Metallic Chains Chains of Metals, Volume 1 (Handbook of Metal Physics)

Metallic Chains/Chains of Metals HANDBOOK OF METAL PHYSICS SERIES EDITOR Prasanta Misra Department of Physics, Univer...

Author: Michael Springborg | Yi Dong

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Metallic Chains/Chains of Metals

HANDBOOK OF METAL PHYSICS SERIES EDITOR Prasanta Misra Department of Physics, University of Houston, Houston, TX, USA

Metallic Chains/Chains of Metals MICHAEL SPRINGBORG Physical and Theoretical Chemistry, University of Saarland, 66123 Saarbru¨cken, Germany

YI DONG Physical and Theoretical Chemistry, University of Saarland, 66123 Saarbru¨cken, Germany

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 The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK

First edition 2007 Copyright r 2007 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-13: 978-0-444-51380-9 ISBN-10: 0-444-51380-9 ISSN: 1570-002X

For information on all Elsevier publications visit our website at books.elsevier.com

Printed and bound in The Netherlands 07 08 09 10 11 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. Owing 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 selfcontained 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 as well as electronically which will be available in Science Direct. 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 vii

viii

Preface

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. The availability of the Book Series electronically (in addition to hard copy) would make this job even much simpler. 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 their comments to me. 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. In fact, I have been greatly impressed by the quality of this ﬁrst volume written by Professors Michael Springborg and Yi Dong. 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. It was also her idea to publish the Book Series electronically as well as in hard cover. 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 endeavour 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.

Preface

ix

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 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 endeavour. My fondest dream is that when my grandchildren Annika and Millan attend college in 2,021, 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

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Contents

Preface

vii

Chapter 1. Metals and Chains?

1

Chapter 2. Single-Particle Properties 2.1. A simple model

3 3

2.2. Extending the simple model 2.3. Transmission and complex band structures

6 11

2.4. Conduction 2.5. Conclusions

14 19

References

19

Chapter 3. Many-Body Properties 3.1. The electronic Schro¨dinger equation

21 21

3.2. Hartree–Fock approaches

23

3.3. Density-functional theory approaches 3.4. Single-particle models

25 27

3.5. Many-particle models 3.6. The Hubbard and the extended Hubbard models for a chain

28 28

3.7. The Luttinger liquid

31

3.8. Conclusions References

34 35

Chapter 4. The Jellium Model 4.1. Chains of jellium

37 38

4.2. Conclusions

44

References

44

Chapter 5. Gold Chains: The Prototype? 5.1. The structure of a linear chain of Au atoms 5.2. Conduction 5.3. More complicated structures

45 45 48 52

xi

xii

Contents

5.4. Chains containing other atoms 5.5. Gold chains on surfaces – Luttinger liquids?

63 65

5.6. Conclusions

76

References Chapter 6. Chains of other sd Elements 6.1. Ag

77 79 79

6.2. Cu 6.3. Hg, Cd, and Zn

83 86

6.4. Pt 6.5. Pd and Ni

87 90

6.6. Ir, Rh, and Co

91

6.7. Ru 6.8. Nb

91 92

6.9. Zr and Ti 6.10. Conclusions References Chapter 7. Chains of sp Elements 7.1. Al

95 95 96 97 97

7.2. Ga, In, and Tl

103

7.3. C 7.4. Si, Ge, and Pb

105 119

7.5. As and Bi 7.6. S and Se

123 124

7.7. Conclusions

124

References Chapter 8. Chains of s Elements 8.1. Na 8.2. Li, K, Rb, and Cs 8.3. Conclusions References Chapter 9. Mixed Systems 9.1. Order, disorder, and quasi-periodicity 9.2. Alloys and compounds 9.3. Filled nanotubes

127 131 131 137 139 139 141 141 142 146

Contents

xiii

9.4. Decorating chains 9.5. Guest–host systems

150 151

9.6. Conclusions

155

References Chapter 10. Crystalline Chain Compounds 10.1. CaNiN

155 159 159

10.2. SN 10.3. MX2 chains

162 164

10.4. Metal trichalcogenides 10.5. Metal tetrachalcogenides

167 171

10.6. Metal oxides: spin-chain and spin-ladder compounds

173

10.7. Incommensurate elemental crystals 10.8. CH3 BiI2

181 183

10.9. PtðCNÞ4 -based chain materials 10.10. Conclusions References Chapter 11. Mixed-Valence MX Chain Compounds and Related Systems 11.1. The MX chain compounds

184 185 187 191 191

11.2. The MMX chain compounds

199

11.3. Magnus’ green salt 11.4. Conclusions

200 201

References

202

Chapter 12. Synthetic Metals: Conjugated Polymers 12.1. The prototype: polyacetylene

203 204

12.2. Other carbon-based conjugated polymers 12.3. Incorporating heteroatoms

214 218

12.4. Incorporating metal atoms 12.5. Applications

226 227

12.6. Conclusions

230

References Chapter 13. Charge-Transfer Salts 13.1. General properties 13.2. The TTF–TCNQ family 13.3. The TMTSF2 -X and ET2 -X families

231 235 236 241 245

xiv

Contents

13.4. The TTF–CA family 13.5. Conclusions References

248 250 251

Chapter 14. Concluding Remarks Reference

253 255

Subject Index

257

Chapter 1

Metals and Chains?

Conventionally, metals are understood as materials that can conduct charge, and very often this charge transport is accomplished through mobile electrons or holes. Thus, the electronic orbitals are assumed being delocalized over the complete material. Moreover, these delocalized orbitals are supposed to be the building block for the metallic bonds that hold the atoms together, which is to be contrasted with the directional, covalent bonds found for many semiconductors and the electrostatic interactions found for many ionic insulators. Accordingly, the structure of metallic materials can often be explained through packing arguments: the atoms are packed closely together and the electrons ﬂow around everywhere in between. With this in mind, ‘chains’ and ‘metals’ seem to be self-contradictory: the existence of chains appears to require directional bonds that occur more in covalently bonded materials in contrast with the fact that close packing and delocalized electrons are the central properties of metals. Of course, one may imagine and study a one-dimensional world where electrons are conﬁned to one dimension, and, as we shall see, many studies have been devoted to such systems. Nevertheless, our world is three-dimensional and also individual atoms are three-dimensional objects so even when forming a chain of atoms, the electrons will be moving in a threedimensional world that at most can be called quasi-one-dimensional. The object of the present volume is to study the properties of real existing materials that somehow are quasi-one-dimensional. We shall often make connection to the results of theoretical studies where truly one-dimensional systems have been examined in order to explore whether quasi-one-dimensional systems can be considered truly one-dimensional or rather as being three-dimensional systems for which the interactions in one direction are considerably stronger than those in the other two. Most of the materials we shall study are produced in the laboratory, but some are also found in nature. Among the former are chains of metal atoms deposited on various surfaces, where it is hoped (but sometimes questioned) that the interactions between the chains and the surface can be ignored. Chains produced by narrowing a junction between two metal tips also belong to this class. Here, the chains are often fairly short so that it is not obvious that these systems can be considered extended chains. In other cases, crystalline materials containing channels may host chains, wherein an obvious question is whether the host–chain interactions are important.

1

2

Chapter 1. Metals and Chains?

Yet other systems are formed by highly anisotropic crystals that may be considered as consisting of weakly interacting chains. Also here, an interesting issue is to identify the role of the different dimensionalities. Alternatively, ‘metallic chains’ may be formed from non-metals. As an extreme case we shall to some extent discuss those carbon-based, organic, polymeric materials that have been given the name ‘synthetic metals’. These materials have been at the centre of an intensive research since about quarter of a century, in the beginning mainly due to their property as possible good electrical conductors (with a conductivity comparable to those of more conventional metals), but later mainly due to their properties as semiconductors. Finally, the electronic orbitals of chains of molecules are relatively strongly localized to the molecular units but may nevertheless interact and form bands that permit conduction. However, due to the strong localization, electronic interactions (i.e., many-body or correlation effects) may be important, and in many cases they lead to phase transitions to charge- or spin-density wave states or even to superconducting states. An interesting example in this context is the charge-transfer salts. In this volume, we shall review the properties of these different types of materials. The authors are theoreticians and the approach will be that of theoreticians. Furthermore, the scientiﬁc work of the authors is related to density-functional studies of electronic and structural properties of materials, and this background will form the main viewpoint for our presentation. We shall therefore start out with summarizing the main theoretical approaches, both for weakly interacting (singleparticle) systems and, subsequently, for strongly interacting (many-body) systems and in both cases we will concentrate on truly one-dimensional systems. Subsequently, the various systems brieﬂy outlined above will be discussed. It shall be emphasized that the authors have a background in broader disciplines between those of chemistry and physics, which also should be taken into account when reading our presentation. Simultaneously, it should be mentioned that exactly those materials that we are going to study very often are produced and studied by scientists from both disciplines. We expect that the reader has an overall interest in materials, being, e.g., a physicist, a chemist, or a material scientist, without necessarily being an expert in the materials of our work, but rather being interested in obtaining a general overview of existing materials that are approximately one dimensional, i.e., quasi one dimensional. We emphasize that the enormous wealth of high-quality research work on quasi one dimensional systems makes it absolutely impossible to present anything but a small, subjectively chosen part. Therefore, the omission of other works does not imply that these should be considered as being of lower quality, but rather that the restricted space puts natural bounds on what can be presented.

Chapter 2

Single-Particle Properties

In order to discuss the properties of metallic chains, we shall in this section review various aspects related to single-particle descriptions of the materials. In the next section, we shall discuss many-particle properties, i.e., what happens when we no longer can consider the electrons as being independent particles that move in some average ﬁeld created by all the other particles. Later in this presentation we shall relate experimental observations on speciﬁc systems with the general discussion presented in this section.

2.1.

A simple model

As a starting point we shall discuss a very simple model in some detail. The model is related to the model of Su et al. [1–3] for polyacetylene that we shall return later (Chapter 12), but neither the details of the model, nor the special material, polyacetylene, is of further relevance here. We consider an inﬁnite, periodic, linear chain of identical atoms as that of Figure 2.1. We shall allow for the bond lengths to alternate, as shown in the lower part of Figure 2.1. We shall, moreover, assume that each atom has one electron and that there is one atomic orbital per site. These orbitals are assumed to be orthonormal. Assuming, furthermore, that only on-site energies and nearest-neighbour hopping integrals are non-vanishing, we end up with the following Hamiltonian for the electrons: X H^ p ¼ ½n c^yn c^n þ tn;nþ1 ð^cynþ1 c^n þ c^yn c^nþ1 Þ, (2.1) n

Figure 2.1. A linear chain of identical atoms with (upper part) non-alternating and (lower part) alternating interatomic distances. 3

4

Chapter 2. Single-Particle Properties

where c^yn and c^n are the creation and annihilation operator, respectively, for the function of the nth site, and n and tn;nþ1 are the on-site energy and hopping integral, respectively. When the bond lengths alternate, we can describe their values through the single parameter u; i.e., the bond lengths are d 0 u: If ju=d 0 j51; we may let the hopping integrals depend linearly on u; i.e., tn;nþ1 ¼ t0 au.

(2.2)

Moreover, the on-site energies are independent of n; n ¼ 0 .

(2.3)

That part of the total energy that is not contained in H^ p is written to lowest order in u as a harmonic function, K H^ s ¼ 2N u2 , 2

(2.4)

where N is the number of two atomic units. The total Hamiltonian is, accordingly, H^ tot ¼ H^ s þ H^ p .

(2.5)

For t0 ¼ 2; 0 ¼ 0; K ¼ 1; a ¼ 1; and a lattice constant a ¼ 1; the two contributions to the total energy per two-atomic unit vary as a function of u as shown in Figure 2.2. It is clearly seen that the total energy has two equivalent minima for structures with alternating shorter and longer bonds (the two structures are related through an interchange of those two types of bonds). This symmetry lowering is related to an opening up of a gap at the Fermi level, as shown in Figure 2.3, and is nothing else than the well-known Peierls distortion [4]. It shall be emphasized that the distortion is general and found for any values of the parameters of the model. Thus, E s is ﬂat, but upward curved at u ¼ 0; whereas the downward curved E p has a non-zero derivative at u ¼ 0: Thus, an inﬁnitesimally small value of u will invariably lead to a reduction of the sum of the two energy contributions, but for larger values of u; E p depends weaker than quadratic on u; so that there will be some ﬁnite value of u for which the total energy has a minimum. Finally, a schematic representation of the orbitals closest to the Fermi level (Figure 2.4) can explain the reduction of E p when increasing u: For ua0; the shorter bond between nearest neighbours get stronger due to an in-phase interaction between the atom-centred functions for the highest occupied orbital, whereas the longer bonds become weaker. This leads to a stabilization of this orbital. A similar destabilization is observed for the lowest unoccupied orbital, which, however, is not occupied and, therefore, does not contribute to the total energy. The model and results we have discussed here in some detail is unrealistic, simple, and well known. Nevertheless, we shall use it as a basis for discussing various results and have, therefore, presented the detailed analysis here.

2.1. A simple model

5

Figure 2.2. The variation in the total energy (E tot ), the single-particle energy (E p ) and the elastic energy (E s ) per two atoms as a function of the parameter u that describes the size of the bond-length alternation.

Figure 2.3. The band structures for the structure (left part) without a bond-length alternation and (right part) with the optimized bond-length alternation.

6

Chapter 2. Single-Particle Properties

Figure 2.4. Schematic representation of (upper part) the highest occupied orbital and (lower part) the lowest unoccupied orbital for k ¼ p for the chain with a bond-length alternation. Here, circles of the same (different) colour represent contribution of the atomic orbitals that have the same (opposite) sign, i.e., are bonding (antibonding).

Figure 2.5. Schematic representation of two interacting chains.

2.2.

Extending the simple model

In the preceding section we have discussed a model that is so simple that it in many cases is too simple to yield realistic results. There are many ways of extending the model so that it becomes more realistic, but here we shall consider only some simple cases. The model we have considered is that of an isolated, inﬁnite, periodic, linear chain with identical atoms and one atomic function and electron per atom. For this, a metallic state was found to be unstable, and the systems would instead spontaneously distort into a structure with alternating bond lengths and, simultaneously, a gap at the Fermi energy. However, any attempt of experimentally realizing such a system leads to one that at most approximately resembles the ideal system. Thus, very often the chain is not completely isolated but may be deposited on a surface of some crystal, be conﬁned to the interior of some channels of a host, or having more orbitals and electrons per atom. Through some simple extensions of the simple model of the preceding section we can study how these changes will affect the properties of the system. We start with considering the system of Figure 2.5. Here, the chain of our interest, i.e., the one of the black atoms, interacts with another chain, the one of the white atoms. This situation can model both a linear chain deposited on some surface, a chain with a more complicated structure (in that case the black and white atoms could be identical), atoms with more orbitals (then, the black and white atoms would represent two different types of orbitals), and a linear chain conﬁned in some channel (whereby the white atoms would represent the host). With the

2.2. Extending the simple model

7

results of the preceding section in our mind we shall explore whether the system of Figure 2.5 may show a behaviour different from that of the system of Figure 2.1. We shall only consider the electronic part of the total energy, i.e., H^ p ; that we shall write as H^ p ¼

X

½n;1 c^yn;1 c^n;1 þ tn;nþ1;1;1 ð^cynþ1;1 c^n;1 þ c^yn;1 c^nþ1;1 Þ

n

þ n;2 c^yn;2 c^n;2 þ tn;nþ1;2;2 ð^cynþ1;2 c^n;2 þ c^yn;2 c^nþ1;2 Þ þ tn;n;1;2 ð^cyn;1 c^n;2 þ c^yn;2 c^n;1 Þ,

ð2:6Þ

where c^yn;i and c^n;i are the creation and annihilation operator, respectively, for the function of the nth site of the ith chain, and n;i the corresponding on-site energy, whereas tn;m;i;j is the hopping integral for the nth and mth functions on the ith and jth chain, respectively. We shall restrict ourselves to periodic chains without bond-length alternation and have, accordingly, n;i ¼ 0;i tn;nþ1;i;i ¼ ti tn;n;1;2 ¼ tn;n;2;1 ¼ t0 .

ð2:7Þ

We choose 0;1 ¼ 0 and t0 ¼ 2; as in the example of the preceding section, but set this time the interatomic distance along one chain equal to a ¼ 1: In Figure 2.6, we show some typical results of the calculations with this model. In all cases the Fermi level is crossing through the band with its major contributions from the chain of our interest. The four cases of the upper row simulate the case where the chain is interacting weakly with the ‘white’ chain, either so that this interaction is provided by unoccupied orbitals of the ‘white’ chain (two left cases) or provided by occupied orbitals of the ‘white’ chain, but the energetic difference between the orbitals of the two chains is large (we set it equal to 5 units), and the orbital interaction between the two chains is fairly small (0:5 units). This situation resembles that of a chain of metal atoms deposited on some surface with a fairly large gap at the common Fermi level. For each of the two cases we consider both the situation where t2 and t1 have the same sign, so that there is a large gap between the band from the chain of our interest and that from the ‘white’ chain for every single k point, whereas the other situation correspond to having t2 and t1 with opposite sign so that the bands interact in some parts of the Brillouin zone. We have set t2 ¼ 1:5 units and t0 ¼ 0:5: Nevertheless, we see that the situation of a half-ﬁlled band is recovered in all these cases, suggesting (using the same rationale as in the preceding section) that the chain will be stabilized upon a bond-length alternation that opens up a gap at the Fermi level. Thus, in the case that the chain of interest possesses only relatively weak interactions with the surrounding, the fundamental properties of the isolated chain (in this case it will lower its symmetry and simultaneously open up a gap at the Fermi level) are retained.

8

Chapter 2. Single-Particle Properties

Figure 2.6. Band structures for different cases of two interacting chains.

However, when the interactions between the two chains are stronger and, simultaneously, the orbital energies of the functions of the two types of atoms are not too different, then the situation becomes markedly different. There will be strong hybridizations between the orbitals of the two different chains, and no band can be characterized as belonging mainly to either the ‘black’ or the ‘white’ chain. This is exempliﬁed in the lower row of panels, where we have set t2 ¼ 1:5; t0 ¼ 2:5 and gradually reduced 2 from 5.0, via 2.0 and 1.0, to 0.0. In these cases, the Fermi level crosses the band at two different places, so that a bond-length alternation may open up a gap but not at the Fermi level. This means that the ‘conventional’ Peierls distortion, which is the one discussed in the previous section, is not energetically favoured and will, therefore, not take place. However, Berlinsky [5] has shown that even in the case when two bands cross the Fermi level, a symmetry-lowering distortion that leads to the occurrence of a gap at the Fermi level can be found. The length of the repeated unit will be larger (may even be very large), but can, in principle, be identiﬁed. This last set of examples demonstrates that, fundamentally, the properties of a chain can be altered through the interactions with other systems. Speciﬁcally, we have shown that the Peierls distortion can be suppressed and replaced by either a metallic state or by a distortion of a completely different symmetry. As we shall

2.2. Extending the simple model

9

Figure 2.7. A two-dimensional square lattice of parallel chains whose chain axes are perpendicular to the plane of the ﬁgure.

demonstrate below, other interactions can also modify the properties of the chains similarly. As a model for a material containing weakly interacting parallel chains, we consider the system of Figure 2.7 that is constructed as periodically repeated, parallel chains of the type shown in Figure 2.1. We shall neglect any bond-length variation along the chains and, moreover, assume that only nearest neighbours of different chains interact. Letting ðl; m; nÞ denoting the nth atom of the ðl; mÞth chain (notice that the individual chains of Figure 2.7 can be uniquely identiﬁed through two integers) the electronic Hamiltonian can be written as H^ p ¼

X

½0 c^yl;m;n c^l;m;n þ t0 ð^cyl;m;nþ1 c^l;m;n þ c^yl;m;n c^l;m;nþ1 Þ

l;m;n

þ t0 ð^cylþ1;m;n c^l;m;n þ c^yl;m;n c^lþ1;m;n þ c^yl;mþ1;n c^l;m;n þ c^yl;m;n c^l;mþ1;n Þ.

ð2:8Þ

The ratio jt0 =t0 j gives the importance of interchain interactions, or, alternatively expressed, whether the material can be considered as consisting of essentially independent chains or, instead, as being a more or less anisotropic three-dimensional crystal. For this model, k~ space is three-dimensional. Therefore, we do not show the complete band structures but instead cuts through the Fermi surface in a ðkx ; kz Þ plane for constant ky : This is shown in Figure 2.8 for different values of t0 =t0 and by assuming that the lattice constant in all three directions equals 1. If t0 =t0 ¼ 0 the chains are non-interacting and we are having a situation like that of Figure 2.3, i.e., for any value of ðkx ; ky Þ the Fermi surface is found for

10

Chapter 2. Single-Particle Properties

Figure 2.8. A cut through the Fermi surface for ky ¼ 0 (upper panels) and ky ¼ p=2 (lower panels) for (from left to right) t0 =t0 ¼ 0, 0.05, 0.25, and 0.5.

kz ¼ ðnz þ 12Þ p with nz an integer. The important point is now that with the socalled nesting vector k~n ¼ ð0; 0; pÞ it is possible to connect two points on the Fermi surface for a ‘large part’ of this surface (actually, in the present case, for the complete Fermi surface). As a consequence, it can be shown that a modulation of the system with a wave that has k~n as a wave vector may result in the opening up of a gap at the Fermi energy. In the preceding section (cf., e.g., Figure 2.3) we saw one example of such a modulation, i.e., a structural distortion where the atoms were shifted alternately in one or the other direction along the chain. This type of modulation is often called a bond-order wave, but also other types of modulations can be found like, e.g., the occurrence of a charge-density wave, where, in our case, atoms or bonds with a larger electron density alternate with atoms or bonds with a reduced electron density. Yet another type of modulation is the existence of a spin-density wave where atoms with a majority of spin-up electrons alternate with atoms with a majority of spin-down electrons, i.e., the chains show a magnetic ordering. Also the onset of superconductivity will lead to the opening up of a gap at the Fermi energy and can, accordingly, also occur. When t0 =t0 a0; but still is small, the Fermi surface changes slightly from consisting of parallel planes, as exempliﬁed in Figure 2.8 for the case t0 =t0 ¼ 0:05: This means that there is not a perfect nesting, but the nesting vector k~n ¼ ð0; 0; pÞ only approximately joins different parts of the Fermi surface for a large part of this. Therefore, a distortion with this wave vector will open up a gap that only approximately is found at the Fermi energy. Nevertheless, this distortion may stabilize the system, but whether it actually occurs depends critically on the detailed properties of the system, most notably on the size of the gap due to the distortion in relation to the size of the warping of the Fermi surface. For even larger values of t0 =t0 the warping of the Fermi surface increases, the occurrence of the modulations becomes less likely, and the system changes

2.3. Transmission and complex band structures

11

increasingly into an anisotropic three-dimensional crystal. This is exempliﬁed through the last two parameter values of t0 =t0 of Figure 2.8. As the ﬁgures clearly demonstrate, it is then no longer possible to identify a nesting vector that can deﬁne a structural distortion that is accompanied with the opening up of a gap at the Fermi energy.

2.3.

Transmission and complex band structures

The concept of complex band structures has become useful when analysing transmission of electrons through a chain. We shall now describe the concept through the simple model of equation (2.1). Any eigenfunction can be found from H^ p c ¼ c.

(2.9)

The basis functions of the nth atom is denoted wn ; and c is written as a linear combination of those, X c¼ xn wn (2.10) n

with xn being the expansion coefﬁcient to wn : Using that ð^cyp c^q Þðxn wn Þ ¼ dn;q xn wp ,

(2.11)

Equation (2.9) can be rewritten as X H^ c c ¼ ½n xn wn þ tn;nþ1 xn wnþ1 þ tn;nþ1 xnþ1 wn n

¼

X ½n xn þ tn1;n xn1 þ tn;nþ1 xnþ1 wn n

¼

X

xn wn .

ð2:12Þ

n

Since the basis functions are orthonormal, we obtain n tn1;n xnþ1 ¼ xn xn1 , tn;nþ1 tn;nþ1

(2.13)

and xn ¼

n1 tn2;n1 xn1 xn2 . tn1;n tn1;n

Inserting equation (2.14) into equation (2.13), we have ð n Þð n1 Þ tn1;n ð n Þtn2;n1 xn2 . xn1 xnþ1 ¼ tn1;n tn;nþ1 tn;nþ1 tn;nþ1 tn1;n Equations (2.14) and (2.15) can be written in matrix form ! ! xn1 xnþ1 ¼ Tn xn xn2

(2.14)

(2.15)

(2.16)

12

Chapter 2. Single-Particle Properties

where T n is the transfer matrix [6], 0

ð n Þð n1 Þ tn1;n B tn1;n tn;nþ1 tn;nþ1 B Tn ¼ B n1 @ tn1;n

1 ð n Þtn2;n1 tn;nþ1 tn1;n C C C. tn2;n1 A tn1;n

(2.17)

We shall now restrict ourselves to the case of the periodic structures of equations (2.2) and (2.4) with, moreover, the two values of the hopping integrals denoted t and tþ : Then T n is independent of n; 0

ð 0 Þ2 t B tþ B t t Tn ¼ T ¼ B þ 0 @ t

1 0 C t C C. tþ A t

(2.18)

T describes how an electron (or rather its wavefunction) is propagated through an inﬁnite, periodic system. Its eigenvalues, in particular, gives information on decay or increase of the wavefunctions as one moves from one pair of atoms to the next. It can easily be veriﬁed for the model above that detðTÞ ¼ 1; which is a consequence of the general result that for any decaying wavefunction there is a corresponding increasing wavefunction. This result is not limited to the present model but holds for any chain independently of the range of the interactions, the number of basis functions per repeated unit, and the number of electrons per unit, as long as one stays within a single-particle approach and the system is periodic. A special case is that of orbitals belonging to periodic states, i.e., to eigenvalues of T with a modulus of 1. The eigenvalues l of T of equation (2.18) obey ð 0 Þ2 t tþ l2 l þ 1 ¼ 0. (2.19) tþ t tþ t Choosing l ¼ eiy

(2.20)

with y being real, we arrive at ¼ 0 ½t2 þ t2þ þ 2t tþ cosðyÞ1=2

(2.21)

and can, accordingly, identify y ¼ ka,

(2.22)

a being the lattice constant, and k the conventional one-dimensional momentum of the Bloch functions. We may, however, also seek for solutions for non-real values of y of k as long as the corresponding energies remain real. The solutions correspond then to

2.3. Transmission and complex band structures

13

Figure 2.9. The complex band structures for the model of equation (2.1).

wavefunctions that are decaying or growing inside the chain. Such solutions are relevant when considering a long but ﬁnite chain suspended between two electrodes. Then, electrons of a given energy may tunnel through the chain if the decay length of the wavefunction at that particular energy is sufﬁciently long [i.e., Im(k) sufﬁciently small]. Moreover, it has to be assumed that the presence of the electrodes does not perturb the chain too much in the regions closest to the electrodes. Also when studying ‘surface’ states of ﬁnite chains (i.e., states localized to the regions at the ends) these solutions are relevant. Ultimately, when applying equation (2.22) we may describe these solutions with a complex k which then leads to the concept of complex band structures. As a demonstration we show in Figure 2.9 the complex band structures for the above model with 0 ¼ 0 and a ¼ 1; as well as for two different values of ðtþ ; t Þ; i.e., ðtþ ; t Þ ¼ ð2:1; 1:9Þ and ðtþ ; t Þ ¼ ð2:5; 1:5Þ: Figure 2.9 shows how the bands for complex k connect the bands that we ﬁnd for real k: In particular, below the lowest band as well as above the highest band, ImðkÞ increases rapidly as a function of the energy difference to the band-structure energies, showing how these orbitals decay in the chains. Also in the energy region of the gap between the two regular bands for real k the orbitals are decaying functions with a decay constant that increases the more we move away from the band edges. Although, admittedly, the results we have presented here are those of an extremely simple model, they do grasp the essential features of the complex band structures [7]. Thus, also for systems with more orbitals and electrons per repeated unit and for which the orbital interactions are of a longer range, we will have a conventional band structure that is characterized by bands separated by gaps. The complex band structures will form loops that connect tops and bottoms of the bands across the band gaps in the complex k plane. And ImðkÞ gives information on the decay behaviour of those orbitals.

14

2.4.

Chapter 2. Single-Particle Properties

Conduction

With the concepts of the preceding section we are entering the ﬁeld of transmission of electrons through a chain, i.e., that of conduction. Landauer [8] studied the conduction through a chain like that of Figure 2.10, where a ﬁnite, but long chain is connected via perfect leads to two electron reservoirs (see also, e.g., Refs. [9–15]). As an idealization one may consider the situation of Figure 2.11, where from an incoming wave, a fraction, R; is reﬂected and the remaining part, T; is transmitted. What is special about the system of Figure 2.11 is that for any incoming wave of energy there is exactly one channel that can transfer the wave from the left side to the right side. Thus, in some sense the system resembles the one we have discussed in great detail throughout this section, i.e., one for which the electronic orbitals of the chain are formed from exactly one function per atom. For this type of system Landauer [8] found a conductance given by G¼

e2 T . p_ R

(2.23)

Thus, for the conductance the important ingredient is the quotient of the transmission coefﬁcient to the reﬂection coefﬁcient, both quantities that are related to the transfer matrix of the preceding subsection. When the applied voltage over the chain is small, all quantities have to be calculated for being the Fermi energy of the chain.

Figure 2.10. A ﬁnite chain connected to two perfect leads.

Figure 2.11. An idealization of the system of Figure 2.10 with an incoming wave that is partly reﬂected and partly transmitted.

2.4. Conduction

15

In order to obtain the result of equation (2.23) the starting point is that of a single electron moving in an electromagnetic ﬁeld [11]. For this, the Hamiltonian is 1 e ~2 ~ H^ ¼ þ V ð~ rÞ, (2.24) p A 2m c ~ r; tÞ the electromagnetic vector potential, and where m is the mass of the electron, Að~ V ð~ rÞ the scalar potential. We now consider a one-dimensional system of length L that is connected at both ends to a matching ‘wire’ of total length L0 : The DC conductance of this system is deﬁned as I G ¼ lim lim , (2.25) 0 o!0 L !1 V where o is the frequency of the external electro-magnetic ﬁeld, I the current through the system, and V ¼ EL

(2.26)

is its voltage. Finally, E is the electric ﬁeld which is taken to be uniform over the system. Keeping only linear terms in the electromagnetic ﬁeld and applying linearresponse theory [16] one obtains, after a longer derivation (see, e.g., Ref. [11]), for the real part of the current IðzÞ [9], Z occ unocc X pe2 _ X 0 0 0 ½dðo oai Þ dðo þ oai Þ. IðzÞ ¼ W ðzÞ W ðz ÞEðz Þ dz ia ia 2om2 i a (2.27) Here, the z coordinate runs along the chain, and the z0 -integral is over the whole region of the system (i.e., both the wire itself as well as the leads). Moreover, W ia ðzÞ ¼ ci ðzÞ

@ca ðzÞ @c ðzÞ ca ðzÞ i @z @z

(2.28)

is related to the current operator. The a-summation is over all unoccupied orbitals for the system without the external electromagnetic ﬁeld, whereas the i-summation is over all occupied orbitals. Finally, oai ¼

a i . _

(2.29)

At the extreme left and right end of the total system (i.e., far away from the central wire), the normalized eigenfunctions can be written as cL ðzÞ ¼ Aeikz þ Beikz , cR ðzÞ ¼ Feikz þ Geikz ,

ð2:30Þ

16

Chapter 2. Single-Particle Properties

where A and B are related to F and G through a transfer matrix involving the transmission and reﬂection amplitudes t and r of the system, obeying jtj2 ¼ 1 jrj2 ¼ T.

(2.31)

Here, T is the transmission coefﬁcient. The transfer matrix that relates ðA; BÞ to ðF ; GÞ becomes then 0 1 1 r B t tC C (2.32) M¼B @r 1 A. t

t

Similarly, F and G are related to A and B through a transfer matrix involving the transmission and reﬂection amplitudes t0 and r0 of the ‘wire’. Combining all these identities, one can show that [9] 1 A ¼ pﬃﬃﬃﬃﬃﬃﬃ0 , 2L rt B ¼ pﬃﬃﬃﬃﬃﬃﬃ0 . 2L

ð2:33Þ

In the limit o ! 0; for z at the center of the system, one arrives then at [9] jW ia ðzÞj ¼

2kF pﬃﬃﬃﬃ T. L0

(2.34)

Economou and Soukoulis [9] considered two special cases. When the external ‘wire’ is a perfect one-dimensional conductor, they obtain G¼

e2 T, p_

(2.35)

whereas when connecting the system to a current source, they obtain G¼

e2 T . p_ 1 T

(2.36)

This last equation is the Landauer formula. The detailed derivation is beyond the scope of the present discussion, and, therefore, we have here just brieﬂy sketched how one arrives at it. The interested reader is referred to, e.g., [9,11] for more details. We shall, however, use the results when discussing experiments on various metallic wires. Bu¨ttiker et al: [14] generalized the approach of Landauer to the case where there are more channels that can be used in transmitting the electrons through the chain, which essentially means that they considered the case of more orbitals per atom giving rise to a situation where more bands cross the Fermi level for the chain. The

2.4. Conduction

17

Figure 2.12. The many-channel generalization of the system of Figure 2.11 with an incoming wave for channel i that is partly reﬂected in channel j and partly transmitted in channel j:

derivation of their ﬁnal formula is non-trivial and, therefore, shall not be repeated here (the interested reader is referred to the original literature, Refs. [14,15]). With N channels the matrix S of Figure 2.12 is a 2N 2N matrix of the form, r t0 S¼ , (2.37) t r0 with r and t being N N matrices that describe the reﬂection and transmission, respectively, of waves coming from the left lead towards the right lead, whereas r0 and t0 are the same quantities for the waves coming from the right lead. The transmission and reﬂection probabilities T ji and Rji of Figure 2.12 are given through T ji ¼ jtji j2 , Rji ¼ jrji j2 .

ð2:38Þ

For the temperature equal to 0, the ﬁnal expression for t G¼

e2 Trðty tÞ, p_

(2.39)

which is the generalization of equation (2.35). By diagonalizing ty t (notice that this matrix depends on the energy of the incoming wave, most often taken as the Fermi energy) and denoting the eigenvalues ti ; an alternative expression is G¼

e2 X ti , p_ i

(2.40)

i.e., we may interpret the conductance as resulting from a superposition of transmitting electrons through certain channels (the so-called eigenchannels, corresponding to the eigenvectors of ty t). The channels that provide perfect transmission (i.e., ti ¼ 1) contribute to the conductance with one quantum of conductance, G0 ¼

e2 2e2 ¼ , p_ h

(2.41)

18

Chapter 2. Single-Particle Properties

whereas other channels contribute to only some fraction thereof, i.e., 0 ti 1.

