Hydrodynamic theory is concerned with the slow, long-wavelength degrees of freedom, modes for which the frequency ! vanishes as some power of the wavevector q in the limit q ! 0, i.e., in the long-wavelength limit: !ðqÞ qn

as

q ! 0;

ð5:2Þ

If we take ! as a complex quantity, then both the real and imaginary parts of the frequency should vanish in the long-wavelength limit. For example, in the case of acoustic modes in a crystal the real part vanishes linearly with q and the imaginary part (which is associated with damping), vanishes quadratically. Basically, there are two separate categories of hydrodynamic variables. The first category involves variables which obey a conservation law. In a closed system, the total mass or the total number of particles is conserved. Since particles cannot be created or destroyed locally, a density fluctuation of wavelength l can only develop if particles move over distances of the order of l- or l/2- and that will require a characteristic time that should scale as some power of l, as ! 1 (or, equivalently, ! = 2/ should scale as some power of q = 2/l, as in Eq. (5.2)). For a monoatomic system, the variables associated with a conservation law are normally: – the mass density – the three components of the momentum density – the energy density i.e., five variables altogether. In a simple fluid, which is the simplest possible type of system, with the highest possible symmetry, that is all there is to say and hence, in that case, one expects five hydrodynamic modes: – longitudinal sound which is propagating and therefore counts for two modes (one mode propagating to the right and one, to the left, the two being related by timereversal symmetry) – 2 diffusive shear waves (the equivalent of transverse sound but with nonpropagating, diffusive character) – the heat or entropy mode which is also diffusive The second category of hydrodynamic variables concerns ordered systems for which the ordered state breaks a continuous symmetry of the Hamiltonian or of the free energy (if we are dealing with a macroscopic system in thermal equilibrium).

112


Let us illustrate this by considering the case of an ordinary crystal. In addition to the same five conserved variables, we now must include three more hydrodynamic variables corresponding to the position vector of the center of mass of the crystal. The reason as to why this is so is summarized as follows: – the free energy of the crystal is invariant under any translation operation (by ‘any’ one means any arbitrary translation, nothing to do with the periodicity of the crystal). – so we have a continuous symmetry or a continuous degeneracy of the ground state. – the ground state breaks that symmetry, as the system eventually decides to be there and not somewhere else. – it is possible to distinguish between these various possible degenerate groundstates: for example, the time-averaged electronic density (r) is changed when the system is moved by some arbitrary translation T (except if T happens to be a multiple of the crystal’s own periodicity). Note that in the case of a fluid it is not possible to make the same distinction: if one looks at some specific spot within the bulk of a fluid, one will not be able to tell whether someone has been moving the container or not, as, in all cases, particles will be seen moving in and out of the small observation volume, and the time-averaged electronic density will be unchanged. So, going back to the case of the crystal, one may argue as follows: since a uniform translation does not cost energy, then a local displacement field T(r) cannot cost much energy, as long as it is a continuous and slowly-varying function of space coordinates. Furthermore, that energy should vanish in the long-wavelength limit, i.e. it should scale as some power of q = 2/l where l defines the characteristic length scale over which T(r) varies. Thus T(r) qualifies as a hydrodynamic variable and we must add it to the set of five conserved quantities, which brings us to a total of eight variables, eight because T(r) is a 3-component vector. The solution of the set of eight coupled hydrodynamic equations [3] yields the eight hydrodynamic modes of the crystal, namely: – three pairs of propagating acoustic modes !i = –vi q; (i = 1,3), – the heat diffusion mode, – a vacancy diffusion mode. This last mode allows one to decouple the transport of mass from the motion of the crystal’s frame. In fact there are as many such modes as there are atomic species in the system, since the total number of particles of each species is separately conserved. So that’s the essence of the so-called Goldstone theorem: each continuous degeneracy of the free energy, which is broken by the thermodynamic groundstate, brings in a new hydrodynamic variable and a new hydrodynamic mode.


113

Several remarks are in order at this point: – Goldstone theory is based on a continuity argument: in the example above, one assumes that the free energy associated with the transformation T(r) is an analytic function of the atomic displacements involved in the transformation and that, in the long-wavelength limit, it can be expressed as a power series of the spatial gradients of these displacements. One can already anticipate that the analyticity condition will limit the applicability of the theory in the case of quasiperiodic structures. – because of the complex structure of the system of coupled hydrodynamic equations [3], the addition of a new hydrodynamic variable does not necessarily give rise to a new independent hydrodynamic mode. Quite often, an existing mode is promoted from diffusive to propagating, where it is counted as a pair of timereversal-degenerate modes. This is the case for the isotropic ferromagnet and for the isotropic antiferromagnet, where spin diffusion modes of the paramagnetic phase transform into propagating spin waves in the ordered phase, although the detailed story is quite different for the two types of systems. This is also the case for crystals where we saw that the three new positional variables give rise to only one new mode (the vacancy diffusion mode) while two diffusive modes become propagating (the two transverse acoustic modes). A similar situation arises in superfluid 4He, where the heat diffusion mode becomes propagating (second sound) below the l-point, when the quantum phase of the superfluid condensate is included as a new hydrodynamic variable. Fortunately, this latter type of complication does not arise in the case of quasiperiodic systems, which we shall now address explicitly.

5.1.4.

Internal space translations

As any other physical system, quasiperiodic systems are degenerate with respect to translations in physical space and this degeneracy is as physically meaningful as in the case of the crystal, i.e., one can distinguish between the various degenerate groundstates. Hence one expects to get the same eight hydrodynamic modes as in the crystal’s case, including three pairs of propagating acoustic phonons. In addition, quasiperiodic structures are degenerate with respect to translations in internal space and hence one may expect to get as many Goldstone phason branches as there are dimensions in internal space. However, as pointed out before, the applicability of Goldstone theory to translations in internal space is a priori questionable. The reason is simply that although the translation of the cut may be thought of as a continuous transformation in internal space, the corresponding atomic motions may well be discontinuous in physical space. In that case it is not clear in which space the continuity criterion should be applied. For displacively modulated crystals, as long as the modulation function is smooth and continuous one is safe because the atomic motions are continuous in physical space. But in all the other cases, the situation is less clear.

114


Unmodulated chain Modulated chain

b a

λ

λ

λ Figure 5.3. The Frankel-Kontorova model: (a) a chain of particles of natural periodicity b is adsorbed on a sinusoidally corrugated substrate of strength l and periodicity a, incommensurate with b; (b,c) chain configurations in the unlocked and locked regimes.

One general way to formulate the problem is to note that during the translation of the cut, individual particles have to overcome local potential barriers. This is true whether the atomic motions are continuous or jump-like, it is simply more obvious in the latter case. Depending upon the strength of the inter-particle interactions, these local barriers may or may not translate into a collective potential barrier. If they do, as expected in the limit of weak interactions, the translation of the cut will be pinned, and the system will be said to be in a locked state. In the opposite limit, the motion of the cut will be free and the system, unlocked. This problem was first treated quantitatively by Aubry [2,4] in the context of the Frankel-Kontorova (FK) model [5,6], which deals with the motion of a chain of atoms adsorbed on a periodically corrugated substrate. The atoms are connected by harmonic springs, as shown in Figure 5.3, and there is an irrational ratio between the natural periodicity of the chain (i.e., the periodicity it would have on a flat substrate) and the period of the substrate. When one tries to set the chain in motion relative to the substrate by, say, pulling on the end atom, Aubry has shown that there is a well-defined phase transition as a function of substrate potential strength: – for weak potentials the chain is in the unlocked state, i.e., there is no finite force to be applied to set the chain in motion. The chain slides freely over the hills of the substrate because the springs are strong enough to transport the momentum from the atoms going downhill to those going uphill, which need a push. – if the hills get too high the chain enters a locked state where the springs are more and more strained and the atoms tend to localize in the valleys of the substrate. In that case there is a finite force to be applied to unpin the chain. At the same time


115

the function which describes the modulated atomic positions becomes discontinuous: no atoms want to sit near the top of the hills when the chain is at rest. For that reason, when one tries to pull on the chain some atoms have to get over the top of the hills and for an infinite chain this happens all the time. This means that the chain has to go through configurations which are energetically unfavorable and therefore there is a collective energy barrier to be overcome. Note in this context that introducing temperature in the FK model would only renormalize the substrate potential value at which the locked/unlocked transition occurs, but pinning of the chain would still occur precisely at the point when the atomic modulation function begins to develop discontinuities. So that is the essence of the so-called transition by breaking of analyticity, which has since been shown to exist in displacively modulated and composite crystals by Janssen and coworkers [7,8]. In all cases the answer is the same: as soon as the static modulation functions become discontinuous the collective motion is pinned. What about quasicrystals? If the same results can be transposed to the case of quasicrystals, where atomic domains are strongly believed to be discontinuous, then one may be in the locked non-analytic regime all the time and hydrodynamic theory, phason elasticity, etc., may not apply at all. This precise question was already on the mind of Lubensky and coworkers [9], when they wrote their seminal 1985 paper on the hydrodynamic theory of icosahedral quasicrystals. At the time, the microscopic model of quasicrystals was not sufficiently developed to provide a reliable answer to this question and they actually had to assume that quasicrystals in the unlocked state would have to exist, in order for their work to be at all relevant. We reproduce below a few sentences taken from their work: ‘One piece of speculation is in order. In a one-dimensional incommensurate system (the discrete Frankel-Kontorova model), Aubry and Peyrard have shown that anharmonic effects can pin the sliding mode (analogous to the phasons here), rendering it non-hydrodynamic. If a similar pinning transition were to take place for the phasons in a quasicrystal, the dynamics, especially of dislocations, would change dramatically. We do not know if such a transition is possible. All statements in this paper apply only when this transition has not occurred.’ In fact, this point has been examined further by a number of authors since and both experiment and numerical simulations show that there exists indeed an analytic regime at high temperatures where phasons appear to be hydrodynamic and phason elasticity, applicable. With respect to simulations, the result does not depend on whether the simulation is based on a random tiling [10] or a matching-rule [11] model: in the high-temperature limit, the answer is the same in both cases. Note however that melting of the quasicrystal also occurs in the high-temperature limit, and the possibility that melting may occur before the unlocked regime is reached, cannot be excluded on the basis of the simulations alone. Thus, quasicrystals seem to be different from the other classes of quasiperiodic systems, in the sense that discontinuous atomic domains appear to coexist with hydrodynamic phasons. (Note that continuous atomic surfaces model with icosahedral symmetry have been considered by Levitov [12], and Wirth [13]. However such models were only obtained for the non-centrosymmetric icosahedral Y space groups, and correspond to a very peculiar atomic distribution in real space).

116


The conclusion is then that the physics of the FK model is simply not transposable to the case of quasicrystals. The origin of this difference is most probably related to the fact that the energetics of quasicrystals is less directly related to the shape of the atomic domains than for the other classes of quasiperiodic systems. We elaborate further on this point in Section 5.4, in the framework of the entropystabilized random-tiling model of quasicrystals.

5.2. 5.2.1.

Modulated crystals General

We now focus on the class of quasiperiodic structures for which the properties of phasons are best understood, namely the incommensurate modulated crystals. As mentioned earlier, these systems can be viewed as resulting from a continuous phase transition at, say, some finite temperature Ti. As illustrated in Figure 5.4, the modulated phase is stable, typically, between a high-temperature parent periodic phase, which plays the role of the high-symmetry phase, and a low-temperature commensurately modulated phase, which is again periodic (the corresponding sequence for quasicrystals would be liquid – quasicrystal – crystalline approximant). In the quasiperiodic state there is at least one new periodicity, the modulation wavevector qs, and each Bragg reflection of the high temperature periodic phase becomes a fundamental reflection and is accompanied by a series of satellite reflections, m = 1,2,3, . . . for first-, second-, and third-order satellite reflection. The nature of the modulated variable may be a number of different physical quantities. One may have a displacive structural phase transition at Ti and modulated atomic positions below. One may have an order/disorder-type transition and modulated site occupation probabilities below Ti. One may have a modulated chemical ordering process as in minerals and alloys, or modulated magnetic moments, or mixtures of several types of modulations. In Figure 5.5 we show the example of sodium nitrite, with a simple orthorhombic structure in which the V-shaped nitrite ions have two alternate orientations, up or down, corresponding to opposite electric dipole moments. At low temperature (T < Tc) all the electric dipoles point in the same direction, downward in the figure, and we have a ferroelectric phase. At high temperature (T > Ti) the up and down

Fourier components m=1 m=2 Tc Comm.

m=3

Ti Inc. Wavevector qs

T

Periodic

Figure 5.4. A typical phase diagram for a modulated crystal. The temperature evolution of the modulation wavevector qs and of the m = 1,2,3 Fourier components is shown schematically.

117

5.2. Modulated crystals

T < Tc

T > Ti

a

Tc < T < Ti

Na

0

N

b c

a Modulation period = 9.22 a

Figure 5.5. Mixed displacive-order/disorder modulation in NaNO2: Na+ and NO–2 ions are found in two alternate positions (full and broken lines) with occupation probabilities proportional to shaded areas. Probabilities and ionic shifts vary sinusoidally along a (bold arrow) with a temperature-dependent periodicity of 8–10 a; after Kucharczyk et al. [14].

orientations are occupied randomly (with some short-range order, of course) and in between these two phases (Tc < T < Ti), there is a modulated phase with mixed displacive-order/disorder character [14]. Along the a-direction (bold arrow in the figure) there is a rather long-wavelength shear wave ( e9.2 unit cells) with displacements along b, coupled to an orientational probability wave for the nitrite ions, in phase with the shear displacement wave. The dynamics of such a system is dominated by the large amplitude orientational jumps of the nitrite ions with a typical energy barrier of 0.4 eV e10 kT, at T eTi e 450 K. Therefore, the dynamics is of the slow relaxational type, closer to what is found in quasicrystals than to the usual soft phonon behaviour observed near displacive phase transitions. So what are the predictions of hydrodynamic theory for modulated crystals, in general? Let u(r) denote the modulated variable: uðrÞ ¼ expfi½qs

r þ ðrÞ g;

ð5:3Þ

where is a temperature-dependent amplitude and (r) is the local value of the modulation’s phase. Since the free energy is independent of the phase, when the phase is uniform, we have a situation of continuous degeneracy and (r) is a potential hydrodynamic variable below Ti. Furthermore, since the modulation amplitude vanishes continuously as T ! Ti, one may expect the analytic regime to hold over a finite temperature range below Ti, and, over that range, long-wavelength hydrodynamic phasons should be observable. Using Landau theory, we may also address the practical question of where to go in Fourier space to observe these long-wavelength phasons? The answer is: we

118


should look near first-order satellite reflections because phasons are part of the fluctuation spectrum of the order parameter and that is where these fluctuations are visible above Ti. That defines the scattering geometry for an inelastic neutron scattering measurement – or for an X-ray diffuse scattering experiment, if one is interested in the energy-integrated phason response: Q ¼ ki kf ¼ Ghklm ¼ 1 þ q,

ð5:4Þ

where Q is the scattering vector, ki and kf are the incident and scattered radiation wavevectors, Ghklm=–1 = Ghklm=0 – qs is the scattering vector for Bragg scattering on a first order satellite reflection and q is the excitation’s wavevector.

5.2.2.

Displacive modulations in the quasi-harmonic approximation

Another important question, to which Landau theory provides an answer, concerns the connection between the dynamical response function from phasons below Ti and the order parameter fluctuation spectrum above Ti. To be specific, let us consider the case of a displacive modulation, for which the order parameter fluctuation spectrum above Ti is of the soft-mode-type. In a simple mean-field, quasi-harmonic approximation, one can characterize this response by a soft-mode frequency, which has a minimum at the critical wavevector. The frequency at the minimum is temperature-dependent and vanishes continuously as T ! Ti+: !SM 2 ðq; T Þ ¼ aðT Ti Þ þ bq2 ;

ðb > 0Þ,

ð5:5aÞ

Figure 5.6 shows how the soft-mode dispersion may look like in a real system, namely ThBr4 [15]. Here the entire optic branch softens on cooling and, roughly speaking, touches the horizontal axis at the point 0.31 c* along the tetragonal 4-fold axis, at T = Ti = 95 K. Below Ti, a first-order satellite Bragg reflection develops at that point. What Landau theory predicts is that near that satellite reflection, two excitations will be observable: One, which has the character of a fluctuation in the amplitude of the condensed modulation, with a frequency minimum at the satellite position; but the frequency at the minimum quickly increases on cooling, because the thermal equilibrium value of the modulation amplitude grows and it costs more and more energy to deviate from that equilibrium value: !A 2 ðq; T Þ ¼ 2aðTi T Þ þ bq2 ;

ðb > 0Þ,

ð5:5bÞ

and the second, one which has the character of a phase fluctuation and which has zero frequency at the satellite position, independent of temperature, and a velocity related to the curvature b of the soft mode dispersion curve at the minimum: ! 2 ðq; T Þ ¼ 0 þ bq2 ;

ðb > 0Þ:

ð5:5cÞ

119


Frequency (THz)

T = 300 K T = 150 K T = 120 K T = 101 K

1

LA

TA

5

5

qs

[0, 0, ζ]

Figure 5.6. Dispersion curves from the longitudinal acoustic (LA), transverse acoustic (TA) and soft optic branches in ThBr4; after Bernard et al. [15].

Modes at q = 0

A

ω2

SM

0

φ Ti

T

Figure 5.7. Displacively-modulated crystal: temperature dependence of the q = 0 soft mode (SM), amplitudon (A) and phason (f) frequencies.

This is summarized schematically in Figure 5.7, for the modes at the critical wavevector, which becomes the first-order satellite position in the modulated phase (q = 0 in Eqs. (5.4,5.5)). Figure 5.8 shows how the dispersion curves look like below Ti for ThBr4 [16]. Instead of one optic branch we now have two branches: – the amplitudon branch with a finite frequency at the satellite position – the phason branch with a linear dispersion, somewhat like an acoustic mode emanating from a first-order satellite position, but with a different slope. The two branches are well defined only in a neighborhood of the satellite. Moving away from the satellite position one of the two branches disappears gradually while the other picks up strength, and eventually one recovers the picture of a single optic branch as seen in the high-T phase – the broken curve in Figure 5.8 shows the optic mode dispersion at some temperature T > Ti.

120


0.6

T = 81 K

Frequency (THz)

0.5 0.4 0.3 0.2 0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

[0, 0, ξ]

Figure 5.8. Phase- (open circles) and amplitude- (closed circles) mode dispersions in ThBr4 at 81 K. The broken curve shows the soft-mode dispersion at 150 K; after Bernard et al. [15].

Phason q ≠ 0

Amplitudon q ≠ 0

Amplitudon q = 0

Phason q = 0 Figure 5.9. Displacement patterns associated with q = 0 and q 6¼ 0 phase and amplitude fluctuations for a continuum model of a modulated crystal.

So, what has happened is that the original optic branch, which has mirror symmetry about the fundamental reflection – due to time-reversal symmetry – is replaced by two branches which have mirror symmetry about the satellite reflection (the symmetry refers to frequencies, not to intensities). That is how the system ‘tries’


121

to take account of the fact that the satellite position has become some sort of new Brillouin zone center, about which time-reversal symmetry should be fulfilled. Figure 5.9 gives an idea of the displacement patterns associated with the modes of the modulated phase: the central frame shows a blown-up version of the dispersion curves below Ti; the top frame shows the q = 0 amplitude mode with the fluctuating displacements in phase with the static average modulation (solid line); the bottom frame shows the q = 0 phason with the fluctuating displacements 90 out of phase with the average modulation; in the right frame a finite-wavelength amplitudon is shown, the wavelength being equal to the distance between two consecutive maxima of the envelope curve; similarly, the finite-wavelength phason (left) might be viewed as a time-dependent frequency modulation of the static average modulation, the latter playing the role of a sort of static carrier wave. Note that the modes we have been discussing so far and which are shown in Figure 5.9 above, are linear excitations, i.e., small amplitude fluctuations. But for the q = 0 phason, for which the restoring force vanishes, it makes sense to consider a large amplitude version of the same displacements. The name sliding mode is usually reserved to the mode in which the phase of the modulation shifts indefinitely. In such a motion, the fluctuating atomic displacement amplitudes are equal to the full static modulation amplitudes. In charge-density-wave (CDW) systems [17], it is possible to excite the sliding motion of the CDW with respect to the underlying atomic lattice, by applying a small – but finite – dc voltage across the sample. The voltage-induced unpinning of the CDW contributes an additional collective charge transport channel, the non-linear CDW conductivity [18].

5.2.3.

Displacive modulations: anharmonic effects

It is important, at this point, to realize that we have a serious problem with respect to hydrodynamic theory, because a pair of propagating phasons, as drawn in Figure 5.8, is one too many hydrodynamic mode: the modulated state introduces only one scalar hydrodynamic variable, (r) , and hence we only expect one new hydrodynamic mode. The way out of this apparent contradiction is to include anharmonicity and to note that the atomic motions involved in the soft-mode vibration above Ti and in the amplitude and phase modes, below, are all very similar at the microscopic level and optic-like in character. By ‘optic’ one means that near-neighbor atoms do not necessarily move in the same direction and by almost the same amount, as in a longwavelength acoustic mode. Consequently, one expects their damping coefficients G to be all very similar and slowly varying with wavelength and temperature: GSM GA G :

ð5:6Þ

For phasons, this implies a damping coefficient that is slowly varying in q and T and finite at q = 0. Hence phasons, unlike acoustic modes, appear to be hydrodynamic with respect to the real part of the frequency but not with respect to damping.

122


One can readily show, however, that a pair of ‘half-hydrodynamic’ phasons has the same response function as a single diffusive hydrodynamic mode, or, in other words, that in the limit of large overdamping – i.e., when the real part of the frequency vanishes while the imaginary part stays finite – a damped-harmonicoscillator (DHO) is equivalent to a diffusive mode. This is detailed below, with the phason dynamical susceptibility written in standard DHO form: ðq; !Þ ¼ f!2 v2 q2 i!G g1 :

ð5:7Þ

In the low q, low ! limit it is perfectly justified to drop the inertial !2 term: ðq; !Þ ¼ fv2 q2 i!G g1 :

ð5:8Þ

Using fluctuation/dissipation theory, one takes the imaginary part of the dynamical susceptibility (q,!), times the Bose population factor – which we write classically as (kT /!)– and one obtains the scattering function S (q,!) in Lorentzian form: S ðq; !Þ ¼ ðkT=!ÞIm ðq; !Þ ðkT=G Þ=f!2 þ D2 q4 g,

ð5:9Þ

where the phason diffusion constant D is equal to the square of the phason velocity over the soft-mode damping coefficient: D ¼ v2 =G :

ð5:10Þ

The order of magnitude for D is e10-5 m2/s which is quite fast (a few mm/sec.). With X-rays, one obtains a phason thermal diffuse scattering signal, varying as 1/q2, around each first-order satellite reflection: Z S ðqÞ ¼

S ðq; !Þd! ¼ kT=ðv qÞ2 ¼ kT=ðG D q2 Þ:

ð5:11Þ

The inclusion of anharmonicity is thus essential to reconcile the picture emerging from Landau theory with the hydrodynamic mode-counting arguments. The prediction then, is that of a propagating phason at large enough q, becoming progressively overdamped – and eventually undistinguishable from a diffusive mode – in the low q limit.

5.2.4.

Displacive modulations: pinning by defects

The hydrodynamic regime is expected to hold over a finite temperature range below Ti. Below that temperature range, one expects the phason branch to develop a finite frequency gap at q = 0, either because the modulation wavevector qs becomes commensurate – in which case the energy and even the symmetry of the system depends on the value of the phase variable – or because qs remains incommensurate but the atomic modulation functions become non-analytic.

123


The transition between the two incommensurate regimes may be observed experimentally, provided the gap frequency !G is large enough on the scale of the instrumental resolution width D!R and on the scale of the mode damping coefficient G . More precisely, the expected Lorentzian width !2G/G must be at least a tenth of the instrumental width D!R, whence the condition: !G 0:3ðD!R G Þ1=2 :

ð5:12Þ

In practice, the detailed study of the phason response in the low q limit and the detection of a possible gap at q = 0, is a challenging task, requiring the best energy resolution available from a cold-neutron three-axis spectrometer. Such a study was attempted by Ollivier et al. [19,20] in the case of the modulated organic compound bis-4-chlorophenyl-sulfone (BCPS) – for which the experimental conditions were particularly favorable – with the partly surprising result that a gap (!G 100 GHz) was found to be present at all temperatures, even at Ti, as shown in Figure 5.10. The origin of the gap is believed to be connected with the presence of defects or impurities which pin the phase of the modulation even at Ti. More generally, in real systems, the response of low-frequency optic modes – such as the soft-mode

500

Frequency (GHz)

TI 400

Amplitudon

300

Soft mode 200

Phason 100

(a) 0 250

FWHM (GHz)

200

Phason

150

Soft mode

100 50 0

Amplitudon 0

50

100

(b) 150

200

250

300

Temperature (K) Figure 5.10. Mode-frequencies (top frame) and damping coefficients (bottom frame) of the soft phonon (T > Ti = 150 K), phason and amplitudon (T < Ti) in BCPS. The dotted line are extracted from Raman results by Ecolivet et al. [20]. The dashed area indicates the temperature range where both the soft phonon and the central mode are present in the experimental spectra; after Ollivier et al. [19].

124


above Ti or the phase and amplitude modes, below – is modified by the presence of frozen or slowly-relaxing defects which couple linearly with the optic-mode atomic motions [21]. Looking at the top frame in Figure 5.10, we see that the soft-mode frequency does not vanish, as could have been expected, at Ti: we have a finite soft-mode saturation frequency and, at the same time, there is either a purely elastic (! = 0) or a narrow unresolved quasielastic component which grows critically on approaching Ti, starting about 50 K above Ti. This last feature is called the central peak or central mode [22,23] in the folklore of structural phase transitions and it is observed systematically near displacive transitions, whether the critical wavevector of the instability is incommensurate or not. There is substantial experimental evidence that it is not an intrinsic feature but that it is due to impurities or defects which couple linearly to the soft-mode displacements, and models have been developed which account simultaneously for both aspects of the phenomenon, i.e., the central peak and the finite soft-mode saturation frequency [24]. The important point which Figure 5.10 brings out, is the fact that, both the q=0 phason and q = 0 amplitudon have a finite frequency at Ti and that this gap frequency is very comparable, within experimental accuracy, with the saturation frequency of the soft-mode above Ti, suggesting that the same physical mechanism is operating above and below Ti. The bottom frame in Figure 5.10 shows the damping coefficients of the three modes obtained via the same DHO analysis of the experimental spectra. It confirms that the three damping coefficients are of comparable magnitude, in agreement with Eq. (5.6) above. Thus, the conclusion concerning the evidence for hydrodynamic phasons in displacively modulated crystals is somewhat more qualified than a priori expected: in ideal systems, long-wavelength phasons would be possibly hydrodynamic but with diffusive rather than propagative character, due to anharmonic effects; in real systems, the presence of defects induces a q = 0 gap which partly destroys the hydrodynamic character of the phason branch. Why only partly? From a spectroscopist’s standpoint, the presence of a gap implies that the mode is no longer hydrodynamic. However the energy-integrated phason response still diverges at q = 0, when the central component is included – as in the case of the soft-mode response at Ti. Hence the phason thermal diffuse scattering does vary as 1/q2, in the neighborhood of a first-order satellite reflection, which is a perfectly valid criterion on the basis of which one may argue that the mode still has hydrodynamic character.

5.2.5.

Order/disorder modulations

We shall briefly discuss the case of modulated order/disorder systems. There the modulated variable is an atomic or molecular site occupation probability – the up or down orientation of the nitrite radical in Figure 5.5. This probability is given by a statistical average over an atomic plane of essentially infinite extent, normal to qs.


125

Consequently, the modulation function can be perfectly continuous, even though the atomic motions involved in an internal space translation are jump-like, i.e. discontinuous. Thus, in principle, one expects to observe an analytic regime characterized by hydrodynamic phasons. Here the order-parameter fluctuations are entirely relaxational, not just in the long-wavelength limit, as in the displacive case, but at all wavelengths. Nevertheless, the results from the displacive case can be transposed to the order/disorder case by simply replacing (!2/G)phonon by the relaxation rate or inverse relaxation time 1. So, above Ti, one observes the critical slowing-down of the order-parameter fluctuations, characterized by a T- and q-dependent relaxation rate with a parabolic minimum at the critical wavevector: 1 ðq; TÞ ¼ aðT Ti Þ þ bq2 ;

ðb > 0Þ,

ð5:13aÞ

and below Ti one gets two relaxation rates 1 A and 1 : 1 A ðq; TÞ ¼ 2aðTi TÞ þ bq2 ;

ðb > 0Þ,

ð5:13bÞ

and: 1 ðq; TÞ ¼ 0 þ bq2 ;

ðb > 0Þ,

ð5:13cÞ

with a diffusive, T-independent phason response. The parameter b, which determines the curvature of the relaxation rate dispersion above Ti, becomes the phason diffusion constant below Ti (typical values are in the range 1010 m2/s). In Figure 5.11 we show the results obtained by Durand et al. [25] on sodium nitrite using the neutron spin echo (NSE) technique. The upper right frame shows a series of spectra taken at the critical wavevector qs as a function of temperature for T ! Ti+. The corresponding relaxation rates are shown in the bottom frame. One notices that the relaxation rate does seem to extrapolate to zero at Ti, not quite linearly – as would be expected in a strictly mean-field picture – but there is no indication of a gap at Ti. By continuity this implies that one does not expect to find a phason gap in the modulated phase just below Ti. Direct measurements in the modulated phase are more difficult to interpret because the NSE technique is less accurate when several relaxation rates are present. But the data in Ref. [25] are certainly consistent with a diffusive, temperatureindependent phason relaxation rate. To summarize, this type of order/disorder modulated material seems to exhibit hydrodynamic phasons, in spite of the discontinuous nature of the atomic motions involved. It is unfortunate that the incommensurate phase in sodium nitrite is only stable over a 1.5 K temperature range around 450 K, which precludes a more detailed study of a possible transition from the analytic to the non-analytic regime, within the incommensurate phase.

126


(a)

(b)

0.5 1.0

t–1(q)

PNSE (qδ, T)

0.4

T–Ti (K)

(T = Ti + 0.55 K) 0.3

0.2

0.55 1.65 4.00 8.00 14.00

PNSE(qs, T)

0.5

qs

0.1

00

0.1 qδ

0

0.2

0

0.5

1.0

1.5

t (ns)

qx /a* (r.u.) (c)

1/τ(qd) (ns–1)

2

t–1 (qs, T)

1

0

0

5

10

15

T–Ti (K) Figure 5.11. Critical slowing-down of NO2– reorientations in NaNO2: (a) Relaxation rate 1(q) for q along a* and T = Ti+0.3 K () neutron data, (D) ultrasonic attenuation data from Hu et al. [111], (&) dielectric relaxation data from Hatta [112]; (b) NSE spectra at the critical wavevector qs as a function of temperature; (c) Temperature dependence of -1(qs); after Durand et al. [25].

5.3.

Binary composites

Binary composite systems consist of two interpenetrating sublattices with incommensurate periodicities in one direction. Examples are: – the well-known mercury-chain compound Hg3-AsF6 [26,27]. – the alkane-urea inclusion compounds [28]. – the Nowotny phases [29] which are semiconducting compounds involving a transition metal tetragonal sublattice and a sublattice of Si, Ge, Ga or Sn dumbbells, also tetragonal and with four atoms per unit cell (Figure 5.12). The two sublattices have the same basal plane lattice parameter a, but different c-axis

127

0

Ghklm = h a∗ + k b∗ + lc Mn ∗ + m csi∗

c/2

Mn27Si47

c

Mn11Si19

5.3. Binary composites (110)

(220)

Si(111) Mn(112)

Si(002) Mn(004)

Si(113)

a=b cSi

cMn ≈ 1.73 cSi

a=b

cMn

Figure 5.12. Structure of the Nowotny phases TXx (T = Ti, Zr, V, Mo, Cr, Mn, Rh; X = Si, Ge, Sn). Top left: sketch of the unit cell for two stoichiometric compounds in the MnSix series (x = 1.73 – 0.02); top right: sketch of the corresponding diffraction pattern with a few fundamental reflections indexed; bottom: basic subcells associated with the Si and Mn sublattices; after De Ridder [29].

parameters, the ratio of the c-axis parameters being directly related to the stoichiometry of the compound. What about hydrodynamics and broken translational symmetries? We have the three ordinary translations and possibly the relative translation (or anti-translation) of one subsystem against the other. We must say possibly because there is no guarantee that the analytic regime exists at all, as discussed explicitly by Radulescu and Janssen [8]: there is no continuous structural phase transition which we may invoke here, as was done in the case of modulated crystals, and the two sublattices may be in a strongly modulated non-analytic state throughout the stability range of the compound. It is important to realize that the anti-translation is not a rigid motion of each sublattice, as shown in Figure 5.13: what moves to the right is the A sublattice together with the modulation which the A sublattice induces on the B sublattice, and vice versa for sublattice B. Each modulation is thus phase-shifted with respect to the relevant average periodic sublattice: comparing this with the phason displacements in a modulated crystal, we might say that in binary composites the anti-translation involves two coupled phason-like motions, one on each subsystem. Again the hydrodynamic phason can only be diffusive in the long-wavelength limit because we only have one new broken symmetry since the direction of the

128


A

B Figure 5.13. The anti-translation in binary composites: sublattice A moves to the right together with the modulation which it induces on B, while B moves to the left together with the modulation it induces on A.

anti-translation is uniquely determined by the structure. As for modulated crystals, the diffusive character of the anti-translation arises from the anharmonic overdamping associated with the optic-like atomic motions involved in the relative displacement of the two sublattices. Several authors [30–33] have considered the propagation of longitudinal sound along the direction of incommensurability. At a phenomenological level one expects two distinct regimes: – at short wavelengths, i.e., for q larger than some crossover value qCO, one expects longitudinal sound to propagate independently on each sublattice, i.e., one expects to observe two longitudinal acoustic (LA) branches with different acoustic velocities – at long wavelengths (q < qCO), the incommensurability between the two sublattices becomes irrelevant and the two sublattices become coupled through various macroscopic mechanisms. In that regime one should observe a single, propagating, LA mode and an overdamped, eventually diffusive phason branch – if the modulations are analytic – or a phason branch with a gap at q = 0, if they are not. It is not easy to estimate the order of magnitude of qCO, of course, but clearly inelastic neutron scattering should be able to access the uncoupled regime, while ultrasonics should measure the true (coupled) LA mode. Whether Brillouin scattering probes the coupled or uncoupled regime is more difficult to say. As mentioned earlier, the experimental situation is not clear at the moment, mostly because samples of appropriate spectroscopic quality are difficult to prepare. For example, the reported observation of a propagating phason branch by Brillouin scattering in n-heptadecane/urea [34] has been later refuted [35] as an artefact due to sample imperfections. In Figure 5.14 we show some recent inelastic neutron scattering data on the c-axis longitudinal modes in the MnSi1.725 Nowotny compound [36]. There is good evidence for the existence of two acoustic-like branches, which one might tentatively ascribe to independent sound propagation on the two subsystems. However, in that case, one would expect one branch to be observable near the fundamental reflections of one sublattice, and conversely for the other branch. This is not at all the case here: both modes are observed simultaneously on both types of fundamental reflections – and even on some satellite reflections. So it seems that even in the large q regime the two sublattices are coupled, in which case one could – more plausibly – interpret the two modes as one coupled LA mode and one propagating phason (which is allowed at large q). A careful analysis of the mode intensities is required in order to confirm that interpretation.

129

5.4. Hydrodynamics of icosahedral phases 7

Frequency (THz)

6 5 4 3 2 1 0 0.0

0.2

0.4

0.6

0.8

1.0

qeff (A–1) Figure 5.14. Longitudinal sound propagation along the direction of incommensurability in MnSi1.725. The symbols refer to different neutron scattering geometries. The two branches can be interpreted either as two sound waves propagating on each sublattice independently, or as one ‘coupled’ sound wave + one propagating phason branch; after Bourgeois et al. [36].

5.4.

Hydrodynamics of icosahedral phases

5.4.1.

Fundamental hypotheses

We present in this section a brief outline of the hydrodynamic and phason elasticity theory of icosahedral quasicrystals and its relationship with the measured diffuse scattering. We also give some details on phase transitions and stability of the phases. A more detailed presentation can be found in the review papers by Lubensky [37] and by Henley [10]. As already explained in the introduction, the basis for the symmetry breaking and mode counting analysis is the invariance of the free energy of the quasicrystal with respect to a rigid translation of the cut space along the perpendicular (or internal) direction, in the high dimensional description. In addition, the Landau theory of quasicrystals allows one to derive the pertinent elastic coefficients by using a phenomenological approach which incorporates all the relevant symmetries of the real system. In this approach, the order parameters which distinguish a quasicrystal from a homogenous fluid are the Fourier components of the average mass density defined by: hðxÞi ¼

XD

E Qpar eiQpar :x ,

ð5:14Þ

Qpar

where Qpar is the parallel or physical space component of the 6D vector of the periodic reciprocal lattice defined as Q6D = (Qpar, Qper). The simplest atomic density with icosahedral symmetry is obtained by the summation of six mass density waves of equal amplitudes, with six Qpar components pointing towards the six vertices of an icosahedron. When minimizing the free energy of the system at thermal equilibrium, it can be shown that for the icosahedral structure, six density wave phases are not fixed

130


by the equilibrium conditions. These six phases are parameterized by displacements along parallel space, upar, and along complementary (perpendicular) space uper with the relation: fn = upar . Qpar + uper . Qper where the index n refers to the six reciprocal vectors pointing to the six vertices of an icosahedron. Because the displacements upar and uper are associated with respectively, phonon and phason modes, they are frequently referred to as displacements along the ‘phonon’ or ‘phason’ variables. The above formulation in term of density waves is identical to the 6D picture, in which the cut space is translated by a uniform translation (upar, uper). Deriving the elasticity relations of the icosahedral phase requires two hypotheses: (1) The free energy of the system is invariant under a uniform increment of upar and uper. (2) An analytic expansion of the free energy is possible. Because we are at equilibrium, the first term in the expansion will be quadratic in the spatial gradient of upar and uper, i.e: F = a|Hupar| 2+b|Huper| 2. We have already commented on the applicability of such hypotheses to real systems. In particular, because of the shape of the atomic surfaces describing the atomic structure of icosahedral phases, a uniform increment of uper results in atomic re-arrangements. In the 2D-1D Fibonacci example of Figure 5.15, this will for instance lead to some LS to SL re-arrangements. The two 1D structures obtained before and after the shift (black and grey lines in Figure 5.15) are undistinguishable: this means that they are not identical, but it is always possible to find a physical space translation that will bring in coincidence any portion of finite length R of the two structures. It is thus reasonable to consider that the hypothesis (1) is fulfilled. The second hypothesis is more questionable and will be discussed in more detail later on. For the time being let us assume that the two hypotheses are fulfilled. The two terms Hupar and Huper are often named ‘phonon’ and ‘phason’ strains as they are coupled to the ‘phonon’ and ‘phason’ variables. The ‘phonon’ strain is the ‘usual’ elastic strain encountered in any solid state medium. The ‘phason strain’ term might be viewed as a rotation of the cut space, away from the orientation defining the quasicrystal. Whereas it is possible to produce physically (by compression or shear)

Epar

Eper L

S

Figure 5.15. Illustration of the 2D embedding of the Fibonacci chain. The Fibonacci chains obtained as cut by the dark and grey lines are indistinguishable.

5.4. Hydrodynamics of icosahedral phases

131

a ‘phonon’ strain, the phason strain is a more hypothetical one and can not be produced directly. Using group theory arguments it can be shown that the elastic free energy is the sum of three terms F = Fpar þ Fper þ Fpar per. The first term is proportional to the squared ‘phonon’ strain. It depends on the two Lame´ coefficients l and m; because of the high symmetry of the icosahedral point group, this term is analogous to what is obtained for an isotropic solid. The second term is proportional to the squared ‘phason’ strain and depends on two phason elastic constants K1 and K2. Finally the third term depends on both phonon and phason strain and on a phonon-phason coupling constant K3. Five elastic constants are thus required to account for the elasticity of icosahedral phases [9,38–40]. Once the elasticity equations are derived it is possible to deduce some general properties on modes and their dispersions, on the equilibrium thermal fluctuations, on the diffuse scattering and the phase stability.

5.4.2.

Phonon and phason modes

Hydrodynamics deals with excitations in the long-wavelength limit. When going from the homogeneous fluid to the quasicrystal, hydrodynamic theory predicts six ‘new’ modes in the long-range ordered quasicrystalline state, corresponding to the translational symmetry breaking along parallel and perpendicular space. Together with conservation laws, this leads to one longitudinal and two transverse acoustic propagative phonon modes (which are only diffusive shear waves in the fluid), a vacancy diffusive mode, and three diffusive ‘phason’ modes. Acoustic phonons have a linear dispersion relation: ! ¼ Vq,

ð5:15Þ

where q is the phonon wavevector, ! its frequency, and V the sound velocity. For a given wavevector q, there are three phonon modes having different polarizations e: a longitudinal (parallel to q) and two transverse ones (orthogonal to q), the latter being degenerated due to the icosahedral symmetry. There are thus two independent elastic phonon constants, the l and m Lame coefficients. The sound velocities are obtained as eigenvalues of the hydrodynamics matrix and reads VL = [(2m+l)/r]1/2 the longitudinal and VT = (m/r)1/2 for the transverse sound velocities, with r the mass density. Phasons are also hydrodynamic modes, but diffusive. In the long-wavelength limit, the ‘dispersion relation’ of a phason mode with wavevector q writes: ! ¼ i Dphason q2 :

ð5:16Þ

Once Eq. (5.16) is plugged into the equation of motion one obtains an exponential time decay exp(–t/) with a characteristic time proportional to q2 or, equivalently, proportional to the square of the phason wavelength: 1= ¼ Dphason q2 ,

ð5:17Þ

132


where Dphason is the ‘phason’ diffusion constant. The diffusion associated with phason fluctuations is a slow dynamical process so that Dphason is believed of the same order of magnitude as for the diffusion of vacancies. The polarization of a phason mode is somewhat more complex to define than in the case of phonons. In the case where the phonon-phason coupling term K3 can be neglected, for each wavevector q in physical space there are three phason modes with 3D polarization vectors eiper (i = 1,3) lying purely in the perpendicular space. The polarization vector is in that case related to the displacements of the atomic surfaces in complementary space and only indirectly to the geometry of atomic jumps themselves (when a precise atomic model is known). Contrary to the phonon case, there is no simple relationship between the direction of q and the polarization vectors in perpendicular space. However, for a wavevector q lying along a 2-fold (or 3-fold, or 5-fold) high symmetry axis in the physical space, one of the phason mode can be shown to have its polarization vector eper lying along a 2-fold (or 3-fold, or 5-fold, respectively) high symmetry axis in perpendicular space [41]. (Remark: in the case where K3 is non-zero, a coupling occurs between phonon and phason modes so that atomic displacements are now induced by both dynamical processes. It leads to 6D polarization vectors with two 3D components defined in the physical and in the complementary space). Another characteristic feature of the phason diffusion constant Dphason is its high spatial anisotropy. More precisely, the value of 1/ phason and polarization vectors eper of a phason mode are proportional to eigenvalues and eigenvectors of the phason hydrodynamic matrix Cper,per(K1,K2,q) which depends on K1 and K2 and on the direction of the phason wavevector q.

5.4.3.

Equilibrium thermal fluctuations and scattered intensity

At a given temperature and at thermodynamic equilibrium there is a distribution of phonon and phason modes in the quasicrystal. These modes lead to changes of atomic positions with respect to the ‘ideal’ (i.e., without any fluctuations) structure, and have thus consequences on the scattered intensity in a diffraction experiment. In general the scattered intensity may be decomposed into two components: SðQÞ ¼ SBragg ðQÞ þ SDiff ðQÞ,

ð5:18Þ

The first term is the Bragg component, which remains a set of delta functions if fluctuations are not diverging, the second term corresponding to the diffuse scattering observed in between Bragg reflections. The calculation of these two terms can be carried out following a procedure analogous to what is done for the calculation of the diffraction pattern of a periodic crystal with phonons at equilibrium: in this case it is well known that the Bragg peak intensity remains a delta function but with an intensity decreased by the Debye-Waller factor, while thermal diffuse scattering (TDS) appears in between the Bragg reflections [42]. In the case of a quasicrystal the same is true for phonons but an additional component arises from phason fluctuations [41,43,44].


133

The Bragg component is related to the Fourier transform of the time and ensemble average of the density < r(R)> , whereas the diffuse scattering component is related to the Fourier transform of the 2-body density correlation function < r(ri)r(rj)> - < r(r)> 2. The Bragg component is computed in considering that the average structure is obtained by the convolution of Gaussian fluctuations with the ‘ideal’ (i.e., without fluctuations) decorated lattice. In the case of quasicrystals, the Bragg component is easily calculated in the high-dimensional periodic lattice decorated by atomic surfaces. Phonon fluctuations will lead to a Gaussian broadening of the atomic surfaces along the parallel direction, with a mean squared amplitude < u2par> . Phason fluctuations lead to a Gaussian broadening in the perpendicular direction with a mean squared amplitude < u2per> . Indeed a phason fluctuation may be described equivalently either by an undulation of the cut space or by a fluctuation uper(R) of atomic surfaces along the perpendicular direction. The Bragg peak intensity measured on a point of the reciprocal lattice with a vector Qpar is givem by: SBragg ðQpar Þ ¼ Sideal ðQpar Þ: expð < upar 2 > Qpar 2 Þ: expð < uper 2 > Qper 2 Þ,

ð5:19Þ

where Sideal(Qpar) is the Bragg component of the ideal quasicrystal without any fluctuations and where we have neglected the phonon-phason coupling term. The values of < u2par> and < u2per> can be computed from the hydrodynamic matrix C(q). When the phonon–phason coupling term is neglected the matrix C(q) is block diagonal with two 3 3 matrices acting on the parallel and perpendicular component of the reciprocal vector Q and noted Cpar,par(q) and Cper,per(q) (see Annex). D E u2par ¼

1 3ð2Þ3

Zqmax 3 Tr½C1 par,par ðqÞ d q,

ð5:20Þ

0

The value of < u2par> is obtained as the integral, in which the summation is carried out up to a cut-off wavevector qmax which is of order 2/d where d is equal to several first neighbour interatomic distances. The same relation is obtained for < u2per> by replacing the par indices by per indices in the above relation. An order of magnitude of this mean square displacement can be given if we note that C(q) is of the order Kq2/kBT, where K is the phonon (phason) elastic constant, kB is the Boltzmann constant and T the equilibrium temperature: D E u2par

Zqmax

1 ð2Þ

3

kB T qmax kB T , 4q2 dq ¼ Kq2 22 K

ð5:21Þ

0

and a similar relation for phason fluctuations. From this result we conclude that the parallel and perpendicular mean squared fluctuations remain bounded in a 3D quasicrystal when phonon and phason fluctuations are at equilibrium and hence the quasiperiodic long range order is preserved. (Note that this would not be the case in 1 and 2 dimensions, where

134


fluctuations diverge, and no long range order exists. It is a similar result as the one obtained for periodic crystals). Thus Bragg peaks of icosahedral quasicrystals remain delta functions with their intensity reduced by both a ‘phonon’ and a ‘phason’ Debye-Waller term. The diffuse scattering measured at the point Qpar+q is also computed using the hydrodynamical matrix C. We obtain: ð5:22Þ SDiff ðQpar þ qÞ ¼ SBragg ðQpar Þ < Q6D C1 ðqÞQ6D > where the Bragg peak position has coordinates Q6D = (Qpar, Qper) and the measurement is carried out at the position Qpar+q. In the case where the K3 phononphason coupling term can be neglected this expression can be written as: SDiff ðQpar þ qÞ ¼ SBragg ðQpar Þ < Qpar Cpar,par 1 ðqÞQpar > þ SBragg ðQpar Þ < Qper Cper,per 1 ðqÞQper >: ð5:23Þ The first term of this expression corresponds to the usual thermal diffuse scattering (TDS), whereas the second term corresponds to ‘phason’ diffuse scattering. The TDS component is isotropic and presents the same shape around all Bragg peaks: it is an ellipsoid elongated in the transverse direction. The well known characteristics of the TDS component are summarized as follows. Along a given direction a Bragg reflection, the TDS intensity decays as q2. The term q from 1 < Qpar Cpar,par ðqÞQpar > induces a selection rule as a function of the relative orientation of q (or e) and Qpar: for instance if the two vectors are parallel, only longitudinal phonons will contribute to the signal, whereas only transverse phonons contribute when the vectors are orthogonal. In these two directions the TDS intensity is inversely proportional to the longitudinal and transverse squared sound velocities V2L and V2T, and since VL is larger than VT the TDS intensity is smaller along a direction collinear with Qpar than for a direction perpendicular to Qpar , giving rise to a characteristic ellipsoidal intensity distribution around each Bragg reflection. This is illustrated Figure 5.16 (left panel). Finally the TDS intensity is proportional to the Bragg peak intensity, and to Q2par. The TDS is better measured at high Qpar and around strong Bragg peaks. The same reasoning can be followed up for the phason part of the diffuse scattering. When measured along a given direction q from the a Bragg reflection, diffuse scattering intensity decays as q2. The term < Qper Cper,per 1 ðqÞQper > contains the product between the matrix Cper,per 1 ðqÞ and the perpendicular component of the Bragg scattering vector Qper, leading to a ‘selection rule’ as a function of the relative orientation of ePer (the polarization of the phason mode) and Qper. Different phason polarizations are thus enhanced in the measured diffuse scattering signal for different q and Qper orientations. The term which drives the anisotropy of the signal is the ratio K2/K1 whose value lies between –0.75 and +0.75 (see below). As a consequence, the phason diffuse scattering intensity can display strong anisotropies, even for reflections which have similar Qpar directions, if their Qper components are different. This is illustrated Figure 5.16 (right panel), where the phason part of the diffuse scattering has been simulated in a 2-fold scattering plane as it would appear

135


2.5

Phason contribution K2/K1 = –0.6

Qy [2Pi/a unit]

Qy [2Pi/a unit]

TDS contribution

1.0

1.0

2.5

Qx [2Pi/a unit]

2.5

1.0

1.0

2.5

Qx [2Pi/a unit]

Figure 5.16. Comparison between the phonon (TDS, left panel) and the phason (right panel) contributions to the diffuse scattering as deduced from elasticity theory. The shape of the intensity distribution of the TDS contribution is similar around every Bragg peak, whereas strong anisotropies are observed for the phason contribution. The simulation of the phason contribution has been carried out with a ratio K2/K1 of –0.6.

around a few Bragg reflections of an icosahedral quasicrystal, for a K2/K1 ratio equal to –0.6. The two reflections on the 5-fold axis and nearby show a diffuse scattering anisotropy having elongations almost orthogonal to each other, as a result of their Qper components being almost orthogonal: this is a characteristic signature of phason diffuse scattering very different from TDS (Figure 5.16, left panel). Finally, the phason diffuse scattering intensity scales as the Bragg peak intensity and as Q2per. This relation can be easily checked by measuring the diffuse scattering around several reflections lying along the same high symmetry axis. As a conclusion, and as illustrated on Figure 5.16, diffuse scattering measurement is a very efficient tool to evidence and characterize long-wavelength phason fluctuations.

5.4.4.

Hydrodynamic and thermodynamic instabilities

As already explained above, the shape of the phason diffuse scattering depends on the ratio K2/K1 when the K3 coupling constant is neglected. A detailed study of the hydrodynamic matrix Cper,per (Eqs. (5.20) and (5.23)) shows that for a given set of K1 and K2 phason elastic constants, K3 = 0, the eigenvalue of the phason elastic tensor is minimum when q points along one of the high symmetry directions. By varying the values of the K1 and K2 elastic constants, conditions on the value of the ratio K2/K1 can be found, for which the eigenvalue is zero [41,44]. These K2/K1 values correspond to the so called hydrodynamic instability: in this case phason fluctuations (Eq. (5.20)) diverge and the quasiperiodic long range order is destroyed. The hydrodynamic instability occurs for a ratio K2/K1 = –0.75 or +0.75, so that the ratio K2/K1 should lie in between these bounds. When K2/K1 approaches this hydrodynamic instability, the structure factor is dominated by the softening of this

136


phason mode leading to streaks along the wavevector q of that specific mode. When K2/K1 is close to –0.75, the soft phason mode has wavevector along the 3-fold direction and a polarisation in perpendicular space along a 3-fold direction: as a result the diffuse scattering displays streaks along directions parallel to a 3-fold axis. When K2/K1 is close to 0.75, the soft phason modes have wavevectors along the 5-fold direction and a polarisation along a 5-fold direction in perpendicular space: in this case the diffuse scattering displays streaks along directions parallel to a 5-fold axis. Finally, when K3 is non negligible, one gets an instability along 2-fold directions. However these hydrodynamic instabilities can not be reached because a first-order transition occurs before. Indeed, in the framework of the Landau theory, the stability of the quasicrystal imposes constraints on the value of the elastic constants. In particular the quasicrystal must be stable with respect to all uniform phason strains of infinitesimal or finite amplitude. A uniform phason strain of finite amplitude transforms the icosahedral quasicrystal into a crystalline approximant, or possibly into an aperiodic crystal with a lower symmetry. The possible phason strains and the lower symmetries are imposed by Landau and group theory. In the case of the icosahedral quasicrystal the two most important symmetry breakings are transitions toward a rhombohedral or a pentagonal phase, with linear phason strains directed along 3-fold or 5-fold directions in perpendicular space [45,46]. These are first order transitions, whose stability criterion is given by K2/K1 > –0.6 and K2/K1 < 0.75 when K3 is neglected. When including K3 it has been shown that the first-order transition always arises before the hydrodynamic instability [41,44].

5.4.5.

Random tiling and matching rule models

So far we have only discussed the hydrodynamic theory, which is a continuum theory, and did not consider much the microscopic origin or the energetics of phason fluctuations. This is particularly important however if we want to discuss the validity of hypothesis (2) in section IV.1, namely that an analytic expansion of the free energy is possible with F = a|Hupar| 2+b|Huper| 2. We may ask for instance ‘what is the restoring force’ which brings back the system to the quasicrystalline state when a phason strain is applied? What are the interatomic forces which play a role in this process? Of course answering these questions requires modeling at the atomic scale. We present in the following the two ‘toy’models which are frequently opposed: the random tiling and the matching rule tiling.

5.4.5.1.

Random tiling model

The model for which the most extensive theoretical and numerical studies have been carried out is the random tiling model [10,47]. The main assumption of this model is that the quasicrystalline structure can be described by a decorated quasiperiodic tiling. The simplest one is the 3D Penrose or Amman tiling which is constructed with two rhombi, a fat and a thin one. This tiling is then decorated by atomic clusters, so that the edge length is at least of the order of the inter-cluster distance. A more complex tiling scheme has been proposed by Henley which he names the canonical


137

cell tiling. Whether a real quasicrystal can be described by a decorated tiling is however still an open question. Indeed, if most of the recent icosahedral phase models can be mapped onto decorated tilings, it has not been proven yet that the decoration scheme is uniquely defined: it seems that a certain amount of chemical disorder has to be introduced which changes the decoration from one tile to the other (See the chapter by Yamamoto in this volume). This being said let us consider what has been obtained so far for random tiling models. The effect of a local distortion of the cut space on the resulting 3D structure depends on the geometry of the atomic surfaces. We have seen that in the case of a Fibonacci sequence this results in permutations LS to SL in the sequence. For the 3D Penrose tiling this results in a change inside a rhombic dodecahedron made of two fat and two thin rhombuses: the two possible sites are related by mirror symmetry along a 2-fold direction (see Figure 5.17), and are called dodecahedron sites hereafter. The simplest 3D Penrose random tiling is obtained by a randomization of each dodecahedron site by a Monte-Carlo move (it has been called maximally random tiling by Henley, corresponding to the high T limit). When performing these moves, new dodecahedron sites will appear and other disappear. In this model, the only energy coming into play is the configurational entropy. It has been shown numerically that the free energy of this model random tiling is minimum in the absence of phason strain and increases quadratically with the amplitude of such strain, thus establishing the applicability of phason elasticity theory [48]. In particular the diffuse scattering intensity can be modeled by using a positive elastic constant ratio K2/K1. An example of a simulated diffraction pattern is shown Figure 5.18: whereas the ideal 3D Penrose tiling only displays Bragg peaks, shown as vertical bars, the random tiling diffraction pattern has both a Bragg and a diffuse scattering component. It can also be observed that most of the weak Braggs reflections of the ideal tiling have disappeared in the diffraction pattern of the random tiling. This is because weak reflections have a large Qper component and have thus their intensity strongly diminished by the perpendicular Debye-Waller factor (Eq. (5.19)). A measure of the ‘degree’ of disorder present in the random tiling model is given by the mean squared perpendicular ‘fluctuations’ of the atomic surfaces < u2per > whose value is of the order of half the average radius of the atomic surface describing the 3D Penrose tiling (a triacontahedron) [48,49]. In this random tiling model, the restoring force is the configurational entropy. It means that the quasicrystalline state has the largest configurational entropy

Figure 5.17. Illustration of the two equivalent sites (or ‘flip’ sites) inside the rhombic dodecahedron of the 3D Penrose tiling. A 3D random Penrose tiling is obtained by a Monte-Carlo randomization of all these dodecahedral sites.

138


3D Penrose tiling I Random Tiling

1

I Ideal Tiling

Intensity

0.1 0.01 0.001 0.0001 10–5 10–6

0

1

2

3

4

5

6

7

Q Figure 5.18. Simulated diffraction pattern along a 2-fold axis for a perfect 3D Penrose tiling (vertical bars) and the corresponding 3D random Penrose tiling (continuous line). Most of the weak peaks have disappeared whereas a diffuse scattering contribution occurs between the Bragg reflections.

and any homogeneous or long-wavelength phason distortion driving the system away from the quasicrystalline state (as for instance toward a periodic approximant) will have smaller entropy: roughly speaking, the density of dodecahedron sites is maximal in the quasicrystal allowing more flexibility for configurational rearrangements. It might seem surprising at first sight that the elasticity theory applies to this model. Indeed the elementary process involved in this randomization is discontinuous, and the geometry of the high dimensional description is rather analogous to the Frankel-Kontorova model in the regime of strong coupling between the atomic chain and the substrate, in which the free energy is non-analytic. However this comparison may be misleading, because there is no direct link between the high dimensional random tiling picture and its entropy variation with strains, contrary to the case of the Frankel-Kontorova model where the modulation function directly describes the coupling between the two subsystems. In the case of the random tiling quasicrystal the link with the hydrodynamics is done via a coarse graining of the fluctuating ‘surface’ representing the cut space (see Henley’s review).

5.4.5.2.

Matching rules model

The applicability of hydrodynamic theory has also been numerically studied on energetically stabilized quasicrystals or ‘matching rules’ model [50]. This model assumes that the quasicrystal is a ground state at 0 K. A way to implement this hypothesis on a tiling model is to consider the ‘matching rules’, defined as the set of local environments which uniquely characterize the quasicrystal. In this model any violation of the matching rule has an energy cost. By considering a simple Hamiltonian on a 3D Penrose tiling model, Dotera and Steinhardt [11] succeeded in favouring a quasicrystalline ground state at 0 K. As the temperature is increased,


139

flipping inside the dodecahedron sites (Figure 5.17) becomes possible and produces a matching rule mismatch. The temperature study of this simulation showed that hydrodynamic theory only applies above a critical temperature Tc [11]. Although not proven, it has been conjectured that below Tc the free energy is non-analytic with respect to phason strain and would vary as |Huper| instead of |Huper|2. This transition from the non analytic to the analytic regime bears some similarities with the lock-unlocked transition studied by Aubry in the Frankel-Kontorova model. But again, contrary to the case of the Frankel-Kontorova model, there is no direct connection between the shape of the atomic surfaces in the quasicrystal and the analyticity breaking. The analyticity breaking is most likely related to the spatial distribution of dodecahedron sites inside the tiling as the temperature is raised, which might be more ‘homegeneous and continuous’ above a given temperature. Indeed, the associated perpendicular coordinate and its variation as a function of site position represents a surface in high-dimensional space, which would be continuous above a given T, allowing all long wavelength phason fluctuations to develop. In any case, for temperature larger than Tc, the situation is similar to the one of the random tiling model and the ‘restoring force’ with respect to phason strain comes from the tile configuration entropy. It should be kept in mind that the tile configuration entropy is not the only possible ‘restoring force’. Indeed the only requirement for hydrodynamic theory to apply concerns the free energy F = U – TS. Koschella et al. have recently carried out simulations on a 2D binary tiling using pair potential interactions that stabilise a quasicrystalline ground state at 0 K [51,52]. They have investigated the variation of the energy (U) of the system and found that U increases quadratically with phason strain. This can be understood if we note that a phason strain will imply a change both in the density of the different local environments and in the local chemical composition of the system, which in turn will change the total energy of the system. In this simulation, the ‘restoring’ force is strongly related to the ‘chemistry’ of the system. A more realistic model with an adapted Hamiltonism was also proposed by Mihalkovic and Henley [53] (see hereafter). We will see in the following that there are strong experimental evidences pointing towards the interpretation of phason modes in term of the system chemistry.

5.4.6.

Temperature dependence of elastic constants

We turn now to the predicted T dependence of the phason elastic constants as a function of T and its consequences on the diffuse scattering. We compare the two simple models which have been put forward to explain the stability of quasicrystals: the energy [11] or the entropy [10] stabilised models, which predict a quite different T dependence for the phason elastic constants. In the first model, the quasicrystal is considered as being a ground state at 0 K and is energetically stabilized. Whether this can be achieved using only finite range interatomic interactions or by using infinite range interactions is still an open question [54]. Since phason fluctuations are increasing with temperature, we should thus observe an increase of the diffuse scattering when the temperature increases.

140


The shape of the diffuse scattering provides a clue on the applicability of the hydrodynamic theory. We already discussed the 3D ‘matching rules’ Penrose model which presents a locked-unlocked phase transition [11]. Although the diffuse scattering intensity has not been computed below Tc, it would most likely resemble a broad Lorentzian, whose width would decrease as the temperature is raised, as phason fluctuations build up. Above Tc we would recover the 1/q2 decay of the diffuse scattering, whose intensity should increase proportionally to kBT. The entropy stabilized model, has been developed in the random tiling scenario. It postulates that the quasicrystal is stabilized by the configurational entropy which is essentially associated to short wavelength phason fluctuations. In this scheme the quasicrystal is only stable at high temperature and undergoes a phase transition towards a crystalline phase as the temperature is lowered. This transition is accompanied by some degree of pretransitional phason softening and hence the first order phase transition should be preceded by changes in the phason elastic constants and corresponding changes in the diffuse scattering. The soft phason branch to be considered depends on the nature of the first order transition and the corresponding phason strain involved: for instance we expect a soft phason branch along 3-fold axes for an icosahedral to tetrahedral transition. This T dependence has been verified on a simple canonical cell model: large variations of the elastic constants are observed in the simulation when going from the high to the low T region, the ratio K2/K1 going from positive to negative in the low T region, the 0 K ground state being a crystalline phase [53]. To summarize, the two models can be distinguished by the relative weight they give to the energy and entropy terms stabilising the quasicrystal. Both models predict a high temperature region, where phason fluctuations are hydrodynamic and the diffuse scattering can be computed. However the two models predict a different temperature dependence of the diffuse scattering. In the case of the energetically stabilised scenario, the diffuse scattering should increase as the temperature is increased and phasons are hydrodynamic only above a critical temperature Tc. In the random tiling scenario, there should be a low temperature crystalline phase, the quasicrystal being stable only in the high T region. Starting from the high temperature quasicrystalline phase, and going down in temperature, soft phason modes should appear leading to an increase of the diffuse scattering.

5.4.7.

‘Phason’ jump, phason strain, phason modes

Up to now we have only considered the case of equilibrium long wavelength phason fluctuations in a quasicrystal. There are other experimental situations in which the term ‘phasons’ is also used. Unfortunately the terminology used in the quasicrystalline community is often rather confusing. We give a brief definition of each one [55]. The local change of configuration equivalent to the LS to SL change is usually named ‘phason’ jump (or tile flip, or atomic jump): it is really a local process involving a single or a few atoms. A uniform phason strain will break the icosahedral symmetry of the structure. For appropriate values of the strain it results in a crystalline periodic phase. The effect on the diffraction pattern is easily calculated and gives rise to peak shifts, whose magnitude is proportional to Qper.

141


There are situations for which there is a distribution of uniform phason strains in the quasicrystal: this is sometime called a ‘random’ phason strain or simply phason strain. In this case small domains, each one with a different uniform phason strain, build up the quasicrystal. Because there is no correlation from one domain to the other, the perpendicular space fluctuation of the atomic surfaces diverges as R goes to infinity, and the quasiperiodic long range order is destroyed. Instead of delta peaks the diffraction pattern consists of broadened Bragg reflections, whose width scales linearly with Qper [56]. Finally it might occur that a single, finitewavelength, phason mode is frozen in, leading to a modulated quasicrystal whose diffraction pattern presents icosahedral Bragg reflections and additional satellite reflections [57]. Some of these realizations are schematically represented in Figure 5.19. What is really surprising is that all these transformations which are defined in high dimensional space have been observed experimentally. We turn now to the experimental study of long wavelength phason fluctuations and its interpretation in the framework of hydrodynamic theory. Most of the 25

25

'E

20

20

R par

15

Ry

15

Ry 10

R par

ε

10

R per

5

R per

5 )

0 h(R par

0

0

'

ine

a om

Rx

0

10

20

30

Rx

'd

(a)

(b) 25 20

R par E

Ry

15 10

R per

5 0

0

10

20

30

Rx (c) Figure 5.19. Schematic illustration of a quasicrystal with a random phason strain distribution (a), longwavelength phason fluctuations (b) and with a single sine wave modulation (c). In the case of random phason strain distribution fluctuations are diverging, and Bragg peak are no longer d functions, whereas long range quasiperiodic order is preserved in the two other cases.

142


experimental results were obtained in the i-AlPdMn phase which we discuss in the next section.

5.5.

Phason modes in the icosahedral AlPdMn quasicrystal

High quality, centimeter size, single grains of the i-AlPdMn quasicrystal can be grown from the melt by the Czokralski method [58,59]. This has allowed a large number of experiments to be carried out, including in situ temperature studies of the diffuse scattering and of the dynamics of phason fluctuations. It is the only system up to now for which there is a complete experimental study of phason fluctuations. The high structural quality of the i-AlPdMn quasicrystalline phase is best evidenced by its diffraction pattern. First, the Bragg peaks widths are almost resolution limited when measured with a high resolution setting. Although there is a slight linear dependence of the Bragg peak width with Qper, indicative of a distribution of uniform phason strains, it can be significantly reduced by sample annealing, as shown in Figure 5.20. In this figure Bragg peaks display a spiky structure, because a coherent beam has been used. After annealing, the strong 2-fold Bragg reflection 20/32 (where we use N and M indices following the convention in [60]) is resolution limited with a longitudinal full width at half maximum of 104 A˚1 and a rocking curve width of 0.005 [61,62]. When measuring Bragg peaks with a larger Qper component (bottom panel) a small residual broadening is observed, whose contribution can be written as G = bQper with b = 0.0005. This is by far the smallest value observed for all studied icosahedral phases. Another indication of the high

Phason component Qp Qp er er =1 =0 .5 .28

Annealing

(a)

(c)

As

An

nea

gro

led

wn (b)

(d)

Figure 5.20. Intensity distribution in a 2D image of two 2-fold Bragg reflections in an as grown (a, c) and annealed (b, d) single grain sample of the i-AlPdMn phase. Each image is a slice in reciprocal space of approximate size 5.103 5.103 A˚2. The spiky structure of the broad Bragg reflection is due to the use of coherent X-ray. The as grown sample shows a Bragg peak broadening proportional to Qper. In contrast, the reflection in (b) displays only a single speckle peak, indicating a much higher structural quality of the annealed sample.

5.5. Phason modes in the icosahedral AlPdMn quasicrystal

143

structural quality of the i-AlPdMn quasicrystal is the occurrence of dynamical effects in scattering experiments: for instance the anomalous transmission Borman effect has been observed [63] and X-ray standing waves [64] could be set up, both indicative of a breakdown of the kinematical X-ray scattering approximation. The atomic structure is also relatively well understood. The most accurate model has been proposed by Yamamoto [65], using complex atomic surfaces (see Chapter 2 in this book) in an improvement of the crude spherical model initially proposed by Boudard et al. [66]. The structure contains atomic clusters which are quasiperiodically packed. Two different topological clusters have been identified: a pseudo-Mackay cluster, 0.96 nm in diameter, and a smaller one of the Bergman dodecahedron type. The Mackay cluster occurs with two main chemical decorations: the outer shell of the first one consists of a Mn icosahedron and an Al icosidodecahedron whereas the second one has an Al icosahedron and a Pd icosidodecahedron. An alternative description can be given in term of packing of dense atomic planes [66]: in the case of the i-AlPdMn phase the densest planes are packed perpendicularly to a 5-fold axis. A portion of such a plane is shown Figure 5.21: traces of a few Mackay clusters are highlighted with circles. A local distortion of the cut space leads to atomic re-arrangements analogous to the SL to LS permutation in the Fibonacci chain. Such a re-arrangement is shown in the Figure 5.2, right panel: the two sites at the intersection of the two circles are too close to be occupied simultaneously as this would lead to too short interatomic distances of 1 A˚. Only one of the two positions is thus occupied in the ideal quasicrystal: however, as can be seen in the figure, the two positions have similar local environments, so that the total energy of the system is almost identical for the two positions.

ZA5 = 4.562(Å) 40

0

40 –40

0

40(Å)

[τ, 1, 0]// Figure 5.21. Left panel: example of an atomic plane, perpendicular to a 5-fold axis, as deduced from the structural model (Boudard et al.) of the i-AlPdMn phase. Traces of the Mackay clusters are visible as 10-fold rings, some of which are highlighted (after [66]). Right panel: enlarged part of the left panel. The two sites at the intersection of the two circles are too close to be occupied simultaneously. They have however almost the same local environment, and are possible ‘hopping’ sites, analogous to the dodecahedral sites of Figure 5.17.

144

5.5.1.


Local atomic ‘hopping’

Looking at the previous figure, one can guess that there should be a temperature range for which the atoms can ‘hop’ easily from one position to the other. Such temperature studies are best achieved by quasielastic incoherent neutron scattering experiments, a technique which is mainly sensitive to local atomic motions. A complete temperature study has been performed by Lyonnard et al. using both powder and single grains of the i-AlPdMn phase [67,68]. Above 600C, a quasielastic signal has been observed. From the temperature dependence of the width and intensity of this signal, Lyonnard et al. concluded that the hopping process is an assisted one: indeed the width of the quasielastic signal, and thus the jump rate associated with the hopping process, does not change as the temperature is increased. Instead, it is the intensity of the quasielastic signal which displays an Arrhenius behavior as the temperature increases. A more detailed study on a single grain quasicrystal showed that the quasielastic signal displays anisotropies in reciprocal space: from these measurements Coddens et al. [69] deduced that the hopping process occurs mainly along 3-fold axes over a distance of 3.8 A˚. The microscopic description of the associated assisting process is not yet understood. Note that a single jump over a distance of 3.8 A˚ is highly unlikely and that the 1 A˚ jump, which should be the most frequent from the structural model, has not been observed, probably because it does not fall in the (Q,E) ‘window’ of the experiment.

5.5.2. Phason modes and room temperature diffuse scattering measurements As explained in the previous section, an efficient way to evidence long wavelength phason modes is by means of diffuse scattering measurements. This experimental approach is completely analogous to the one used for the study of phonons in crystals by Laval, Olmer, Currien and Walker in the 1940s and 1950s [70–74]. Data analysis is much more simplified if the TDS contribution can be suppressed. This can be achieved by using inelastic neutron scattering and energy analysis on a three-axis spectrometer. With the proper settings, the measured elastic signal only contains the contribution from static distortions and with ‘slow’ time dependences, the phonon contribution is eliminated. Measurements have thus been carried out around a few Bragg reflections in the neutron diffraction pattern shown in Figure 5.22 [75]. In the case of elastic (phononlike) distortions, the diffuse intensity is expected to display the same shape around all Bragg reflections, because the icosahedral phase is elastically isotropic. This will not be the case if phason fluctuations are present: depending on the ratio of K2/K1, different shape anisotropies are expected around the different Bragg reflections as already shown in Section 5.4.3. Since the phason diffuse scattering scales as Q2per (Eq. (5.23)), reflections having different perpendicular components were studied. They are labelled by their N/M indices, which is a short hand notation for the full 6D indexing. The 18/29, 15/24, and 20/32X reflections lie on a 5-, 3-, and 2-fold axis, respectively (Figure 5.22).

145


A5

Qy (Å–1)

4

18/29 16/24

2

A3 15/24 A 20/32

0

0

2 Qx (Å–1)

4

Figure 5.22. Neutron scattering intensity distribution in a 2-fold plane for the i-AlPdMn phase. The area of the dots is proportional to the intensity of the reflections. A few of them are indexed with their N/M indices of [60].

The results of these measurements are shown Figure 5.23, where iso-intensity contours are shown. The same intensity range has been selected for all four reflections, so that the different panels can be directly compared [75]. As can be seen, the intensity distribution displays strong anisotropies. Several observations point

(c)

A3

A3

(d) A3

1.6

1.8

2

(f)

–0.2

2.4

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0

1

2

20/32X 0.2

(b)

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1.2

(a) 2.6

15/24

16/24

18/29

2.4

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2.8

3

3.2

–1

Qx (Å ) Figure 5.23. Top panel (a-d): isointensity contour plots of the diffuse scattering signal measured by elastic neutron scattering around different Bragg peaks in the i-AlPdMn phase at room temperature. The indices are given in the N/M notation and refer to Figure 5.22. All contour lines are on the same scale. Bottom panel (e–h): simulation of the diffuse scattering using as a single parameter the K2/K1 ratio set to –0.52 (from [75]).

146


toward a phason contribution to this diffuse scattering. First the 16/24 and 15/24 reflections have almost the same Bragg peak intensities, but different values of the Qper components of the scattering vector, the 16/24 one being about three times larger than the 15/24. As can be observed in Figure 5.23(b, c), there is a much larger diffuse scattering intensity around the 16/24 reflection as would be expected from Eq. (5.23). Second, the 18/29 and 16/24 reflections, are very close to each other in reciprocal space: any elastic (phonon-like) distortion would lead to a very similar shape of the diffuse scattering whereas the experiment displays almost orthogonal anisotropies. This can be understood if we look at the perpendicular component of these two reflections: they are almost orthogonal, and will thus lead to a very different scalar product (Qper eper) ‘selection rule’ in Eq. (5.23). Finally, measuring two collinear reflections, it is possible to show that the intensity of the diffuse scattering decays as q2 and scales with the product IBragg Q2per. These results are confirmed by a computation of the diffuse scattering using Eq. (5.23). If we consider that the diffuse scattering is due mainly to phason fluctuations and if we neglect the coupling constant K3, then the shape anisotropy depends only on the ratio K2/K1. Using the measured Bragg peak intensities it is then possible to simulate the diffuse scattering. This is what has been carried out in Figure 5.23 (e–h), using a ratio K2/K1 = –0.52 [75]. As can be seen in the figure there is a good agreement between simulation and experiment. A quantitative agreement was also achieved by comparing the intensity decay observed along different directions and various Bragg peaks [76]. If the diffuse scattering is measured on an absolute scale it is also possible to get the individual values of the two elastic constants K1 and K2. This kind of measurement is completely analogous to the determination of elastic constants or phonon dispersions using X-ray diffuse scattering. From the experimental point of view, an absolute scale measurement is more demanding, so that it has been only carried out in a few systems. Using both X-ray and neutron diffraction leads to similar results, i.e., to values of K1/kT and K2/kT equal to 0.1 and –0.052 atom1, respectively [76]. An example of the results obtained from absolute X-ray diffuse scattering measurements along a 2-fold axis is shown Figure 5.24. The different contributions to the signal are indicated, namely the phason diffuse scattering component (only the contributions from the strong 20/32 and 8/12 reflections are shown), the Compton scattering and the TDS contribution originating from the 20/32 reflection. It is clear from this figure that the contribution from TDS in X-ray measurements is indeed much smaller than the phason one, which makes the X-ray data easy to analyse in terms of phason diffuse scattering. However this cannot be inferred a priori, and neutron elastic measurements or absolute scale X-ray measurements are crucial for the characterisation of the phason diffuse scattering. The 1/q2 decay of the diffuse scattering is also well reproduced, as shown in Figure 5.25 which displays a log-log plot of the diffuse scattering measured around the 20/32 reflection: although not perfect, the 1/q2 decay extends over a rather wide wavevector range, roughly from 0.03 to 0.6 A˚1, corresponding to a phason mode wavelength between 200 and 10 A˚. In fact high-resolution measurements showed that the lower q bound for which the 1/q2 decay is observed is 0.01 A˚1, corresponding to about 600 A˚. Absolute scale measurements of the diffuse scattering are also important for any quantitative comparison of the amount of diffuse scattering present in different samples.

147


2-fold Q-scan

i-AlPdMn I (e.u./Å3) 20/32 8/12 summ TDS 20/32 Compton

I (e.u./Å3)

107 105 1000

10

0.1

1

1.5

2

2.5

3

3.5

Q (Å–1) Figure 5.24. X-ray diffraction pattern measured along a 2-fold direction in the i-AlPdMn phase. The intensity is set on an absolute scale and the various contributions to the signal are highlighted (Compton and TDS parts). The solid line is the result of the simulation for the sum of all contributions: the phason contribution dominates the diffuse scattering intensity (from [76]).

Intensity (e.u./Å3)

1000

100

10

1 0.1

1

q (2π /a6D unit) Figure 5.25. Log-log plot of the diffuse scattering intensity as a function of q. The dependence is almost linear with a slope equal to –2, as shown by the solid line and as expected for a 1/q2 decay of the diffuse intensity.

In particular it was shown that the amount of diffuse scattering is independent of the sample annealing treatment for i-AlPdMn single grains [76]. As grown samples have a larger Qper dependence of their Bragg peaks width when compared to the annealed ones (Figure 5.20), indicating that the amount of ‘phason strain’ is larger in the as grown samples. Yet both samples have exactly the same amount of diffuse scattering which demonstrates that the observed phason fluctuations are thermal equilibrium fluctuations and are not related to a distribution of structural defects (as dislocations for instance) giving rise to the broadening. As we will show further with

148

Chapter 5. Phason Modes in Aperiodic Crystals 0.5

1.0

1.5

2.0

2.5

3.0

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–2.0 0.0

0.5

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1.5

2.0

Qx [2Pi /a units]

2.5

3.0

Qy [2Pi/a units]

Qz [2Pi/a units]

0.0 2.0

–2.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Qx [2Pi /a units]

Figure 5.26. Diffuse scattering in the i-AlPdMn phase. Left panel: diffuse scattering measured around the strong 5-fold 18/29 reflection, in a plane perpendicular to the 5-fold axis. Measurements were performed on a rotating anode (Cu target) four circle diffractometer. Right panel: simulation of the diffuse scattering taking into account the contribution from in and out of plane Bragg reflections. Both the shape and the intensity distribution of the diffuse scattering are well reproduced as for instance the lobe-like distribution seen on the outer part of the diffraction pattern.

the temperature dependence of the diffuse scattering intensity, it is because phason fluctuations are indeed an intrinsic property of the i-AlPdMn quasicrystal. Phason fluctuations also allow to explain the observed diffuse scattering when a wide survey of reciprocal space is carried out. Figure 5.26, displays such a measurement in a diffracting plane orthogonal to a 5-fold axis and passing through the strong 18/29 reflection. The diffraction pattern (left panel), displays characteristic distributions of the intensity with arcs of diffuse scattering for instance, and a 5-fold symmetry. Such an intensity distribution is fairly well reproduced (Figure 5.26, right panel), if the diffuse scattering arising from different Bragg reflections, and in particular out of plane reflections, is added up together. With such a calculation, phason fluctuations can explain 90% of the observed diffuse scattering.

5.5.3.

Temperature dependence of the diffuse scattering

The room temperature measurements do not give any hint on the nature of the microscopic mechanisms associated with phason fluctuations. Indeed, at room temperature, these fluctuations are most likely frozen in. Remembering that the two simple models of the energetically and entropically stabilised quasicrystal predict a different T dependence for the diffuse scattering, in situ temperature studies of the diffuse scattering are of crucial importance. The temperature dependence of the diffuse scattering has been studied by neutron scattering between 25 and 770C, using a three-axis spectrometer to suppress the phonon contribution [77]. This is particularly important, since the mean square

149


(b)

2.2

2.2

1.4

1.6

1.8

(c)

1.4

1.6

1.8

1.4

1.6

1.8

(d) 2.6

A3

A3

2.2

2.2

2.4

2.4

2.6

Qy (Å–1)

A3

2.4

2.6

A3

2.4

2.6

(a)

1.4

1.6

1.8

Qx (Å–1) Figure 5.27. Temperature dependence of the diffuse scattering measured around the 18/29 5-fold reflection at 200C (a) and 770C (b). The evolution is well reproduced by changing the phason elastic constant ratio K2/K1 from –0.52 (c) to –0.4 (d) (from [77]).

atomic displacement due to phonons is roughly proportional to T, so that the TDS contribution might not be negligible in the high T region. Figure 5.27 displays iso-intensity contours of the diffuse scattering measured around the 18/29 reflection at 200C (Figure 5.27(a)) and at 770C (Figure 5.27(b)). Two observations can be made: first there is a clear decrease of the diffuse scattering intensity at high temperature; second the shape of the diffuse scattering is more ‘circular’ at high temperature. The decrease of the diffuse scattering intensity when T is increased has been checked on systematic scans taken around different reflections as a function of the temperature: almost no intensity change is observed up to 500C; from this temperature and up to 770C there is a continuous decrease of the diffuse scattering intensity, by a factor of almost 2 for some directions [77]. In the mean time, the intensity of some weak high Qper reflections increases. All these results can be interpreted if we consider a change of the ratio of the phason elastic constants as T is increased. It would go from K2/K1 = –0.52 at room temperature to K2/K1 = –0.4 at 750C. Such an evolution reproduces quite well the observed diffuse scattering, in particular the more ‘circular’ part of the diffuse scattering in the high T region, as shown on Figure 5.27 (c, d). In fact we can even give a more detailed evolution of the phason elastic constants, since all data are measured on the same scale. Using Eq. (5.23), we can deduce that K2/kBT is T independent and only K1/kBT decreases as T is decreased.

150


We thus have a softening of the K1 phason elastic constant as the temperature is decreased [77]. These results are clearly in agreement with the random tiling scenario. However in this scheme we should observe a transition towards a crystalline phase at low temperature, which is not the case experimentally. The evolution of the phason elastic constants points towards a 3-fold type instability. From the Landau theory of phase transitions this should lead to a phase with D3d symmetry. One possible explanation is that the transition is frozen-in because of kinetics reason. The ground state of the system would be a crystalline phase with tetrahedral symmetry, but because atomic diffusion becomes too slow below 500C, the system is blocked in a metastable icosahedral state with frozen in phason fluctuations. Some hint that this is indeed the correct interpetation is given by a temperature study of the phase transition occurring for icosahedral phases with different Pd/Mn contents. If the Mn concentration in the single grain goes from 9% to 8%, the icosahedral phase is only stable at high temperature. At about 700C, there is a first order transition towards a phase of tetrahedral symmetry of extreme complexity, named F2M. Although the symmetry is D3d, the F2M phase is quasiperiodic [78–80]. It can be described as a super-structure on the icosahedral phase, with satellite reflections along directions parallel to 3-fold axes. An example of the diffraction pattern measured around a strong high temperature 2-fold reflection is shown Figure 5.28. In the high-temperature phase only the reflection labelled i-ref

sf2 3.00

2s1

dir // A3

s1

s1 sf2 2.95

Qy (2π / ap unit)

(220) s1 s1 (220)

i-ref

2.90

sf2 s1 2.85

0.50

0.55

0.60

0.65

Qx (2π / ap unit) 5

10

log (Intensity)

Figure 5.28. Diffraction pattern of the F2M phase measured around the 20/32 2-fold reflection, labelled i refl. It is the only Bragg peak that survives in the high temperature icosahedral phase. The supplementary satellite reflections, characteristic of the F2M phase, are labelled as S1 and SF2 (from [113]).


151

survives. The position of the reflections S1 and SF2 with respect to the main reflection can be expressed as rational values of reciprocal lattice vectors of the parent high-temperature phase, having a wavevector qpar of 0.0369 and 0.120 A˚1, respectively. A detailed atomic description of this F2M phase is not yet achieved, but the intensity distribution of the supplementary S1 superstructure reflections is relatively well reproduced by considering a cosine wave distortion of the high temperature icosahedral phase with a wavevector pointing along a 3-fold axis in the parallel (physical) space and a polarization along a 3-fold axis in the perpendicular (phason) space. From the measured intensity it can be shown that the amplitude Uper of this phason wave is of the order of 0.55 A˚, the wavelength of the modulation being equal to 170 A˚ [78]. This F2M phase may thus be considered as resulting from a phason driven phase transition, and would be the stable low temperature phase. However, we should keep in mind the complexity of the chemistry, of the phase diagram and of the stabilisation mechanism of quasicrystals. The two models (the energetic and entropic ones) are certainly only first order approximations. From experimental results, it is clear that the entropy term has an important weight on the stability of the i-AlPdMn quasicrystal since the diffuse scattering intensity decreases as the temperature is raised. However, dramatic changes in the phase diagram with small chemical composition changes indicate that both the configurational geometric entropy and the chemical potential (which contains an energy and an entropy term) are playing a role. The observed hydrodynamic behaviour of phason fluctuations would thus be the result of a squared gradient behaviour of the free energy with an entropy part and an energetic part. As we have shown, the observed diffuse scattering can be explained by long wavelength pre-transitional phason fluctuations. The hydrodynamic theory does not give a microscopic interpretation of these long wavelength phason fluctuations. Some hint on the resulting 3D structure can be given by taking the ‘undulating cut’ picture and consider that the 3D structure is obtained as the convolution of an ideal quasicrystal with perpendicular fluctuations. The effect on the resulting 3D structure will be both some partial site occupancy (as the one illustrated in Figure 5.21), but also chemical disorder at specific location: this can be for instance Mn/Pd disorder or Al/Pd disorder, the term disorder being understood by comparison with an ‘ideal’ quasicrystal. Because of the geometry of the high dimensional model, the disorder will not be randomly distributed on the whole structure, but will occur on specific sites.

5.5.4.

Dynamics of phason modes

So far we have only considered the static aspect of phason modes in quasicrystals. As explained in the introduction, phason modes are collective diffusive modes. It means that a phason fluctuation has exponential time decay, with a characteristic time scale t going like q2 where q is the phason wavevector (Eqs. (5.16, 5.17)). Expressed in real space, this is equivalent to a characteristic timescale going like l2. Long-wavelength phason fluctuations are expected to be too ‘slow’, to be visible in the neutron timescale ‘window’. Their dynamical study has thus been carried out

152


using coherent X-ray and photon correlation spectroscopy techniques. These techniques allows one to measure slow dynamics (103 to 1000 s) at the atomic scale [81]. When a coherent X-ray beam is used to measure the diffuse scattering, the q dependence of the intensity does not show any longer a smooth variation (resulting from an ensemble average) but rather displays strong intensity fluctuations called speckles pattern. These speckles are due to constructive and destructive interferences between the scattered waves from the illuminated sample volume. This process is analogous to what is obtained when a disordered medium is illuminated with a Laser light. Reasonable intensity coherent X-ray beams can be obtained at third generation synchrotron sources like ESRF, using high brilliance undulator sources. For instance, using the focusing optics of the ID20 beam line (ESRF), a flux of several 109 photons/s is obtained through a 10 mm pinhole located just before the sample [61]. The diffuse scattering spectrum is recorded using a directly illuminated CCD camera, located 1.85 m from the sample and acting as a 2D photon detector using a droplet algorithm [82]. With such a set up the partial coherence of the beam b was found equal to 5% and 3% for low- and high-angle reflections, respectively. A typical 2D image of the diffuse scattering recorded close to a Bragg reflection, together with its cross section is shown Figure 5.29. Spikes and large intensity fluctuations are clearly visible. The shape of the diffuse scattering is well reproduced using only phason fluctuations, which ensures that the observed signal is the one which is relevant, i.e., that it is connected to phason modes. A dynamical fluctuation in the atomic distribution r(R) due to phason fluctuations will lead to a different speckle distribution, so that any time evolution of phason fluctuations will result in a time dependence of the speckle pattern. This is best evidenced by computing the intensity correlation function defined as: Fcor ðq; tÞ =hIðQB þ q; t0 Þ IðQB þ q; t0 þ tÞit0 =hIðQB þ q; t0 Þit0 ; 2

ð5:24Þ

where QB is the Bragg peak position and q, the phason wavevector. We have the relation Fcor(q, t) = [1+bg(q, t)], where g(q, t) is the function accounting for the time dependence of a phason mode with wavevector q. The i-AlPdMn sample was placed in a furnace under secondary vacuum and in situ measurements have been carried out between RT and 650C. Measurements have been carried out around the 5-fold 7/11 reflection, and for different position of the detector whose central spot probed phason modes with wavevectors along the (–1 0) direction [83]. When performing the measurement below 500C, the correlation function does not show any time dependence up to timescales of the order of 30 mn, confirming that phason modes are ‘frozen-in’ on this timescale. Above 500C the correlation function did show a slow time evolution. Results obtained at 650C and for two values of the wavevector q are displayed Figure 5.30. The time decay of the correlation function is clearly visible, and has been fitted to an exponential law g(q,t) = exp(–t/(q)), from which a characteristic time is extracted. The top and bottom parts of Figure 5.30 correspond to a wavevector q of 0.064 and 0.13 nm1, respectively. As can be seen in the figure the characteristic time is much shorter for the larger wavector, in agreement with the expected diffusive character of phason


153

350 300

Counts

250 200 150 100 50 0

–0.004

–0.002

0

q

0.002

0.004

(Å–1)

Figure 5.29. Top: 2D CCD image of the diffuse scattering measured close to a Bragg reflection using a coherent X-ray beam. Bottom: section through the diffuse scattering. The ‘spikes’ are much larger than the statistical uncertainties, as shown by the error bars: they are due to the coherence of the X-ray beam (from [61]).

modes. The diffusive character of the phason mode is evidenced in a more quantitative way in Figure 5.31, where the characteristic time measured along the direction (–10) for different wavectors is plotted as a function of q2. As can be seen there is a linear dependence, from which a phason diffusion constant defined as q2 = Dphason is evaluated and found equal to 2.21018 m2s1 [83,84]. A few measurements were also carried out at 600C. At this temperature phason fluctuations have much larger time decays. An estimate of the phason diffusive constant is difficult, because characteristic times are now close to the maximum value which can be measured. Nevertheless, we find that characteristic times are between 5 and 10 times larger than at 650C. Assuming an Arrhenius law for the variation of the diffusion constant as a function of the temperature, we estimate that the activation energy of phason modes is of the order 3 (–1) eV. Such results are compatible with what is known for atomic diffusion in these systems. Mn, which is the slow diffuser in the i-AlPdMn phase, has a diffusion constant equal to 1014 m2s1 at 650C and an activation energy equal to 2 eV. We found a smaller phason diffusion constant: this might be understood as the result of

154

Chapter 5. Phason Modes in Aperiodic Crystals 1.11 1.1

q = 0.064 nm–1 τ = 60 s

1.09

F cor

1.08 1.07 1.06 1.05 1.04

q = 0.13 nm–1 τ = 19 s

1.04

F cor

1.03

1.02

1.01

1

0

100

200

300

Time [s] Figure 5.30. Time dependent intensity correlation function measured at 650C in the i-AlPdMn phase. The solid line is a fit to an exponential decay. The top and bottom figures correspond to two different wavevectors: the decay is much faster for the larger wavevector (from [83]).

a relaxation process which involves ‘exchange’ of chemical species on neighbouring sites, and thus with a longer time scale. The phason diffusion constant has been measured for wavevectors along a single direction (–10). As noted in Section 5.4.3, for each wavevector there are three phason modes with different polarizations. In the above analysis we made the assumption that a single mode, i.e., a single polarization, is picked up because of selection rules. Measurements along other directions should display different characteristic times, and thus different diffusion constant values. This anisotropy in the diffusion constant is given by the variation of the eigenvalues of the Cper-per matrix as a function of the eigenvectors. In the case of the i-AlPdMn phase, it is expected that 3-fold phason modes (with a polarization along a 3-fold axis) are the slowest ones. In some sense this is equivalent to the case of phonon dispersion relations for which longitudinal modes have a different sound velocity than transverse ones. Figure 5.31 is thus the ‘dispersion’ relation of one phason mode. Phason fluctuations are believed to play an important role in the growth process and in the mechanical properties of quasicrystals. In the i-AlPdMn quasicrystal, plastic deformation is only possible for temperatures larger than 600C and occurs via

155

5.6. Phason modes in other quasicrystals 80 70 60

τ (s)

50 40 30 20 10 0 0

2 10–15

4 10–15

6 10–15

(2π /q)2

8 10–15

1 10–14

1.2 10–14

(m2)

Figure 5.31. Evolution of the characteristic time as a function of l2 measured at 650C, along the (, –1, 0) direction. The solid line is a linear fit which yields the phason diffusion constant (from [83]).

dislocation motions [85]. Because of the quasiperiodic long range order, the distortion field around a dislocation contains both a phonon and a phason part (See Chapter on mechanical properties in this book). When a dislocation is moving, a highly localized phason strain (named phason wall) is left behind [86]. In situ high resolution TEM images have shown that these phason walls annealed out very rapidly above 650C. A detailed study of this dynamics has been carried out using in-situ TEM images [87]. There is a contrast in the images, which is due to the phason wall. This contrast vanishes exponentially with time, from which a characteristic time is extracted. The time scale at 640C is of the order 140 s for a sample thickness of the order 200 nm, i.e., the time necessary to anneal the phason wall on 100 nm (half the sample thickness) is about 140 s. This is in very good agreement with the value of the phason diffusion constant report above at 650C, where for a wavelength of 100 nm one finds a time scale of the order 70 s (see Figure 5.31). The authors have also determined the temperature dependence of the characteristic time which follows an Arhenius law. The activation energy extracted from this dependence is equal to 4 eV, in agreement with the coherent X-ray measurements. The high activation energy of the phason modes allows to explain why the i-AlPdMn quasicrystal is brittle below 600C: in this case the time necessary to relax the localized phason strain is too long, and dislocations are pinned. This set of experiments on phason strain relaxation in the vicinity of a dislocation agrees thus nicely with those obtained by coherent X-ray diffraction.

5.6.

Phason modes in other quasicrystals

In this section we give an overview of the results obtained in other icosahedral and decagonal phases.

156

5.6.1.


AlPdRe, AlCuFe, and CdYb icosahedral phases

Phason modes have been studied using room temperature measurements of the diffuse scattering in the i-AlPdRe, i-AlCuFe, i-CdYb, and i-ZnMgSc phases [88–90]. The first two icosahedral phases are interesting since they are iso-structural with the i-AlPdMn one. The CdYb icosahedral phase is the first stable binary quasicrystal. Moreover it contains atomic clusters which are quite different from the one observed in the i-AlPdMn phase quasicrystal. We will show in the following that all these phases present diffraction patterns with a Qper dependent diffuse scattering pointing to the presence of phason fluctuations. Single grains of the i-AlPdRe phase have been obtained so that accurate diffuse scattering measurements could be carried out both on laboratory sources and at synchrotron facilities. The diffraction pattern of the i-AlPdRe phase displays a significant amount of diffuse scattering around Bragg reflections. Its shape anisotropy is however significantly different than the one observed in the i-AlPdMn phase. That the diffuse scattering is most likely related to phason fluctuations could be demonstrated by comparing the diffuse intensity around two Bragg peaks having a different Qper component and lying on a same 2-fold axis. The observed intensity decays as q2 and is proportional to IBraggQ2per, as expected for phason fluctuations. However, the shape anisotropy has not yet simulated using elasticity theory. A comparison of the absolute scale measurements of the i-AlPdMn and i-AlPdRe phases shows that both phases display the same ‘amount’ of diffuse scattering. This is illustrated in Figure 5.32, which compares the 2-fold diffraction pattern of the two phases. The amount of diffuse scattering around the 20/32 reflection is almost identical in both phases [88]. Both room temperature and in-situ temperature measurements have been carried out for the i-AlCuFe phase. In the vicinity of the composition Al63.5Cu24Fe12.5 it has been shown that the icosahedral phase is only stable at high temperature. When slowly cooled from 820C, the icosahedral phase undergoes a phase transition towards a crystalline multi-domains structure of rhombohedral symmetry

8/12

I (e.u./Å3)

107

20/32

i-AlPdRe i-AlPdMn

105

1000

10

0.1

1

1.5

2

2.5

3

3.5

Q (2π /ap unit) Figure 5.32. Comparison of the diffuse scattering measured on an absolute scale in the i-AlPdMn and i-AlPdRe quasicrystal along a 2-fold axis. There is the same amount of diffuse scattering in both phases (from [88]).

5.6. Phason modes in other quasicrystals

157

(aR = 32.18 A˚, a = 36) [91,92]. The icosahedral to rhombohedral phase transition is achieved via an intermediate modulated icosahedral state whose diffraction pattern displays first order satellite reflections around the icosahedral ones. The modulation has a wavevector q lying along 5-fold icosahedral axes, so that each i-phase reflection is surrounded by a cloud of 12 satellite reflections. Depending on the heat treatment, the modulation wavelength varies from 150 to 400 A˚ [93]. The intensity distribution of the satellite reflections is well reproduced by considering a simple cosine wave propagating along a 5-fold axis in physical space and having a polarisation along a 5-fold axis in perpendicular space, with an amplitude equal to 0.35 A˚, to be compared to the 10 A˚ radius of the large atomic surfaces describing the high dimensional structure of the i-AlCuFe phase [57]. This modulated state is most likely a metastable state but it points out the importance of phason fluctuations during the phase transition (phason driven phase transition) [94,95]. The diffuse scattering of the high temperature icosahedral phase has been studied both on quenched samples at room temperature and in situ. Again, there is a clear distribution of diffuse scattering intensity around Bragg reflections, with a distinct anisotropy related to phason fluctuations. Its shape anisotropy is best reproduced using a value of the phason elastic constant K2/K1 close to 0.5, i.e. with a sign opposite to the one obtained in the i-AlPdMn phase [89]. This has to be related to the observed phase transition in i-AlCuFe, whose symmetry breaking is consistent with a 5-fold type instability i.e., a positive K2/K1 ratio. Again, this points to a strong connection between the phenomenological phason elastic constants and the chemical potential describing the chemical stability of the system. The recently discovered i-CdYb phase [96–98] is interesting for at least two reasons: (i) it is the first stable binary quasicrystal, (ii) its atomic structure is made up of clusters completely different from all previously known icosahedral quasicrystals, i.e., neither a Mackay nor a Bergmann one. Moreover, a cubic 1/1 approximant can be obtained for a slightly different chemical composition. A similar structure has also been obtained in the ZnMgSc system. Figure 5.33 shows the intensity distribution around some Bragg reflections in the i-CdYb phase [88]. The shape anisotropy is related to phason fluctuations, with a tendency toward a 3-fold instability, as evidenced by the diffuse streaks along directions parallel to 3-fold axes. A detailed study of the diffuse scattering has been carried out in the ZnMgSc system, where measurements have been performed on both the 1/1 crystal approximant and the quasicrystal [90]. Whereas the 1/1 approximant diffraction pattern only displays thermal diffuse scattering (TDS) the quasicrystal has a significant supplementary contribution, which is interpreted as resulting from phason fluctuations. This nicely demonstrates that phason modes are really specific and a characteristic signature of the aperiodic long range order. The diffuse scattering has been measured on an absolute scale in the i-ZnMgSc phase: the amount of diffuse scattering is found to be about 5 times less (all other parameter being equal) than in the i-AlPdMn phase (see Ishimasa chapter). This point to the very high structural quality of the i-ZnMgSc sample. Moreover, since the diffuse scattering intensity is roughly proportional to 1/K1 this indicates that the magnitude of the phason elastic constant is much larger in the i-ZnMgSc than in the i-AlPdMn phase.

158


4.0

A5 A3

3.5

K 3.0

2.5

0

0.5

1.0

1.5

2.0

2.5

H Figure 5.33. Isointensity contours for the diffuse scattering measured in a 2-fold scattering plane of the i-CdYb phase. Anisotropic intensity distributions are clearly visible, in particular along directions parallel to 3-fold axes.

5.6.2.

Decagonal phases

Intensive structural studies have been carried out for the d-AlNiCo and d-AlCuCo decagonal phases (see [99] for a recent review). Since their structure is described as a periodic stacking of quasiperiodic planes, both ‘microscopic’ techniques such as X-ray and neutron diffraction and local ones such as HREM images obtained by transmission electron microscopy can be used. However, the phase diagram in these systems is extremely complex, with a wide existence domain of the decagonal phase at high temperature. Depending on the Ni/Co or Cu/Co content, several phase transitions occur when the temperature is lowered. It can be either a transition towards a crystalline phase (in the AlCuCo system) or towards ‘superstructure’ phases corresponding to integer multiple of the basic high-dimensional cell. All these decagonal phases (except the Ni rich AlNiCo one), present a large amount of diffuse scattering, pointing to a high degree of structural/chemical disorder [100]. In general the period c (along the periodic axis) of the decagonal phases is 4 A˚. The diffuse scattering is observed both in the quasiperiodic planes (located at nc*) and at positions (m+1)c*/2 where it forms sheets of diffuse scattering, narrow along the c* direction and with broad features in the plane. It is a general trend that the intensity of the diffuse scattering diminishes as the temperature is raised [101,102]. This is a strong indication that the decagonal phase is stabilised by entropic terms. Several models have been proposed to explain both the inplane and in-between plane observed diffuse scattering. They are related to the organisation of columnar clusters which are observed in structural studies. The first evidence that long-wavelength phason fluctuations might be present in decagonal phases was published by Edagawa et al. [103]. They performed an in situ study of the d-AlNiCo phase using transmission HREM images. When looking at thick slices of their specimen, they could map the contrast observed on the images to


(a)

159

(b)

A

(c)

(d)

B

B

(f)

(e)

A

A

B

Figure 5.34. In situ transmission HREM images measured at 1123 K by Edagawa et al. in the d-AlCuCo phase. (a) to (f) are time sequences of the same area taken at 0, 5, 8, 110, 113, and 115s. The scale bar indicates 2 nm. Some column jumps are highlighted. The bottom panel indicates the tile ‘flip’ configuration which may be overlayered on the top panels (from [103]).

a Penrose tiling with an edge length of 2 nm (Figure 5.34). In this case the observed image contrast is not on an atomic scale, but most likely corresponds to ‘cluster’ columns with a diameter of order 1 nm. When taking a series of images at 1123 K, there are variations in the distribution of Penrose tiles which corresponds to ‘flip’ from one configuration to another (bottom panel). These changes in the images correspond to atomic columns containing several tens of atoms because images were taken on a thick specimen (more than 200 A˚). Moreover the ‘jump’ distance is of order 10 A˚, and can by no means be related directly to a single atomic jump. A more detailed analysis of the time dependence of the images has been carried out. By analysing changes in the tiling pattern, the spatial and temporal evolutions of the phason field could be extracted [104]. There is a decay which bears some resemblance with the predictions of the random tiling model: however a large phason strain seems to remain in the sample. Local atomic jumps have been revealed by temperature in-situ studies of the Al72Ni20Co8 decagonal phase. Using in-situ atomic-resolution annular dark field

160


images, Abe et al. have observed an anomaly in atomic vibrations at 1100 K [105]. This anomaly occurs at specific sites of the structure, and is interpreted as the result of atomic jumps between two sites separated by 1 A˚. These sites correspond to the border of atomic surfaces in the high-dimensional description of the decagonal phase, which is precisely the theoretically expected location for easy jump sites. As stated above, diffuse scattering phenomena and phase transitions are extremely complex in decagonal phases. However, most phase transitions can be explained by the application of a linear phason strain onto the high temperature decagonal phase. In a recent work, Kobas et al. made an attempt to interpret the observed diffuse scattering in the framework of the phason elasticity of decagonal phases [106]. Figure 5.35 (a,b,d,e) shows two experimental surveys of the l = 0 and l = 2 [Å–1] –0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

J (h, k, 0) 0.04

0.04

0

0

–0.04

–0.04

0.6

0.6

0.4

0.4 –0.75

–0.7

–0.75

–0.65

–0.7

–0.65

0.2

0.2

(b)

Jpf (h, k, 0)

(a)

(c)

Jp (h, k, 0)

(d)

(f)

Jp (h, k, 2)

0

0

Jpf (h, k, 2)

(e)

–0.2

–0.2

–0.4

–0.4 –0.6

–0.6

–0.65

–0.65

–0.7

–0.7

–0.6

–0.6

–0.05

–0.8

–0.6

0

0.05

–0.4

J (h, k, 2) –0.2

–0.05

0

0.2

0.4

0

0.6

0.05

0.8

[Å–1] Figure 5.35. Comparison between the measured and calculated diffuse scattering in the d-Al71.5Ni14.6Co13.9 decagonal phase (From Kobas et al. [106]). (a, b) and (d, e) correspond to measurements carried out at 1120 K for the Qz = 0 and 2 quasiperiodic ‘Bragg’ layers respectively. (a) and (d) are the original data, whereas (b) and (e) are obtained with specific data treatment. (c, f) are the corresponding simulations including the phonon and phason contribution to the diffuse scattering (see text).