(2.42)

We shall illustrate the results through those of a simple theoretical model that was studied by Yamaguchi et al. [17]. They considered the system of Figure 2.13 that has many similarities with those that we shall discuss later in our presentation of real systems. In the system of Figure 2.13, we recognize a narrow constriction between two wider junctions. This situation can, e.g., be obtained by ﬁrst creating a narrow junction in some metal ﬁlm and, subsequently, pulling the two parts apart or, alternatively, by ﬁrst creating physical contact between the tip of a scanning– tunnelling microscope and some surface and, afterwards, slowly removing the tip. When a voltage is applied across the system of Figure 2.13, current (electrons or holes) may ﬂow between the two ends, and the transmission of the ﬁnite chain describes the conductance. Using an approach as the one we have outlined in this section, Yamaguchi et al. [17] found results like those of Figure 2.14. It is remarkably that the conductance depends very strongly on the number of atoms: in their study Yamaguchi et al. found a clear even–odd oscillation. Moreover, the precise form of the junction is

VL

VR

tS tR

tR

Figure 2.13. Schematic representation of the model system used in studying conduction quantization through a ﬁnite chain. Reproduced from Ref. [17].

Conductance (units of e 2/h)

T= 4.2 K

T = 300 K

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0 2 4 6 8 10 12 14 16 18 20 (a) Number of atoms

(b)

0

2

4

6 8 10 12 14 16 18 20 Number of atoms

Figure 2.14. The conductance for the system of Figure 2.13 as a function of length of the chain and for (a) a shorter and (b) a longer chain-tip distance (i.e., larger and smaller values of the hopping integrals V of Figure 2.13). Reproduced from Ref. [17].

References

19

also very critical. For a weak junction, as in Figure 2.14(b), the conduction becomes reduced, at least when working at slightly elevated temperatures.

2.5.

Conclusions

In this section, we have brieﬂy outlined the main scenario for single-particle descriptions of the properties of quasi-one-dimensional systems. We have seen that distortions that open up a gap at the Fermi energy are very important and will ultimately change any truly one-dimensional metallic system into a semiconductor or an insulator. In this respect, the Peierls distortion, i.e., a structural distortion, is the most prominent example, but modiﬁcations of this can also be found. In this context, we mention the spin-Peierls distortion, where, e.g., a half-ﬁlled band for both spin components changes into two bands, one for each spin direction, with one being completely ﬁlled and the other completely empty. Also the occurrence of a charge- or a spin-density wave is a possible distortion that can open up a gap at the Fermi level. In order to suppress these distortions, interactions with the surroundings could be important. Here, it may be a matter of convention whether one considers the resulting system as still being quasi-one-dimensional or whether it rather should be considered as (highly anisotropic) two- or three-dimensional. In the next section, we shall discuss how also many-body effects can modify the conclusions we have found above, depending on the relative strengths of single- and many-body interactions. Finally, we also discussed brieﬂy the theoretical approach for studying conductance through quasi-one-dimensional systems, i.e., the Landauer theory. Since conductance measurements are becoming increasingly important in characterizing these systems as well as in being relevant when attempting to construct electronic devices with sizes at the molecular level, this theory represents an important theoretical framework for quasi-one-dimensional system.

References [1] W.P. Su, J.R. Schieffer, and A.J. Heeger, Phys. Rev. Lett. 42, 1698 (1979). [2] W.P. Su, J.R. Schieffer, and A.J. Heeger, Phys. Rev. B 22, 2099 (1980); Erratum: Phys. Rev. B 28, 1138 (1983). [3] A.J. Heeger, S. Kivelson, J.R. Schieffer, and W.-P. Su, Rev. Mod. Phys. 60, 781 (1988). [4] R. Peierls, Quantum Theory of Solids (Oxford University Press, Oxford, 1955). [5] A.J. Berlinsky, J. Phys. C 9, L283 (1976). [6] M. Springborg, H. Kiess, and P. Hedega˚rd, Synth. Met. 31, 281 (1989). [7] J.K. Tomfohr and O.F. Sankey, Phys. Rev. B 65, 245105 (2002). [8] R. Landauer, IBM J. Res. Dev. 1, 223 (1957). [9] E.N. Economou and C.M. Soukoulis, Phys. Rev. Lett. 46, 618 (1981). [10] D.S. Fisher and P.A. Lee, Phys. Rev. B 23, 6851 (1981).

20

[11] [12] [13] [14]

Chapter 2. Single-Particle Properties

A.D. Stone and A. Szafer, IBM J. Res. Dev. 32, 384 (1988). E. Emberly and G. Kirczenow, Nanotechnology 10, 285 (1999). A.D. Stone, J.D. Joannopoulos, and D.J. Chadi, Phys. Rev. B 24, 5583 (1981). M. Bu¨ttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B 31, 6207 (1985). [15] D.C. Langreth and E. Abrahams, Phys. Rev. B 24, 2978 (1981). [16] J. Callaway, Quantum Theory of the Solid State (Academic Press, New York, 1974). [17] F. Yamaguchi, T. Yamada, and Y. Yamamoto, Solid State Commun. 102, 779 (1997).

Chapter 3

Many-Body Properties

In the preceding section we discussed brieﬂy single-particle properties of quasione-dimensional systems. This means that we assumed that the electrons were so delocalized that smaller redistributions of some of the electrons were not felt by the others and, accordingly, we could consider the individual electron as moving in some kind of averaged ﬁeld from the others. More speciﬁcally, we did not include effects that describe energetics related to processes where electrons are localized to regions of space (e.g., on atoms or bonds) where other electrons are present. In this section, we shall describe such many-body or correlation effects. We ﬁnally add that Giamarchi [1] has presented a more detailed discussion of the theoretical foundations for single- and many-body effects in quasi-one-dimensional systems.

3.1.

The electronic Schro¨dinger equation

In this section we shall brieﬂy describe the foundations for ab initio electronicstructure calculations as they often are used in theoretical studies of properties of materials. Moreover, they also can be considered as forming the foundation for more approximate approaches based, e.g., on model Hamiltonians as discussed in the previous section as well as for single-particle models that have been extended with many-body effects. For a more detailed description of the methods the reader is referred to Ref. [2]. The starting point is the exact time-independent Schro¨dinger equation for the system of interest. We assume that we have M nuclei and N electrons. Moreover, we will denote the masses of the nuclei M k ; k ¼ 1; 2; . . . ; M; whereas those of the electrons are m. The positions of the electrons are ~ ri ; i ¼ 1; 2; . . . ; N and those of the ~k ; k ¼ 1; 2; . . . ; M: Finally, the charges of the nuclei will be denoted nuclei R eZ k ; k ¼ 1; 2; . . . ; M and those of the electrons e with e being the elemental charge. Then the Hamiltonian for the total system becomes

H^ ¼

þ

M N M X X _2 2 _2 2 1 X 1 Z k 1 Z k 2 e2 rR~ r~ri þ ~k1 R ~k2 j 2 k ak ¼1 4p0 jR 2M k k 2me i¼1 k¼1 1 2 N M X N X 1 X 1 e2 1 Z k e2 . ~k ~ 2 i ai ¼1 4p0 j~ ri 1 ~ ri2 j k¼1 i¼1 4p0 jR ri j 1

2

21

ð3:1Þ

22

Chapter 3. Many-Body Properties

Here, we have introduced a short-hand notation for the gradient-operators like (in Cartesian coordinates) ~~ri ¼ r

@ @ @ ; ; . @xi @yi @zi

(3.2)

We split H^ into ﬁve terms, H^ ¼ H^ k;n þ H^ k;e þ H^ p;nn þ H^ p;ee þ H^ p;ne ,

(3.3)

i.e., the kinetic-energy operator for the nucleus, the kinetic-energy operator for the electrons, the potential-energy operator for nucleus–nucleus interactions, the potential-energy operator for electron–electron interactions, and the potentialenergy operator for nucleus–electron interactions. The solution C to the time-independent Schro¨dinger equation ^ ¼ EC HC

(3.4)

depends on the spin and position coordinates of all electrons, i.e., on r2 ; s2 ; . . . ;~ rN ; sN Þ ð~ x1 ; ~ x2 ; . . . ; ~ xN Þ ~ x, ð~ r1 ; s1 ;~

(3.5)

and on the spin and position coordinates of all nuclei, ~1 ; S1 ; R ~2 ; S2 ; . . . ; R ~M ; SM Þ ðX ~1 ; X ~2 ; . . . ; X ~M Þ X ~. ðR

(3.6)

Accordingly, ~; ~ C ¼ CðX xÞ,

(3.7)

and the Schro¨dinger equation takes the form ~; ~ ~; ~ ^ ¼ ðH^ k;n þ H^ k;e þ H^ p;nn þ H^ p;ee þ H^ p;ne ÞCðX xÞ ¼ E CðX xÞ. HC (3.8) We shall group the Hamilton operators into two parts ~; ~ ~; ~ ½ðH^ k;n þ H^ p;nn Þ þ ðH^ k;e þ H^ p;ee þ H^ p;ne ÞCðX xÞ ¼ E CðX xÞ.

(3.9)

The ﬁrst part depends solely on the nuclear coordinates whereas the second part also depends on the electronic ones. Very often one invokes the Born–Oppenheimer approximation which is accurate when the electrons move much faster than the nuclei. Then, one may assume that whenever the nuclei are displaced, the electrons adjust themselves ‘immediately’ to ~; the electrons the new structure so that for any given structure, characterized by X will always have a certain distribution independent of whether the nuclei are in the

23

3.2. Hartree-Fock approaches

middle of some movement or are staying ﬁxed at this structure. This means that the total wavefunction is factorized, ~; ~ ~Þ Ce ðX ~; ~ CðX xÞ ¼ C n ð X xÞ.

(3.10)

Here, the nuclear part, Cn ; depends solely on the structure of the system, whereas the electronic part, Ce ; depends only parametrically (and, accordingly, not functionally) on the structure implying that a new arrangement of the positions of the nuclei will lead to a new electronic wavefunction. As a second part of the Born–Oppenheimer approximation the kinetic-energy term of the nuclei in H^ is ignored, H^ k;n ¼ 0.

(3.11)

Then, inserting C of equation (3.10) into the Schro¨dinger equation, (3.8), leads to two equations ~; ~ ~; ~ ~ÞCe ðX ~; ~ ðH^ k;e þ H^ p;ee þ H^ p;ne ÞCe ðX xÞ H^ e Ce ðX xÞ ¼ E e ð X xÞ

(3.12)

and E¼

3.2.

M 1 X 1 Z k 1 Z k 2 e2 ~Þ. þ E e ðX ~k1 R ~k2 j 2 k ak ¼1 4p0 jR 1 2

(3.13)

Hartree–Fock approaches

Except for the absolutely simplest systems, that are of no relevance to the present work, the electronic Schro¨dinger equation, (3.12), cannot be solved exactly and one has to restore to approximations. The theoretical foundation for those is the variational principle from which one can arrive at hFjH^ e jFi E ð0Þ e , hFjFi

(3.14)

^ where E ð0Þ e is the ground-state energy for H e ; and F is an approximation to the ð0Þ corresponding wavefunction Ce : A very common approximation is that of Hartree and Fock, according to which f1 ð~ x1 Þ f2 ð~ x1 Þ f ð~ x Þ f2 ð~ x2 Þ 1 1 2 Fð~ x1 ; ~ x2 ; . . . ; ~ xN Þ ¼ pﬃﬃﬃﬃﬃﬃ . . .. N! .. f1 ð~ xN Þ f2 ð~ xN Þ

... ... .. . ...

fN ð~ x1 Þ fN ð~ x2 Þ , .. . fN ð~ xN Þ

(3.15)

24

Chapter 3. Many-Body Properties

where each electron occupies one orbital. Minimizing the expectation value of equation (3.14) by varying the orbital wavefunctions fk leads to the Hartree–Fock equations F^ fk ¼ k fk ,

(3.16)

where F^ ¼ h^1 þ

N X

ðJ^ i K^ i Þ

(3.17)

i¼1

is a sum of a single-electron operator M _2 2 X Z l e2 h^1 ¼ r þ , ~l j 2m rR l¼1 j~

(3.18)

that is independent of the solutions ffk ; k ¼ 1; 2; . . . ; Ng; and two-electron operators Z

e2 f ð~ x1 Þfk ð~ x2 Þ d~ x2 j~ r2 ~ r1 j i Z e2 f ð~ J^ i fk ð~ x1 Þ ¼ fi ð~ x2 Þ x1 Þfi ð~ x2 Þ d~ x2 , j~ r2 ~ r1 j k x1 Þ ¼ K^ i fk ð~

fi ð~ x2 Þ

ð3:19Þ

that do depend on the solutions to the Hartree–Fock equations, (3.16). As a further approximation, one uses the approach of Roothaan [3], i.e., one expands the single-electron orbitals fl ð~ xÞ in a set of pre-chosen, ﬁxed, basis functions, fl ð~ xÞ ¼

Nb X

wp ð~ xÞcpl ,

(3.20)

p¼1

where only the expansion coefﬁcients cpl are optimized. These are determined through the secular equation F cl ¼ l O cl ,

(3.21)

where F contains the Fock matrix elements F pm ¼ hwp jh^1 jwm i þ

Nb N X X

cni cqi ½hwp wq jh^2 jwm wn i hwq wp jh^2 jwm wn i

(3.22)

i¼1 n;q¼1

and O contains the overlap matrix elements Opm ¼ hwp jwm i.

(3.23)

3.3. Density-functional theory approaches

25

We have here deﬁned h^2 ð~ x1 ; ~ x2 Þ ¼

e2 . j~ r1 ~ r2 j

(3.24)

The Hartree–Fock(–Roothaan) approach neglects correlation effects, i.e., any electron moves in the mean ﬁeld of all the others. Going beyond that, i.e., including correlation effects, requires that the many-electron wavefunction is written as a superposition of many Slater determinants, and not only the single one corresponding to the ground state of equation (3.15). This is, however, beyond the scope of the present work, as it hardly has been applied to the systems of our interest.

3.3.

Density-functional theory approaches

An alternative to solving the electronic Schro¨dinger equation, (3.12), more or less accurately has been provided by Hohenberg, Kohn, and Sham [4,5]. Hohenberg and Kohn [4] showed that, e.g., E e of the ground state, E ð0Þ e ; is a functional of the electron density of the ground state, rð0Þ ð~ rÞ; ð0Þ ð0Þ E ð0Þ rÞ, e ¼ E e ½r ð~

(3.25)

although they were not able to give the precise form for the functional. Kohn and Sham [5] showed subsequently that useful approaches could be obtained by separating the total energy into four contributions, ð0Þ E ð0Þ e ¼ E k þ E ext þ E C þ E xc ,

(3.26)

where the ﬁrst term is the kinetic energy for the electrons when assuming that they are non-interacting, the second term is the energy due to the external interactions (which in most cases will be the Coulomb interactions of the nuclei, but which also may, e.g., contain contributions from external electrostatic potentials), the third term is the classical Coulomb energy for the interactions of the electrons with themselves, and, ﬁnally, the fourth term is the so-called exchange-correlation energy that contains everything that is not included in the ﬁrst three terms. This term is often written in the form Z E xc ¼

xc ð~ rÞrð~ rÞ d~ r,

(3.27)

where xc often is approximated through some function of the electron density and (some of) its gradients at the point of interest, ~ rÞj; r2 rð~ xc ð~ rÞ ¼ xc ½rð~ rÞ; jrrð~ rÞ; . . ..

(3.28)

26

Chapter 3. Many-Body Properties

Furthermore, even spin-polarized approximations exist, where dependence on both the spin-up and spin-down densities are included. In addition to the real system, Kohn and Sham [5] considered also a model system of non-interacting particles with the same total energy and density as the electrons of the real system, and for this system they calculated the orbitals by solving the so-called Kohn–Sham equations, ½h^1 þ V C ð~ rÞ þ V xc ð~ rÞfi ð~ rÞ ¼ i fi ð~ rÞ,

(3.29)

with h^1 being the same as in the Hartree–Fock approach, i.e., without external ﬁelds it is given through equation (3.18). Moreover, V C and V xc are the potentials corresponding to E C and E xc above, respectively. Thus, X K^ i , rÞ ¼ (3.30) V C ð~ i

with K^ i given in equation (3.19) and the sum running over all occupied orbitals, and V xc ð~ rÞ ¼

d E xc . drð~ rÞ

(3.31)

As above, the solutions, fi ; are expanded in a basis set, and the task of solving the Kohn–Sham equations becomes that of solving a matrix–eigenvalue problem, heff cl ¼ l O cl ,

(3.32)

where heff contains the matrix elements ðheff Þpm ¼ hwp jh^1 þ V C þ V xc jwm i

(3.33)

and O contains the overlap matrix elements Opm ¼ hwp jwm i.

(3.34)

Accordingly, just as for the Hartree–Fock–Roothaan approach, the matrices depend on the solutions to the eigenvalue equations and, therefore, in both cases one solves the equations iteratively. The Hartree–Fock–Roothaan and the Kohn–Sham equations are very similar. Formally, they have identical contributions to the kinetic energy and from the external potential as well as from the Coulomb energy of the electrons. But they differ in the treatment of exchange and correlation interactions. In the Kohn–Sham approach both parts are included, and they are treated via an approximate treatment, whereas in the Hartree–Fock–Roothaan approach only exchange interactions are included that, moreover, are treated exactly. Nevertheless, experience has shown that for many purposes the outcome of the two approaches are similar, although there are fundamental differences that sometimes are very important. One of those is that the Hartree–Fock approximation is not able to

3.4. Single-particle models

27

describe metallic situations but will always ultimately lead to a vanishing density of states at the Fermi level. We shall here, however, not discuss these issues further. Finally, the Hohenberg–Kohn–Sham [4,5] approach is in principle exact, but, unfortunately, it has not been possible to obtain exact expressions for E xc ½rð~ rÞ or for V xc that, accordingly, have to been approximated. There exist various approximations to those, including the local-density, the local-spin-density, and the generalized-gradient approximations. The interested reader is referred to other works where the similarities, foundations, differences, and performances of those have been discussed in detail, see, e.g., [6–8].

3.4.

Single-particle models

Independent of whether we use a Hartree–Fock–Roothaan or a Kohn–Sham approach, we end up with a matrix eigenvalue problem H ci ¼ i O ci ,

(3.35)

where O contains overlap matrix elements, whereas H contains matrix elements between the basis functions and the single-particle ‘energy’ operator that may be either the Fock or the Kohn–Sham operators. The important point is that in all cases H depends on the complete electronic distribution, either in the Hartree–Fock case, in form of all the occupied electronic orbitals or in the Kohn–Sham case, on the total electron density. This means that the matrix element ^ mi H lm ¼ hwl jHjw

(3.36)

depends not only on wl and wm as well as their relative positions, but also on all other orbitals in the system of our interest. However, in many cases it may be considered a useful approximation to assume that H lm depends only on wl and wm and their relative positions, but not on the other orbitals, at least as long as one consider ‘similar’ situations. Thus, for the two basis functions being centered on two atoms, A and B, that are assumed to be close to each other, H lm takes the same value as long as the A–B pair is embedded into similar surroundings. Moreover, it may be assumed that H lm is independent of the precise electronic distribution around the A–B pair, so that even if some electron is being excited the value of the matrix element does not change. With these approximations, the secular equation, (3.35), is independent of its solutions and can, therefore, be solved in a non-iterative way. This simpliﬁes the solution of the electronic Schro¨dinger equation considerably. The next step is to assume that the basis functions fwk g constitute a complete set of orthonormal basis functions, at least for the problem of interest. Then the overlap matrix O becomes the unit matrix and, furthermore, the Hamilton operator H^ can be written as H^ ¼

X lm

H lm c^yl c^m ,

(3.37)

28

Chapter 3. Many-Body Properties

where c^yl and c^m are the creation and annihilation operators for the basis functions wl and wm ; respectively, that obey c^m jwk i ¼ dk;m j0i, c^yl j0i ¼ jwl i.

ð3:38Þ

Here, j0i is the vacuum state. With these approximations we arrive at electronic Hamilton operators like those discussed in the preceding section, except that there we used tlm for H lm :

3.5.

Many-particle models

The next step is to realize that it may be a too inaccurate approximation to assume that H lm is completely independent of the distribution of all the other electrons. In the most general case we will modify the Hamilton operator of equation (3.37) to H^ ¼

X

tlm c^yl c^m þ

lm

X

V klmn c^yk c^yl c^m c^n .

(3.39)

klmn

Here, tlm contains only matrix elements for the single-particle operator, e.g., for h^1 of equation (3.18), tlm ¼ hwl jh^1 jwm i,

(3.40)

whereas V klmn ¼ hwk wl jh^2 jwm wn i,

(3.41)

h^1 and h^2 may be the operators from the Hartree–Fock–Roothaan or the Kohn– Sham theory above, but may also differ from those. Ultimately, this is less important as all information about those is assumed to be contained in the matrix elements of equations (3.40) and (3.41).). More important is that it is assumed that the basis functions fwk g are supposed to be orthonormal and localized so that the matrix elements tlm and V klmn are non-vanishing only when all the involved basis functions are sitting on sites that are close to each other.

3.6.

The Hubbard and the extended Hubbard models for a chain

In order to illustrate the consequences of including many-body effects, i.e., the parameters V klmn above, we shall study a chain containing identical atoms and one orbital per spin direction and atom. Accordingly, we study a system similar to the one we have discussed in [Section 2.1], but extended with many-body effects. We shall, furthermore, limit ourselves to models related to the celebrated one of Hubbard [9]. Thus, the above orbitals can be characterized through the atom and the spin direction, so that we will let wk" and wk# be the two orbitals for the kth site

29

3.6. The Hubbard and the extended Hubbard models for a chain

and differing in the spin direction. The operators n^ k" ¼ c^yk" c^k" , n^ k# ¼ c^yk# c^k# , n^ k ¼ n^ k" þ n^ k# ,

ð3:42Þ

give the number of electrons with spin-up, spin-down, and in total for the kth atom, respectively. Thus U n^ k" n^ k#

(3.43)

describes the energy for adding one electron (for instance with spin-up) to the site k when there already is another electron with the opposite spin (i.e., in the example this electron has spin-down). This energy is denoted U and is accordingly a measure for the strength of the electronic repulsion. We will expect this energy to be the largest one related to the inter-electronic interactions. However, we may also ascribe the process of adding an electron to a site k if there are electrons on the neighbouring sites. In general, we introduce operators like X X Vl (3.44) n^ k n^ kþl . l

k

Keeping only Ua0 leads to the Hubbard model, whereas setting also V 1 ¼ V 1 V a0 and all other V l ¼ 0; la 1 results in the extended Hubbard model. However, further terms can also be included at the cost of computational complexity and the risk of decreased insight, but with the gain of increased ﬂexibility. Some simpliﬁcations are due to Ohno [10] Vl ¼

U ½1 þ ðd l =a0 Þ2 1=2

,

(3.45)

as well as due to Mataga and Nishimoto [11] Vl ¼

U 1 þ ðd l =a0 Þ

(3.46)

where d l is the distance to the lth nearest neighbour, and a0 is a constant. Staying within the (extended) Hubbard model we shall study the model at the beginning of the preceding section. Thus, we generalize H^ p of equation (2.1) to H^ p ¼

XX s

k

tk;kþ1 ð^cykþ1;s c^k;s þ c^yk;s c^kþ1;s Þ þ U

X k

n^ k" n^ k# þ V

X

n^ k n^ kþ1 ,

k

(3.47) where the s summation is over the two spin directions. Compared with equation (2.1) we have set the on-site energies equal to 0, and we shall moreover assume that

30

Chapter 3. Many-Body Properties

the hopping integrals alternate as in equation (2.2). Finally, we shall add a harmonic term to account for the remaining parts of the total energy that are not included in H^ p ; just as in equations (2.4) and (2.5). In the preceding section we gave a simple example for the Peierls distortion, where a linear chain with one orbital and electron per atom was found to be unstable against a distortion resulting in an alternation of the bond lengths. We also saw that modiﬁcations of the model, still within the independent-particle approximation, could modify this conclusions, for instance through the presence of more orbitals per atom, the interactions with a substrate or a surrounding medium, or the presence of more chains. Also the inclusion of many-body effects may inﬂuence the occurrence of a bondlength alternation. Naively one may suggest that since the parameters U and V quantify the energy costs for localizing the electrons to nearest neighbours, the inclusion of these terms will favour situations with delocalized electrons, e.g., those that occur for structures without bond-length alternation. We shall here consider this issue within the extended-Hubbard model and show in Figure 3.1 results from model calculations for a closed ring with 10 sites [12]. Since it is no longer possible

0.08 V=0 V=1 V=2 V=3

0.07

|ΔE (δ)|

0.06

0.05

0.04

0.03

0.02

0

1

2

3

4 U/t0

5

6

7

8

Figure 3.1. The electronic energy difference between a dimerized and an undimerized periodic ring of 10 sites as a function of the Hubbard parameters U and V. U and V are given in units of t0 : Reproduced with permission of Springer-Verlag from Ref. [12].

3.7. The Luttinger liquid

31

to derive closed analytical results even for the simplest models, we have to restore to numerical calculations. The importance of the many-body interactions can be quantiﬁed through the parameter U=t0 and whether a bond-length alternation is favoured can be described through a parameter that gives the electronic energy gain upon such an alternation (i.e., only the contribution from H^ p is considered). Such a parameter, denoted jDEðdÞj; is shown in Figure 3.1 for various values of U and V. The scenario we discussed in the previous section corresponds to U ¼ V ¼ 0; i.e., according to Figure 3.1 to jDEðdÞj ’ 0:05: The ﬁgure shows, accordingly, that the presence of many-body interactions enhances the bond-length alternation, at least for V

U , 2

(3.48)

and for U=t0 not being very large. U=t0 can become very large if the units are essentially non-interacting, which can be the case if the units are (charged or neutral) molecules, so that we are considering a chain of molecules. In that case many-body effects become dominating and other phases (e.g., superconducting, magnetic, incommensurate, . . .) may compete with the occurrence of a bond-length alternation. Later in this report we shall see some examples of this.

3.7.

The Luttinger liquid

Whenever materials properties are studied experimentally, one perturbs the system of interest somehow and measure the response of the system. Through knowledge of the perturbation and the response, information of the properties of the system, which is sought, is obtained. A very important class of perturbations is that of excitations. And here there are fundamental differences between excitations of interacting electrons in one dimension and those of interacting electrons in two and three dimensions. Without many-body effects the excitations would involve moving an electron from an occupied orbital to an unoccupied one, and since the other orbitals are not affected by this redistribution, the excitation energy would essentially be the energy difference of the two orbitals. However, including many-body effects all orbitals would, in principle, be affected by an excitation like the one above, and the excitation energy should accordingly include contributions both from the orbital energy difference and from the relaxations of all the other orbitals. Nevertheless, in two and three dimensions the well-known Fermi-liquid theory shows that the excitation energies have sharp features when the excitation energies are equal to the orbital-energy differences, as long as these are not too large. For larger energies the features get more and more smeared out. A simple picture may be invoked to rationalize this result: in two and three dimensions the electrons can move around each other without coming in conﬂict with the Pauli principle. This situation is different in one dimension. Here, the electrons are conﬁned to move along the chain and cannot surpass each other. Instead of having excitations

32

Chapter 3. Many-Body Properties

Figure 3.2. Cartoon showing the consequences of photoexcitation of a chain of atoms with electrons of alternating spin directions.

involving the quasiparticles above, which possess the main characteristics of electrons (i.e., they have both charge as well as spin ¼ 12), new types of excitations occur. In order to study this problem, one often considers the so-called Luttinger model [13–20] which we will not discuss here in mathematical details. Instead we shall brieﬂy discuss the main results of the analysis of the model. In Figure 3.2 we show schematically a chain of atoms with one electron per atom. The electrons are supposed to have alternating spin. Figure 3.2(a) shows the arrangement of the electrons for the ground state. Upon photoexcitation one may remove one electron, which gives rise to a situation as in Figure 3.2(b). We see that we have a vacancy where both spin and charge of the electron is missing. This vacancy may propagate along the chain and, accordingly, lead to a situation as in Figure 3.2(c). We see here that the spin of the vacancy in Figure 3.2(b) has moved in one direction (to the left), whereas the charge has moved in the opposite direction. Thus, spin and charge have been separated and we are left with some sort of quasiparticles that either carry spin but no charge or carry charge but no spin. These quasiparticles are called spinons and holons, respectively, and they may propagate with different speeds. It can, moreover, be shown that these spinons and holons are bosons, and not fermions like the electrons. This charge–spin separation is special for one-dimensional systems and is a characteristics of the excitations of a Luttinger liquid. However, it is not possible directly to see the holons and spinons. Therefore, in order to explore whether a Luttinger liquid exists or not, other, more or less direct, experimental methods have to be employed. To this end, momentum- and energyresolved spectroscopy is the most appropriate choice. Then, information on the spectral function rðq; oÞ is obtained. rðq; oÞ describes the intensity with which the system of interest can absorb a momentum and energy transfer of q and o;

3.7. The Luttinger liquid

33

Figure 3.3. The band structures of Figure 2.3 with different energy- and momentum-dependent excitations. In the right panel we show only vertical transitions (momentum transfer q ¼ 0), and in the left panel only non-vertical transitions (qa0). Notice that the details of the band structures are not important, so that vertical transitions are not restricted to systems with a band gap!

respectively, and it can be probed, e.g., with angle-resolved photo-electron spectroscopy when the sample consists of ordered sub-systems. For an independent-particle system the band structures may be like those of Figure 2.3, repeated in Figure 3.3. Possible excitations are those that connect an occupied band orbital with an unoccupied one, as shown in Figure 3.3. rðq; oÞ will then have peaks if particularly many occupied and unoccupied orbitals differ exactly by Dk ¼ q and D ¼ o; where Dk and D are the difference in k and energy, respectively, between the occupied and unoccupied orbitals. This is the case when ﬂat parts of the bands are connected, e.g., for k ¼ q ¼ 0 for both panels in the ﬁgure and also for k ¼ p; q ¼ 0 in the right panel. Notice that for all those arguments we have neglected matrix-element effects. When the particles are interacting they are no longer independent, and it is not possible to excite the individual particles. In that case it has been shown [20] that rðq; oÞ ðo vs qÞa1=2 jo vr qjða1Þ=2 ðo þ vr qÞa=2 ,

(3.49)

where vs and vr are parameters describing the spinons and holons, respectively. It may be noted that a is a constant that depends on the strength of the interactions between the particles. The derivation of equation (3.49) is far from trivial and shall

34

Chapter 3. Many-Body Properties

ρ

α=0.125

ρ

2

α=1.5

500

0.000 −0.001 −0.002 −0.003 −0.004 q −0.005 0.010

0 −0.2 −0.4

1

x 10

0 −0.010

−0.005

0.000

ω

0.005

0

−0.6 −1

−0.5

q

−0.8 0

ω

0.5

1

−1

Figure 3.4. The spectral function rðq; oÞ of a Luttinger liquid for two different values of the constant a: The interactions between the particles is stronger for larger a: Reproduced with permission from Ref. [21].

therefore not be repeated here. Instead we want to stress the most important prediction of this equation: that rðq; oÞ ! 0 as some power for q; o ! 0: This prediction is illustrated in Figure 3.4. Thus, for a Luttinger liquid we will expect a vanishing signal for vanishing energy and momentum transfer and, moreover, that the signal goes to 0 as some power of momentum and energy transfer. This is in marked contrast to the ﬁndings for a conventional two- or three-dimensional metal and have, accordingly, been sought to be observed in several studies devoted to proving the existence of Luttinger liquids. We will return to this issue when we will discuss experimental studies on real materials.

3.8.

Conclusions

Electrons are charged particles and will, therefore, interact with each other. In many cases these interactions can be treated in an averaged way in which case single-particle models can be adequate. However, a detailed description of the materials properties requires an accurate description of these interactions. The starting point will then be parameter-free electronic-structure methods, as we have brieﬂy outlined in this section. From those one may derive simpler models, either without or with the inclusion of many-body effects. Whether the many-body interactions are crucial or not, can in some cases be discussed. We saw that the symmetry-lowering (i.e., Peierls distortion) that was discussed in the previous section, could quantitatively be modiﬁed through their inclusion, although the qualitative picture remained. However, in other cases, completely new phenomena showed up due to the many-body interactions. We demonstrated this in the present section through the occurrence of Luttinger-liquid behaviour with properties in marked difference to those of the more conventional Fermi-liquid theory that is relevant for quasi-two- and three-dimensional systems. Ultimately, only accurate calculations or experiment on real systems can tell whether single- or many-body effects are dominating. Such studies are, moreover, the only way to identify (quasi-)one-dimensional behaviours and to answer the question, whether metallic chains or chains of metals really exist. Therefore, we shall now slowly turn our attention to the studies of real(istic) systems.

References

35

References [1] T. Giamarchi, Chem. Rev. 104, 5037 (2004). [2] M. Springborg, Methods of Electronic-Structure Calculations (Wiley, Chichester, UK, 2000). [3] C.C.J. Roothaan, Rev. Mod. Phys. 23, 69 (1951). [4] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). [5] W. Kohn and L.J. Sham, Phys. Rev. 140, A1133 (1965). [6] M. Springborg, in Specialist Periodical Reports: Chemical Modeling, Applications and Theory, Vol. 1, ed. A. Hinchliffe, p. 306 (Royal Society of Chemistry, Cambridge, UK, 2000). [7] M. Springborg, in Specialist Periodical Reports: Chemical Modeling, Applications and Theory, Vol. 2, ed. A. Hinchliffe, p. 96 (Royal Society of Chemistry, Cambridge, UK, 2002). [8] M. Springborg, in Specialist Periodical Reports: Chemical Modeling, Applications and Theory, Vol. 3, ed. A. Hinchliffe, p. 69 (Royal Society of Chemistry, Cambridge, UK, 2004). [9] J. Hubbard, Proc. Roy. Soc. (London) A276, 238 (1963). [10] K. Ohno, Theor. Chim. Acta 2, 219 (1964). [11] N. Mataga and K. Nishimoto, Z. Phys. Chem. 13, 140 (1957). [12] D. Baeriswyl, D.K. Campbell, and S. Mazumdar, in Conducting Polymers, ed. H. Kiess, p. 7 (Springer, Heidelberg, 1992). [13] J. So´lyom, Adv. Phys. 28, 201 (1979). [14] S. Tomonaga, Prog. Theor. Phys. 5, 544 (1950). [15] J.M. Luttinger, J. Math. Phys. 4, 1154 (1963). [16] D.C. Mattis and E.H. Lieb, J. Math. Phys. 6, 304 (1963). [17] I.E. Dzyaloshinkiıˇ and A.I. Larkin, Sov. Phys. JETP 38, 202 (1974). [18] F.D.M. Haldane, Phys. Rev. Lett. 45, 1358 (1980). [19] F.D.M. Haldane, J. Phys. C 14, 2585 (1981). [20] J. Voit, J. Phys. Cond. Matt. 5, 8305 (1993). [21] J. Voit, cond-mat/0005114.

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Chapter 4

The Jellium Model

For some metals the valence electrons are very delocalized and do not participate in the formation of directional bonds between nearest neighbours. This is the case for s metals like Na, K, and Cs, but less for Au and Ag, and certainly not the case for metals with 3d or 4f valence electrons. When the valence electrons are very delocalized, the precise positions of the nuclei become less important and one may, accordingly, as an approximation study the hypothetical case that the nuclei and the core electrons are smeared out to a homogeneous medium with a constant density, the so-called jellium, in which the valence electrons move. In the beginning of the 1980s it was shown [1–3] that this model was able to explain experimentally the observed mass abundances of sodium clusters. It was assumed that the clusters were spherical, and that the core electrons and nuclei could be approximated through a jellium that had the same density as in the inﬁnite, periodic crystal. Then, the solutions to the Hartree–Fock or the Kohn–Sham equations can be written as a radial part Rnl ðrÞ times an angular part given by harmonic functions Y lm ðy; fÞ: The radial parts were found by numerically solving the resulting one-dimensional Kohn–Sham equations, _2 2 lðl þ 1Þ r þ þ V ðrÞ Rnl ðrÞ ¼ nl Rnl ðrÞ, r2 2m

(4.1)

where V ðrÞ is the potential that includes both the Coulomb potential from the jellium, that from the electrons, and the exchange-correlation potential from the electrons. It was found that the experimental mass abundance spectra could be explained through an electronic shell-ﬁlling effect, i.e., whenever the jellium sphere (i.e., the cluster) was so large that all ðn; lÞ shells were completely ﬁlled or completely empty, then the cluster was particularly stable. Later, these studies have been extended to explain many other properties as well as generalized to structures of lower symmetry (see, e.g., [4,5] and references therein), but it shall be stressed that it is not always accurate. For instance, when the electrons cannot be considered completely delocalized but are localized to the atoms or to nearest-neighbour bonds, the approximation may fail. Second, particular stability may also be a result of packing effects, i.e., if one may form highly symmetric structures with N closed-packed atoms, these structures may be particularly stable, which often is the case for structures containing 13 atoms (i.e., an icosahedron) or 55 atoms (an icosahedron, a tetrahedron, or a cuboctahedron). 37

38

4.1.

Chapter 4. The Jellium Model

Chains of jellium

Similarly, for certain chains of metals, the jellium model may also provide a ﬁrst useful insight into their properties. Thus, assuming that these systems have an overall cylindrical shape, one might speculate that particular stability would result if a gap between occupied and unoccupied orbitals could be created, or at least if the density of states at the Fermi level is reduced. Zabala et al. [6] studied this proposal using a special version of the above-mentioned jellium model, called the stabilized jellium model. Owing to the translational symmetry along the chain axis as well as the rotational symmetry about it, the electronic orbitals may be written as eikz z eimf csmnkz ðr; f; zÞ ¼ pﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ Rsmn ðrÞ, L 2p

(4.2)

where L is the (very large) length of the chain, and s the spin. Accordingly, we also allow for a spin polarization, i.e., the case that the spin-up and the spin-down electrons have different orbitals and energies. A ﬁrst issue is whether there exists diameters for which the chains are particularly stable, in similarity with the above-mentioned results for clusters. For the clusters we have mentioned that it was found that particular stability was related to the closing of electronic shells. This means that studying the energy of the highest occupied orbital, or the Fermi energy, would directly give information about particularly stable structures: for those the Fermi energy would be discontinuous as a function of size. In Figure 4.1 we show the Fermi energy as well as the energy of the different bands [i.e., the lowest band energies in equation (4.2) for different values of ðm; nÞ] as a function of radius of the chain for Na. Also here, it is clearly seen that the Fermi energy has local maxima whenever a new band starts becoming occupied, but in contrast to the case for the clusters, the Fermi energy is here a continuous function of size of the system. The reason is that compared with the ﬁnite clusters, where we ﬁnd discrete electronic levels, the chains are inﬁnite and, therefore, the electronic levels are broadened into bands that may even overlap. We also notice that the calculations predict that some of the structures are magnetic, which actually happens for diameters slightly larger than those for which a new band has started being occupied. This suggests that the system attempts to lower its energy by pushing one of the two bands (for the different spin directions) above the Fermi energy and, thereby, lowering the density of states at the Fermi energy. The parameter rs given in Figure 4.1 is the so-called electron-gas parameter. It gives the radius of a sphere containing exactly one electron for the inﬁnite, periodic crystal. Thus with the average valence-electron density in the crystal denoted r; ¯ rs is given by r¯ ¼

3 . 4pr3s

(4.3)

Zabala et al. [6] studied also the surface energy for the cylindrical wires as a function of their radii and considered, moreover, the three metals Al, Na, and Cs.

39

4.1. Chains of jellium

−1 rs=3.93 a0

Subband bottom energy (eV)

Fermi level −2

(1,1) (2,1) (3,1) (1,2)

(0,2)

(4,1)

(2,2)

−3 (0,1) −4

−5 (lml,n) −6

2

4

6

8

10

12

14

Wire radius R (a0) Figure 4.1. Energy eigenvalues of a Na wire as a function of radius of the wire. The thick curve shows the Fermi level, and the thinner curves show the bottom of the different bands in the case of lacking spin polarization. The dark circles mark results for the systems where the spin polarization leads to a lowering of the total energy. Reproduced with permission of American Physical Society from Ref. [6].

The surface energy is given through 1 3 2 E tot ½r" ; r# =L pR r¯ F ðrÞ , sðRÞ ¼ ¯ þ xc ðrÞ ¯ 2pR 5

(4.4)

where E tot ½r" ; r# the total energy of the system, F ðrÞ ¯ the Fermi energy relative to the energy of the lowest occupied orbital of the inﬁnite, periodic, homogeneous electron gas with a density of r; ¯ and xc ðrÞ ¯ the exchange-correlation energy density for this system. This quantity is shown in the left part of Figure 4.2. It is seen that this shows regular, damped oscillations. Moreover, it has minima for radii that are slightly smaller than those for which new bands start becoming occupied in Figure 4.1, once again conﬁrming that electronic-shell effects determine the stability of these wires. Finally, Zabala et al. [6] also studied the force for elongating the wires, F ¼

dE dsðRÞ ¼ pRsðRÞ þ pR2 . dL dR

(4.5)

This quantity is shown in Figure 4.2, too, and this also shows a clear correlation between band ﬁlling (and, hence, stability) and force. Yannouleas and Landman [7] studied the conductance through such a jellium wire of constant, ﬁnite radius. Figure 4.3 shows their results for a chain of Na. The oscillating behaviour of the Fermi energy as a function of radius of the wire, cf.

40

Chapter 4. The Jellium Model 4 A I rs=2.07 a0

35 30 25 20

(a)

Na rs=3.93 a0

Planar surface

1

1.5

(b)

2 2.5 R / rs

3

3.5

0.8 0.6 0.4 0.2 Planar 0 −0.2 −0.4 −0.6 −0.8 0.5 1

Na rs=3.93 a0

1.5

0.2 0.1

Cs rs=5.62 a0

1.8

Force (nN)

Surface energy (meV/a02)

(c)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 R / rs

(b)

2

1.6 1.4 1.2 Planar surface

1

1.5

2 2.5 R / rs

3

(c)

2.5

3

3.5

3

3.5

Cs rs=5.62 a0 planar

0

−0.2

0.5

3.5

2 R / rs

−0.1 −0.3 −0.4

1 0.8 0.5

−2 −6

Force (nN)

Surface energy (meV/a02)

10 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 R / rs 6 5.5 5 4.5 4 3.5 3 2.5 0.5

0

−4

15 Planar surface

(a)

A I rs=2.07 a0

2 Planar Force (nN)

Surface energy (meV/a02)

40

1

1.5

2 R / rs

2.5

Figure 4.2. (Left part) Surface energy for (a) Al, (b) Na, and (c) Cs chains as a function of wire radius and (right part) the force for elongation the wires also as a function of wire radius. The dark circles mark results for the systems where the spin polarization leads to a lowering of the total energy. For the sake of reference, the values for the planar surface (corresponding to an inﬁnite wire radius) are shown, too. Reproduced with permission of American Physical Society from Ref. [6].

Figure 4.3(a), is a manifestation of the increased number of bands that become (partially) occupied the thicker the wire is. In parallel with this also an increased number of bands cross the Fermi level, and, accordingly, also the conductance increases as a function of wire radius, cf. Figure 4.3(b) and the discussion of Section 2.4. Brandbyge et al. [9] have studied the conductance through a narrow constriction in an otherwise extended nanowire and, in particular, examined the conductance quantization as a function of the width and shape of the narrowest parts of the junction. We shall not discuss their work further here but add, however, that it has been argued [10] that for ﬁnite chains suspended between two junctions/tips one has to include the discrete atomic structure in order to obtain a qualitatively and quantitatively correct description of conduction as a function of applied voltage. Thus, it is obvious that the jellium model has limitations.

41

4.1. Chains of jellium

(a) εF (eV)

−2.3 −2.5 −2.7 −2.9 24 G(g0)

(b) 12

0

4

12 R (a.u.)

20

Figure 4.3. Variation in (a) the Fermi energy and (b) the conductance as a function of radius of an inﬁnite Na wire described with the jellium model. The conductance is given in units of G0 of equation (2.41). Reproduced with permission of American Chemical Society from Ref. [7].

In a later paper, Ogando et al. [8] studied breaking processes of such thin wires. In many experimental studies (of which some will be discussed later) one produces a narrow junction, e.g., by etching, in a thin ﬁlm of the material to be studied. Subsequently, this junction is made increasingly thin (this can, e.g., be achieved by pulling the two parts at each side of the junction apart) whereby, ultimately, a junction that is as thin as individual atoms is obtained before the junction breaks. Various properties, including the conduction through the junction, can be measured during the breaking process. When the material of interest can be treated with a jellium approximation, one may study the breaking of the junction using the approach of Ogando et al. [8] that is schematically shown in Figure 4.4. In this study the so-called ultimate jellium model was used according to which the electron density and the background density in every single point of space is identical so that only the exchange-correlation potential is felt by the electrons. Ogando et al. considered a nanowire with cylindrical symmetry. First, they considered one for which the radius was constant. Subsequently, it was elongated and the radius, as in Figure 4.4, was allowed to vary, although it was required that the volume of the cylinder stayed constant. The deformation was conﬁned to a ﬁnite region, whereas outside this region it was required that the wire had the structure of the undisturbed wire. As above, Ogando et al. found in this case also that there are certain nanowire radii for which the nanowire is particularly stable. Considering a metal with rs ¼ 4:18a0 ; they found particularly stable wires for radii of 4:3; 7:3; 10:3; 13:6; 17:8; 20:7; . . . atomic units (a.u.). Subsequently, they considered the breaking of

42

Chapter 4. The Jellium Model

Lcell = L0 + Δ L

left

left

periodic

frozen

image

contact

UJ CONSTRICTION

l

z edge

right

right

frozen

periodic

contact

image

r

z edge

Figure 4.4. Schematic view of the model system for the breaking of a ﬁnite nanowire suspended between two leads. Reproduced with permission of American Physical Society from Ref. [8].

60

40

Z (a0)

20

0

−20 −40 −60 −20 (a)

0 20 −20 Radius (a0) (b)

0

20 −20 (c)

0

20 −20 (d)

0

20

Figure 4.5. Electron-density plots from the simulations of the breaking of a nanowire, as shown schematically in Figure 4.4. DL equals 7.9, 19.8, 20.8, and 25:8 a0 ; respectively, and the constriction contains about 8 electrons. a0 is the Bohr radius. Reproduced with permission of American Physical Society from Ref. [8].

a wire with some chosen radius. Figure 4.5 shows examples of the results, where a wire with radius of 10.7 a.u. was considered. Moreover, various properties during the pulling process are shown in Figure 4.6 for two different values of the radius of the initial wire (10.7 and 20.7 a.u.). The middle panels show an effective

4.1.

43

Chains of jellium

Conductance (G/G0)

Conductance (G/G0)

25 20 15 10

7 6 5 4 3 2 1 0

0

5

10 15 20 Elongation (a0)

25

5

10 15 20 Elongation (a0)

25

5 (a) Effective radius (a0)

12

Effective radius (a0)

20 18 16 13.6 a0

14

10 8 7.3 a0 6

2 0

12

4.3 a0

4

0

10.3 a0

10 7.3 a0

8 (b)

−0.3 −0.4

Elongation force (nN)

Elongation force (nN)

−0.2

−0.5 −0.6 −0.7 −0.8 −0.9

(c)

0

10

−0.05 −0.1 −0.15 −0.2 −0.25 −0.3 −0.35 −0.4 −0.45

0

5

20

10 15 20 Elongation (a0)

30

25

40

Elongation (a0)

Figure 4.6. Conductance, effective radius of the constriction, and elongation force for a wire of initial radius equal to 20.7 a.u. (insets: 10.0 a.u.) as functions of elongation. Reproduced with permission of American Physical Society from Ref. [8].

constriction radius at the centre of the constriction which is seen to show some tendency towards plateaus for the ‘magic radii’ mentioned above. Also the elongation force (lowest panel) is a non-regular function of elongation, once again being partly dictated by the electronic properties through the existence of the ‘magic radii’. Finally, similar behaviours are found for the conductance shown in the upper panel, i.e., during the elongation the ‘magic radii’ occur more frequently than other radii.

44

4.2.

Chapter 4. The Jellium Model

Conclusions

In total the jellium studies have shown that the properties of the nanowires may depend critically on the electronic properties. First of all, the occurrence of ‘magic radii’, i.e., radii for which the nanowires are particularly stable, is within this model a purely electronic effect. These ‘magic radii’ could be recovered also in the development of the properties for wires that were elongated until breaking. However, the jellium model can be considered only as a ﬁrst approximation: it completely neglects the precise structure of the material of interest. Therefore, after having spent the ﬁrst chapters on more or less purely theoretical issues, it is now time to turn to real materials.

References [1] W.D. Knight, K. Clemenger, W.A. de Heer, W.A. Saunders, M.Y. Chou, and M.L. Cohen, Phys. Rev. Lett. 52, 2141 (1984). [2] W. Ekardt, Phys. Rev. B 29, 1558 (1984). [3] D.E. Beck, Phys. Rev. B 30, 6935 (1984). [4] W.A. de Heer, Rev. Mod. Phys. 65, 611 (1993). [5] M. Brack, Rev. Mod. Phys. 65, 677 (1993). [6] N. Zabala, M.J. Puska, and R.M. Nieminen, Phys. Rev. B 59, 12652 (1999). [7] C. Yannouleas and U. Landman, J. Phys. Chem. B 101, 5780 (1997). [8] E. Ogando, T. Torsti, N. Zabala, and M.J. Puska, Phys. Rev. B 67, 075417 (2003). [9] M. Brandbyge, K.W. Jacobsen, and J.K. Nørskov, Phys. Rev. B 55, 2637 (1997). [10] A. Levy Yeyati, A. Martı´ n-Rodero, and F. Flores, Phys. Rev. B 56, 10369 (1997).

Chapter 5

Gold Chains: The Prototype?

In Chapters 2 and 3 we have discussed brieﬂy some (but certainly not all) general theoretical aspects related to quasi-one-dimensional systems without taking too much into considerations whether the models we were discussing had any relation to real materials. As a ﬁrst step towards realistic studies we studied in the preceding section simple metals that could be modelled through a simple jellium model where the precise positions of the atoms are of only secondary importance. But the validity of this model can also be questioned. From now on we shall consider real materials and compare experimental and theoretical studies. We shall then often compare with the results of the Chapters 2 and 3. As a starting point we shall consider chains containing only gold atoms, being one of the most studied systems in the context of quasi-one-dimensional metals.

5.1.

The structure of a linear chain of Au atoms

Two papers in 1998 [1,2] led to a large number of studies of the structure of a linear chain of gold atoms. Ohnishi et al. [1] used two different techniques in producing atomically thin wires of gold. First, they used a scanning-tunnelling microscope (STM) tip made of gold that was dipped into a small gold island and subsequently removed again, whereby a bridge of gold atoms was formed between the two parts. They measured the conductance through this bridge. Second, they used an electron beam to etch two holes in a thin gold ﬁlm. Ultimately, the two holes are separated by just an atomically thin bridge whose structure was studied. In the other work, Yanson et al. [2] studied the current through a mechanically produced break junction of gold. In this experiment a gold wire containing a very thin constriction was slowly elongated whereby in particular the thin constriction is becoming increasingly thinner until it ultimately becomes atomically thin before it breaks. Both papers found that the conductance for the thinnest wires was equal to 2e2 =h; i.e., the value of equation (2.41). Both studies suggested that the interatomic distance for the linear chain of Au ( which is considerably larger than what atoms was fairly large, about 3.5–4:0 A; ( Later, it was reported [3] that the is found for, e.g., crystalline gold, i.e., 2:88 A: Au–Au bond length reported in the work of Yanson et al. [2] was overestimated ( Nevertheless, the two reand, instead, the correct value should be around 2:6 A: ports above led to a large activity in theoretical studies of the possible reason for the 45

46

Chapter 5.

Gold Chains: The Prototype?

large bond length (see, e.g., Refs. [4–16]). Some of those [4,6,8,14] considered isolated, inﬁnite, periodic chains, whereas others were devoted to ﬁnite, isolated chains [8,13,16] and in yet other studies ﬁnite chains between two tips, modelling, e.g., a STM and a substrate or the thin junction in a breaking-junction experiment were studied [4,5,7,10–12,15]. As an example we show in Figure 5.1 the variation in the total energy as a function of bond length for an inﬁnite, periodic chain of gold atoms [8]. In this theoretical study the effects of inclusion of the (computationally more complicated) spin–orbit couplings were also studied but it is seen that the structure of the lowest total energy does not depend signiﬁcantly on their inclusion. As in all other theoretical studies, this one predicts that the lowest total energy is found for bond ( lengths around 2:6 A: In an experiment, however, the chains are being pulled. This means that the chains will be exposed to some force, being related to the slope of the total-energy curve. Thus, a chain will try to somehow react against that force. Increasing the force with which the chain is being pulled will gradually lead to an increase in the bond length as in Figure 5.1 until we reach the turn-over point where the force decreases as a function of increased bond length, i.e., the inﬂation point where d 2 E tot =dd 2 ¼ 0 (with d being the nearest-neighbour bond length). At this point, the

Figure 5.1. Total energy for an inﬁnite, periodic, linear chain of Au atoms as a function of nearestneighbour bond length both (squares) without and (triangles) with the inclusion of spin–orbit couplings. Reproduced from Ref. [8].

5.1. The structure of a linear chain of Au atoms

47

Figure 5.2. Band structures for a linear chain of Au atoms with a bond length slightly larger than that of the lowest total energy. In (a,b) periodic, inﬁnite chains are considered, whereas the orbital energies for ﬁnite chains are shown in (c). In (b) spin–orbit couplings have been included in the calculations, and in (c) N gives the number of atoms in the chain. Reproduced from Ref. [8].

wire will break. But even when taking these considerations into account, it is found ( that the chains will not be stable above a bond length of slightly more than 3 A: Moreover, as shown in Figure 5.2, for expanded chains we have a situation with exactly one band crossing the Fermi level. This suggests, as discussed in Chapter 2, that the chain will seek to lower its total energy by letting the bond lengths alternate. The calculations do, actually, support this suggestion. In addition, the orbital energies for the ﬁnite and the inﬁnite chains are so similar that this conclusion most likely is independent of the precise surroundings of the chain, i.e., whether it is ﬁnite or inﬁnite, or whether it is placed between two tips. In these studies only the static properties of a chain were considered. However, the formation of the chain may be a dynamic process which could lead to surprising structures. In Figure 5.3 we show snapshots of the structure of a gold nanowire during the breaking process. These snapshots have been obtained with the use of molecular-dynamics calculations using a parameterized tight-binding model for the calculation of orbital and total energies. This picture shows clearly that atomically thin nanowires which largely are linear are formed. Nevertheless, the surprisingly large bond length that was reported by Ohnishi et al. [1] was observed using STM and has so far not been contradicted. Therefore, it is still considered a puzzle as to how these large bond lengths could be realized. Gold is a heavy element and, therefore, it has been suggested that in the microscope you may see only gold atoms whereas other atoms, that could be present, are not resolved. Therefore, if the chain contains extra atoms the long Au interatomic distance could be explained. These extra atoms could be those of SCH3 units, carbon atoms, or hydrogen atoms [5,13,14,16]. In other studies it was suggested that

48

Chapter 5.

Gold Chains: The Prototype?

Figure 5.3. The atomic conﬁguration of a gold nanowire at selected elongation states. Reproduced with permission of the American Physical Society from Ref. [12].

the chains are not linear but have a zigzag symmetry so that only every second atom is seen in the STM image leading to an apparently too large bond length [6,9,10,14]. The ﬁnal answer to the puzzling question of the reason for the large bond lengths seen by Ohnishi et al. [1] has, however, not been give so far.

5.2.

Conduction

Chains of gold atoms, as well as of many other elements as we shall see later, have been studied in the context of quantum conductance. van Ruitenbeek et al.

49

5.2. Conduction

L u

1 2 3 4 5 Figure 5.4. Schematic top and side view of a mechanically controlled break junction (1), two ﬁxed counter supports (2), the bending beam (3), and piezo elements. By moving the support (5) upwards and the counter supports (2) downwards, the junction (4) is elongated and ultimately broken. Reproduced with permission from Ref. [17].

(see, e.g., Ref. [17]) have used the method of mechanically controlled break junction, schematically shown in Figure 5.4. From the material that is to be studied a narrow junction has been created. Subsequently, the sample is glued on a substrate so that the junction is placed between the two drops of glue. With the help of a three-point bending conﬁguration (formed by the two supports and the piezo element in Figure 5.4) the sample is bent so that the junction is elongated until it ultimately is broken. By reducing the force on the substrate the two parts of the broken junction can be brought into contact again. Through a careful mechanical set up it is possible to determine the elongation of the junction very precisely, and by simultaneously applying a voltage between the two ends and measuring the current that ﬂows through the junction, its resistance can be measured. Finally, this experiment can be repeated very many times so that also statistics can be applied to the results. In Figure 5.5 we show one example of such an experiment. At some point the junction becomes so thin that the electrons behave as being conﬁned to a onedimensional wire, although we stress that this wire may not be atomically thin. At this point, electrons moving through the wire will propagate essentially in a quasione-dimensional system so that the considerations of Chapter 2 become applicable. The resistance will be dominated by the thinnest part of the junction, and from Section 2.4 we learn that the conductance may then be given through G¼

X e2 Tr ti , p_ i

(5.1)

50

Chapter 5.

Gold Chains: The Prototype?

8 25 Return distance (Å)

7

Conductance (2e2/h)

6

5

20 15 10 5

4

0 3

4 8 12 16 Plateau length (Å)

20

2 Plateau length 1 Return distance 0 0

4

8

12 16 20 24 Electrode displacement (Å)

28

32

Figure 5.5. Conductance as a function of the displacement of the two gold electrodes. The insert shows the length of the return distance as a function of the plateau length from several measurements. Reproduced with permission from Nature from Ref. [2].

where 0 ti 1

(5.2)

describes the transmission for the various conduction channels that, when the applied voltage is small, to some extent can be related to the bands that cross the Fermi level for an inﬁnite system. The perfect channels (i.e., those for which ti ¼ 1) contribute with one quantum of conductance, G0 ¼

e2 2e2 ¼ , p_ h

(5.3)

to the conductance. The thicker the wire is, the more bands cross the Fermi level and, consequently, contribute to the total conductance, so that the conductance then is relatively large and becomes unstructured as a function of wire thickness. But for the thin wires, where only few bands cross the Fermi level, one may resolve the individual channels, in particularly when the wire is made progressively thinner and thinner, so that a decreasing number of channels contribute to the conductance. This is the effect that is seen in Figure 5.5.

51

5.2. Conduction

Number of times each length is observed

Just before breaking, at which point the conductance drops to 0, the conductance shows a long plateau at a value close to one times the quantum conductance G 0 : Comparing with the simulations of Figure 5.3 it can be suggested that this situation corresponds to that of a ﬁnite, essentially linear chain of Au atoms suspended between the two parts of the junction. We may, moreover, assume that the chain is somehow under strain so that the band structures of Figure 5.2 are realistic for this situation. Accordingly, we see that we have exactly one band crossing the Fermi level, giving an excellent explanation for the observed conductance. Once the chain/junction is broken, the two loose ends will relax so that when slowly releasing the bending force and bringing the two ends closer, they will not join each other at the same point as where the junction was broken. For example, in Figure 5.5 one can see that after the chain is broken, the two ends will tend to relax and become shorter. Therefore, the return distance in Figure 5.5 is longer than the plateau length. Yanson et al. [2] have repeated the measurements very many times. In Figure 5.6 we show their original results from roughly 100 000 measurements. Assuming that the last plateau in Figure 5.5 is due to the formation of a ﬁnite, essentially linear chain of gold atoms that upon pulling becomes increasingly long (see also Figure 5.3) the length of this plateau gives information on how many atoms this ﬁnite chain may contain. The results of Figure 5.6 show peaks that appear with some ( This spacing can be interpreted as being due to the regular spacing of about 3:6 A: formation of chains with 1; 2; 3; 4; . . . atoms before breaking and suggests, accord( However, as we have ingly, that the interatomic distance in the chains equals 3:6 A: discussed above, it has later been discovered that the length scale in Figure 5.6 (as ( bringing the well as in Figure 5.5) is wrong and, instead, the spacing is around 2:6 A;

200

x10

150

100

14

16

18

20

22

8 10 12 14 16 Length of the last plateau (Å)

18

20

22

50

0

0

2

4

6

Figure 5.6. The number of times a certain length of the last plateau was observed in the mechanically break junction experiments on gold nanowires. Reproduced with permission of Nature from Ref. [2].

52

Chapter 5.

Gold Chains: The Prototype?

observed interatomic distance close to the calculated values that were reported in Section 5.1.

5.3.

More complicated structures

In the preceding sections we discussed the properties of chains whose structure essentially was that of a linear chain. However, we also saw that, e.g., during the formation of such a chain with the break-junction technique, other structures are also formed, as can be seen in Figure 5.3. In this section we shall accordingly discuss the properties of gold chains with slightly more complicated structures. The surprisingly large bond length that was reported to be observed for the gold nanojunctions and discussed in Section 5.1 led to the suggestion that other structures than the linear one could be responsible for the observed large interatomic distance. The occurrence of zigzag, and not linear, chains was one suggestion [6,9]. In Figure 5.7 we show results from calculations on zigzag chains [6]. The ﬁgure shows that the zigzag chain indeed has a lower total energy than the linear chain. 180 α (deg)

160

a

140

α

120 100

r

80

x

60 3.0

r (Å)

2.9

b

2.8 2.7 2.6 2.5

E (eV/atom)

−2.0 −2.1

c

−2.2 −2.3 −2.4 −2.5 1.5

2.0

2.5

3.0

Wire length x (Å /atom) Figure 5.7. (a,b) Structural parameters and (c) total energy for a zigzag chain of gold atoms. In (c), the open circles show the results for a linear chain for comparison (in this case a ¼ 180 and r is undeﬁned). Reproduced with permission of American Physical Society from Ref. [6].

53

5.3. More complicated structures

However, the next nearest-neighbour distance at the optimized structure, i.e., 2 x of Figure 5.7(c), which is 4:6 A˚ at the minimum, is too large to explain the experimental results. The band structures, shown in Figure 5.8, show that the linear chain at its optimized structure indeed has more than one band crossing the Fermi level, whereas the zigzag chain has a structure with only one band crossing the Fermi level. For the linear chain we have seen above, cf. Figure 5.2, that for a slightly expanded structure only one band remains at the Fermi level, which is in accord with the results of the conductance measurements. Thus, in principle the zigzag structure may explain parts of the experimental observations. However, despite its obvious energetic preference, to our knowledge there has so far been no unambiguous observation of the zigzag structure for the gold chains. But in other experimental studies gold nanojunctions have been produced that are not atomically thin. As an example, in Figure 5.9 we show high-resolution transmission-electron-microscope pictures of gold nanowires that have been produced by focusing an electron beam on different sites of a self-supported gold ﬁlm [18]. The beam produces holes that are allowed to grow until a nanometre narrow neck (the junction) is produced. A schematic representation of the possible structures of the junctions is shown in Figure 5.10. Subsequently, the conductance was measured, but this shall not be discussed further here. The fact that for wires that have a cross section of only few atoms, the individual atoms are important can be seen not only in the mechanical properties, but also in the electronic properties. For the latter, in particular the conductance depends ﬁrst

a

6

b

4

Energy (eV)

2

0 −2 −4 −6 −8

Γ

X

Γ

X

Figure 5.8. Band structures for the optimized (a) linear and (b) zigzag chain of gold atoms. Reproduced with permission of American Physical Society from Ref. [6].

54

Chapter 5.

Gold Chains: The Prototype?

Figure 5.9. High-resolution transmission-electron-microscope images of gold nanowires. The atomic positions appear as dark. (a), (b): nanowires along the [100] and [110] directions, respectively; (c)–(e): the temporal evaluation of a nanowire is shown. Reproduced with permission of American Physical Society from Ref. [18].

of all on the thinnest part of the junction, but for mechanical properties the complete arrangement of all atoms making up the junction is important. Maybe most clearly this can be seen in molecular-dynamics simulations of which we in Figure 5.11 show one example [19]. A junction formed by ﬁve layers of alternatingly 24 and 25 atoms are been elongated slowly. The initial situation is shown as A in Figure 5.11(b). Upon stretching, the interlayer distance is increased which only can be accomplished through the application of a growing force, cf. Figure 5.11(a). At some point, however, a part of this force is released by a structural rearrangement of the atoms which ultimately leads to the occurrence of an extra layer. This extra layer is most easily seen in Figure 5.11(c) as the sudden occurrence of not ﬁve but six roughly straight lines. These six layers can also be seen in B in Figure 5.11(b). This process is continued until, at a tensile strain of slightly more than 1.3, the junction is broken. That it is broken can easily be seen in Figure 5.11(c). But it is very clear that the atomic rearrangements can be recognized in the tensile-force curve of Figure 5.11(a), which, in principle, can be measured. The chain-like structures we have considered so far are all ones that were produced by creating a narrow junction that ultimately was made atomically thin.

55

5.3. More complicated structures

a

2nd layer

1st layer

[111]

Top view

[100]

b

Top view

3rd layer

α = 63°

α = 54.7°

[211]

[110] α = 90°

α = 70.5°

[010]

[011]

[110]

[110]

8/6 [110]

4/3 or 2/2 rod

c

4/3

2/2

Figure 5.10. Schematic representation of possible atomic arrangements of gold nanowires along the (a) [111], (b) [100], and (c) [110] direction. Both top and side views are shown. Reproduced with permission of American Physical Society from Ref. [18].

However, there may also exist other quasi-one-dimensional structures made of only gold. In particular several theoretical studies have been devoted to this issue and here we shall now discuss a few of those studies. Bilalbegovic´ [20–22] has used molecular-dynamics calculations in predicting that multi-shell nanowires may exist. A snapshot of a such is shown in Figure 5.12, both from the side and as a cross section. It is seen that it consists of essentially concentric shells of atoms. Experimental results of Kondo and Takayanagi [23] have suggested that more such thick gold nanowires may exist. Related to the predictions of the previous section, where we mentioned that – as for metal clusters – there exists special sizes (i.e., radii) for which the system is particularly stable, Kondo and Takayanagi analysed transmission-electron-microscope images and found that these showed

56

Chapter 5.

Gold Chains: The Prototype?

B

A

C

40

(b) A

20

2.5 B 10

2.0 C

1.5 1.0

0

0.5 −10 (a)

0 0

0.5 1 Tensile Strain

1.5 (c)

0.2 0.4 0.6 0.8 1 1.2 1.4 Tensile Strain

Atomic Position (nm)

Tensile Force (nN)

30

0.0

Figure 5.11. Results from a molecular-dynamics calculation of the effects of the elongation of a gold nanojunction. (a) The tensile force as a function of the tensile strain; and (b) Three snapshots of the atomic arrangements for the structures marked in (a); (c) The variation of the atomic coordinate along the strain direction as a function of the strain. Reproduced with permission of The Physical Society of Japan from Ref. [19].

Figure 5.12. (a) Side and (b) top view of a nanowire of 588 gold atoms with a length of 4 nm and an average radius of 0.9 nm. The trajectories show the movements of the atoms over a time span of approximately 7 ps. Reproduced with permission of the American Physical Society from Ref. [20].

5.3. More complicated structures

57

Figure 5.13. (A) A part of a triangular network sheet that is rolled up to an ðn ¼ 7Þ-membered tube, so that the upper and lower row of atoms are identical. Depending on the way this is done, different helical nanotubes, characterized by two indices ðn; mÞ; can be generated, that ultimately can be characterized by a shear strain : (B) The translational period as a function of and n: Reproduced with permission of the American Association for the Advancement of Science from Ref. [23].

that various helical gold nanowires were formed in their experiments. In those, they used electron beams to create holes in a thin gold ﬁlm until the holes were separated through a thin junction. By stretching the ﬁlm the junction becomes almost atomically thin, and this junction is then studied with the transmission-electron microscope. In order to interpret their results they studied the construction of Figure 5.13. This shows a triangular network that can be rolled up to form a tube. Several different such tubes can be created, whose strain energy increases as the radius of the tube decreases. Finally, the helical symmetry may lead to a very long

58

Chapter 5.

Gold Chains: The Prototype?

Figure 5.14. Various multishell gold nanotubes, to the left as model images, and to the right as transmission-electron-microscope images. The tubes are, from the top, the 7 – 1, 11 – 4, 13 – 6, 14 – 7 – 1, and 15 – 8 – 1 ones, where the integers give the n values for the different shells, starting from the outermost one. n is described in Figure 5.13. Reproduced with permission of the American Association for the Advancement of Science from Ref. [23].

translational period, as also shown in this ﬁgure. Ultimately, by inserting more of those tubes concentric inside each other various multi-shell structures, as shown in Figure 5.14, can be generated. Ono and Hirose [24] calculated the conductance for the systems of Figure 5.14. It is interesting to notice that the conductance is not proportional to the number of

59

5.3. More complicated structures

(5,0)

(p,q)

a2

a1

(6,0) (m,n)

(0,0)

(8,4) (7,3)

Figure 5.15. Starting from a layer of gold atoms in the (111) plane, this layer is folded into a cylinder by putting the atom at the ðm; nÞ position on top of the one at the origin of the coordinate system. For different values of ðm; nÞ different nanowires result, as shown. Reproduced with permission of the American Association for the Advancement of Science from Ref. [25].

atomic chains one may imagine. Thus for the 1, 7–1, 11–4, 13–6, 14–7–1, and 15–8–1 wires they found conductances of 0:96G 0 ; 5:19G0 ; 9:08G 0 ; 11:97G 0 ; 13:82G 0 ; and 14:44G 0 ; whereas the number of atoms per cross section is 1, 8, 15, 20, 22, and 24, respectively. This once again shows that conductance is related in a complicated way to the band structures of the material of interest. In another work, Tosatti et al. [25] studied theoretically the stability of more different nanowires. The notation they used in describing the wires is related to the one that is commonly used for carbon nanotubes (that we shall discuss later in this work) as well as to the one of Figure 5.13 and is brieﬂy outlined in Figure 5.15. As shown in this ﬁgure, different values of the two indices ðm; nÞ lead to different

60

Chapter 5.

Gold Chains: The Prototype?