161


quasiperiodic layers of reciprocal space, obtained with synchrotron radiation, on the d-Al71.5Ni14.6Co13.9 decagonal phase. Measurements were carried out at 1123 K. There is a large amount of diffuse scattering, which displays characteristic anisotropic shapes. Part of this anisotropy is well accounted for by a superposition of phonon and phason contributions. As pointed out by Ishii, only a phonon–phason coupling term can introduce anisotropy in the diffuse scattering of decagonal phases [107]. Right panels in Figure 5.35 are simulations using both the phonon elastic constants as determined by ultrasonic measurements and phason elastic constants K1, K2, and R set to a value of 0.02, 0.4, and 0.1, respectively, where R represents the phonon–phason coupling term. Although not perfect, there is a reasonable agreement between simulation and experimental data. The same authors also proposed an atomic model to interpret their results. As a first approximation they considered a fivefold orientational disorder of the columnar cluster shown Figure 5.36. This orientational disorder, obtained by successive 72 rotation of the initial cluster, produces two kinds of disorder: occupational disorder (Al-Ni/Co for instance), and displacive ‘split’ positions as shown by arrows and labels in the figure. Although the simulated diffuse scattering obtained with this model is only approximately similar to the observed one, it is an interesting step toward a better understanding of phason fluctuations at the atomic scale. To conclude this overview of results obtained in various quasicrystals, phason fluctuations have been evidenced in many different systems. However their microscopic interpretation is far from being achieved and certainly requires elaborated

A layer

B layer

3 1 4

5

6 7

2

1

: TM-atom, original position : Al-atom, original position : TM-atom, rotated position : Al-atom, rotated position Figure 5.36. Local disorder phenomena in the d-AlNiCo phase induced by a 72 rotation of the initial cluster. The two layers A and B are shown in the top and bottom diagram respectively. The different kinds of disorder phenomena are labelled: ‘flips’ of Al atoms (1), occupational disorder (3,4) and ‘split’ positions between transition metal TM and Al (5) (from [106]).

162


simulations at the atomic scale. All temperature studies carried out so far lead to the same results: there is either a phase transformation at low temperature toward a crystalline phase, and/or the diffuse scattering increases as the temperature is lowered. This is a strong indication of an entropy term stabilising the quasicrystalline structure.

5.7.

Conclusion

The hyperspace description of quasiperiodic structures together with the hydrodynamic mode analysis provide a useful approach to understanding the common aspects of phason dynamics in the different structural classes of aperiodic crystals. In particular, it makes it clear as to why the concept of propagative phason must necessarily break down in the long wavelength limit, where it is a diffusive mode. Moreover, a phason mode is, in this approach, characterized by its wavevector and a polarization vector which lie in the perpendicular space. Phason dynamics has been observed in all the different classes of aperiodic crystal, and has been shown to be in agreement with the hydrodynamic theory. This unifying picture should not mask the deep physical differences between these various classes of systems. One such difference concerns the respective roles of energy and entropy terms in the stabilization of the aperiodic state. In quasicrystals, the observed T-dependence of the phason elastic constants indicates a general tendency for a low-T instability toward a ‘‘commensurate’’ state of lower rotational symmetry, which often cannot be reached due to kinetic limitations. The same trend exists in modulated crystals – for which low-T kinetics is much less of a problem – where the modulated state is frequently observed at an intermediate temperature range, illustrating the role of the entropy term. However it is far from being systematic, as many systems are known to be stable in the modulated state at 0 K. The dominant role of entropic terms is also at the origin of the applicability of phason elasticity theory to quasicrystals in spite of the discontinuous character of the atomic domains in that case. This point, however, still needs to be confirmed through detailed 6D structural modeling as well as accurate diffuse scattering experiments. On the microscopic side, there is a wide gap in our level of understanding for modulated and composite systems on one hand and quasicrystals on the other. Fairly realistic models have been proposed, based on competing microscopic interactions, which are able to account for the occurrence of commensurate and incommensurate modulated phases in a wide variety of systems. The best known are the Axial-Next-Nearest-Neighbor-Interaction (ANNNI) model [1], the DiscreteFrustrated-f4 (DIFFOUR) model [7], and the Double-Chain (DCM) model [8] in the case of composites. The mean-field version of the DIFFOUR model, in particular, has been quite successful in predicting model phase diagrams, which compare well with a wide range of experimental observations. The predictions of the same DIFFOUR model serve also as a useful guide for dynamical studies, within the limits of the quasiharmonic approximation. For quasicrystals on the other hand, many open questions remain, concerning the microscopic aspects of phason dynamics. The low value of the phason diffusion

5.7. Conclusion

163

constant – one or two orders of magnitude lower than for atomic self-diffusion – seems to indicate that the elementary microscopic mechanism – the local phason jump – is more complex than just a vacancy-assisted atomic hoping process. This is illustrated by the timescale gap between these local processes – a few picosec – and the timescale for long-wavelength collective processes – 10 to 100 s for l = 1000 A˚. This ratio of thirteen orders of magnitude seems hardly physical. For comparison, the ratio between the elementary nitrite ion reorientation time and the collective relaxation time for a l = 1000 A˚ phason fluctuation, in sodium nitrite, is only of order 105. Another approach is to use the value of the phason diffusion constant obtained from the data in Figure 5.31, to calculate an extrapolated value for at q = 2/a, where a is of the order of an interatomic distance ( e 0.3 nm). One would then expect to obtain an absolute lower bound for the single flip characteristic time. The extrapolated characteristic time is found to be e6104 s, much slower than the picosec single-flip timescale reported [67]. Understanding these differences in timescale would certainly provide a useful lead on how to further improve our description of phasons in quasicrystals. For icosahedral phases, three phason modes with different polarizations in the perpendicular space are associated to each q value. The relaxation rate of these different phason modes is theoretically directly linked to the observed anisotropy of the diffuse scattering. Further experiments are necessary to check this hypothesis. Diffuse scattering related to phason modes has been evidenced for all known quasicrystals, pointing to the importance of such fluctuations in these phases. For icosahedral phases, the observed phason fluctuations does not seem to be related to a geometric configurational entropy alone, since phases with a similar atomic structure but with different chemistry have phason elastic constants with opposite signs. Moreover, phason-driven phase transition are observed in some compositional range of the i-AlPdMn and i-AlCuFe phases. Finally significant differences in the magnitude of experimentally determined phason elastic constant are observed. All these observations point to an interpretation of phason modes both in term of configurational entropy but also in term of chemical potential, and chemistry. Indeed if a local phason strain is applied to the 6D structure it generates chemical disorder on specific sites (either by generating split atomic positions or by producing atom type exchanges), changes in the number and density of local configurations, and local changes in the stoechiometry of the phase. All these parameters contribute to the free energy, which in turn may vary quadratically with phason strain. In this perspective, phason modes would be a result of the very specific chemical order or disorder in quasicrystalline phases. To proceed further in the understanding of this point requires temperature studies on other quasicrystalline phases such as the binary CdYb icosahedral phase. Realistic simulation at the atomistic scale is a key point. Most of the simulations related to phason elasticity in quasicrystals were carried out so far on toy models, which are certainly far from being a proper representation of real systems. There is now a good understanding of the atomic structure and elaborated models are available for several quasicrystals, so that simulations can be carried out on firm

164


grounds. With the current status of computing power, ab-initio calculations can handle systems containing at most 500 atoms and allowing to study relatively short ‘physical’ timescale. It is thus still necessary to parametrize the Hamiltonian with some functional such as pair potentials, to reach ‘long’ time scales on large system size. Based on ab-initio calculations, elaborated simulations have been achieved for decagonal phases [108]. Evaluations of the entropy gain due to tile configuration or chemical disorder are available. A decagonal phase has even been predicted by such a calculation, in the BMgRu system [109]: the simulation shows that the quasicrystal would be favored with respect to the crystal by the configurational entropy gained in the quasicrystalline state. We have already pointed out the importance of phason fluctuations for the mechanical properties of quasicrystals. Further detailed and quantitative studies in this field, with a link to what is known from the diffuse scattering experiments are certainly of importance. Finally, all quasicrystals are metallic alloys, and most of the time closed-packed structures. In that respect a link with studies of order/disorder transitions in metallic alloys with the occurrence of long-period structures would certainly be enlighting for the field of quasicrystals. Phason mode studies remain thus one of the most fascinating and challenging problems in aperiodic systems. The high dimensional space approach provides a unifying frame. The large amount of experimental and theoretical data obtained in different classes of aperiodic crystals form a firm basis for our further understanding of phason modes and its consequences on physical properties of aperiodic crystals.

ANNEXE We give below the convention used for the definition of the hydrodynamic matrix [44,110] which is a 9 9 matrix which decomposes in 4 (3 3) block matrices expressed in Fourier space as: Cparpar Cparper C¼ Cperpar Cperper The matrix elements are related to the elastic constant and to the components of the wavevector q=(qx,qy,qz) of the mode to which it refers. I is the identity matrix. The Cpar-par matrix, is related to phonon elastic constants as: 0

qx 2

B B ! B Cparpar q ¼ q2 I þ ðl þ ÞB B qx qy B @ qx qz

qy qx qy 2 qy qz

qz qx

1

C C C qz qy C C: C A 2 qz

165

References

The Cper-per matrix is related to phason elastic constants as: 0

1 1 q2 qy 2 þ qz 2 B 3 B ! B C?,? q ¼ K1q2 I þ K2B 2 2 qx qy B B @ 2 2 qx qz

2 2 qy qx

2 2 qz qx

1 1 q2 qz 2 þ qx 2 3

2 2 qz qy

2 2 qy qz

1 1 q2 qx 2 þ qy 2 3

1 C C C C: C C A

The Cpar-per matrix couple phonon and phason distortions and is related to the K3 coupling constant: 0

C==;?

! q ¼ C?;==

B B ! B q ¼ K3B B B @

1 qx 2 qy 2 þ qz 2

2 q q y x

2qz qx

2qx qy

1 qy 2 qz 2 þ qx 2

2 qq z y

2 q q x z

2qy qz

1 qz 2 qx 2 þ qy 2

1 C C C C: C C A

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Chapter 6

Electronic Structures and Stability Mechanism of Quasicrystals Yasushi Ishii1 and Takeo Fujiwara2 1 Department of Physics, Chuo University, Tokyo 112-8551, Japan 2 Department of Applied Physics, University of Tokyo, Tokyo 113-8654, Japan

6.1.

Introduction

Quasicrystals (QCs) are long-range ordered materials with rotational symmetry incompatible with translational one [1]. Icosahedral QCs are alloys with noncrystallographic icosahedral symmetry and decagonal ones are those with twodimensional decagonal symmetry where aperiodic atomic layers are stacked periodically along the 10-fold axis. Icosahedral and decagonal QCs have been discovered so far in more-than one hundred compounds and are considered as one of universal phases of matter. Diffraction pattern of QC consists of densely distributed sharp Bragg spots with non-crystallographic symmetry and each Bragg spot is indexed with integers more-than the spatial dimension. For 3-dimensional icosahedral QCs, six integers are needed for indexing. This suggests that the quasiperiodic structure is described with higher-dimensional lattice. Methods generating icosahedral and decagonal quasiperiodic structures from higher-dimensional lattices have been established and even realistic atomic structures of QCs are described in higher-dimensional space [2]. Electronic structures of QCs have been studied mainly from two different points of view. First, quantum-mechanical motion of electrons in a quasiperiodic potential is of special interest. Difficulty comes from the fact that the Bloch theorem for electrons in a periodic potential cannot be adopted here. Numerical and analytical studies have been done on wave functions and eigenvalue spectra for Penrose lattice and other model systems. We leave this subject to published review articles [3] and will not discuss here in detail. The second point of view concerns the electronic structures of real materials. Questions to be answered are why alloy phases with such complicated structures are stabilized, what novel physical properties are expected in such non-periodic systems, and so on. We shall review current understandings on the electronic structures of QCs HANDBOOK OF METAL PHYSICS ISSN 1570-002X/DOI 10.1016/S1570-002X(06)02006-4 171


172

Chapter 6. Electronic Structures and Stability Mechanism of Quasicrystals

from the second point of view. Anomalous transport property is one of the most characteristic ones in QCs and shall be discussed by Mayou [4] in this volume. Equilibrium QCs have been obtained for many kinds of alloy systems and are considered as universal phases of matter. Icosahedral QCs obtained so far are classified into two families [5]: one is a family of aluminum-based transition-metal (TM) alloys, which will be called the Al-TM family. The other one is a family of alloys without transition elements having similar composition to that of the so-called Frank-Kasper (FK) phase, which is an alloy phase with tetrahedrally closed-packed structures. This will be called the FK family. The FK family involves Al–Cu–Li, Zn–Mg–RE (RE = rare earth), and so on. Atomic structures of QCs are usually understood by considering atomic clusters with non-crystallographic rotational symmetry as building blocks. The two families of QCs were first distinguished by referring their cluster structures [6]. In QCs belonging to the Al–TM family, a building unit is an icosahedral cluster called Mackay icosahedron (MI) and some glue atoms filling space between clusters are needed. More recent structural studies on QCs in the Al–TM family with face-centered ordering and related alloys clarify more complex cluster structures [2,7,8]. In QCs belonging to the FK family, a building unit is a triacontahedral (TC) cluster sharing atoms with each other. The two families were sometimes referred to as MI-type and TCtype but we prefer the names of the Al–TM and FK families because they express materials more directly. Newly discovered Cd- and Zn-based QCs [9] are sometimes categorized as a new family because a building block is an icosahedral cluster different from the MI and TC ones and constituent elements are quite different from the others. Not only the cluster structures but also other quantities are distinguishable in these two families. It is empirically known that stable QCs are obtained if the average number of valence electrons per atom, which is denoted as e/a, is close to 1.7 and 2.2 for the Al–TM and FK families, respectively [5]. Here the number of valence electrons for transition metals is estimated by using Raynor’s values [10]. It is also known that a ratio of an average atomic diameter d to the quasilattice constant aR, which is obtained from diffraction experiments, is about 0.61 and 0.57 for the Al–TM pffiffiffi and FK families, respectively [5]. If one assumes the atomic density as a ¼ 2 d3 , which is that of face centered cubic lattice with an atomic diameter being the nearest neighbor distance, and calculates the Fermi wavenumber k3 from the electron density by using n ¼ ðe=aÞ a ¼ F2 , a diameter of the Fermi 3 sphere (FS) is expressed as 2kF ¼ f

2 ; aR

with a dimensionless numerical factor f being f¼

pffiffiffi!1=3 3 2 ðe=aÞ1=3 ðe=aÞ1=3 1:1 : ðd=aR Þ ðd=aR Þ

ð6:1Þ

Using experimental values for e/a and d/aR, the factor f is estimated as 2.2 and 2.5 for the Al–TM and FK families, respectively. As will be discussed later in more

6.2. Stability mechanism – Hume-Rothery versus Hybridization

173

detail, one can assume that QCs of good quality are obtained if the FS diameter coincides with particular reciprocal lattice vectors, which may give strong Bragg scatterings. In this article, we shall review our current understandings of the electronic structures of QCs and related materials with complex structures with emphasis on stability mechanism. In Section 6.2, we discuss two basic ideas on stability, Hume-Rothery mechanism and hybridization effects. First-principles calculation of electronic structures is a powerful tool to understand physical properties of real materials but conventional technique of the band calculation cannot be applied to QCs straightforwardly because of lack of periodicity. In Section 6.3, we explain a concept of periodic approximant of QCs and ab initio methods calculating electronic structures of QCs. In Section 6.4, we discuss electronic structures of individual systems separately for the Al–TM compounds, the Bergman phase, Zn–Mg–RE compounds and Cd- and Zn-based systems. Concluding remarks will be given in Section 6.5.

6.2.

Stability mechanism – Hume-Rothery versus Hybridization

Stable QCs are obtained by controlling the average number of the valence electrons per elements. This suggests that the system is stabilized electronically. As we shall see later for individual cases, the Fermi level of QC-related compounds is located at minimum of the electronic density of states (DOS), or a pseudogap, to minimize the band energy. Formation mechanism of the pseudogap is considered to be exactly stability mechanism of QCs and hence has been discussed by many authors. A diameter of the Fermi sphere coincides with particular reciprocal lattice vectors for stable QCs. This experimental facts seem to support strongly the conjecture that the pseudogap is a consequence of the Brillouin-zone(BZ)–Fermi-sphere(FS) interaction (the Hume-Rothery (HR) mechanism) [11–14]. In the nearly-free electron ~ induces splitting model, a strong interference of electronic waves wiht k~ and k~þ G ~ ~ ~ of energy bands at k = G/2, where G is the 3-dimensional reciprocal lattice vector ~ 2kF is satisfied, this giving strong Bragg scattering. If the nesting condition |G| splitting reduces the band energy of electrons and stabilize a particular system with ~ Reduction of the band energy is also interpreted in real space. corresponding G. Atoms embedded in free-electron gas interact with each other by oscillatory interaction given by ðrÞ

cosð2kF rÞ ; r3

ð6:2Þ

where r denotes an interatomic distance. If the atoms are arranged to occupy the minima of the potential, the system is periodic with a reciprocal lattice vector G satisfying the nesting condition |G| 2kF. The nesting condition in normal crystals is, however, satisfied only at several points on the 3-dimensional FS and hence the interference effect on the DOS is not remarkable enough. In one-dimensional solids where the Fermi surface is point-like, the nesting condition is complete and the energy gap always reduces the band

174


energy. Similar situation is expected in QCs where the reciprocal lattice vectors are distributed densely with icosahedral symmetry. Then the nesting condition is satisfied over large area on the FS and more remarkable pseudogap is expected. A model calculation for a hypothetical QC of Al by using pseudopotentials indeed shows a pronounced modification of the DOS from that of free electrons due to the BZ–FS interaction [15]. In approximant crystals, which are crystalline phases with similar composition and local atomic arrangements to those in QCs, on the other hand, the interference effect is considered to be less efficient than in QCs because of sphericity of distribution of G. Diameters of the FS, in unit of 2/aR, estimated from the electron density of stable QCs are 2.2 and 2.5 for those in the Al-TM and FK families, respectively (Eq.(6.1)). In this range of the wavevector, we have the reciprocal lattice vectors for icosahedral QCs, ~ ~ G211111 2:12 2=aR ; G221001 2:23 2=aR ; ~ G2221000 2:53 2=aR ;

~ G311111=222110 2:62 2=aR ;

where the reciprocal lattice vectors are 3-dimensional real-space components of the 6-dimensional ones. Subscripts 211111 etc denote indexing integers for Bragg scatterings from icosahedral QCs and the indexing scheme for primitive quasilattice is used [16]. The Jones zone or the pseudo Brilloiun zone, which is a polyhedral zone constructed from bisection planes of a set of the reciprocal lattice vectors giving strong Bragg scatterings, is shown in Figure 6.1. One can see sphericity of the pseudo-BZ. As is also confirmed from calculation of the diffraction intensity of a simple 3-dimensional Penrose lattice, the Bragg scatterings indexed as (211111) and (221001) are strong for icosahedral QCs [16]. So we expect that the deep pseudogap due to the BZ–FS interaction is obtained for QCs in the Al–TM family. Although the Bragg scatterings indexed as (222100) or (311111)/(222110) are not so strong, the fact that e/a for the stable QCs is pinned at a particular value in a discrete fashion seems to support the HR idea also for the FK family. To consider the geometric and chemical effects on the electronic stability mechanism more precisely, let us expand the scattering potential for electrons in Fourier

(a)

(b)

Figure 6.1. Pseudo-Brillouin zone constracted from (211111) and (221001) recciprocal vectors (a) and that from (222100) and (311111)/(222110) vectors.

6.2. Stability mechanism – Hume-Rothery versus Hybridization

series as Vð~ rÞ ¼

P ~ G

175

~

vG~eiG~r . The Fourier component is divided into the structure and

the form factors for individual atomic species as vG~ ¼

X

ð Þ ð Þ

G~ vG~ ;

ð6:3Þ

where denotes an atomic species. The structure and form factors are defined as ð Þ

G~ ¼ ð Þ

vG~ ¼

X

~~

eiGRi ;

ð6:4aÞ

i¼ sites

1 V

Z

~

d~ r v ð~ rÞeiG~r ;

ð6:4bÞ

where v ð~ r Þ is a pseudopotential for the atomic species . Summation is taken over atomic positions for atom and integration is carried out over entire volume V of the system. Obviously the structure factor represents geometric effects whereas the form factor represents chemical effects and selectivity of particular elements. In the most naive application of the Hume-Rothery idea, the electronic structure is treated within a rigid-band model and effects of alloying are considered only through a change of the number of valence electrons. In this sense, the HR mechanism is interpreted as of geometric origin and a drastic change in the atomic form factor is primarily neglected there. This might be justified for combination of similar elements such as Cu and Zn. However, in alloys of elements with different chemical natures such as valence, electro-negativity, atomic size and so on, the atomic form factor plays an essential role and one should treat differences of pseudopotentials correctly. In transition metals, sd hybridization induces the energy-dependent pseudopotential for s electrons near the d levels as r

Dv ðr Þ

V2sd ; " "d

ð6:5Þ

where Vsd and d represent the sd hybridization and the energy of the d states, respectively [14]. If the Fermi energy is close to the d level, the electronic states near the Fermi level are strongly modified by the energy-dependent pseudopotential. As we shall see later, the hybridization effect is important not only for the Al–TM family but also for the systems without transition elements. Roles of the degenerate d levels are interpreted in a different way as well. When transition elements with unoccupied d-bands form alloys with Al, which has smaller electro-negativity than the transition elements, valence electrons of Al fill the d-band and eventually the transition elements behave as acceptors of electrons. Such effects of charge transfer control the Fermi level to be pinned at the pseudogap and hence contribute to the cohesion. For alloys containing a small concentration of transition elements, Raynor noticed that a volume of the FS matches with that of the BZ derived from the reciprocal lattice vectors of strong Bragg scatterings if one assumes a charge

176


transfer from the conduction band to the d band [10]. The number of electrons transferred to transition elements (effective negative valence) was estimated to compensate the unpaired spins as 4.66, 3.66, 2.66, 1.66 and 0.66 for Cr, Mn, Fe, Co and Ni, respectively. For example, Fe accepts 2.66 electrons/atom and Al3Fe gives the valence electron density, (3 3–2.66)/4 = 1.58 electron/atom. The valence electron density, e/a, for QCs in the Al–TM family is, in fact, estimated by using Raynor’s valence for transition elements. The valence electron density is an important factor controlling phase stability in the HR scenario. But even more important question would be why Raynor’s valence gives reasonable estimates for predicting stability of Al–rich TM alloys or how we can construct reliable database of effective valence for other elements.

6.3.

Ab initio methods for calculating electronic structures of QCs

First-principles calculation of electronic structures is a powerful tool to understand physical properties of real materials, but is difficult for non-periodic solids because the Bloch theorem cannot be applied. Several approaches have been adopted to elucidate characteristic features in the electronic structures and stability mechanism of QCs. Among them, the first-principles electronic-structure calculations for ‘approximant crystals’ or ‘crystalline approximants’ give us reliable information. The approximant crystals are crystalline phases with complex structures and often found at composition very close to the QC [17]. The approximant crystal involves typically more than 100 atoms in a unit cell with the atomic clusters with non-crystallographic icosahedral symmetry. Hence it is believed that the local atomic arrangement in the approximant is similar to that in QCs and their electronic structures share common characteristics. Atomic structures of QCs are described as periodic structures in higher-dimensional space and obtained by selecting a subset of atoms in the higher-dimensional lattice. Irrational numbers are inherent in the scheme generating the 3-dimensional atomic arrangements from the higher-dimensional structures and specify the non-crystalline rotational symmetry and the quasiperiodicity of QCs. For the icosahedral and decagonal with 5-fold rotational symmetry, the irrational number is the golden mean pQCs ffiffiffi ¼ ð 5 þ 1Þ=2 1:618 L. If the irrational number is replaced with a rational one in the scheme generating the 3-dimensional atomic positions, one obtains a periodic structure, which is the crystalline appoximant. Such modification corresponds to introducing ‘strains’ into the periodic structure in higher-dimensional space and the introduced strains are called ‘phason’ strains. One of the best rational approximations for the golden mean is given by Fn+1/Fn where Fn is the Fibonacci number satisfying a recurrence formula Fn+1 = Fn+Fn–1 with F0 = F1 = 1. By using a better approximation in this series of 1/1, 2/1, 3/2 etc, one obtains the approximant with a larger unit cell, which is expected to represent the atomic structures of QCs more nicely. For the icosahedral QCs, the approximant phases with cubic symmetry are often studied but the approximants with other rotational symmetry can be obtained by modifying the generating scheme of the 3-dimensional structures properly [18]. More precisely, if ‘atoms’ are placed at lattice points of the 6-dimensional simple cubic lattice, a 3-dimensional Penrose lattice consisting of prolate and oblate rhombohedra is generated. Similarly, from the 2- and 5-dimensional simple cubic lattices, 1-dimensional Fibonacci and 2-dimensional Penrose lattices are generated, respectively.

177

6.3. Ab initio methods for calculating electronic structures of QCs

L

S

L

L

S

L

L

S

L

L

L

S

S

L

L

L

S

S

L

Periodic

L

L

Quasiperiodic

Figure 6.2. Cut-and-projection method generating 1-dimensional quasiperiodic Fibonacci lattice and periodic approximant. A dotted square is a unit cell of 2-dimensional square lattice.

In Figure 6.2, we show how to generate quasiperiodic Fibonacci lattice and its periodic approximant. The quasiperiodic lattice is obtained from a cut parallel to the (,1) direction of the 2-dimensional square lattice whereas the periodic one is obtained by a cut parallel to the (2,1) direction. Such quasiperiodic lattice models of QCs are used to study characteristic aspects for quasiperiodic systems such as eigenvalue spectra, wavefucntions and others [3]. A series of approximants is often used because one can expect to obtain properties of a quasiperiodic system in the limit of large unit cells. The atomic structures of real QCs are also analyzed with the higher-dimensional lattice although ‘atoms’ in the higher-dimensional space have complicated structures [2]. Once the atomic structure in the higher-dimensional space is obtained, one can generate atomic positions for the approximant crystal with a unit cell of arbitrary size. Alternatively the atomic structure of the complex crystalline phase found at composition very close to the QCs can be used for the first-principles calculation of the electronic structures. Experimental atomic structures usually involve fractionally occupied sites and such uncertainty in the structure might be essential in QCs. To adopt the first-principles method of band calculation, however, one has to introduce additional assumption to avoid the fractional occupation. It is desirable to study effects of chemical or structural disorder by going beyond the conventional band calculations. The tight-binding linear-muffin-tin-orbital (TB-LMTO) method [19] is one of the powerful methods for a system with a large unit cell. The TB-LMTO method with the atomic-sphere-approximation (ASA) has been used in the electronic structure

178


calculation of many approximants. In the TB-LMTO method, the wave functions expanded with the muffin-tin orbitals RL centered at atomic positions as ci ¼

X

ci;RL RL ;

ð6:6Þ

RL

where R and L denote the atomic position and the angular momentum quantum number (l, m), respectively. The basis functions are given as a linear combination of a solution of the Schro¨dinger equation with a fixed energy Ev , RL , satisfying 1 r2 þ U E RL ¼ 0; 2

ð6:7Þ

inside the atomic sphere at R and its energy derivative _ RL . Here U is an effective spherical potential involving electrostatic field due to an ionic core and effects of the electron-electron interaction. In the frozen-core approximation, only one orbital is considered for each atom and each angular momentum l. If one considers the orbitals with l = 0 (s), 1 ( p), 2 (d ) for each atom, the number of the basis functions is 9 times the number of atoms in a unit cell. So a system with several hundreds of atoms in a unit cell can be treated with exact diagonalization of the Hamiltonian matrix. In the TB-LMTO method, the whole space is treated as a sum of overlapping atomic-spheres within the ASA and correction for the sphere overlapping is included within subscription called the combined correction. The ASA is generally considered to be reasonable for metallic systems with densely packed atomic structures but more elaborate methods are needed for accurate estimate of the total energy and structural optimization. Tight-binding formalism is convenient for analysis of bonding/anti-bonding nature of the electronic eigenstate, which is characterized by nodes of the wavefunction between. To study such bonding nature in solid state, we introduce energy-resolved bond order matrix as pRL;R0 L0 ð"Þ ¼

X

ci;RL ci;R0 L0 ð" "i Þ;

ð6:8Þ

i

where "i is the energy eigenvalue for ci . The bond order matrix is, however, not sufficient to describe the bonding nature solely because its element can be either positive or negative for the bonding states depending on the angular momentum. This inconvenience can be treated by weighting with Hamiltonian or overlap matrix elements [20]. The crystal orbital Hamilton population (COHP) and the crystal orbital overlap population (COOP) are defined with weight of Hamiltonian and overlap matrix elements, respectively, as hRL;R0 L0 ð"Þ ¼ pRL;R0 L0 ð"Þ hRL jHjR0 L0 i;

ð6:9aÞ

oRL;R0 L0 ð"Þ ¼ pRL;R0 L0 ð"Þ hRL jR0 L0 i:

ð6:9bÞ

6.4. Electronic structures of QC-related compounds

179

For a simple example of ss hopping, the hopping integral (the off-diagonal Hamiltonian matrix element) is negative and c i;RL ci;R0 L0 is positive for the bonding state. From such simple arguments, one can see that a negative value for COHP and a positive one for COOP imply the bonding states. Plane-wave expansion with the first-principles pseudopotentials has been developed in 1990s and is available for complex systems with more than 100 atoms in a unit cell [21]. The wavefunctions are expanded with the plane-wave basis as ci ¼

X ~ G

h i ~ ~ r ; ciG ~exp iðk þ GÞ ~

~ is a reciprocal lattice vector for where k~ is a wavevector in the Brillouin zone and G an approximant crystal. Plane-waves with kinetic energy less than a cut-off energy Ec are used to expand the wavefunctions and Ec is typically chosen as 10–20 [Ryd] for approximant crystals with more-than 100 atoms in a unit cell. Pseudopotentials representing effective interaction between valence electrons and ionic cores are determined from the first principles by using the modern technique such as the norm-conserving or the ultra-soft pseudopotentials [22]. The plane-wave method enables to calculate forces acting on each atom with reasonable accuracy. Hence this method can be used for structural optimization of complex systems including the approximant crystals. Since the TB-LMTO within ASA cannot give accurate evaluation of the forces acting on atoms, the plane-wave expansion can be used as a complimentary method. Effects of the electron-electron interaction are treated within the local-densityapproximation (LDA) in the density-functional theory [23]. The LDA in the density-functional theory is a unique practical approach in the non-empirical electronic structure calculations for solids. In particular, the cohesive properties of metals, semiconductors and insulators are well described even quantitatively. It turns out, however, that the applicability of LDA is limited for some classes of material, where the electron correlation is strong. Some of QCs involves rare-earth elements with localized f electrons, for which the electron correlation is important, and some caution may be needed for, for example, spectral properties such as position of the f band and magnetism. Nevertheless we believe that the LDA approach gives quantitatively correct results, in particular, for the stability mechanism.

6.4. 6.4.1.

Electronic structures of QC-related compounds Al–TM compounds

The electronic structure of an idealized approximant was first studied by Fujiwara for a-Al–Mn–Si [24]. The model structure is the 1/1 cubic approximant, with 114 Al and 24 Mn atoms in a cubic cell and the space-group symmetry Pm-3. The structure consists of icosahedral clusters of Mackay-type (MI clusters) placed at a vertex and a body-center of cubic unit cell with a lattice parameter 12.68 [A˚] and a few more Al/Si atoms as glue atoms filling space between the clusters.

180


States/(Ry.Unit cell)

3000 EF 2000

1000

0

–0.8

–0.6 –0.4

–0.2

0

0.2

0.4

Energy (Ry) 80

10 Al-site

Local density of states (states/site Ry)

Local density of states (states/site Ry)

Mn-site

70 60 50 40 30 20 10 0 –1

8 6 4 2 0

–0.5

0

Energy (Ry)

0.5

–1

–0.5

0

0.5

Energy (Ry)

Figure 6.3. Total and local DOS for a-Al–Si–Mn where Al/Si sites are assumed to be occupied by Al [22,2].

The innermost shell of the MI cluster is an icosahedral shell of twelve Al atoms and twelve Mn atoms are on the vertices of the Al icosahedron. On the edges of the Mn icosahdron, Al atoms are placed to form a third shell. Favorable positions of Si atoms are not known experimentally and all Si atoms are substituted by Al atoms in the calculation. The calculated total DOS (Figure 6.3(a)) reveals a pseudogap at the Fermi energy with a width of 0.5 [eV] and consists of a dense set of spikes with a width of 10–50 [meV] or even narrower. A large peak at E ¼ –0.2 –0.05 [Ryd] comes from the Mn 3d states and a deep tail of the DOS mainly from the Al 3s and Mn 4s states. The narrow Mn-d band is due to absence of neighboring Mn-Mn pairs in the model structure. One can see a typical resonance shape of the Mn-d states embedded in a continuous spectrum formed by Al and Mn s and p states and the pseudogap corresponds to the resonance gap (Figure 6.3(b)). Qualitative arguments on the TM-d states are proposed within Anderson model for magnetic impurity [14]. The local DOS for the TM-d states is calculated from the local Green function Gd (z) as 1 nd ðEÞ ¼ Im Gd ðE þ iÞ;

ð6:10Þ


181

where the local Green function Gd (z) is given by Gd ðzÞ ¼ ½z "d d ðzÞ 1 :

ð6:11Þ

If one neglects the electron-electron interaction in the d states, the self-energy d (z) is calculated in terms of the sp-d mixing as d ðzÞ ¼

X

jVsd j2 ; z "

ð6:12Þ

where summation is taken over the sp electron states labeled by a. If one neglects a real part of the self-energy, the local DOS is calculated as nd ðEÞ ¼

1 Im d ðE þ iÞ ; ðE "d Þ2 þ ðIm d ðE þ iÞÞ2

ð6:13Þ

and D E Imd ðE þ iÞ ¼ NðEÞ jVsd j2 ; E

ð6:14Þ

with the average sp-d mixing at " ¼ E and the density of states of the sp electrons N(E ). If an imaginary part of the self-energy is almost constant near the atomic level for the TM-d, the local DOS for the TM-d state is Lorentzian as is well-known for the virtual bound state. If N(E ) is strongly varying because of the Bragg scattering of the sp electrons as is demonstrated for a hypothetical QC of Al [15], the local DOS is significantly modified from that of the virtual bound state yielding a deeper pseudogap. A shape of the resonance gap above the Fermi level for a-AlMn seems to be consistent with such modification of the virtual bound Mn-d state. The position of the pseudogap and the Fermi energy measured from the bottom of the wide Al-sp bands for a-AlMn corresponds to kF = 1.6 [A˚–1]. A quasilattice constant aR for QCs of the Al-TM family is measured as 4.6 [A˚] and hence a unit of the reciprocal lattice vector 2/aR is 1.4 [A˚–1]. The value of 2kF fairly coincides with the momentum transfer of strong Bragg scatterings indexed as (211111) and (221001). Therefore it is reasonable to conclude that the formation of the pseudogap is due to the FS–BZ interaction and the system is stabilized electronically. The Fermi energy in the calculated DOS is located slightly below the minimum of the DOS. In the stable structure, a few Al sites are actually occupied by Si atoms and the Fermi level would shift towards the minimum of the DOS. This would be the stability enhancing mechanism of Si substitution. The Al-p states are strongly mixed with the Mn-d ones. The Mn-d states are pushed down and the Al-p states are pushed up due to the p-d hybridization. The resultant Fermi energy shifts downwards by about 2 [eV] to be pinned at the resonance gap, but the bottom of the conduction bands does not change. Hence the p-d hybridization induces trends of charge transfer from Al to Mn, leading to effective negative valence for Mn in Al. The contribution of the Al-d states may play some roles in the pseudogap formation. The Al-d states push down the Al-p states because of the orthogonality in

182


the same atomic region (or, more specifically in the LMTO method, atomic sphere) and this makes the hybridization of the Al-p and Mn-d states more effective. A dense set of the sharp spikes is another characteristic feature in the DOS of the approximants. For an approximant crystal with a large unit cell, the Brillouin zone is folded and many band branches cross, mix with each other and split into separated bands. Overlap of the bands is quite small and each band is very flat. Then the van Hove singularities in the DOS form a set of spikes in the DOS. The character of the wave functions within one band may vary very rapidly by changing the wave vector because the flat band does not imply a dispersionless localized state. Many experimentalists tried to observe the spikes in the DOS but there have never been experimental evidence for the fine structure in the DOS. Probably thermal noise and self-averaging due to structural disorder, which is inevitable in real samples, may smear the fine structures. Although the sharp spikes in the DOS do not contribute to stabilization of the system, they are expected to affect low-energy response such as transport properties in a system free from chemical and structural disorder. Al–Cu–Fe is a system forming stable icosahedral QC [25]. This system is known as face-centered ordered phases with highly perfect structural order. Electronic structure calculations for these stable QCs have been done by several groups. In Figure 6.4, the DOS for a hypothetical 1/1 cubic approximant of Al–Cu–Fe is shown [26]. A pseudogap with a width of 0.5 [eV]. is obtained near the Fermi level. A peak at –0.35 [Ryd] comes mainly from the Cu-3d states and is fully occupied. Less remarkable peaks just below the pseudogap at –0.16 and –0.22 [Ryd] come from the Fe-3d band strongly hybridized with Al-3p states. Two peaks are attributed to inequivalent Fe atoms. Detailed analysis of the local p- and d-DOS leads to conclusion that chemical bonds between Fe and neighboring Al are formed. Trambly de Laissardie`re et al. studied electronic structures of several Al–TM alloys, which are not approximant crystals but found in similar Al-rich composition to QCs

4000

DOS (States/Ry.Unit cell)

EF 3000

2000

1000

0

–0.8

–0.6

–0.4

–0.2

0

0.2

Energy (Ry) Figure 6.4. Electronic DOS for 1/1 approximant of Al–Cu–Fe [26].


183

[27]. They calculated the DOS without sp-d hybridization by suppressing the corresponding matrix elements in the LMTO Hamiltonian and found that valleys in the DOS become less pronounced and disappear for some alloys by suppressing the sp-d mixing (Figure 6.5). Although a charge transfer from one element to another in compounds cannot be estimated reliably by counting the number of electrons in the

2

(a)

1.5 1 0.5 0

(g) 5

Total sp DOS (states/(eV unit cell)

4 3 2 1 0

(h) 12

8

4

0

( j) 8 6 4 2 0 –12

–8

–4

0

4

Energy (eV) Figure 6.5. Partial sp-DOS for Al3Ti (a), Al6Mn (g) Al7Cu2Fe (h) and Al6Co2( j) calculated with (full curve) and without (dotted curve) sp-d mixing. Vertical dashed line indicates the Fermi energy [27].

184


atomic sphere, Trambly de Laissardie`re et al. calculated difference in the number of the sp electrons with and without the sp-d mixing. They stressed that a charge transfer from the conduction band to the transition metal d-state does not take place actually because of a too large electrostatic energy, but an increase of the sp electrons around TM atoms is what Raynor called the negative valence. In fact, a difference in the number of the sp electrons agrees quantitatively with Raynor’s value. Among the Al–TM alloys studied by Trambly de Laissardie`re et al., Al3Ti and Al3V (DO22) are interesting systems with a very deep pseudogap in the DOS. The DO22 structure is an ordered structure on a tetragonal lattice where an atomic layer consisting of only Al atoms (the Al layer) and that consisting of Al and TM (the Al– TM layer) are stacked alternatively along the (001) direction. If the stacking of layers along the (001) direction is slightly modified, one obtains different ordered structure called L12. Krajcˇı´ and Hafner [28] calculated electronic structures of Al3V with different types of ordering, DO22 and L12. A deep pseudogap near the Fermi level is obtained for DO22 but not for L12 (Figure 6.6). They found an enhanced charge density along the Al–V bond for the DO22 structure and claimed that such charge density distribution is characteristic for covalent bonding. One can show that the psudogap formation is due to strong mixing of TM-d states with Al-3p states [29]. As the strong Al–TM covalent bond for the DO22 structure is between atoms in the neighboring layer along the (001) direction, termination of the bulk structure at the Al–TM plane may induce dangling bond states. In Figure 6.7(a), the local DOS at the surface Al–TM layer obtained by slab calculation for Al3V is shown. A peak at the Fermi level, which is not seen in the bulk DOS, demonstrates formation of the dangling bond. If one terminates the crystal at the Al layer, noticeable dangling bond states do not appear but the charge transfer to the second Al–TM layer takes place as demonstrated by a shift of the local DOS near the Fermi energy to lower energies (Figure 6.7(b)). In Al-rich alloys with late transition elements, there is a trend of the charge transfer from Al to TM. Although the trend is not so remarkable for V and early transition elements, a small amount of charge transfer is still expected. If the Al–TM layer above the Al layer is missing, the valence electrons are transferred backwards to the bulk Al–TM layer. Recently clean surfaces are investigated under ultra-high vacuum for Al–Pd–Mn and other stable QCs [30]. It is known

10 A13V (DO_22)

DOS (states/eV cell)


10 8 6 EF

4 2 0 –12

–10

–8

–6

–4

–2

Energy (eV)

0

2

4

A13V (L1_2)

8 6 EF

4 2 0 –12 –10

–8

–6

–4

–2

0

2

4

Energy (eV)

Figure 6.6. Electronic DOS for Al3V with DO22 and L12 ordered structures, which are shown in insets. Large black spheres denote V atoms [29].

185


(a)

(b) 20

20 Al surface



Al-TM surface 15 EF 10

5

0 –12

–10

–8

–6

–4

–2

Energy (eV)

0

2

4

15 EF 10

5

0 –12

–10

–8

–6

–4

–2

0

2

4

Energy (eV)

Figure 6.7. Local DOS at the surface layer of Al3V slab for different terminations. Top layer is the Al-TM layer (a) and the Al layer (b). Dotted curve shows the DOS for a bulk layer [29].

experimentally that the Al-rich surfaces are usually obtained but the surface with many TM atoms are not. If the local covalent bonding is realized near TM sites, the TM-rich surface would be less stable because of formation of the dangling bonds. Theoretical work on the surface electronic structures of QCs is certainly expected.1 Another interesting example is binary Al2Ru (C54), which is a semiconducting alloy with a real energy gap at the Fermi energy. Krajcˇı´ and Hafner studied electronic structures of this semiconducting Al–TM alloys extensively [32,33]. They calculated the COOP for neighboring Al–TM and Al–Al pairs and found that the bonding nature of the electron eigenstates changes from bonding to antibonding at the Fermi energy. Formation of covalent bond is also confirmed by the enhanced charge density between atoms. It is also interesting to note that Al4MnCo alloys derived from the Al2Ru structure, where Mn and Co are chosen so that total valence equals to two Ru atoms, are also semiconducting with a real energy gap at the Fermi level. This implies that substitution of transition elements does not induce a substantial change in the shape of the DOS near the pseudogap though the d-states for Co and Mn are located slightly deeper and shallower, respectively, compared with that of Ru. It is desired to investigate such a rigid-band picture from a viewpoint of the charge transfer from Co to Mn. Higher-dimensional structural models for Al–TM family of QCs have been proposed by several groups [8,34]. Approximant crystals can be derived from the higher-dimensional models although further modification is needed to treat the fractionally occupied atomic sites for electronic structure calculations. For the 1/1 approximant with a few hundreds of atoms in a unit cell, structural optimization using the plane-wave-based method is feasible. Krajcˇı´ and Hafner [35] performed optimization of the atomic positions for the 1/1 approximant Al88TM40 (TM = Ir) derived from the six-dimensional model proposed by Katz and Gratias and 1

Recently, Krajcˇı´ and Hafner have reported calculations of slab models for 10-fold surfaces of the Al–TM QCs [31]. There are still some difficulties in calculations because a size of a unit cell for the approximant for 10-fold surfaces and thickness of vacuum layer cannot be taken large enough. Nevertheless their work is the first attempt to study QC surfaces by the first-principles calculation.

186


Boudard et al. [34]. They found that the narrow energy gaps open in the DOS for the unrelaxed structure but the structural relaxation results in closing one gap and widening substantially the other one. If one assumes a rigid-band behavior, it is expected to obtain the semiconducting 1/1 approximant, for which the Fermi energy is exactly located in the energy gap, by substituting transition elements. The semiconducting approximants are thus predicted for Al88Pd24Ir16, Al88Pd36W4, Al88Au12Ir28 and others. When one introduces defects by substitution of Al and TM atoms, localized defect states appear in the energy gap. If one goes beyond the 1/1 approximants, the ab inito structural relaxation is not feasible and the electronic states are calculated only for the ideal positions obtained by a six-dimensional cut projection. The band gap does not appear as clearly as in the relaxed 1/1 approximant though the pseudogap or the narrow gap in the DOS is certainly formed in the DOS. From these observations it has been concluded that the complete ordering of Al and TM atoms together with the structural relaxation may realize the semiconducting QC. In the Al–TM compounds with simple crystal structures, formation of the covalent bonding is demonstrated by analysis of the charge density distribution and the COOP [28,32,33]. In the 1/1 approximant Al88Pd24Ir16, the charge density is enhanced between Al–TM and Al–Al atoms. From these observations, we can infer that a deep pseudogap or a real energy-gap in Al–TM systems is a consequence of covalent bondings between Al–TM and Al–Al atoms. Recently Kirihara et al. [36] have obtained electron density distribution experimentally by using the maximum entropy method with synchrotron radiation powder data for a-AlMnSi. They found significantly anisotropic charge distribution, which is claimed to be evidence of directional covalent bonding in a-AlMnSi, whereas rather uniform charge distribution is obtained for metallic Al12Re, in which Al12 icosahedral clusters with Re at the center are placed at a corner and a body-center of a cubic cell. Fujimori and Kimura [37] carried out semiempirical molecular orbital calculation for icosahedral Al clusters with and without Al at the center. They found that, when an external atom is added to the cluster, it is bound at a tetrahedrally closed-packed site for the cluster with the center whereas a on-top site is favored for the cluster without the central Al. From this observation, they concluded that metallic bonding favoring close-packed structures is realized for the cluster with the center whereas covalent bonding favoring porous network structures with directional bonds is realized in the cage cluster. Although selection mechanism of the close-packed and porous network structures is not clear, such coexistence of metallic and covalent natures in trivalent Al may provide a key for understanding covalent bonding in Al-rich alloys. Another interesting observation concerning the covalent nature in Al–TM QCs was done by Terauchi et al. [38]. They measured the Al L-shell excitation spectra from Al–Ni–Co and Al–Ni–Rh decagonal QCs by a high-resolution electronenergy-loss spectra (EELS) microscope and found that the binding energy of the Al L-shell electrons is greater than that of pure Al by 0.1–0.4 [eV]. They conjectured that the increase in the binding energy indicates that the number of valence electrons at Al atom sites in QCs is smaller than that in pure Al and hence the covalent nature in the bonding in QCs is more remarkable than that in pure Al.

187


m5

B Bands

(b)

Energy (Ry)

0.0

(a)

m3 (c)

m2 m4

–0.1 m1

Γ

S

P Γ

Z Γ

C Bands

A Bands

Z

Figure 6.8. Energy dispersion for decagonal Al–Cu–Co [39].

Decagonal QCs are found in several Al–TM compounds such as Al–Mn, Al–Cu–Co, Al–Co–Ni and so on. For decagonoal QCs, electron microscopy gives direct information on quasiperiodic arrangements of atoms and clusters in a plane perpendicular to the periodic axis. Hence several atomic models have been proposed and available for studies of the electronic structures. Trambly de Laissardie`re and Fujiwara [39] calculated electronic structures of decagonal Al–Cu–Co for a model approximant proposed by Burkov [40]. Structural anisotropy causes a large anisotropy in the electronic structure. The energy dispersion similar to the free-electron band is obtained along the periodic direction whereas flat bands with large effective masses are obtained within the quasiperiodic plane (Figure 6.8). In spite of such anisotropy, the pseudogap in the DOS is obtained near the Fermi level. It is claimed, however, that a shape of the DOS and the pseudogap are very sensitive to the structural models, in particular, to the positions of Cu and Co. This fact certainly indicates importance of the sp-d hybridization. Krajcˇı´ et al. [41] have demonstrated the sensitiveness of a shape of the DOS with several variants of the Burkov model and concluded that position of the Co DOS is crucial. If spectroscopic data are available, one can use them for prediction of optimal ordering of transition elements in QCs by comparing photoemission data with calculated DOS. Krajcˇı´ et al. [42] applied this idea to chemical ordering in columnar clusters in decagonal Al–Co–Ni.

6.4.2.

Bergman phases

Al–Cu–Li QC is identified as the T2-phase in equilibrium phase diagram and thus thermodynamically stable [43]. The cubic crystalline phase called the R-phase is found at composition very close to the T2-phase [44] and regarded as the 1/1 approximant of QC. The Al–Cu–Li R-phase is isostructural to the (Al,Zn)49Mg32 Bergman phase [45], which is a typical Frank-Kasper alloy with tetrahedrally closed

188

Chapter 6. Electronic Structures and Stability Mechanism of Quasicrystals Table 6.1. Atomic position and site occupation of Bergman phase.