Figure 5.16. The calculated string tension as a function of radius of a gold nanowire with the structure as outlined in Figure 5.15. Only the values for the smallest and largest values of n for ﬁxed value of m for the ðm; nÞ wires are shown. Reproduced with permission of the American Association for the Advancement of Science from Ref. [25].

structures of the nanowire. In Figure 5.16 we show their calculated string tension for different nanowires as a function of the radius of the nanowire. It is clear that exactly the (7,3) nanowire is the one of the lowest tension and, therefore, is of particular stability as also found in experiment [23]. Tosatti et al. [25] also explored the reason behind the particular stability of this system. In contrast to the results for the jellium wires of Chapter 4, they found that all systems were metallic and, accordingly, the stability was not related to a situation where certain electronic bands were completely ﬁlled or empty. Instead they suggested that it was related to mechanisms that were also responsible for surface reconstructions of clean, perfect surfaces of the otherwise inﬁnite crystal. For large values of ðm; nÞ the wires discussed by Tosatti et al. [25] will be hollow, whereas those of Bilalbegovic´ [20–22] consist of more concentric layers. It may not be clear which structure is the one of the largest stability, i.e., of the lowest total energy. However, Hui et al. [26] performed a theoretical, largely unbiased study where essentially nothing was assumed about the structure of the nanowire except for ﬁxing its length and its number of atoms. Figure 5.17 shows some of the obtained structures that clearly are seen to be fairly compact, and in Figure 5.18 we show the total energy per atom as a function of the diameter of the nanowire. Both ﬁgures demonstrate that nanowires at most may be metastable but that ultimately gold atoms prefer a high coordination which is obtained in a purely three-dimensional structure like that of an inﬁnite crystal: chains of metals represent at most a metastable situation. Experimentally, it has been shown [27] that thicker wires also possess speciﬁc properties. In experiments similar to those of Figure 5.5, Mares et al. [27] simply

61

5.3. More complicated structures

Wire 1

Wire 2

Wire 3

Wire 4

Wire 5

Wire 6

Figure 5.17. Structures of different gold nanowires. Reproduced with permission of the American Institute of Physics from Ref. [26].

Figure 5.18. Average binding energy of different gold nanowires like those of Figure 5.17 as a function of their diameter. Reproduced with permission from the American Institute of Physics from Ref. [26].

counted the number of times a certain value of the conductance G in terms of G 0 [equation (2.41)] was observed when repeating the experiment very many times. Results of this are shown in Figure 5.19. In particular, the right part of this ﬁgure shows that the conductance possesses some regular behaviour and, moreover, that

Chapter 5.

Gold Chains: The Prototype?

8 kFR

4

Number of counts [10 ]

25

Au

7

10

6

20

5

5

15 10

5

10 peak index

3

Na

2

5 0 0

4

1 20

40 G/G0

0 60

Number of counts [103]

9

30

Normalized number of counts

62

1.0 (a) 0.5 (b)

0.0

0

10

20

30 G/G0

40

50

60

Figure 5.19. The left part shows the conductance histogram for gold (thick curve, right axis) and sodium (thin curve, left axis). The insert shows the peak positions as a function of peak index for gold. In the right part further results for gold are shown. Reproduced with permission from the American Physical Society from Ref. [27].

for G=G 0 larger than roughly 15, this regularity is changed. The central issue is now to extract information from this ﬁgure that is related to other properties of the system. To this end, Mares et al. [27] suggest that the peaks reﬂect particularly stable structures that, due to the stability, more frequently occur in the experiment. When assuming that the wire is cylindrical with electrons carrying the current, one can obtain the following approximate expression for the conductance [27,28]: " # kF R 2 kF R 1 þ , G ’ G0 (5.4) 2 2 6 where kF is the Fermi wave vector kF ¼

ð9p=4Þ1=3 rs

(5.5)

with rs being the electron-gas parameter of equation (4.3). This expression is obtained [28] by studying the conduction through a narrow constriction in a threedimensional electron gas, where the constriction is assumed having a spherical cross section that at its narrowest point has a radius of R: Moreover, the derivation is based on the Landauer formula that we discussed in Section 2.4. Equation (5.4) makes it possible to relate the conductance to a certain radius of the junction. Next, we need to identify the reason behind the particularly stable structures. To this end, Mares et al. [27] consider a semiclassical model for electrons moving in a cylindrical system and ﬁnd that the particularly stable radii occur with a spacing of kF DR ¼ 1:21 0:02 (5.6) when it can be assumed that electronic effects are responsible for the stability. The value of equation (5.6) agrees fairly well with the slope of the insertion the left part of Figure 5.19, which is 1:02 0:04: However, as seen in Figure 5.20, for the larger peak indices the slope becomes considerably smaller. This suggests that there is a transition from thinner wires

5.4. Chains containing other atoms

63

Figure 5.20. The positions of the peak positions for gold chains. For details, see the text. Reproduced with permission from the American Physical Society from Ref. [27].

where electronic effects determine stability to thicker ones where some other effect is the main driving force for the increased stability. Mares et al. [27] have suggested that for the thicker wires atomic packing is the driving force. Gold crystallizes in the fcc structure, and by cutting out a wire along the (110) axis one can obtain a particularly dense wire. It can be constructed so that it contains four (111) facets and two (100) facets. kF DR ¼ 2:85 between wires for which one extra atomic layer has been added to all facets, but assuming that special stability occurs when an extra layer has been added to only one facet, one obtains kF DR ¼ 0:476; which is so close to the slope in Figure 5.20 that the authors were convinced that for thicker wires atomic-shell effects determine particular stability.

5.4.

Chains containing other atoms

In the previous section we have discussed the two surprising reports of experimental observations of chains of gold atoms with an unexpected large bond length. The interpretation of the experimental results of one of the two reports was later modiﬁed bringing the bond length into agreement with expectations as well as with theoretical results, whereas the other report has, to our knowledge, not been conﬁrmed or modiﬁed. A number of suggestions for the long bonds was presented, as was discussed above, including that the observed structure was not that of a linear chain of only gold atoms but of a chain with some other structure, as well as the suggestion that the chain contains other species. Since then experimental studies of gold chains containing heteroatoms have also been reported and, therefore, we shall here discuss the consequences of adding other (groups of) atoms to the gold chains. The fact that sulphur can form strong bonds with gold has been known for a long time and is very often utilized to create chemical bonds with gold. Therefore, one

64

Chapter 5.

Gold Chains: The Prototype?

Figure 5.21. The calculated structures of a ﬁnite gold chain suspended between two gold contacts and, in the middle and right panel, containing a SCH3 molecule. Reproduced with permission from the American Chemical Society from Ref. [5].

Figure 5.22. Snapshots at four different states of the process of pulling the terminal carbon atom of an ethylthiolate molecule initially bonded to a gold atom of the surface of a gold crystal. The arrows indicate the pulling direction, and the insets magnify the contact close to the surface. Reproduced with permission of The American Physical Society from Ref. [29].

obvious suggestion is that the (ﬁnite) gold chains forming a nanojunction have been contaminated with sulphur-containing species. Ha¨kkinen et al. [5] have theoretically studied short gold chains suspended between two gold contacts and containing methylthiol molecules, SCH3 : Figure 5.21 shows their calculated structures. It is obvious that the gold chains readily accommodate the SCH3 molecules. Moreover, the Au–S bonds are relatively short, and the Au–S–Au bond angle is less than 180 ; ( not so that the Au–Au distance across the SCH3 unit becomes slightly less than 4 A; too far from the experimental Au–Au bond lengths mentioned above. The strong sulphur–gold bond can also be utilized in producing short gold chains, as demonstrated theoretically by Kru¨ger et al. [29,30]. They deposited an ethylthiolate molecule, CH3 –CH2 –S, on a stepped gold surface, whereby an S–Au bond was formed. Subsequently, they pulled the molecule away from the surface. In contrast to what one may have expected, the Au–S bond is not broken but instead more Au–Au bonds are broken, so that a short chain ultimately is pulled out of the surface as shown in Figure 5.22. However, the presence of sulphur atoms may modify the electronic properties of the chains signiﬁcantly. This was shown in an experimental and theoretical study of

5.5.

Gold chains on surfaces – Luttinger liquids?

65

Figure 5.23. Illustration of the mechanism of pulling a gold chain with a hydrogen molecule. Reproduced with permission from Ref. [33].

Mehrez et al. [31]. By studying the current through a chain either without or with sulphur atoms as a function of applied voltage they found that the I–V curves depend critically both on the precise structure of the contact between the chain and the rest of the system and on the presence of, e.g., sulphur atoms. The latter leads to a tunnelling barrier and a non-linear I–V curve. Other elements have also been assumed incorporated into gold chains, most notably hydrogen. Thus, Csonka et al. [32,33] studied the conductance using the mechanically controlled break-junction technique of a gold junction in a hydrogen atmosphere. Compared with the results obtained in vacuum they found a signiﬁcant decrease in the conduction, indicating that the hydrogen molecules interact electronically with the gold chains. Moreover, they found that the gold chains upon stretching could be made longer before breaking and they suggested, therefore, the mechanism of Figure 5.23. Instead of hydrogen molecules, hydrogen atoms were considered by Skorodumova and Simak [9] who showed that a stable structure with particularly large Au–Au distances could be obtained for an inﬁnite chain with alternating Au and H atoms, cf. Figure 5.24. They have also analysed the band structures and found one band ( Thus, crossing the Fermi energy for the AuH chain with Au–Au distances of 3:6 A: the chain should be conducting as a pure Au chain. The large Au–Au distances could, according to the theoretical study of Legoas et al. [16], be explained also as being due to the presence of other heteroatoms, for instance carbon atoms. These authors also showed that incorporating a single hydrogen atom into a ﬁnite gold chain and subsequently pulling the chain would ultimately lead to the breaking of one of the Au–H bonds, i.e., the presence of the heteroatom leads to a weakening of the chain.

5.5.

Gold chains on surfaces – Luttinger liquids?

The experiments we have discussed so far have focused on the mechanical/energetic properties as well as transport properties of gold chains. These gold chains were more or less free standing, but often suspended between two contacts. Other experiments have also been devoted to the electronic properties of gold chains, but in

66

Chapter 5.

Gold Chains: The Prototype?

a

Ebinding (eV)

−0.5

−1.5

−2.5

Ebinding (eV)

b

2

2.5

3 3.5 Au − Au distance (Å)

4

4.5

−3

−4

−5

−6 2.5

3

3.5 4 Au − Au distance (Å)

4.5

Figure 5.24. Binding energy curve for (a) an inﬁnite Au chain and (b) an inﬁnite AuH chain. The thin curves are the results when assuming all Au–Au nearest-neighbour distances to be identical, whereas this is not the case for the results shown with the ﬁlled circles. Finally, the horizontal bars show the range of existence of a linear Au chain. Reproduced with permission from The American Physical Society from Ref. [9].

these experiments the chains are most often deposited on some surface. It is the purpose of this section to discuss what has been learned from those experiments. Nilius et al. deposited gold atoms on a NiAl(110) surface [34,35]. Subsequently, using the tip of a STM the atoms were moved along the surface so that ﬁnite chains of up to 20 atoms were gradually built up. Examples of the structures are shown in Figure 5.25. By placing the tip of an STM above such a chain at some position along it, and applying a voltage between the tip and the substrate (in this case, NiAl), charge may tunnel from the tip, through the chain, and into the substrate (or vice versa). Speciﬁcally, the derivative of the current I with respect to the applied voltage V ; i.e., dI=dV ; is proportional to the local density of electronic states (LDOS) evaluated on

5.5. Gold chains on surfaces – Luttinger liquids?

67

Figure 5.25. (A) A schematic picture of an Au5 chain on the NiAl(110) surface, whereas (B)–(F) show STM images of various stages in the construction of the Au20 chains. Reproduced with permission of The American Association for the Advancement of Science from Ref. [34].

the chain at the position of the tip apex and for an energy that is related to the applied voltage relative to the Fermi energy of the chain [36]. This means that by applying different voltages and moving the tip along the chain, it is possible to get information about the wavefunctions with that particular energy. An example for the results for a chain with 11 atoms is shown in Figure 5.26 [35]. As may be recognizable in the ﬁgure, the images show density oscillations that become increasingly dense with increasing applied voltage. Hence, the higher the energy the orbital has, the more oscillating it is in space. This is a behaviour that is well known for even the simplest possible systems like, e.g., a particle in a box. Wallis et al. [35] tried to analyse the results within such a model and found indeed that it described very well their results. Thus, one conclusion was somewhat surprising: the electrons in a gold chain behave as free, non-interacting particles conﬁned in a one-dimensional box. The particles were found to have an effective mass of 0:4 0:1 times that of a free electron. This proposal was further analysed by Mills et al. [37], who suggested that instead of considering a strict one-dimensional system, it is more realistic to assume that the electrons of the gold chains are conﬁned to a three-dimensional cylinder of ﬁnite length. They studied this model analytically and compared the results with those obtained from density-functional calculations on ﬁnite gold chains. Figure 5.27 shows examples of their study for Au8 ; and it is indeed seen that the model gives a simple and highly realistic description of the density-functional results. The conclusions from these studies are that the electrons in gold chains are essentially non-interacting and, in addition, moving in a three-dimensional,

68

Chapter 5.

Gold Chains: The Prototype?

Figure 5.26. Sixteen dI=dV images of the Au11 chain taken at 0.1 V intervals between 1.0 and 2.5 V. Reproduced with permission from the American Physical Society from Ref. [35].

conﬁned region. This means that the many-body effects that we may expect for onedimensional systems and that we discussed in Chapter 3 – in particular in relation to the possible occurrence of Luttinger-liquid behaviours – may not be relevant for gold chains. However, some experiments have suggested the opposite. The approach of Nilius et al. [34,35] in producing ﬁnite gold chains of welldeﬁned length is cumbersome and not simple. Instead, one may try to use other methods in producing linear chains of gold atoms with more or less well-deﬁned structure. In Figure 5.28 we show, schematically, a cubic crystal for which the structures of two surfaces are indicated. Both surfaces are seen to lead to terraces (perpendicular to the plane of the picture), but for the (110) surface these terraces consist of just a single row of atoms, whereas for the (150) surface they consist of ﬁve rows. Thus, by choosing or creating the surface carefully, it is possible to obtain fairly broad terraces. Subsequently, another type of atoms may be

69

5.5. Gold chains on surfaces – Luttinger liquids?

n, m

E0, eV

n, m

0.26 5.0

4.59

0.52 1.18

6.0

E0, eV

2.1

4.60

1.37 5.02 2.49

7.0

2.36

3.1

3.99 8.0

5.57

3.51 4.93

9.0

4.99

4.1

5.53

4.82 6.28 4.31 5.1

1.1

6.22

4.37

Figure 5.27. Each panel shows the electron density of the individual orbitals from (upper part) densityfunctional calculations on Au8 and (lower part) particles in a ﬁnite cylinder. The quantum numbers ðn; mÞ characterizing the orbitals of the latter are also given, as are the orbital energies; for the densityfunctional calculations, relative to the energy of the highest occupied orbital. Reproduced with permission of The American Institute of Physics from Ref. [37].

Figure 5.28. Schematic representation of two different surfaces, (110) and (150), of a cubic crystal. The line between the black and white atoms deﬁne the (150) surface, whereas the other line deﬁnes the (110) surface.

70

Chapter 5.

Gold Chains: The Prototype?

deposited on the surface, and due to the different surroundings of the surface atoms, the adsorbed atoms may prefer a certain position on the terraces, ultimately leading to rows of adsorbed atoms that are lying parallel to the steps of the terraces. Here, the substrate will dictate the organization of the adsorbed atoms. The width of the terraces will determine the distance, and hence interactions, between the chains. This idea has been utilized by Hasegawa et al. [38,39] and Shibata et al. [40]. As substrate they used a silicon crystal, and as adsorbants gold atoms. As one example, we show in Figure 5.29 results from the work of Shibata et al. [40] where the creation of gold chains clearly is recognizable. On the other hand, it is also clear that the idealized picture of Figure 5.28 only partly is recovered in Figure 5.29 where, instead, a more heterogeneous structure with different phases on the surface is observed. Segovia et al. [41] managed to create very well-deﬁned samples with gold chains on a Si(557) surface. They reported that their sample contained terraces with the (111) surface, that the terraces are essentially ﬁve substrate lattice spacings wide, ( and that, accordingly, the gold chains have an interchain spacing or roughly 20 A; cf. Figure 5.30. Therefore, there should be no interaction between the chains, and since silicon is a semiconductor with a gap around the Fermi level of more than 1 eV, any electronic effects closest to the Fermi level should be due to the electrons of the gold chains. Thus, this system should be a perfect system for studying electrons conﬁned to a quasi-one-dimensional chain, although, as we saw above, it is not clear whether the chains can be treated as purely one-dimensional or rather as narrow three-dimensional objects. Since the chains are well ordered, angle-resolved photoelectron spectroscopy, ARPES, can be used to probe the electronic properties of the chains. The sample is irradiated with photons that have a well-deﬁned energy, polarization, and direction, and electrons whose energy is measured are been emitted from the sample as a function of direction. By varying the angle that the photons make with the surface of the sample, the momentum transfer q can be varied, cf. Section 3.7. Such experiments were carried through by Segovia et al. [41] and in Figure 5.31 we show some of their results. In their spectra we observe a peak that approaches the Fermi energy, E F ; as the angle ye ! 0 (corresponding to momentum transfer q ! 0), but whose intensity is reduced or even vanishing for ye sufﬁciently close to 0. The momentum transfer can also be related to the k number parallel to the chain direction, kk ; so that the experimental results ultimately can be presented as the band structures for the occupied bands. This is done in Figure 5.31, too, and from this curve it is seen that the experimental results are in accordance with the existence of one band crossing the Fermi level at some non-zero kk : Therefore, if the material was a ‘conventional’ two- or three-dimensional material, we would expect to see the peak approaching the Fermi level for arbitrarily small ye : The fact that the peak vanishes is, accordingly, taken as an indication of a Luttinger-liquid behaviour, as shown schematically in Figure 3.3 and discussed in Section 3.7. Therefore, Segovia et al. [41] have concluded that the gold chains on the Si substrate represent an experimental realization of a Luttinger liquid. We add, ﬁnally, that the band

71

5.5. Gold chains on surfaces – Luttinger liquids?

0

5×2

7×7

5×8 3 1 P

7×7

5×2

7×7

1 2 3

2

P P

P

31

3

5×2 2

P P

[112] [110] P

P

P

P 7×7

5×2

7×7 2

P

3

1

A R Au Au Au P R Au P R Au B Adatom Top layer atom 2nd layer atom Adatom on terrace Bright protrusion Gold atom [112]

A [110]

[10+20]m

B P P [5+23]m [5+23]m

0° 0°

Figure 5.29. Top panel a STM image of the Au-adsorbed vicinal Si(111) surface. The conﬁguration of the terraces and steps is shown schematically on the top and the numbers indicate the type of the Auadsorbed terraces. Finally, the arrows at P indicate the rows of gold atoms. Bottom panel the corresponding structural model. Reproduced with permission of The American Physical Society from Ref. [40].

structures in Figure 5.31 agree well with those of theoretical studies on gold chains with a lattice constant slightly larger than the optimized one, e.g., Figures 5.2 and 5.8. This conclusion has, however, been questioned. The lack of intensity at the Fermi level may actually have more reasons of which one simply is a small cross section. Segovia et al. [41] used photons with an energy of hn ¼ 21:2 eV; and the cross section depends on this energy. Losio et al. [42] performed very similar experiments, but used different photon energies. They did indeed found that for hn ¼ 21:2 eV

72

Chapter 5.

Gold Chains: The Prototype?

a

[111]

[112]

b

[110]

[112] Figure 5.30. Side and top view of the Si(557) surface with (111) terraces separated by regular steps. The gold atoms (larger symbols) on the terraces are also shown. Reproduced with permission of Nature from Ref. [41].

the intensity at the Fermi energy is very low, if not vanishing, but changing the energy to, e.g., hn ¼ 27 or 34 eV they found markedly different results, as shown in Figure 5.32. The middle panels show clearly that the intensity goes not as some power-law to zero as the energy approaches the Fermi energy i.e., as ðE2E F Þa ; but instead abruptly becomes zero at the Fermi energy. Moreover, there is a clear non-zero intensity at E F ; and even a band splitting near this energy can be identiﬁed. The clean silicon may not possess any signals in the energy range of Figure 5.32, and a single linear gold chain with a slightly increased interatomic distance compared with the optimized structure would have a single band crossing the Fermi level. Nevertheless, when depositing the gold chain on the silicon surface, there will be some interactions between silicon orbitals and gold-chain orbitals, cf. Section 2.2. In the experimental study of Losio et al. [42] they found that the unit cell contained two separate chains although they could not determine whether the chains were due to gold or silicon atoms. Further information was obtained from a theoretical study of Sa´nchez-Portal et al. [43,44]. Some of the results from the ﬁrst study [43], which was the ﬁrst theoretical study that gave a qualitatively correct description of the experimental

73

5.5. Gold chains on surfaces – Luttinger liquids?

a P-pol Hel 12 K e k|| Intensity (arbitary units)

16° 14° P-pol Hel 12 K

12° − π 2a

e −6°

0° Γ

−16°

π −24° a

−0.6

b

−0.4 −0.2 Energy (eV)

EF

0.6

−10° Intensity (arbitrary units)

−12° π 2a −14°

−8°

−12° −12.5° −13° −13.5°

0.4

−14°

k|| (A-1)

0.2 0.0

−14.5°

−0.2

−15°

−0.4

−16°

−0.6

−18° -0.4

EF 0.4 Energy (eV)

0.8

−0.8

−0.6

−0.4 −0.2 Energy (eV)

EF

Figure 5.31. Angle-resolved photoemission spectra of the gold chains on the Si surface for a k~ along the chain direction. The polar emission angle ye is deﬁned with respect to the surface normal. The right part shows a more detailed presentation of the results in the upper panel in the left part. The lower panel in the left part shows the band structures that are extracted from the experimental results. Reproduced with permission of Nature from Ref. [41].

ﬁndings, are summarized in Figure 5.33. As shown in Figure 5.33(a), the surface with gold atoms contains a large number of dangling bonds that cause signiﬁcant structural relaxations, cf. Figure 5.33(b). The dangling bonds as well as the gold atoms are responsible for the occurrence of many extra orbitals that are localized to the surface region and whose energies lie in the gap of the band structures for crystalline silicon. These orbitals are marked with the thick symbols in Figure 5.33(d). Here, the stars mark dangling bonds of the silicon atoms at the step edge and the open symbols are for dangling bonds of the silicon atoms on the terraces. There are two bands with essentially vanishing dispersing perpendicular to the chain direction, i.e., along M 0 –G; with signiﬁcant contributions from the gold atoms but also with contributions from the neighbouring silicon atoms. Thus, to some extent the situation we have discussed in Section 2.2 seems to be observed

E = EF

Chapter 5.

Gold Chains: The Prototype?

34eV

Intensity

74

27eV

0

E (eV)

0.0 0.2 −0.1

Intensity

E = EF−0.6 eV

−0.6

0 1.1

1.3

1.1

1.3

k|| parallel to the chains [Å−1] Figure 5.32. Results from photoemission studies on gold chains on Si(557). The left and right panels correspond to two different photon energies, 34 and 27 eV, and the middle panels give the measured intensities as a function of energy and kk : The upper and lower panels, ﬁnally, show the intensities at two different energies, marked with the arrows on the right. Reproduced with permission of The American Physical Society from Ref. [42].

here. Moreover, the theoretical results are largely in agreement with the experimental results of the group of Himpsel [42] and supporting the consensus that the gold chains on the silicon surfaces can not be considered isolated (quasi-)onedimensional chains but instead parts of a complex Au–Si material. In the most recent theoretical study of Sa´nchez-Portal et al. [44] the authors showed that when allowing the structure to relax and including all relativistic effects, not only a qualitatively correct but also quantitatively correct description of the experimental results is obtained. Then, of the bands crossing the Fermi level, only one can be ascribed to the gold chains, when spin–orbit couplings are excluded in the calculations. Upon their inclusion, two bands from orbitals on the gold chains occur are found around the Fermi level, and a qualitatively correct description of the experimental results is obtained. In subsequent studies, the group of Himpsel studied experimentally gold chains on more different silicon surfaces [45–48]. They found that they could partly control the electronic properties by varying the surface, and also that the fact that the chain is deposited on the surface indeed suppresses the Peierls distortion, similar to what we have discussed in Chapter 2.

5.5. Gold chains on surfaces – Luttinger liquids?

75

Au

(a)

Au

(b) Edge

1

2

3

4

Step

(c)

K K Γ

M′

Step edge

M

(d)

2

Energy (eV)

1

0

−1

−2

Γ

K

M′ Γ

MK

Figure 5.33. (a) The structure of the Si(557)-Au surface before relaxation and (b) after relaxation. The arrows in (a) mark the dangling bonds. (c) A top view of the same system and in (d) the calculated band structures are shown. Here, the symbols are described further in the text. Reproduced with permission of The American Physical Society from Ref. [43].

However, even this conclusion has been questioned. Ahn et al. [49] have studied the Au/Si(557) system using ARPES experiment. They varied the temperature and observed that the two bands crossing the Fermi energy show different behaviour as a function of temperature. One of the two bands show the opening up of a gap when the temperature is decreased, which is typical for a Peierls distortion with a

76

Chapter 5.

Gold Chains: The Prototype?

low energy gain. The other band, on the other hand, does not possess such a gap even for the lowest temperatures of the study. The ﬁrst result conﬁrms once more that the original conclusions of Segovia et al. [41] that they had observed a Luttinger-liquid behaviour of the gold chains are not correct. The different behaviour of the two bands crossing the Fermi level may suggest that they have different origin, for instance coming from different parts of the system. Thus, at the moment of writing the exact origin of the experimentally observed bands is still an open question.

5.6.

Conclusions

Gold chains constitute one of the most studied systems within the concept of metallic chains/chains of metals. Therefore, it was natural to start the description of real systems with the discussion of gold chains. Since the end of the 1990s several reports have led to a signiﬁcant interest in this system. Free-standing chains of gold atoms can be produced either through etching or in break-junction experiment. We saw that ultimately, in one case, the initially reported large bond lengths for this system have subsequently been found to be unrealistic and that the bond lengths for such chains are comparable to what one would expect when comparing crystalline gold and the Au2 molecule. On the other hand, the reasons behind the other report of large bond lengths remain unexplained. When the wire becomes thinner, the conductance becomes inﬂuenced by quantum effects, i.e., the electrons experience that they are conﬁned in two dimensions and have only a smaller ﬁnite number of conduction channels in the third dimension. Indirectly, also structural properties can be observed in the conductance measurements: in break-junction experiments the atoms move irregularly so that sudden larger rearrangements of many atoms lead to sharp narrowings of the junctions. This narrowing changes the number of conductance channels discontinuously which ultimately shows up in the conductance experiments as steps. These rearrangements of the atoms also give rise to irregularities in the force vs. stretching. The conductance experiments and their interpretation in terms of Landauer’s conductance formula suggested that a single-particle description of the electrons in gold chains is appropriate. This was further supported by studies of the orbitals of ﬁnite gold chains deposited on a surface, whereas, on the other hand, photoelectron experiments on gold chains on vicinal Si surfaces were at ﬁrst interpreted as showing a Luttinger-liquid behaviour, i.e., that many-particle interactions were very important. Later, however, it has been found that these experiments more likely show that such gold chains are not isolated but, instead, interacting with the substrate so much that the experiments become obscured by these. Finally, we saw that more complicated chains, including tubes, can exist and, moreover, that other elements can also be incorporated into the chains or be used in the controlled variation of the properties of the chains. We discussed in particular hydrogen whose presence clearly affected the conductance experiments, as well

References

77

as sulphur that always has been important when manipulating gold due to the relatively strong Au–S bond.

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Chapter 6

Chains of other sd Elements

In the preceding section we have discussed chains of gold atoms in considerable details. Gold, a noble metal, is most likely the most studied element in the context of low-dimensional systems, explaining our detailed discussion of this element. However, in order to obtain further degree of freedom for controlled variation of the materials properties (which also may lead to a further understanding of the properties of chains of metals), one may consider other elements. Accordingly, chains of also other elements have been studied both experimentally and theoretically. In this and the following two sections we shall discuss some of those. Gold has almost ﬁlled s and d valence shells, as we also saw in the preceding section. It is also a heavy element so that relativistic effects are important. Therefore, the ﬁrst elements we shall discuss are the other coinage elements, Ag and Cu, that have a similar valence-orbital conﬁguration, but for which the relativistic effects are less important. Subsequently, we shall consider the elements for which the s and d valence shells are completely ﬁlled, i.e., Hg, Cd, and Zn, and, afterwards, we shall consider the elements that have increasingly less electrons in the s and d valence shells. This means starting with Pt, Pd, and Ni, we continue to Ir, Rh, and Co. Subsequently, we consider Ru, Nb, and Zr, and the last element we shall consider is Ti. It may be noticed that for the elements of the same valence-electron count, we discuss ﬁrst the heaviest one (i.e., the one closest to Au) and then pass on to the lighter ones.

6.1.

Ag

For reference we ﬁrst brieﬂy discuss the structures of Figure 6.1 that were studied theoretically by Springborg and Sarkar [1]. The calculated band structures for the optimized structures are shown in Figure 6.2. It is not surprising that for this system the relativistic effects are not very important. This is exempliﬁed in Figure 6.2 through the very small band splittings that occur when spin–orbit couplings are included in the calculations. Moreover, in particular the linear chain shows the existence of exactly one band crossing the Fermi level – in contrast to the case for the linear gold chains, this situation is obtained for silver chains without increasing the bond length beyond the optimized value. As we have discussed in Chapter 2, the occurrence of exactly one band crossing the Fermi level suggests that the total energy may be lowered upon a lowering of the symmetry. This issue was studied theoretically by McAdon and Goddard [2] for 79

80

Chapter 6. Chains of other sd Elements

Figure 6.1. Schematic representation of different structures of inﬁnite, periodic chains. The ﬁgure shows the linear, the zigzag, the double zigzag, and the tetragonal chain. Reproduced with permission of The American Physical Society from Ref. [1].

various closed rings of Cu, Ag, Au, Li, and Na using the Hartree–Fock approximation. They found indeed that within the restricted Hartree–Fock approximation, where it is assumed that the spin-up and the spin-down orbitals are identical, a bond-length alternation would lower the total energy. However, when removing this restriction, a state of even lower total energy could be found. This state contains a spin-density wave, i.e., can be considered as showing a magnetic ordering, whereas the bond lengths did no longer alternate. This was found for all metals studied by them, but has to the best of our knowledge not been conﬁrmed in later studies on any of the materials they considered. In particular, Delin and Tosatti [3] examined whether linear chain of 4d metals could be magnetic. Although some of the elements of their study (see later) were found to show a spin polarization, silver did not belong to that category. Break-junction experiments have also been performed for Ag (see, e.g., Ref. [5]). The results show that the smallest value of the conductance is close to 1G 0 ; indicating that exactly one electronic band crosses the Fermi level for the thinnest possible systems. This is in accord with the formation of (ﬁnite) linear chains just before breaking the junction. This issue was addressed theoretically by Bahn and Jacobsen [4], who studied the possible chain formation in breaking a junction for six different elements, Cu, Ag, Au, Ni, Pd, and Pt. We shall return to their results in the next section but some of their key results are shown in Table 6.1. The table shows, not surprisingly, that when the coordination is reduced, i.e., when passing from the crystal to the linear chain, the bonds get stronger and shorter. However, most relevant for the experiment is the break force F 0 : When pulling a chain and assuming that it will be elongated regularly over the complete chain, the structure will be that for which the total energy as a function of bond length has exactly the slope as given by the pulling force. Thus, the maximal slope deﬁnes the largest possible force with which the chain can be pulled. If the force exceeds this maximal value, the break force, the chain will break.

6.1. Ag

81

Figure 6.2. Band structures for (a)–(d) linear, (e)–(h) zigzag, (i)–(j) double zigzag, and (k)–(l) tetragonal chains of Ag. In (b), (d), (f), (h), (j), and (l), spin–orbit couplings were included, but not in the other panels. Moreover, the results of (c), (d), (g), and (h) were obtained with a generalized-gradient approximation within density-functional theory, whereas the other ones were based on a local-density approximation. k ¼ 0 and k ¼ 1 marks the centre and the edge of the ﬁrst Brillouin zone, respectively, and the horizontal dashed lines mark the Fermi level. Reproduced with permission of The American Physical Society from Ref. [1].

In a break-junction experiment, however, the system can, as an idealization, be considered as consisting of both crystalline parts (forming the junctions) and a chain. Thus pulling the complete system, it will ultimately break ‘somewhere’. This ‘somewhere’ may be in the chain if this is not ‘sufﬁciently’ strong compared with the crystalline parts. Therefore, most important is the relation between the break forces for the chain and the break force for the crystal. For this, Bahn and Jacobsen

82

Chapter 6. Chains of other sd Elements

Table 6.1. Calculated bond length, d; binding energy, E 0 ; per bond, and break force, F 0 ; for various elements either in the crystalline fcc structure or as linear chains. From [4]. System Fcc

Chain

Property

Cu

Ag

Au

Ni

Pd

Pt

d (A˚) E 0 (eV) F 0 (eV/A˚) d (A˚) E 0 (eV) F 0 (eV/A˚)

2.59 0.59 0.44 2.33 1.63 1.18

2.93 0.44 0.34 2.65 1.25 0.90

2.96 0.51 0.44 2.62 1.59 1.31

2.49 0.86 0.64 2.16 2.23 1.60

2.82 0.63 0.54 2.52 1.20 1.10

2.83 0.93 0.77 2.41 2.83 2.45

predicted that ﬁrst of all chains of Pt and Au can be formed, although, cf. Table 6.1, there is not too large a difference between the various elements. However, as we shall see below for Cu, the authors found in simulations that Au but not Cu or Ag formed chains. The lack of stable wires in break-junction experiments was reported by Smit et al. [6], who argued that chains are not observed for Ag, Cu, Rh, and Pd, but for Ir, Pt, and Au. They suggested that the difference could be related to the way the surfaces of the inﬁnite crystals reconstruct and to a competition between s and d bonding. But break-junction experiments have been performed on Ag, suggesting nevertheless that at least very thin wires of Ag are been formed in the experiment. In one of those experiments, Rodrigues et al. [7] observed a conductance value of 1G 0 : They performed in addition calculations using a simple Hu¨ckel-like model on different types of wires. They found for an atomically thin wire (i.e., a linear chain) that the conductance is indeed very close to the value of 1G0 ; whereas it for thicker wires it was 2G 0 and upwards. This consensus is further supported from the results of Zhao et al. [8]. They studied theoretically the conductance for the tetragonal chain of Figure 6.1 and found it to exceed 2G 0 : Thus, so far no structural model except for the linear chain has been able to reproduce the experimentally observed conductance value of 1G0 ; suggesting that the linear chain indeed is observed in the experiment. In Section 2.4, we discussed the conductance and its relation to the transmission coefﬁcients, describing how electronic orbitals are progressing through a chain-like structure. We saw also, cf. Figure 2.14, that the conductance could show a clear dependence on the length (i.e., the number of atoms) of a chain. This result has later been conﬁrmed theoretically using more accurate methods by, e.g., Lee et al. [9]. Their results show, cf. Figure 6.3, that when looking at conductance at the linear chains there is not much difference between the different elements. If a linear chain is less likely to occur, one may ask as to which structure a nanowire of silver has. Ribeiro and Cohen [10] have addressed this question for a whole series of metals, Au, Al, Ag, Pd, Rh, and Ru, and in Table 6.2 we summarize their main ﬁndings. It is seen that a zigzag chain for all metals is more stable than a linear chain. These authors also ﬁnd that there is a range of strains for which the linear chain is stable without breaking or distorting, with Ru being the only exception. Thus, most studies agree that linear chains can be observed in experiments on Ag.

83

6.2. Cu

L3

(a)

L2

L1

TL

TR 1

2

3

4

R1

R2

R3

Conductance (G0)

1.00 0.95

Na

0.90 0.85 Cs

0.80

(b)

Conductance (G0)

1.00 0.99 0.98 0.97

Cu Ag Au

0.96 0 (c)

1

2 3 4 5 6 7 8 Number of atoms in chain, N

9

10

Figure 6.3. (a) The model used in the calculation of the conductance, whereas (b), (c) show the calculated conductance as a function of number of atoms in the chain part in (a) and for different elements. Reproduced with permission of The American Physical Society from Ref. [9].

Finally, in the previous section we also discussed gold chains on vicinal Si surfaces, in particular in the context of whether these chains provide physical realizations of a Luttinger liquid. Similar experiments have also been performed for Ag deposited on a Si(5 5 12) surface [11]. In agreement with the now accepted conclusion for gold on Si surfaces, the experimental results for silver on Si surfaces have been interpreted as showing the existence of Si dangling bonds as well as indicating some interaction between the adsorbant (here, silver) and the substrate.

6.2.

Cu

When passing from Au to Ag we observed some quantitative differences in the results, but the overall behaviour of the two elements was very similar. Equivalently,

84

Chapter 6. Chains of other sd Elements Table 6.2. Calculated lengths (per atom) along the chain axis and energies for zigzag and linear chains of various elements.

Property

Au

Al

Ag

Pd

czz;1 (A˚) czz;2 (A˚) czz;3 (A˚) cln (A˚) cln;L (A˚) cln;T (A˚) czz;max;1 (A˚) czz;max;2 (A˚) czz;max;3 (A˚) cln;max (A˚) E zz;1 (eV) E zz;2 (eV) E zz;3 (eV) E ln (eV)

1.29 2.26

1.21 2.34

1.33

1.24

2.49 2.83 2.71 1.48

2.39 3.32 2.93 1.33 2.73

2.57 2.89 2.67 1.54

2.37 2.86 2.42 1.48

2.96 2.87 2.31

2.52 2.69 1.88

3.06 2.15

2.79 2.69

2.08

1.82

1.56

1.73

Rh 1.19 1.83 2.10 2.25 2.57 2.44 1.51 1.96 2.29 2.66 4.18 3.49 3.45 3.27

Ru 1.22 1.80 2.21 2.33 2.55 1.43 2.16 2.66 5.54 5.17 4.58

Note: For some systems, more different stable zigzag chains were found, in which case they are distinguished through extra indices. czz is the length per atom along the chain, and E zz is the corresponding total energy per atom (relative to that of the isolated atoms) for the zigzag structures. cln and E ln give the same quantities for the linear chain. Moreover, cln;L is the length at which a transition to a linear structure with alternating bond lengths occurs, and cln;T is the length where a transition to a zigzag chain occurs. Finally, cmax is the maximum length of a chain before it breaks. From [10].

one may expect that once again only smaller differences are found when passing on to Cu. And in fact, the results of various experimental and theoretical studies conﬁrm this simple prediction. Thus, for Cu, McAdon and Goddard [2] predicted that the lowest total energy for a ﬁnite, closed ring is found for a spin-density structure without bond-length alternation. In this case there exists experiment with which their prediction can be compared. Dallmeyer et al. [12] deposited Cu atoms on a vicinal Pt(997) surface and performed subsequently angle-resolved photoelectron spectroscopy (ARPES) experiment on the sample. Similar experiments on Co on the Pt(997) surface revealed a band due to the Co atoms that was split into two, suggesting an exchange splitting and the occurrence of local magnetic moments. A similar feature was not found for Cu, which may be taken as an indication of the lack of a local magnetic moment. Moreover, Bahn and Jacobsen [4] predicted that Cu is one of the elements that would not form chains in a break-junction experiment and showed snapshots from a molecular-dynamics simulation on Au and Cu junctions that are pulled apart. Figure 6.4 shows these results and it is here indeed seen that the Cu junction is broken immediately, whereas the Au junction leads to the formation of a chain. That chains of Cu are not formed in break-junction experiments was also reported by Smit et al. [6]. However, break-junction experiments have been performed for Cu and these have, as for Au and Ag, shown that the smallest conduction is found for a value of 1G0 : As we saw for Au and Ag, this value is best interpreted as being due to the existence of a linear chain between the two junctions, so that we conclude that linear chains for Cu indeed can be formed in the experiments and that these are the ones that are responsible for the conductance of 1G 0 that is observed

85

6.2. Cu

Figure 6.4. Snapshots from simulations of breaking gold (upper row) and copper contacts (lower row). Reproduced with permission of The American Physical Society from Ref. [4].

80 70

Na

Cu alat=6.76

60

DOS

50

alat=9.60

40

(3,2) (3,1) (2,3) (1,3)

(2,1) (1,2) (2,2)

30

(3,2) (2,3)

20 (3,1) (2,2) (1,3)

(1,1) 10 0

(1,1) 0

(2,1) (1,2) 0.1

0.2

EF

0.3

Energy (Ry) Figure 6.5. Density of states in states/Ry for 6 6 wires of Cu and Na. Reproduced with permission of The American Physical Society from Ref. [13].

experimentally. Moreover, as we saw above in Figure 6.3, Lee et al. [9] found that the conductance for linear chains of Au, Ag, and Cu are very similar. We ﬁnally mention that Opitz et al. [13] studied theoretically isolated chains of Cu that were somewhat thicker. They compared the results with those for Na, which, to a good approximation, may be considered as a free-electron metal. In Figure 6.5 we show the calculated density of states for the two systems for wires

86

Chapter 6. Chains of other sd Elements

with a square cross section of 6 6 atoms. In the low-energy region, the freeelectron behaviour leads to nearly the same density of states (except for an energy scaling) and, in particular, the sequence of the peaks is the same for both materials. The different energy scale may be ascribed different lattice constants. However, closer to the Fermi level the occurrence of many extra features in the curve for Cu is a result of the many 3d orbitals that are far from free-electron like.

6.3.

Hg, Cd, and Zn

Au, Ag, and Cu each misses one electron in order to ﬁll the s and d valence orbitals. With this extra electron one arrives at Hg, Cd, and Zn, respectively, that, accordingly, are closed-shell atoms that may exhibit less interesting behaviours: the interatomic interactions of closed-shell atoms are weak and essentially undirected. On the other hand, Hg, Cd, and Zn are not inert gases and do participate in bonding. However, to our knowledge there is no experimental report on chains of those atoms. In a series of theoretical studies, Kim and coworkers [14–16] have used theoretical methods in studying the properties of linear chains of various elements. By comparing these elements, they found that Hg is unusual. In Figure 6.6 we show the calculated nearest-neighbour bond lengths and cohesive energy per atom for the chains relative to the same values for the crystalline compounds. It is seen that Hg does show an abnormal behaviour. First, the cohesive energy for the chain is very small and, moreover, the bond length is larger for the chain than for the crystal. In particular the second observation is in conﬂict with normal chemical intuition, that the bonds get shorter when the coordination is lower. On the other hand, the calculated cohesive energy for linear chains of Hg is very small (0.10 eV/atom). For 1.0

0.8 Fractional den Fractional Ec 0.6

0.4

0.2 Fe Ru Os Co Rh Ir Ni Pd Pt Cu Ag Au Zn Cd Hg Group 8

Group 9

Group 10

Group 11 Group 12

Figure 6.6. The fraction of the cohesive energy per atom E c for the linear chain relative to that for the crystal as well as the fraction of the nearest-neighbour bond length d nn of the linear chain relative to that for the crystal for various elements. Reproduced with permission of The American Physical Society from Ref. [16].

6.4. Pt

87

Zn and Cd it is larger (0.48 and 0.32 eV/atom, respectively), but here also the binding energies are small; cf., e.g., Tables 6.1 and 6.2 for other elements. These small values may explain why such chains have not been produced in experiment. According to the work of Kim et al. [16] Hg possesses one further peculiarity, i.e., the linear chain is semiconducting with a band gap of about 1.5 eV. Hg is a heavy element, and as we shall see later for Pb, relativistic effects may modify the band structures markedly, but the semiconducting behaviour of Hg chains is independent of the inclusion of relativistic effects. Passing to layers of Hg atoms, the system becomes metallic. To summarize, chains of Hg, Cd, and Zn seem to pose a challenge to experimentalists that has not yet been met. It may, however, be worthwhile to attempt to produce these systems, as they may have peculiar properties.

6.4.

Pt

When passing from Au to Hg, we ﬁlled the valence s and d orbitals and, accordingly, the interatomic interactions were weak and chains can only be formed, if at all, with great difﬁculties. Moreover, linear chains of Hg were found to be semiconducting, meaning that the conductance is essentially zero. On the other hand, when going to the left in the periodic table, i.e., when passing from Au to Pt, the s and d valence shells get less ﬁlled and the Fermi level may appear at an energy where not only one band (as in slightly stretched Au chains) but more bands are found. Each band crossing the Fermi level contributes to the conductance (although not necessarily with as much as 1G 0 ; but in most cases with only a fraction thereof), so that we may expect ﬁrst of all that Pt chains can be formed (as a consequence of stronger interatomic interactions) and second that the conductance may be larger than the conductance of Au chains. In fact, chains of Pt atoms do form one of the more studied systems within the context of chains of metals. Furthermore, break-junction experiments give that the smallest value of conductance is somewhere between 1:5G 0 and 2:5G 0 [5], supporting that more bands cross the Fermi level. Both theoretical [4] and experimental [6] studies suggest that, besides gold, platinum is the material for which linear chains can be formed most easily. In Figure 6.7 we show an example of the measured conductance for chains of Au, Pt, and Ir [17]. The results were obtained by averaging over very many measurements. The ﬁgure shows clearly that the conductance for Pt is larger than that for Au. In analysing the results, Smit et al. [17] observed moreover that the conductance as a function of length of the chain does show some indications of an oscillatory behaviour. This was interpreted as an experimental veriﬁcation of the odd–even oscillations that we have discussed in Section 2.4 on the basis of simple model calculations and also seen in Figure 6.3 for different metals. So far we have seen that Pt and Au possess similar properties although with material-dependent differences in the details. But, it has been shown that the two materials also can show signiﬁcantly more different behaviours. First, Nielsen et al. [18] studied the current as a function of applied voltage in a break-junction

88

Chapter 6. Chains of other sd Elements

1.1 1.05

Au

1

conductance (2e2/h)

0.95

2

Pt

1.5

2.2 Ir 2

1.8

1.6

0

0.2

0.4

0.6 0.8 length (nm)

1

1.2

1.4

Figure 6.7. The thick curves show the conductance in units of G0 as a function of length of the chains from break-junction experiments on Au, Pt, and Ir. The curves are the results of averaging over very many measurements. In the background histograms of the plateau lengths for the same experimental data are shown. Reproduced with permission of The American Physical Society from Ref. [17].

experiment for a certain length of the chain and studied both Au and Pt. Representative results are shown in Figure 6.8. Subsequently, the results were ﬁtted with a curve of the form IðV Þ ¼ GV þ G 0 V 2 þ G 00 V 3 ,

(6.1)

i.e., also non-linear conductivities were included. Setting G 0 ¼ 0; Nielsen et al. [18] obtained the results of Figure 6.9. Focusing on only the abscissa we recover that G for Pt is typically above 1:5G 0 ; whereas for Au G has 1G 0 as its lowest value. But it is more interesting to observe that G00 is very close to 0 for Au, but clearly non-zero for Pt, although highly scattered. By combining the experimental study with theoretical calculations, Nielsen et al. [18] were also able to offer an explanation for the non-Ohmic behaviour of Pt compared with the Ohmic behaviour of Au. For gold only one orbital per atom, formed by s; pz ; and d z2 functions on the atoms (with the chain lying along the z-axis), contributes to

89

6.4. Pt

400

(a)

300

I −V on Au at 4.2 K I −V curves

200

4.2G0 3.1G0 2.0G0

3rd order fit I (μA)

100

1.0G0

0 -100 -200 -300

200

-1.0

-0.5

-0.4

-0.2

0.0 0.5 Voltage (Volt) 0.0

0.2

100

0.4

0.6

3.0G0

I −V on Au at 4.2 K I −V curves

2.3G0

3rd order fit

50 I (μA)

1.5

4.1G0

(b) 150

1.0

1.7G0

Low bias G 0 -50

-100 -150

Figure 6.8. Representative I–V curves of thin wires of (a) Au and (b) Pt. The inset in (b) shows the experimental setup, and the dashed and dotted curves show a 1st and 3rd order ﬁt, respectively. Reproduced with permission of The American Physical Society from Ref. [18].

G′′(G0 V -2)

0.5 0.0 -0.5 -1.0 -1.5

Au 1240 I−V Pt 834 I−V Pt mean 1

2

3

4

G (G0)

Figure 6.9. G00 as a function of G for Au and Pt chains. Reproduced with permission of The American Physical Society from Ref. [18].

90

Chapter 6. Chains of other sd Elements

Figure 6.10. The left part shows high-resolution transmission-electron microscope images of (upper panel) Pt and (lower part) Au nanowires. The right part shows simulated images of different Pt helices. Reproduced with permission of The American Physical Society from Ref. [20].

the conductance. For most other sd metals, other atomic functions also contribute and there is, in addition, more than one conduction channel. This results in a signiﬁcantly more complex behaviour and, ultimately, in a non-linear dependence of the current on the applied voltage. Delin and Tosatti [19] have predicted that inﬁnite, periodic, linear chains of Pt atoms are magnetic and, furthermore, that the magnetism increases upon stretching the wire. This possibility was not included in the considerations above by Nielsen et al. [18] but would, of course, increase the complexity of this material. Finally, Oshima et al. [20] have observed nanowires of Pt that are regular but thicker than single atoms. By analysing high-resolution transmission-electron microscope images of the nanowires, they concluded that the structure was that of helical multi-shell wires, similar to what the authors have observed earlier for gold [21] (which we discussed in Section 5.3). In Figure 6.10 we show their structures for Pt nanowires.

6.5.

Pd and Ni

We have mentioned earlier that both the theoretical work of Bahn and Jacobsen [4] and the experimental work of Smit et al. [6] have predicted that chains of the heavier element Au could be formed, but not of the lighter ones, Ag and Cu. The same studies also predicted that similar results are obtained when Pt is compared with Pd and Ni, i.e., that chains of Pt but not of Pd or Ni can be formed. However, as for Ag and Cu, later experimental studies have contradicted this prediction for Pd and Ni; see, e.g., Ref. [5]. Furthermore, the smallest conductance that is measured is around 2:5G 0 ; making these materials similar to Pt at least when considering conductance through a linear chain. On the other hand, the large value of the conductance suggests that the band structures around the Fermi energy is complicated and many bands cross the Fermi level. Later theoretical calculations, by Ribeiro and Cohen [10], have actually found that chains of Pd are stable and should exist also in break-junction experiments; cf. Table 6.2, and, ﬁnally, Delin and Tosatti predicted that Pd chains should be magnetic [8].

6.7. Ru

6.6.

91

Ir, Rh, and Co

Compared to Pt, Pd, and Ni, the three elements Ir, Rh, and Co have each one less valence electron. Therefore, the Fermi level is expected to lie in the middle of the manifold of bands created from the valence s and d orbitals. This in turn may suggest that the smallest conductance, for an atomically thin wire, will be even larger compared to the value for Au, Ag, and Cu, as was the case for Pt, Pd, and Ni. On the other hand, the large number of both ﬁlled and vacant valence s and d orbitals may lead to a high reactivity of these systems, so that chain structures may not be formed. In one work, Smit et al. reported [17] that the conductance for the thinnest Ir chains indeed is around 2:3G 0 ; which is larger than the equivalent value for Au and Pt. Since any band crossing the Fermi level cannot contribute more than 1G0 to the conductance, the experimentally measured value of Smit et al. [17] indicates that several bands are crossing the Fermi level. This is in accordance with the ﬁrst expectation above. However, in another work Smit et al. reported [6] that just as for Ag/Au and Pd/Pt, there is a clear difference between Rh and Ir: Ir can form linear chains, just as Au and Pt, whereas Rh does not, just as Ag and Pd. Notice, however, that for Ag the original suggestion that linear chains will not be formed later was contradicted by experiment and, therefore, the same may be the case for Rh. On the other hand, in their theoretical work Ribeiro and Cohen [10] found that both linear and zigzag chains of Rh should be stable over a certain range of strains, i.e., both structures may be found for Rh in a break-junction experiment. They also presented the band structures for the optimized structures of linear and zigzag chains of various elements. These are reproduced in Figure 6.11. In this ﬁgure we clearly see how the number of bands crossing the Fermi level increases when, e.g., passing from Ag via Pd to Rh. This may be most clearly recognized for the linear chain but can also be seen for the zigzag chain. We ﬁnally mention that, just as for Pd, Delin and Tosatti [19] found in their theoretical work that the linear chain in its ground state structure should have a non-zero magnetic moment. For Co wires deposited on the vicinal Pt(997) surface a non-vanishing magnetic moment has indeed been observed experimentally [12]. According to the theoretical work of Sabirianov et al. [22] slightly thicker wires of Co are also magnetic.

6.7.

Ru

Ru is an element in the middle of the transition metals. It has an approximately half-ﬁlled d shell as well as an half-ﬁlled s shell and, accordingly, we expect many bands to cross the Fermi level. This is, in fact, also what is found in the theoretical calculation of Ribeiro and Cohen [10] as shown in Figure 6.11. Moreover, the conductance for a linear chain of Ru should take a fairly large value, because of the large number of bands crossing the Fermi level. Delin and Tosatti [19] predicted a value for the conductance of Ru comparable with that of Rh. We are not aware of any experimental studies of Ru chains; maybe this material indeed is one for which linear chains are not stable, not even under strain. This

92

Chapter 6. Chains of other sd Elements

[linear chains @ equilibrium]

2

Energy [eV]

0 −2 −4 −6 Al

Au

Ag

Pd

Rh

Ru

Au

Ag

Pd

Rh

Ru

−8

[Zigzag chains @ equilibrium]

2

Energy [eV]

0 −2 −4 −6 Al −8

Γ

π/a Γ

π/a Γ

π/a Γ

π/a

Γ

π/a

Γ

π/a

Figure 6.11. Band structures of the optimized linear (upper panel) and zigzag (lower panel) chains for different elements. In all cases the Fermi level is put at the energy zero. Reproduced with permission of The American Physical Society from Ref. [10].

conclusion was actually obtained by Ribeiro and Cohen [10] in their theoretical study. This means that any distortion may lower the total energy. This could for instance be the structural distortions considered by Ribeiro and Cohen (i.e., the formation of a zigzag structure). Another type of distortion is the occurrence of a spin-density wave, i.e., of magnetic type. This was studied by Delin and Tosatti [19] who found that out of the four 4d elements they studied (Ru, Rh, Pd, and Ag), a linear chain of Ru possessed the largest magnetic moment, both in the ground-state structure and when being strained.

6.8.

Nb

Nb has been the subject of break-junction experiments, cf., e.g., Ref. [5]. Moreover, a relatively early theoretical study of Cuevas et al. [23] on the conductance through a single atom focused on Nb. We shall discuss here this study in some detail.

93

6.8. Nb

Cuevas et al. [23] considered the system of Figure 6.12. The left and right lead are approximated through a jellium model, whereas for the junction between the two leads a tight-binding model is used. Applying a voltage of V between the leads, the current ﬂowing is given through I¼

2e h

Z TðE; V Þ½f L ðEÞ f R ðEÞ dE,

(6.2)

where f L and f R are the for Fermi-distribution functions of the left and the right lead, respectively, and TðE; V Þ the transmission probability that depends on both energy and the applied voltage. The conductance G ¼ I=V may subsequently be approximated through (see Ref. [23] and Section 2.4) G¼

2e2 Tr½tty h

(6.3)

where t is a transmission matrix that depends on the energy of interest. The eigenvalues of tty are real and lie between 0 and 1. Perfect transmission is obtained for channels with eigenvalues equal to 1, but for most channels the eigenvalues are smaller and often only very few channels contribute to the conduction. Cuevas et al. [23] depicted the eigenvalues of tty as a function of energy. The results are shown in Figure 6.13. It is seen that there are six channels contributing to the conduction in an energy range of 1 eV from the Fermi level and that two pairs give the same contributions. The six channels correspond to the ﬁve 4d and one 5s orbitals that are the band orbitals in that range. Moreover, for a linear chain along the z direction (see Figure 6.12), the d xz and d yz as well as the d x2 y2 and the d xy orbitals

Layers

Left Lead

Right Lead

z Central Region

Figure

6.12. Idealized geometry for an one-atom contact. Reproduced with permission of The American Physical Society from Ref. [23].

94

Chapter 6. Chains of other sd Elements

Figure 6.13. Transmission eigenvalues as a function of energy for a Nb one-atom contact for the structure of Figure 6.12. Reproduced with permission of The American Physical Society from Ref. [23].

5–1

6–1

9–4

12–6–1

17–12– 6 –1

Figure 6.14. Structures of Ti nanowires with diameters from 0.75 to 1.71 nm. The left and right part shows, respectively, a top view and a side view. The helical multishell structures are from the top to the bottom the 5 – 1, 6 – 1, 9 – 4, 12 – 6 – 1, and 17 – 12 – 6 – 1 ones. Here, the integers give the number of atoms in the different, concentric shells of atoms. Reproduced with permission of The Institute of Physics from Ref. [24].

6.10. Conclusions

95

are degenerate, thus explaining their identical contributions to the conduction. The ﬁgure shows very clearly that the perfect conduction only very rarely is observed and, moreover, that the conduction depends critically on orbital and energy.

6.9.

Zr and Ti

Compared to Au, Ag, and Cu, the elements Zr and Ti are in the opposite side of the periodic table and have close to empty d valence shells. This suggests that in a break-junction experiment one would again observe only few conduction channels and, accordingly, a relatively low conductance. There is, however, to our knowledge no experimental studies of this issue. In fact, chains of either Zr or Ti have been the focus of only few experimental or theoretical investigations. Here, we shall in particular emphasize the theoretical works of Wang et al. [24–26] who used parameterized methods in studying the electronic and structural properties of Zr and Ti nanowires. They found, similar to what we have discussed for Pt and Au, that helical multishell wires can exist and are stable. Some representative examples are shown in Figure 6.14. It turns out that the binding energy is surprisingly high; up to around 90% of the values found for the crystalline materials.

6.10.

Conclusions

No element possesses identically the same properties as any other element. Nevertheless, when comparing the results of this section with those for the prototype, gold, that was discussed in the preceding section, we can identify many general trends. First, many, if not all, elements can form atomically thin chains of shorter or longer length, for instance in break-junction experiments. However, experimental or theoretical predictions have suggested that this is not the case for certain elements. Second, in many cases helical, slightly thicker wires can be formed. For Pt, as also for Au, these have been observed experimentally. Third, the conductance for the thinnest wires increases as more bands cross the Fermi level. However, this general conclusion should be taken with much caution, as the contribution of each band to the conduction varies strongly, both as a function of type of underlying orbital and as a function of energy. Fourth, some chains show a magnetic behaviour, which is in clear contrast to the ﬁndings for gold. Also here, magnetism is expected to be largest for elements in the middle of the transition-metal series. Fifth, the search for Luttinger-liquid behaviour has hardly been extended to other sd metals than gold. To summarize, by varying the element, a new degree of freedom for controlled variation of system properties is obtained. This degree of freedom can be exploited to obtain general knowledge about the properties of atomically thin wires.

96

Chapter 6. Chains of other sd Elements

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

M. Springborg and P. Sarkar, Phys. Rev. B 68, 045430 (2003). M.H. McAdon and W.A. Goddard, III., J. Chem. Phys. 88, 277 (1988). A. Delin and E. Tosatti, J. Phys. Condens. Matter 16, 8061 (2004). S.R. Bahn and K.W. Jacobsen, Phys. Rev. Lett. 87, 266101 (2001). N. Agraı¨ t, A.L. Yeyati, and J.M. van Ruitenbeek, Phys. Rep. 377, 81 (2003). R.H.M. Smit, C. Untiedt, A.I. Yanson, and J.M. van Ruitenbeek, Phys. Rev. Lett. 87, 266102 (2001). V. Rodrigues, J. Bettini, A.R. Rocha, L.G.C. Rego, and D. Ugarte, Phys. Rev. B 65, 153402 (2002). J. Zhao, C. Buia, J. Han, and J.P. Lu, Nanotechnology 14, 501 (2003). Y.J. Lee, M. Brandbyge, M.J. Puska, J. Taylor, K. Stokbro, and R.M. Nieminen, Phys. Rev. B 69, 125409 (2004). F.J. Ribeiro and M.L. Cohen, Phys. Rev. B 68, 035423 (2003). J.R. Ahn, Y.J. Kim, H.S. Lee, C.C. Hwang, B.S. Kim, and H.W. Yeom, Phys. Rev. B 66, 153403 (2002). A. Dallmeyer, C. Carbone, W. Eberhardt, C. Pampuch, O. Rader, W. Gudat, P. Gambardella, and K. Kern, Phys. Rev. B 61, 5133 (2000). J. Opitz, P. Zahn, and I. Mertig, Phys. Rev. B 66, 245417 (2002). T. Nautiyal, S.J. Youn, and K.S. Kim, Phys. Rev. B 68, 033407 (2003). T. Nautiyal, T.H. Rho, and K.S. Kim, Phys. Rev. B 69, 193404 (2004). W.Y. Kim, T. Nautiyal, S.J. Youn, and K.S. Kim, Phys. Rev. B 71, 113104 (2005). R.H.M. Smit, C. Untiedt, G. Rubio-Bollinger, R.C. Segers, and J.M. van Ruitenbeek, Phys. Rev. Lett. 91, 076805 (2003). S.K. Nielsen, M. Brandbyge, K. Hansen, K. Stokbro, J.M. van Ruitenbeek, and F. Besenbacher, Phys. Rev. Lett. 89, 066804 (2002). A. Delin and E. Tosatti, Surf. Sci. 566–568, 262 (2004). Y. Oshima, H. Koizumi, K. Mouri, H. Hirayama, K. Takayanagi, and Y. Kondo, Phys. Rev. B 65, 121401 (2002). Y. Kondo and K. Takayanagi, Science 289, 606 (2000). R.F. Sabirianov, A.K. Solanki, J.D. Burton, S.S. Jaswal, and E.Y. Tsymbal, Phys. Rev. B 72, 054443 (2005). J.C. Cuevas, A. Levy Yeyati, and A. Martı´ n-Rodero, Phys. Rev. Lett. 80, 1066 (1998). B. Wang, S. Yin, G. Wang, and J. Zhao, J. Phys. Condens. Matter 13, L403 (2001). B. Wang, G. Wang, and J. Zhao, Phys. Rev. B 65, 235406 (2002). L. Hui, B.L. Wang, J.L. Wang, and G.H. Wang, J. Chem. Phys. 120, 3431 (2004).

Chapter 7

Chains of sp Elements

In the last two sections we have studied elements with partly ﬁlled valence s and d shells. Often the crystal structures of these elements are closely packed and the interatomic interactions are largely undirectional. Therefore, the valence electrons are often either very delocalized or, for the 3d transition metals, well localized to the atomic cores. The situation is different when we pass to the sp elements. Here, the interatomic interactions are often due to the directional bonds from p orbitals or sp3 ; sp2 ; or sp hybrid functions. Moreover, in some cases the elements are not metallic in the crystalline phase but may be so when considering lower-dimensional structures like surfaces and chains. In this section we shall discuss these elements.

7.1.

Al

For the sd elements gold may be considered the prototype that is at the centre of the scientiﬁc studies. Equivalently, aluminium may be considered the prototype for the sp metals and has, therefore, been studied often and in detail. For this reason we shall start our discussion of the sp elements with Al. Before discussing break-junction experiments we shall discuss the properties of isolated, inﬁnite, periodic, atomically thin Al wires. These have been studied in more theoretical works [1–5]. In the Table 6.2 and Figure 6.11, we presented results from the theoretical study of Ribeiro and Cohen [1] on linear and zigzag chains of various elements, including Al. The linear chain was found to be metallic with one band crossing the Fermi level. Passing to the zigzag chain, the Al–Al–Al bond angle is found to be fairly large but, nevertheless, there are some marked changes in the band structures compared to those of the linear chain. Ribeiro and Cohen did not directly exploit the zigzag symmetry of the system and have, hence, a translational repeated unit containing two atoms. Therefore, their bands for the zigzag chain all meet pairwise at the boundary of the Brillouin zone. In Figure 6.11 this is most clearly seen for Al. One of these pairs of bands crosses the Fermi level twice. As we discussed in Chapter 2, the existence of a partly ﬁlled band suggests that the total energy can be lowered upon some distortion that increases the size of the repeated unit and simultaneously opens up a gap at the Fermi level. One possibility is a structural distortion; another one being a magnetic distortion. Ayuela et al. [5] 97

98

Chapter 7. Chains of sp Elements

found, however, that a linear chain remains metallic despite the occurrence of a magnetic order and that it will not lower its total energy upon a bond-length alternation (i.e., a doubling of the unit cell). Later, Ono and Hirose [3] extended this study to include more different structural and magnetic orderings. Figure 7.1 summarizes some of their main ﬁndings. The ﬁgure shows that the lowest total energy is found for a fairly complicated structure, containing both a magnetic and a dav − Δd

(A)

dav + Δd

dav−Δd

dav + Δd

Lz 2(dav − Δd) dav + 2Δd

(B)

2(dav − Δd) dav + 2Δd

Lz 3(dav − Δd)

(C)

dav + 3Δd

3(dav − Δd)

dav + 3Δd

Lz (D) (A)PM (B)PM (C)PM

(A)FM (B)FM (C)FM

(A)AFM (B)AFM (C)AFM

Total energy (meV/atom)

40

0

−40

0.0

0.2

0.4 Δd (a.u.)

0.6

0.8

Figure 7.1. (A–C): Models of dimerized, trimerized, and tetramerized linear chains of Al atoms. (D): The calculated total energy as a function of the symmetry-lowering parameter for paramagnetic (PM), ferromagnetic (FM), and antiferromagnetic (AFM) ordering. Reproduced with permission of American Physical Society from Ref. [3].

99

7.1. Al

structural distortion leading to a trimerized chain. However, despite this low symmetry, the chain remains metallic according to the calculations of Ono and Hirose [3]. However, Sen et al. [4] and Zheng et al. [2] found that the linear chain is unstable compared with other types of chains. This is actually in agreement with the results of Ribeiro and Cohen [1] that a linear chain can be formed only upon stretching. In particular, Sen et al. [4] considered several different forms for chains and optimized their structure. Their results are shown in Figure 7.2. They found moreover that all structures were metallic. The fact that a linear chain has one band crossing the Fermi level suggests that in a mechanically controlled break-junction experiment on Al, the smallest conductance would be at most 1G 0 ; if not somewhat below this value. In fact, measurements (see Ref. [6]) have given a value around 0:8G 0 ; in perfect agreement with the consensus that linear chains are formed just before breaking. However, the experiments [7] give clear signals of larger values of conductances, suggesting that not only a linear chain can be formed but also thicker ones with more atoms per cross section. The theoretical study of Jelı´ nek et al. [8] illustrates this clearly. They performed a molecular-dynamics-simulation study of stretching a junction formed by aluminium. The structure of the junction at different steps during the process is shown in Figure 7.3, where it clearly is seen how the structure gradually becomes increasingly thinner. The conductance was also calculated by Jelı´ nek et al. [8] during the s=d=2.41 A

−1.6

L α

−2.0

d=2.51 A

d=2.53 A

α

W α

ET (eV/atom)

s=1.26 A

−2.4

s=2.37 A

T

α

h

S

d

−2.8 s=1.28 A

−3.2

d=2.79 A d

C

h=4.15 A Bulk Al

−3.6 1.0

d

d

1.5

2.0 s(A)

2.5

Top view

3.0

Figure 7.2. Calculated total energy for various structures of an inﬁnite, periodic chain of Al atoms. The energies are given relative to that of a free Al atom. Reproduced with permission of The American Physical Society from Ref. [4].

100

Chapter 7. Chains of sp Elements

5

1

5 1

2

5 1

4 2

5 1

4 2

3

3

3

(A)

5

5

4

(B)

(C)

4 2

(D)

2

2

3

4

1

4

1 3

3

(E)

(F)

Figure 7.3. Structure of an Al nanowire for different steps during a stretch process. Reproduced with permission of The American Physical Society from Ref. [8].

4

G [2∗e^2/h]

3

2

1

0 0.0

0.4

2.4 2.0 1.6 1.2 [Ang] displacement Stretching 0.8

2.8

3.2

Figure 7.4. Total conductance (thick curve) and contributions from different conductance channels during the stretching of the Al nanowire of Figure 7.3. Reproduced with permission of The American Physical Society from Ref. [8].

stretching, and is shown in Figure 7.4. When comparing the two ﬁgures we see that the number of conductance channels decreases whenever the narrowest part of the junction gets thinner, clearly demonstrating how band-structure effects determine the conductance, i.e., how the number of bands crossing the Fermi level for a chain with a structure similar to that of the thinnest part of the junction is of paramount importance for the conductance. Thygesen and Jacobsen [9] observed an interesting aspect in their theoretical study of the conductance through a ﬁnite, linear chain of Al atoms. In Figure 6.3 we have shown that the conductance through ﬁnite chains of Na, Cs, Cu, Ag, and Au shows a remarkable even–odd oscillatory behaviour. For Al a similar, but

101

7.1. Al

Conductance (2e2/h)

2.5 First principles Model

2

1.5

1

0.5

1

2

3

4 5 6 7 Number of atoms (N)

8

9

Figure 7.5. The conductance through a ﬁnite, linear Al wire as a function of number of atoms in the chain. Reproduced with permission of The American Physical Society from Ref. [9].

nevertheless different behaviour is observed, i.e., the conductance shows a fouratom periodicity, cf. Figure 7.5. A ﬁnite chain is, of course, not an inﬁnite chain. Therefore, for a ﬁnite chain not all band energies that exist for an inﬁnite chain are found, but only a ﬁnite subset of those. When coupling the ﬁnite chain to two electrodes and sending a current through it, the conductance is large when the ﬁnite chain has an orbital with an energy close to the Fermi level of the complete system (chain and electrodes). In the limit of a very long chain there is no net charge per Al atom on the wire and the Fermi level is then that of the inﬁnitely long chain. However, the Fermi level is ﬁxed by the macroscopic electrodes, so that for a shorter chain the Fermi level is also as it is for the inﬁnite chain. For the ﬁnite chains, Thygesen and Jacobsen [9] found that orbitals with energies close to the Fermi level were found for chains of 3; 7; . . . atoms, thus giving a four-atom periodicity in the conductance. The fact that linear chains of Al at most are metastable may be related to either the occurrence of directional bonds or to a tendency of forming closed-packed structures. Therefore, other types of chains of Al may also exist. We have noted that zigzag chains form one possibility but this is not the only one (Figure 7.2). One might therefore suggests that the fcc crystal structure of Al is retained for wires of a ﬁnite width. This possibility was explored by Taraschi et al. [10] in a theoretical study. They found indeed that starting with the inﬁnite fcc crystal structure and subsequently cutting out a wire along the (001) direction with a square cross section one obtains a stable structure. However, to our knowledge there has been no direct experimental observation of this kind of structure. Very different types of structures were considered by Gu¨lseren et al. [11] both for Al and for Pb. They compared thin wires with a fcc-like structure with what they found in a molecular-dynamics simulations. They found that for very (but not atomically) thin wires a different class of structures was the most stable ones. These structures were found to have a low symmetry and was by the author called weird wires. We show their results in Figure 7.6.