Site – Al(1) Li(1) Li(2) Al(2) Al(3) Al(4) Li(3)

Wyckoff symbol

x

y

z

2a 24g 16f 24g 24g 48h 12e 12e

0 0 0.1874 0 0 0.1574 0.4037 0.1985

0 0.944 0.1874 0.3047 0.1802 0.1896 0 0

0 0.1544 0.1874 0.1171 0.3150 0.4059 0.5 0.5

Al–Cu–Li

Al–Zn–Mg

Cd–Pb–Na

– Al/Cu Li Li Al/Cu Al/Cu Al Li

(Al) Al/Zn Mg Mg Al/Zn Al/Zn Mg Mg

– Cd/Pb Na Na Cd Cd/Pb Cd/Pb Na

packed structure and can be interpreted as a 1/1 approximant of icosahedral QC [46]. The structure of the R- or Bergman-phase is interpreted as a body-centered cubic lattice where an atomic cluster of an icosahedral shape (Bergman cluster) is placed at a corner and a body-center of a unit cell. Space group is Im-3 and a lattice constant is 13.91 [A˚] [47] and 14.16 [A˚] [45] for Al–Cu–Li and Al–Zn–Mg, respectively. In Table 6.1., the structural data for the Al–Li–Cu R-phase and atomic species at the corresponding site in the (Al,Zn)49Mg32 Bergman phase are shown. The 2a site was originally assigned as to be occupied by Al, but more recent observation shows that it is an empty site. One can see that most of Al sites are fractionally occupied by either trivalent Al or monovalent Cu for Al–Cu–Li and divalent Zn for Al–Zn–Mg. This suggests that the number of the valence electrons is controlled by changing Al–Cu and Al–Zn compositions. Also one can see that the Li sites in Al–Cu–Li are occupied by Mg in Al–Zn–Mg except for the Al(4) site. A building block of the Bergman alloy is an icosahedral cluster placed at a vertex and a body-center of a cubic unit cell. The innermost shell of the cluster is an icosahedron formed by twelve Al/Cu atoms at Al(1) site. Twenty Li atoms are placed on a face of the innermost icosahedron (Li(1) and Li(2) sites) to form a dodecahedral shell, and then twelve more Al/Cu atoms are placed on a face of the Li dodecahedron (Al(2) site). The 4th shell is a truncated icosahedron of Al/Cu at Al(3) and Al(4). The outermost Li(3) sites form an interpenetrated triacontahedron together with atoms at the Li(1) and Li(2) sites. Fujiwara and Yokokawa [48] investigated the electronic structure of the R-phase of Al–Cu–Li. The occupation probability of Al in the Al(2) site is nearly 50% whereas the other Al/Cu sites are mainly occupied by Al. If one assumes that all the Al/Cu sites are occupied only by Al, the DOS is calculated as in Figure 6.9. A deep pseudogap above the Fermi level is very remarkable. The position of the pseudogap measured from the bottom of the Al-sp band corresponds to the momentum kF = 1.64 [A˚–1]. The value of 2kF coincides with the reciprocal lattice vectors (222100) and (311111)/(222110) and it seems reasonable to assume that the pseudogap is formed due to strong interference effects with the particular diffracted waves. If one assumes ordering such that the Al(2) sites belonging to a cluster at the corner of a cubic cell is differently decorated from that at the body-center, the symmetry is reduced to Pm-3. A shape of the DOS is essentially unchanged except for the shallow occupied Cu-d states. Main effects of the Cu substitution are to

189


(a)

0

Density of states (states/Ry. cell)

0

(b)

700

200

0

0

(c)

1400

Integrated density of states (states/cell)

200

700

400

0

0 –1.0

0

0.5

Energy (Ry) Figure 6.9. Electronic DOS of Al–Cu–Li. Partial Al/Cu sites are occupied by Li (a), Cu (b) and Li and Cu (c) [48].

reduce the total number of the valence electrons and eventually to shift the Fermi energy toward the minimum of the DOS. Similar effects are also observed in Al–Si–Mn as discussed above. This is again indirect evidence that the pseudogap and the stability are obtained due to the BZ–FS interaction. The formation mechanism of the pseudogap in this system, however, may not be so simple. The DOS for the hypothetical binary Bergman alloys Al–Li and Al–Mg are shown in Figure 6.10. Here all the Al/Cu and Al sites in Al–Cu–Li are occupied by Al and the Li sites are occupied by Li or Mg. So these two systems are completely isostructural but Mg atoms in one system are replaced with Li in the other. Nevertheless there is a very deep pseudogap in the DOS for Al–Li but that for Al–Mg is much shallower. Therefore choice of chemical species is essential not only for controlling the valence electron density but for the pseudogap formation. Sato et al. have arrived at different conclusion [49]. They calculated the electronic DOS within the nearly-free electron model with the model pseudopotentials and reported that a deep pseudogap is obtained both for Al–Cu–Li and Al–Zn–Mg. They

190

Chapter 6. Electronic Structures and Stability Mechanism of Quasicrystals 60

60


Al-Li (Bergman phase)

Al-Mg (Bergman phase) EF

EF

40

40

20

20

0

0 –10

–5

0

Energy (eV)

5

–10

–5

0

5

Energy (eV)

Figure 6.10. Electronic DOS for isostructural Al–Li (a) and Al–Mg (b) Bergman phases.

concluded that the site configurations around Al atoms are essential in the Al-3p hybridization and the pseudogap formation. There is a family of intermetallic compounds called a Zintl phase [50]. These compounds are composed of alkali (and sometimes alkaline-earth) metals with posttransition elements. Alkali elements are usually electro-positive and donate electrons to electro-negative post-transition elements. Then the post-transition elements are bonded to satisfy a so-called octet rule [51], which states that bonded atoms share their eight electrons with neighbors. Al–Li is known as a typical Zintl compound. In Al–Li, lithium donates one electron to aluminum and so aluminum has four electrons and prefers tetrahedral coordination to satisfy the octet rule. In fact, structure of Al–Li is of Na–Tl type (B32), which is an ordered structure on a bodycentered cubic lattice consisting of two diamond sublattices. Space group is Fd-3m and a lattice parameter is 6.3757 [A˚] [52]. The electronic DOS and band structure of the Al–Li Zintl phase are shown in Figure 6.11 [53]. The Fermi level is located in the pseudogap in the DOS and one can see similarity in the band structure of this system to Si diamond, where the pockets near the X point shift to higher energies to open a real energy gap. So we conclude that this pseudogap is caused by a similar mechanism for the real band gap in Si diamond and the covalent bonding in Al–Li is induced by the charge transfer from Li to Al. To confirm this idea, the charge density map is shown in Figure 6.12 [53]. Enhancement of the bond charge between Al atoms is clearly observed. For a hypothetical isostrutural Al–Mg compound, for which a lattice parameter is chosen to minimize the total energy as 6.67 [A˚], the electronic structures are quite different and there is no energy gap near the Fermi energy. This is because the ionic potential of divalent Mg is deeper than that of monovalent Li and hence the sp band of Mg shift to lower energies near the Fermi level. Such a trend is also interpreted in terms of difference in the electronegativity: Mg is less electro-positive than Li and hence the charge transfer to Al is less effective in Al–Mg. Cd–Pb–Na compounds are other examples to form both Bergman and Zintl phases. In the Cd–Pb–Na Bergman phase (a = 15.992 [A˚]) [54], the Al sites in Al–Cu–Li are occupied mainly by Cd and the valence electron density seems to be controlled by substitution of Pb for Cd. A noticeable pseudogap is obtained in the DOS for the model structure where all the Al sites are assumed to be occupied by Cd

191


DOS (states/Ry. cell)

40

40 Al-Li Zintl phase

Al-Mg Zintl phase

30

30 EF

20

20

10

0 –0.8

EF

10

–0.6

–0.4

–0.2

0

0.2

0 –0.8

0.4

–0.6

Energy (Ry.)

Energy (Ry.)

–0.4

–0.2

0

0.2

0.4

Energy (Ry.)

0.4

0.4

0.2

0.2

0

0

–0.2

–0.2

–0.4

–0.4

–0.6

–0.6

–0.8 L

Γ

X

W

L

Γ

K

–0.8

Γ

L

X

W

L

K

Γ

Figure 6.11. Electronic DOS and band dispersions for isostructural Al–Li and Al–Mg (hypothetical) Zintl phases (B32) [53].

Li Al

Figure 6.12. Contour plot of the total charge density on (110) plane of Al–Li Zintle phase [53].

(Figure 6.13) [53]. Here the states at about –10 [eV] are the Cd-4d band, which is fully occupied. In the Cd–Pb–Na2 Zintl phase, Na atoms are placed on one diamond sublattice and Cd/Pb atoms are placed on the other to form a zincblend structure. Space group is F-43m and a lattice parameter is 7.39 [A˚] [55]. Electropositive sodium

192

Chapter 6. Electronic Structures and Stability Mechanism of Quasicrystals Cd-Na (Bergman phase)

70

50

EF

40 30 20 10 0

Cd-Pd-Na (Zintl phase)

80



60

60 50

EF

40 30 20 10

–10

–5

0

5

Energy (eV)

0 –1

–0.8

–0.6

–0.4

–0.2

–0

0.2

Energy (Ry.)

Figure 6.13. Total DOS for Cd–Na Bergman phase (left) and Cd–Pb–Na2 Zintl phase (right) [53].

atom donates one electron to the Cd/Pb sublattice, which has an average valence, 3. The DOS of the Cd–Pb–Na2 Zintl phase is actually similar to that of the Al–Li Zintl compound except for splitting of the valence band and a very narrow peak at –0.9 [Ryd] coming from the Cd-4d states. Although stabilization mechanism of this family of Cd–Pb–Na compounds may not be the same as those for recently discovered Cd- and Zn-based QCs, which are alloys of Cd and Zn with divalent Ca/Yb or trivalent early-transition elements, a charge transfer between post-transition elements (Zn, Cd) and electro-positive elements should be studied in more detail in future.

6.4.3.

Zn–Mg–RE Compounds

Zn–Mg–RE (RE = Y or rare-earth elements) is a first stable QC in the FK family with the face-centered ordering [56]. Zn–Mg–RE alloys form also a series of unique hexagonal crystals in a composition range close to icosahedral QC with almost the same periodicity along the c-axis, c = 8.6 [A˚]. We refer to these hexagonal crystals with lattice parameters a = 14.6, 23.5 and 33.6 [A˚] as S-, M-, and L-phases, respectively. Hexagonal Laves (C14) phase, which is a typical FK alloy with the tetrahedrally closed-packed structure, Zn2Mg, with a = 5.2 [A˚] and c = 8.6 [A˚] may be involved in this family. Crystal structures of the hexagonal phases have been studied by Takakura et al. [57], Sugiyama et al. [58], and Abe et al. [59] It is interesting to note that the periodicities of the S-, M-, and L-phases in the hexagonal plane are scaled approximately with the golden mean. Space group of the hexagonal phases is P63/mmc and there are 92 and 248 atoms in a unit cell for the S- and M-phase, respectively. In the S-phase, there is one fractional site occupied by either Mg or Y with probability 0.85 for Mg [57]. In the electronic structure calculations presented below, we assume that this site is fully occupied by Mg. Although the structural model for the M-phase is proposed for Zn–Mg–Sm [59], the same atomic coordinates are assumed for Zn–Mg–Y. Compositions of model structures are Zn60Mg26Y6 and Zn162Mg60Y26 for the S- and M-phases, respectively.

193

6.4. Electronic structures of QC-related compounds 1000

Zn-Mg-Y

Total


800

600

EF

400

200

0 800

d-DOS


400

0 800

p-DOS

400

0 800

s-DOS

400

0 –1

–0.8

–0.6

–0.4

–0.2

0

0.2

Energy (Ry.) Figure 6.14. Total and partial DOS for hexagonal S-phase of Zn–Mg–Y [60].

The total, partial p- and d-DOS for the S-phase of Zn–Mg–Y are shown in Figure 6.14 [60]. The Zn-3d band appears at about –0.7 [Ryd], which is about 0.6 [Ryd] below the Fermi level and does not contribute cohesion of the present systems. One can see deep pseudogap near the Fermi energy and enhancement of the DOS for the states below and above the pseudogap. The states below the pseudogap are mainly made from the Zn-4p states but a small amount of the d components is involved. Similar separation of the occupied p- and the unoccupied d-states is observed for cubic Cd–M (M = Yb and Ca), as shown later. For cubic Cd–M systems, Ishii and Fujiwara [61] concluded that the low-lying unoccupied d-states in Yb/Ca are important for the pseudogap formation in the Cd-based compounds. It would be reasonable to consider that the pseudogap in Zn–Mg–Y

194


(a)

(b) 1000 Zn1 to vacancy



Y to Mg 800 EF 600

400 200 0 –1

–0.8

–0.6

–0.4

–0.2

Energy (Ry.)

0

0.2

800 EF 600

400 200 0 –1

–0.8

–0.6

–0.4

–0.2

0

0.2

Energy (Ry.)

Figure 6.15. The total DOS for the modified S-phase. (a) Y is replaced with Mg, which has no low-lying unoccupied d-states. (b) Zn at Zn(1) sites is removed [60].

is also induced by the sp-d hybridization [62]. However, the pseudogap formation in the hexagonal Zn–Mg–Y is not entirely due to the hybridization effects of the Y-4d states. The DOS for a hypothetical S-phase where Y atoms are replaced with Mg, which has no low-lying unoccupied d-states, still has a shallow pseudogap (Figure 6.15(a)). It is safe to say that the sp-d hybridization enhances the pseudogap to some extent. A diameter of the FS, 2kF, for the hexagonal S-phase is estimated as 3.0–3.1 [A˚–1]. A quasilattice constant for Zn–Mg–RE is about 5.2 [A˚] and hence 2/aR is 1.2 [A˚–1]. A bunch of Bragg peaks for the hexagonal crystals and also the (222100) and (311111)/(222110) Bragg peaks for icosahedral Zn–Mg–RE are found in this range of wavevectors. So the BZ–FS interaction seems also important for the gap formation. This is consistent with the above observation that the pseudogap does not disappear when Y atoms are replaced with Mg because the structure factor does not change very much by substitution. Recently Suchodolskis et al. [63] showed that the pseudogap obtained in photoemission experiments for icosahedral Zn–Mg–Y is explained in the framework of the HR mechanism by assuming appropriate values for particular Fourier transform of the pseudopotentials for electrons V311111 and V222100. But we suspect that the DOS may be modified more drastically if one considers all the Fourier components with the amplitude consistent with the assumed ones. To demonstrate that the pseudogap formation is not entirely due to the HR mechanism, we show the DOS for a modified S-phase where twelve Zn atoms, which are designated as Zn(1) in Ref. [57], are replaced with vacancies. Although the structure factor does not change very much by this substitution, the pseudogap seen for the perfect crystal almost disappears (Figure 6.15(b)). This suggests that the electronic states near particular atomic sites are responsible for the pseudogap formation. The DOS is enhanced below the pseudogap. It is reasonable to interpret this enhancement as due to formation of bonding orbitals by hybridization. To observe spatial distribution of the bonding orbitals, we calculate the valence electron density made-up from the states in an energy window below the pseudogap, –0.24 < E < –0.16 [Ryd] for the S-phase [60]. The contour plot for the z = c/4 plane (the flat layer)


195

S-ZnMgY (z = c/4)

Zn1

Figure 6.16. The contour plot of the valence charge density made up from just below the pseudogap on the flat layer at z=c/4 for the S-phase of Zn–Mg–Y. Dashed curve are the contours for higher density. A dotted line indicates a unit cell [60].

is shown in Figure 6.16. Intervals of contours are 0.0002 [a.u.]. The highest electron density is found between neighboring Zn(1) sites for the S-phase as indicated by an arrow in Figure 6.16. Distance between the neighboring Zn(1) sites is about 2.5 [A˚], which is as small as a covalent bond length for Zn. We speculate that this bonding is important for stabilizing the present compounds. To analyze the bonding nature of the electronic states furthermore, we evaluate the COHP for the S-phase of Zn–Mg–Y [64]. The ‘total’ COHP is a sum of the COHP for neighboring pair of atoms with multiplicity in a unit cell. The COHP are calculated for all pairs of atoms with distance less than 3.0, 3.2, and 3.4 [A˚] for Zn–Zn, Zn–Mg, and Zn–Y pairs, respectively. This yields 8–12 neighbors for Zn atoms. The total COHP changes its sign near the pseudogap and the unoccupied states above the pseudogap are strongly anti-bonding (Figure 6.17). Such crossover in the bonding nature is usually seen in semiconductors. From energy dependence of the COHP for Zn–Zn and Zn–Y pairs, one obtains further information about chemical bonding. We show here the COHP for Zn–Zn and Zn–Y bonds separately for Zn–Mg–Y. One can see the COHP for Zn–Zn bonds changes its sign almost exactly at the Fermi energy. On the other hand, the COHP for Zn–Y bonds shows that the states in the pseudogap and those just above it are bonding ones. This implies that the pseudogap is not the bonding-anti-bonding splitting of orbitals centered at Zn and Y sites. These observations are followed by conjecture that the Y-4d and the Zn-4p states make pd-hybridized orbitals and the pseudogap is the bonding-anti-bonding splitting of such orbitals.

196

Chapter 6. Electronic Structures and Stability Mechanism of Quasicrystals 40

20

Zn-Zn

20

–COHP (/cell)

–COHP (/cell)

Total

0

EF

–20

–40 –1

–0.8

–0.6

–0.4

–0.2

0

10

0

EF

–10

–20 –1

0.2

–0.8

Energy (Ry)

–0.6

–0.4

–0.2

0

0.2

Energy (Ry)

6

Zn-Y

–COHP (/cell)

4 2 0 –2

EF –4 –6 –1

–0.8

–0.6

–0.4

–0.2

0

0.2

Energy (Ry)

Figure 6.17. Total COHP (a), COHP for Zn–Zn bonds (b) and COHP for Zn–Y bonds in the S-phase of Zn–Mg–Y. Values of the COHP multiplied by –1 are plotted [64].

The hexagonal Zn–Mg–Y phases are not exactly approximant ones because the structures are not generated simply by introducing phason strains into quasiperiodic structures. Nevertheless we have a series of unique hexagonal phases with lattice constants scaled with the golden mean. The electronic DOS for the hexagonal M-phase, which contains 248 atoms in a unit cell, and that for the hexagonal Laves phase Zn2Mg are shown in Figure 6.18 [65]. The DOS for the hexagonal Laves phase does not have a distinct gap at all and no enhancement of the p-DOS near the Ferni energy is observed. This may be consistent with the above conjecture that the pd-hybridized states made from the Y-4d and the Zn-4p states are essential in the gap formation. The DOS for the M-phase seems similar to that for the S-phase although the pseudogap is a bit shallower for the M-phase. If the pseudogap in QCs and approximants is induced by the FS–BZ interaction, one expects more remarkable pseudogap for QCs than approximants and for larger approximants than smaller ones because of sphericity of the pseudo BZ. The above trend in the DOS for the S- and M-phases of Zn–Mg–Y, however, suggests that the expectation is not necessarily realized in the present systems. Structure of the M-phase consists of two kinds of atomic columns, one of which is the same as that in the S-phase [57,59]. Such difference in the columnar structure may be responsible for quantitative differences in the electronic DOS for the S- and M-phases. In fact, the Zn–Zn bond, which is

197

6.4. Electronic structures of QC-related compounds 100 Total

ZnMgY 2000



2500

EF

1500 1000 500

EF

60 40 20 0

0 d-DOS

2000

d-DOS

80 40


1000


Total

Laves-ZnMg 80

0 p-DOS

2000 1000 0

s-DOS

2000

0 p-DOS

80 40 0

s-DOS

80 40

1000 0 –1 –0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

Energy (Ry.)

0 –1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

Energy (Ry.)

Figure 6.18. Total and partial DOS for the M-phase of Zn–Mg–Y (left) and those for the hexagonal Laves Zn2Mg (right) [60,65].

discussed above as responsible for the pseudogap formation, is located at the surface of the column in the S-phase and the number of this bond is smaller for the M-phase.

6.4.4.

Cd- and Zn-based compounds

Stable binary icosahedral phases have been discovered in Cd–Yb and Cd–Ca alloys [66,67]. The quasicrystalline phases are identified as unknown phases in the phase diagrams, Cd5.7Yb and Cd17Ca3. At the composition very close to the QC, cubic crystalline phases, Cd6M (M ¼ Yb, Ca), are obtained [68,69]. Although the isostructural cubic alloys are also obtained for other systems with M ¼ Sr, Y and most of rare-earth elements [70], the icosahedral QC is realized only for M ¼ Yb and Ca. Icosahedral QCs have been obtained also in ternary Zn–M–Sc alloys with various kinds of third element M such as simple metals (M ¼ Mg) [71], noble metals (M ¼ Pd, Pt, Ag, Au) [72] and transition elements (M ¼ Mn, Fe, Co, Ni) [73]. In these Zn-based ternary QCs, the third elements may be substituted for Zn. Different ternary QCs Zn–M–Mg (M ¼ Hf and Zr) are also discovered and it is pointed out that Sc sites are replaced with Hf/Zr and Mg atoms [74]. This family of Zn-based QC may be closely related to the binary Cd–Yb and Cd–Ca QCs because of

198


similarity in chemical species and also existence of isostructural cubic approximant phases Zn17Sc3 and Cd6M (M ¼ Ca, Yb and others). Crystal structure of Cd6M has been studied for M ¼ Yb [68]. Cd6M is a body centered cubic crystal where an atomic cluster of icosahedral shape is placed at the corner and the body center of a cubic cell. A core of the cluster is an atomic shell of non-icosahedral symmetry as in a pseudo-Mackay cluster in Al–Pd–Mn [34]. Although recent careful experiments conclude that the structure of the cluster core is disordered in a complicated fashion [75], the simplest model is such that four Cd atoms are placed at vertices of a small cube with occupancy probability 0.5 [68]. Because Cd atom is not small enough to occupy neighboring vertices, it is reasonable to assume that the central shell of Cd atoms is of tetrahedral shape. The second and third atomic shells are a dodecahedron of 20 Cd atoms and an icosahedron of 12 M atoms, respectively. The fourth shell is a Cd icosidodecahedron obtained by placing 30 Cd atoms on the edge of the M icosahedron. A few more Cd atoms are shared by neighbouring clusters and form an incomplete triacontahedral shell [75]. Structure of Zn17Sc3 is composed of icosahedral clusters similar to those in cubic Cd6M (M ¼ Ca, Yb) but the innermost core of the cluster was reported as vacant. More recent observation, however, concludes that cubic Zn–Sc is isostructural to Cd6M and a core of the cluster is an atomic shell of non-icosahedral symmetry [76]. The electronic structures of cubic Cd6M crystals are calculated for several divalent elements [61]. To avoid the fractional occupation of Cd atoms in the electronic structure calculations, a tetrahedral cluster of four Cd atoms, instead of a cubic one, is placed at the corner and the body-center of the cubic unit cell. Space group is then I23. Atomic positions obtained for Cd6Yb are assumed for other cubic approximants as well. Lattice parameters are assumed to be 15.638, 15.680, 16.040 and 15.22 [A˚], respectively for M ¼ Yb, Ca, Sr, and Mg. where cubic Cd6Mg is hypothetical and the lattice parameter is chosen to give a minimum of the total energy. In Figure 6.19, a narrow band in the total DOS at about –0.8 [Ryd] is the Cd-4d band and does not contribute to cohesion. Another narrow peak just below the Fermi level for Cd6Yb is attributed to the Yb-4f states. Since the narrow 4f band is almost filled, Yb is divalent as is Ca. This is consistent with measurements of magnetic susceptibility [68]. A shallow dip in the DOS is seen near the Fermi level for M ¼ Yb, Ca, and Sr whereas it vanishes for M ¼ Mg. By checking decomposition of the DOS into partial waves, we find that the occupied states below the dip are predominantly made from the Cd-5p states. This is a qualitatively different feature from pure Cd metal, in which the s and p states contribute equally to the states near the Fermi Level. An unoccupied peak at 0.0–0.1 [Ryd] is made from the d states of Yb, Ca, and Sr whereas magnesium has no low-lying unoccupied d-states near the Fermi level. Therefore hybridization of the d states near the Fermi level is essential for the dip formation in the Cd-based compounds. Existence of the sp-d hybridization has been recently proved experimentally by resonance photoelectron spectroscopy and other techniques [77]. Cohesive energies for the cubic Cd6M compounds are calculated as 1.68, 1.75, 1.69, and 1.47 [eV/atom] for M ¼ Yb, Ca, Sr, and Mg, respectively. The cohesive energy for Cd6Mg at the optimal lattice parameter is similar to that for pure Cd (1.48 [eV/atom] for fcc with a ¼ 4.45 [A˚]) and smaller by about 15% than those for the other Cd6M compounds with the almost unoccupied d band near the Fermi level.

199


(a)

(b) 1000

1000


Cd-Yb

Cd-Ca

800

800

EF

600

600

400

400

200

200

0 –1

–0.8

–0.6

–0.4

–0.2

0

0 –1

0.2

(c)

EF

–0.8

–0.6

–0.2

0

0.2

0

0.2

(d) Cd-Sr


–0.4

Cd-Mg

800

800

EF

600

600

400

400

200

200

0 –1

–0.8

–0.6

–0.4

–0.2

Energy (Ry.)

0

0.2

0 –1

EF

–0.8

–0.6

–0.4

–0.2

Energy (Ry.)

Figure 6.19. Total and partial DOS for cubic Cd6M (M ¼ Yb (a), Ca(b), Sr(c) Mg(d)). Full, dotted and dashed curves represent the total, p- and d-DOS, respectively [61].

Therefore the hybridization of the d state with a wide sp band certainly contributes to stabilizing the cubic Cd6M compounds. A quasilattice parameter for the icosahedral Cd–Yb is 5.7 [A˚] and a diameter of the FS calculated from the electron density is very close to the (222100) and (311111)/(222110) Bragg scatterings and hence the ~ is satisfied also for the Cd-based compounds. Nevernesting condition 2kF |G| theless there is neither distinct dip (or pseudogap) in the DOS nor additional stabilization due to alloying for isostructural and isovalent Cd6Mg, which has a similar structure factor to the other Cd6M compounds. Thus we should say that the interference effect associated with the strong Bragg peaks in the structure factor is not of primary importance for the stability of the Cd-based compounds. For the FK family of QC without transition elements, e/a is usually close to 2.1. This empirical rule for e/a is not satisfied for the Cd-based compounds because all the elements are divalent and hence e/a is exactly 2. The Cd- and Zn-based compounds with trivalent elements are also obtained as cubic approximants. The DOS for the cubic Cd6Y and Zn17Sc3 are shown in Figure 6.20 [78]. Here the structure of Zn17Sc3 is assumed to be that without the tetrahedral core at the cluster center. The pseudogap is observed again at the Fermi level in both of the systems. Splitting of the p- and d-symmetric components below and above the pseudogap is similar to that in the other cubic Cd6M. Thus the sp-d hybridization is also important in stability of Cd6Y and Zn17Sc3. Since Y and Sc are trivalent elements, the Fermi

200


(a) 1000

Total


Zn-Sc 800 EF 600

400

200

0


800

d-DOS

400

0 800

p-DOS

400

0 –1

–0.8

–0.6

–0.4

–0.2

0

0.2

Energy (Ry.) Figure 6.20. Total and partial DOS for cubic Zn–Sc (a) and Cd6Y (b) [78].

energy is shifted towards a minimum of the DOS and we obtain larger cohesive energies, 2.15 and 2.48 [eV/atom] for Cd6Y and Zn17Sc3, respectively. Although the electronic energy could gain by the substitution of trivalent elements, the binary Cd-based QC is obtained only with divalent Yb and Ca. The atomic radii of the trivalent rare-earth elements and Y are considerably small (1.75–1.87 [A˚]) in comparison with the divalent elements, Yb and Ca (1.96–1.97 [A˚]), whereas that of Sr is large (2.15 [A˚]). Matching of the atomic size is certainly important for the QC formation in the binary systems. We should point out here that an exceptional case is divalent Eu, which does not form Cd–M QC in spite of a similar atomic size to Yb and Ca. For the Zn–M–Sc QCs, atomic radii of Zn and Sc are about 10% smaller than those of Cd and Yb/Ca, respectively, but those of the third elements are scattered in a wide range around the one for Zn. The Fermi levels of the Cd6M with divalent elements are pinned at the shoulder of the occupied band, not at the minimum of the DOS. Nevertheless reasonable amount of energetic stabilization is obtained for the cubic Cd6M compounds although the Fermi level is shifted to the minimum of the DOS, yielding the larger cohesive energy, by substituting trivalent atoms for Yb and Ca. This is because the

201


(b) 1000


Cd-Y

EF

Total

800

600

400

200

0


800

d-DOS

400

0 800

p-DOS

400

0 –1

–0.8

–0.6

–0.4

–0.2

0

0.2

Energy (Ry.) Figure 6.20. (Continued)

occupied states just below the dip are bonding orbitals, whose levels are lowered by alloy formation. To check this idea, we calculate the COHP for Cd6Ca [79]. The COHP is calculated for neighboring sites within atomic distances 3.2 and 3.6 [A˚] for Cd–Cd and Cd–Ca, respectively. This choice corresponds to assumeing 8–12 coordination numbers at Cd sites. In Figure 6.21, we show the COHP for pairs of sites in the different atomic shells in Cd–Ca. A label ‘S2-S4’ means that the COHP is summed over neighboring pairs of a site in the second dodecahedral shell and one in the fourth triacontahedral one. For pairs of Cd atoms between different atomic shells (Figures 6.21(a) and 6.21(c)), the COHP changes its sign and the bonding nature is changed at the Fermi level from bonding to anti-bonding. Such a trend is typical for covalent semiconductors with a band gap and differs from simple metallic behavior. The sharp crossover of the bonding nature for the Cd–Cd bonds is a consequence of hybridization of Cd-4p states with low-lying unoccupied d states. Signs of the COHP for the Cd–Ca bonds (Figures 6.21(b) and 6.21(d)) are not changed at the Fermi level and the Cd–Ca bonds are weakly bonding in the energy range of interest. These trends in bonding nature for the Cd–Cd and Cd–Ca bonds are similar to those observed for hexagonal Zn–Mg–Y discussed above.

202

(a)


(b) 10

20

Cd-Ca: S2-S4

–COHP (/cell)

15

Cd-Ca: S3-S4 5

10

5 0

0 –5

EF

–10

–5

EF

–10 –20

–1

(c)

–0.8

20

–0.6

–0.4 –0.2 Energy (Ry)

0

0.2

–10 –1

–0.8

–0.4 –0.2 Energy (Ry)

0

0.2

0

0.2

(d) 10

Cd-Ca: S4-S5

Cd-Ca: S3-S5

15

–COHP (/cell)

–0.6

5

10

5 0

0

–5

EF

–10

EF

–5

–10 –20 –1

–0.8

–0.6

–0.4

Energy (Ry)

–0.2

0

0.2

–10

–1

–0.8

–0.6

–0.4

–0.2

Energy (Ry)

Figure 6.21. COHP for neighboring bonds in cubic Cd6Ca. Values of the COHP multiplied by –1 are plotted [79].

So we presume a picture that the Cd/Zn p states make pd-hybridized orbitals with the d states of early transition elements and the pseudogap is the bonding–antibonding splitting of such orbitals could be a universal scenario of stability of Cdand Zn-based alloys. Recently, it is found that the cubic approximant Cd6M (M ¼ Yb, Ca) exhibits anomalous temperature dependence in the electric resistivity and specific heat at about 100 K [80]. Tamura et al. concluded that the anomalies are caused by an order–disorder transition originated from a change of orientation of the innermost first shell of the cluster, which is assumed to be a Cd4 tetrahedron. According to the specific heat measurements, entropy released at the transition is about kB log2 per tetrahedron. If we assume the simplest model, in which four Cd atoms are placed at vertices of a small cube with occupancy probability 0.5, the Cd4 tetrahedra are oriented at random either of two orientations at high temperatures and ordered at lower temperatures. X-ray and electron diffraction measurements at low temperatures suggest that the orientation pffiffiffi pffiffiof ffi the tetrahedra is ordered alternatively along (110) direction to form a 2a 2a a orthorhombic lattice [81]. Similar transitions have been found in cubic Zn–Sc [82] and Cd–M under high pressure [83]. First-principles total energy calculations for cubic Cd6Ca with different orientations of the tetrahedral core show that the anti-parallel configuration, in which the tetrahedron at a vertex of a cubic cell is in a opposite orientation of that at a body center, is more favorable than the parallel one [84,85]. Assigning two different orientations as Ising spin, one may construct a lattice gas model for the order–disorder


203

transition. The interaction for the nearest neighbor pair favors the anti-parallel configuration but the first-principles calculation with a cubic unit cell is not sufficient to obtain interaction parameters for further pairs. Widom and Mihalcovic [84] evaluate the interaction parameter for the next neighbor interaction from a system with a 2a a a unit cell. However, the Ising model thus obtained failed to predict the transition temperature correctly and it was suggested that fairly longrange interaction is needed to explain the experimentally observed transition temperature. First-principles total-energy studies with the core structure proposed by Gome´z and Lidin [75] have been done by Nozawa and Ishii [85]. In the model proposed by Gome´z and Lidin, a vertex of a tetrahedron is split into three partial sites to form a distorted tetrahedron. Six symmetry-related structures of the Cd4 distorted tetrahedron are obtained as optimal ones and the anti-parallel orientation for the clusters at a vertex and body-center of a unit cell is found to be more favorable. Detailed comparison among the anti-parallel orientations of the Cd4 shows that four configurations have the almost same total energy and this result suggests that there is still some disorder in orientation at low temperatures. Nozawa has also pointed out significant effects of structural relaxation because a distance between an atom in the (distorted) tetrahedron and one in the second shell is unphysically small. These results altogether suggest that the order–disorder transition observed in cubic Cd6M is more complicated than a simple picture of Ising-like degrees of freedom for the orientation of the tetrahedral clusters.

6.5.

Concluding remarks

The electronic structures of QCs and related intermetallic compounds are discussed with special emphasis on stability and formation mechanism of the pseudogap in the electronic DOS. For the Al-TM family, the TM d-states play important roles for pseudogap formation and stability. The charge transfer from electro-positive Al to electro-negative TM are induced by the sp-d hybridization in this family of compounds and this trend is important to control the effective valence electron density. The charge transfer induces enhanced charge density between Al and TM, which is characteristic in covalent bonding. Formation of such covalent bonds is demonstrated in several Al-based alloys. Bergman phase is an alloy phase, which contains no transition elements. It is demonstrated that the pseudogap formation is strongly affected by choice of chemical species. Al–Cu–Li and Al–Zn–Mg are Al-based alloys, in which Al is more electro-negative than Li and Mg contrary to the Al–TM family. Charge transfer from alkali and alkari-earth elements to Al induces covalent bonding between Al in Zintl compounds. There is no direct evidence for the covalent bonding induced by charge transfer in Bergman alloys at the moment and further careful studies should be desirable. The Zn–Mg–RE compounds form unique series of hexagonal crystals, which are not exactly approximants. Instead of icosahedral atomic clusters, the structure consists of characteristic columnar units. Unoccupied d-states of RE enhance the pseudogap but do not play decisive roles in the pseudogap formation. In newly discovered Cd- and Zn-based QCs and their approximants, almost unoccupied d-states of Yb, Ca, and Sc are indispensable for

204


the pseudogap formation and stability of these compounds. Analysis of the bonding nature of the electronic states leads us to conjecture that the d-states of Yb/Ca/Sc and the Cd/Zn p-states make pd-hybridized orbitals and the pseudogap is the bonding-anti-bonding splitting of such hybridized orbitals. Similar behavior of the bonding nature is suggested for the hexagonal Zn–Mg–Y where the d-states come from Y. The Hume-Rothery idea has been extensively used as a guiding principle for searching new material. With the Fermi wavenumber kF estimated from the valence ~ 2kF is satisfied for many known QCs, electron density, the nesting condition |G| where the reciprocal lattice vectors responsible for the FS–BZ interaction are those indexed as (211111) and (221001) for QCs in the Al–TM family, and (222100) and (311111)/(222110) for QCs in the FK family and presumably the Cd- and Zn-based QCs. Partial replacement of the constituent atoms, which can adjust the valence electron density to satisfy the nesting condition more strictly, leads to improvement of quality of QCs. New QCs are often obtained by replacing one element with others, if they are chosen so that the number of the valence electrons is unchanged. These facts are indirect but strong evidences supporting the statement that QCs are electronic compounds stabilized by the HR mechanism. Nevertheless we have emphasized in this article non-geometric effects in the electronic structures of QC-related compounds such as hybridization effects, induced covalent bonds, trends of charge transfer and so on. This is because the stabilization mechanism of QCs and related compounds is certainly more involved than the HR idea in a naive sense. For the thermodynamically stable QCs of high quality, we notice different trends in the electro-negativity for the Al–TM compounds and others. In Table 6.2, we show two different parametrizations of electro-negativity used in a field of chemistry [86,87]. One can see that the electro-negativity of Al is smaller than those of the late 3d transition elements (Fe, Co, Ni) and Cu. Thus, in the Al–TM compounds, which are combination of electro-positive Al and electro-negative late-transition elements, the sp-d hybridization induces charge transfer from Al to TM and formation of bond charge between Al and TM. For the Zn–Mg–RE and the Cd- and Zn-based compounds, on the other hand, Zn and Cd have larger electronegativity than Yb, Ca, Mg, Y, and Sc, which form QCs by alloying with Zn/Cd. Although a charge transfer is not remarkable in QCs and related compounds in this family, we have seen the charge transfer effect is very remarkable in the Zintl compound of Al–Li, where the charge transfer from electro-positive alkali metals to Al induces covalent bonds between Al. It is speculated that a deep pseudogap in the Al–Cu–Li Bergman phase and hence stability of Al–Li–Cu QC are consequences of such covalent nature induced by the charge transfer. In chemical bonding of atoms in molecules and condensed matter, coexistence and crossover between covalent and ionic natures are often observed. Fujimori and Kimura [37] have pointed out crossover between metallic and covalent bonding for Al–Al bonds depending on atomic arrangements. Also we have stressed here effects of the charge transfer, which may induce partly ionic configurations in the QC-related intermetallic compounds. In this respect, we conclude that the stabilization mechanism of QCs and related compounds is more involved than the HR idea in a naive sense and chemical effects controlling metallicity, covalency and ionicity of material are certainly important.

Table 6.2. Electronegativity of elements. Two different parametrizations are shown: Upper and lower lows represents Pauling’s electronegativity [86] and Allred-Rochow’s electronegativity [87], respectively.

1 H 2.20 2.20 Li 0.98 0.97 Na 0.93 1.01 K 0.82 0.91 Rb 0.82 0.89 Cs 0.79 0.86

2

Be 1.57 1.47 Mg 1.31 1.23 Ca 1.00 1.04 Sr 0.95 0.99 Ba 0.89 0.97

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

C 2.55 2.50 Si 1.90 1.74 Ge 2.01 2.02 Sn 1.96 1.72 Pb 2.33 1.55

N 3.04 3.07 P 2.19 2.06 As 2.18 2.20 Sb 2.05 1.82 Bi 2.02 1.67

O 3.44 3.50 S 2.58 2.44 Se 2.55 2.48 Te 2.10 2.01 Po 2.00 1.76

F 3.98 4.10 Cl 3.16 2.83 Br 2.96 2.74 I 2.66 2.21 At 2.20 1.96

Er 1.24 1.14

Tm 1.25 1.11

Yb

Lu 1.27 1.14

Sc 1.36 1.20 Y 1.22 1.11 La 1.10 1.08

Ti 1.54 1.32 Zr 1.33 1.22 Hf 1.30 1.23

V 1.63 1.45 Nb 1.60 1.23 Ta 1.50 1.33

Cr 1.66 1.56 Mo 2.16 1.30 W 2.36 1.40

Mn 1.55 1.60 Tc 1.90 1.36 Re 1.90 1.46

Fe 1.83 1.64 Ru 2.20 1.42 Os 2.20 1.52

Co 1.88 1.70 Rh 2.28 1.45 Ir 2.20 1.55

Ni 1.91 1.75 Pd 2.20 1.35 Pt 2.28 1.44

Cu 1.90 1.75 Ag 1.93 1.42 Au 2.54 1.42

Zn 1.65 1.66 Cd 1.69 1.46 Hg 2.00 1.44

B 2.04 2.01 Al 1.61 1.47 Ga 1.81 1.82 In 1.78 1.49 Tl 1.62 1.44

La 1.10 1.08

Ce 1.12 1.06

Pr 1.13 1.07

Nd 1.14 1.07

Pm

Sm 1.17 1.07

Eu

Gd 1.20 1.11

Tb

Dy 1.22 1.10

Ho 1.23 1.10

1.07

1.01

1.10

1.06

206


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Chapter 7

Quantum Transport in Quasicrystals and Complex Metallic Alloys Didier Mayou1 and Guy Trambly de Laissardie`re2 1 Institut Neél, CNRS and Universite´ Joseph Fourier, B^ at D, B.P. 166, 38042 Grenoble Cedex 9, France 2 Laboratoire de Physique Theórique et Mode´lisation, CNRS and Universite´ de Cergy–Pontoise, St Martin, 95302 Cergy–Pontoise, France

7.1.

Introduction

Immediately after the discovery by Shechtman et al. [1] of quasiperiodic intermetallics one major question was raised about the physical properties of phases with this new type of order. In particular, one expected that the electronic and thermal properties could be deeply affected. Indeed the description of electrons or phonons in periodic phases rests on the Bloch theorem which cannot be applied to a quasiperiodic structure. Within a decade a series of new quasiperiodic phases and approximant were discovered and intensively studied. These investigations indicated that indeed the electrons and the phonons properties could be deeply affected by this new type of order. The first quasiperiodic alloys AlMn where metastable and contained many structural defects. As a consequence they had conduction properties similar to those of amorphous metals with resistivities in the range 100–500 cm. In 1986 the first stable icosahedral phase was discovered in AlLiCu. This phase was still defective and although its resistivity was higher (800 cm) it was still comparable to that of amorphous metals. The real breakthrough came with the discovery of the stable AlCuFe icosahedral phase, having a high structural order. The resistivity of these very well ordered systems where very high, of the order of 10 000 cm, which gave a considerable interest in their conduction properties. Within a few years several important electronic characteristics of these phases were experimentally demonstrated. The density of states in AlCuFe was smaller than in Al, about one third of that of pure Al, but still largely metallic. The conductivity presented a set of characteristics that were either those of semi-conductors or those of normal metals. In particular, weak-localization effects were observed that are typical of amorphous HANDBOOK OF METAL PHYSICS ISSN 1570-002X/DOI 10.1016/S1570-002X(06)02007-6 209


210

Chapter 7.

Quantum Transport in Quasicrystals and Complex Metallic Alloys

metals. Yet the conductivity was increasing with the number of defects just as in semi-conductors. Optical measurements showed that the Drude peak, characteristic of normal metals, was absent. In 1993 another breakthrough was the discovery of AlPdRe which had resistivities in the range of 106 cm [2–4]. This system gave the possibility of studying a metal-insulator transition in a quasiperiodic phase. There are still many questions concerning electronic transport in AlPdRe phases. One difficulty concerns the homogeneity and the quality of samples which are crucial for transport properties but are difficult to determine exactly. Since the discovery of Shechtman et al. [1] our view of the role of quasiperiodic order has evolved. For electronic or phonon properties of most known alloys it appears that the medium range order, of one or a few nanometers, is the real length scale that determines properties. This observation has lead the scientific community to adopt the point of view that quasicrystals are an example of a larger class. This new class of Complex Metallic Alloys contains quasicrystals, approximants and alloys with large and complex unit cells with possibly hundreds of atoms in the unit cell. In this chapter we shall concentrate on ‘the way electrons propagate’ in a quasicrystal or in a complex metallic alloy. The main objective is to show that the nonstandard conduction properties of some quasicrystals and related complex metallic alloys result from purely quantum effects and cannot be interpreted through the semi-classical theory of transport. This is of great importance since the semiclassical Bloch-Boltzmann theory is at the heart of our understanding of conduction in solids, ranging from metals to semi-conductors. This new type of quantum transport is related to the specific propagation mode of electrons in these systems. Indeed in quasicrystals and related complex phases the quantum diffusion law deviates from the standard ballistic law characteristic of perfect crystals in two possible ways. In a perfect quasicrystal the large time diffusion law is a power law instead of a ballistic one as in perfect crystals. In a complex crystal the diffusion law is always ballistic at large time but it can deviate strongly from the ballistic law at sufficiently small times. It is this specific character that provides a basis for the interpretation of the strange conduction properties of AlCuFe, AlPdMn and probably also for those of AlPdRe. This chapter is organized as follows. Section 7.2 is the most technical part but it can be skipped by readers not interested by mathematical aspects. Section 7.3 presents a detailed physical interpretation of anomalous diffusion and low frequency conductivity laws in crystals and quasicrystals. Section 7.4 presents evidence of anomalous diffusion in experimental quasicrystalline and approximant phases. In Section 7.2 we give some definitions and present the mathematical relations that exist between the low frequency conductivity, including the dc conductivity, and quantum diffusion. We consider also the relaxation time approximation (RTA) that allows to treat the role of disorder on quantum diffusion and conductivity. We demonstrate general formulas for quantum diffusion and low frequency conductivity (within the RTA) in periodic and quasiperiodic models of potential. In Section 7.3 we focus on the physical interpretation and consequences of the formulas derived in Section 7.2. On a general ground we discuss the limitations of the RTA and the possibility of a metal-insulator transition. We apply this to a general theory of low frequency conductivity and metal-insulator transition in crystals and quasicrystals.

7.2. Quantum formalism for electronic transport

211

In Section 7.4 we present briefly the experimental transport properties of phases such as AlMnSi, AlPdMn, and AlCuFe or AlPdRe. These experimental transport properties indicate a conduction mode which is neither metallic nor semiconducting. For the -AlMnSi phase, recent ab-initio computations are presented, which confirm the existence of an anomalous diffusion and allow for a semi-quantitative ab-initio computation of conductivity. Concerning AlCuFe and related quasiperiodic phases, which cannot be addressed by band structure calculations, we present a phenomenological model. This model based on anomalous quantum diffusion provides a coherent interpretation of the strange electronic transport of these systems. We conclude by a short summary. We discuss also briefly the link with other problems such as phonons in quasiperiodic systems or electrons in heavy Fermions systems or in polaronic systems.

7.2.

Quantum formalism for electronic transport

In this section we give some definitions and recall general properties of the conductivity. We present the mathematical relations that exist between the low frequency conductivity, including the dc conductivity, and quantum diffusion. We also consider the relaxation time approximation (RTA) [5] that allows to treat the role of disorder on quantum diffusion and conductivity. We demonstrate general formulas for quantum diffusion and low frequency conductivity (within the RTA) in periodic and quasiperiodic models of potential.

7.2.1.

Impulse response and analytical properties of the conductivity

Let us consider a system, at thermodynamical equilibrium, submitted to an impulse of electric field EðtÞ ¼ EðtÞ;

ð7:1Þ

where ðtÞ is the Dirac function. The resulting current density is JðtÞ (JðtÞ ¼ 0 for t < 0) and the response jðtÞ is defined by jðtÞ ¼

JðtÞ ; jðtÞ ¼ 0 for t < 0: E

ð7:2Þ

Then the complex conductivity ð!Þ and the response jðtÞ ¼ JðtÞ=E are related through Z1 ð!Þ ¼

ei!t jðtÞdt: 0

ð7:3Þ

212

Chapter 7.


From Eqs. (7.2) and (7.3) one obtains Re ð!Þ ¼ Re ð!Þ;

ð7:4Þ

and 1 jðjtjÞ ¼

Zþ1 ei!t Re ð!Þ: d!

ð7:5Þ

1

In Eq.(7.3) the integral over the time t runs over (t > 0) only due to causality (jðtÞ ¼ 0 for t < 0). This implies that the conductivity ð!Þ is an analytical function of frequency ! in the upper half of the complex plane. From the analyticity of ð!Þ the Kramers-Kro¨nig relations, that relate the real part and the imaginary part of the conductivity, one obtains 1 PP

Im ð!Þ ¼

Z1 1

Re ðuÞ du; !u

ð7:6Þ

and 1 Re ð!Þ ¼ PP

Z1 1

Im ðuÞ du: !u

ð7:7Þ

where PP means the principal part of the integral. This implies also the following spectral decomposition for z in the upper half complex plane: i ðzÞ ¼

Zþ1 1

Reð!0 Þ d!0: z !0

ð7:8Þ

Finally we recall that the conductivity obeys sum rules. For example the response jðt ¼ 0Þ is independent of the quantum character of electrons. It depends only on their concentration n, mass m and charge e through jðt ¼ 0Þ ¼

ne2 : m

ð7:9Þ

Combining with Eqs.(7.4) and (7.5) one obtains Zþ1 Reð!Þ d! ¼ 0

ne2 2m

ð7:10Þ

213


7.2.2. Relation between low frequency conductivity and quantum diffusion The quantum diffusion of states having an energy E is defined as DX2 ðE; tÞ: DX2 ðE; tÞ ¼ h½XðtÞ Xð0Þ 2 iE ;

ð7:11Þ

where < A >E means an average of the diagonal elements of the operator A over all states with energy E. XðtÞ is the position operator along the axis x expressed in the Heisenberg representation. The velocity operator is defined as Vx ðtÞ ¼ dXðtÞ= dt, its correlation function CðE; tÞ is defined as CðE; tÞ ¼ hVx ðtÞVx ð0Þ þ Vx ð0ÞVx ðtÞiE ¼ 2 Re hVx ðtÞVx ð0ÞiE ;

ð7:12Þ

and is related to quantum diffusion [6] through d DX2 ðE; tÞ ¼ dt

Zt

CðE; t0 Þ dt0 :

ð7:13Þ

0

As shown in [6], the real part of the low-frequency conductivity is related to quantum diffusion. Indeed from the Kubo-Greenwood formula the real part of the conductivity is given by Z Re ð!Þ ¼ h!

dE FðE; !Þ; h!

ð7:14Þ

where is the chemical potential. In (7.14) the Fermi-Dirac distribution function is taken equal to its zero temperature value. This is valid provided that the electronic properties vary smoothly on the thermal energy scale kT. For finite temperature, the effect of the Fermi-Dirac distribution function on the transport properties has been studied in the literature [7–10]. But, theses analysis could not explain the unconventional conduction of quasicrystals and related alloys (very high resistivity at low temperature, and conductivity that increases strongly when defects or temperature increases). Therefore in the following, the Fermi-Dirac distribution function is taken equal to its zero temperature value. But the effect of defects and temperature (scattering by phonons) on the diffusivity is taken into account via the relaxation time approximation (Section 7.2.3). The central quantity FðE; !Þ is given by FðE; !Þ ¼

2 he2 TrhðE HÞVx ðE þ h! HÞVx i;

ð7:15Þ

where is the volume of the system and Tr means the Trace of an operator. Expressing the operator ðE HÞ as the Fourier transform of the evolution operator eiHt one can show that

214

Chapter 7.

2FðE; !Þ ¼ e2 nðEÞ


Z1 dt ei!t hVx ðtÞVx ð0ÞiE ;

ð7:16Þ

1

and 2FðE h!; !Þ ¼ e2 nðEÞ

Z1 dt ei!t hVx ð0ÞVx ðtÞiE ;

ð7:17Þ

1

where nðEÞ is density of states per unit volume (summed over up and down spins which are assumed to have the same transport properties here). Then one finds 2 Re ~ðE; !Þ ¼ FðE; !Þ þ FðE h!; !Þ;

ð7:18Þ

where nðEÞ ~ðE; !Þ ¼ e 2

Z1

2

ei!t CðE; tÞ dt:

ð7:19Þ

0

We note that the function ~ðE; !Þ is analytical in the upper half of the complex plane. For large ! : ~ðE; !Þ / 1=! and the Kramers-Kro¨nig relations are valid. Finally the usual sum rule is valid Z1 Re ~ðE; !Þ d! ¼

e2 nðEÞ e2 n CðE; t ¼ 0Þ ¼ ; 2 2m

ð7:20Þ

0

where m is the effective mass and n the density of conduction electrons. If the variation of FðE; !Þ with energy is small in the interval ½EF h!; EF þ h!

of values of E, one deduces from the previous set of equations that nðEF Þ Re Re ð!Þ ’ e 2

Z1 ei!t CðEF ; tÞ dt:

2

ð7:21Þ

0

Eq.(7.21) is valid at sufficiently small values of !. In particular at zero frequency the dc conductivity is given by the Einstein relation ð0Þ ¼ e2 nðEF ÞDðEF Þ;

ð7:22Þ


215

with DðEF Þ ¼ lim

1d

t!1 2 dt

ð7:23Þ

DX2 ðEF ; tÞ:

Finally there is a simple relation between the velocity correlation function at the Fermi energy and the impulse response jðtÞ. Indeed comparing Eq.(7.21) and Eq.(7.3) one derives the following equivalence at large time jðtÞ ’ e2

7.2.3.

nðEF Þ CðEF ; tÞ: 2

ð7:24Þ

Relaxation time approximation (RTA)

Within the relaxation time approximation one assumes that the response currents with disorder jðtÞ and without disorder j0 ðtÞ are related through jðtÞ ¼ j0 ðtÞejtj= ;

ð7:25Þ

where is the relaxation time. So the relaxation time approximation (RTA) allows to treat the effect of disorder on quantum diffusion and conductivity. We derive the relations satisfied by conductivity and quantum diffusion in this approximation. The conditions of validity of the RTA are discussed in Section 7.3. Using Eq.(7.3), and within the RTA, the conductivity with disorder ð!; Þ and without disorder 0 ðzÞ are related by ð!; Þ ¼ 0

i !þ :

ð7:26Þ

The real part of conductivities with defects Re ð!; Þ and without defects Re 0 ð!Þ are related simply. Using Eq.(7.3), it is straightforward to obtain 1 Re ð!; Þ ¼

Zþ1 1

Re 0 ð!0 Þ ð!

!0 Þ 2

þ

1 2

d !0 ; :

ð7:27Þ

which allows to compute the real part of the conductivity with defects. We discuss now the RTA from the point of view of quantum diffusion. In all cases we consider that the influence of disorder is much stronger on the quantum diffusion than on the density of states. We thus neglect the variation of nðEÞ with disorder. From Eq.(7.24) one deduces that, for not too large disorder i.e., for sufficiently large relaxation time , the RTA is equivalent to CðE; tÞ ¼ C0 ðE; tÞejtj= ;

ð7:28Þ

216

Chapter 7.


where CðE; tÞ and C0 ðE; tÞ are respectively the velocity correlation functions with and without disorder. After Eq.(7.13) one deduces that the long time propagation is diffusive with a diffusion coefficient defined as 1 DðEÞ ¼ 2

Zþ1

C0 ðE; tÞejtj= dt;

ð7:29Þ

0

which is equivalent to DðEÞ ¼

1d DX2 ðE; tÞ if t : 2 dt

ð7:30Þ

At zero frequency the diffusivity can be written in the useful form DðEF ; Þ ¼

L2 ðEF ; Þ : 2

ð7:31Þ

Using the t ¼ 0 conditions DX2 ðE; t ¼ 0Þ ¼ 0 and forming two integrations by part, we obtain Zþ1 L2 ðEF ; Þ ¼

DX20 ðEF ; tÞet= dt

0

Zþ1

d 2 dt DX ðE; t

¼ DX2 ðEF ; tÞ ;

¼ 0Þ ¼ 0 and per-

ð7:32Þ

et= dt

0

where :: is a time average on a time scale . DX0 ðE; tÞ is the spreading of states of energy E, in the perfect system, i.e., without disorder. More generally at low frequency, using Eq.(7.21) one can define a frequency dependent diffusivity DðEF ; !Þ such that Re ð!Þ ’ e2 nðEF ÞDðEF ; !Þ;

ð7:33Þ

and: 1 DðEF ; !Þ ¼ Re 2

Z1 ei!t CðEF ; tÞ dt:

ð7:34Þ

0

Within the RTA, Eq. (7.34) can be written as 1 DðEF ; !Þ ¼ Re 2

Zþ1 0

eði!1=Þt C0 ðE; tÞ dt:

ð7:35Þ


217

It is convenient to use the equivalent form which expresses the frequency dependent diffusivity DðEF ; !Þ in terms of the quantum diffusion without disorder DX20 ðE; tÞ: 8 9 2 Zþ1 E means an average of the diagonal elements of the operator A over all states with energy E. XðtÞ is the position operator along the x axis in the Heisenberg representation. The velocity operator Vx ðtÞ ¼ dXðtÞ= dt has a correlation function CðE; tÞ defined as E D E D CðE; tÞ ¼ Vx ðtÞVx ð0Þ þ Vx ð0ÞVx ðtÞ ¼ 2 Re Vx ðtÞVx ð0Þ ; E

E

ð7:97Þ

and is related to quantum diffusion through d DX2 ðE; tÞ ¼ dt

Zt

CðE; t0 Þ dt0 :

ð7:98Þ

0

The low-frequency conductivity satisfies Re ð!Þ ’ e2 nðEF ÞDðEF ; !Þ:

ð7:99Þ

with nðEÞ the total density of states and 1 DðEF ; !Þ ¼ Re 2

Zþ1 ei!t CðEF ; tÞ dt:

ð7:100Þ

0

In the limit of large time, jðtÞ and CðEF ; tÞ are related by jðtÞ ’ e2

nðEF Þ CðEF ; tÞ; 2

ð7:101Þ

where EF is the Fermi energy and nðEF Þ is the density of states at the Fermi energy (summed over spin up and down). Within the relaxation time approximation (RTA) one assumes that jðtÞ and CðE; tÞ with disorder are related to j0 ðtÞ and C0 ðE; tÞ without disorder through jðtÞ ¼ j0 ðtÞejtj= ;

ð7:102Þ

CðE; tÞ ¼ C0 ðE; tÞejtj= :

ð7:103Þ

and

Here the Fermi-Dirac distribution function is assumed to be equal to its zero temperature value. This is valid provided that the electronic properties vary smoothly on the thermal energy scale kT. But in the RTA, the effect of defects and temperature (scattering by phonons . . .) is taken into account through the relaxation time . The diffusivity is given by

230

Chapter 7.