102

A1

a) Al wires

−1.2

A2 −2.4

A3

A8

A9

b) Pb Wires

B2 B3

A5

−1.4

−2.6

B1

B5

A6 A7

−3.0

A16

A10 A11

A13

Energy (eV/atom)

Energy (eV/atom)

−2.8

B4 −1.6

B8

B6 B7

−1.8

A14

−3.2

−1

bulk −3.4 0.0

0.1

−1 Rc

0.2

A12 0.3 R−1 (Å−1)

0.4

0.5

−2.0 0.0

Rc bulk 0.1

n=1,2,3

n=1,2

0.2 −1

0.3

n=1 0.4

−1

R (Å )

Figure 7.6. Total energy per atom for (left panel) Al and (right panel) Pb wires as a function of the inverse radius. A selection of morphologies is shown, too, and the solid and dashed curves represent a ﬁt to the results for fcc-derived and weird wires, respectively. At Rc the weird wires are the most stable ones. Reproduced with permission of The American Physical Society from Ref. [11].

Chapter 7. Chains of sp Elements

A4 A15

7.2. Ga, In, and Tl

103

The study of Gu¨lseren et al. [11] was focused on structural and energetic properties. Subsequently, Di Tolla et al. [12] studied the electronic properties of the structures found by Gu¨lseren et al. [11]. Di Tolla et al. [12] found (maybe not surprising) that the number of bands crossing the Fermi level decreases as the wire radius becomes smaller. Simultaneously, the results of Gu¨lseren et al. suggested that for wires with radii below a certain threshold, Rc ; their structure is that of a weird wire. Correlating these two ﬁndings, Di Tolla et al. could predict that Al wires with a conductance below 11G 0 would not have a fcc-like but a weird structure. In a further study, Makita et al. [13] found that wires like those of Figure 7.6 may grow by adding atoms to the walls of thin wires and thereby becoming thicker. This would, of course, require that the wires are formed in some kind of equilibrium situation and given the opportunity to grow. Whether this will become possible is at the moment of writing an open question. In Chapters 5 and 6 we also studied chains that were deposited on surfaces. For sd metals one may assume that there is no dangling bonds, neither on surfaces nor on chains, so that the surfaces of slightly thicker wires of sd metals may not be highly reactive. The situation is different for the materials of this section, i.e., the sp metals. Using theoretical methods, Gupta and Batra [14] studied the properties of chains of Al, Ga, and In deposited on a dihydrogenated Si(001) surface. First they found that the naked surface as such has a fairly large gap at the Fermi level. Placing Al atoms on the surface and identifying the structure of the lowest total energy led to a structure where the Al atoms occupy two different types of positions and, moreover, the system has still a gap at the Fermi level, although small. Thus, in this case the metallic property of the Al chains that we have discussed above is suppressed on the surface, indicating that in this case we do not really have a chain deposited on the surface but a new system comprising of both Si, H, and Al atoms.

7.2.

Ga, In, and Tl

Ga, In, and Tl belong to the same group as Al but have been studied much less intensively within the context of quasi-one-dimensional structures. Break-junction experiments on Ga (see, e.g., Ref. [6]) have given a smallest conductance around 1G 0 ; i.e., comparable with the value for Al. Moreover, the theoretical work by Gupta and Batra [14] showed that Ga and In, when deposited on a dihydrogenated Si(001) surface, may be metallic but the substrate modiﬁes the chain properties markedly. The fact that Ga and In can also be deposited on a silicon surface has been shown in experiment. Below we shall discuss in more detail the case of In, but ﬁrst we brieﬂy mention the very detailed work of Gonza´lez et al. [15] who used both experimental and theoretical methods in characterizing Ga wires on the Si(112) surface. The most stable structure was that of a zigzag arrangement of the Ga atoms that led to a passivation of all dangling bonds on the silicon surface. This emphasizes once more that the role of the surface is often more active than simply providing a medium on which the chains can lie without interactions between the chains and the substrate.

104

Chapter 7. Chains of sp Elements

Figure 7.7. Scanning-tunnelling-microscopy images of the (a) room-temperature and (b) the lowtemperature phase of In on Si(111). The black dashed lines mark the 4 1 and 400 200 unit cell in the two cases, respectively. White areas mark the In atoms, and in (b) the white dashed lines mark the phase boundary between two different charge-density-wave conﬁgurations. Reproduced with permission of The American Physical Society from Ref. [16].

There is, however, one exception. Thus, indium chains on the Si(111) have been at the centre of some discussion. Yeom et al. [16] reported that indium chains selforganize on the Si(111) surface giving a highly regular structure that, what may be the most interesting feature, shows a phase transition as a function of temperature. This transition was monitored with scanning-tunnelling-microscopy experiments, as shown in Figure 7.7. In Figure 7.7 we see a change when passing from room temperature to a lower temperature. This change is recognized as the occurrence of alternating darker and lighter regions along the In chains. Yeom et al. [16] argued that in the hightemperature phase of the higher symmetry there are three bands crossing the Fermi level. As temperature is decreased, a symmetry lowering takes place, as we have discussed in Chapter 2, and a gap opens up at the Fermi level. The symmetry lowering is in this case a charge-density wave that gives rise to alternating higher and lower electron densities, as also seen in the ﬁgure. The occurrence of a gap at the Fermi level may have many reasons, including the creation of a charge-density wave. However, a Luttinger-liquid behaviour may also be proposed, but this possibility has been excluded on the basis of angle-resolved photoelectron spectroscopy by Yeom et al. [17]. Theoretical methods have also been applied for In chains on Si(1 1 1). Here, we brieﬂy discuss the results of Cho et al. [18]. In Figure 7.8 we show their optimized structures. Here, it is clearly seen that the indium atoms are integrated into the surface and, accordingly, cannot be considered as lying on the surface with only weak interactions between In and Si. Moreover, the 4 1 and 4 2 structures both contain zigzag chains on/in the surface. In Figure 7.9 we show the calculated band structures for the 4 2 structures. The unit cell of the 4 2 structure is twice that of

7.3. C

105

Figure 7.8. Calculated structure of In chains on the Si(111) surface. (a)–(b) The 4 1 structure from (a) the side and (b) the top; and (c) the 4 2 structure from the top. Dark and white circles mark In and Si atoms, respectively. Reproduced with permission of The American Physical Society from Ref. [18].

the 4 1 structure, as a consequence of the lower symmetry of the former structure. This leads to some interactions of the bands and the opening up of smaller gaps, as shown in (b) and (c) in Figure 7.9. Therefore, depending on the precise position of the Fermi level (see Figure 7.9), it may appear at an energy of some of those small gaps, i.e., at an energy of a reduced spectral intensity. Therefore, in an experiment it may look as if the intensity is vanishing, although it may just be (signiﬁcantly) reduced. Tl is the heaviest element in this group and, accordingly, relativistic effects should be most pronounced for this element. In Figure 7.10 we show calculated [19] band structures for some of the structures of Figure 6.1. The lowest bands are due to 5d orbitals so that their splitting gives an immediate estimate of spin–orbit couplings. Thus, these amount to some 2–3 eV for Tl and its adjacent elements. However, also Tl has been found to form chains when being deposited on a Si surface, this time the Si(100) surface [20]. As for In, a temperature-dependent transition is observed with a low-symmetry structure at low temperatures. In this case the experiments give indications of couplings between the chains through the substrate, so that for this system it is certainly not possible to talk about largely isolated chains of metals.

7.3.

C

Carbon is extremely well known for its importance in organic and biological chemistry, but is only rarely brought into connection with metal physics. Nevertheless, we shall here include a short description of some pure-carbon systems that are quasi-one-dimensional chains and that show properties that resemble those of chains of metal atoms. We add that due to the diversity of carbon chemistry, it is not possible here to provide anything but just a ﬂavour of this ﬁeld. Moreover, later, in Chapter 12, we shall discuss some other materials based on carbon that also can show metallic behaviour. The importance of carbon in chemistry can be traced back to its ability to form directed bonds through hybridization. From the 2s and 2p valence orbitals of the

106

Chapter 7. Chains of sp Elements

(a)

Energy [eV]

1.0 0.5 m1 0.0

A A

B′ B

m2 m3

−0.5

C D x′ k′ M

−1.0

Γ k x

k

Γ

k′

x′

Γ

(c)

(b) 0.5 A′ 0.19 A 0.0

En

B′ 0.24

En En B

Figure 7.9. The band structures for the 4 2 structure of Figure 7.8 for In on Si(111). Only the surface bands are shown, and (b), (c) blow-ups of parts of the band structures from (a). By moving the Fermi level slightly upwards, it may occur at an energy of a low density of states. Reproduced with permission of The American Physical Society from Ref. [18].

isolated atom, one may form spn hybrids with n ¼ 1; 2; or 3. These hybrid orbitals can participate in directional covalent (s) bonds with neighbouring atoms. These bonds are strong and, accordingly, they appear at deep energies. The equivalent unoccupied antibonding orbitals appear at high energies and a large energy gap separates them. However, when na3; 3 – n additional p orbitals exist that may participate in weaker ðpÞ bonds between the atoms. These bonds are weak and only a relatively small gap separates occupied and unoccupied orbitals. For such systems the gap may even (approximately) vanish and the system is close to be a metal. As an illustration we show in Figure 7.12 band structures for the linear chains of carbon atoms that are shown in Figure 7.11. The chemical bonds between the atoms of a linear chain of carbon atoms are formed from sp hybrids that form strong, energetically deep s orbitals as well as double degenerate p orbitals. In Figure 7.12 it can be seen that the s orbitals give rise to the bands between 14 and 21 eV; whereas the p orbitals are found at energies above 10 eV: As shown in Figure 7.11, one may consider two different structures for a linear chain of carbon atoms, i.e., either one of the constant bond lengths or one of the alternating bond lengths. Figure 7.12 shows that the former results in a vanishing gap

107

7.3. C

0

Energy (eV)

−5

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

−10

−15

−20 0

Energy (eV)

−5 −10 −15 −20 −25 0

Energy (eV)

−5 −10 −15 −20 −25 −30

k

k

k

Figure 7.10. Band structures for (left column) linear, (middle column) zigzag, and (right column) double zigzag chains of (top row) Tl, (middle row) Pb, and (bottom row) Bi. Reproduced from Ref. [19].

at the Fermi level, whereas the latter gives rise to a gap there. In fact, the lowest total energy is found for a structure of alternating bond lengths (see, e.g., Ref. [21]) and due to the simultaneous occurrence of a gap at the Fermi level, one may interpret this as a manifestation of a Peierls distortion as we have discussed in Section 2.1.

108

Chapter 7. Chains of sp Elements

Figure 7.11. Structure of a linear chain of carbon atoms with (upper part) non-alternating (double) bonds between the atoms and (lower) part alternating (single and triple) bonds between the atoms.

0

Energy (eV)

−5 −10 −15 −20 −25

0

0.5 k

1 0

0.5 k

1

Figure 7.12. Band structures for the linear chains of carbon atoms of Figure 7.11. The bond lengths were non-alternating in the left panel and equal to 2.6 a.u., and in the right panel alternating and equal to 2.5–2.7 a.u. k ¼ 0 and k ¼ 1 mark the centre and the edge of the ﬁrst Brillouin zone, and the horizontal dashed line represents the Fermi level.

However, although much effort has been invested in trying to synthesize inﬁnite chains of carbon atoms (see, e.g., Ref. [22]), it has so far not been possible to produce extended chains. In fact, theoretical studies have shown that for larger values of n, Cn will prefer other, energetically lower structures like quasi-two-dimensional sheets or real three-dimensional clusters [23]. The transition is found to lie for n ’ 20: But for smaller values of n such chains can be produced and examined. Furthermore, in contrast to many of the elements we have discussed so far, carbon wires are not produced in break-junction experiments. Instead, carbon nanowires of ﬁnite length form a popular playground for ﬁrst of all theoretical studies of organic molecules that may be the smallest possible semiconducting devices that can be produced. As discussed above, molecules based on sp2 -bonded carbon atoms have a relatively small energy gap between the occupied and unoccupied orbitals which, in turn, are formed largely by fairly delocalized p electrons. Therefore, to some extent, such molecules have properties similar to those of semiconductor devices. In order to construct a semiconductor device based on these organic molecules, the ﬁrst step is to be able to address the molecules, i.e., to attach

7.3. C

109

leads to them. Accordingly, much effort is invested in studying such molecules when they are placed between two metallic tips that are forming the leads and, subsequently, sending a current through the molecules by applying a voltage between the leads. Although the immediate goal is similar to what we have studied for many other elements, i.e., to study the conductance through an atomically thin junction, the ultimate goal as well as the production route differ markedly. For the organic molecules, one wants to construct semiconductor devices. Moreover, often the organic molecules are prepared in some solution and sought attached to the two metal tips through careful chemical synthesis of special organic molecules containing special groups that can form a strong bond to the metal tips. This could be, e.g., sulphur-containing groups (for instance thiols), when gold is the tip material. Despite the fact that the present work focuses on metal systems, we shall here also brieﬂy discuss the carbon chains, as many of the results for those are both relevant and also related to those that we have discussed for more ‘real’ metal chains. Broglia [24] as well as Lang and Avouris [25] studied theoretically the charge transfer through a ﬁnite, linear chain of carbon atoms as a function of length of the chain and of the applied voltage. Figure 7.13 shows results from the study of Lang and Avouris [25]. As discussed above, an inﬁnite linear chain of carbon atoms with non-alternating bond lengths has one doubly degenerate band crossing the Fermi level, whereas a bond-length alternation leads to a gap at the Fermi level. This may suggest the occurrence of either two or none conduction channels. However, the results of Lang and Avouris for a chain of non-alternating bond lengths show that the conductance is somewhere between 1G0 and 2G0 and that it shows a clear even–odd alternation similar to what we have seen for other elements (cf., e.g., Figure 6.3). That the conductance depends critically on the precise structure of the chain, was further demonstrated by building in a 90 bend in the middle of the chain. Intuitively one may expect that the electrons then less easily are transferred through the system and, actually, the conductance is reduced. The authors found also that the bonding of the chain to the metal tips is accompanied by a transfer of electrons from the tips to the chain that increases when the chain length is increased, although slower than proportional to the chain length. Subsequently, Lang and Avouris [25] studied in detail what happened when such a chain, i.e., the C7 wire, was exposed to an external electrostatic potential as would be the case when using the wire as an active component in a semiconductor device. Figure 7.14 shows some of their results, where they have compared both the isolated free wire and the wire linked to the two metals. The two uppermost panels of Figure 7.14 demonstrate that the metal reservoirs participate actively in the response to the external ﬁeld. Moreover, when the carbon wire is inserted between the metals, the potential drop is ﬁrst of all more structured but also spread out over a much larger part of the system. The linear carbon chains provide a simple model system for exploring carbonbased systems based on sp- or sp2 -bonded carbon atoms with p orbitals closest to the Fermi level. Whereas sp bonds are relevant for the linear carbon chains, most experimentally realized systems possess sp2 -bonded carbon atoms, and in many

110

Chapter 7. Chains of sp Elements

Conductance (2e2 /h)

(a) 2.0 1.8 1.6 1.4 1.2 1.0 0.8 3

4

8

1.3

0.30

1.2 0.25

1.1 1.0

0.20

0.9 0.15

0.8 3

4

5

6

7

8

9

Transferred charge (e/C-atom)

Transferred charge (e)

9

Length (a.u.) 10 12 14 16 18 20 22

(b) 6

5 6 7 8 Number of C-atoms

Number of C-atoms Figure 7.13. (A) The conductance through a linear chain of carbon atoms as a function of chain length. The triangles mark results for which the chain has a 90 bend at the central atom. (B) The amount of transferred charge from the metal reservoir to the carbon chain, either per atom or in total as a function of chain length. Reproduced with permission of The American Physical Society from Ref. [25].

cases also other types of atoms. There exists, however, one type of system that is quasi-one-dimensional, based on only carbon atoms, and have interatomic bonds formed by sp2 hybrids. This system, carbon, or fullerene, nanotubes, has been at the centre of research since its discovery in 1991 [26]. A large number of experimental and theoretical studies have been devoted to fullerene nanotubes (see, e.g., Refs. [27–36]), but here we shall discuss just some few aspects, i.e., mainly those that are related to the discussion in the remaining parts of this presentation. Often one think of the carbon nanotubes as being formed by rolling up a graphene sheet, as shown in Figure 7.15. A graphene sheet consists of regularly placed hexagons with carbon atoms at the apexes. Figure 7.15 shows such a pattern. The hexagons can be enumerated with two integers ðn; mÞ; cf. Figures 7.15 and 7.16. The sheet is now rolled up so that one of the hexagons (except for the one at the origin) is placed directly on top of the hexagon at the origin, cf. Figure 7.14. The two indices ðn; mÞ of the graphene sheet are then used in characterizing the resulting nanotube. Actually, the materials of the earliest experimental studies contained microtubules consisting of more, essentially concentric, nanotubes with an intertubulus distance similar to the one that is found between the graphene sheets in ( crystalline graphite, i.e., around 3:4 A:

111

7.3. C

Distance (a.u.)

4 2 0 −2 −4 −10

−5

0 Distance (a.u.)

5

10

−10

−5

0 Distance (a.u.)

5

10

−10

−5

0 Distance (a.u.)

5

10

Distance (a.u.)

4 2 0 −2 −4

0.0

Potential (eV)

−0.5 −1.0 −1.5 −2.0 −2.5 −3.0

Figure 7.14. (A) The difference in the electron density of a free C7 wire exposed to an external ﬁeld of strength 6 V/nm and that of the same system without the ﬁeld. (B) The same difference when the C7 wire is linked to two metal reservoirs. (C) The differences in the electrostatic potential between the metal reservoirs without and with an applied electrostatic ﬁeld of 3 V. Here, the solid curve are the results without the C7 wire, whereas it is included in the results for the dashed curve. Reproduced with permission of The American Physical Society from Ref. [25].

Structurally different nanotubes occur only for m n; and two special types occur for m ¼ 0 and for m ¼ n: These two types are both shown in Figure 7.17 and are the only ones that have a translational symmetry. All other ones possess a screw-axis, or helical, symmetry. For the planar graphene sheet the interatomic bonds are due to strong s orbitals formed by the carbon sp2 orbitals and weaker p orbitals formed by carbon

112

Chapter 7. Chains of sp Elements

(a) Roll-up

Graphene sheet

SWNT

(b)

T

(n,0)

a1

Ch

a2 (n,n)

Figure 7.15. (a) A schematic representation of a two-dimensional sheet that is rolled up to form a nanotube; (b) The numbering scheme used for classifying the nanotubes. Reproduced with permission of The American Chemical Society from Ref. [30].

• [4,4] • [3,3] • [3,2]

• [2,2] R2

• [0,0]

•

• [2,1]

• • [1,1] R1 • [1,0]

• [2,0]

• [3,0]

• [3,1]

• [5,3]

• [4,3] • [4,2]

• [4,1] • [4,0]

• [5,4]

• [5,2] • [5,1]

• [5,0]

• [6,3] • [6,2]

• [6,1] • [6,0]

• [7,1] • [7,0]

Figure 7.16. The numbering scheme for a graphene sheet. Reproduced with permission of The American Physical Society from Ref. [37].

p functions. This separation in s and p orbitals is in the strict sense only possible for the purely planar system, whereas for the curved nanotubes it is no longer completely true. However, for not too small values of n (i.e., n 10), the curvature, or radius, of the nanotubes is so large that neglecting mixing of the s and p orbitals is a good approximation. For smaller values of n, this is no longer a good approximation, but these nanotubes have a very large strain, which may as to why for a long time only the larger nanotubes were found in experimental studies tell us.

7.3. C

113

Figure 7.17. Armchair (right) and zigzag (left) nanotubes. Reproduced with permission of American Institute of Physics from Ref. [38].

When the nanowire is not very small and the s=p separation provides a good approximation, the electronic structure of a nanotube can be related to that of a graphene sheet. The latter is periodic in two dimension so that any electronic orbital ~ When being rolled up, only a strip can be characterized through a two-dimension k: ~ of ﬁnite width is used and, accordingly, k can only take certain values. The values lie on a ﬁnite set of lines in the two-dimensional Brillouin zone of the graphene sheet; which lines depends on the values of n; m: A graphene sheet is a semi-metal with a zero-band gap at the Fermi level for exactly one set of symmetrically equivalent k~ points in the ﬁrst Brillouin zone. When this point is lying along one of the abovementioned lines that are relevant for a given carbon nanotube, this nanotube will have zero band gap, too; if not, the nanotube will have a band gap at the Fermi level. Analysing this in detail [37,39,40] gives that metallic nanotubes are obtained for n m ¼ 3q;

q ¼ 0; 1; 2; . . . .

(7.1)

This aspect is illustrated in Figure 7.18 that shows the band structures for two zigzag and one armchair nanotube. Furthermore, the two-dimensional Brillouin zone of a graphene sheet is shown too, as well as the lines that describe the allowed k~ values for a given nanotube. For an inﬁnite graphene sheet, the unoccupied and occupied bands meet at the K point and, therefore, a nanotube is metallic only when this k~ point lies on one of the lines.

114

Chapter 7. Chains of sp Elements

Tubule axis

E (eV)

Unit K

(b)

M r

4

3

3

2

2

1

EF

0

(d)

5

4

−1

1 −1 −2

−3

−3

−4

−4 −5

X

r

EF

0

−2

−5

(a)

(c)

5

E (eV)

(a)

r

X

(e) Tubule axis

5 4 3 2

Unit

(b)

E (eV)

1 K

EF

0 −1 −2

r

M

−3 −4 −5

Γ

X

Figure 7.18. The upper part shows the band structures for two zigzag carbon nanotubes (i.e., m ¼ 0) of which one is metallic, the other not. The lower part shows the band structures for an armchair carbon nanotube (i.e., n ¼ m). Reproduced with permission of The American Physical Society from Ref. [40].

The ﬁrst samples of carbon nanotubes that were generated experimentally contained multiwall nanotubes, i.e., nanowires consisting of several concentric tubes. Moreover, even the smallest of the nanotubes were so thick that the above-mentioned separation into s and p orbitals was considered an adequate approximation. Since then, an enormous wealth in experimental and theoretical improvements has appeared of which we shall review just a few. A very important development was the possibility to produce single-wall nanotubes [41]. As mentioned above, the band gap at the Fermi level of such nanotubes depends critically on the two parameters ðn; mÞ: Therefore, as individual nanotubes became available it became possible to study the electrical and optical properties of well-deﬁned nanotubes without those being obscured by interactions between different nanotubes. Moreover, theoretical speculations about the consequences of

7.3. C

115

various modiﬁcations also became highly relevant for the interpretation and understanding of the experimental results. These modiﬁcations include, e.g., the ﬁnite length (in contrast to the inﬁnite, periodic systems that usually are studied), the way a ﬁnite nanotube is being truncated, the growth processes of these materials, the occurrence of carbon heptagons and pentagons (i.e., structural defects) replacing the otherwise regularly repeated hexagons, mechanical deformations like bents, heterojunctions between nanotubes of different ðn; mÞ (this may lead to semiconductor/metal, metal/metal, or semiconductor/semiconductor junctions), Y-shaped junctions, decorations of the walls with various elements or molecules, ﬁlling of the tubes with smaller particles (we shall return to this issue later in the special case that metallic chains then are created inside the nanotubes), mechanical devices consisting, for instance, of a moveable, shorter, thicker tube outside a longer, thinner one, and many others. It is beyond the scope of this presentation to discuss all this variety of studies. Instead, we shall focus on those issues that are of direct relevance to ‘metallic chains’. Above, we gave in equation, (7.1) a simple rule for determining when a carbon nanotube is metallic. A zigzag nanotube, ðn; 0Þ; is metallic if n is an integral multiple of 3, for instance 15. Moreover, any armchair nanotube, ðn; nÞ; is metallic. Ouyang et al. [42] studied both types using tunnelling microscopy at low temperatures. In Figure 7.19 some of their results are reproduced, both for metallic n ¼ 9; 12, and 15 zigzag nanotubes and for a metallic n ¼ 8 armchair nanotube. The atomically resolved images show that most often the nanotubes occur in bundles. Measuring the current I through an individual nanotube as a function of applied voltage V and subsequently displaying dI=dV as a function of V gives a curve that is related to the local density of states at the position of the current. These ﬁgures are also shown in Figure 7.19. The ﬁgure shows clearly that only in one case, the isolated armchair nanotube, all systems show a small gap (below 0.1 eV) around the Fermi level. The authors interpreted this as being due to small, but obviously non-negligible, interactions between the different nanotubes making up a bundle. Thus, in order to obtain metallic behaviour it may be crucial to have isolated nanotubes. After it became possible to produce single-wall carbon nanotubes it also became possible to study these individually. In one of the earliest experiments in this direction [43] it was shown that such single-wall carbon nanotubes could act as a genuine quantum wire for which electrical conduction occurs through well separated, discrete electron states that are coherent over the whole wire. The same group also used scanning-tunnelling microscope and spectroscopy on individual single-wall carbon nanotubes in getting information on the electronic density of states, as shown in Figure 7.20 [44]. In this ﬁgure one can easily recognize sharp features that resemble the van Hove singularities that are well known and most pronounced for quasi-one-dimensional systems. Moreover, by studying a number of different nanotubes whose chirality [i.e., ðn; mÞ] was also determined, they found indeed that the gap at the Fermi energy depends critically on n; m: A number of experimental and theoretical studies has been devoted to the issue of determining how small the carbon nanotubes actually can be. For very small diameters one would expect strong strains that may make them unstable. On the other hand, for those the separation into s- and p-like orbitals may not at

116 Chapter 7. Chains of sp Elements Figure 7.19. Each (left and right) part shows atomically resolved images of carbon nanotubes, either (left part) zigzag or (right part) armchair tubes. In the right part not only bundles of tubes but also an individual one is shown. Moreover, dI=dV curves for different systems are shown. Reproduced with permission of American Association for the Advancement of Science from Ref. [42].

117

(dI/dV)/(I/ V )

4

0.2

DOS (a.u.)

5

dI/dV (nA V−1)

7.3. C

0.1 0.0

−1

1 0 Vbias (V)

−1

0 1 E (eV)

3

2

1

0

−1

0 Vbias (V)

1

Figure 7.20. The left insert shows a measured dI=dV curve for a single-wall carbon nanotube, whereas dI the right insert shows a calculated density of states. Finally, the main ﬁgure shows the measured VI dV curve. Reproduced with permission of Nature from Ref. [44].

all be adequate and, therefore, new effects may show up when this separation breaks down. Theoretically it has been shown (see, e.g., Ref. [45]) that for very small carbon ( the strain costs so much nanotubes with a diameter smaller than roughly 3:5 A; energy that these nanotubes are even less stable than a ﬁnite graphene sheet of the ( this energy difference exceeds same number of atoms. For diameters less than 2:5 A; even 1 eV/atom. However, Zhao et al. [46] have reported the experimental obser( These tubes were found vation of nanotubes with a diameter of not more than 3 A: as the innermost nanotubes in multi-wall carbon nanotubes. Also Sun et al. [47] have reported the synthesis of very thin carbon nanotubes. From a theoretical point of view the ﬁnding of such thin nanowires is surprising. It has been shown [48] that compared with just a rolled-up graphene sheet these nanotubes experience strong structural relaxations which ﬁrst of all result in an increase in their diameter through an increase in the lengths of the bonds lying along the perimeter of the tubes, whereas the bonds parallel to the tube axis are shortened. The band structures deviate signiﬁcantly from those of the graphene sheet that we have discussed above so that in many cases several bands cross the Fermi level, and equation (7.1), which gives those values of ðn; mÞ for which the nanotubes are metallic, is no longer satisﬁed. Finally, interactions between s and p orbitals and even across the inner part of the nanotubes lead to new orbitals and binding situations. The latter is illustrated in Figure 7.21 where we show some of the orbitals closest to the Fermi level for some very thin carbon nanotubes [48]. Carbon nanotubes have also been studied in the context of the possible existence of a Luttinger-liquid behaviour. Bockrath et al. [49] have studied the voltage and temperature dependence of the conduction through individual nanotube ropes

118

Chapter 7. Chains of sp Elements

Figure 7.21. The electron density for some of the frontier orbitals for some of the thinnest possible carbon nanotubes either perpendicular or parallel to the nanotube axis. The size of each plane is 10 10 a:u: Reproduced from Ref. [48].

(i.e., several nanotubes rolled-up forming a rope). They found that these dependencies were in accord with a Luttinger-liquid behaviour. Furthermore, Ishii et al. [50] performed photoelectron spectroscopy experiments on metallic carbon nanotubes and analysed in particular the spectra nearest to the Fermi level. For low temperatures they observed a vanishing spectral intensity at the Fermi energy and attributed this to a Luttinger-liquid behaviour. At the moment of writing it is, however, not clear whether these systems indeed form a Luttinger liquid. As we have seen repeatedly throughout this presentation, any interaction between the chain and some other system (other chains, a substrate, . . .) may provide alternative explanations. In some of the experiments we have discussed above, the conductance properties of carbon nanotubes were studied. These have also been the focus of several theoretical studies of which we shall just brieﬂy review some few.

119

7.4. Si, Ge, and Pb

Left lead

Device region

V1

V4 z1

Right lead

V z1

Figure 7.22. Schematic representation of a junction between a semi-inﬁnite Al(100) metal electrode and a semi-inﬁnite (4,4) armchair carbon nanotube. Reproduced with permission of The American Physical Society from Ref. [51].

Taylor et al. [51] considered a semi-inﬁnite nanotube that was attached to a semiinﬁnite metal lead as shown schematically in Figure 7.22. They developed a theoretical method for calculating the electrical transport through such a molecularelectronics device and applied the method ultimately on, e.g., a carbon nanowire attached to an Al electrode, as shown in Figure 7.22. They did not consider structural relaxations and have, accordingly, two semi-inﬁnite ideal systems in contact. It turned out, however, that the junction between the two is very critical to the conductance. By simply varying the distance between the lead and the nanotube, the calculated current I as a function of applied voltage V can be varied signiﬁcantly, cf. Figure 7.23. The slope of the I – V curve is the conductance that in the two examples of Figure 7.23 is 1G 0 and 0:39G 0 ; roughly. Taylor et al. [51] pointed out that for an ideal, inﬁnite, metallic carbon nanotube the conductance should be 2G 0 ; but that the measured values for such nanotubes attached to metals were signiﬁcantly below this value. The discrepancy may, accordingly, be due to the junction between the two systems. A similar dependence of conductance through a carbon nanotube on the contact geometry was also found by Palacios et al. [52]. They considered the two geometries of Figure 7.24. For the sake of completeness we should add that the lower panel in Figure 7.24 may provide a better description of most experimental set-ups. Often in the latter, carbon nanotubes are simply deposited more or less randomly on a patterned metal surface and through simple search one ﬁnds nanotubes that happen to form a bridge between two different metal leads.

7.4.

Si, Ge, and Pb

In the periodic table Si, Ge, and Pb are below C and, accordingly, many similarities between carbon and the other elements exist. There are, however, important

120

Chapter 7. Chains of sp Elements

20 15

d = 2.0 d = 2.5

10

1 (μA)

5 0 −5 −10 −15 −20 −25 −0.3

−0.2

−0.1

0

0.1

0.2

0.3

V1-VI (V) Figure 7.23. Calculated current I as a function of applied voltage V for the system of Figure 7.22. Two different nanotube-metal distances, 2.0 and 2.5 a.u., have been considered. Reproduced with permission of The American Physical Society from Ref. [51].

Figure 7.24. Structure of two contact geometries for a ﬁnite carbon nanotube in contact with two metal leads. Reproduced with permission of The American Physical Society from Ref. [52].

121

0.6

0.6

0.55

0.55

0.5

Energy (eV/atom)

Energy (eV/atom)

7.4. Si, Ge, and Pb

sh1

0.45

cd2 f-ful5

0.4

5

6

sh4 cd1

f-ful4

Ge

0.35

7

8

9

10

11

Nanowire Diameter (Å)

12

0.5

sh1

0.45

5

sh4 cd1

f-ful4

Ge

0.35 13

cd2

f-ful5

0.4

6

7

8

9

10

11

12

13

Nanowire Diameter (Å)

Figure 7.25. Total energy per atom relative to that of the crystalline diamond structure for Si and Ge nanowires as a function of their diameter. They have been labelled according to the scheme of Figure 7.26. Reproduced with permission of American Physical Society from Ref. [55].

differences. Thus, the richness of organic chemistry can be ascribed to carbons ability to participate in interatomic bonds formed by sp, sp2 ; and sp3 hybrids. A similar ﬂexibility does not exist for the atoms below C. Si has a strong tendency to occur in four-fold coordination with sp3 hybrids forming the interatomic bonds. A similar tendency is found for Ge where, however, d functions also start playing a role in the interatomic bonds. At the bottom of this row we ﬁnd Pb that is so heavy that realistic effects are important and can lead to noticeable modiﬁcations in the properties. Accordingly, Si, Ge, and Pb are systems of their own right, but less intensively studied in the context of chain systems. Theoretical studies [53–55] have conﬁrmed the consensus above, that quasi-onedimensional structures of Si are at most metastable. As one example, we show in Figure 7.25 results from the work of Kagimura et al. [55], who considered both Si and Ge nanowires. The total energies are seen to increase with decreasing diameter of the nanowire. It should, however, be mentioned that this is also found for carbon nanotubes, implying that neither these should exist from an energetic point of view. That they after all do exist shows that energetics is not the only important factor, but also kinetics can lead to materials that at ﬁrst should not occur. But in contrast to the case of C, the (meta-)stable structures for Si and Ge nanowires are not tubuli but instead more closed-packed, three-dimensional objects. In Figure 7.26 we show the structure of some of those. Okano et al. [56] compared the conductance through Al and Si nanowires as a function of their structure. Al and Si are neighbouring elements in the periodic table. Accordingly, a simple electron-counting argument may suggest that these two materials behave quite differently with respect to conduction, which, however, only is partly the case. Figure 7.27 illustrates this, where we show the conductance of a linear chain of Al or Si atoms as a function of chain length for linear chains varying from 1 to 8 atoms. For an inﬁnite, linear chain of either Al or Si two bands are crossing the Fermi level, of which one is doubly degenerate. Thus, if each band provides full conductance, a maximum conductance of 3G 0 can be expected, which indeed is the case in Figure 7.27. It should be added that the calculations were performed under the assumption of perfect junctions between chain and electrodes. On the other hand, Landman et al. [57] studied the conductance through somewhat

122

Chapter 7. Chains of sp Elements

(a) (a) cd1

(b) cd2

(c) sc

(d) -tin

(e) sh1

(f) sh4

f-ful5

f-ful4

(b)

Figure 7.26. The left part shows cross sections and the right part shows side views of selected Ge nanowires together with their labelling. Reproduced with permission of American Physical Society from Ref. [55].