DðEF ; !Þ ¼

1 Re 2


Zþ1

eði!1=Þt C0 ðEF ; tÞ dt:

ð7:104Þ

0

It is convenient to use the following equivalent form which expresses the frequency dependent diffusivity DðEF ; !Þ in term of the quantum diffusion in the system without disorder DX20 ðE; tÞ, 8 9 2 Zþ1 . However this is not the case in many disordered systems which present elastic scattering. For example in the case of free electrons scattered by static defects at random in a three dimensional system there are interferences between several scattering paths. This is represented in Figure 7.4. As a consequence of these interferences the long time behavior of the response current jðtÞ, in the absence of a magnetic field, is jðtÞ ¼ At3=2 with A > 0:

ð7:108Þ

Thus interferences effects in the diffusive regime cannot be in general described properly by the RTA. Indeed in the RTA all quantum correlations are lost beyond the


231

Figure 7.4. Interference between a path (solid line) and its time reversal image (dashed line) for an electron which diffuses through a static disordered potential. The interference that occurs between two different paths after several scattering events cannot be described by the RTA.

scattering time scale . Figure 7.4 shows a type of interference that occurs on a time scale greater than the scattering time and thus cannot be described in the RTA. Ultimately these interferences can lead to a localization of the states, provided that the disorder is sufficiently strong. This Anderson localization is schematically represented in Figure 7.5. Of course a correct theory of the Anderson localization is also beyond the RTA. For a 3-dimensional system the importance of quantum interferences depends on the ratio between the dc-conductivity of the system dc and the Mott value Mott ’ 600ðcmÞ1 = where is the mean-free path expressed in Angstro¨ms. If R ¼ dc =Mott 1, the effect of the quantum interferences on dc is small. In that case the quantum interferences in the diffusive regime change only slightly the value of the conductivity and the RTA can describe the role of elastic scattering. If the ratio R is closer to one, then the RTA cannot be used to describe the role of elastic scattering.

Density of states Localized states Extended states

Energy Figure 7.5. Schematic band structure of an Anderson insulator, with increasing disorder from bottom to top. At zero temperature the system is metallic if the Fermi energy is in the extended states region. The system is insulating if the Fermi energy is in the localized states region.

232

Chapter 7.


The case of inelastic scattering is different. It is generally assumed that inelastic scattering with a scattering time in destroys quantum interferences on a time scale t > in . So the relaxation time approximation is expected to be valid in the case of inelastic scattering. Yet a condition is that the system contains a sufficiently large number of states so that the electron can find a state to scatter into close to the scattering event. For this reason the hopping regime between localized states (either short range hopping or variable long range hopping) is not described by the relaxation time approximation. Indeed each hopping process requires an exchange of energy with phonons that provides to the electron the difference in energy between the initial and final localized states.

7.3.1.2.

General case according to the scaling theory

We discuss now the case where there is both elastic scattering (characterized by el ) and inelastic scattering scattering (characterized by in ) and assume that in > el . This often happens since in diverges at low temperature whereas el is temperature independent at sufficiently low temperature. In the case where in < el the elastic scattering is expected to have a minor effect on transport, and one is back to the case of pure inelastic scattering. Let us define tðLÞ as the time needed for a wavepacket to spread on a length scale L and DðLÞ ¼ L2 =2tðLÞ, the diffusivity DðLÞ at length scale L. Let us define also Lin which is such that in ¼ tðLin Þ. According to the scaling theory the diffusivity depends on the length scale due to the quantum interferences in the diffusive regime. One can distinguish three steps. (1) Define the diffusivity at the length scale of the elastic mean free path Le . This can be done by using the RTA with a relaxation time equal to the elastic scattering time. This is valid because the quantum interferences do not act on length scales smaller than the elastic mean free path Le . (2) Consider the conductance gðLÞ of a cube with size L. gðLÞ ¼ e2

nðEÞ L3 ¼ e2 nðEÞDðLÞL; 2 tðLÞ

ð7:109Þ

and apply the scaling relation on length scales L such that Lin > L > Le . d logðgÞ ¼ ðgÞ: d logðLÞ

ð7:110Þ

ðgÞ is represented schematically in Figure 7.6. The relation in Eq.(7.110) defines entirely the quantum diffusion and thus the velocity correlation function as a function of the length scale L or also as a function of time scale tðLÞ. Thus it allows to compute Cscaling ðE; tÞ where Cscaling ðE; tÞ is the velocity correlation function corresponding to the purely elastic scattering.

233


β(g )

β(g ) = s Log(g /gc) d=3

1

0

Weak localisation regime β(g ) = 1 − 1/π2 g d=2 gc

−1

Log(g )

d=1

Figure 7.6. Schematic representation of the ðgÞ function for systems of dimension d.

(3) One stops the renormalization procedure at the length scale Lin ¼ Lðin Þ, which means that the macroscopic diffusivity is simply D ¼ DðLin Þ. Equivalently the role of inelastic scattering is simply to destroy the velocity correlation on the time scale in , i.e., one obtains CðE; tÞ ¼ Cscaling ðE; tÞet=in :

ð7:111Þ

The renormalization procedure determines entirely Cscaling ðE; tÞ and thus through Eq.(7.111) determines entirely the quantum diffusion of the system with inelastic scattering.

7.3.2.

Phenomenon of backscattering

Let us consider the response jðtÞ to an electric field EðtÞ applied to a system. During the impulsion the dynamics is dominated by the effect of the electric field. This implies that jð0Þ is independent of the atomic forces applied to the electrons and is given by the classical response jð0Þ ¼ ne2 =m. In particular jð0Þ is positive. In the classical Drude model jðtÞ ¼ jð0Þexpðt= Þ tends to zero at larger times but is always positive. This is illustrated by Figure 7.7. Quantum effects can lead to a counter intuitive situation where jðtÞ becomes negative in some time interval. This is the phenomenon of backscattering. This phenomenon occurs in a recurrent manner in the physics of quasicrystals and related complex intermetallics. So we give here an account of its characteristics and its consequences on conductivity. We illustrate first the phenomenon of backscattering in two cases concerning disordered systems: the weak-localization and the strong localization regimes.

234

Chapter 7.


j (arbitrary unit)

1 0.8 0.6 0.4 0.2 0

0

10

20

30

40

50

Time (arbitrary unit) Figure 7.7. Response current jðtÞ within the Drude model. Without disorder (solid line) the response is constant j ¼ ne2 =m. With disorder (dashed line) the response decays exponentially on the time scale .

7.3.2.1.

Inelastic scattering in the weak localization regime

We assume that the elastic scattering time el is much shorter than the inelastic scattering time in . On the time scale el < t < in , the interferences are insensitive to inelastic scattering and jðtÞ ¼ At3=2 (for a 3-dimensional system), but on the time scale t > in the interferences are destroyed. Thus a general expression of jðtÞ for t > el is (Figure 7.8): jðtÞ ¼ At3=2 et=in with A > 0:

ð7:112Þ

In that case using Eq.(7.95) one finds for the dc conductivity: Z1 dc ¼ 0

pffiffiffi 2A jðtÞ dt ¼ 0 þ pffiffiffiffiffi ; in

ð7:113Þ

where 0 is the dc-conductivity in the absence of inelastic scattering. That is inelastic scattering tends to increase the conductivity. This paradoxal result is directly related to the backscattering phenomena that occurs with static disorder. Indeed in the

j (arbitrary unit)

1 0.8 0.6 0.4 0.2 0 0

10

20

30

40

50

Time (arbitrary unit) Figure 7.8. Response function jðtÞ in a disordered system with weak-localization effect, and thus backscattering at large time. Without inelastic scattering (solid line) and with inelastic scattering (dashed line).


235

integral in Eq.(7.113) with jðtÞ ¼ At3=2 et=in , in can be considered as a cut-off which supresses the negative contributions of jðtÞ ¼ At3=2 at t > in . The role of a small frequency ! is comparable to that of inelastic scattering at a time scale in ’ 1=!. It acts also as a cut-off. For example in the case of the weak localization one has for in ! 1: pffiffiffiffiffiffiffiffiffi Re ð!Þ ¼ 0 þ A 2!:

ð7:114Þ

This means that Re ð!; Þ increases with frequency at small frequencies. This is at the opposite of the standard behavior of metals, where the low frequency conductivity is dominated by the Drude peak. The role of frequency can also be deduced from the relation in Eq.(7.27) between the real parts of conductivity with scattering and without scattering. Indeed this relation (7.27) shows that dc is the average of Re0 ð!Þ by a Lorentzian of width 1= centered at zero frequency. dc can increase with 1= only if Re0 ð!Þ increases with !.

7.3.2.2.

Inelastic scattering for localized states: the Thouless regime

Let us consider a system which is an insulator. In that case the zero frequency conductivity is zero: Z1 ð! ¼ 0Þ ¼

jðtÞ dt ¼ 0:

ð7:115Þ

0

j (arbitrary unit)

Since the integral is zero, and since the small time response jðtÞ > 0 it means that there are times interval where jðtÞ < 0. Thus in an insulating system the backscattering phenomenon necessarily occurs. For Anderson localization one expects that jðtÞ > 0 at small times and jðtÞ < 0 at large time. Beyond a characteristic time Tc , the response jðtÞ is essentially zero (see Figure 7.9).

τ=∞

0.8

τ1 τ2 (τ2 < τ1)

0.4

0

–0.4 0

1

2

3

Time (arbitrary unit) Figure 7.9. Response function jðtÞ for an insulator. Beyond a characteristic time (Tc ’ 2 here) the response j0 ðtÞ is1essentially zero. Without inelastic scattering (solid line) the conductivity is zero and the integral R dc ¼ j0 ðtÞ dt ¼ 0. With inelastic scattering (dashed lines) the system is conducting and the integral 0 R1 dc ¼ j0 ðtÞet=in dt > 0. 0

236

Chapter 7.


Provided that the RTA is applicable one obtains Z1 ð! ¼ 0; Þ ¼

j0 ðtÞet= dt ¼ 0:

ð7:116Þ

0

If > Tc then the exponential differs significantly from 1 only in the large time region t > Tc . In that case one can replace expðt=Þ by 1 t= in the range (t < Tc ) where jðtÞ 6¼ 0. This gives 1 ð! ¼ 0; Þ ¼

Although

R1

Z1 j0 ðtÞt dt ¼ 0:

ð7:117Þ

0

j0 ðtÞ dt ¼ 0, the integral

0

R1

j0 ðtÞt dt in Eq.(7.117) is negative because

0

j0 ðtÞ is negative at large times and positive at small times. The regime described here corresponds to the Thouless regime, that can be described by analyzing the quantum diffusion. In the Thouless regime, the physical picture is that of electrons spreading during a time in between two inelastic scattering events and then loosing completely their phase at each inelastic scattering event. According to the Thouless scenario, in the limit of large inelastic scattering time in , the spreading of the electron wavefunction between two inelastic scattering events is bounded by the localization length E . Since the electron must wait till the next inelastic scattering event to loose phase memory and spread again, the diffusivity is given by DThouless ðEÞ ¼

E2 : 2in

ð7:118Þ

Within the RTA the diffusivity which is the square of the spreading during in divided by in is given by DðEÞ ¼

L2 ðE; in Þ ; 2in

ð7:119Þ

with the mean free path L2 ðE; in Þ given by Eq.(7.107) which is equivalent to L2 ðE; in Þ ¼

X

ðE E 0 Þ2

E 0 6¼E

E 0 Þ2

ðE

þ ð h=in Þ

2

X2E;E 0 ;

ð7:120Þ

with XE;E 0 ¼ hEjXjE 0 i:

ð7:121Þ

At large inelastic scattering time in one gets L2 ðEÞ ¼ limin !1 L2 ðE; in Þ which can be written as


L2 ðEÞ ¼ hEjðX hEjXjEiÞ2 jEi ¼ E2 ;

237

ð7:122Þ

where E is the localization length of the state of energy E. From Eq.(7.119) the diffusivity is DðEÞ ¼ E2 =2in , in agreement with the argument of Thouless. One thus recovers the typical dependence on the scattering time , i.e., dc / 1= that was deduced from the analysis of the response jðtÞ and the backscattering.

7.3.3. Anomalous quantum diffusion and conductivity of periodic systems 7.3.3.1.

Conductivity within the RTA

The semi-classical theory of conduction in crystals is based on the concept of a charge carrier wave-packet propagating at a velocity V ¼ ð1=hÞ@En ðkÞ=@k, where ‘‘En ðkÞ’’ is the dispersion relation for band n and wavevector k. The validity of the wave-packet concept requires that the extension Lwp of the wave-packet of the charge carrier is smaller than the distance V of traveling on the time scale . On the contrary, if V < Lwp , the semi-classical model breaks down. The quantum formalism presented here allows to treat on the same footing the standard regime where the semi-classical approach is valid and the small time regime V < Lwp . As shown in Section 7.2 the spreading of states with energy E in crystals is given by DX20 ðEF ; tÞ ¼ V2 t2 þ DX2NB ðEF ; tÞ:

ð7:123Þ

The first term in the right-hand side of Eq.(7.123), V2 t2 , corresponds to the Boltzmann contribution. This term dominates at large times and describes the intercell ballistic propagation of wavepackets on long time scale in crystals. The physical origin of this term is the coupling between successive unit cell that allows the electron to travel in the whole crystal. The second term DX2NB ðEF ; tÞ is the NonBoltzmann contribution. It describes the intracell spreading of the electron. Indeed as shown in Section 7.2 this spreading is bounded by a term of the order of the square of the unit cell size. In a standard crystal the Boltzmann term dominates at the time scale relevant for transport, i.e., the scattering time due to disorder. In approximants of quasicrystals the Non-Boltzmann term can dominate. Let us anticipate on the ab initio calculations which show that DX2NB ðEF ; tÞ is nearly constant DX2NB ðEF ; tÞ ’ DX2NB except at very small times t . Then the Eq. (7.105) leads to DðEF ; !Þ ’

V2 DX2NB : þ 2 1 þ !2 2

ð7:124Þ

Thus the frequency dependent diffusivity DðEF ; !Þ is the sum of a Drude like contribution (first term on the right-hand side of Eq.(7.124)) and a contribution independent of frequency which increases with disorder. As we show from ab initio calculations (see below) the Drude like contribution can be small in some periodic

238

Chapter 7.


100

Diffusivity D /(V 2τ∗)

10

D DB

1

DNB

0.1

0.1

1

10

100

τ∗ /τ Figure 7.10. Variation of the normalized zero frequency diffusivity D=V2 with scattering time for the expression (7.124). For ¼ the Boltzmann and non-Boltzmann contributions are equal.

approximants of quasicrystals. This explains why in these systems the optical conductivity presents no Drude peak and why the dc-conductivity increases with disorder. We define as the time for which the Boltzmann and Non-Boltzmann contributions to the dc-diffusivity are equal. Thus: ¼

DXNB pffiffiffi : V 2

ð7:125Þ

In that case the dc-diffusivity can be represented as in Figure 7.10. The acdiffusivity is represented in Figure 7.11.

7.3.3.2.

Backscattering

The Non-Boltzmann contribution to the diffusivity DNB ¼ DX2NB =ð2Þ is formally similar to the Thouless expression for localized states DThouless ¼ E2 =ð2Þ. Indeed in the Thouless regime the quantum diffusion between two inelastic scattering events is limited by the localization length E . For crystals, it is the Non-Boltzmann contribution to quantum diffusion which tends to DXNB . Let us recall that DXNB is itself limited by a term of the order of the unit cell size. As discussed previously (see Section 7.3.2) the increase of conductivity with disorder is a direct consequence of the backscattering. Indeed we shall find in Section 7.4 that ab initio calculations prove the existence of backscattering. Finally we emphasize an important difference with the Thouless regime. In the Thouless regime it is the inelastic scattering that destroys the localization produced


239

τ = 3 τ∗ τ = τ∗ τ = τ∗/3

10

D /DNB

8

6

4

2

0

0

1

2

3

ωτ∗ Figure 7.11. Variation of the normalized frequency dependent diffusivity D=DNB . For > the transport is dominated by the Boltzmann term and the Drude peak is well defined. For < the transport is dominated by the Non-Boltzmann term and the Drude peak is absent.

by the elastic scattering. Here, provided that the RTA is valid, it is either the elastic or the inelastic scattering that destroys the localization induced by the potential due to the atomic structure in a unit cell.

7.3.3.3.

Metal–Insulator transition

Let us discuss now the role of quantum interferences according to the scaling theory of localization. As explained in Section 7.3.1, the central quantity is the conductance of a cube with a size equal to the elastic mean free path LðEF ; Þ. g ’ e2 nðEF ÞDðEF ; ÞLðEF ; Þ

ð7:126Þ

Since the quantum interference effects do not have the possibility to operate at smaller length scale than LðEF ; Þ then this quantity can be computed with the RTA according to Eq.(7.107). We still assume that DXNB ðEF ; tÞ is nearly constant equal to DXNB except at the smallest time (see below the ab inito results on an approximant of -AlMnSi). Thus the typical propagation length LðEF ; Þ on a time scale , i.e., the mean-free path, is L2 ðEF ; Þ ¼ DX2NB þ 2V2 2 :

ð7:127Þ

240

Chapter 7.


We introduce g0 , which is characteristic of the perfect crystal and is defined by ð7:128Þ

g0 ¼ e2 nðEF ÞDX2NB V

We also introduce a non-dimensional value ~ of the scattering time defined by ~ ¼

V ¼ pffiffiffi : DXNB 2

ð7:129Þ

We define the function fðxÞ: fðxÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þx ð1 þ 2x2 Þ: 2x

ð7:130Þ

Then one obtains Þ: g ¼ g0 fð~

ð7:131Þ

After the scaling theory a three-dimensional system is insulating (metallic, respectively) if g < gc (resp. g > gc ) where gc is the value of the universal critical con Þ it is equivalent to say that the ductance in the scaling theory. Using g ¼ g0 fð~ Þ and metallic if g0 =gc > 1=fð~ Þ. This is illustrated system is insulator if g0 =gc < 1=fð~ in Figure 7.12. Note that g0 =gc is characteristic of the perfect crystal whereas 1=~ measures the scattering rate 1/ in units of V=DXNB . A first remarkable property of this phase diagram is that if g0 > Rgc with R ¼ ð2=3Þ3=2 ’ 0:54 then the system is always metallic whatever the value of the scattering rate (phase (a) in Figure 7.13). This is not the case for normal metals

Metal

0.1

g0 /gc

0.6 0.5

Metal 0.05

g0 /gc

Insulator 0 0

0.4

0.2

~ 1/τ

0.3

0.4

Insulator

0.2 0.1 0

0

2

4

6

~ 1/ τ

8

10

12

pffiffi Figure 7.12. Metal-Insulator phase diagram as a function of the two parameters g0 =gc and 1=~ ¼ 2 . The insert represents the limit of a normal metal, i.e., for fixed and V the limit of a small DXNB . After . Eq.(7.128) and Eq.(7.129) this limit is in the region of the phase diagram at small g0 =gc and small 1=~


241

phase (a)

Metal

g0 /gc

0.5

phase (b)

Insulator

0

0

2

4

6

8

10

~ 1/ τ Figure 7.13. Two types of systems can exist. Phases of type (a) are always metallic whatever the value of the scattering time . Phases of type (b) can be either metallic or insulating depending on the value of the scattering time. For phases of type (b) the Metal–Insulator transition that occurs at the highest value of 1=~ is unconventional since the metallic state is obtained by increasing, i.e., by increasing disorder.

which always become insulating at sufficiently small scattering time (i.e., at sufficiently large disorder). If g0 < Rgc the system is metallic at large and small scattering rates and insulator in an intermediate zone (phase (b) in Figure 7.13). This means that if the system is in the large 1=~ metallic region it will become insulating by decreasing 1=~ that is by decreasing disorder! This is just the opposite of the standard conditions for the occurrence of the Anderson localization transition. The other insulator–metal transition is normal in the sense that the metallic state is obtained by decreasing disorder. Note that the case of a normal metal corresponds to the limit DXNB ! 0. In that case one Þ for large ~ namely pffiffiffi uses the asymptotic form of the function fð~ fð~ Þ ’ 2~2 . One then recovers the standard criterion for free-like electrons.

7.3.4. Anomalous quantum diffusion and conductivity of quasiperiodic systems 7.3.4.1.

Conductivity within the RTA

A generalized Drude formula (7.82,7.83) for the low-frequency conductivity is derived in Section 7.2.5. Yet it is interesting to derive it from simple physical arguments. One first notes that at zero frequency the dependence on the scattering time is easy to establish. Indeed the diffusivity is

242

Chapter 7.

DðEF ; Þ ¼


L2 ðEF ; Þ ; 2

ð7:132Þ

where L2 ðEF ; Þ is the mean-free path i.e. a typical distance of propagation in the perfect structure during the scattering time . Assuming that in the perfect quasiperiodic structure the spreading of a wavepacket is DX2 ðtÞ ’ At2 one obtains L2 ðEF ; Þ ’ A 2 ;

ð7:133Þ

and DðÞ ’

L2 ðEF ; Þ A 21 ’ : 2 2

ð7:134Þ

We also note that the conductivity depends on scattering time and frequency only through the combination =ð1 i!Þ. This stems directly from the RTA formula (7.105) which is expressed as a Fourier-Laplace integral with ð1 i!Þ ¼ ð1 i!Þ=. Thus the frequency dependent diffusivity is Dð!; Þ ’

A 21 : Re 2 1 i!

ð7:135Þ

Except for a numerical factor ð2 þ 1Þ this formula is equivalent to the Generalized Drude formula (7.82), (7.83) which we recall here: Re ðEF ; !Þ ¼ Re ~ðEF ; !Þ;

ð7:136Þ

with ~ðEF ; !Þ ¼

21 e2 nðEF Þ : Að2 þ 1Þ 2 1 i!

ð7:137Þ

The behavior of the conductivity depends on the value of compared to 0.5. The frequency dependence is represented in Figure 7.14. If > 0:5 the behavior is similar to that of a normal metal. The dc-conductivity decreases when disorder increases and the low frequency conductivity presents a peak at low frequency, somewhat similar to the Drude peak. If < 0:5 the behavior is not that of a metal. In the absence of disorder the system is insulating, and the dc-conductivity increases when disorder increases [24]. The real part of the conductivity increases when frequency increases. Instead of a Drude peak there is a dip. One also notes that even in the absence of scattering, i.e., for ! 1 the real part of the conductivity is non zero in the limit of small frequency. This means that there is absorption of electromagnetic energy by the system. This is not the case in a normal metallic crystal. Here the absorption of energy can be understood by considering approximants with large unit cell. In perfect approximants the

243


β = 0.5

1

β = 0.2

R(ω)

R(ω)

3

β = 0.6 0.5

2

β = 0.4 β = 0.8 β = 0.5

1 0

5

10

β=1 15

20

0

0

ωτ

5

10

15

20

ωτ

Figure 7.14. Dissipative part of the conductivity Rð!Þ normalized by the zero frequency value (Rð!Þ ¼ Reð!Þ=ð! ¼ 0Þ), as a function of ! (frequency normalized by 1=). The left panel shows that the conductivity increases with frequency when < 0:5. The right panel shows that there is a low frequency peak if > 0:5. The case ¼ 0.5 is an intermediate case for which the conductivity is independent of frequency.

absorption of energy is made through interband transitions. For a given frequency the absorption of energy by interband transition can occur with sufficiently large unit cell because the bands become very narrow and very close in energy allowing for interband transitions.

7.3.4.2.

Backscattering

For < 0:5 the behavior of conductivity with frequency and disorder is not that of a metal. This can be attributed to the phenomenon of backscattering. Indeed after the relation (7.98) between quantum diffusion and velocity correlation one gets C0 ðE; tÞ ¼

d2 DX20 ðE; tÞ ’ A2ð2 1Þt22 ; dt2

ð7:138Þ

if the quantum diffusion law is DX20 ðE; tÞ ’ At2 in the perfect quasiperiodic system. From Eq.(7.138) it appears that the velocity correlation function is negative at large time if < 0:5. The response jðtÞ is then also negative. The phenomenon of backscattering then implies that the conductivity increases with disorder or with frequency, as discussed for weak-localization regime or for the Thouless regime (see Figure 7.15).

7.3.4.3.

Metal–Insulator transition

Let us discuss now the role of quantum interferences according to the scaling theory of localization. As explained in Section 7.3.1 the central quantity is the conductance of a cube with a size equal to the elastic mean free path LðEF ; Þ.

244

j (arbitrary unit)

Chapter 7.


0.8

β = 0.8

0.4

0

β = 0.2 0

10

20

30

40

50

Time (arbitrary unit) Figure 7.15. Typical behaviour of the response jðtÞ for a quasiperiodic system. Without disorder (solid and solid dotted lines) and with disorder (dashed and dashed dotted lines). For > 0:5 the response jðtÞ is positive but for < 0:5 the response is positive at small times and negative at long times (backscattering).

g ’ e2 nðEF ÞDðEF ; ÞLðEF ; Þ:

ð7:139Þ

Since the quantum interference effects do not have the possibility to operate at smaller length scale than LðEF ; Þ, this quantity can be computed with the RTA. We assume that DX20 ðEF ; tÞ is nearly equal to the asymptotic form At2 . Then the typical propagation length LðEF ; Þ on a time scale is of the order of: L2 ðEF ; ÞA 2 ;

ð7:140Þ

and the diffusivity in Eq.(7.105) is of the order of: DðEF ; Þ ’

A 21 : 2

ð7:141Þ

From Eqs.(7.139)–(7.140), one obtains that the conductance of a cube of size LðEF ; Þ is approximately given by g ’ e2

nðEF Þ 3=2 31 A 2

ð7:142Þ

From this expression one concludes that if < 1=3 the conductance tends to zero at large . This means that the system becomes insulating when the disorder decreases. This is what happens in the case of crystals if the Non-Boltzmann contribution to transport dominates (see Section 7.3.3). Yet one must note that due to the Guarneri inequality [6] the spectrum becomes singular continuous for a three dimensional system with < 1=3. In that case the density of states cannot be considered as a constant in the perfect system. Thus we have to assume that the density of states is sufficiently smooth when averaged on the energy scale given by the inverse scattering time 1/.


245

7.4. Evidence of anomalous quantum diffusion in quasicrystals and approximants In this part we present briefly the experimental transport properties of phases such as AlMnSi, AlPdMn, and AlCuFe or AlPdRe. These experimental transport properties indicate a conduction mode which is neither metallic nor semiconducting. For the -AlMnSi phase, recent ab-initio computations are presented, which confirm the existence of an anomalous diffusion and allow for a semi-quantitative ab initio computation of conductivity. Concerning AlCuFe and related quasiperiodic phases, which cannot be addressed by band structure calculations, we present a phenomenological model. This model based on anomalous quantum diffusion provides a coherent interpretation of the strange electronic transport of these systems.

7.4.1. Experimental transport properties of icosahedral and related approximant phases Quasicrystals of high structural quality reveal unusual transport properties [21–23,25] (Figure 7.16). For instance, one of the main features is the low conductivity 4K ¼ 100 200 cm1 for icosahedral AlPdMn and AlCuFe and 4K < 1cm1 for AlPdRe [2–4,26–29], although the DOS still has a metallic character. This means that the high resistivity is due mainly to a small diffusivity of electrons. Experimental measurements show that approximants phases like -AlMnSi [23], 1/1 AlCuFeSi [23], R-AlCuFe [23], 1/1 AlReSi [31,32] etc., have transport properties similar to those of quasicrystals AlPdMn and AlCuFe. This suggests that the local atomic order on the length scale of the unit cell, i.e., 10–30 A˚, determines their

1e+5

Doped semi-conductors

ρ Mott

“Perfect ” stable quasicrystals (i-AlCuFe and i- AlPdMn)

Resistivity ρ (μΩcm)

1e+4

1000

“Imperfect ” stable quasicrystals (i-AlLiCu)

100

Metastable quasicrystals (i-AlMn), Amorphous alloys 10

Metallic alloys

1 4K

300 K

Temperature Figure 7.16. Order of magnitude and schematic temperature dependencies of the resistivity of icosahedral quasicrystals compared to amorphous and metallic crystals. From [25].

246

Chapter 7.


transport properties. As atomic medium-range order of quasicrystals and approximants are similar, it should also be important in the understanding of transport properties of quasicrystals. This remark is confirmed by the fact that AlTM crystals with a small unit cell (typically less than 50 atoms in a unit cell) does not exhibit such particular transport properties. The resistivity, ¼ 1=, of crystals with a small unit cell increases as temperature T increases and generally follows the Mathiessen rule: ðTÞ ¼ 0 þ DðTÞ:

ð7:143Þ

In contrast, the resistivity of some quasicrystals and approximants (AlPdMn, AlCuFe) decreases when temperature increases, and their conductivity follows approximatively an ‘inverse Mathiessen rule’ [33,23]: ðTÞ ¼ 0 þ DðTÞ:

ð7:144Þ

Besides, after annealing a sample, with a consequent reduction of the structural defects, the resistivity of quasicrystals and approximants increases. The relation between the particular transport properties of these phases and their structure is still debated. For AlPdMn quasicrystals, J.J. Pre´jean et al. [34] found that local defects might be related with the occurrence of Mn atoms with localized magnetic moment. Thus, magnetic properties, transport properties and structural quality are intimately linked for those complex phases. Another remarkable experimental result is the linear energy dependence of the optical conductivity of AlCuFe and the absence of Drude peak [35,36]. The icosahedral AlPdRe is the most resistive known quasicrystalline material [27,28]. This material displays a strong decrease of the conductivity when the temperature is reduced and the conductivity value can fall below 1 ðcmÞ1 at 4 K. Although the behavior depends strongly on the composition and the preparation of the sample, many authors [2–4,26–29] have reported that AlPdRe quasicrystal are very close to the metal-insulator transition. Three successive regimes are revealed [28] as the temperature is increased to room temperature: a low temperature variable range hopping-like behavior, followed by a Thouless regime and a high temperature critical regime. Experimentally a low density of states (DOS) at the Fermi energy EF is usually measured in quasicrystals and their crystalline approximants. For instance, a density of states at EF reduced by 1/3 from its free electrons value is measured in i-AlCuLi and R-AlLiCu approximant [22,37–39]. The presence of the pseudogap in these phases is confirmed by photo-emission measurements [40] and NMR experiments [41]. For icosahedral phases containing transition metal (TM) elements, specific heat measurement indicate a DOS at EF of 1/3 of the free electron value for i-AlCuFe and 1=10 for i-AlCuRu [42] and i-AlPdRe [43]. From photo-emission spectroscopy the pseudogap in the DOS is confirmed for many icosahedral quasicrystals in the systems: AlMn (metastable) [44], AlMnSi [44,45], AlCuFe [46–51], AlCuFeCr [49], AlPdMn [52–55], AlCuRu [56], AlPdRe [54]. The pseudogap has been also

DOS (states/eV.cell)


12

EF

Al6Mn

8

4 0 –12

DOS (states/eV.cell)

247

–10

–8

–6

–4

–2

0

2

4

2

4

EF 120

α-Al69.6Si13Mn17.4

80

40

0 –12

–10

–8

–6

–4

–2

0

Energy (eV) Figure 7.17. LMTO total DOS of Al6 Mn [62] and -Al69:6 Si13:0 Mn17:4 .

measured in many approximants of quasicrystals. For instance R-AlCuFe [41,23], 1/1 AlCuFeSi [30], -AlMnSi [23,45], 1/1 AlCuRuSi [57,58], 1/1 AlReSi [32] have a DOS at EF reduced by a similar factor as in i-AlCuTM and i-AlPdMn. A pseudogap near EF has been also confirmed by ab-initio calculations of the electronic structure in many icosahedral approximants (Figure 7.17, see below and see also the chapter by Ishii and Fujiwara in this book). It has also been shown experimentally that transition metal elements have a important role on the unusual transport properties of quasicrystals and related phases [59–63].

7.4.2. Ab initio electronic structure and quantum diffusion in perfect approximants 7.4.2.1.

Density of states

Electronic structure determinations have been performed in the frame-work of density functional formalism in the local density approximation (LDA) by using the self-consistent Tight-Binding (TB) Linear Muffin Tin Orbital (LMTO) method in the Atomic Sphere Approximation (ASA) [64]. The LMTO DOS of an -AlMn idealized approximant (Elser-Henley model [65,66]) has been first calculated by T. Fujiwara [68,69]. This original work shows

248

Chapter 7.


the presence of a Hume-Rothery pseudo-gap near the Fermi energy, EF , in agreement with experimental results [22,23] (see also Figure 7.17). The role of the transition metal (TM) element in the pseudo-gap formation has been shown from ab initio calculations [73] and experiments [62]. Indeed the formation of the pseudo-gap results from a strong sp–d coupling associated to an ordered sub-lattice of TM atoms. Just as for Hume-Rothery phases a description of the band energy can be made in terms of pair interactions. It has been shown that a medium-range Mn–Mn interaction mediated by the sp(Al)–d(Mn) hybridization plays a determinant role in the occurrence of the pseudo-gap [67–76]. It is thus essential to take into account the chemical nature of the elements to analyze the electronic properties of approximants. The electronic structures of simpler crystals such as orthorhombic Al6 Mn, cubic Al12 Mn, present [73] also a pseudo-gap near EF which is less pronounced than in complex approximants phases. Zijlstra and Bose [79] show that Si atoms are in substitution with some Al atoms in the -phase. The main effect of Si is to shift EF in the pseudo-gap in agreement with the Hume-Rothery mechanism that minimizes the band energy.

7.4.2.2.

Role of clusters

As for the local atomic order, one of the characteristics of the quasicrystals and approximants, is the occurrence of atomic clusters on a scale of 10–30 A˚ [80]. Nevertheless the clusters are not well defined because some of them overlap, and in addition there are a lot of glue atoms. The role of clusters has been much debated in particular by Janot [83] and Trambly de Laissardie`re [84]. Janot considers as a reference clusters that are isolated in vacuum but it is more realistic to consider a model of clusters that are not isolated but are embedded in metallic medium. The model [84,85] is based on a standard description of intermetallic alloys. Considering the cluster embedded in a metallic medium, the variation DnðEÞ of the DOS due to the cluster is calculated. For electrons, which have energy in the vicinity of the Fermi level, transition atoms (such as Mn and Fe) are strong scatters whereas Al atoms are weak scatters. Then, following a standard approximation, the potential of Al atoms was neglected in reference [84]. In Figure 7.18, DnðEÞ due to different clusters are shown. The Mn icosahedron is the actual Mn icosahedron of the -AlMnSi approximant. As an example of a larger cluster, we consider one icosahedron of Mn icosahedra, which appeared in the structural model proposed by Janot [83]. DnðEÞ of clusters exhibits strong deviations from the Virtual Bound States (1 Mn atom) [86,87]. Indeed several peaks and shoulders appear. The width of the most narrow peaks (50–100 meV) are comparable to the fine peaks of the calculated DOS in the approximants. Each peak indicates a resonance due to the scattering by the cluster. These peaks correspond to states ‘localized’ by the icosahedron or the icosahedron of icosahedra. They are not eigenstate, they have finite lifetime of the order of h=E, where E is the width of the peak. Therefore, the stronger the effect of the localization by cluster is, the narrower is the peak. A large lifetime is the proof of a localization, but in the real space these states have a quite large extension on length scale of the cluster or the cluster of clusters. The physical origin of these states can be understood as follows. Let us consider incident electrons, with energy E closed to EF , scattered by the cluster. In AlMn


249

8

7

Δn (E ) (states/(eV transition atom))

1 Mn atom

δE

6

5

1 Mn icosahedron

4

3

1 icosahedron of 12 Mn icosahedra

2

1

0

6

7

8

9

10

11

12

13

14

15

Energy E (eV) Figure 7.18. Variation DnðEÞ of the DOS due to 1 Mn atom (Virtual Bound State), 1 Mn icosahedron, and 1 icosahedron of Mn icosahedra obtained after an inflation by a factor ’2 of an initial Mn icosahedron. EF ’ Ed ¼ 10:88 eV. From [84].

alloys EF ’ Ed , where Ed is the energy of the d-orbital. In this energy range, the potential of the Mn atom is strong and the Mn atoms can roughly be considered as hard spheres with radius of the order of the d-orbital size (0.5 A˚). By an effect similar to that of a Faraday cage, electrons can by confined by the cluster provided that their wavelength satisfies > l, where l is the distance between two hard spheres. In the case of -AlMnSi approximant, ’ 0.5 A˚ (if we assume a free electron band and EF ¼ 10:33 eV) and the distance l is about 3.8 A˚. Consequently, we expect to observe such a confinement. This effect is a multiple scattering effect, and it is not due to an overlap between d-orbitals because Mn atoms are not first neighbor. We have also shown that these resonances are very sensitive to the geometry of the cluster [85]. For instance, they disappear quickly when the radius of the Mn icosahedron increases.

7.4.2.3.

Quantum diffusion in perfect crystals

In the following we present calculation of the quantum diffusion in perfect crystalline systems. Some works have already been done from ab initio calculation and give indication of none ballistic diffusion [17,88,89]. We consider the -AlMnSi approximant and compare it with simpler crystals orthorhombic Al6 Mn, and cubic Al12 Mn [90–92]. For the -AlMnSi phase, we use the experimental atomic structure [93] and

Chapter 7.

n (states/(eV.unit cell))

250


150

(a) Total DOS

100

50

0

D (cm2 s–1)

–0.6

9000

–0.4

0

0.2

0.4

0.6

0

0.2

0.4

0.6

0

0.2

0.4

0.6

σ

(b)

6000 3000 0

–0.6

–0.4

0.15

–0.2

D DB DNB

(c)

D (cm2 s–1)

–0.2

0.1

0.05

0 –0.6

–0.4

–0.2

EF – EF (LMTO) (eV) Figure 7.19. (a) LMTO total DOS n, (b) conductivity , and (c) diffusivity D, in the cubic approximant -Al69:6 Si13:0 Mn17:4 . From [90].

the Si positions proposed by Ref. [79] with composition: -Al69:6 Si13:0 Mn17:4 . In Figure 7.19, the total DOS n of the -AlMnSi phase is presented versus the energy. The total density of states is characterized [68] by a pseudogap near the Fermi energy EF . Following the Hume-Rothery condition, it is expected that the most realistic value of EF in the actual -phase corresponds to the minimum of the pseudo-gap, i.e., EF EFðLMTOÞ ¼ 0:163 eV for our calculation. We compute the velocity correlation function CðE; tÞ for crystals (complex approximants and simple crystals). In Eqs.(7.12) and (7.37), the average is obtained ~ such as by taking the eigenstates for each k~ vector with and energy En ðkÞ 1 ~ < E þ 1 DE: E DE < En ðkÞ 2 2

ð7:145Þ

DE is the energy resolution of the calculation. The calculated CðE; tÞ is sensitive to the number Nk of k~ vectors in the first Brillouin zone when Nk is too small.


251

2e+12

C NB CB

C(E F, t ) /12

1e+12

0

–1e+12

–2e+12

0

1e–14

2e–14

3e–14

t (s) Figure 7.20. Velocity correlation function CðEF ; tÞ (m2 s2 ) in -Al69:6 Si13:0 Mn17:4 versus large time t. The dashed lines are the corresponding Boltzmann velocity correlation function CB ðEF ; tÞ ¼ 2v2F . From [91].

Therefore Nk is increased until CðE; tÞ does not depend significantly on Nk . For Al, Al12 Mn and -Al69:6 Si13:0 Mn17:4 , DE is equal to 0.272, 0.272, and 0.272 eV, respectively, and Nk is equal to 1203, 403 and 323, respectively. CðEF ; tÞ for the cubic approximant -AlMnSi is shown in Figure 7.20. After Eqs.(7.12) and (7.37), CðEF ; tÞ is the sum of a constant Boltzmann term CB ðE; tÞ and a Non-Boltzmann term containing oscillating terms that average to zero on long time scale: CðE; tÞ ¼ CB ðE; tÞ þ CNB ðE; tÞ;

ð7:146Þ

where

CB ðE; tÞ ¼ 2 V2x E ; Z1 lim

!1

CNB ðE; tÞet= dt ¼ 0;

ð7:147Þ

ð7:148Þ

0

and V2x is the square of Boltzmann velocity (intra-band velocity) along the X direction at the Fermi energy: vF ¼ 9:4 107 , 3:6 107 , and 2:7 106 cm.s1 , for Al, Al12 Mn and -Al69:6 Si13:0 Mn17:4 , respectively. This last result is very similar to the original work of Fujiwara et al. for the -Al114 Mn24 (with the atomic structure model of Elser-Henley) [69], for a model of icosahedral approximant AlCuFe [81]. The strong reduction of vF in the approximant phase with respect to simple crystal phases shows the importance of a quasiperiodic medium-range order (up to distances equal to 12–20 A˚). This leads to a very small Boltzmann conductivity for approximants [69,81]. In the case of a decagonal approximant AlCuCo, a strong anisotropy has been found between vF in the ‘pseudo’ quasiperiodic directions and vF in the periodic direction [82].

252

Chapter 7.


3e+12

C (EF , t ) /2

Al (f.c.c) 2e+12

1e+12

0

0

2e–16

4e–16

6e–16

8e–16

1e–15 1.2e–15

8e–16

1e–15 1.2e–15

C (EF , t ) /2

8e+12

Al12Mn 4e+12 0 –4e+12 0

2e–16

C (EF , t ) /2

8e+12

4e–16

6e–16

α –Al69.6Si13Mn17.4

4e+12 0 –4e+12 0

2e–16

4e–16

6e–16

8e–16

1e–15 1.2e–15

t (s) Figure 7.21. Velocity correlation function CðEF ; tÞ (m2 s2 ) versus small time t. The dashed lines are the corresponding Boltzmann velocity correlation function CB ðEF ; tÞ ¼ 2v2F . From [91].

On small time scale t (Figure 7.21), CðEF ; tÞ and CB ðEF ; tÞ differ, and there is a new difference between approximant and simple crystal. In the case of Al (f.c.c.) phase, CðEF ; tÞ is always positive, and the Boltzmann value is reached rapidly when t increases. But for some values the velocity correlation function CðEF ; tÞ is negative for Al12 Mn and -Al114 Mn24 . That means that at these times the phenomenon of backscattering occurs. The transport properties depend on the average value of CðEF ; tÞ on a time scale equals to the scattering time [6,96] (see for instance Eq.(7.29)). A realistic value of has been estimated to about 104 s [33]. For the simple crystals Al12 Mn, CðEF ; tÞ is mainly positive when t > 21015 s. But for the complex approximant -Al114 Mn24 , a lot of t values correspond to CðEF ; tÞ < 0, even when t is close to or larger (Figure 7.20). Therefore, in the case of Al12 Mn, the backscattering (negative value of CðEF ; tÞ) should have a negligible effect on the transport properties, whereas this effect must be determinant for the approximant.

253

7.4. Evidence of anomalous quantum diffusion in quasicrystals and approximants 600

ΔX 2 ΔX 2B ΔX 2NB

ΔX 2 (Å)

400

200

0

0

2e–14

4e–14

6e–14

8e–14

1e–13

t (s) Figure 7.22. Square spreading DX2 of electrons states with Fermi energy EF versus time t, in the cubic approximant -Al69:6 Si13:0 Mn17:4 . DX2 ¼ DX2B þ DX2NB (see text). From [90].

As discussed in Section 7.3 the phenomenon of backscattering is associated to unusual quantum diffusion. It is illustrated on the plot of the average spreading of states DX2 versus time t (Figure 7.22). After Eq.(7.40), DX2 results in two term: DX2 ðE; tÞ ¼ VB ðEÞ2 t2 þ DX2NB ðE; tÞ;

ð7:149Þ

a Boltzmann term VB ðEÞ2 t2 and a non-Boltzmann term. The non-Boltzmann contribution, DX2NB , which comes from the non-diagonal matrix element Eq.(7.44), increases very rapidly and saturates to a maximum value of the order of the square size of the unit cell. In the -approximant, at small time t, DX2B is smaller than in Al due to a very small velocity VF of the electron with energy EF . The calculated VF is equal to 2:7107 cm.s1 , which is about 30 times smaller than aluminum values. Thus -AlMnSi is a non-conventional metal at these time scale, i.e., when the scattering time is < where ’ 3 1014 s. In a normal crystal, the DX2NB ðtÞ term is negligible with respect to the Boltzmann term DX2B ðtÞ. On the contrary, in the approximant both terms have the same magnitude at the realistic scattering times scale, typically a fews 1014s.

7.4.3.

Ab initio RTA model for the conductivity of approximants

Within a relaxation time approximation the diffusivity DðE; !Þ is calculated. At low frequency one obtains Re ðE; !Þ ¼ e2 nðEÞDðE; !Þ:

ð7:150Þ

DðE; !Þ ¼ DB ðE; !Þ þ DNB ðE; !Þ;

ð7:151Þ

Here,

254

Chapter 7.


where DB ðE; !Þ ¼

V2 ; 1 þ !2 2

ð7:152Þ

and DNB ðE; !Þ ¼

8 , and the insulating like regime (Non-Boltzmann regime) at small scattering time, < . ¼ 3:071014 s is defined as the time for which the Boltzmann and non-Boltzmann contributions are equal. As expected from our model, NB is almost proportional to 1/. Therefore, in the Non-Boltzmann regime, the conductivity increases with disorder as observed experimentally. For realistic, values, < [33], NB dominates and increases when 1/ increases i.e. when defects or temperature increases. varies from 250 ( cm)1 for ¼ 3:31014 s, to 2000 ( cm)1 for ¼ 105 s. This is consistent with experimental results in -AlMnSi: ð4KÞ ’ 200 ( cm)1 and ð900KÞ ’ 2000 ( cm)1 and with standard estimates for the scattering time in these systems [23]. Furthermore for equal to a few 104 s, i.e. when the Boltzmann term is negligible, the mean free path is given by the square root of the saturation value of DX2NB and is of the order of 15 A˚. This is in agreement

Conductivity σ (Ω cm)−1

1000 800

σ σB σNB

600 400 200 0 1e+14

2e+14

3e+14

4e+14

1/τ (s−1) Figure 7.23. Ab initio dc conductivity in cubic approximant -Al69:6 Si13:0 Mn17:4 versus inverse scattering time.


255


(a)

1000

σ σB σNB

100

10

1e+13

1/τ∗

1e+14

1e+15

1/τ (s−1)

(b) Drude (Al)

σ (Ω cm)−1

1e + 06

Al (f.c.c.)

1e + 05

Al12Mn Al6Mn

10000

1000

100

α-Al69.6Si13.0Mn17.4

1e + 13

1/τ∗

1e + 14

1/τ (s−1) Figure 7.24. Ab initio electrical conductivity ðEF Þ (logarithmic scale) versus inverse scattering time 1= (logarithmic scale). (a) In cubic approximant -Al69:6 Si13:0 Mn17:4 : ðEF Þ is the sum of a ballistic term (Boltzmann term), B ¼ e2 nðEF ÞV2F , and a non-ballistic term (non-Boltzmann term), NB . (b) In pure Al (f.c.c.), the Boltzmann term dominates, and the model is compatible with a simple Drude model (dashed line). In cubic Al12 Mn and orthorhombic Al6 Mn crystal, the model predicts also a Boltzmann behavior as expected experimentally. From [90].

with estimates in the literature [23]. As discussed in Section 7.3 this means also that the systems is far from the Anderson transition despite its low conductivity. From the ab initio calculations the estimated value of the ratio g0 =gc for the -AlMnSi phase is about 2–3. This means that this system is always metallic as discussed in Section 7.3.3. According to Figure 7.13 the -AlMnSi phase is a phase of type (a).

7.4.3.1.

Optical conductivity

Within the relaxation time approximation used here, the optical conductivity ð!Þ is the sum of two terms. The Boltzmann contribution (B ð!Þ, diagonal elements of the

256

Chapter 7.


velocity operator) gives rise to the so-called Drude peak and the Non-Boltzmann conductivity gives rise to a nearly frequency independent contribution. This is a consequence of the fact that DX2NB ðE; tÞ is nearly constant on the time scale of (see Section 7.3). Fujiwara et al. [69] have also estimated the optical conductivity from the LMTO band dispersion of a -Al114 Mn24 (Figure 7.25). This calculation reproduces the linearity and the peak position observed experimentally. Our ab initio calculation (Figure 7.26) confirms that a Drude peak can be identified in the Boltzmann regime, > , whereas in the non Boltzmann regime, < , the Drude peak disappears. The role of transition metal elements (TM = Fe, Mn, Co, Pd, Re) in the electronic structure of quasicrystals and related phases as been often discussed in the literature [67–78]. Because of their strong scattering potential with respect to Al(Si) atoms, TM elements play a crucial role in the formation of the HumeRothery pseudogap that contributes to the stability of these phases. This effect is related to an effective medium range interaction between TM atoms mediated by a strong sp(Al)-d(TM) hybridization [73]. TM elements have also a very important role on the transport properties. As an example, it is shown in the previous paragraph how a Mn-cluster can ‘localize’ electrons [84,85]. To evaluate the effect of TM elements on the conductivity calculation in the RTA, we have considered an hypothetical -Al69:6 Si13:0 Cu17:4 constructed by putting Cu atoms in place of Mn atoms in the actual -Al69:6 Si13:0 Mn17:4 structure. Cu atoms have almost the same number of sp electrons as Mn atoms, but their d DOS is very small at EF . Therefore in -Al69:6 Si13:0 Cu17:4 , the effect of sp(Al)-d(TM) hybridization on electronic states with energy near EF is very small. As a result, the pseudogap disappears in total DOS, and the dc-conductivity is now ballistic (metallic) as shown on Figure 7.27.

Optical conductivity (Ω cm)−1

1000

800

600

400

200

0

0

1

2

3

4

5

6

hω (eV)

Figure 7.25. Inter-band optical conductivity, NB ð!Þ, in -Al114 Mn24 calculated from the LMTO results, with a relaxation time equals to infinity. From Fujiwara et al. [69].


257

500

Optical conductivity σ (Ω cm)−1

no Drude peak 400

300

τ = τ ∗/3 τ = τ∗ τ = 3 τ∗

Drude peak

200

100

0

0

0.02

0.04

0.06

hω (eV)

Figure 7.26. Ab initio optical conductivity ð!Þ in cubic approximant -Al69:6 Si13:0 Mn17:4 for three values. ! is the pulsation. For ¼ 3 , the non Boltzmann conductivity NB is smaller than Lorentzian of the Boltzmann conductivity B , B ð!Þ ¼ B ð0Þ=ð1 þ !2 2 Þ. For ¼ ¼ 3:07 1014 s, NB ð0Þ ¼ B ð0Þ. For ¼ =3, the Non-Boltzmann conductivity NB ð!Þ dominates.


4e+05

3e+05

σ σB σNB

2e+05

1e+05

0 1e+14

2e+14

3e+14

4e+14

1/τ (s−1) Figure 7.27. Ab initio dc conductivity in an hypothetical cubic approximant -Al69:6 Si13:0 Cu17:4 versus inverse scattering time.