Conductance (2e2/h)

3 2.5 2 1.5 1 0.5 0

Al Si 0

5

10 15 20 25 30 35 Electrode distance {a.u.}

40

45

Figure 7.27. The conductance at the Fermi level of ﬁnite linear chains of Al or Si as a function of length for chains with 1–8 atoms. Reproduced with permission of The American Physical Society from Ref. [56].

thicker silicon nanowires that were connected to Al leads and for which, moreover, hydrogen atoms were passivating the surface of the nanowires. Some of the systems are shown in Figure 7.28. For these they found a signiﬁcantly reduced conductance compared to the systems studied by Okano et al. [56]. In some cases, the conductance was even vanishing. As mentioned above, relativistic effects are important for Pb. This is illustrated most clearly through the properties of a linear chain of Pb atoms. Band-structure calculations [19] without and with the inclusion of spin–orbit couplings show (cf. Figure 7.29) that the spin-orbit couplings lead to the opening up of a gap at the Fermi level so that the system changes into a semiconductor. We add that this was the only Pb system for which this was observed. Finally, as for Al, Gu¨lseren et al. [11] showed that Pb wires may also possess the so-called weird structures when being sufﬁciently thin, as shown in Figure 7.6.

123

7.5. As and Bi

(i) Si94NW

(ii) Si24NW

(iii) Si96NW

(a) Si96NW

(b) Si96Al5NW

Figure 7.28. Structure of different silicon nanowires attached to Al electrodes with H atoms passivating dangling bonds on the surface. Reproduced with permission of The American Physical Society from Ref. [57].

0 (b)

(a)

Energy (eV)

−5

−10

−15

−20

−25

k

k

Figure 7.29. Band structures of a linear chain of Pb atoms without (left part) and with (right part) the inclusion of spin–orbit couplings. The interatomic distance was set equal to 6 a.u. Reproduced from Ref. [19].

7.5.

As and Bi

Of the elements of the next group, in particular Bi has been the topic of several studies within the concept of chain materials. Much more scarce is the number of studies on As. This material has ﬁrst of all been studied when being deposited on some surface, e.g., a H-terminated Si(100) surface [58]. Not surprisingly, signiﬁcant interactions between the substrate and the chain are observed, so that in this case one can hardly consider the system as consisting of non-interacting chains on a substrate.

124

Chapter 7. Chains of sp Elements

Costa-Kra¨mer et al. [59] performed break-junction experiments on Bi nanowires and found conduction plateaus at 1G 0 and 2G0 : The former value is in excellent agreement with the occurrence of one band crossing the Fermi level for a linear chain of Bi atoms, as has been found in theoretical calculations [19] (see Figure 7.10). Finally, Heremans et al. [60] studied thicker Bi nanowires, i.e., with diameters between 25 and 80 nm. Without an external magnetic ﬁeld the electrons feel the spatial conﬁnement due to the ﬁnite width of the nanowires, but when a magnetic ﬁeld parallel to the chains is turned on, there will be some cross-over at a certain magnetic-ﬁeld strength where the electrons do no longer feel the spatial conﬁnement. This critical magnetic-ﬁeld strength will depend on the wire thickness. Actually, Heremans et al. [60] did observe such a cross-over that was interpreted as a transition from one- to three-dimensional localization.

7.6.

S and Se

The group VI elements, to which S and Se belong, show a range of structures that are unique for this group of elements [61]. Many of these structures consist of parallel chains or cyclic molecules [61], and have been of particular interest in the study of liquid sulphur and amorphous selenium. Here, we shall, however, brieﬂy discuss their electronic properties. Figure 7.31 shows the band structures [62] for the various conﬁgurations of S and Se chains that are shown in Figure 7.30. The lowest total energy is in both cases found for a structure that is helical and is, moreover, semiconducting. However, a planar zigzag structure has a total energy that is only slightly higher than that of the helical structure [62]. This structure is, moreover, metallic with two bands crossing the Fermi level. On the other hand, a linear chain has a considerably higher total energy, but is also metallic. A further ﬁnding from the theoretical calculations [62,63] is that both chains are extremely soft (see, e.g., Figure 7.32), meaning that it costs very little energy to change their structure. Accordingly, one may expect that such chains, when for instance being deposited on some surface or inserted into some channels, easily adapt a structure according to that of the host.

7.7.

Conclusions

The sp elements are clearly different from the sd elements which we have discussed in the two preceding sections. The sp elements often form structures with directional bonds, partly due to s–p hybridization, whereas the structures of the sd elements are dictated by close packing. Therefore, lower-dimensional systems (clusters, chains, surfaces) of sp elements may be much more reactive than lower-dimensional systems of sd elements due to the occurrence of dangling bonds on the surfaces for the former. On the other hand, the fact that the structure is dictated more by directional bonds and less by close packing, may result in an increased stability of lowdimensional structures like (solid or hollow) nanowires. Au may be considered the prototype of the sd elements. Similarly, Al and C can both be considered prototypes of sp elements with, however, signiﬁcantly different

125

7.7. Conclusions

(b)

(a)

Figure 7.30. A helical and a zigzag chain.

0

(c)

(b)

(a)

−0.5

EF

εi (Ry)

3p

−1.0 3s −1.5

S π/v 0

0

(a)

0

π/v 0 (c)

(b)

EF

4p

εi (Ry)

−0.5

π/v

−1.0 4s −1.5

Se 0

π/v 0

π/v 0

π/v

Figure 7.31. The band structures of (a) the helical structure, (b) the zigzag structure, and (c) the linear structure of (upper part) a S chain and (lower part) a Se chain. Reproduced with permission of The American Institute of Physics from Ref. [62].

126

Chapter 7. Chains of sp Elements

0.8

(4.41,80)

4.54,80

0.6

4.61,82

0.6

5.12,90.5

(4.48,97)

0.4

4.85,87

5.65,99

ΔE (eV)

ΔE (eV)

(4.79,85)

5.15,92 5.50,96

5.02,88

5.34,93 5.23,93

(4.44,110)

(4.40,931)

4,57,80

0.4

5.00,88

0.2 0.2

4.59,80

4.93,87

(4.06,82) (4.37,91)

4.84,84.5

(4.44,98)

(4.07,85)

80

90

100

αsss (deg)

4.59,81

0

(4.22,86,51)

0

110

120

4.73,82.5

90

100 110 α (deg)

120

Figure 7.32. Total energy per atom for (left part) S helices and (right part) Se helices as a function of bond angle. The numbers on the curves give the values of the bond lengths (in a.u.) and dihedral angles (in deg.). Reproduced with permission of The American Institute of Physics from Ref. [62] and of The American Physical Society from Ref. [63].

properties. Al forms structures with a clear three-dimensional character due to the sp3 hybrids. In contrast, C can also form linear chains (with sp hybrids) as well as hollow tubes (with sp2 hybrids). In fact, this ﬂexibility of carbon to participate in many bonding and structural situations may be considered one main reason for the richness of organic chemistry. This difference between Al and C is also responsible for the differences in the occurrence of quasi-one-dimensional structures. Stable, isolated nanowires of Al can hardly be produced, whereas such wires of C are being produced routinely today. Break-junction experiments on sp elements are not as common as is the case for sd elements, maybe partly because of the above-mentioned reactivity, and also maybe because these elements in many cases are semiconductors and not metals. Nevertheless, for Al we found an interesting four-period oscillatory behaviour of the conductance as a function of chain length (measured in the number of atoms) that could also be ascribed directly to band-structure effects for the absolutely smallest chains. The carbon nanotubes are worth a whole book themselves, but here they have been discussed brieﬂy. Their stability together with the possibility of controlling their properties simply by varying their helicity make them excellent systems for all kind of studies, within both basic and applied science. For the heavier elements we saw that relativistic effects can be very important. Most notably, this was seen for a linear chain of Pb atoms that due to relativistic effects is a semiconductor.

124

Chapter 7. Chains of sp Elements

Costa-Kra¨mer et al. [59] performed break-junction experiments on Bi nanowires and found conduction plateaus at 1G 0 and 2G0 : The former value is in excellent agreement with the occurrence of one band crossing the Fermi level for a linear chain of Bi atoms, as has been found in theoretical calculations [19] (see Figure 7.10). Finally, Heremans et al. [60] studied thicker Bi nanowires, i.e., with diameters between 25 and 80 nm. Without an external magnetic ﬁeld the electrons feel the spatial conﬁnement due to the ﬁnite width of the nanowires, but when a magnetic ﬁeld parallel to the chains is turned on, there will be some cross-over at a certain magnetic-ﬁeld strength where the electrons do no longer feel the spatial conﬁnement. This critical magnetic-ﬁeld strength will depend on the wire thickness. Actually, Heremans et al. [60] did observe such a cross-over that was interpreted as a transition from one- to three-dimensional localization.

7.6.

S and Se

The group VI elements, to which S and Se belong, show a range of structures that are unique for this group of elements [61]. Many of these structures consist of parallel chains or cyclic molecules [61], and have been of particular interest in the study of liquid sulphur and amorphous selenium. Here, we shall, however, brieﬂy discuss their electronic properties. Figure 7.31 shows the band structures [62] for the various conﬁgurations of S and Se chains that are shown in Figure 7.30. The lowest total energy is in both cases found for a structure that is helical and is, moreover, semiconducting. However, a planar zigzag structure has a total energy that is only slightly higher than that of the helical structure [62]. This structure is, moreover, metallic with two bands crossing the Fermi level. On the other hand, a linear chain has a considerably higher total energy, but is also metallic. A further ﬁnding from the theoretical calculations [62,63] is that both chains are extremely soft (see, e.g., Figure 7.32), meaning that it costs very little energy to change their structure. Accordingly, one may expect that such chains, when for instance being deposited on some surface or inserted into some channels, easily adapt a structure according to that of the host.

7.7.

Conclusions

The sp elements are clearly different from the sd elements which we have discussed in the two preceding sections. The sp elements often form structures with directional bonds, partly due to s–p hybridization, whereas the structures of the sd elements are dictated by close packing. Therefore, lower-dimensional systems (clusters, chains, surfaces) of sp elements may be much more reactive than lower-dimensional systems of sd elements due to the occurrence of dangling bonds on the surfaces for the former. On the other hand, the fact that the structure is dictated more by directional bonds and less by close packing, may result in an increased stability of lowdimensional structures like (solid or hollow) nanowires. Au may be considered the prototype of the sd elements. Similarly, Al and C can both be considered prototypes of sp elements with, however, signiﬁcantly different

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Chapter 8

Chains of s Elements

From the sd elements with their largely undirected interatomic interactions via the sp elements with their often directed interatomic bonds we now turn our attention to the s elements. s orbitals are per construction isotropic so that we for these elements once again will expect that the interatomic interactions are undirected. Moreover, the s elements we shall discuss, i.e., Li, Na , K, and Cs, all have a ﬁlled shell plus just one single valence s orbital that, accordingly, is fairly delocalized in space and little localized to the atom. Therefore, for the s elements the jellium model, that we discussed in Chapter 4, is often believed to constitute a good approximation. Actually, we saw in Chapter 4 that for clusters of the s elements the jellium model can describe the experimental observation of the so-called magic numbers, i.e., cluster sizes for which the clusters are of particularly high stability. This has been explained as being due to an electronic shell-closing effect, i.e., when all orbitals for the largely delocalized electrons are either ﬁlled or empty, then the cluster is particularly stable. But also simple packing effects that may be very important for systems with undirected interatomic interactions may dictate structures of particularly high stability. In Chapter 4 we saw that the jellium model also has been applied to chain-like systems and we discussed some of the outcomes of these studies. In the present section we shall discuss the ‘real’ systems, but simultaneously discuss whether the jellium model or other simple models can explain the observations.

8.1.

Na

As Au for the sd elements and Al (and with some justiﬁcation also C) for the sp elements can be considered the prototype of the relevant class of elements within the chains class of materials, Na may be considered the prototype for the s elements. In Figure 6.5 we presented results from the theoretical study of Opitz et al. [1] who considered chains of Cu and Na that were not atomically thin but had a square-like cross section. The density of states in Figure 6.5 shows that the electrons of Na chains indeed behave roughly as they would according to the jellium model. The curve is that of a free-electron gas but superposed by some extra features (these are indicated in the ﬁgure) that can be related to the ﬁlling of electronic shells, as we discussed in Chapter 4 (see in particular Figure 4.1). This has also been observed experimentally [2]. Yanson et al. [2] measured the conductance through a nanometre thin junction in Na using the break-junction 131

132

Chapter 8. Chains of s Elements

technique. Repeating the experiment very many times they obtained a histogram of number of times a certain conduction was measured. This histogram is shown in Figure 8.1 and possesses clearly regularly spaced maxima, i.e., there exists values of conductances that appear more frequently than other values. In order to relate these values of the conductance to a size, i.e., a radius of the nanojunction, they used a formula like equation (5.4),

G ’ G0

2 pR lF 1 , lF pR

(8.1)

10

8 Counts

Number of counts [a.u.]

Na 6 4 2

1

20 40 60 80 100 120 G/G0

(a) 0.1 0

20

40 60 80 Conductance [2e2/h]

100

120

25

kFR

20

(b)

15 10 5 0

2

4

6

8 10 12 14 Shell or peak number

16

18

Figure 8.1. (a) Shows a histogram of the number of times a certain conductance for Na wires were measured. The insert shows the full curve, whereas the background has been subtracted for the larger curve. (b) Shows the square root of the peak conductances vs. its index. Reproduced with permission of Nature from Ref. [2].

8.1. Na

133

where lF is the Fermi wave length lF ¼

2p 2prs ¼ . kF ð9p=4Þ1=3

(8.2)

The open squares in Figure 8.1(b) were obtained from Figure 8.1(a) by simply indexing the peaks and subsequently plotting ðG=G 0 Þ1=2 as a function of this index. Alternatively, one may study the behaviour of electrons conﬁned to a region with a radius R that can be obtained from equation (8.1). This radius may, e.g., be that of a spherical system or that of a cylindrical system. In both cases there will be certain values of R for which the system has a particularly high stability when focusing on electronic effects, as we are doing here. These values of R can be estimated through a semiclassical analysis of the electronic paths. This gives (see Ref. [2]) _kF L ¼ nh;

n ¼ 1; 2; 3; . . . ,

(8.3)

where L is the length of the path. Analysing the different paths for either a spherical or a cylindrical geometry leads to certain values of R for which the system is particularly stable and, subsequently, the corresponding conductance can be evaluated through equation (8.1). This leads to the results of Figure 8.1(b) for either spherical (ﬁlled circles) or cylindrical (open triangles) geometries. Taking into account the crude approximations that have been used in determining those system sizes (i.e., values of R) for which the system is particularly stable, the agreement between observed and estimated results in Figure 8.1(b) is remarkable. Later, the same authors [3] extended the analysis of the conductance histograms that were obtained in break-junction experiments. Their data are shown in Figure 8.2 that indeed resemble those of Figure 8.1(a) with, however, some additional features explicitly marked with arrows. As before, they correlate the particularly stable nanowire radii R with the occurrence of certain closed electronic orbits according to the semiclassical Bohr–Sommerfeld quantization condition, equation (8.3). These orbits, shown in Figure 8.3, are also marked in Figure 8.2(a). However, the special feature of Figure 8.2(b) is that the regular pattern of peaks at certain places, marked with arrows, are interrupted. Such quantum-beat effects have been predicted theoretically for clusters within the jellium model [4] and shortly later observed experimentally for clusters of Na atoms [5]. However, for the clusters these so-called supershells were found ﬁrst for quite large systems (i.e., clusters containing some 1000 atoms), whereas for the nanowires here they occur for relatively small sizes. The results of Figure 8.2 show also that the conductance mainly peaks at 1G 0 ; 3G 0 ; 5G0 ; and 6G 0 ; whereas the values 2G 0 and 4G 0 largely are absent. This is a general ﬁnding for the s metals (see, e.g., Ref. [6]) and is considered a conﬁrmation of the jellium-like behaviour of these metals. For a very thin jellium cylinder, no conduction channel exists, but for a slightly thicker one a single channel opens up, giving the conductance value of 1G0 : For an even slightly thicker jellium chain, the two next channels are energetically degenerate and will, accordingly, open up simultaneously. Therefore, the conductance jumps from 1G 0 to 3G0 : Increasing the thickness the number of conductance channels will again increase by two leading to

134

Chapter 8. Chains of s Elements

6

10

a kFR

Counts [arb.un.]

5 4

21.8

8 6

8.74 5.04

11.7

2.85

4

3

16.1

1

2 4 6 3 5 Peak or shell number

2 1 0 0

Counts [arb.un.]

4

20

40 60 80 Conductance [2e2/h] 1.2 1.0 0.8 0.6 0.4 0.2 0.0

b

2

0

5

10

100

15

20

15

20

120

kFR Figure 8.2. Similar to Figure 8.1, but with a more detailed analysis. Reproduced with permission of American Physical Society from Ref. [3].

a conductance of 5G0 : Since the orbitals of the subsequent channel are of s symmetry, the next conductance value will be 6G 0 ; and so on. This scenario has been conﬁrmed by the theoretical study of Nakamura et al. [7]. They simulated the stretching of a wire of sodium atoms by step-by-step elongating a ﬁnite wire between two leads. At each step they also calculated the conductance. The results, shown in Figure 8.4, conﬁrm the qualitative predictions from the jellium model, which we have discussed above for the conductance values of 1G 0 ; 3G 0 ; and 6G 0 : Barnett and Landman [8] have used theoretical methods in studying the timeevolution of a junction during a break-junction experiment for Na. Similar to what we have seen repeatedly also for other systems during this presentation, they found that the thinning of the junction occurs in steps that lead to extra layers of atoms along the nanowire. This can be recognized in Figure 8.5. An interesting aspect for slightly thicker nanowires is that the authors could identify a Na13 building block (see Figure 8.5) as an important ingredient. For free isolated clusters, the

135

8.1. Na

...

(2,1)

(3,1)

(4,1)

(5,1)

... (4,2)

(5,2)

etc.

(6,2)

Figure 8.3. The semiclassical orbits inscribed inside a circular cross-section. The orbits are labelled with two numbers ðM; QÞ; where M is the number of vertices and Q the winding number. The scheme is applicable both for clusters (spheres) and for nanowires (cylinders). Reproduced with permission of The American Physical Society from Ref. [3].

L=28.0Å

L=21.2Å

L=22.0Å

Conductance (G0)

6 L=26.0Å

5 4 3 2 1

L=30.0Å

L=24.0Å

|τn|2

1.0 0.5 0.0 16

18

20 22 24 26 28 Unit cell length (Å)

30

Figure 8.4. The left part shows the atomic structure for a ﬁnite Na chain during an elongation process. The length of the chain is given on the top. The right part shows the conductance during this elongation (top panel) as well as the eigenchannel transmission probabilities (bottom panel; these sum up to the total transmission). Reproduced with permission of The American Physical Society from Ref. [7].

icosahedral Na13 is particularly stable due to geometric packing effects, and according to the results of Barnett and Landman [8] it is also an important building block in nanowires. Some of the studies we have discussed earlier in connection with our discussion of other elements have also included Na. Among those is the study of Lee et al. [9] who examined the conduction through a ﬁnite, linear chain of atoms. They found, cf. Figure 6.3, a clear even–odd oscillation when looking at the conduction as a function of the number of atoms forming the chain. This oscillation was actually smallest for Na among all the materials they studied.

136

Chapter 8. Chains of s Elements

Figure 8.5. Development of the structure of a Na nanowire during stretching. (c) Shows the stable Na13 building block. Reproduced with permission of Nature from Ref. [8].

Mirror boundary r z

α

jellium cone rs = 3.93

Na atoms

Figure 8.6. Geometry of a chain of Na atoms placed between two metal leads that are modelled as jellium cones. Reproduced with permission of The American Physical Society from Ref. [10].

Havu et al. [10] studied the conductance through a small, ﬁnite, linear chain of Na atoms that were placed between two metals. The metal leads were approximated as cones of jellium, thereby mimicking the structure of two tips, as shown in Figure 8.6. It is interesting to notice that these authors found a remarkable sensitivity of the conductance not only on the number of atoms in the chain, but also on the angle deﬁning the cone. This ﬁnding is illustrated in Figure 8.7. This shows that the measured conductance depends critically on all parts of the system: structure and material of the leads, structure and size of the nanowire itself, and the interphases between nanowire and leads.

137

8.2. Li, K, Rb, and Cs

1

0.5

Conductance (2e2/h)

67° 0 1

0.5 68° 0 1

0.5 69° 0 1

2 3 Number of Na atoms in the wire

4

Figure 8.7. Conductance through the system of Figure 8.6 as a function of number of Na atoms of the chain and of the jellium cone angle. The latter is given in the panels. Reproduced with permission of The American Physical Society from Ref. [10].

Finally, as for Cu, Ag, and Au, McAdon and Goddard [11] predicted theoretically that chains of Na atoms will contain a spin-density wave. This prediction has, to our knowledge, not been studied or conﬁrmed by others.

8.2.

Li, K, Rb, and Cs

Of the other s elements, K is the one that has been studied most intensively. Yanson et al. [12] studied K in break-junction experiments as a prototype of an alkali metal nanowire. In a large number of experiments they measured the conductance during a break-junction experiment and depicted subsequently the conductance in a histogram similar to the ones we have shown in Figures 8.1 and 8.2 above for Na. As we saw for Na it is possible to identify a large number of maxima that can be indexed, as shown in Figure 8.8. For larger values of the conductance the nanowires have to be thicker than just some few atoms. As shown in Figure 8.9 the authors could not describe the peak sequence of Figure 8.8 when assuming that closed electronic shells dictate particular stability. Instead, the authors suggested that from a certain critical radius and onwards geometric effects, i.e., the packing of atoms that are largely spherical, are responsible for the occurrence of particularly stable

138

Chapter 8. Chains of s Elements

(kFR) 15

20

25

30

35

electronic shells

1

10

50

45

40

35

30

25

20

1000

15

10

Counts [a. u.]

5

2000

atomic shells 0

2

4

6

8

10 (G/G0)1/2

11

12

13

Figure 8.8. Histogram of the number of times a certain conductance for K wires was measured as a function of the square root of the conductance. Reproduced with permission of The American Physical Society from Ref. [12].

18

Peak position [(G/Go)1/2]

hexagonal facets

full atomic shells

16 14

K

electronic shells

12 10 8 6 4 2

0

10

20

30 Peak number

40

50

60

Figure 8.9. The positions of the peaks in Figure 8.8 as a function of peak index. The insert shows a structure of a hexagonal nanowire with the axis along the [111] direction and the bulk bcc stacking. Reproduced with permission of The American Physical Society from Ref. [12].

References

139

nanowires. In the insert in Figure 8.9 such a wire is shown, and by analysing the construction of such closed-atomic-shell nanowires, Yanson et al. [12] were able to explain the peak sequence of Figure 8.8 as shown in Figure 8.9. They assumed that a wire like the one that is shown in Figure 8.9 is constructed by adding atoms to one facet thereby gradually making the wire thicker and thicker. When one facet is completed, a particularly stable structure results, and atoms will then be added to a new facet. Thus, these experiments extend the ones above for Na nanowires by predicting a transition from electronic shells to atomic shells as a function of nanowires thickness. Different periodic structures of K, Rb, and Cs were studied by March and Rubio [13]. They predicted that not linear but zigzag chains should be the stable structure when, e.g., depositing chains of these elements on surfaces, although to our knowledge no experimental conﬁrmation of this prediction exists. This is also the case for the theoretical prediction of McAdon and Goddard [11] that Li chains should possess a spin-density wave. Finally, Lee et al. [9] found that the conduction of ﬁnite, linear chains of Cs atoms depends much stronger on chain length than is the case for Na and for the sd metals Cu, Ag, and Au; cf. Figure 6.3.

8.3.

Conclusions

In this section we have studied chains of atoms that have a single s valence electron per atom. Owing to the delocalized nature and lack of directional dependence of these orbitals, one might expect a jellium-like model to provide a good description of the properties of these chains. This was indeed found to be the case, with the results of the conductance experiments of Na chains being the most prominent demonstration of this. Here, we saw clearly that electronic-shell effects were experimentally observable. However, for even thicker chains we saw for K that geometric packing effects also could dictate the properties. Thus it is clear that also for these elements there is a competition between packing effects and electronic-shell effects.

References [1] J. Opitz, P. Zahn, and I. Mertig, Phys. Rev. B 66, 245417 (2002). [2] A.I. Yanson, I.K. Yanson, and J.M. van Ruitenbeek, Nature 400, 144 (1999). [3] A.I. Yanson, I.K. Yanson, and J.M. van Ruitenbeek, Phys. Rev. Lett. 84, 5832 (2000). [4] H. Nishioka, K. Hansen, and B.R. Mottelson, Phys. Rev. B 42, 9377 (1990). [5] J. Petersen, S. Bjørnholm, J. Borggreen, K. Hansen, T.P. Martin, and H.D. Rasmussen, Nature 353, 733 (1991). [6] N. Agraı¨ t, A.L. Yeyati, and J.M. van Ruitenbeek, Phys. Rep. 377, 81 (2003). [7] A. Nakamura, M. Brandbyge, L.B. Hansen, and K.W. Jacobsen, Phys. Rev. Lett. 82, 1538 (1999). [8] R.N. Barnett and U. Landman, Nature 387, 788 (1997).

140

Chapter 8. Chains of s Elements

[9] Y.J. Lee, M. Brandbyge, M.J. Puska, J. Taylor, K. Stokbro, and R.M. Nieminen, Phys. Rev. B 69, 125409 (2004). [10] P. Havu, T. Torsti, M.J. Puska, and R.M. Nieminen, Phys. Rev. B 66, 075401 (2002). [11] M.H. McAdon and W.A. Goddard, III., J. Chem. Phys. 88, 277 (1988). [12] A.I. Yanson, I.K. Yanson, and J.M. van Ruitenbeek, Phys. Rev. Lett. 87, 216805 (2001). [13] N.H. March and A. Rubio, Phys. Rev. B 56, 13865 (1997).

Chapter 9

Mixed Systems

In the preceding sections we have discussed, almost element by element, quasi-onedimensional structures of single elements. We have seen that the properties could be varied through variation of the thickness of the nanowires but also through variation of the element. Even more ﬂexibility is, of course, possible if one combines more elements, although this may pose considerable challenge to the experimental realization of such systems. Some of the systems we shall consider are quasi-onedimensional chains where two or more types of units are placed in a more or less regular sequence, but that hardly have been realized experimentally. In some sense these materials may be considered as being quasi-one-dimensional alloys. Other systems consist of two parts of which maybe only one is the quasi-one-dimensional chain of our interest. This can be carbon nanotubes ﬁlled with some other elements, organic chains decorated with a metal, or guest–host systems where the metallic chains are placed inside channels of a host. We stress that we have already above considered examples of such systems (for instance the Bi wires in the study of Heremans et al. [1] that were synthesized inside an Al2 O3 host), and that chains deposited on a surface in some sense can also be interpreted as constituting a guest– host system. Here, we shall consider some few further examples of metallic chains inside crystalline hosts, but no further examples of chains on a surface.

9.1.

Order, disorder, and quasi-periodicity

In Chapter 2 we studied the electronic properties of a linear chain of repeated identical units. In Figure 2.3 we plotted the band structures for a linear chain of identical atoms with one orbital per atom and either without or with a bond-length alternation. The latter would open up a gap and, accordingly, split the band into two branches. Instead of having a bond-length alternation one may also consider the case that two types of atoms alternate along the chain. In equation (2.1) this can be modelled by letting the on-site energies n alternate between two values. The band structures would then look like the ones of Figure 2.3(b), even in the case of a vanishing bond-length alternation. For the present discussion it is, however, most important that despite the increased heterogeneity of the chain, any electronic orbital is delocalized over the complete chain. Experimentally produced systems are rarely perfect but contain impurities and perturbations of various kinds. A central issue is then, to which degree such 141

142

Chapter 9. Mixed Systems

disorder can be accepted by the system without disturbing the properties it would have had, were it perfect. It is well known that two- and three-dimensional systems possess some ﬂexibility, i.e., as long as the disorder is not too strong, the electronic orbitals will remain delocalized until a certain threshold is reached at which point they will localize. In a model like that of equation (2.1) one may let the on-site energies n and/or the hopping integrals tn;nþ1 take random values with a certain spread s; with s characterizing the disorder. The situation is markedly different in one dimension. Here, any non-zero value of s leads to a localization of all electronic orbitals [2–4]. This would imply that no chain with any degree of disorder could conduct as long as it is sufﬁciently long. There are, however, interesting exceptions to this rule. First, it is based on a singleparticle description, so that many-body effects may lead to different behaviours. Second, it has been shown [5,6] that if one introduces disorder in a slightly different way, i.e., letting two neighbouring on-site energies always have the same value, some of the states may be delocalized. This kind of disorder can be relevant for some of the synthetic metals we shall discuss later. Third, deviations from pure one dimensionality may change the results. Fourth, deterministic, quasi-periodic, but nevertheless non-periodic, structures may contain delocalized states. We shall brieﬂy discuss here the last case. Deterministic, quasi-periodic systems have been at the centre of some theoretical interest (see, e.g., Ref. [7,8]) although experimental realizations are scarce. We consider a chain consisting of two types of atoms, A and B. A periodic structure could be AABAABAABAAB ; i.e., we construct it by following a speciﬁc prescription (here, repeating units of two A atoms and one B atom). A quasi-periodic structure can also be constructed in a deterministic way, but is not periodic. A popular example is the so-called Fibonacci chain. We construct the two shortest members of a certain sequence of Fibonacci chains, e.g., Dð1Þ ¼ A and Dð2Þ ¼ AB: Then all following members are constructed through Dðn þ 1Þ ¼ DðnÞDðn 1Þ; i.e., Dð3Þ ¼ ABA; Dð4Þ ¼ ABAAB; Dð5Þ ¼ ABAABABA; Dð6Þ ¼ ABAABABAABAAB; and so on. There are other rules leading to other types of quasi-periodic systems, e.g., the Thue–Morse chains, and, moreover, by deﬁning the two ﬁrst members differently, other Fibonacci chains can be constructed. As mentioned, the electronic orbitals of such systems behave differently from those of periodic or disordered chains, making it is of great interest to produce and understand such systems. However, experimental realizations of purely quasi-onedimensional chains of this type, to our knowledge, do not exist. The closest approximation is that of sequences of different layers of crystals stacked on top of each other, like GaAs–AlAs [7–9] and Nb–Cu [10]. Since these systems cannot in any way be considered as ‘chains of metals’ or ‘metallic chains’ we shall not discuss them further here, except for emphasizing that such structures possess unusual properties and, therefore, it is worthwhile attempting to produce them.

9.2.

Alloys and compounds

There exists only few examples of break-junction experiments on metallic alloys or compounds (see, e.g., Ref. [11]). The problems with the interpretation of the

143

9.2. Alloys and compounds

experiments are obvious. Consider for example the case of Au with 5% Co. In a break-junction experiment the junction at the last steps just before breaking consists of just some few 10s of atoms. This is so few that it is not clear as to whether it will contain any Co atoms at all, or whether some segregation will lead to no (or, alternatively, relatively many) Co atoms in the junction. Thus, a clear identiﬁcation of structure/composition–property relations is not easy. Alternatively, theoretical studies on idealized systems may help, although being scarce. Geng and Kim [12] studied AuZn and AuMg alloy wires. They found that the most stable structures for Au, Mg, and Zn chains correspond to zigzag arrangements (see Figure 9.1) but that for the mixed systems stable linear chains exist. Thus, alloying leads in this case to a stabilization of the linear chain. The band structures (Figure 9.2) show that those for the mixed systems are not simply a superposition of those for the individual components. This is indeed an important result as it shows that by varying composition new phenomena may occur. Similar results were recently obtained for AuPt nanowires [13]. On the other hand, it has been shown that it is possible to produce one-dimensional wires of one material on the surface of another. This is, e.g., the case for NiAl wires on AlAs surfaces that have been studied for instance by Katsumoto et al. [14]. They measured the temperature dependence of the conductance through such a nanowire and found indications that the nanowire indeed was quasi-one-dimensional, i.e., largely decoupled from the underlying substrate. Another example of mixed systems is that of modifying the carbon nanotubes that we discussed in Section 7.3. Here, the fact that boron nitride also forms hexagonal layers that are very similar to those of graphite has ultimately led to mixed nanotubes of the form Cx ðBNÞ1x (see, e.g., Refs. [15–17]). For instance, Miyamoto

Cohesive energy (eV/cell)

0

Zn

−1 Mg AuZn

−2

−3 AuMg −4

Au d/2

d

−5 2

3

4

5

6

7

Wire length d (Å/cell) Figure 9.1. Cohesive energy of different A and AB chains as a function of wire length. Reproduced with permission of The American Physical Society from Ref. [12].

144

Chapter 9. Mixed Systems

Zn

AuZn

Au

AuMg

Mg

0 Energy (eV)

−2 −4 −6 −8 −10 −12 0

Energy (eV)

−2 −4 −6 −8 −10 −12

Γ

X Γ

X Γ

X Γ

X Γ

X

Figure 9.2. Band structures for the structures of Figure 9.1. The upper panel shows those for a linear chain, the lower panel those for zigzag chains. Reproduced with permission of The American Physical Society from Ref. [12].

et al. [15] studied nanotubes of the composition BC2 N but found, in contrast to the case for the pure carbon nanotubes, that they were all semiconducting and not metallic. These systems are also interesting because even with the same number of atoms different structures can be obtained, as shown for instance in Figure 9.3. The (possible) existence of carbon and Cx ðBNÞ1x nanotubes is often related to the possibility of creating planar sheets of the material that subsequently is thought to be rolled-up as a tube, cf. Section 7.3. Generalizing this idea, one may suggest that any material that consists of planar sheets can also form nanotubes. This is, e.g., the case for WS2 that, however, has a somewhat more complicated structure. Thus, the WS2 sheets are not planar but consist of one planar sheet of W atoms with S atoms both above and below the sheet of W atoms. Therefore, the formation of nanotubes of WS2 is connected with an increased strain. Nevertheless, such nanotubes have been produced [18] and also have been the subject of theoretical studies [19]. Figure 9.4 shows clearly how the rolling-up of a WS2 sheet leads to an increased strain. A different kind of wire was produced by Venkataraman and Lieber [20]. They made wires of Mo6 Se6 with a structure like that of Figure 9.5. Through scanningtunnelling microscopy experiments they could show that these wires indeed were metallic with clear indications of van Hove singularities in the density of states, as would be expected of quasi-one-dimensional systems. Finally, the existence and properties of other types of nanotubes have been the focus of a number of theoretical studies. This includes, e.g., the study of Coˆte´ et al.

9.2. Alloys and compounds

145

Figure 9.3. Two different (4,4) BC2 N nanotubes. Reproduced with permission of The American Physical Society from Ref. [15].

Figure 9.4. Zigzag (22,0) WS2 (left) and armchair (10,10) WS2 (right) nanotube. Reproduced from Ref. [19].

146

Chapter 9. Mixed Systems

a

0.45 nm

b

Height (nm)

1.2 0.8 0.4 0.0

−0.4 0

100

200 300 Length (nm)

400

500

Figure 9.5. (a) The structural model of a Mo6 Se6 nanowire. (b) An atomic-force-microscopy image of a such wire on a mica surface. Here, the inset shows the image across the sample at the position of the white line. Reproduced with permission of The American Physical Society from Ref. [20].