7.4.4. Phenomenological model for the low frequency conductivity of AlCuFe quasicrystals The ab initio calculations which rest on the Bloch theorem are applicable to approximants only. Here we present a phenomenological model of the optical conductivity of AlCuFe QCs, which should be approximately valid also for the

258

Chapter 7.


related QC phases AlPdMn or AlFeCr [94,95]. This anomalous diffusion model allows to derive an analytical expression for the conductivity that fits experiments very well. In particular the model explains quantitatively the main experimental facts: – the increase of conductivity with disorder – the ‘inverse Mathiessen rule’ [23,24], i.e., the fact that the increases of conductivity due to different sources of scattering are additive – the absence of the Drude peak One needs first a model of conductivity of the perfect system at all frequencies. For low frequencies, according to the discussion in Section 7.2 we assume: 12 j!j Re 0 ð!Þ ¼ 0 !1

for j!j < !1 :

ð7:154Þ

A small value of ( 1 ) is imposed by the nearly linear experimental variation of Re ð!Þ at ! < 8000 cm1 (see Figure 7.28). This means that the system without defects would be insulating. In Eq.(7.154) we take ¼ 0, !1 ’ 8000 cm1 and 0 ’ 6000 ðcmÞ1 . At higher frequencies one uses other analytical expressions. For 8000 cm1 < ! < 25 000 cm1 a polynomial of ! reproduces the experimental value. For ! > 25000cm1 , we take the Drude expression according to [35]. Let us note that the experimental uncertainty on the high frequency conductivity [35] has essentially no effect on the results presented here. Within the RTA (Eqs. (7.3) and (7.25)) one has for the optical conductivity: Zþ1 Re ð!; Þ ¼ 1

Re 0 ð! !0 Þ d!0 ; ð!02 þ 1= 2 Þ

ð7:155Þ

6000

Re σ (Ωcm)−1

5000 4000 3000 2000 1000 0

0

1000 2000

3000 4000 5000

6000 7000

8000

ω (cm–1)

Figure 7.28. Real part of the conductivity as a function of the frequency for different cases. Line with circles: experimental conductivity of an AlCuFe QC [9]. Thin line: conductivity of the model without defect Re0 ð!Þ ¼ 1 ðj!j=!1 Þ. Thick line: conductivity of the model with defect, for ¼ 31014 s. From [96].


259

i.e., the real part Re ð!; Þ of the conductivity of the system with defects is the convolution of Re 0 ð!Þ of the perfect system and of a Lorentzian of width 1/. As shown in Section 7.2 for ! < !1 the conductivity is well represented by 2 Re ’

0

1

3

A 6 !1 B C 7 ’ 4 þ log@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ !Arctgð!Þ5; 2 1 þ ð!Þ

ð7:156Þ

where A ¼ 20 =!1 The analytical expression (7.94) with ’ 0:7, h!1 ’ 1eV and 1 ’ 6000ðcmÞ1 describes well the electronic conductivity in Figures 7.28 and 7.29. Let us focus on the low frequency conductivity (! < !1 ) which is the real test of the model (7.155–7.156). Figure 7.28 gives a comparison of the experimental Re exp ð!Þ for AlCuFe [35] with the theoretical Reð!; Þ. The scattering time is chosen to reproduce the experimental dc-conductivity dc ’ 350 ðcmÞ1 . One finds ’ 31014 s which is rather long, in agreement with the high structural quality of these systems. The fit is good except for the peak in the experimental curve around 200cm1 . This peak is attributed to the conductivity of phonons [35] which is not incorporated in the model. The mean-free path is related to the scattering time and to the diffusivity D through D ¼ 2 =3. One estimates [23,34] D 0:2 cm2 =s and since ’ 3 1014 s, one has 15 20 A˚. From Eqs.(7.155) and (7.156) one obtains the dc-conductivity dc ðÞ as a function of the relaxation time (see Figure 7.29). dc ðÞ increases with 1/ and varies nearly linearly with 1= on a large range of values of dc ðÞ i.e. dc ’ A þ B=. For two independent sources of scattering characterized by scattering times 1 and 2 it is common that the inverse relaxation times add. Then 1= ’ 1=1 þ 1=2 and dc ’ A þ B= ’ A þ B=1 þ B=2 . Thus each source of disorder gives its contribution to the conductivity in agreement with the ‘inverse Mathiessen rule’ [23]. The present phenomenological model treats the disorder within the relaxation time approximation (RTA). Indeed, as shown now, the RTA is applicable to AlCuFe QCs, at least for T 200 300 K. A first indication is that quantum

700

σdc (Ωcm)−1

600 500 400 300 200 100 0

0

10

20

30

40

50

x = 1000 /ω1τ Figure 7.29. Thick line: Variation of dc with x ¼ 1000=!1 . is given by ¼ ð6:6=xÞ1013 s. The straight thin line shows that dc varies nearly linearly with in the range dc ¼ 150700 ðcmÞ1 . From [96].

260

Chapter 7.


interferences have been found for T 200 300K [23]. They indicate that the main scattering sources are elastic in this temperature range. Indeed, if the dominant scattering were inelastic, the coherence of the electron wavefunction would be lost at each scattering event. In that case there would be no interferences in the diffusive regime. In addition, the experimental fits [23,34] show that the quantum interferences and the electron-electron interaction give only a correction to the conductivity. Therefore, as the elastic scattering dominates and as quantum interferences are weak, the RTA is a good approximation for the AlCuFe QC studied in [35] at least at T 200 300K. In particular, a scenario of hopping between localized critical states, such as proposed by Janot [83] is not consistent with the present analysis. Note that the model is consistent with the observed weak-localization effects. Indeed in the context of the scaling theory of localization [97] the importance of quantum interferences depends on the ratio between the dc-conductivity of the system dc and the Mott value Mott ’ 600ðcmÞ1 =, where is the mean-free path expressed in Angstro¨ms. If R ¼ dc =Mott 1, the effect of the quantum interferences on dc is small. Here R ¼ dc =Mott ’ 5 10 and the localization effects are only corrections. To conclude, the dynamics of electrons in AlCuFe quaiscrystals and related systems such as AlPdMn [94], AlCrFe [95] quasicrystals is not free electron like. The minimum of optical conductivity at low frequency and the increase of dcconductivity with disorder (in the RTA scheme) are intimately related to the backscattering (Figure 7.30) or equivalently to an anomalous diffusion law with < 0:5 (Ref. [6]). The low value of implies a slow anomalous diffusion LðtÞ / t , which suggests the proximity to a localized state. This value ’ 0 is consistent with the results on approximant phases. Indeed the non-Boltzmann term in the quantum diffusion DX2NB ðtÞ which saturates very quickly is analogous to a quantum diffusion with ’ 0.

7

j(t) (× 10–17 Ω–1.cm–1.s–1)

6 5 4 3 2 1 0 −1

0

1

2

3

t (× 10–15 s) Figure 7.30. Value of jðtÞ deduced from the experimental conductivity. The negative value of j(t) at large times indicates backscattering. From [96].

7.5. Conclusion

7.5.

261

Conclusion

In this chapter we concentrated on quantum diffusion and electronic conduction properties in quasiperiodic and periodic systems. We found that deviations from the standard ballistic propagation exist either in quasiperiodic or even in periodic systems. This anomalous diffusion mode has deep consequences on the conduction properties at zero and low frequency. The anomalous diffusion mode is related to a tendency to localization and to a phenomenon of backscattering which is well known in disordered systems. The phenomenon of backscattering is the fact that an impulse of electric field creates a current density JðtÞ which is opposite to the electric field at large time. Backscattering is associated with an increase of conductivity with frequency and disorder. The physics of phonons in quasicrystals could be affected by the anomalous diffusion phenomenon. In particular it has been argued that the heat conductivity could be sensitive to this effect [98]. The concepts developed here also open a new insight in the physics of correlated systems. Indeed recent studies of some heavy fermions or polaronic systems [99– 101], where charge carriers are also slow, show that their conduction properties present a deep analogy with those described here. In particular a transition from a metallic like regime at low temperature where scattering is weak to an insulating like regime at higher temperature with a stronger scattering is observed.

Acknowledgements The work presented in the review paper have been done since the 1990s. Our work owes much to the discussions with Prof. T. Fujiwara, Prof. J. Bellissard, Prof. J. Friedel and Prof. N.W. Ashcroft. We are very grateful to many colleagues with whom we had collaborations during this time: C. Berger, F. Cyrot–Lackmann, J. Delahaye, T. Grenet, J.P. Julien, T. Klein, L. Magaud, J.J. Pre´jean, S. Roche and F. Triozon. We also thanks F. Hippert, R. Mosseri, J. Vidal and C. Sire for fruitful discussions.

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Chapter 8

Elastic and Plastic Properties of Quasicrystals S. Takeuchi1 and K. Edagawa2 1 Tokyo University of Science, Kagurazaka Shinjuku-ku, Tokyo 162-8601, Japan 2 Institute of Industrial Science, The University of Tokyo, Meguro-ku, Tokyo 153-8505, Japan

8.1.

Introduction

Since the establishment of a new class of material ‘the quasicrystal’ in 1984, physical properties of quasicrystals have attracted much attention by solid-state physicists and materials scientists. The mechanical property is one of them. Due to the specific structure of the quasicrystal, i.e., the quasiperiodic structure different from the periodic structure in crystalline matter, mechanical properties of quasicrystals exhibit specific properties different from those of crystalline matter. Concerning the elastic property, quasicrystals have a special type of elastic degrees of freedom, termed phason degrees of freedom. Quasicrystals are accompanied by the phason elastic field in addition to the phonon (conventional) elastic field. The generalized elasticity of quasicrystals is described in terms of the two types of elastic fields. Within linear elasticity, elastic free energy of quasicrystals consists of three types of quadratic terms: phonon-phonon, phason–phason and phonon–phason coupling terms. Correspondingly, elasticity of quasicrystals comprises the three parts: pure phonon elasticity, pure phason elasticity and phonon–phason coupling. As for the plastic property, while the crystalline matter can be plastically deformed by conservative slip processes along crystallographic planes, due to the lack of periodicity in quasilattices, no conservative slip can be allowed in quasicrytals, which makes quasicrystals hard and brittle. However, perfect dislocations can be defined also in the quasilattice as in crystals, but they accompany not only the phonon strain (conventional elastic strain) but also the phason strain around them. Since the relaxation of the phason strain requires phason flips which are the thermally activated process, the migration of a dislocation in quasicrystal is possible at high temperatures. Actually, it has been clarified that quasicrsytals are plastically deformable at high temperatures and that the deformation is brought by a dislocation process. In recent years, it has been shown that the high temperature HANDBOOK OF METAL PHYSICS ISSN 1570-002X/DOI 10.1016/S1570-002X(06)02008-8 267


268

Chapter 8.

Elastic and Plastic Properties of Quasicrystals

deformation of quasicrystals is produced not by the glide motion but by the climb motion of dislocations. In the present chapter, we review the present understandings of characteristic features of elastic and plastic properties of quasicrystals both from the theoretical and experimental point of views. We should note that our present understandings of the above properties are far from perfect; e.g., the phason elasticity, the elementary thermally activated process of the dislocation motion, the mechanism of the pronounced work-softening phenomenon, etc. have yet to be investigated further to achieve deeper understandings.

8.2.

Elastic properties

8.2.1.

Phonon and phason degrees of freedom

In general, the diffraction intensity function IðqÞ (q: wave vector) of a solid is given by IðqÞ jSðqÞj2 ; where Z SðqÞ ¼

ðrÞ expð2iq rÞdr:

ð8:1Þ

Here, ðrÞ is the atomic-density function in real space. The function IðqÞ observed experimentally for a quasicrystal has the following characteristics [1–3]: (1) It consists of –functions. (2) The number of basis vectors necessary for indexing the positions of the -functions exceeds the number of dimensions. (3) It shows a rotational symmetry forbidden in the conventional crystallography. Conversely, we define the quasicrystal as the material with a ðrÞ which gives a diffraction intensity function IðqÞ satisfying these conditions. The condition (1) implies that this material has a kind of long-range translational order. The conditions (2) and (3) indicate that the order is not periodicity. The translational order defined by the conditions (1) and (2) is called quasiperiodicity. According to the definition, every d-dimensional quasiperiodic atomic-density function ðrÞ can be expressed as X m1 ;:::;mN exp 2 iGm1 ;:::;mN r ; ð8:2Þ ðrÞ ¼ mi 2I

where m1 ; :::;mN is the Fourier component associated with the reciprocal vector N P Gm1 ;:::;mN ¼ mn qn . Here, {qn } (n ¼ 1,2,...,N ) (N > d ) are the reciprocal basis n¼1

vectors. Now, let us define an N-dimensional function h ðx1 ; :::; xN Þ as

269


ðx1 ; :::; xN Þ ¼ h

X

" m1 ;:::;mN exp 2 i

mi 2I

N X

# mn xn :

ð8:3Þ

n¼1

This function is periodic in every xn (n ¼ 1,2,...,N ) with the period of unity. Comparing Eqs. (8.2) and (8.3), we find ðrÞ ¼ h ðx1 ; :::; xN Þ;

ð8:4Þ

where xn ¼ qn r ðn ¼ 1; 2; :::; NÞ:

ð8:5Þ

This indicates that the d-dimensional quasiperiodic function ðrÞ can be described as a d-dimensional section of an N-dimensional periodic function. As an example, Figure 8.1(a) presents a typical one-dimensional quasiperiodic structure known as a Fibonacci lattice, which is described as a one-dimensional section of a two-dimensional periodic function (d ¼ 1 and N ¼ 2). Here, Ek and E? denote the physical space and the complementary space perpendicular to it, respectively. In this case, the two-dimensional periodic structure consists of a periodic arrangement of a line segment extending in the direction of E? . The line segment is called atomic surface. A point sequence is obtained on the Ek section, comprising an arrangement of two spacings L and S. Here, the slope of Ek with respect to the

(a)

(b)

E⊥

E⊥

U L S L

L S

L S L

L S

O

(c)

L S L

L S L

E

(d)

E⊥

E

O

E⊥

u⊥ Ou

E

O

E

Figure 8.1. A Fibonacci lattice (a), and the structure resulting from a displacement of U (b), that of u (c) and that of w (d).

270

Chapter 8.


pffiffiffi two-dimensional lattice is an irrational number (the golden mean: ð1 þ 5Þ=2). In this case, the resultant sequence of the two spacings becomes equivalent to the Fibonacci sequence and therefore it is called a Fibonacci lattice. The irrational slope indicates lack of periodicity in the arrangement of L and S. In the description of the Fibonacci lattice in Figure 8.1(a), let us consider a translation of the two-dimensional periodic structure by a vector U with respect to the origin of the physical space Ek (Figure 8.1(b)). It can be shown that the two structures on Ek before and after the displacement U shown in Figures 8.1(a) and (b) can be overlapped out to arbitrarily large finite distances by a finite translation in Ek . Two structures satisfying this condition are said to belong to the same local isomorphism class (LI class) [1–3]. Thus, the displacement U represents the degrees of freedom of generating a series of structures belonging to the same LI class. Obviously from the definition of the LI class, a series of structures in the same LI class are geometrically indistinguishable on any finite scale and thus they are also physically indistinguishable: they give the same diffraction intensity function IðqÞ and have the same energy. The vector U can be decomposed into u in Ek and w in E? : U ¼ u þ w:

ð8:6Þ

Here, u represents the degrees of freedom of d-dimensional translation in physical space, which conventional crystals also possess, and w represents ðN d Þ degrees of freedom characteristic of quasiperiodic system. As shown in Figures 8.1(c) and (d), while u results in translation of Fibonacci lattice in Ek , w generates a rearrangement of L and S. The two kinds of degrees of freedom are called phonon and phason degrees of freedom, and u and w phonon and phason displacements, respectively [3,4]. When these displacements vary spatially, the gradients of them yield a strain. More specifically, while the gradient of u gives the conventional elastic strain, that of w yields a strain called the phason strain. Figures 8.2(b) and (c) illustrate, respectively, a phonon-strained and a phason-strained structure of the Fibonacci lattice in Figure 8.2(a). In Figure 8.2(b), a uniform phonon strain is introduced by a compression deformation of the two-dimensional structure. On the other hand, a uniform phason strain is introduced by a shear deformation of the two-dimensional structure in Figure 8.2(c). It is noteworthy that the phonon and phason stain fields should have quite different dynamical properties. As described above, a phason displacement results in a local rearrangement of points (atoms) such as LS $ SL, which is called the phason flip. Examples of the phason flip in two dimensional Penrose lattice, which is known as a typical two-dimensional decagonal quasicrystal, are shown in Figure 8.3. In these cases, a lattice point in a hexagon makes a transition between two (meta-) stable positions. Introduction or annihilation of phason strain requires a combination of phason flips. Generally, the phason flip is a thermally activated process and thus relaxation of a phason strain whose elementary process is the phason flip must proceed relatively slowly like atomic diffusion in solids [3–5]. This is in sharp contrast to the conventional strain, which can be relaxed instantaneously via displacive phonon modes. Experimentally, thermally induced phason flips have been investigated by neutron scattering [6–9], by Moessbauer spectroscopy [10], by

271


(a)

(b)

E⊥

L S L L S L S L L S L S L L S L O

E

O

(c)

O

Figure 8.2. A Fibonacci lattice (a), and its phonon-strained (b) and phason-strained (c) structures.

Figure 8.3. Examples of phason flips in the two-dimensional Penrose lattice.

NMR [11,12] and specific-heat measurements [13,14]. Direct observations of thermally induced phason flips have been made by in situ high-temperature, high-resolution transmission electron microscopy [15–18]. The activation enthalpy of a phason flip or collective phason flips has been estimated to be 1eV in an Al–Pd–Mn icosahedral quasicrystal by radiotracer diffusion experiments [19]. In general, to decompose properly the N total degrees of freedom into d phonon and ðN d Þ phason degrees of freedom, or equivalently, to embed properly a given d-dimensional quasiperiodic structure into an N-dimensional hypercrystal, the information on the point group symmetry of the system is needed. In the following, the method of embedding is briefly reviewed for the two most important classes of quasicrystals: icosahedral and decagonal quasicrystals [3,20–23].

272

Chapter 8.


To describe the structure of a three-dimensional icosahedral quasicrystal, we use a six-dimensional hypercubic lattice spanned by di (i ¼ 1,...,6): di ¼ Mij ej 2 6 6 6 2 12 6 M ¼ a 2 þ 2 6 6 6 6 4

1

1 1 0

0

3

1

0

0

0 1

0 1

0 1 0 0 1 0 1 0

7 7 7 7 7; 1 7 7 7 1 5 0 1

ð8:7Þ

pffiffiffi where is the golden mean ( ¼ ð1 þ 5Þ=2) and a is the lattice constant. ei (i ¼ 1,...,6) are the orthonormal unit vectors of the six-dimensional space. The icosahedral point group Y is generated by a fivefold rotation C5 and a threefold rotation C3 . The lattice spanned by di (i ¼ 1,...,6) is invariant under these operations. The actions of these operations on di (i ¼ 1,...,6) are given, respectively, as 2 6 6 6 6 ðC5 Þ ¼ 6 6 6 6 4

1

0

0

0

0 0

0 0

1 0

0 1

0 0 0

0 0 1

0 0 0

0 0 0

3

2

0 07 7 7 0 07 7; 1 07 7 7 0 15

6 6 6 6 ðC3 Þ ¼ 6 6 6 6 4

0 0

0 0

0

0

0

0

0

1 0

0 0

0 0

0 0

0 1

0 0 0

0 0 1

1 0 0 1 0 0

0 0 0

1

3

7 7 7 7 7: ð8:8Þ 07 7 7 05 0 0 0

The character table for the icosahedral goup Y is presented in Table 8.1. Here, the irreducible representations of the group Y are a one-dimensional 1 , two distinct three-dimensional 2 and 3 , a four-dimensional 4 , and a five-dimensional 5 . The six-dimensional representation in Eq. (8.8) is reducible and can be decomposed into the sum of the irreducible representations as ¼ 2 þ 3 . The subspace spanned by e1 , e2 , and e3 , and the subspace spanned by e4 , e5 , and e6 are the eigenspaces of the representations 2 and 3 , and correspond to the physical space Ek and its complementary space E? , respectively. Because the six-dimensional hypercubic lattice is invariant under the inversion Ci , it has the maximum icosahedral point group symmetry Yh ¼ Y Ci . By placing atomic surfaces spreading in E? (in this case Table 8.1. Character table for an icosahedral group Y. Y

E

12C5

12C25

20C3

2 3 4 5

1 3 3 4 5

1 1 1 0

1 1 1 0

1 0 0 1 1

1

15C2 1 –1 –1 0 1

273


they are three-dimensional objects) at appropriate positions in the six-dimensional lattice, a three-dimensional icosahedral quasicrystalline structure with the symmetry Y or Yh is obtained as a section on Ek . Here, to preserve the symmetry, each of the atomic surfaces must satisfy the site symmetry of its position in the corresponding symmetry group. Similar to the case illustrated in Figure 8.1, a six-dimensional displacement U can be defined, which can be decomposed into u 2 Ek and w 2 E? , representing the three-dimensional phonon and the three-dimensional phason displacements. To describe a three-dimensional decagonal quasicrystal, we use a five-dimensional lattice spanned by di (i ¼ 1,...,5): di ¼ Mij ej ; 2 a pffiffi ðc1 1Þ 5 6 a 6 pffiffi ðc2 1Þ 6 5 6 M ¼ 6 paffiffi5 ðc3 1Þ 6 6 paffiffi ðc 1Þ 4 5 4

paffiffi s1 5 paffiffi s2 5 paffiffi s3 5 paffiffi s4 5

0

paffiffi ðc2 5 paffiffi ðc4 5 paffiffi ðc1 5 paffiffi ðc3 5

1Þ 1Þ 1Þ 1Þ

0

0

paffiffi s2 5 paffiffi s4 5 paffiffi s1 5 paffiffi s3 5

0

3

7 07 7 7 0 7; 7 07 5 c

0

ð8:9Þ

where a and c are the lattice constants, and ci ¼ cosð2 i=5Þ and si ¼ sinð2 i=5Þ. ei (i ¼ 1,...,5) are the orthonormal unit vectors of the five-dimensional space. The character table for the decagonal group C10v is shown in Table 8.2. The generators of this group are a tenfold rotation C10 and a mirror v . The lattice spanned by di (i ¼ 1,...,5) is invariant under these operations. The actions of these operations on di (i ¼ 1,...,5) are given, respectively, by 2

0 6 0 6 6 ðC10 Þ ¼ 6 0 6 4 1 0

1 1

0 0

0 0

3 0 07 7 7 ; 07 7 05

0

0

0

1

1 1 0 1 0 1

2

0 60 6 6 ðv Þ ¼ 6 0 6 41

0 0

0 1

1 0

0 0

3 1 0 0 07 7 7 : ð8:10Þ 0 07 7 0 05

0

0

0

0

1

Table 8.2. Character table for a decagonal group C10v . C10v

E

2C10

2C5

1 2 3 4 5 6 7 8

1 1 1 1 2 2 2 2

1 1 1 1 1 1

1 1 1 1 1 1

2C310

2C25

C2

5v

5d

1 1 1 1 1 1

1 1 1 1 1 1

1 1 1 1 2 2 2 2

1 1 1 1 0 0 0 0

1 1 1 1 0 0 0 0

274

Chapter 8.


This five-dimensional representation is reducible: ¼ 5 þ 7 þ 1 . The subspace E1k spanned by e1 and e2 , the subspace E? spanned by e3 and e4 , and the subspace E2k along e5 are the eigenspaces of the representations 5 , 7 and 1 , respectively. E1k is parallel to the quasiperiodic plane and E2k is along the tenfold periodic directions in the physical space Ek ¼ E1k þ E2k . Because the lattice spanned by di (i ¼ 1,...,5) in Eq. (8.9) is invariant under the inversion Ci , it has the maximum decagonal point group symmetry D10h ¼ C10v Ci . There are seven point groups in the decagonal system: C10 , C5h , C10h , C10v , D10 , D5h , and D10h . By placing atomic surfaces spreading in E? at appropriate positions in the five-dimensional lattice, a three-dimensional decagonal quasicrystalline structure with any of those decagonal point group symmetries can be constructed. Here, to preserve the symmetry, each atomic surface must satisfy the site symmetry of its position in the given point group. A fivedimensional displacement U can be defined, which can be decomposed into u 2 Ek and w 2 E? , representing the three-dimensional phonon and the two-dimensional phason displacements.

8.2.2.

Elastic free energy

i As described in the preceding section, the spatial variation of u, i.e., ruðrÞ ¼ @u @rj @wi yields the phonon (conventional) strain while that of w, i.e., rwðrÞ ¼ @rj yields a strain called the phason strain. Here, u and w are the functions of only r 2 Ek and @r@ i denotes a spatial derivative in Ek . More the phonon strain should be specifically, @u i defined as a symmetrical form uij ¼ 12 @rij þ @u , to remove the component of the @rj rigid rotation which does not change the elastic free energy. The elastic free energy density f is given as a function of the phonon strain uij and the phason strain wij . The function f can be expanded into the Taylor series in the vicinity of uij ¼ 0 and wij ¼ 0. In the regime with juij j; jwij j L is generally satisfied. Thus, as in the case of the kink-pair formation, the jog-collisionless case is realized. For L < lj , the dislocation climb velocity Vc is determined by the rate of jog-pair formation on the segment length L and is given by Vc ¼

H jp ð Þ DHs =2 4LDs Va exp ; kB T a2 kB T lnð z=bjj Þ

ð8:27Þ

where Va is the atomic volume, H jp is the activation enthalpy of the jog-pair formation. Writing Ds ¼ a2 D expðHs =kB T Þ (Hs : the activation enthalpy of the self-diffusion; D : the Debye frequency) and approximating Va ¼ a3 , Eq. (8.27) is rewritten as H jp ð Þ þ ðHs DHs =2Þ 4La3 D exp : Vc ¼ kB T kB T lnð z=bjj Þ

8.4. 8.4.1.

ð8:28Þ

Mechanical properties Hardness

The hardness is the most easily measurable quantity related to the mechanical property of materials. Since quasicrystals are brittle, the reliable hardness value can only be obtained under a low applied load without cracks. Vickers hardness measurements have been made for a number of icosahedral quasicrystals and also for single decagonal quasicrystals [98–110]. Hardness values at room temperature of Mackay-icosahedron type quasicrystals are as high as 1000, which is comparable to the hardness value of covalent Si at room temperature. In Figure 8.12, we plot the hardness value (HV ) measured at room temperature for various quasicrystals against Young’s modulus (E ). We find that HV is correlated with E by HV =E 0:05, suggesting that the hardness of quasicrystals is governed by a common deformation mechanism. It is interesting to note that the relation HV =E 0:05 for quasicrystals is close to the relation HV =E 0:06 0:005 that has long been established for metallic glasses [111,112]. The hardness values at elevated temperatures have also been reported for icosahedral quasicrystals [101,102,104,106,110] and Al–Ni–Co decagonal quasicrystal [110]. The results are summarized in Figure 8.13. The results show that the hardness decreases gradually up to about 600 K and then rapidly above 600 K with increasing temperature, suggesting a transition of the deformation mechanism at around 600 K. The effect of high temperature plastic deformation by compression on the hardness value has been measured for icosahedral Mg–Zn–Y [107]. Corresponding to the work-softening behavior in compression tests (mentioned later), the hardness value also decreases with increasing plastic strain applied at a high temperature. This softening phenomenon is in striking contrast to the work hardening phenomenon commonly seen in crystalline matter.

292

Chapter 8.

11

i -Al-Pd-Mn i -Al-Cu-Fe i -Al-Li-Cu i -Mg-Zn-Y d -Al-Co-Ni (basal) d -Al-Co-Ni (prism)

10 9

Vickers hardness, Hv (GPa)


8 7

Hv /E = 0.05

6 5 4 3 2 1 0

0

20

40

60

80

100 120 140 160 180 200

Young’s modulus, E (GPa) Figure 8.12. Hardness value plotted against Young’s modulus for i-Al–Pd–Mn [98], i-Al–Cu–Fe [100,103], i-Al–Li–Cu [98], i-Mg–Zn–Y [105] and d-Al–Cu–Ni [98].

8.4.2. 8.4.2.1.

High temperature plasticity Compression test of icosahedral quasicrystals

Since bulk quasicrystalline samples have become available in the late 1980s, high temperature compression experiments have been performed for a number of poly-quasicrystalline as well as single-quasicrystalline samples of various alloy systems. The first compression test at high temperature was performed by Shibuya et al. for i-Al–Cu–Ru [113]. Then, compression experiments have been performed for i-Al–Pd–Mn [102, 115–124], i-Al–Cu–Fe [125–134], i-Al–Li–Cu [135,136], i-Mg–Zn–(Y, Ho, Dy) [136–138] and i-Cd–Yb [139]. Among them, single quasicrystals of i-Al–Pd–Mn have been most extensively studied with detailed electron microscopy observations. The results of compression tests revealed the following common features for high temperature plasticity. (1) Undeformed samples can be plastically deformable without fracture only at high temperatures above 0.75Tm (Tm : the melting point). Only under a confined pressure or after pre-strained to a large strain at a high temperature, samples could be deformed below 0.75Tm . (2) At high temperatures, a gradual yielding takes place with the yield stress rapidly decreasing with increasing temperature.

293

8.4. Mechanical properties i -Al70Pd20Mn10 (2-, 3-, 5-fold plane) i -Al72Pd20Mn8 i -Al70Pd21Mn9 i -Al63.5Cu24Fe12.5 i -Mg30Zn60Y10 i -Mg36Zn56Y8 d -Al73Ni14Co13 (10-, 2-fold plane)

12 11 10

Vickers hardness (GPa)

9 8 7 6 5 4 3 2 1 0

300

400

500

600

700

800

900

1000

Temperature (K) Figure 8.13. Temperature dependence of the Vickers hardness of i-Al20Pd20Mn10 [102], i-Al72Pd20Mn8 [101], i-Al70Pd21Mn9 [110], i-Al63.5Cu24Fe12.5 [106], i-Mg30Zn60Y10 [110], i-Mg36Zn56Y8 [104] and d-Al Ni–Co [110].

(3) After the gradual yielding, the flow stress decreases continuously until levelingoff. The decrease of the flow stress, i.e. work softening, is more pronounced at lower temperatures and almost disappears at high temperatures where the yield stress becomes quite low. The flow stress decrease is often as large as 80% of the yield stress [121,140], i.e. the flow stress at high strain becomes only one-fifth of the yield stress. Such a large work softening up to large strains without showing work hardening is the most unique feature of the plasticity of icosahedral quasicrystals. Figure 8.14(a) shows true-stress versus true-strain curves at various temperatures for single Al–Pd–Mn and poly-CdYb icosahedral quasicrystals. (4) As a result of the above pronounced work softening at lower temperature, the temperature dependence of the leveling-off stress is rather weak. The temperature dependence of the yield stress of the samples pre-deformed at a high temperature to the leveling-off stage is also quite weak [140]. (5) In some poly-quasicrystalline samples, the deformation is localized to form large steps on the specimen surfaces [113,114], but in most of the samples including single-quasicrystalline samples, the deformation undergoes rather

294

Chapter 8.

(a) 250

1000

Cd-Yb 698 K

200

True stress (MPa)


Al-Pd-Mn

800 963 K

150

600

723 K

993 K 100

400 748 K

50

200

1013 K

773 K 0

0

0.2

0.4

0.6

0.8

0

0

0.2

True strain

0.4

0.6

0.8

True strain

(b)

Activation volume (nm3)

5

5

Cd-Yb

4

Al-Pd-Mn

4 698 K 3

3

2

2

993 K

723 K 1

963 K

1 1053 K

0

773 K 0

20

40

60

Shear stress (MPa)

80

0

0

50

100

150

200

250

Shear stress (MPa)

Figure 8.14. (a) true-stress versus true-strain curves for i-Al–Pd–Mn [124] and i-Cd–Yb [140]. (b) shear stress dependence of the activation volume for i-Al–Pd–Mn [124] and for i-Cd–Yb [140] obtained by the stress relaxation test at various temperatures.

homogeneously. In the homogeneous deformation no clear deformation markings are observed on the surfaces. (6) The stress relaxation tests showed that the flow stress decreases almost exponentially with time towards a quite low stress, indicating that the flow stress is strain-rate sensitive. On the assumption of the constant mobile dislocation density, the activation volume of the dislocation motion has been obtained as a function of the stress by the equation

@ ln "_ ðÞ ¼ kB T @

: T

ð8:29Þ

295

8.4. Mechanical properties 1000 Al63.5Cu24Fe12.5

900 800

Al70.5Pd21Mn8.5

Stress (MPa)

700 Al70.4Pd20.8Mn8.8

600 Zn46Mg51Dy3

500 Mg36Zn56Y8

400 300 200

Mg36Zn56Gd8

Cd85Yb15

Al60Li30Cu10

100 0

Al68Ru15Cu17

400

500

600

700

800

900 1000 1100 1200

Temperature (K) Figure 8.15. Temperature dependence of the upper yield stress for icosahedral quasicrystals of i-Al–Ru–Cu [113], Al–Pd–Mn [122,123], Al–Cu–Fe [130], Zn–Mg–Dy [139], Cd–Yb [140], Mg–Zn–Y [137], Mg–Zn–Gd [137] and Al–Li–Cu [136].

The obtained activation volume is a hyperbolically decreasing function of the stress, and the value at high stress is of the order of 0.1 nm3, which corresponds to about four atomic volumes. Figure 8.14 (b) shows the applied stress dependence of the activation volume at various temperatures obtained by the stress relaxation test; the shaded zones indicate the scatter of the data. In Figure 8.15 are plotted upper yield stresses as a function of the temperature for seven icosahedral quasicrystals. From the figure, the temperature range for the plastic deformation is largely different among different alloy systems. Open marks are for the data of single-quasicrystals. Corresponding closed marks are for the data of poly-quasicrystalline samples of the same alloy. We find that no essential difference in the yield stress between single- and polyquasicrystalline samples. Since icosahedral quasicrystals possess a highly isotropic structure and have quite many equivalent glide and climb systems, the grain boundaries do not act as an effective barrier for propagation of plastic deformation. The plastic deformation at lower temperatures under a confined pressure to suppress brittle fracture has been reported for some icosahedral quasicrsytals [108, 141]. The results are shown in Figure 8.16. We find a break of the temperature dependence of the yield stress from a steep temperature dependence to a gentle temperature dependence, suggesting a change of the deformation mechanism at the breaking point. Such a break may correspond to the break of the temperature dependence of the hardness shown in Figure 8.13.

296

Chapter 8.


2000

Yield stress (MPa)

1500

1000

Al63.5Cu24.0Fe12.5

Mg36Zn56Y8

500

0 200

300

400

500

600

700

800

900

1000

Temperature (K) Figure 8.16. Temperature dependence of the yield stress of i-Mg–Zn–Y [141] and i-Al–Cu–Fe [108] under a confined pressure.

8.4.2.2.

Compression test of decagonal quasicrystals

In contrast to icosahedral quasicrystals, the structure of the decagonal quasicrystals is anisotropic and hence the high temperature plasticity has been studied only for single-quasicrystals in an attempt to reveal plastic anisotropy which reflects the anisotropic structure [142–149]. As in the case of icosahedral quasicrystals, plastic deformation is possible at temperatures higher than 0.75Tm . The yield and flow stresses are strongly temperature and strain-rate dependent. The orientation of the compressive axis has been selected either in the tenfold direction (k-specimen), in a twofold direction (?-specimen) or in a direction making 45 to the tenfold axis (45-specimen). Since the structure is periodic in the tenfold axis, one may expect that the 45-specimen would be most easily deformable due to dislocation glide on prismatic planes in the tenfold direction. However, the plastic anisotropy is not large and also specimen dependent for the same alloy system. Figure 8.17 shows temperature dependence of the yield stress for d-Al–Cu–Co [144] and for d-Al–Ni–Co single-quasicrystals obtained by two groups [145,146]. Only in 45-specimens, rather clear deformation markings parallel to twofold planes have been observed. Stress relaxation experiments to obtain activation volumes have also been performed for decagonal quasicrystals with the similar results as those of icosahedral quasicrystals.

297


45° d -Al72Ni12Co16

1000

d -Al70Ni15Co15 d -Al65Cu20Co15

σy (MPa)

800

600

400

200

0

950

1000

1050

1100

1150

Temperature (K) Figure 8.17. Temperature dependence of the yield stress of d-Al65Cu20Co15 [143], d-Al72Ni12Co16 [146] and Al40Ni15Co20 [146].

8.5. 8.5.1.

Deformation mechanisms Electron microscopy

To elucidate the microscopic deformation mechanism, electron microscopy is an essential technique to visualize directly the microscopic processes occurring during deformation, as has long been utilized for studying the plasticity of crystalline matter. The presence of dislocations in as-grown quasicrystals was shown earlier by electron microscopy, but detailed studies of deformation structure have become possible after the discovery of high quality, stable quasicrystals. Transmission electron microscopy of plastically deformed samples of bulk, thermodynamically stable quasicrystals of i-Al–Pd–Mn revealed an increase of the dislocation density to the order of 108/cm2 from 106/cm2 in the as-grown state [150], suggesting a dislocation mechanism for plastic deformation. In i-Al–Cu–Fe, the twin-like structure and planar faults were observed after plastic deformation [126,128,129], but later high density dislocations and stacking faults were also observed in this alloy [95] as in i-Al–Pd–Mn. The most direct evidence for the dislocation mechanism of plastic deformation has been obtained by an in-situ stretching experiment on a high temperature tensile stage in an electron microscope [94, 95]. Under an applied stress, straight dislocations have been observed to move steadily and continuously, just like dislocation glide process observed in-situ by transmission electron microscopy for

298

Chapter 8.


bcc metals at low temperatures and for semiconducting crystals at high temperatures. In the following, we concentrate mainly on the dislocation mechanism of plastic deformation; twining-like mechanism may be involved in the deformation of i-Al–Cu–Fe but no detailed investigation has been made. Microstructural investigations have been made most extensively for single-quasicrystals of i-Al–Pd–Mn by several groups. The results of the deformation microstructures are summarized below for high temperature deformation and low temperature (from the room temperature to the ductile-brittle transition temperature) deformation.

8.5.1.1.

High temperature deformation of i-Al–Pd–Mn

Transmission electron microscopy observation has been made post-mortem for deformed samples in many cases, but in-situ observation at high temperature has also been performed. (1) The density of dislocation increases by two-orders of magnitude after a compressive strain of the order of 10% [116,150,151], but the density does not increase monotonically with increasing strain but rather decreases with further increase of the strain accompanying work softening [151]. The density of dislocation is higher for lower temperature deformation [151]. (2) Dislocations distribute rather homogeneously without forming cell-structures. Dislocations often consist of straight segments [95,93,152]. (3) Dislocations in specimens deformed at relatively low temperatures trail planar fault or phason fault [152,153]. (4) bjj component of dislocations is parallel to two-fold directions in most cases [116,150,154–156]. jb? j=jbjj j has been analyzed and this ratio of the dislocations introduced by plastic deformation increases and then decreases with increasing plastic strain [116]. (5) The planes of the dislocations introduced are most frequently five-fold planes [116]. (6) Fault vectors of the phason faults on two-fold and five-fold planes are perpendicular to the respective fault planes [153]. (7) In-situ straining experiments have revealed that dislocations move steadily keeping a straight form either on three-fold plane or five-fold plane [94,157]. The dislocation motion trailing phason fault has been observed [158]. In situ heating experiments have shown that dislocations moving in five-fold planes have the bjj component in the two-fold direction perpendicular to the plane of the motion, indicating the climb motion [152]. Climb motion has also been suggested for as-grown dislocations, where bjj component in a two-fold direction is perpendicular to the trailing fault plane [93,156]. (8) An in-situ annealing experiment in an electron microscope has shown that a dislocation dipole is formed by climb motion but no subsequent annihilation by glide occurs, indicating that the glide mobility is much lower than the climb mobility [159].

299


8.5.1.2.

Low temperature deformation of i-Al–Pd–Mn

(1) Indentation at room temperature produces fragmentation into small grains below the indenter without producing dislocations [160]. (2) Low temperature deformation under a high confining pressure produces long straight bands parallel to five-fold planes of narrowly spaced dislocations with bjj in a two-fold direction [161–163]. (3) In the specimen deformed at 573 K under a high isostatic pressure, all the dislocations trail phason fault and have the bjj perpendicular to the fault, indicating that all the dislocations have climbed [164].

8.5.2. 8.5.2.1.

Microscopic deformation mechanism High temperature deformation in icosahedral quasicrytals

In earlier studies, the dislocation mediated deformation at high temperatures was believed to be due to dislocation glide process. The molecular dynamics computer modeling of deformation process for realistic model icosahedral quasicrystals under shear stresses indicated the slip deformation via glide of dislocations, as mentioned in 3.3. Various deformation models have been put forward based on thermally activated dislocation glide overcoming obstacles such as Mackay clusters [93,116] and quasiperiodic Peierls potentials [114,165], and on phenomenological mobility equation [166]. However, recent electron microscopy observations have indicated that climbing process is the dominant deformation process over the gliding process as mentioned in a previous subsection. How can we justify the fact that the climb mobility is larger than the glide mobility? From Eqs. (8.24) and (8.28), the ratio of Vg =Vc is given by fH kp ð Þ H jp ð Þg þ fH0 ðHs DHs =2Þg Vg lnðz=bÞ exp ¼ Vc 8 kB T

ð8:30Þ

For a typical case of T ¼ 1000 K, Hs ¼ 2eV and DHs ¼ Hs =2, the pre-exponential factor of Eq. (8.30) is of the order unity. H0 may be comparable to the migration enthalpy of vacancy which is about Hs =2, and hence the second term of exponent is small. Thus, the ratio Vg =Vc is determined essentially by the relative magnitude of H kp and H jp or the relative magnitude of the kink energy Ek and the jog energy pffiffiffi Ej . Let P =G ¼ , the kink energy is given by Ek 0:5 Gdb2 [167]. Ej ¼ Gb2 h=f4ð1 Þg 0:1Gb2 h ½97 : Assuming ¼ 101 e 102 , Ek d ¼ ð0:5e1:5Þ : Ej h

ð8:31Þ

Since the kink height d (period of the Peierls potential) is generally larger than the jog height h (atomic spacing), Ek can be larger than Ej , and hence Vc can be larger than Vg .

300

Chapter 8.


6

Al-Cu-Fe

5

σy /E (10–3)

Al-Pd-Mn 4 Mg-Zn-Y

Cd-Yb

3

2

Al-Li-Cu

1

0

0

1

2

3

4

5

–3

T/(Ea /kB) (10–3) Figure 8.18. Normalized upper yield stress versus normalized temperature replotted from Figure 8.12.

Although the climb controlled deformation has been confirmed only in i-Al–Pd–Mn icosahedral quasicrystal, we assume that high temperature plasticity of any icosahedral quasicrystals is governed by a common dislocation climb process. Then, we expect that some homologous relation should hold, as already established in bcc metals [168] and tetrahedrally coordinated crystals [169]. In Figure 8.18, we replotted the upper yield stress versus the temperature relations in Figure 8.15 to the normalized upper yield versus the normalized temperature relations for five different icosahedral quasicrystals, where the upper yield stress is normalized with respect to Young’s modulus E and the temperature with respect to the material parameters of Young’s modulus times cube of the average atomic diameter divided by the Boltzmann constant, i.e., E a3 =kB . As seen in Figure 8.18, a homologous relation holds approximately. For DHs ¼ Hs =2, the climb velocity is written as H jp þ 3=4Hs 4L a3 D Vc ¼ exp : kB T lnðz=bÞ kB T

ð8:32Þ

Assuming DHs Hs/2, Hs ¼ 3 eV, a ¼ bk=0.3 nm, L ¼ 10 mm, and D ¼ 1014 s1, the pre-exponential factor at a low stress of ¼ 10 MPa ( 104 E ) and at T 1000 K is estimated to be 7 107 ms1. Since the mobile dislocation density at the upper yield point is of the order of 109 cm2 [146], the dislocation velocity at the yielding stage is 3 108 ms1. At the yielding stage at high temperature and low stress, the value of exponent of Eq. (8.32) is estimated to be 35.


301

Activation enthalpy of high temperature deformation has also been obtained experimentally by use of temperature dependence of the flow stress and stressrelaxation tests according to the equation H ¼ kB T2

@ @T

"_

@ @ ln "_

:

ð8:33Þ

T

Experimentally obtained activation enthalpies are, however, not necessarily reliable due to effects of work softening, recovery process during the relaxation test and the temperature dependent phason dragging stress. The exponent values experimentally obtained are largely scattered around 40: 40 [114], 60 and 43 [117], 45 [132], 65 [123], 20 [124], and 35 [140]. Using the value of 35 for the exponent and the homologous temperature for a3 =kB in Figure 8.18, the total activation enthalpy at y 0 (To) of about 0.0042E 0 is experimentally obtained as H0 ¼ 35kB T0 ¼ 0:15Ea3 :

ð8:34Þ

Then, how can we rationalize theoretically the above H0 value? From Eq. (8.32), 3 H0 ¼ 2Ej þ Hs : 4

ð8:35Þ

the energy of a jog with the height h is Ej ¼

Gb2 h Eb2 h : ¼ 4ð1 Þ 8ð1 2 Þ

ð8:36Þ

Approximating b a and h a, and using ¼ 0.22 [172], Ej 0.04 E a3 . There a3 exists a correlation between the activation enthalpy of self diffusion Hs and E 3 value for metallic crystals. In Figure 8.19, we plot E a value against Hs for 16 cubic metals, showing a correlation E a3 =Hs ¼ 7:5 except two transition metals V and Nb. Assuming the same correlation for icosahedoral quasicrystals, from Eq. (8.35) H0 ¼ 0:08E a3 þ 0:1E a3 ¼ 0:18E a3

ð8:37Þ

giving a reasonable agreement with experimental value of Eq. (8.34). In conclusion, about a half of the activation enthalpy of deformation by dislocation climb is due to the jog-pair energy and the other half to the jog migration enthalpy via vacancy flow.

8.5.2.2.

Mechanism of work softening

At first, we should note that the work softening is due neither to a change of the mobile dislocation density nor to an effect of non-equilibrium vacancy

302

Chapter 8.


W 50

Mo 40

–3

Ea /Hs = 7.5

Ea (eV)

30 –3

Ta Cr Pt

Ni

Fe

20

Pd Cu V

Ag Au

Al

10

Nb

Pb K 0 0

1

2

3

4

5

6

7

Hs (eV) Figure 8.19. The relation between E a3 and the activation enthalpy of self-diffusion Hs for cubic metals.

supersaturation effect. The yield drop phenomena observed in covalent crystals at high temperature, bcc metals at low temperature and solid-solution alloys at high temperature have long been interpreted by an increase of mobile dislocation density with an increase of plastic strain, since the theory of yielding by Gilman and Johnston [170,171]. However, in quasicrystals the dislocation density rather decreases with increasing plastic strain in the work softening region. By annealing for a day in the unloaded state of a work-softened sample at the deformation temperature does not recover the softened flows stress. These results together with the difference of the activation enthalpy at the yielding stage and at a work-softened stage mentioned previously clearly indicate that the softening is a result of a structural change with plastic deformation, which is most probably an increase of the phason defects in the deformed sample. The quasicrystalline structure does not consist of equally spaced latticepplanes but ffiffiffi typically of two different spaces with the ratio of the golden mean ð ð 5 þ 1Þ=2Þ. In an undefected quasicrystalline structure, dislocations lying in Peierls potential valleys are straight and undergo climb motion by jog-pair formation followed by jog motion successively in two kinds of lattice spacing, quasiperiodically arranged.


303

3 2, 3 2 1

1

Figure 8.20. Quasiperiodic lattice stacking containing phason defects.

Since the jog energy is proportional to the lattice spacing, the climb velocity of the dislocation is determined by the jog-pair formation process at the wide lattice spacing. On the other hand, after a considerable amount of plastic deformation by dislocation motion in a number of climb systems leaving behind a high density of re-tiling faults, the lattice planes become quite jaggy and dislocations lying in Peierls potential valleys are no longer straight but jaggy, as illustrated in Figure 8.20. In such a situation, jog-pair formation can be done at jog sites and the jog-pair formation energy is reduced to only about 40% of that at a wide spaced region. As a result, the activation enthalpy determining the climb velocity in phason defected lattice is reduced by about 30% compared with that in the undefected lattice, leading to a drastic decrease of the flow stress.

8.5.2.3.

Low temperature deformation in icosahedral quasicrystals

In an indentation or a pressing deformation at low temperatures, the deformation is produced by a drastic fragmentation into nano-particles without producing dislocations [160,172,173]. A phase transformation to a crystalline state beneath the indenter has also been observed [173]. Since no martensitic transformation is expected to occur in quasicrystals, transformation to crystalline state must have been produced by a heating effect at the contact surface. On the other hand, confining pressure experiments at low temperatures produces planar bands of dense dislocations [162,163,174]. These bands are possibly produced by collective glide of dislocations. Thus, the low temperature deformation mechanism is either the mode III fracture process or the collective glide of a group of dislocations, and the latter process can occur only when the stress concentration at the contact surface allows the nucleation of dislocations from the surface.

304

Chapter 8.


Dislocation climb process has been reported to occur even near the room temperature [162,174], but they are probably caused only by quenched-in excess vacancies.

8.5.2.4.

Deformation mechanism of decagonal quasicrystals

Microstructures in Al–Ni–Co decagonal single quasicrystals have been studied in detail. Unlike icosahedral quasicrystals, optical microscopy of specimen surfaces has revealed rather clear slip lines parallel to two-fold prismatic planes in specimens [146,150] and corresponding slip steps on the surfaces [149]. Deformation markings have also been observed along pyramidal planes for k and ? specimens [145]. Electron microscopy observations of deformed specimens have revealed elongated rectangular-shaped dislocations parallel to the tenfold direction or to the Burgers vector [145,148]. These results clearly indicate that deformation of 45 specimens is governed by dislocation glide on twofold planes in the periodic direction and the glide is governed by the Peierls potential for the screw dislocation. In deformed d-Al–Ni–Co, in addition to dislocations with bjj parallel to the tenfold direction those with bjj in the quasiperiodic plane and those with mixed character have been observed [147,173,175]. In k specimens, a high density of prismatic dislocation loops with the Burgers vector in the periodic direction have been observed and hence the deformation is brought by climb process of edge dislocations absorbing vacancies [148]. In ? specimens, the observed dislocation structures have indicated that the interactions of dislocations with the Burgers vector parallel to the ten-fold direction and those with mixed dislocations produce dislocation network and the deformation is brought by both glide and climb of the dislocations [148]. Thus, the microscopic deformation mechanisms for k, ? and 45 specimens are different among them. It is interesting to note that although the deformation mechanisms are different among the three orientations, the temperature dependences of the yield stress of the three kinds of orientation of d-Al–Ni–Co are not much different as seen in Figure 8.17. This may be fortuitous because the activation enthalpy for dislocation glide and that for dislocation climb consist of different physical quantities. In fact, in a d-Al–Cu–Co, the plastic deformation is quite anisotropic [143].

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Chapter 9

Ab-Initio Studies of Quasicrystalline Surfaces M. Krajcˇı´ 1 and J. Hafner2 Institute of Physics, Slovak Academy of Sciences, Du´bravska cesta 9, 84511 Bratislava, Slovakia 2 Faculty of Physics and CCMS, Universita¨t Wien, Sensengasse 8/12, A-1090 Wien, Austria 1

9.1.

Introduction

One of the peculiar properties of quasicrystals is the smoothness of their surfaces. It is remarkable that despite the complex aperiodic atomic structure of quasicrystals their surfaces can be atomically flat. Quasicrystalline surfaces and coatings exhibit high hardness, good tribological properties such as low surface friction, high-oxidation resistance, i.e., properties important for technological applications. Surfaces of quasicrystals have also unusual chemical reactivity and can be used as templates for molecular adsorption [1]. During the past ten years significant progress has been made in the preparation and characterization of surfaces of quasicrystals. Surfaces of stable icosahedral Al–Pd–Mn and decagonal Al–Co–Ni quasicrystals have been studied most intensively. Important advances have been particularly made in the determination of quasiperiodic structure of the fivefold Al–Pd–Mn surface [2–5]. In 1994, a scanning tunneling microscopy (STM) experiment conducted by Schaub et al. [6] revealed a set of atomically flat terraces with steps of different heights in a succession forming a Fibonacci-like sequence. High-resolution STM images revealed fivefold motifs and a quasiperiodic arrangement on the surface of the Al–Pd–Mn sample. Using low-energy electron diffraction (LEED), Gierer et al. [7] deduced that the structure of the 5-fold Al–Pd–Mn surface is essentially the same as that obtained by cleavage of the bulk Al–Pd–Mn structure. Since then, intensive efforts have been made in the surface preparation of quasicrystals and atomically flat terraces have been formed routinely. STM images of very high quality have been recorded, permitting a comparison between experimental observations in real space and the atomic arrangement in planes of the bulk structure [8–12]. Papadopolos et al. [11] mapped high-resolution STM images of a fivefold surface of i-Al–Pd–Mn onto a planar tiling derived from a geometrical model of icosahedral Al–Pd–Mn. The experimentally HANDBOOK OF METAL PHYSICS ISSN 1570-002X/DOI 10.1016/S1570-002X(06)02009-X 313


314

Chapter 9. Ab-Initio Studies of Quasicrystalline Surfaces

derived tiling of the surface matches a geometrical tiling derived from the structural model. It was found that the atomic positions on the terraces correspond exactly to cuts across the basic atomic clusters of the bulk structure, the so-called Bergman and pseudo-Mackay clusters. This work has led to a better understanding of how to dissect bulk models to generate surface planes [11–13]. In addition, by considering density rules one can now predict which planes should be stable at the surface of the 5-fold Al–Pd–Mn sample [14]. Zheng et al. [5] have investigated the structure of the fivefold surface of the i-Al–Pd–Mn using X-ray photoelectron diffraction (XPD). They have analyzed the structural relaxation at the quasicrystal surface and found differences in the interlayer spacing of surface layers in comparison to the spacing of atomic layers in the bulk. Similar findings were reported also by LEED analysis [7,17]. Fourneé et al. studied the electronic structure of quasicrystalline surfaces using Auger electron and photoelectron spectroscopies [15]. It was found that in i-Al–Pd–Mn the pseudogap characteristic for the electronic density of states (DOS) of the bulk is not leveled out at the surface, but it can be restored by annealing at sufficiently high temperatures. A summary of the progress in experimentally determining the structure and properties of quasicrystalline surfaces can be found in Refs. [16] and [17]. Irrespective of the progress in the experimental characterization of quasicrystalline surfaces, a deeper understanding of the structure of quasicrystalline surface at an atomic scale remains an open problem. STM measurements of quasicrystalline surfaces with atomic resolution provide the most detailed information about the structure of the surface. Figure 9.1 presents a typical high-resolution STM image of the fivefold i-Al–Pd–Mn surface. In the image one can recognize certain

Figure 9.1. 140 A˚ 140 A˚ high resolution STM image of fivefold i-Al–Pd–Mn (It =0.14 nA, Vb =+0.37 V). In the image the characteristic structural features – the white flower (WF) and the dark pentagonal hole, known as a dark star (DS), marked by larger and smaller circles, respectively – can be recognized. Figure is from J. Ledieu and R. McGrath [20].

9.1. Introduction

315

characteristic structural motifs with pentagonal symmetry. These characteristic structural features have been labeled as the white flower (WF) and the dark star (DS). The atomic structure of these structural motifs is not evident from information deduced from the STM image. STM does not image the atomic structure of a surface, but rather the electron density distribution in a certain energy interval below or above the Fermi level and at a certain distance above the surface, depending on the position of the tip of the scanning microscope, the voltage between tip and sample, and the magnitude and polarity of the tunnel current. Experience with STM on complex crystalline surfaces has taught the lesson that hand-weaving interpretations can be quite misleading. On the other hand it has been demonstrated that STM experiments combined with ab initio electronic structure calculations can be very fruitful – a recent example is the analysis of the surface reconstruction of ultrathin alumina films [18]. The progress in computational methods and the increasing computational power of the available computers allows us to obtain interesting physical information from ab initio calculations on highly realistic structural models. When a structural model of the surface is constructed it is possible to simulate STM images of the surface. A direct comparison of the calculated and the experimental STM images provides the missing information about the positions and the chemical identity of the individual atoms on the surface. In the present review it will be demonstrated that the ab initio electronic structure calculations combined with information from STM experiments represent an extremely powerful tool for the structure analysis of complex surfaces. As example, form the study of the structure and electronic properties of the fivefold surface of i-Al–Pd–Mn quasicrystal and the tenfold surfaces of d-Al–Co–Ni quasicrystal, one can appreciate the capability of the present advanced ab initio methods to contribute to the understanding of the structure of quasicrystalline surfaces. The present review is based on our previous ab initio studies [19–21] of surfaces of i-Al–Pd–Mn and d-Al–Co–Ni quasicrystals. A prerequisite for an ab initio calculation is a structural model. A structural model of the quasicrystalline surface is obtained from a model of the bulk quasicrystal by cleaving it at a proper plane. A problematic point is that often no satisfactory structural model of the bulk quasicrystal is available. Although it is generally argued that the skeleton of the quasicrystalline structures can be reasonably well described on the basis of cut-and-projection models from higher-dimensional space the positions of the atoms and their local coordination depend on the details of the acceptance domains in perpendicular space which have been accurately determined using diffraction data in only a small number of cases [22]. In certain favorable cases the precise atomic positions of the atoms can be inferred from the known crystal structures of low-order periodic approximants. Fortunately, the structure of i-Al–Pd–Mn quasicrystals is one of the best understood quasicrystalline structures. Several sophisticated structural models have been proposed. We shall show that a rather simple model of the i-Al–Pd–Mn quasicrystal can reproduce the characteristic structural features seen in the high-resolution STM images. In comparison with icosahedral quasicrystals, the structure of decagonal quasicrystals with only two-dimensional quasiperiodicity seems to be easier to resolve. However, the situation with the structural model of the decagonal Al–Co–Ni quasicrystal is far more complicated than in the case of the icosahedral Al–Pd–Mn quasicrystal.

316


Quasicrystalline Al–Co–Ni has a very complex phase diagram with several different phases in a rather narrow range of composition and temperature [23]. Up to now no satisfactory and generally accepted model of a decagonal Al–Co–Ni quasicrystal exists. In our study we have used as a model of the d-Al–Co–Ni quasicrystal the structure of the W-(AlCoNi) phase that is considered as a low-order periodic approximant to the decagonal Al–Co–Ni quasicrystal. The positions of the atoms in this phase have been experimentally resolved and further refined by the total energy calculations. In Section 9.2 the computational method is briefly described. In Section 9.3 and 9.4 results of the studies of the structure of the fivefold surface of i-Al–Pd–Mn quasicrystal and the tenfold surfaces of d-Al–Co–Ni quasicrystal are presented. The results are summarized in Section 9.5.

9.2.

Computational method

The theoretical background of our ab initio calculations is density-functional theory within the local-density-approximation. The Vienna ab initio simulation package VASP [24,25] has been used to perform electronic structure calculations. The wave functions are expanded in plane-waves. The Hamiltonian is based on pseudopotentials derived in framework of the projector-augmented-wave (PAW) method [25]. VASP performs an iterative diagonalization of the Kohn-Sham Hamiltonian. The plane-wave basis allows to calculate Hellmann-Feynman forces acting on the atoms. The total energy may by optimized with respect to the positions of the atoms within the cell. The calculations were performed within the generalized-gradient approximation (GGA) to the exchange-correlation functional [26]. VASP has also been used to calculate charge-distributions. The projector-augmented-wave version [25] of VASP calculates the exact all-electron potentials eigenstates and charge densities, hence it produces very realistic valence-electron distributions. More information about the method can be found in Ref. [27]. The possibility to calculate the interatomic forces allows us not only to relax the idealized geometrical positions of atoms from the 6D projection and thus to obtain a more realistic model of the bulk, but also to investigate possible surface relaxations or reconstructions. It is possible to study the change of the positions of atoms at and near the surface relative to the bulk termination in response to the broken bonds at the surface.

9.3.

Fivefold surface of i-Al–Pd–Mn

A structural model of the fivefold surface of i-Al–Pd–Mn quasicrystal is obtained from a model of the bulk quasicrystal by cleaving it at a proper plane. The construction of the structural model of i-Al–Pd–Mn quasicrystal is described in next Section 9.3.1. While for ordinary crystals the choice of the cleavage plane is usually obvious, in the case of icosahedral quasicrystals the choice of the cleavage plane is a non-trivial task and requires a careful analysis of atomic planes in the quasicrystal. The choice of the cleavage plane is discussed in Section 9.3.2.

9.3. Fivefold surface of i-Al–Pd–Mn

9.3.1.

317

Structural model of bulk i-Al–Pd–Mn

The icosahedral Al–Pd–Mn phase belongs to the F-type (face centered) icosahedral quasicrystals. A structural model of this class of quasicrystals was first proposed by Cornier et al. [28] on the basis of diffraction data for i-AlCuFe. In the literature this model is mostly referred to as the model of Katz and Gratias [29] (KG-model). The model consists of three types of atomic surfaces decorating the vertices of a 6D hypercubic lattice. Boudard et al. [30] and de Boissieu et al. [31] have successfully applied the model to the description of the structure of icosahedral Al–Pd–Mn. They have proposed a shell structure of the atomic surfaces defining the chemical ordering of the aluminum and transition metal atoms in the quasicrystal. As we consider the correct chemical ordering to be of great importance, we denote this model of i-Al–Pd–Mn as the model of Katz-Gratias-Boudard (KGB-model). Gratias et al. [32] have presented a detailed study of the atomic clusters encountered in F-type quasicrystals. Significant contributions to the understanding the real-space atomic structure of i-Al–Pd–Mn were made by Elser [33], Papadopolos et al. [34], and Kramer et al. [35]. Elser recognized that the KG model of i-Al–Pd–Mn can be interpreted as a threedimensional Penrose tiling with the vertices decorated by Bergman and pseudoMackay clusters. On the basis of Elser’s ideas Papadopolos, Kramer and coworkers proposed a tiling model of i-Al–Pd–Mn. The authors have labeled the model as the M-model. They decorated the 3D Penrose tiling in accordance with the F character of the 6D lattice. In its 6D representation the M-model has also three atomic surfaces like the KG model, but their shapes are somewhat more complex. A purely 6D model of i-Al–Pd–Mn was proposed also by Yamamoto [36,22]. His model consists of three rather complex multiply connected atomic surfaces at the same nodes of the 6D lattice as in the KG and M models, but in addition there is also an atomic surface at the midedge positions of the hypercubic lattice. Unfortunately, in this sophisticated model the chemical ordering is not well defined. Most of the atomic surfaces in the model of Yamamoto have mixed Al/TM occupations. A computer-simulation of the structure of i-Al–Pd–Mn, based on ab initio density functional calculations of the total energies and interatomic forces was presented by Quandt and Elser [37]. However, their model consists of only 65 atoms and is not large enough to be representative for any surface modeling. The KGB model agrees well with the experimental diffraction data, density and stoichiometry [28,30,38,39]. The stoichiometry of the KGB model, Al0:7073 Pd0:2063 Mn0:0864 , is in very good agreement with the experimentally determined composition of Al0:711 Pd0:202 Mn0:087 (Ref. [10]) or Al0:705 Pd0:21 Mn0:085 (Ref. [40]). The KGB model also provides very good agreement of the calculated photoemission spectra with the experimental ones [41]. For the purpose of electronic structure calculations this model is the most suitable. The reason is not only its satisfactory description of the chemical order, but also the simplicity of this model in 6D space which allows us to impose a linear phason strain and thus to construct well defined finite approximants. The KGB model of i-Al–Pd–Mn starts from a six-dimensional (6D) hyper-cubic lattice [42]. The lattice nodes are decorated by three kinds of triacontahedral atomic surfaces: large triacontahedra at the ‘even’ nodes (n0 ¼ ½000000 ) and the ‘odd’ nodes (n1 ¼ ½100000 ) and a smaller triacontahedron

318


n1

n0 bc1

Figure 9.2. In the KGB model of the i-Al–Pd–Mn quasicrystal the nodes of a 6D hyper-cubic lattice are decorated by three kinds of triacontahedral atomic surfaces: large triacontahedra at the ‘odd’ nodes n1 and the ‘even’ nodes n0 and a small triacontahedron at the body-centered bc1 positions. The atomic surface at the n0 node is truncated by its intersections with its 12 images displaced by 3 along the fivefold axes (indicated by the dotted triacontahedron). The internal structure of atomic surfaces determine the chemical ordering: Al – white outer shells of both n0 and n1 nodes, Pd – gray shell at n0 node and entire domain at the bc1 node, Mn dark cores of the domains at both n0 and n1 nodes.

at the body-centered bc1 positions (bc1 ¼ 12 ½111111 ), (see Figure 9.2). The bc0 positions (bc0 ¼ 12 ½011111 ) are empty. The atomic surface at the n0 node is truncated by itspintersections with its 12 images displaced by 3 (with the golden ffiffiffi mean ¼ ð1 þ 5Þ=2) along the fivefold axes. The radii measured along the fivefold directions of the large triacontahedra at n0 and n1 are and the radius of the smaller one at bc1 is 1. The inequivalence of the atomic surfaces centered at the even and odd nodes and at the bc0 and bc1 nodes breaks the simple cubic symmetry of the 6D lattice and leads to a face-centered superstructure. The atomic surfaces have an inner shell structure determining the chemical order of the quasicrystal which has a significant influence on the electronic properties of the structural model. The large triacontahedra at the even nodes n0 and the odd nodes n1 contain small triacontahedra in the center occupied by Mn atoms. At the n1 node the Mn core is surrounded by an outer Al shell. At the even node n0 the Mn core is surrounded by an inner Pd shell and an outer Al shell. In our version of the KGB model the fivefold radii of the inner shells containing transition metal atoms are 2 1 and 2 2 at n0 and 1 at n1 . The body-centered bc1 positions are decorated with Pd atoms. For any numerical calculation one needs well-defined finite models. In the case of quasicrystals a systematic approach to the construction of such models is available, based on the concept of quasicrystalline approximants. Periodic quasicrystalline approximants are constructed by imposing a linear phason strain, i.e., a certain deformation of the atomic surfaces in the 6D hypercubic lattice. This procedure is equivalent to a replacement of the golden mean in the corresponding formalism by a rational approximant given in terms of the Fibonacci numbers Fn , n ¼ Fnþ1 =Fn with Fnþ1 ¼ Fn þ Fn1 and F0 ¼ 0; F1 ¼ 1, n ¼ 0,1,2, . . . . The effect of this procedure is the replacement of the irrational vectors, e.g., h; 1; 0i, in

319


Table 9.1. Structural data of approximants to i-Al–Pd–Mn and of the slab models for the fivefold surface. Cell dimensions di , i ¼ x, y, z in A˚, number of atoms Na and chemical composition. Model

dx

dy

dz

Na

N(Al)

N(Pd)

N(Mn)

2/1 bulk 2/1 MS 2/1 M

20.31 23.88 23.88

20.31 20.31 20.31

20.31 6.60 4.08

544 205 136

372 137 96

124 54 36

48 14 4

3/2 bulk 3/2 MS 3/2 M

32.86 39.40 39.40

32.86 32.86 32.86

32.86 6.60 4.08

2292 535 357

1612 364 251

472 132 92

208 39 14

5/3 bulk 5/3 MS 5/3 M

53.17 62.51 62.51

53.17 53.17 53.17

53.17 6.60 4.08

9700 1401 930

6844 965 663

2012 340 236

844 96 31

perpendicular space by a rational approximant to , hFnþ1 =Fn ; 1; 0i. We note that this replacement is performed only in perpendicular space; in parallel (physical) space the vector components remain irrational. In physical space the approximants thus have exact fivefold local axes. The unit cells of all approximants have cubic shape, p the ffiffi edge dn of the unit cell of the n-th approximant is dn ¼ ð2= ð þ 2ÞÞ nþ1 aqc , where aqc is the quasilattice constant equal to 4.56 A˚. The 1/1 approximant to i-Al–Pd–Mn has 128 atoms/cell, the 2/1, 3/2, and 5/3 approximants have 544, 2920, and 9700 atoms/cell, respectively, see Table 9.1. The space-group symmetry is simple cubic (P21 3) for all approximants. The real-space structure of quasicrystals is frequently discussed in terms of atomic clusters possessing icosahedral symmetry. Their identity and stability, however, is a subject of controversial discussions. The F-type icosahedral phases are often designated as the ‘Mackay-type’ structures as opposed to the ‘Bergman-type’ structures like Al–Cu–Li or the ‘Tsai-type’ such as Cd–Yb. However, it has been already several times pointed out [29,33,34] that F-type icosahedral structures contain both Mackay and Bergman-type clusters. They are not independent structural units, as the clusters are mutually interpenetrating. We note that the ‘Mackay’ and ‘Bergman’ clusters in i-Al–Pd–Mn are a little different from those encountered in various complex intermetallic crystalline structures. Instead of using the term pseudo-Mackay or pseudo-Bergman we shall, in agreement with Gratias et al. [32], denote the clusters as M and B clusters. The B cluster consists of 33 atoms with a Pd atom in the center. The M clusters are centered by Mn atoms. The first atomic shell of the M cluster is very irregular. Seven or eight atoms occupy vertices of a small dodecahedron, but their spatial arrangement around the central Mn atom is irregular with respect to the icosahedral symmetry. The clusters and their distribution in the F-type quasicrystals have been analyzed in the work of Gratias et al. [32]. It was stressed that the B clusters are extremely robust. Any model based on two main atomic surfaces at the even and odd nodes and a small surface at bc generates B clusters as a natural consequence of the geometry of 6D lattice. Gratias et al. have suggested to use the B clusters as guidelines for discussing the atomic structure and physical properties of the F-type quasicrystals. They consider the B clusters as the best candidates for identifying the mean structural features of

320


these quasicrystals. In particular, the positions of a terraces observed in STM images of i-Al–Pd–Mn surfaces are matched to layers characterized by a density of the B clusters and their distribution pattern.

9.3.2.

Choice of the cleavage plane

The position of the plane bounding the quasicrystal should correspond to a plane of high atomic density [14]. The average atomic density of a fivefold surface has been determined experimentally [7] as 0.136 atoms per A˚2 . This is a rather high value comparable with the atomic density of an Al(111) plane which is 0.141 atoms per A˚2 . Figure 9.3(a) shows a projection of the position of atoms in the 5/3-approximant onto the (x,y) plane. The orientation of one of the 5-fold axes lying in this plane is indicated by the arrow. The slope of the projected pffiffiffiffiffiffiffiffiffiffiffifivefold planes with respect to the x-axis is ¼ 58:28 degrees, sinðÞ ¼ = þ 2. The projection displays the atomic density in the 5-fold planes perpendicular to the (x,y) plane. One can observe that planes with high atomic density are separated by gaps. These ‘empty streets’ can be easily seen by viewing the figure from a low angle. We note that these gaps have a

S M

L

M S M M

S M

L

M S M (a)

L

M S M M (b)

Figure 9.3. Left: (a) Projection of the atomic positions of a 5/3-approximant to icosahedral Al–Pd–Mn onto the (x,y) plane. The circles represent the projected positions of atoms: Al – open circles, Pd – shades circles, Mn – small solid circles. The arrow shows the direction of one of the pentagonal axes parallel to the plane. The structural model can be decomposed into a sequence of slabs of three different thicknesses marked by S, M, and L, cf. text. A model of the surface is obtained by cleaving the structure along a plane, located in the density gap, marked by the thick straight line. Right: (b) The atomic structure of the surface obtained by cleaving the 5/3-approximant. The structure of the surface can be interpreted in terms of Bergman (B) and (pseudo)Mackay (M) clusters. Two such clusters are marked by circles. The regular decagonal rings of Pd atoms correspond to the B clusters. A regular arrangement of atoms with pentagonal symmetry is observed also around most Mn atoms – the centers of the M clusters.


321

meaning only in a geometrical sense, if atoms are considered as points. They are well resolved only because the other atomic layers have a dense spacing. The characteristic spacing between atomic layers on both sides of the gap is 1.56 A˚, while the interlayer distance between the next two layers of is only 0.48 A˚. A layer spacing of 1.56 A˚ is not extraordinarily large. For comparison, the interlayer spacing of < 111 > atomic planes in fcc Al is 2.34 A˚. Nevertheless, the gaps between highdensity planes are natural cleavage planes of the quasicrystal [14]. The distances between the atomic density minima form a Fibonacci sequence. One can recognize three different distances: s ¼ 2.52 A˚, m ¼ 4.08 A˚, and l ¼ s+m ¼ 6.60 A˚. The theoretical values of these distances were derived and reported by Papadopolos et al. [11]. The sequence of the distances s, m, l corresponds to the sequence of terraces in the STM images measured by Schaub et al. [6] . While Schaub et al. observe only a Fibonacci sequence of terraces with heights m and l, Shen et al. [8] have also found a lower terrace height comparable with s. A quasicrystalline approximant can be thus decomposed into a sequence of slabs of three different thicknesses. According to their thickness s, m, and l, we denote these slabs as S, M, and L, respectively. A model of a quasicrystalline surface is created by cleaving the approximant at a plane separating the slabs. The width of the gap between the slabs is not the same. It is natural to chose the cleavage plane between the slabs where observed density gap is the largest. One can find such a large gap between the slabs S and M. As a model of a surface of the Al–Pd–Mn quasicrystal we have chosen the surface of the M slab separated from the neighboring S slab by the widest density gap. This choice corresponds to one of the most frequently reported terminations of i-Al–Pd–Mn perpendicular to a fivefold axis. Figure 9.3(b) shows the atomic structure of the pentagonal surface. The structure of i-Al–Pd–Mn can be interpreted in terms of Bergman (B) and (pseudo)Mackay (M) clusters [32]. In the figure one can easily recognize a regular structure of decagonal rings of Pd atoms. These rings correspond to the B clusters. Each B cluster is centered by a Pd atom. In addition a regular arrangement of atoms with pentagonal symmetry is observed also around most of the Mn atoms. These are centers of the M clusters. In the F-type quasicrystal there exist two possible links between the B and M clusters. The most frequent link is along the fivefold direction. As one observes the surface perpendicular to the fivefold axis, such links between the clusters cannot be seen at the surface. The other possible link is along the threefold direction. All links between the B and M clusters seen at five-fold surface are links along three-fold axes. While the B and M clusters linked along the fivefold axis share an outer face and do not overlap, the B and M clusters linked along a threefold axis are mutually interpenetrating. An outer part of a B cluster is shared with a M cluster and vice versa. This overlap of the clusters leads to a conflict between their building principles and chemical decorations, eventually causing a substitutional defect or incompleteness of one of the clusters. We note that the atomic structures of both surfaces of the M slab, front and rear, are almost identical. They differ only in some details related by symmetry operations. The M slab cut from the 5/3-approximant consists of 930 atoms. For electronic structure calculations such a model is too big and on the other hand its thickness 4.08 A˚ does not sufficiently represent the structure of the bulk quasicrystal. A more tractable model is a model derived from the 3/2-approximant. We shall consider two


S

M

322

M

S

M

M

S

M

L

Figure 9.4. Side view of the MS and the M models of the i-Al–Pd–Mn surface derived from the 3/2-approximant. The MS model consists of two slabs – an M slab and the adjacent S slab. The M model consists of the M slab only. The dashed line in the middle of the M slab indicates the positions of centers of the B clusters. The radius of one B cluster is indicated by the circle. The surface plane dissects the B clusters. The MS model includes the complete bottom part of the B clusters. The small circles represent projected positions of the atoms, cf. Figure 9.3.

variants of the model. The first one consists of two slabs – the M slab and the adjacent S slab. The thickness of this model is 6.6 A˚ and we find it to be sufficiently large to support the surface. The model includes 535 atoms. We shall also consider a second model consisting of only the M slab which includes 357 atoms. We designate these models as MS and M, respectively. In Figure 9.4 the construction of the models is presented. The dashed line in the middle of the M slab indicates the positions of the centers of B clusters. The radius of one B cluster is indicated by a circle. It is seen that the MS model representing the bulk termination dissects the B cluster. While the complete B cluster consists of 33 atoms, in the truncated cluster 6 atoms are missing. On the other hand the thickness of the model of 6.6 A˚ is just sufficient to include the complete lower part of the B clusters. From the figure it is possible to see that the surface of M slab is Al-rich. It is also remarkable that in the M slab the Mn atoms are located just at the surface. The structural parameters of the considered models are listed in Table 9.1. The computational cell has an orthorhombic shape. It includes a layer of atoms with a thickness of 6.6 A˚ (in the case of the MS model) and a 6 A˚ thick vacuum layer. Although the slabs forming the MS model cut from the cubic approximants have a monoclinic geometry, it is easy to adapt them to an orthorhombic

323


computational cell. The fivefold axis is parallel to the c-axis. The length of the b edge is equal to the lattice parameter of the cubic approximant, b=dn , where n is the order of the approximant, the length of the cell along the a-axis is given by a ¼ pffiffiffiffiffiffiffiffiffiffiffi b þ 2=.

9.3.3.

Atomic structure and charge density distribution at the surface

Figure 9.5(a) shows the atomic structure of the surface of the 3/2-M model derived from the KGB model of bulk i-Al–Pd–Mn. The surface is covered by a periodic approximant of the quasiperiodic P1 tiling [11]. The edge of the P1 tiling is 7.76 A˚. The tiling consists of three different tiles – a regular pentagon, a pentagonal star and

Mn

Mn

M

n

Al

Mn

Mn Mn

C (a)

n

M

SV

C′

(b)

Figure 9.5. The atomic structure of the surface of i-Al–Pd–Mn as represented by a M slab cut from the 3/2 approximant. (a) The surface is covered by a P1 tiling. The pentagons may be classified as ‘left’ and ‘right’, as marked by circles and crosses, respectively, cf. text. The positions of atoms are displayed by circles: Al – open circles, Pd – shaded circles, Mn black circles. The right part (b) shows the electronic charge density distribution of the same model. The contour plot presents the valence charge density distribution in a plane intersecting the top atomic layer, cf. text. The most striking features are the charge depletions inside some of the pentagonal tiles. They correspond to surface vacancies (SV). A side view on the model cut along the CC0 line is presented in Figure 9.6.

SV Z

n

M

Z′

Mn

Figure 9.6. Side view of the M model of the i-Al–Pd–Mn surface derived from the 3/2-approximant. The line ZZ0 marks the plane for which the electron density is shown in Figure 9.5. Dashed and dash-dotted circles mark the B and M clusters centered by Pd and Mn atoms, respectively.

324


Table 9.2. Number of tiles in the approximants to the quasiperiodic P1 tiling. Pt , Pb – pentagons with orientations designated as top or bottom, respectively, B – boat, S – star, R s , R f – skinny and fat rhombi, respectively. Model 2/1 3/2 5/3

N(Pt )

N(Pb )

N(B)

N(S)

N(Rs )

N(Rf )

2 4 13

1 6 12

1 0 4

0 1 1

0 1 5

1 0 0

a thin golden rhombus. A general quasiperiodic P1 tiling consists of four species of tiles, in addition to the three listed above a boat can occur. In the 2/1 approximant one can observe also a thick golden rhombus (see Table 9.2). The appearance of this motif (which does not appear in the quasiperiodic P1 tiling) is a consequence of the phason strain leading to the formation of a periodic pattern. The P1 tiling is a planar tiling which can be obtained by projection from a 6D hyper-space. The occupation domain of the P1 tiling is a decagon. The cut-and-projection constructions of the KGB model for the structure of bulk i-Al–Pd–Mn and that of the P1 tiling of a fivefold surface are of course closely interrelated. The occupation domain of the planar tiling is a cross-section through the bc1 domain creating centers of the B clusters in the icosahedral structure. As the surface termination has been chosen at the most densely occupied plane of the quasicrystal, the acceptance domain of the tiling is the maximal cross-section of the domain producing the quasiperiodic lattice. For the triacontahedral domain this is just a decagon. Figure 9.5(a) demonstrates that most vertices of the P1 tiling coincide with the positions of the Pd atoms in the centers of the truncated B clusters. On the other hand the pentagonal tiles are located at the positions of the M clusters centered by the Mn atoms. An interesting information about the atomic structure of the surface can be derived from the electronic charge density distribution. Figure 9.5(b) represents the charge density distribution in the plane of the top atomic layer. The top layer is occupied only by Al and a few percent of Mn atoms [5,7,14] (see Table 9.3). The surface consists of two closely spaced atomic planes separated by a vertical distance of only 0.48 A˚. The figure shows that the Pd atoms from the next layer located 0.48 A˚

Table 9.3. Number of atoms Ns , partial numbers of atoms (in percent) at the surface of the approximants and surface atomic density (in atoms/ A˚2 ). Ns

Ns (Al)

1st 2nd together

41(100) 24(100) 65(100)

39(95.12) 9(37.50) 48(73.84)

0 15(62.25) 15(23.08)

2(4.88) 0 2(3.08)

0.085 0.049 0.134

3/2 3/2 3/2

1st 2nd together

105(100) 66(100) 171(100)

101(96.19) 23(34.85) 124(72.52)

0 40(60.60) 40(23.39)

4(3.81) 3(4.55) 7(4.09)

0.081 0.051 0.132

5/3 5/3 5/3

1st 2nd together

283(100) 161(100) 444(100)

263(92.93) 74(45.96) 337(75.90)

0 87(54.04) 87(19.60)

20(7.07) 0 20(4.50)

0.085 0.049 0.134

Model

Layer

2/1 2/1 2/1

Ns (Pd)

Ns (Mn)

ns


325

below the top layer also contribute to the surface charge density. The Mn atom in the center of the pentagonal star is also located in this subsurface layer. The surface is thus composed of the atoms from the two top most layers. The chemical composition of the surface of the models is given in Table 9.3. The total surface atomic density of the model derived from the 3/2-approximant is ns =0.132 atoms/A˚2 . For two other models derived from the 5/3 and 2/1-approximants the total surface atomic density is ns =0.134 atoms/A˚2 . These values are in very good agreement with the experimental value of 0.136 atoms/A˚2 reported by Gierer et al. [7]. Figure 9.5(b) displays charge density minima at the vertices of the P1 tiling occupied by Pd atoms, surrounded by a complete or incomplete pentagon of Al atoms. The Pd atoms in the centers of the truncated B clusters are at positions deeper in the slab and their electrons do not contribute to the surface charge density. However, the most striking features of the surface charge density distribution are large charge density minima inside some of the pentagonal tiles. These charge depletions correspond to surface vacancies. These vacancies are the consequence of the irregular structure of the first atomic shell of the M clusters. In a regular Mackay cluster the first atomic shell consists of 12 atoms forming a small icosahedron. In the M cluster in i-Al–Pd–Mn the central Mn atom has a low coordination of 7–8 atoms [42], as confirmed by EXAFS experiments [43]. These atoms are distributed on the sites of a small dodecahedron, but their spatial arrangement breaks the local icosahedral symmetry. The surface vacancies exist in the model of idealized coordinates. The existence of vacancies in the structure of i-Al–Pd–Mn has been confirmed experimentally by Sato et al. [44]. The change in the atomic arrangement around the vacancies under structural relaxation is studied in Section 9.3.4. The surface vacancies and their behavior under thermal treatment have been studied experimentally by Ebert et al. [45]. The shape of the surface vacancies depends on the position of atoms in the low-coordinated shell around the central Mn atom in the center of the M cluster. Some of the vacancies are large and exhibit fivefold symmetry. Such vacancies appear in the ‘bottom’ pentagonal tiles only. The background of the vacancy is formed by the deep-lying Mn atom surrounded by five atoms out of the 7–8 atoms forming the first low-coordinated shell. If the remaining 2–3 atoms are located deeper in the bulk, then the vacancy at the surface has the shape of a pentagonal hollow. Figure 9.6 shows a section perpendicular to the surface displaying the valence charge density distribution at the position of the largest surface vacancy. The contour plot represents a cut through the charge density distribution in a plane perpendicular to the surface at the position of lines CC’ (see Figure 9.5 (b)). The section goes also through the centers of two B clusters. The B clusters are marked by dashed circles, M clusters by dot-dashed circles. From the contour plot in Figure 9.6 the depth of the surface vacancy can be estimated to be 2.2 A˚ while the depth of the charge density minima at the centers of the Bergman cluster is 0.7 A˚ only. The surface vacancies can also be well recognized in STM images of the surface (see Section 9.3.6). The positions of the pentagonal tiles correspond to the positions of the M clusters. The centers of the M clusters are projected onto the centers of the pentagonal tiles. In the P1 tiling the pentagonal tiles adopt two different orientations. The orientation of the pentagonal tiles is related to the vertical position of the M clusters. According to this vertical position of the centers of M clusters we designate

326


the pentagonal tiles as top and bottom. In the top pentagon the center of the M cluster is at the top surface of the M slab. In the bottom pentagon the center of the M cluster is at bottom surface of the M slab. We note that any two neighboring pentagons that share one edge have always opposite orientations. In Figure 9.5 the central Mn atoms in the bottom pentagons are near the bottom of the M slab and therefore they do not contribute to the surface charge density. This contributes to the formation of the pronounced charge density minima associated with the surface vacancies. On the other hand in the top pentagons, the central Mn atoms are in the top atomic layer (together with Al atoms) are visible in the surface charge density distribution. Figure 9.5(b) shows that not only the internal arrangement of the pentagons is irregular, but the thin rhombi also have irregular decorations. Although the pentagonal star in the figure exhibits a relatively high regularity, an analysis of other approximants reveals that the decoration of the pentagonal star is in general quite irregular. It is possible to conclude that the quasiperiodic order at the surface is represented by the P1 tiling, but on the other hand the decoration of the tiles is rather irregular. We emphasize that these irregularities exist in the model with idealized positions of atoms and therefore are not a consequence of the relaxation of the atomic positions or of a surface reconstruction (see Section 9.3.4) which naturally leads to deviations of the positions of the atoms from their ideal positions and hence introduces another degree of irregularity. The spatial modulation of the ideal quasiperiodic structures resulting from long-range interatomic forces has been discussed in detail in our earlier work on higher-order approximants of i-AlZnMg and i-AlCuLi [46,47]. The observed irregularity and corrugation of the i-Al–Pd–Mn surface have great influence on the structure of monolayers adsorbed on the quasicrystalline surface [48].

9.3.4.

Relaxation of atomic positions and surface reconstruction

In general, surfaces of solids can adopt a different structure than the bulk. The existence of unsaturated bonds at the surface can lead to a rearrangement of the positions of the atoms at the surface with a periodicity larger than that of the bulkterminated surface. Such surface reconstructions are well known in the case of C, Si, Ge, and many other covalently bonded solids where the recombination of dangling bonds leads to the formation of surface-dimers stabilized by double-bonds and a ð2 1Þ surface reconstruction [49,50]. Eventually Jahn-Teller distortions cause an asymmetry of the surface dimers and a longer periodicity of the surface reconstruction [51]. In metals, surface reconstruction is observed mainly on the more open surfaces. Prominent examples are the long period reconstructions of the (100) surface of the face-centered cubic 5d metals: a ð1 5Þ reconstruction has been reported for Ir(100) [52–54], a long-period pseudohexagonal reconstruction for Au(100) [55,56]. Here the driving force of the surface reconstruction is the stabilization of the surface by adopting a more close-packed pseudohexagonal arrangement of the atoms. Since the surface termination we have chosen for the fivefold surface of i-Al–Pd–Mn exposes the most close-packed layer, in principle no similar reconstructions are to be expected. This question was examined experimentally [10]. The fact that the high-resolution STM image can be interpreted in terms of a tiling derived from a


327

model of the bulk structure [10,11] indicates that in the case of i-Al–Pd–Mn the surface structure does not significantly differ from the bulk structure and does not undergo a large reconstruction. Ab initio methods make it possible to approach the problem of surface reconstruction or relaxation in more detail. Using VASP one can calculate the forces acting on individual atoms. One can therefore relax the model and find the equilibrium position of each atom. Unfortunately, the 3/2-approximant is already too big for such calculations where the electronic structure must be recalculated after each move of the atoms. One has to restrict the relaxation studies to the smaller 2/1-approximant. One has to distinguish between the relaxation of atomic positions in the bulk resulting from the different local environments in an aperiodic structure and the surface reconstruction which is a response to the broken bonds at the surface. We distinguish three variants of our structural model. In model with idealized atomic coordinates the positions of the atoms are obtained directly from the 6D-projection. The surface models presented so far were cut out from models of 5/3 and 3/2 approximants with idealized coordinates of the atoms. More realistic are models with relaxed coordinates of atoms. The position of each atom is shifted to its equilibrium position with respect to the forces acting from the neighboring atoms. There are two possibilities as to how to perform such relaxation. One possibility is to relax atomic coordinates in the bulk and cut the slab representing the surface from such a bulk-relaxed model. The other and apparently the most realistic variant of the model is obtained if the relaxation is performed on the slab. However, the problem with the last variant is that the model must be thick enough to represent the underlying bulk structure. A relaxation of the bottom surface of the slab should have no influence on the relaxation of the top surface. One frequently used possibility that we have also adopted in our calculations is to fix the positions of the atoms in the bottom part of the slab. Figure 9.7 shows projections of the positions of the atoms in one unit cell of the 2/1-approximant. The positions of atoms are projected along the z-axis on the (x,y) plane. Part (a) of the figure shows the positions in model with idealized coordinates, part (b) presents the result of the structural relaxation. The figure demonstrates the fact also known from other studies [57] that in aluminum-transition-metal quasicrystals the transition metal atoms occupy positions close to the ideal sites. It is seen that the positions of Mn and Pd atoms change only very little. As expected, the largest displacements are observed for Al atoms from the irregular first shell of M clusters. The maximal displacement of such an Al atom is 1.71 A˚. The average displacement of the atoms is only a small fraction of their diameter. The average displacements of the Mn and Pd atoms are 0.15 A˚ and 0.14 A˚, respectively. Displacements of the Al atoms are larger, their average value is 0.38 A˚. The effect of the surface relaxation is demonstrated in Figures 9.8 and 9.9. In Figure 9.8 the charge density distribution in the fivefold surface of the 2/1-approximant with idealized coordinates (a) and with a relaxed surface (b) are presented. The contour plots represent cuts of the charge density distribution at the position of the top layer of Al atoms, see line ZZ0 in Figure 9.9(a). The B clusters are marked by dashed circles, M clusters by dot-dashed circles. The charge density distributions for intersections perpendicular to the surface along lines AA’ and CC’ marked in Figure 9.8(a) are presented in Figure 9.9.

328


(a)

(b)

Figure 9.7. Projections of the positions of atoms in one unit cell of the 2/1-approximant, Al – open circles, Pd – gray circles, Mn black circles. Part (a) of the figure show the positions in the model with idealized coordinates, part (b) present the result of the structural relaxation. The figure demonstrates that the positions of Mn and Pd atoms are only little affected by relaxation. The largest displacements are observed for Al atoms from the irregular first shell around the center of the M clusters. The maximal displacement of an Al atom is 1.71 A˚. We note that the circles in the figure represent position of the atoms, the actual size of the atoms is 4 times larger. An average atomic displacement is thus only a small fraction of its diameter, cf. text. The straight line shows the position of the cleavage plane.

Mn

Mn

A′

A

C

Al

Mn

(a)

C′

Mn

(b)

Figure 9.8. The valence charge density distribution in the surface of the 2/1-approximant with idealized coordinates (a) and with relaxed surface (b) are presented. The contour plots represent cuts through the charge density distribution at the position of the top layer of the Al atoms, see line ZZ0 in Figure 9. 9(a). The B clusters are marked by dashed circles, the M clusters by dot-dashed circles. The charge density distributions in planes perpendicular to the surface intersecting along lines AA0 and CC 0 marked in Figures 9.8(a) are presented in Figure 9.9. The transition metal atoms in the plane create a high charge density – black circles. The Mn atoms are marked explicitely, the remaining black circles are the Pd atoms. The positions of the Al atoms can be recognized as small circular islands of local density minima; one Al atom near the center of figure is marked explicitely.

329


Al Z´

Z

Mn Mn

(a)

(c)

Mn Mn

(b)

(d)

Figure 9.9. Sections perpendicular to the surface displaying the valence charge density distribution with idealized coordinates of atoms (a,c) and with relaxed positions (b,d). The contour plots represent cuts of charge density distribution at the position of lines AA0 (a,b) and CC 0 (c,d), i.e., through the centers of the B and M clusters, respectively, see Figure 9.8(a). The B clusters are marked by dashed circles, M clusters by dot-dashed circles. The charge density distributions for intersections along AA0 and CC 0 are presented. Transition metal atoms contribute by a high charge density – black circles. The Mn atoms are marked explicitely, the remaining black circles are the Pd atoms. The positions of the Al atoms can be recognized as small circular islands of local density minima; two Al atoms in the top atomic layer are marked explicitly. Vertical arrows mark surface charge density minima at the positions of the vertices of the P1 tiling.

The thickness of the MS model of 6.6 A˚ is smaller than the thickness of the slabs used in analogous studies of crystalline surfaces. The total thickness of the slab should be large enough to stabilize the bulk structure. Our results for the surface charge density in the relaxed models can be affected by the insufficient thickness of our MS model. We attempted to estimate the effect of the slab thickness on the surface charge density distribution by considering two variants of the model. In the first variant, during the relaxation the coordinates of atoms in the S slab at the bottom of our model were fixed, and only atoms in M slab were allowed to move, in the second variant the positions of all atoms in both M and S slabs were allowed to relax. Although certain minimal quantitative differences in the surface charge density distribution between both variants of the MS model can be recognized, the substantial physical features of the relaxed models which we report below are the same for both variants. One observes that the positions of Pd and Mn atoms remain almost unchanged. The charge density distribution is modified predominantly inside the pentagonal tiles, i.e., around the centers of the M clusters. Again one can see that the inside the B clusters the changes are minimal. The pentagonal cluster of Al atoms around the vertices of the P1 tiling are quite stable as well. Similarly as in the case of the 3/2-approximants shown in Figure 9.5(b), the most striking feature in the charge distribution are the large charge depressions

330


corresponding to surface vacancies. One such vacancy is located near the center of the figure in the central part of a pentagonal tile above a Mn atom. In the relaxed model this depression becomes shallower. As is well seen in the perpendicular sections presented in Figures 9.9(c,d), the charge depression becomes partially filled and shallower after relaxation. Another surface vacancy is seen in the pentagonal tile near the left side of the figure. Although in the sections of the charge density presented here the vacancy seems to be somewhat smaller than the one discussed previously, a closer inspection shows that in fact it is substantially deeper. It can even accept an additional Al atom. We can conclude that while the TM atoms retain relatively fixed positions at the surface and thus propagate the quasiperiodic order of the bulk, the Al atoms are more mobile, particularly around the center of the M clusters and thus contribute more to the surface relaxation. The largest degree of relaxation is observed at the positions of the surface vacancies originating from the low coordination of the center of M clusters. The important result is that the structure of the P1 tiling is conserved if the atoms are allowed to react to the creation of a free surface. The Figures 9.9(a–d) provide an impression of the surface corrugation. Vertical arrows mark surface charge density minima at the positions of the vertices of the P1 tiling. Zheng et al. [5] have analyzed the structural relaxation of the quasicrystalline i-Al–Pd–Mn surface using X-ray photoelectron diffraction. They have found differences in the interlayer spacing of near-surface layers compared with the spacing of atomic layers in the bulk. They considered several termination planes and in average the distance between the top two layers is reduced from the ideal value d12 =0.48 A˚ by 0.06 A˚, and the spacing between the second and fourth layer increased form the ideal value of d24 =1.56 A˚ by 0.04 A˚. The results of a similar study for our 2/1-MS model are listed in Table 9.4. A side view of the structural model is shown in Figure 9.10. We relaxed the surface by minimizing the forces acting on the atoms in the M slab, while in the S slab the atoms were fixed at their bulk positions. Contrary to the conclusions from the XPD analysis in our model the surface relaxation leads to an increase of the interlayer spacing between the two top most layer. In our study d12 increased from its ideal value by 0.05 A˚ to 0.53 A˚. This discrepancy need not to be in contradiction with the experimental data, as the reported experimental value is an

Table 9.4. Ideal and relaxed interlayer spacing in the 2/1-MS model (in A˚). The second column presents the spacing of atomic layers in the model with ideal positions of atoms. The third column presents the spacing of atomic layers after a relaxation where positions of atoms in the M slab are relaxed while, in the S slab they are fixed. The fourth column presents the spacing after relaxation of position of atoms in both M and S slabs. The numbering of atomic layers is shown in Figure 9.10. Layers

Ideal

Relaxed M, fixed S

Relaxed M and S

d12 d23 d34 d45 d56 d67 d78

0.481 0.779 0.779 0.481 1.558 0.481 0.481

0.533 0.789 0.547 0.701 1.556 0.481 0.481

0.541 0.574 0.912 0.538 1.839 0.154 0.279

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1

2

3 5 7

M

4 6 8

S

Figure 9.10. Side view of the 21-MS model. Al – open circles, Pd – gray circles, Mn black circles. The model consists of two slabs M and S. The M slab consists of five layers of atoms, the S slab consists of three layers. The interlayer distances are given in Table 9.4.

average over several different terminations. The scatter of the experimental values is considerable. For two out of nine considered terminations the interlayer spacing d12 increased by 0.046 A˚ and 0.067 A˚, i.e., in a reasonable agreement with our value. To demonstrate that the interlayer spacing sensitively depends on the choice of the termination plane we looked at the changes of the interlayer spacing in both free surfaces of the MS slab, top and bottom. When the coordinates of atoms in both M and S slabs of the MS model were relaxed, the interlayer spacing of the two top layers remained essentially the same, d12 increased from its ideal value 0.48 A˚ by 0.06 to 0.54 A˚, but the interlayer spacing of the two bottom layers is substantially reduced, d78 decreased from its ideal value 0.48 A˚ by 0.20 to 0.28 A˚. If one takes an average of these two values the result is a contraction of the interlayer spacing at the surface by 0.13 A˚.

9.3.5.

Surface electronic structure

Figure 9.11 shows the total and partial DOS of the MS and M models calculated for the 2/1 approximant. For comparison we present also the data for the bulk. In the electronic structure of the MS model the deep pseudogap just above the Fermi level has disappeared, at the surface the DOS at Fermi level is higher than in the bulk. This is a significant observation as Fourneé et al. [15] came to a similar conclusion on the basis of their analysis of photoelectron and Auger electron spectroscopies. The partial DOS show that both Mn and Al states contribute to this enhancement. The Pd band exhibits only very small changes in the region around the Fermi level. On the other hand, one observes a significant shift of 0.5 eV of the Pd band to lower binding energies compared to the bulk. Similarly, the Mn-d band is shifted towards the Fermi level, completely covering the pseudogap. The M slab consists of five atomic layers only. The centers of the Bergman clusters are located in the central layer. This layer has a low atomic density. In the M slab of the 2/1 approximant there are only 4 Pd atoms in this layer. The remaining four atomic layers are surface layers. The DOS of the M slab is thus formed predominantly by contributions from the surface atoms. The dashed lines in Figure 9.11 represent the total and partial DOS of the M slab. One observes a significant change of the Mn partial DOS. The Mn d-band is substantially narrowed. The DOS at the Fermi level is greatly enhanced. This enhancement is the consequence of the fact that all Mn atoms of the model are located just at the

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Chapter 9. Ab-Initio Studies of Quasicrystalline Surfaces 2.0

Total DOS 1.5 1.0 0.5 0.0

Al

n(E) [states/eV/atom]

0.5

0.0

Pd 4.0

2.0

0.0

Mn 6.0 4.0 2.0 0.0

–10

–5

0

E [eV] Figure 9.11. Total and partial DOS of the MS (full lines) and M (dashed lines) slabs based on the 2/1 approximant. For comparison the DOS for the bulk (dotted lines) are also presented. In the electronic DOS of the slabs representing the surface the deep pseudogap just above the Fermi level has disappeared, the DOS at Fermi level is increased for the both MS and M models. In comparison with its position in the bulk quasicrystal the Pd d-band in the surface model is shifted towards the Fermi level. Note a significant enhancement of the Mn partial DOS of the M model.

surface. This suggests that the enhancement of DOS at the Fermi level observed in the photoemission studies of the surface could have its origin in the presence of the Mn atoms at the surface.

9.3.6.

STM images of the fivefold surface of i-Al–Pd–Mn

Figure 9.12 shows a simulated STM image of our 3/2 approximant calculated according to the Tersoff-Hamann approximation [58] from the surface charge

333


(a)

(b)

Figure 9.12. STM images of a fivefold i-Al–Pd–Mn surface calculated from the surface charge density distribution of the 3/2 approximant. The calculation was performed for a STM image measured at voltage +0.37 V. This size and polarity of the voltage corresponds to contributions from unoccupied states up to 0.37 eV above the Fermi level. Part (a) simulates a measurement at constant height of the STM tip 2.0 A˚ above the surface plane. The individual atoms are clearly recognized. Part (b) simulates an STM image measured at constant current.

density distribution. The calculation was performed for an STM image measured at a voltage +0.37 V. With this choice of the STM parameters, the tunneling current is dominated by electrons from unoccupied states up to 0.37 eV above the Fermi level. The STM image calculated for occupied states gives a similar result, albeit with a little lower contrast. Part (a) simulates an image taken at constant height of the tip of 2.0 A˚ above the surface plane. The individual atoms are clearly recognized. Unfortunately, in real experiments at such close distances are not feasible. Part (b) is a simulation corresponding to a STM image measured at constant current. This figure better corresponds to the experimental setup and resembles more closely the experimentally measured STM figures (see Figure 9.1). However, the figure is more diffuse and the information about the positions of individual atoms is not so clear. From the comparison of Figure 9.12 with the structural model in Figure 9.5 one can make the following observations. The STM image is dominated by the distribution of the Al atoms. They are seen as bright spots. On the other hand the Pd atoms are almost invisible in the STM image. They manifest themselves as dark spots. The STM image thus provides a quite different view of the surface compared to the charge density distribution shown in Figure 9.5(b). The reason for the low contribution of the Pd atoms to the STM signal is primarily in the low density of Pd states in the vicinity of the Fermi level. Moreover, the Pd d-states are more localized than the s,p-states of the Al atoms. Pd atoms are also deeper (by 0.48 A˚) below the surface than the Al atoms. The Bergman clusters B are seen in the STM image as small pentagons with an edge of 2.96 A˚ surrounding the vertices of the P1 tiling. However, from the comparison with Figure 9.5(b) it is clear that not all such pentagons in the STM image correspond to Bergman clusters. The bright spots corresponding to the five Al atoms forming the pentagons are in most cases (especially in the constant-current image) merged to one bigger spot. In the center

334


of some of these spots one can observe a small dark spot. It is interesting that these small dark spots are darker if the Pd atom in the center of the pentagon is 0.48 A˚ below the surface plane. In the case of Al pentagons corresponding to Bergman clusters, where the central Pd atom is located at 1.26 A˚ below the surface, the spots are less dark. In the experimental STM images, (e.g., Figure 9.1), one can recognize two characteristic features. The first one has been labeled the white flower (WF). The WF is formed by a central Mn atom surrounded by small Al pentagons with an edge measuring 2.96 A˚. These small Al pentagons form the ‘leaves’ of the flower. Such configurations can be seen also in the experimental STM images of our model: they correspond to M clusters. For clarity, one of these M clusters is shown in Figure 9.13 separately and compared with the experimental image. WF’s appear at the position of the ‘top’ pentagonal tiles of the P1 tiling. In an ideal case the skeleton of the WF is formed by 10 small pentagons surrounding the central Mn atom. Out of these 10 small pentagons five belong to the B clusters centered in the vertices of the pentagonal P1 tile. They are marked in Figure 9.13 by black lines. The other five pentagons, marked in by gray lines, share four of their vertices with the neighboring black pentagons. Not all of the remaining vertices of the gray pentagons must be occupied by Al atoms. In Figure 9.13 in one pentagon an Al atom is missing. However, if all vertices of the small gray pentagons are occupied, the skeleton of the WF has a 10-fold symmetry. The atomic structure of a WF is also presented in Figure 9.15. These centers of two WFs are marked explicitely. The structure of a WF can be viewed in terms of individual atoms or in terms of clusters. From the viewpoint of its cluster structure a WF is formed by a central M cluster surrounded by five B clusters. The surface plane dissects the M cluster in the center, at its equatorial position. The central Mn atom is thus exposed at the surface. While the first shell of the M cluster is irregular, the remaining shells are regular. Atoms of the

Figure 9.13. Comparison of the experimental (left) and calculated (right) STM images of a white flower (WF). The area of the STM image (39.5 A˚ 32.9 A˚) is the same as the size of the structural model. The WF corresponds to a M cluster (the central decagon) surrounded by five B clusters (black pentagons). The skeleton of the WF is formed by 5 dark and 5 gray pentagons of edge length 2.96 A˚, as described in the text, see also Figure 9.15. The big pentagon marked by the thin line indicates the position of the ‘top’ pentagonal tile of the P1 tiling. The experimental figure has been provided by Ledieu and McGrath [20].


335

third shell form an icosidodecahedron. This third shell is seen at the surface as a regular decagonal ring of Al atoms. Around each vertex of the top pentagonal tile there is a truncated B cluster manifesting itself at the surface as the small Al pentagon forming a leaf of the WF. Note that five intact B clusters ‘hanging down’ from the surface can also form the ‘leaves’ of the flower [11]. The second well known characteristic feature in the experimental STM images of the 5-fold i-Al–Pd–Mn surfaces is the dark pentagonal hole. The most surprising result of the present comparison between the calculated and experimental STM images is a revised assignment of the origin of the dark pentagonal holes or ‘dark stars’ (DS). So far it has been supposed [59] that they correspond only to truncated Bergman (B) clusters. From the calculated STM image it is clear that they can originate from surface vacancies that exist inside some pseudo-Mackay (M) clusters. Figure 9.14 compares the calculated and experimental STM images of the dark pentagonal hole. From Figure 9.15 it is obvious that the DS is formed by a surface vacancy surrounded by a tenfold ring of atoms: a pentagon of Al atoms separated by 4.79 A˚ and a pentagon of Pd atoms of the same size. The contrast between the bright Al atoms and the dark Pd atoms forms the characteristic pentagonal ‘arms’ of the DS in the STM image. Both characteristic WF and DS patterns can appear in the STM images in a variety of different configurations. In our model the DS appears next to the WF. The distance between the centers of the WF and the DS is 10.73 A˚. This is the distance between the centers of two adjacent pentagonal tiles with opposite orientation. The same configuration of WF and DS is frequently observed also in experimental STM images (see Figure 9.1.). In our calculated STM image in Figure 9.12 one can also recognize an overlap of two WFs. The distance between their centers is 12.62 A˚. It is the distance between the centers of two neighboring pentagonal tiles with the same orientations. Such a configuration is also observed in the experimental STM image presented in

Figure 9.14. Comparison of the experimental (left) and calculated (right) STM images of a dark pentagonal hole or dark star (DS). The area of the images is 39.5 A˚ 32.9 A˚. The DS is formed by a surface vacancy surrounded by a pentagon of Al atoms separated by 4.79 A˚ and a pentagon of Pd atoms of the same size forming in the STM image dark ‘‘arms’’ of the DS. The skeleton shown in the figure consists of one central pentagon with edge length 4.79 A˚ surrounded by five pentagons of 2.96 A˚ (compare with Figure 9.15). The big pentagon marked by thin the line indicates the position of the ‘bottom’ pentagonal tile of the P1 tiling. The experimental figure has been provided by Ledieu and McGrath [20].

336


WF

DS

Figure 9.15. The atomic structure of the fivefold surface of i-Al–Pd–Mn with superposed P1 tiling and skeletons of ‘white flower’ (WF) and ‘dark star’ (DS), cf. text.

Figure 9.1. We note that the DS is not the only possible type of surface vacancy, however, it is the largest one. There can exist a smaller vacancy where the pentagonal hole DS is partially filled by an Al atom. Such a vacancy does not have the characteristic pentagonal shape. Of course, if the hole is filled by two Al atoms one cannot speak about vacancy any more. In the 3/2 approximant only one surface vacancy in the form of a DS is present. Although we have identified the physical origin of the atomic configuration that gives an STM contrast in the form of a DS, it is still possible to object that the atomic configuration could be an artefact of the 3/2 approximant. A calculation of the STM image of a larger approximant is thus desirable. However, the next higher approximant contains 2 2:6 times more atoms. A larger model of a surface based on the 5/3 approximant consists of 930 atoms. This size is already too big for ab initio calculations. Nevertheless, on the basis of the information obtained from the 3/2-approximant, particularly about the visibility of atoms in the STM image, one can calculate a simulated STM image also for larger approximants. The valence charge density distribution in the 5/3 approximant has been approximated by parameterized analytic functions. The proper parameters has been chosen on the basis of the ab initio results for the 3/2 approximant. Figure 9.16 shows a structural model of the surface based on the 5/3 approximant and its simulated STM image. One can recognize three surface vacancies in the form of the DS. As the size of the 5/3 approximant is 62.5 53.17 A˚ we can also estimate the density of the DS vacancies as 0.0009 10–3/A˚2 . However, it has been shown that the density of DS varies with the surface terminations [13,59]. The surface vacancies thus appear also in a larger model and seem to be an intrinsic property of the fivefold i-Al–Pd–Mn surface.

337

9.4. Study of the tenfold surface of d-Al–Co–Ni

(a)

(b)

Figure 9.16. A structural model of the i-Al–Pd–Mn surface based on the 5/3 approximant (a) and its simulated constant-height STM image (b). In part (a) the positions of atoms are displayed by circles: Al – open circles, Pd – shaded circles, Mn solid circles. In part (b) the positions of the DS’s are marked by small white crosses. One WF is marked by a circle.

We have shown that ab initio calculated STM images fit experimental STM images in very fine detail. The structure of the WF motifs is now understood as consisting of a truncated M cluster in the center with five B clusters as petals. The identification of the DS or dark pentagonal holes with the vacancies is another indication [43–45,60] of the existence of such vacant, or more strictly speaking, less densely packed sites in the structure of the i-Al–Pd–Mn quasicrystals.

9.4.

Study of the tenfold surface of d-Al–Co–Ni

A structural model of a quasicrystalline surface of a decagonal Al–Co–Ni quasicrystal is obtained from a model of bulk quasicrystal by cleaving it at a proper plane. The choice of the cleavage plane in the case of decagonal quasicrystals is in principle simpler than that of the surfaces of icosahedral quasicrystals because their structure is periodic in the direction perpendicular to the surface, and therefore only a few terminations are possible.

9.4.1.

Structural model of bulk d-Al–Co–Ni

The modeling of the structure of decagonal Al–Co–Ni quasicrystals has a long history [23,61–71]. On the basis of the experimental information from diffraction studies, a number of structural models for d-Al–Co–Ni alloys have been proposed. The large number of distinct stable decagonal phases in the d-Al–Co–Ni system makes it difficult to model the quasiperiodic structure on a systematic basis. Thermodynamic and structural studies [72–75] revealed the existence of a very complex phase diagram of the pseudobinary AlCo–AlNi system in the stability range of the decagonal phases, showing a large structural variety. Ritsch et al. [75] report eight

338


different structural modifications of decagonal phases. Most stable decagonal phases exist only at high temperatures in the range of 700 to 1100 K. The largest region of quasicrystalline stability exists at compositions close to Al70 Ni15 Co15 . However, the complexity of the phase diagram is not the main reason for the persisting difficulties in the structural modeling of decagonal Al–Co–Ni quasicrystals. It appears that the main obstacle arises from the existence of a certain degree of disorder in the decagonal phases [71]. The electron diffraction patterns observed for Co-rich decagonal phases exhibit features characterized by pentagonal or rhombic tilings forming columnar clusters with a decagonal section of 20 A˚ in diameter. Models with a perfectly deterministic arrangement of atoms, based on large, highly symmetric decagonal clusters, appear to disagree with the available structural data. A significant breakthrough in the structural modeling is due the recent discovery of the W-approximant in the Al–Co–Ni system. The atomic structure of this phase has been resolved by Sugiyama et al. [70] using X-ray diffraction. The structure of the W-(AlCoNi) phase is closely related to that of the decagonal Al–Co–Ni quasicrystal. It provides fundamental information on the atomic arrangement on the columnar clusters commonly observed in Al–Co–Ni decagonal quasicrystals. A preliminary 5D hyperspace model of the quasicrystal structure derived from the W–approximant has been proposed by Deloudi et al. [76]. As at present the W-(AlCoNi) phase is the only approximant to the decagonal quasicrystal with experimentally determined atomic positions it is natural to base our model of the bulk d-Al–Co–Ni quasicrystal on the W-(AlCoNi) phase. The electron probe microanalysis [70] gives the composition of the W-(AlCoNi) phase as Al71:8 Co21:1 Ni7:1 . This composition is close to the Co-rich decagonal phases. As X-ray diffraction does not provide the information necessary for distinguishing the two transition metal (TM) atoms, Ni and Co, the structural model of Sugiyama et al. [70] for the W-phase is in fact a model for an Al–TM approximant with the composition Al71:8 TM28:2 . Moreover, in the refinement of Sugiyama et al. [70] some atomic sites have fractional Al/TM occupancy. For ab initio studies it is mandatory to have a model with an unambiguously defined chemical occupation of the atomic sites and without fractional occupancies. The computational power of present computers makes it possible to perform a refinement of the chemical order of the model via total energy minimizations. Mihalkovicˇ and Widom have performed an extensive ab initio study of the ternary Al–Ni–Co phase diagram [77]. The structure of the Wapproximant was refined by optimizing the total energy, under the constraint of a fixed overall content of the three species Al, Ni, and Co. They successfully resolved the chemical identity of the transition metal sites and determined the occupancies of the mixed Al/TM sites as well. In the case of binary systems the thermodynamic stability of a phase is determined by the position of the total energy of the phase with respect to a tie-line connecting points corresponding to other stable phases. In the case of ternary phases an analogous construction to the tie-line is a tie-plane. The structural model of the W-phase obtained by Mihalkovicˇ and Widom is only 13 meV/atom above the tie-plane defined by the stable phases Al3 Co in the oI96 structure (Pearson notation), Al5 Co2 (hP28) and Al3 Ni2 (hP5). This is an extraordinarily good result as for models of quasicrystalline approximants energy deficits of 40–80 meV/atom are common. Explicit examples can be found in the alloy database of Ref. [77]. The composition of their model of the W-phase is Al71:70 Co20:75 Ni7:55 , i.e., very close to


339

that experimentally determined by electron probe microanalysis [70]. In addition to this model they proposed several other chemical variants. We have decided to use the variant with the composition Al70:78 Co14:61 Ni14:61 . The composition of this approximant is closer to the composition Al71:8 Ni14:8 Co13:4 of the decagonal quasicrystal used for the surface studies reported, e.g., in Ref. [78]. The space group of the W-phase is Cm (No. 8) with the lattice parameters a=39.668 A˚, b=23.392 A˚, and c= 8.158 A˚. As the monoclinic angle is equal to 90.05 the elementary cell has approximately an orthorhombic shape. The W-phase belongs to the decagonal approximant phases with 8 A˚ periodicity along the decagonal axis. We note that the translation period of other decagonal modifications can be equal to 4 A˚. The structure of the W-phase can be described in terms of alternating flat (A) and puckered (B) atomic layers perpendicular to the c axis. The structure of the flat A layer at z=0 corresponds to that at z=0.5 (A0 ) which is shifted by a half period along the a axis. The puckered layers at z=0.25 and z=0.75 are related to each other by mirror planes at z=0 and z=0.5. The structure of the W-approximant can be thus described as a ABA0 B stacking of atomic planes along the c-axis. The orthorhombic elementary cell consists of 534 atoms. The coordinates of atoms can be found in the alloy database accessible on-line, see Ref. [77]. For bulk calculations it is possible to choose a monoclinic primitive cell with half the number of atoms.

9.4.2.

Cleavage planes forming tenfold surfaces

The structural model of the W-approximant naturally offers two possible termination planes. The bulk quasicrystal can be cleaved at the flat A plane or at the puckered B plane. In the present work we shall consider both possible terminations. The quasiperiodic ordering of atoms at the surface can be visualized by covering the surface by a planar tiling. However, the choice of the tiling is not completely unambiguous. One can put on the surface various kinds of tilings. So far no generally accepted tiling has been proposed. Figure 9.17 shows the atomic structure of the two surfaces. In the TM-rich A layer (Figure 9.17(a)), the lines connect TM atoms separated by 4.68 A˚ in the plane, forming a network of edge–sharing pentagons. Pentagon centers are located at the even/odd vertices of a Hexagon–Boat supertiling, suggesting a way to generalize the W-phase structure to higher-order approximants to the quasicrystal. One can distinguish two kinds of pentagons. The decoration of the ‘even’ pentagons is 5–fold symmetric (in some of the pentagons the chemical occupancy breaks this symmetry), while for ‘odd’ pentagons the interior motif breaks the 5–fold symmetry. In the bulk the orientations of pairs of Al atoms near the centers of the ‘odd’ pentagons is the result of a special ordering phenomenon with 8 A˚ periodicity described in terms of ‘channels’ in which the structure accommodates 3 Al atoms per 8 A˚ period [79]. Because of a different content of TM atoms the tilings are different in both planes. The A plane is more TM-rich. The edge of the tiling superposed to the A plane measures 4.68 A˚. In the TM-poor B–layer (see Figure 9.17(b)) the size of tiles connecting TM atoms is 7.58 A˚, i.e., ¼ 1.618 . . . times larger than that in the A plane. However, the tiling on the A surface can not be simply considered as a -inflated tiling on the B surface. The relation of both tilings is presented in Figure 9.18(a). The vertical dashed line represents the plane dividing

340


(a)

(b)

Figure 9.17. A view of the atomic structure of A (a) and B (b) surfaces of W-phase Al–Co–Ni. The positions of atoms are displayed by circles: Al – open circles, Co – dark gray circles, Ni light gray circles. To represent the quasiperiodic ordering the surfaces are covered by planar tilings of pentagons, thin rhombi and boat. The tiling represents a network of the transition metal (TM) atoms in the surface plane. Because of the different content of TM atoms the tilings are different in both planes. The A plane is more TM-rich. The edge length of the tiling superposed on the A plane is 4.68 A˚. The edge of tiles in the B plane is 7.58 A˚.

(a)

(b)

Figure 9.18. The tilings representing the quasiperiodic ordering of the cleavage planes of the W-approximant to d-Al–Co–Ni. Part (a) compares the A (black lines) and B (gray lines) tilings, cf. the previous figure. The vertical dashed line represents the plane dividing the W-approximant into two parts. While the B-tiling has translational symmetry with respect to this plane, the A-tiling is not symmetric. A few phason flips indicated by arrows can make the A-tiling also symmetric. The left and right part of the A-tiling can be thus understood as two phason versions of the same structure. Small crosses mark the position of the centers of the 20 A˚ columnar clusters, the right one is indicated by a dotted circle. Part (b) shows the relation of the Atiling to a larger hexagon-boat (HB) tiling. A more general hexagon-boat-star (HBS) tiling is a possible candidate to represent the quasiperiodic order in the decagonal quasicrystal.

the W-approximant into two parts. While the B-tiling has translation symmetry – the right part of the tiling can be obtained by translating the left part by a/2 – the A-tiling does not possess such symmetry. Nevertheless, by introducing a few phason flips (and 2 more vertices), the A-tiling can acquire the same translation symmetry. After such a rearrangement one would get, e.g., a pentagonal star instead of the boat tile in the center. However, the vertices of such a symmetric tiling do not


341

represent the positions of TM atoms on the A surface only if the phason flip is accompanied by interchanging an Al and a TM atom, the correct TM decoration of the vertices is recovered. The left and right part of the A-tiling can be understood as two phason variants of the same structure. The relation of the tilings to the 20 A˚ columnar clusters is indicated in Figure 9.18(a). The positions of the 20 A˚ clusters are marked by small crosses. The columnar clusters are concentric with the pentagonal tiles of both the A- and the B-tiling. In the right parts of Figure 9.17 and Figure 9.18(a) one can see the pentagonal symmetry and the atomic structure of two layers of the 20 A˚ cluster that Deloudi et al. propose as a candidate for modeling all kinds superstructures in decagonal Al–Co–Ni [76]. In the left part of Figure 9.18(a) one can actually see the geometrical structure of two other layers of this columnar cluster. The superposed A-tiling shows that in the W-phase the perfect pentagonal symmetry of the columnar cluster is partially broken in the A0 plane. The tiling consists of pentagons, thin rhombuses and boat-shaped tiles. In other approximants one could expect also a pentagonal star. Each center of the pentagonal tiles in the A-tiling can be considered as a vertex of a larger tiling. This larger tiling is in fact a part of the HBS (hexagonboat-star) tiling, see Figure 9.18(b). The HBS tiling is thus a possible candidate to represent the quasiperiodic order in larger approximants or in an infinite decagonal quasicrystal. The atoms at the surface have bulk-relaxed positions. The effect of the surface relaxation on the position of atoms and a possible surface reconstruction will be discussed in Section 9.4.2.1. The relaxed positions of TM atoms coincide remarkably well with the vertices of the tiling. The internal decoration of the pentagons in the B plane maintains the pentagonal symmetry. Particularly interesting is the decoration of the interior of the pentagons by pairs of Al atoms. Such a configuration in a decagonal quasicrystal is surprising. The decoration of some other pentagonal tiles is irregular. This is a manifestation of the fine structure present in d-Al–Co–Ni quasicrystals.

9.4.2.1.

Relaxation of atomic positions and surface reconstruction

For the surface relaxation one has to prepare a slab model of sufficient thickness. The number of atomic layers should be large enough to stabilize the bulk structure in the deeper layers. Generally, in the case of quasicrystalline surfaces it is not possible to take as many atomic layers as usual in the studies of surfaces of crystalline solids. The period of the W-approximant perpendicular to the quasiperiodic plane is 8 A˚. This corresponds to five atomic layers. In the case of a termination by A planes one gets an ABA0 BA sequence of layers, for the B termination a BABA0 B sequence. Unfortunately, five atomic layers in the W-approximant already lead to a structural model with more than 660 atoms in the computational cell. This is too big for an ab initio relaxation where the electronic structure must be recalculated after each move of the atoms. We have to restrict our relaxation studies to a smaller model. We have found that it is quite simple to divide the W-approximant into two almost identical parts. Figure 9.17(b) shows that the periodicity of the B layers along the x-axis is a/2. The A and A0 planes are also identical when shifted by a/2 along the x-axis. When skipping one atom in A layer and one atom in A0 it is possible to enforce

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periodicity with a half period along the x-axis. Thus one achieves a structural model of an approximant with half the number of atoms than in the W-approximant. Figure 9.18 shows the position of the plane where we cut the model into two parts. We call this smaller approximant the U-approximant. The number of atoms in the elementary cell of the U-approximant is 265. Such an approximant could in principle also exist in the phase diagram. However, it is clear that nature prefers the W-approximant as it can accommodate 534 atoms, i.e., 4 more atoms than in two elementary cells of the U-approximant. We relaxed the positions of atoms in the U-approximant. As we started the relaxation from already bulk-relaxed positions it is not surprising that the changes of the atomic positions are quite minimal. It is also known from other studies [57] that in aluminum-transition-metal quasicrystals the transition metal atoms have low mobility. We found that the changes in the coordinates of the Al atoms upon relaxation are also very small, below 0.15 A˚. Ferralis et al. [80] analyzed the structural relaxation of the quasicrystalline d-Al–Ni–Co surface using low-energy electron diffraction (LEED). They found differences in the interlayer spacing of near-surface layers compared with the spacing of atomic layers in the bulk. The outermost layer spacing is contracted by 10% relative to the bulk interlayer spacing, while the next layer spacing is expanded by 5%. We relaxed the surface by minimizing the forces acting on the atoms in two outermost layers of both sides of the slab, while the atoms in the central layer were fixed at their bulk positions. Contrary to the conclusions from the LEED analysis, the interlayer spacing between the two top most layer is essentially unchanged in our model. In our study d12 , defined as a distance between the centers of gravity of the positions of the atoms in the B surface layer and the next A layer, increased from its original value of 2.027 A˚ by 0.016 A˚. Similarly the changes of the layer spacing between the next two layers were also small. The interlayer spacing remained essentially unchanged regardless as to whether the relaxed model had the A plane or the B-plane at the surface. We admit that one possible explanation of this discrepancy with the experimental observation of Ferralis et al. [80] could be the limited thickness of our model.

9.4.3.

Electronic structure of d-Al–Co–Ni

The electronic structure represented by the density of electronic states (DOS) is a sensitive probe reflecting the local structure and the chemical short-range order in solids.

9.4.3.1.

Electronic structure of the bulk

Figure 9.19(a) presents the calculated total and partial DOS of the W-approximant. The calculation has been performed for a monoclinic primitive cell containing 267 atoms. Because of the elongated shape of the primitive cell three k-points in the irreducible part of the Brillouin zone proved to be sufficient. The most remarkable feature of the calculated DOS is the existence of a deep minimum or pseudogap at the Fermi level. The composition Al189 Co39 Ni39 gives a position of the Fermi level just at the center of the deep minimum in the DOS. The coincidence of the position

2.0

Surface B

Surface A

1.0

0.0

1.5

1.5

1.0

1.0

0.5

0.5 0.0

0.0

Al

Al

Al

0.4

0.4

0.2 0.0

Co 4.0 2.0

n(E) [states/(eV.atom)]

0.4



2.0

2.0

Bulk

0.2 0.0

Co 4.0

Co 4.0

0.0

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Ni

0.0

2.0

2.0

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Ni

Ni

4.0

4.0

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2.0

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–10

–5

0

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–10

–5

0

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–10

–5

E [eV]

E [eV]

E [eV]

(a)

(b)

(c)

0

Figure 9.19. (a) The total and partial densities of states (DOS) of the W-approximant representing a bulk d-Al–Co–Ni quasicrystal. The most remarkable feature of the calculated DOS is the existence of a deep minimum or pseudogap. The composition Al70:78 Co14:61 Ni14:61 gives a position of the Fermi level just in the deep minimum of the DOS. The thin lines show the contribution of the d-states to the DOS. The surface electronic DOS of the A and B surfaces is shown in part (b) and (c), respectively. The total and partial densities of states at both surfaces are significantly different from the of bulk. The pseudogap is partially filled. In comparison with the position of the Co and Ni d-band in bulk the d-bands in the surface DOS are shifted towards the Fermi level. Note a significant enhancement of the Al partial DOS at EF .

344


of the Fermi level and the deep minimum in the DOS is commonly interpreted as an electronic stabilization in terms of the Hume-Rothery mechanism [81,82]. In the case of nearly-free-electron sp-bonded quasicrystals like i-Al–Cu–Li [47], i-Al–Zn–Mg [84,85], Zn–Mg–Y [83] there is no doubt that this mechanism plays an essential role in the stabilization of these alloys. However, in the case of Al-TM quasicrystals to which most of the decagonal quasicrystals, including d-Al–Co–Ni, belong, the situation is not so clear [86]. In the light of previous results the observation of a deep pseudogap in the DOS at the Fermi level is thus quite remarkable. Our calculations show that the deep pseudogap exists also in the DOS of the model of the W-phase with the experimentally determined composition Al71:8 Co21:2 Ni7:1 (see Section 9.4.1). The pseudogap is here even somewhat deeper, however, the Fermi level falls a little (0.09 eV) below the minimum. This result could indicate that the pseudogap is rather insensitive to the relative compositions of Co and Ni.

9.4.3.2.

Electronic structure at the surface

The DOS at the surface has been calculated for the U-models with a thickness of 8 A˚. Figure 9.19(b,c) shows the total and partial DOS at both surfaces. The pseudogap at Fermi level is almost completely covered at the surface, especially on the A surface. Both Co and Ni d-peaks are shifted by 0.5 eV to lower binding energies. Remarkably, at the A-surface the Ni d-band is still located at higher binding energies than the Co d-band, while on the B-surface the shape asymmetry of the Ni d-band is reversed compared to the bulk with an absolute maximum closer to EF than the Co d-band. The partial DOS of both TM atoms is steeply ascending below the Fermi level while it is almost constant above the Fermi level. The different height of the main peak in the total DOS of both surfaces reflects the different content of TM atoms in the surfaces.

9.4.3.3.

Comparison with the photoemission spectra

Figure 9.20 presents a comparison of the photoemission spectrum of the d-Al0:70 Co0:15 Ni0:15 quasicrystal measured by Stadnik et al. [87,88] with the calculated spectra of the W-approximant with the composition Al70:78 Co14:61 Ni14:61 and of a crystalline Al9 CoNi phase. The ultraviolet photoelectron spectrum (UPS) was measured at low temperature and He II photons with an energy of 40.8 eV. The theoretical curves were calculated by weighting the partial densities of states with the photoionization cross-sections of the elements and broadening by convolution with a Gaussian resolution function with a FWHM of 0.5 eV. The theoretical curves are normalized to a common maximal height. The figure shows that in comparison with the experimental PES the position of the d-bands in the bulk W-approximant is shifted by 0.8 eV to higher binding energies. The shift to higher binding energies is even larger for crystalline Al9 CoNi. The figure also shows the theoretical PES calculated from the DOS at the surface. The curves for both surfaces A and B are almost identical. As the d-peaks of TM atoms in the surface DOS are shifted to lower binding energies, the main peak of simulated PES is now almost coincident with the experimental one. However, this coincidence cannot be simply interpreted as a good agreement with the experimental data. Although PES is a surface-sensitive

345


I(E) (arb. units)

6.0

4.0

2.0

0.0

–10

–5

0

E [eV] Figure 9.20. Comparison of the measured photoemission spectrum (PES) of a d-Al0:70 Co0:15 Ni0:15 quasicrystal (squares, data from Ref. [87]), with simulated spectra of the W-approximant with the Al70:78 Co14:61 Ni14:61 composition. The full line corresponds to the PES spectrum calculated from bulk DOS, the dashed curve shows for comparison the PES calculated for a crystalline Al9 CoNi phase (composition Al64 Co18 Ni18 ) with the crystal structure of Al9 Co2 . The chain and dotted line corresponds to the spectrum calculated from the surface DOS of A and B surfaces, respectively. The curves of the surface DOS’s for both surfaces are almost identical. The theoretical curves are normalized to a common maximal height.

technique the contributions to the photoelectron intensity come not only from the uppermost surface but also from atomic layers deeper in the bulk. Still our results indicate that the surface-induced changes in the DOS are to a large extent responsible for the discrepancy between the bulk spectrum and experiment. A large difference between the calculated bulk spectrum and experimental PES is observed also in the vicinity of the Fermi energy: a pronounced Fermi-edge appears in the experimental spectrum and also in that calculated for the quasicrystalline surface, whereas for the bulk quasicrystalline and crystalline phases the existence of a deep pseudogap is evident. It is clear that the experimental data correspond to substantially higher DOS at the Fermi level than we calculated for the bulk W-approximant. However, from a detailed analysis of the shape of the PES intensity in the vicinity of the Fermi edge Stadnik et al. [87,88] concluded, using a sophisticated fitting procedure, that the observed high DOS is still only 85% of ‘normal’ DOS and hence the experimental data indicate the existence of a shallow Hume-Rothery pseudogap.

9.4.4.

Atomic structure and the charge density distribution

Interesting information about the atomic structure of the surface can be derived from the electronic charge density distribution. Figure 9.21 displays the valence charge density distribution at both surfaces. The position of the TM atoms is characterized by high valence charge densities (black spots in both figures). At the position of the Al atoms one observes small local valence charge density minima. Because of the different TM concentrations on both termination planes it is not

346


(a)

C´

C´

C

C (b)

Figure 9.21. The valence charge density distribution at the surfaces A (a) and B (b). The contour plots represent cuts through the charge density distribution at the position of the top layer, see also Figure 9.22. The transition metal atoms in the plane create a high local charge density – black circles. The positions of the Al atoms can be recognized as small circular islands of local density minima. The charge density distributions in the planes perpendicular to the surface intersecting along lines marked as CC0 are presented in Figure 9.22. The dashed lines represent the plane dividing the W-approximant into two parts, cf. text.

surprising that the overall view of both surfaces is quite different. It is remarkable that in terms of the variation of the local electron density the roughness of the flat A surface is comparable to that of the puckered B surface. Particularly deep are the charge density minima in the pentagonal tiles of the A surface. Inside the pentagons formed by five TM atoms one observes from one to five additional atoms. The charge density minima are the most pronounced if inside the pentagons only one or two atoms are located, but a clear charge density minimum is formed even in the case where five atoms fill the pentagon. At the B surface the internal decoration of the large TM pentagons is regular. One can here also observe various charge density minima between the atoms. The surface charge density minima are important as attractive sites for possible adsorbate atoms. The results of our study of the formation of an adsorbate adlayer on the surface d-Al–Co–Ni quasicrystal can be found in [89]. The view on the surface valence charge density provides also information about interatomic bonding. As in any complex Al–TM alloy one can also expect in the W-approximant a high degree of covalency of the bonding between Al and TM atoms [90–93]. An interesting bonding configuration between Al atoms only is observed at the A surface. At the midpoint between the pairs of Al atoms inside the pentagonal tiles one can recognize an enhanced charge density. This indicates a significant degree of covalency in the bonding between two Al atoms, as is also suggested by the deep pseudogap in the partial DOS. Since such a bonding has also been observed in the charge density distribution of the bulk W-approximant, one cannot identify these bonds as in-plane bonds that are formed due to a surface reconstruction. This suggests that this bonding configuration contributes to the stabilization of the decagonal quasicrystalline structure – as previously reported for crystalline TM-aluminides [90–93].

347


A B A´ B A

(a)

B A´ B A B (b)

Figure 9.22. Sections perpendicular to the surface displaying the profile of valence charge density distribution in the relaxed models of both surfaces A (a) and B (b). The contour plots represent cuts of charge density distribution at the position of lines CC’ in Figure 9.21. The relaxation has only minimal influence on the position of atoms. The figure shows the roughness of both surfaces.

Figure 9.22 represents a contour plot of the valence charge density distribution in a plane perpendicular to the quasiperiodic surface. Bonding with enhanced covalency can also be observed between the atomic planes. By enhanced covalency of bonding we mean the accumulation of bond charge between the atoms with respect to a homogeneous charge density distribution that would correspond to a purely metallic bonding. The bonding between the atomic planes is promoted by the Al–TM interaction. The figure provides an impression of the surface corrugation. In comparison with the surface of i-Al–Pd–Mn here there are no such special structural defects as the surface vacancies observed at the fivefold surface of i-Al– Pd–Mn. Otherwise the roughness of both d-Al–Co–Ni surfaces is comparable with that of the fivefold i-Al–Pd–Mn surface.

9.4.5.

Simulated STM images of the surface

Although the valence charge density distribution provides important information about the structure of the surface and its chemistry, a link to a comparable experimental information is still missing. STM imaging is considered as the most direct experimental tool for obtaining information about the structure of a surface. As the STM image is formed only by a part of the valence charge density distribution corresponding to a narrow energy interval around the Fermi level, the structure of the surface seen in the STM images can significantly differ from that presented by the valence charge distribution. Simulated STM images of the surfaces have been calculated according to the Tersoff-Hamann approximation [58]. The contrast in the STM images reflects the local DOS determined by the tip-position and the average over energy interval around the Fermi level. The width of the energy interval is determined by the voltage between the tip and the surface. Depending on the direction of the tunneling current STM can image occupied or unoccupied electron states. In the usual experimental setup the tip of the device is grounded. If the voltage of the sample to the tip is negative, the electrons flow from the occupied states of the sample to the tip. A positive polarity corresponds to imaging the unoccupied states.

348


(a)

(b)

Figure 9.23. STM images of the surfaces of the W-approximant to d-Al–Co–Ni calculated form the surface charge density distribution. The calculation simulated a STM image measured at a voltage 1.0 V with the tip at constant height 2.0 A˚ above the surface plane. This size and polarity of the voltage corresponds to contributions from occupied states up to 1.0 eV below the Fermi level. Parts (a) and (b) correspond to the A and B surfaces, respectively. The individual atoms are clearly recognized as bright spots. The dotted circles mark the position of the 20 A˚ pentagonal cluster.

Figure 9.23 shows the calculated STM images of both surfaces for a tip-sample voltage of 1.0 eV. This corresponds to the imaging of the electronic density formed by the occupied states down to 1.0 eV below the Fermi level. In the narrow energy interval around the Fermi level the bulk DOS (see Figure 9.19(a)) is quite symmetric. From this one would expect that for unoccupied states (positive voltage) the images are similar. However, the information on the electronic structure of the bulk presented in Figure 9.19(a) can be misleading. In the vicinity of the Fermi level the partial surface DOS of Co and Ni atoms is not symmetric (see Figure 9.19(b,c)) and therefore there will by a clear difference in the contribution of the TM atoms to the STM current from occupied and unoccupied states. From the comparison of the STM images measured for both tip polarities one could thus distinguish between Al and TM atoms. In a STM image taken at positive polarity (imaging unoccupied states) the brightness of spots corresponding to the TM atoms will substantially decrease. Quasicrystals as complex metallic alloys exhibit a rather rich, spiky structure of the DOS. It is important to measure the STM current at rather large voltage such as to integrate over a larger part of the DOS around the Fermi level to obtain representative images of the atomic structure of the surface. In the usual experimental setup STM measurements are performed more frequently for a constant tunneling current than for a constant height above the surface. Both types of measurements provide essentially the same information about the chemistry of the surface. In our calculations we prefer to present STM images for constant height. We found that STM images calculated for a constant height rather close to the surface are less diffuse than those calculated for constant current. The information about individual atoms can be thus clearly recognized without any additional Fourier filtering usually applied on the experimental images to improve their contrast. The STM images presented in Figure 9.23 are calculated for a constant height of 2 A˚ above the surface. To simulate the contrast of the experimental STM image as presented in Figure 9.23, the contrast of the Figure 9.23 has been further 1= increased by the transformation Iout ¼ Iin with the parameter ¼ 0:5.

9.5. Summary

349

The overall features of the STM images of both surface terminations are quite similar. A brief look on the STM images does not reveal any dramatic differences between both surfaces. The electron density distributions on both surfaces presented in Figure 9.21 display much larger differences. However, a closer inspection shows that each surface has its own characteristic structural features. The STM signal is formed predominantly by Al atoms. This is in agreement with the expectation based on the DOS presented in Figure 9.19(b,c). The STM image of the Al-rich B surface is overall a little brighter than that of the TM-rich A surface. Nevertheless, both TM atoms Ni and Co also significantly contribute to the STM current. Because of the localized character of d-states on the TM atoms the contribution of these atoms to the STM current is generally smaller than the contribution of the s,p-electrons of Al atoms. From a somewhat lower partial DOS of the Ni atoms around the Fermi level in comparison with that of the Co atoms(see Figure 9.19(b,c)), one can expect that in the STM image the Co atoms are seen as brighter spots in comparison with the Ni atoms. A detailed inspection of the central pentagon of TM atoms in the A surface confirms that it is indeed the case.

9.5.

Summary

We have reviewed the state-of the art of ab initio density-functional studies of quasicrystalline surfaces. In the direction along a fivefold axis the structure of the icosahedral Al–Pd–Mn quasicrystal can be decomposed into a sequence of atomic planes with different atomic densities. Planes with high atomic density are separated by gaps. These gaps between high-density planes are natural cleavage planes of the quasicrystal. The distances between the gaps form a Fibonacci-like sequence of three different distances s, m, and l=s+m. The sequence of the distances s, m, l corresponds to the sequence of terraces seen in low-resolution STM images of the i-Al–Pd–Mn surface. The atomic planes can be grouped into slabs marked according to their thickness by S, M, and L. In the present study as a model of the surface of the Al–Pd–Mn quasicrystal we have chosen the surface of the M slab separated from the neighboring S slab by the largest observed density gap. This choice corresponds to one of the most frequently experimentally reported terminations of i-Al–Pd–Mn perpendicular to the fivefold axis. The surface consists of two closely spaced atomic planes separated by a vertical distance of 0.48 A˚. The top layer is occupied only by Al and a few percent of Mn atoms. The second atomic plane is Pd-rich. Both planes form the surface with atomic density 0.134 atoms per A˚2 . The quasiperiodic ordering on the fivefold i-Al–Pd–Mn surface can be described by a planar P1 tiling. The origin of this ordering can be seen in the 6D hyperspace. The decagonal acceptance domains of the P1 tiling corresponds to the maximal crosssections of the triacontahedra defining the structural model in 6D hyperspace. At the surface one can recognize regular arrangements of atoms with pentagonal symmetry corresponding to the B and M clusters truncated by the cleavage plane. Most of vertices of the P1 tiling correspond to the centers of B clusters which are occupied by Pd atoms. The centers of the pentagonal tiles correspond to the positions of the M clusters. In the P1 tiling the pentagonal tiles adopt two different orientations. The orientation of the pentagonal tiles is related to the vertical position of the Mn atoms

350


centering the M clusters. According to this vertical position the pentagonal tiles are labeled as ‘top’ and ‘bottom’. In the top pentagon the center of the M cluster is at the top surface plane. In the bottom pentagon the center of the M cluster is at a position 2.56 A˚ deeper inside the slab. The most striking features of the surface charge density distribution are large charge density minima inside some of the pentagonal tiles. These charge depletions correspond to surface vacancies. These vacancies are the consequence of the irregular structure of the first atomic shell surrounding the Mn atoms in the center of the M clusters. The results of a structural optimizations demonstrate that the skeleton of the P1 tiling fixed by the positions of the Pd atoms remains stable. There is no structural reconstruction of the surface. On the other hand, the arrangement of the Al atoms decorating the tiles undergoes certain modification. Particularly large displacements from their idealized positions are observed for Al atoms around the centers of the M clusters. The effect of the structural relaxations is to flatten the charge-density minima associated with the surface vacancies. The STM image is dominated by the distribution of the Al atoms. They are seen as bright spots. On the other hand the Pd atoms manifest themselves as dark spots. The truncated B clusters are seen in the STM image as pentagons with an edge of 2.96 A˚. In experimental STM images one can recognize two characteristic features labeled as white flower (WF) and dark stars (DS). The WF is formed by a central Mn atom surrounded by small Al pentagons with an edge measuring 2.96 A˚. These small Al pentagons form the ‘leaves’ of the flower. The DS is formed by a surface vacancy surrounded by ten-fold ring of atoms: a pentagon of Al atoms separated by 4.79 A˚ and a pentagon of Pd atoms of the same size. The contrast between the bright Al atoms and the dark Pd atoms forms the characteristic pentagonal ‘arms’ of the DS in the STM image. The W-(AlCoNi) phase can be considered as an approximant to decagonal Al–Co–Ni quasicrystals. The positions of atoms in the W-(AlCoNi) approximant have been resolved experimentally. Since so far no other satisfactory structural models of decagonal Al–Co–Ni quasicrystals exist, the W-(AlCoNi) phase provides an unique opportunity for surface studies. The bulk W-(AlCoNi) structure offers two possible terminations labeled as A and B. The A surface is flat, the surface B is puckered. The surface A is TM-rich, the surface B is Al rich. The structure of both surfaces is stable against surface reconstruction. Atoms at the surface remain essentially at their bulk position. There is no surface reconstruction, relaxations by interatomic forces have only minimal effect on the atomic positions. We did not observe any significant changes of interlayer spacing of the surface layers in comparison with the interlayer spacing in the bulk. The electronic structure of the bulk W-(AlCoNi) phase is characterized by a pseudogap at the Fermi level. However, instead of attributing the formation of the pseudogap to a Hume-Rothery-type Fermisurface Brillouin-zone interaction our analysis shows that pseudogap reflects the enhanced covalency in the interatomic Al–TM bonding of the atoms in the quasicrystal. The pseudogap in the surface local DOS is partially covered and the positions of d-bands of both transition metals undergo a substantial shift towards lower binding energies. A similar behavior of the surface DOS was observed also for the i-Al–Pd–Mn quasicrystal. At the surface of the W-(AlCoNi) phase there are no special structural defects as, e.g., the surface vacancies observed at the fivefold surface of i-Al–Pd–Mn.

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Otherwise the roughness of both W-(AlCoNi) surfaces is similar to that of the fivefold i-Al–Pd–Mn surface. In terms of variation of the local electron density the roughness of the flat A surface is comparable to that of the puckered B surface. The simulated STM images of both surfaces show that despite their significantly different atomic structure and chemical composition both planes exhibit quite similar STM images. The question which of the terminations of the W-(AlCoNi) phase is more representative for the surface of decagonal Al–Co–Ni quasicrystal thus remains open.

9.6.

Outlook

Our ab initio studies have provided detailed picture of the atomic and electronic structures of the surfaces of icosahedral and decagonal quasicrystals. Very recently, there have been intense experimental research efforts to elucidate the influence of quasiperiodic ordering on the chemical reactivity of the surfaces [94–96] and the possibility to use quasicrystalline surfaces as templates for growth of quasiperiodic monolayers and ultra-thin films [78,97–99]. The results presented in this review provide a sound basis for complementary theoretical studies. First results on the formation of quasiperiodic monolayers on a fivefold surface of i-Al–Pd–Mn [48] and on a tenfold d-Al–Co–Ni [89] have been already published. These studies demonstrate the unique potential of ab initio calculations to construct realistic atomic structures of such overlayers. The possibility to propagate the quasiperiodic order of the substrate beyond the monolayer limit is studied for alkali-overlayers on a fivefold i-Al–Pd–Mn surface [99,100] – for such a system, ab initio calculations offer probably the only access to details of the atomic structure. Intense efforts are currently directed towards the investigation of molecular and dissociative adsorption of small molecules [101]. Preliminary results point to the existence of a few highly reactive, quasiperiodically distributed centers on an otherwise quite inert surface – offering an exciting perspective to a formation of quasiperiodic molecular arrangements.

Acknowledgment This work has been supported by the Austrian Ministry for Education, Science and Art through the Center for Computational Materials Science. M. K. thanks also for support from the Grant Agency for Science of Slovakia (No. 2/5096/25) and from the Agency for Support of Science and Technology (Grant No. APVV-0413-06 CEX-Nanosmart).

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Index

central mode, 124 central peak, 124 chemical disorder, 137, 151, 158, 163–164, 286 chemical order, 78, 116, 163, 187, 317, 318, 338 cluster, 7, 12, 14, 23, 27–30, 39–89, 143, 157, 161, 172, 179, 186–188, 203, 248, 283, 314, 317, 319 conductance, 232, 239, 240, 243–244 conductivity, 8, 209, 211, 219–220, 225–228, 237–238, 241, 253–260 covalent bond, 184–186, 190, 195, 203–204 crystal orbital Hamilton population (COHP), 178–179 Cu-Ga-Mg-Sc, 57, 61, 70

ab-initio calculation, model, studies, 164, 176, 247, 313 Al-Co-Cu, Al-Cu-Co, 23, 24, 28, 39, 42, 44, 187, 296, 304 Al-Co-Ni, Al-Ni-Co, 24, 29, 39–43, 186, 277, 280, 291, 296, 304, 313, 315–316, 337–351 Al-Cu-Fe, 35, 63, 77, 100, 182, 276, 279, 280, 282, 292, 295, 297, 298 Al-Cu-Li, Al-Li-Cu, 2, 50, 63, 97, 172, 187–200, 203, 204, 276, 281, 292, 319, 344 Al-Mn-Si, 51, 76, 98–102, 179 Al-Pd-Mn, 18, 22, 29, 33–35, 50, 55–57, 64, 77, 184, 198, 271, 276, 279, 280, 292–295, 297–299, 313–337 Al-Pd-Re, 14, 35, 280 Al-TM, 99, 101, 172–176, 179–187, 204, 338, 346, 350 Al-Zn-Mg, Al-Mg-Zn, 101, 188, 189, 203 analyticity breaking, 110–111, 113, 115, 139, 212 Anderson localization, 231, 235, 241 anomalous diffusion, 210–211, 221, 223–225, 245, 258, 260 aperiodic crystals, 11, 15, 107–162 approximant, 6, 23, 36–38, 50–53, 55, 59, 75–103, 116, 136, 157, 173, 179, 182, 185, 196 atomic radius, 67–68, 71, 95, 103 atomic surface, 110, 115, 130, 137, 139, 143, 157, 160, 269, 272, 317–319 average electron concentration, e/a, 69 average structure, 29, 43, 133

decagonal phase, 158–162 defect, 122–124, 186, 209–210, 246, 254, 278, 283, 292, 303, 321 deformation, 7, 154, 267, 270, 277, 285, 291, 295, 297–299 density of states (DOS), 98, 173, 209, 244, 246, 247–248, 250, 314 diffraction, 1, 6, 7, 11–21, 39, 49, 53, 58, 59–63, 75, 90, 99, 101, 108, 132, 137, 140, 156, 171, 338, 342 electron, 53, 55, 61, 81, 84, 99, 202, 313, 338, 342 neutron, 14, 144, 146, 158 X-ray, 14, 28, 56, 58–59, 81, 84, 88, 98, 101, 155, 338 diffuse scattering, 55, 129, 134–140, 142, 144–148, 151–152, 155–158, 160–162, 163–164, 278–80 diffusion, 113, 132, 150, 162–163, 210–211, 215, 217, 223–225, 228, 230, 237, 241, 245, 249, 258, 271, 288, 290 constant, 122, 125, 132, 153, 154–155, 163 discommensuration, 110 dislocation, 115, 155, 267, 282–291, 297–304 disorder, 79–80, 86, 108, 124–126, 137, 151, 161, 182, 202–203, 223, 230, 259, 286 displacive modulation, 118–121 Drude peak, 210, 235, 242, 256, 258

backscattering, 233–237, 238–239, 243–244, 260 Bergman cluster, 50, 77, 188, 319, 325, 331, 333–334 Bergman phase, 187–191 Bergman-type, 50–51, 63, 67–68, 319 Bloch-Boltzmann theory, 210 bonding, 178–179, 184–185, 190, 194, 201–203, 346–347 Burgers vector, 282, 284, 286, 304 Burkov model, 24, 26, 28, 187, 283 Cd-Ca, 68, 70, 86, 87, 90–92, 93, 95, 99, 101, 103, 197, 201 Cd-Mg-RE, 92–95, 103 Cd-Yb, 22, 29, 43, 59, 65, 68, 77–78, 85, 88–104, 197, 276, 280, 292, 319

elasticity, 7, 108, 115, 129–131, 138, 163, 267, 277 electron energy-loss spectroscopy (EELS), 186 357

358 electron microscopy, 12, 158, 187, 271, 278, 288, 292, 297–298, 304 electronegativity, 190, 204 electronic structure, 8, 171–204, 247–248, 256, 314–317, 321, 331–332, 342, 344, 351 electronic transport, 210, 211–228, 245 entropic stabilization, 148, 151, 158, 278 external space, 12, 14–17, 23, 26, 30 Fermi Sphere-Brillouin Zone (FS-BZ) interaction, 69–70, 98, 101, 173, 350 Fibonacci, 3, 5, 53, 130, 143, 176, 269–270, 313, 321 Frenkel-Kontorova model, 114–115, 138, 139 Frank-Kasper phase, 172, 187, 281, 283 Frank-Kasper-type, 283 Friauf polyhedra, 89 friction, 49, 110, 313 F-type, 50, 57, 97, 317, 319, 321 glide motion, 268, 287 Goldstone modes, 107, 111–113 hardness, 291–293, 313 HBS model, 42, 341 high pressure, 79, 83, 202 higher-dimensional lattice, 171, 176, 177 high-resolution transmission electron microscopy (HRTEM), 12, 13, 23, 24, 271, 278 Hume-Rothery rule, mechanism, phase, 68–71, 98, 101, 173, 204, 344–345 hybridization, 100, 101, 173, 175, 181–183, 198, 199, 248, 256 hydrodynamic instability, 135–136, 279 hydrodynamic theory, 111, 115, 117, 121, 131, 136, 138–141, 151 hydrodynamics, 127, 129–142 icosahedral phase, 1, 7, 108, 129–144, 150, 156–157, 197, 209, 246, 280, 282, 319 icosahedral point group, 131, 272 icosahedral symmetry, 1, 6, 22, 49, 52, 55, 75, 77, 79, 115, 129, 131, 140, 171, 174, 176, 198, 277, 319 In-Ag-Yb, 95–98, 101, 103 incommensurate composites, 115, 162 incommensurately modulated phases, 162 inelastic neutron scattering, 118, 128, 144, 277 internal space, 12–22, 24, 27, 29, 113–116 Katz-Gratias model, 13 kink pair, 287–291 Laves phase, 55, 89, 196 linear muffin-tin orbital (LMTO), 177, 183, 247, 256

Index localization, 221, 231, 234–239, 280 low energy electron diffraction (LEED), 313–314, 342 low-density elimination method, 13, 14, 17–18 Mackay cluster, 77, 143, 198, 299, 314, 317, 325 Mackay-type, 50, 63, 67, 179, 319 matching rule, 2, 4, 42, 115, 136, 138–140, 278 Mathiessen rule, 246, 258–259 maximum entropy method, 19–20, 22, 186 metal-insulator transition, 210, 228, 239–241, 243–244, 246 microvoid, 63–64 modulation, 6, 12, 43, 109–111, 113, 115–118, 121–125, 127–128, 138, 151, 157, 280, 326 oblate rhombohedron, 78 occupation domain, 12, 17, 24–28, 30, 32, 34, 36–39, 44, 324 optical conductivity, 238, 246, 255–257, 260 order-disorder, 202–203 orientation (of clusters), 1, 7, 11, 34, 37, 64, 87, 116, 124, 130, 134, 161, 163, 202–203, 296, 304, 324, 326, 339, 349 parallel space, 107, 130 Penrose pattern, 29 Penrose tiling, 1, 49, 88, 90, 159, 283, 317 perpendicular space, 2, 5, 6, 130–132, 136, 157, 163, 315, 319 phase diagram, 75, 80, 90, 92, 96–97, 151, 158, 162, 187, 197 phase transition, 29, 81, 84, 108, 114, 116, 117, 124, 129, 140, 150, 151, 156–158, 160, 163, 281 phason elastic constant, 131, 135, 139, 140, 149, 150, 157, 161, 163, 279, 281 elasticity, 108, 115, 129, 137, 160, 163, 267, 276, 277–280 flip, 29, 267, 270, 271, 276, 340–341 fluctuation, 132, 135, 136, 139–142, 144, 146–148, 150–154, 156–158, 161, 163, 164, 278–9 gap, 111, 125 jump, 140–142, 163, 284, 290 modes, 107–164 strain, 6, 7, 52, 55, 62, 76, 108, 130–131, 136–137, 139–142, 147, 155, 159–160, 163, 176, 196, 267, 270, 274, 276, 278, 280–290, 317–318, 324 phonon elastic constant, 161, 164, 275–277 strain, 130, 131, 267, 270, 274, 280, 281–282

359

Index phonon-phason coupling, 7, 131–133, 161, 267, 274, 276, 279, 280–282 photoelectron, 198, 314, 330, 331, 344–345 photoemission, 187, 194, 317, 332, 344–345 physical space, 2, 4, 11, 109, 113, 129, 130, 132, 151, 157, 269, 270, 272, 274, 282, 319 plane wave, 179, 185, 316 plastic deformation, 154, 285, 291, 295–298, 302–304 prolate rhombohedron, 4, 78, 90 pseudo Brillouin zone, 99 pseudogap, 70, 173–175, 180–182, 184–190, 193–197, 246–247, 256, 314, 331, 342–346, 350 pseudopotential, 174–175, 179, 189, 194, 316 P-type, 50, 57, 58, 59, 61, 79, 97 quantum diffusion, 210–211, 213, 215, 217, 226, 228–260 quantum interference effect, 239, 244 quantum transport, 209–260 quasilattice, 4, 5, 8, 50, 56, 172, 174, 181, 194, 199, 267, 319 quasilattice constant, 5, 8, 172, 181, 194, 319 quasiperiodic lattice, 2–4, 6–8, 49, 50, 177, 283, 285–286, 324 random tiling, 7, 115–116, 136–139, 140, 150, 159, 278 relaxation time approximation, 210, 211, 213, 215–217, 229, 232, 253, 255, 259 resistivity, 8, 49, 101, 202, 209, 213, 245–246 satellite reflection, 43, 116, 118, 120, 122, 124, 128, 141, 150, 157 scanning tunneling microscopy (STM), 313 section method, 12, 15 semiconducting, 126, 185–186, 211, 245, 298 sliding mode, 115, 121 specific heat, 49, 202, 246, 271 structure factor, 3, 14, 15, 17–20, 69, 135, 175, 194, 199 superlattice, 81

superstructure, 29, 39, 79, 80–84, 99, 151, 158, 318, 341 surface energy, 286 reconstruction, 315, 326–331, 341–342, 346, 350 relaxation, 316, 327, 330, 341 surface states, 278–279 synchrotron radiation, 13, 19, 55, 65, 161, 186 Ta-Te, 14, 36, 37, 38, 39, 44 thermodynamic instability, 279 thermoelectric power, 8, 49 topological disorder, 151 triacontahedron, 4, 29, 63, 85, 87, 137, 188, 317 Tsai-type, 50–53, 58, 59, 63, 64–71, 319 12-fold vertex in Penrose pattern, 29 ultraviolet photoelectron spectrum (UPS), 344 valence electron concentration, e/a, 95, 98–100 valence, 69, 95, 98–102, 172, 173, 175, 176, 181, 184–186, 188–190, 192, 194 weak localization, 209, 233–234, 243, 260 work softening, 268, 291, 293, 298, 301–302 X-ray photoelectron diffraction (XPD), 314, 330 X-ray photon correlation spectroscopy, 152 yield stress, 292–293, 295, 296, 300, 304 Zintl phase, 190–192 Zn-Mg-RE, 2, 77, 92–93, 172–173, 192–197, 203, 204 Zn-Mg-Sc, 55–58, 61, 63, 279–280 Zn-Mg-Y, 67, 68, 71, 192–197, 201, 204, 282, 344 Zn-M-Sc (M=Cu, Ag, Au, Pd, Pt), 56, 58–59, 197, 200 Zn-Sc, 58, 64, 65, 68, 69, 198, 202 Zn-T-Sc (T=Mn, Fe, Co, Ni), 59–61, 63, 64


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