[21] who studied GaSe nanotubes like the Figure 9.6. They found, however (see Figure 9.7) that these nanotubes were not very stable due to a large strain compared with that of carbon nanotubes. Moreover, they were not metallic, but semiconducting. Finally, Zhang and Crespi [22] have theoretically studied B2 O and BeB2 nanotubes and found that they would be metallic. However, to the best of our knowledge, there have been no experimental studies on these systems.

9.3.

Filled nanotubes

Instead, we discuss studies that have been devoted to utilizing the carbon nanotubes as hosts for chain-like arrangements of guest atoms or molecules. Not long after the discovery of the carbon nanotubes, Tsang et al. [23] as well as Guerret-Pie´court et al. [24] reported that they had succeeded in ﬁlling the hollow inner parts of the nanotubes with other materials. Tsang et al. [23] observed larger nanoparticles of metaloxides (e.g., NiO and UO) inside the tubes. However, the

147

9.3. Filled nanotubes

Figure 9.6. A (18,0) GaSe nanotube. Reproduced with permission of The American Physical Society from Ref. [21].

Strain energy (eV/atom)

0.6

GaSe nanotubes

0.4

0.2 Carbon nanotubes

0

0

10

20

30

(n,0) Figure 9.7. The strain energy as a function of size of GaSe and C nanotubes. Reproduced with permission of The American Physical Society from Ref. [21].

nanoparticles were relatively large and without interactions with each other. This was also the case for the metalcarbides (i.e., carbides of Ti, Cr, Fe, Co, Ni, Cu, Zn, Mo, Pd, Sn, Ta, W, Gd, Dy, and Yb) that were observed inside the carbon nanotubes in the study of Guerret-Pie´court et al. [24].

148

Chapter 9. Mixed Systems

Even before these studies, theoretical methods were used to address the question as to what would be the properties of linear chains of metals inside a carbon nanotube. Using a simple Hu¨ckel-like model, Gal’pern et al. [25] studied linear chains of Li and K inside the carbon nanotubes. They found that without these guests the carbon nanotubes would be non-metallic with small gaps, but the alkali metals would change the systems into metals. A more accurate study was performed by Miyamoto et al. [26]. They considered systems like that of Figure 9.8, i.e., a ðn; 0Þ carbon nanotube with potassium metals inside. They found that the alkali atoms arrange to form a linear chain and that the largest heat of formation was found for n ¼ 7 (1.12 eV per K atom), whereas for n ¼ 6 it was essentially vanishing, and for n ¼ 8 and n ¼ 9 it was reduced to 1.07 and 0.30 eV per K atom, respectively. Thus, for the ð7; 0Þ nanotubes K atoms should prefer to be incorporated inside the tubes. Furthermore, they found, as Gal’pern et al. [25], that the K atoms changed the system from a semiconductor into a metal, see Figure 9.9. In a theoretical work Fan et al. [27] incorporated I atoms inside single-wall carbon nanotubes. They found (see Figure 9.10) that the atoms form helices inside

Figure 9.8. Schematic picture of a ð7; 0Þ carbon nanotube with K atoms inside. Reproduced with permission of The American Physical Society from Ref. [26].

149

9.3. Filled nanotubes

(b)

(a)

4

4 NFE

2

2

Energy (eV)

EF EF

0

0

−2

−2

−4

−4

−6

Γ

X

−6

Γ

X

Figure 9.9. The band structures for the system of Figure 9.8 without (a) and with (b) the K atoms inside. Reproduced with permission of The American Physical Society from Ref. [26].

Figure 9.10. High-resolution images of iodine chains inside carbon nanotubes. Reproduced with permission of The American Physical Society from Ref. [27].

150

Chapter 9. Mixed Systems

Figure 9.11. Structure model of a 5 nm helix of I atoms inside a (10,10) carbon nanotube. Reproduced with permission of The American Physical Society from Ref. [27].

the nanotubes and based on their theoretical studies they suggested structures like that of Figure 9.11. Finally, using electron-energy-loss spectroscopy, Liu et al. [28] studied the electronic properties of Ba inside carbon nanotubes. They found effects of a considerably electron transfer (leading to Ba2þ ), i.e., indicating that the combined system of Ba atoms inside the carbon nanotube is a new system and not simply the sum of the two individual systems, similar to what were observed for the metallic alloys above. Finally, Rubio et al. [29] found that chains of metals inside C or BN tubes may change the system from semiconducting to metallic. They considered both Al and K as the metals.

9.4.

Decorating chains

Instead of using the interior of quasi-one-dimensional channels as a means for producing quasi-one-dimensional chains, one may also use the surface of quasi-onedimensional materials as a host for producing chains of metals. We shall brieﬂy discuss here this approach.

9.5. Guest–host systems

151

Nishinaka et al. [30] decorated nucleoprotein ﬁlaments with gold nanoparticles thereby obtaining a kind of chain of gold nanoparticles. They showed that the resulting system indeed was metallic. This work may be considered an extension of others where DNA templates have been covered with various metals like Ag, Au, Pt, Pd, and Cu (see, e.g, [30] and references therein). However, the materials are not very well characterized and it is, therefore, not easy to obtain a detailed understanding of their properties. Finally, Gemming et al. [31] studied theoretically chains consisting of a gold core covered with some other element. They found that the most stable structures were those where the gold chain was covered with Ag whereas the gold chain covered with Pd was even less stable than the pure gold chain.

9.5.

Guest–host systems

Our world is three-dimensional, and most elements prefer to adapt three dimensional, more or less closed-packed structures. Thus, the existence of chains is often in conﬂict with a ‘natural’ tendency of the atoms to form more compact structures. Therefore, very often external inﬂuences are needed to produce chain structures. These may be the forces in a break-junction experiment, or the interactions from a substrate on which a chain is being deposited or grown. Or, alternatively, a host containing channels may provide the external conﬁning forces in which chains can be found. Such systems are the subject of the present section. Romanov [32] synthesized a zeolite mordenite containing chains of the heavy elements Tl, Pb, or Bi. In Figure 9.12, we show a schematic representation of the mordenite crystal where the occurrence of channels is clearly recognizable. Romanov found that the combined system of mordenite crystal and chains (see Figure 9.13) had a high symmetry, i.e., the lattice constant of the chains was commensurate with that of the

Figure 9.12. A schematic representation of a mordenite structure with the channels running perpendicular to the plane of the picture. Reproduced with permission of The Institute of Physics from Ref. [32].

152

Chapter 9. Mixed Systems

Figure 9.13. A schematic representation of a mordenite containing chains. We stress that the structure of the chain should not be taken as the correct one, but merely representing ‘some’ chain. Reproduced with permission of The Institute of Physics from Ref. [32].

host, conﬁrming the experimental observation by him that the structure has a high symmetry. On the other hand, little more was known about the structure and the representation in Figure 9.13 was not to be considered as anything else but schematic and speculative. This was the reason for a theoretical study where different structures of isolated chains like those of Figure 6.1 were studied [33]. In Figure 7.10, we showed the band structures for those, and through the results of Figure 7.29 we even emphasized the role of relativistic effects for these heavy elements. Another result of that study is shown in Figure 9.14, i.e., the chains are extremely soft, meaning that the total energy changes only little when the lattice constant is changed. This means that a chain, when being a guest in, e.g., the channels of a crystalline host, easily will adapt a structure that is commensurate with that of the host. On the other hand, as also shown in Figure 9.14, other properties, such as the band gap, may depend strongly on the lattice constant, ultimately leading to the hypothesis that through appropriate choice of the host material, the electronic properties of the guest can be varied in a controlled way. In another study, Hong et al. [34] synthesized Ag nanowires inside the pores of selfassembled calix[4]hydroquinone nanotubes. The structure is shown in Figure 9.15, where both the host material and the guest chains are shown. The chains were found to have a structure that is commensurate with that of the host and are seen to have a structure like the tetragonal chain of Figure 6.1. These chains were also the subject of some theoretical studies [35,34,36]. In one [34] it was found that the structure of the isolated chain is very similar to that of the chain inside the host, suggesting that the host has very little inﬂuence on the chain properties. On the other hand, in another study [35], signiﬁcant differences between the structure of the isolated chain and that

153

9.5. Guest–host systems

3.0

Energy (eV)

2.0

1.0

0.0

−1.0 5.0

6.0

7.0

8.0

9..0

10.0

Bond length (a.u.) Figure 9.14. Total energy and band gap as function of the interatomic distance for a linear chain of Pb atoms. Reproduced from Ref. [33].

Figure 9.15. Structure of Ag chains inside a calix[4]hydroquinone host. (A) The matrix, (B) the structure of the matrix, and (C), (D) the Ag chains inside the matrix. Reproduced with permission of The American Association for the Advancement of Science from Ref. [34].

of the chain in the host were observed. In the latter case the Ag chains may be similar to the Tl, Pb, and Bi chains discussed above, i.e., they can easily adapt themselves to the structure of a host thus offering a way of controlled varying of their properties through careful choice of the host. Li and Mahanti [37] have used theoretical methods in studying chains of alkali metals inside the channels of a zeolite. They found that when passing from Na, via K and Rb, to Cs the chains changed from being almost linear to having a clear zigzag structure (i.e., the bond angles are 175 ; 125 ; 110 ; and 93 ; respectively). Moreover, the zeolite host without chains is semiconducting (see Figure 9.16), but changes to becoming metallic when incorporating the alkali–metal chains. For Na

154

Chapter 9. Mixed Systems

K-ITQ-4

Na-ITQ-4

ITQ-4 5.0

5.0

5.0

4.0

4.0

4.0

3.0

3.0

2.0

2.0

2.0

1.0

1.0

1.0

0.0

0.0

1.0

1.0

1.0

2.0

2.0

2.0

3.0

3.0

EF

0.0

3.0 (a) Γ

Z

EF

(b) Γ

Z

Rb-ITQ-4

(c)

Γ

Z

Cs-ITQ-4

5.0

5.0

4.0

4.0 EF

3.0

2.0

1.0

1.0

0.0

0.0

1.0

1.0

2.0

2.0

3.0

3.0 Γ

Z

EF

3.0

2.0

(d)

EF

3.0

(e) Γ

Z

Figure 9.16. Band structures for the pure zeolite host (ITQ-4) as well as the host with Na, K, Rb, or Cs chains. Reproduced with permission of The American Physical Society from Ref. [37].

there is a clear tendency towards a bond-length alternation (i.e., a Peierls distortion), which is not the case for the other metals. An interesting proposal was made by Yang et al. [38]. Using a theoretical approach, they studied whether it was possible to insert thin carbon nanotubes (with a ( inside the channels of a zeolite host. They found that the interdiameter of 4 A) actions between the host and the carbon nanotubes were weak leading to only minor changes in the structural properties when passing from the isolated carbon nanotube to the nanotube inside the host. As a last example we discuss experimental and theoretical studies of the properties of Se chains inside the channels of mordenite crystal. Tamura et al. [39] characterized Se chains inside synthetic mordenite (Na2 O Al2 O3 20 SiO2 ) and found that the Se–Se bond length was decreased compared with that of crystalline Se. They interpreted that as showing that the Se atoms of their samples had a lower coordination, i.e., showed essentially no interactions with the host. Later experiments [40] have shown that these chains are helical.

References

155

Theoretical studies on isolated chains [41,42] have shown that these helices are very soft against changes in the dihedral angle (see, e.g., Figure 7.32). The isolated helices are semiconducting, but experiments on the chains inside the mordenite channels have shown that upon photoexcitation extra features in the optical gap shows up [43]. It can be argued [44–46] that, when observing (see Figure 7.31) that a planar zigzag chain is metallic, it is realistic to assume that these features are due to local structural changes where the dihedral angles changes towards that of a planar zigzag chain. That is the fact that the band gap closes for the periodic system when passing from the helical to the planar zigzag structure, manifests itself in the occurrence of extra gap states for the local structural distortions towards planarity. On the other hand, substituting the host by hydrocaoncrinite [Na8 Si6 Al6 O24 ðOHÞ2 2H2 O] the Se chains have a different structure and become a part of the total sample, i.e., in that case there are clear interactions between guest and host [47,48].

9.6.

Conclusions

In this section we have concentrated on chains that were produced inside some host material or, in some few cases, deposited on the exterior of another quasi-onedimensional system. Since chains, in particular of metal atoms, at most are metastable at normal conditions, embedding them inside some other material can provide a useful way of stabilizing them. The examples discussed by us demonstrated this clearly, i.e., chains, which in many cases not are found isolated, could be stabilized inside the host. However, it shall be stressed that although the chains to a large extent may be considered isolated from the host material, they are never absolutely free of any interaction with the host. This aspect has also the interesting consequence that through careful choice of the host material, the properties of the chains may be varied in a partly controlled fashion, at least when the chain is sufﬁciently soft. In this section we also saw examples of this including, for instance, the band gap of the linear chains of Pb atoms. Another example is the Se chains where we showed that depending on the host, the chains are either only weakly interacting with the host or strongly interacting with it. Another interesting aspect is that of combining two chain-like structures. This is, e.g., the case for the chains of metal atoms inside the carbon nanotubes. The carbon nanotubes that are inserted into the pores of a host may also fall into this category. Finally, chains of mixed types of atoms, i.e., quasi-one-dimensional alloys, may also form an interesting new way of modifying the materials properties, although here a big challenge is to be able experimentally to control the composition and structure as well as to characterize them.

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[3] B.I. Halperin, Adv. Chem. Phys. 13, 123 (1967). [4] E. Abrahams, P.W. Anderson, D.C. Licciardello, and T.V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979). [5] H.-L. Wu and P. Phillips, Phys. Rev. Lett. 66, 1366 (1991). [6] H.-L. Wu, W. Goff, and P. Phillips, Phys. Rev. B 45, 1623 (1992). [7] K. Schmidt and M. Springborg, J. Phys. Condens. Matter 5, 6925 (1993). [8] J. Todd, R. Merlin, R. Clarke, K.M. Mohanty, and J.D. Axe, Phys. Rev. Lett. 57, 1157 (1986). [9] G. Carlotti, D. Fioretto, L. Palmieri, G. Socino, L. Verdini, H. Xia, A. Hu, and X.K. Zhang, Phys. Rev. B 46, 12777 (1992). [10] D. Huang and G. Gumbs, Solid State Commun. 84, 1061 (1992). [11] N. Agraı¨ t, A.L. Yeyati, and J.M. van Ruitenbeek, Phys. Rep. 377, 81 (2003). [12] W.T. Geng and K.S. Kim, Phys. Rev. B 67, 233403 (2003). [13] A.M. Asaduzzaman and M. Springborg, Phys. Rev. B 72, 165422 (2005). [14] S. Katsumoto, K. Kamigaki, M. Ishida, N. Sano, and S.-i. Kobayashi, J. Phys. Soc. Jpn. 62, 424 (1993). [15] Y. Miyamoto, A. Rubio, M.L. Cohen, and S.G. Louie, Phys. Rev. B 50, 4976 (1994). [16] A. Rubio, J.L. Corkill, and M.A. Cohen, Phys. Rev. B 49, 5081 (1994). [17] K. Suenaga, C. Colliex, N. Demoncy, A. Loiseau, H. Pascard, and F. Willaime, Science 278, 653 (1997). [18] M. Remsˇ kar, Z. Sˇkraba, M. Regula, C. Ballif, R. Sanjine´s, and F. Le´vy, Adv. Mat. 10, 246 (1998). [19] G. Seifert, H. Terrones, M. Terrones, G. Jungnickel, and Th. Frauenheim, Solid State Commun. 114, 245 (2000). [20] L. Venkataraman and C.M. Lieber, Phys. Rev. Lett. 83, 5334 (1999). [21] M. Coˆte´, M.L. Cohen, and D.J. Chadi, Phys. Rev. B 58, 4277 (1998). [22] P. Zhang and V.H. Crespi, Phys. Rev. Lett. 89, 056403 (2002). [23] S.C. Tsang, Y.K. Chen, P.J.F. Harris, and M.L.H. Green, Nature 372, 159 (1994). [24] C. Guerret-Pie´court, Y. Le Bouar, A. Loileau, and H. Pascard, Nature 372, 761 (1994). [25] E.G. Gal’pern, I.V. Stankevich, A.L. Chistykov, and L.A. Chernozatonskii, Chem. Phys. Lett. 214, 345 (1993). [26] Y. Miyamoto, A. Rubio, X. Blase, M.L. Cohen, and S.G. Louie, Phys. Rev. Lett. 74, 2993 (1995). [27] X. Fan, E.C. Dickey, P.C. Eklund, K.A. Williams, L. Grigorian, R. Buczko, S.T. Pantelides, and S.J. Pennycook, Phys. Rev. Lett. 84, 4621 (2000). [28] X. Liu, T. Pichler, M. Knupfer, and J. Fink, Phys. Rev. B 70, 245435 (2004). [29] A. Rubio, Y. Miyamoto, X. Blase, M.L. Cohen, and S.G. Louie, Phys. Rev. B 53, 4023 (1996). [30] T. Nishinaka, A. Takano, Y. Doi, M. Hashimoto, A. Nakamura, Y. Matsushita, J. Kumaki, and E. Yashima, J. Am. Chem. Soc. 127, 8120 (2005). [31] S. Gemming, G. Seifert, and M. Schreiber, Phys. Rev. B 69, 245410 (2004). [32] S. Romanov, J. Phys. Condens. Matter 5, 1081 (1993). [33] K. Schmidt and M. Springborg, Solid State Commun. 104, 413 (1997).

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[34] B.H. Hong, S.C. Bae, C.-W. Lee, S. Jeong, and K.S. Kim, Science 294, 348 (2001). [35] M. Springborg and P. Sarkar, Phys. Rev. B 68, 045430 (2003). [36] J. Zhao, C. Buia, J. Han, and J.P. Lu, Nanotechnology 14, 501 (2003). [37] H. Li and S.D. Mahanti, Phys. Rev. Lett. 93, 216406 (2004). [38] X.P. Yang, H.M. Weng, and J. Dong, Eur. Phys. J. B 32, 345 (2003). [39] K. Tamura, S. Hosokawa, H. Endo, S. Yamasaki, and H. Oyanagi, J. Phys. Soc. Jpn. 55, 528 (1986). [40] V.V. Poborchii, J. Phys. Chem. Sol. 55, 737 (1994). [41] M. Springborg and R.O. Jones, Phys. Rev. Lett. 57, 1145 (1986). [42] A. Ikawa and H. Fukutome, J. Phys. Soc. Jpn. 58, 4517 (1989). [43] Y. Katayama, M. Yao, Y. Ajiro, M. Inui, and H. Endo, J. Phys. Soc. Jpn. 58, 1811 (1989). [44] M. Springborg, Solid State Commun. 89, 665 (1994). [45] M. Springborg, Z. Phys. B 95, 363 (1994). [46] M. Springborg, Int. J. Quant. Chem. 58, 717 (1996). [47] V.V. Poborchii, M. Sato, and A.V. Shchukarev, Solid State Commun. 103, 649 (1997). [48] A.V. Kolobov, H. Oyanagi, V.V. Poborchii, and K. Tanaka, Phys. Rev. B 59, 9035 (1999).

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Chapter 10

Crystalline Chain Compounds

As a natural extension of the discussion of the preceding section, we shall in this section discuss systems that consist of more or less weakly interacting chains. As above, here also a highly relevant question is, as to whether a speciﬁc system may be considered built from chains or, alternatively, just is a highly anisotropic, but nevertheless three-dimensional crystalline material. Most often, it is not possible to give a clear deﬁnition of this separation, and we shall repeatedly present examples where exactly this issue is being discussed.

10.1.

CaNiN

The crystal structure of CaNiN is shown in Figure 10.1. It consists of layers of parallel NiN chains that, moreover, are rotated 90 from one layer to the next. Ca atoms are placed between the layers. CaNiN was originally synthesized by Chern and DiSalvo [1]. One NiN unit has an even number of atoms, suggesting that an isolated NiN chain would have a gap at the Fermi level, although, on the other

Figure 10.1. Structure of crystalline CaNiN. The structure consists of layers of NiN chains (N and Ni are shown as lighter and darker atoms, respectively) separated by Ca atoms. 159

160

Chapter 10. Crystalline Chain Compounds

hand, many bands due to the partly ﬁlled Ni d orbitals may cross the Fermi level. Therefore, the material may experience some kind of symmetry lowering in order to open up a gap at the Fermi level (cf. Chapter 2). However, the fact that the NiN chain is not isolated but part of a three-dimensional crystal may make this symmetry lowering connected with the occurrence of strain in the crystal. Alternatively, a spin-polarization is also a possibility that may lead to a band gap at the Fermi energy. When the presence of the Ca atoms is also taken into account, these may donate electrons to the chains, but the above arguments are still valid. In fact, in their work, Chern and DiSalvo [1] observed a (weak) magnetic signal that could indicate that the structure is magnetic. A later experimental study [2] has, however, not conﬁrmed this suggestion. CaNiN has been the subject of some theoretical studies [3–6] CaNiN has the interesting property that it unites one-, two-, and three-dimensional effects in one material, i.e., it consists of one-dimensional chains, arranged in two-dimensional layers, that ultimately form a three-dimensional crystal. Moreover, on top of this, its properties may also be inﬂuenced by charge-transfer effects. The separation of these was the topic of one of the theoretical studies [6], where the structure was gradually decomposed. Starting with the real structure (either without or with the inclusion of the Ca atoms), and letting c being the lattice constant perpendicular to the layers, the following calculations were done that for: 1. The normal CaNiN crystal structure. Owing to the alternating direction of the chains in different layers, this tetragonal structure is referred to as ‘‘?’’ in Figure 10.2. 2. The normal CaNiN crystal structure with c doubled. This was done to eliminate the inter-plane interactions. This structure is referred to as ‘‘? 2c’’ in Figure 10.2. 3. The normal CaNiN crystal structure except that every second layer is rotated by 90 to make the chains in the different layers parallel. The positions of the chains in each layer were set so that the Ni (N) atoms were placed directly above Ni (N) atoms of the other planes. This structure is referred to as ‘‘k top’’ in Figure 10.2. Because all of the chains now point in the same direction, this crystal structure is now orthorhombic instead of tetragonal. 4. Structure 3, except that chains on alternating planes are offset so that the Ni (N) atoms were placed directly above N (Ni) atoms. These two calculations, when compared with calculation 1, give further information on the inter-plane interactions. This structure is referred to as ‘‘k offset’’ in Figure 10.2. 5. Structure 3, except that we double c: This structure is referred to as ‘‘k top 2c’’ in Figure 10.2. 6. Structure 4, except that we double c: 7. Structure 5, except that we double the interchain distance in the planes, i.e., we set the lattice constant b equal to 2b: The doubling was done in order to attempt to isolate the chains within each layer. This structure is referred to as ‘‘k top 2c 2b’’ in Figure 10.2. 8. Structure 6, except that we also double b: The results of these calculations are summarized in Figures 10.2 and 10.3. By comparing the total energies in the two panels of Figure 10.3, it is clear that the addition of the Ca atoms have a very strong effect, i.e., the total-energy variation

161

10.1. CaNiN

CaNiN

NiN DOS

10

DOS DOS

(2)

2c

(3)

Top

(3)

Top

5 0 10

DOS

(2)

2c

5 0 10

(4)

Offset

(4)

Offset

5 0 10

DOS

(1)

5 0 10

Top 2c

(5)

Top 2c

(5)

Top 2c 2b

(7)

Top 2c 2b

(7)

5 0 10

DOS

(1)

5 0

−5

0 Energy (eV)

5

−5

0

5

Energy (eV)

Figure 10.2. The density of states for CaNiN (right column) and NiN (left column) for some of the eight calculations that are considered in the analysis of the dimensionality effects of CaNiN. The Fermi energy is in each panel set equal to 0. The calculation number (see the text) is given in parentheses, too. Reproduced with permission of the American Physical Society from [6].

increases by a factor of 10–20. On the other hand, the relative order of the chains in neighbouring layers seems to be of essentially no importance for the total energy, although there are clearly interactions between the chains: increasing the distances between them, increases the total energy. The lowest panels in Figure 10.2 show that this structure clearly is one of the essentially non-interacting chains where sharp van Hove singularities can be identiﬁed. However, even for the most compact structures, i.e., those of the top panels in Figure 10.2, these singularities can also be recognized. Thus, although CaNiN is a three-dimensional crystalline material, in particular its electronic properties are largely those of quasi-one-dimensional chains. Moreover, it is far from the only example of a material with dominating interactions in less than three dimensions, with the high-temperature superconductors forming a particularly well-known class of materials that exhibit both quasi-two- and quasione-dimensional behaviours.

162

Chapter 10. Crystalline Chain Compounds

6

Energy (eV/unit)

CaNiN 4

2

0

Energy (eV/unit)

NiN 0.4

0.2

0

1

2 3 4 5 6 7 8 Calculation

Figure 10.3. Variation in the total energy per CaNiN or NiN unit for the eight calculations that are considered in the analysis of the dimensionality effects of CaNiN. Reproduced with permission of the American Physical Society from [6].

10.2.

SN

Another material for which the importance of interchain interactions, dimensionality, and crystallinity has been discussed intensively and that, moreover, has been at the centre of much attention, is sulphur nitride, SN. In 1910 Burt [7] reported the synthesis of ‘a new sulphide nitrogen’. It took more than 40 years before the compound was identiﬁed as a sulphur nitride polymer (SN)x (see, e.g., Ref. [8]) and it was assumed that each chain consisted of a zigzag arrangement of alternating sulphur and nitrogen atoms, cf. Figure 10.4(a). Renewed interest in this material developed in the 1970s. In 1973 it was discovered by Walatka et al. [9] that the material remained metallic down to 4.2 K, which in 1975 was extended to 1.5 K by Greene et al. [10]. In a subsequent work, Greene et al. [11] reported that the material undergoes a transition to a superconducting state at T c ¼ 0:26 0:03 K; cf. Figure 10.5. This discovery was considered the ﬁrst observation of superconductivity in a quasi-one-dimensional material, and much research activity was devoted to understand this behaviour as well as to search for new, but similar, materials with the same property (see Refs. [12,13]). However, a major problem was that the (SN)x

163

10.2. SN

(a)

(b)

(c)

Figure 10.4. (a) The originally proposed form of (SN)x ; (b) the correct one, and (c) the structure of a chain of S2N2 units. Sulphur and nitrogen is represented by closed and open circles, respectively. Reproduced with permission of the American Physical Society from [20].

1.0 H1 = 335G R R(1°K)

H1 = Earth

0.5

0

0.1

0.2

0.3

0.4

0.5

°K)

T(

Figure 10.5. The resistance of (SN)x as a function of temperature both without and with an external magnetic ﬁeld. Reproduced with permission of the American Physical Society from [11].

was ill-deﬁned, since it consisted of only partially parallel ﬁbers each containing almost parallel macromolecules. This made it difﬁcult experimentally to deduce a crystal structure as well as the geometry of a single polymer chain, and not until 1975 the correct structure of each macromolecule was determined for the ﬁrst time [14,15]. The structure is shown in Figure 10.4(b). It was also suggested that polymerization of planar almost square-shaped S2N2 molecules [Figure 10.4(c)] directly leads to the (SN)x polymer [16–18]. Another question that attracted much attention was why the polymer did not undergo a Peierls transition but remained metallic down to very low temperatures and became superconducting at even lower temperatures. Band-structure calculations suggested that the reason was that not one but two bands crossed the Fermi

164

Chapter 10. Crystalline Chain Compounds

(a)

(b)

0.0

0.0 σ5

−5.0

σ5 π2 σ4

Energy (eV)

−10.0

π2

−5.0

σ4

−10.0

π1 σ3

π1

σ3

−15.0

−15.0 σ2

σ1

−25.0

−30.0 0.0

σ2

−20.0

−20.0

0.5 K

−25.0

1.0

−30.0 0.0

σ1

0.5 K

1.0

Figure 10.6. Band structures of the optimized geometries of (SN)x for the structures of Figure 10.4(a) and (b), respectively. The Fermi level passes through the middle of the p2 band in both cases. Reproduced with permission of the American Physical Society from [20].

level, but as shown by Berlinsky [19] this explanation does not hold (see also Section 2.2). Moreover, some of the most recently band-structure calculations [20] (see Figure 10.6) ﬁnd only one band crossing the Fermi level. On the other hand, in their theoretical study Causa` et al. [21] included the interactions between more chains. As one can see in Figure 10.7, these interchain interactions lead to a signiﬁcant broadening of the bands so that one has ultimately to conclude that (SN)x is not a quasi-one-dimensional material but rather a highly anisotropic three-dimensional material. Finally, the attempts to improve the materials properties, most notably the transition temperature for the superconducting state, T c ; were not successful. Attempts to substitute S (partly) by Se, or to dope the material did not lead to the wanted results, and gradually the interest in this material decreased.

10.3.

MX2 chains

There exists a number of crystalline materials that contain charged MX2 chains surrounded by counterions. Here, M is a metal atom like Fe, Co, Pd, or Pt, whereas

165

10.3. MX2 chains

0.5

E (a.u.)

0.0

−0.5

−1.0

−1.5

10

20−(102)

20−(100)

20−(001)

30

K2

Figure 10.7. Band structures for (SN)x when including an increasing amount of interchain interactions (when passing from the left to the right panel). In these calculations, the Fermi energy has been put equal to 0. Moreover, whereas the results of Figure 10.6 were obtained with density-functional calculations, those of the present ﬁgure have been obtained with Hartree–Fock calculations, which explains why the bands are broader in the latter calculations than in the former ones. Reproduced with permission of the American Institute of Physics from [21].

Figure 10.8. Structure of an MX2 chain. M and X atoms are marked with closed and open spheres, respectively. The depicted structure is a planar one with the angle g between neighbouring M2X2 rhombi being 0 : Also non-planar chains with g ¼ 90 exist. Reproduced from [23].

X most often is sulphur but can also be chlorine. Some examples are KFeS2, RbFeS2, CsFeS2, Na3Fe2S4, Co3Fe2S5, Na2PdS2, Na2PtS2, K2PtS2, Rb2PtS2, and a-PdCl2 [22]. The MX2 chains all have structures like those of Figure 10.8. Silvestre and Hoffmann [22] have presented a theoretical study of the whole class of materials, whereas a more recent theoretical study [23] focused on just a single member of this family, PtS2 with two K atoms per formula unit as counterions. We shall brieﬂy review the results of the latter as a representative case.

166

Chapter 10. Crystalline Chain Compounds

In Figure 10.9 we show the band structures for a neutral PtS2 chain, both for the planar structure with g ¼ 0 and for a structure with g ¼ 90 : Both structures are clearly metallic, and it is remarkable to observe that for g ¼ 0 two extra electrons are needed in order to ﬁll the bands so that the system becomes semiconducting (or insulating), whereas for g ¼ 90 four extra electrons are needed. In both cases we have assumed that the band structures do not change upon this addition. In fact, the (planar) PtS2 chains are surrounded by K counterions, so that the stoichiometry becomes K2PtS2, and this material is indeed non-metallic. Assuming also that other MX2 chains have similar band structures (this is indeed a crude approximation), the results do give some suggestions as to why some chains have g ¼ 0 and others have g ¼ 90 ; partly depending on their charge or, alternatively, on the number and types of the counterions. In the theoretical study, however, it was found that the simple addition of two K atoms per PtS2 unit to a single chain led to quite different results. Thus, in that case the rigid-band picture above was not at all applicable and, instead, several bands crossed the Fermi level and the system was metallic. On the other hand, considering the full three-dimensional structure, the rigid-band picture was recovered. We have here, accordingly, a case where the intrinsic electronic properties of the individual chains are determined by the whole three-dimensional crystalline arrangement of the atoms, although direct electronic interactions between the chains or the counterions are negligible. Rather, the long-range Coulomb potentials are signiﬁcant in determining the properties. A further result in support for this consensus is that the charge also determines the structure of the chain. Thus, in Figure 10.10 we show the variation in the total energy for the neutral PtS2 chain. For the neutral chain, g ¼ 90 ; but adding the counterions, g changes to 0 : In total, this example clearly demonstrates the importance of interactions with the surroundings. Since it is difﬁcult to produce isolated, metallic chains, it is important to understand and, accordingly, also to control the inﬂuences any

Εnergy (ev)

(d)

(c)

(b)

(a) 0

0

0

0

−5

−5

−5

−5

−10

−10

−10

−10

−15

−15

−15

−15

−20

−20

−20

−20

−25

0

0.5

1

−25

0

0.5

1

−25

0

0.5

1

−25

0

0.5

1

Figure 10.9. Band structures of the neutral PtS2 chain with (a,b) g ¼ 0 and (c,d) g ¼ 90 : Spin–orbit couplings were included in (b) and (d); not in (a) and (c). The horizontal dashed lines mark the Fermi energy for the neutral chain. Reproduced from [23].

10.4. Metal trichalcogenides

167

Energy (eV/unit)

2

1

0

0

30 60 γ (deg.)

90

Figure 10.10. Calculated total energy per PtS2 unit as a function of g for a neutral PtS2 chain with g varying from g ¼ 0 to 90 : Spin–orbit couplings were not included in the calculations. Reproduced from [23].

surrounding medium has on the intrinsic properties of the (metallic or non-metallic) chains. In the present example we found that long-range Coulomb potentials were important.

10.4.

Metal trichalcogenides

The metal trichalcogenides, MX3, with M being Nb, Zr, or Ta and X being S or Se, belong to a class of materials that are called charge-density wave (CDW) conductors. The occurrence of a CDW is an effect of quasi-one-dimensionality. A chargedensity wave is the simultaneous modulation of the conduction-electron density and of the underlying atomic lattice, that moves very easily under the inﬂuence of an external electric potential, see Figure 10.11. A CDW is characterized by three special features. First, as we discussed in Section 2.1 in the special case of the Peierls distortion, the CDW is caused by one or more bands that cross the Fermi surface. By lowering the symmetry, the total energy is lowered, similar to what we discussed in Section 2.1. Second, also similar to the discussion of Section 2.1, the symmetry lowering is accompanied with the opening up of a gap at the Fermi energy. This is still considered a clear indication of the occurrence of a CDW (see, e.g., Ref. [25]). Third, the wavelength of the CDW is p=kF ; with kF being the Fermi wave number, i.e., the value of k at which the bands cross the Fermi surface. All these aspects are illustrated in Figure 10.12. As Figure 10.11 indicates, the CDW can very easily move, thus leading to a high conductivity. This is seen in Figure 10.13 where one recognizes a small threshold voltage below which there is no current ﬂowing. But above this threshold, the CDW moves with only little resistance, and, actually, before the BCS theory of superconductivity was developed [26], there had been some thoughts that CDWs were responsible for superconductivity [27,28], although these ideas have now been abandoned. Later in this presentation (Chapter 12) we shall discuss synthetic metals (i.e., conjugated polymers) for which trans polyacetylene is the prototype. For this material we shall discuss how the so-called solitons are held responsible for the large

168

Increasing time

Chapter 10. Crystalline Chain Compounds

Figure 10.11. The modulation of the conduction-electron density and the underlying atomic lattice that is moving under the inﬂuence of an external ﬁeld. Reproduced with permission of the American Institute of Physics from [24].

Energy E(κ)

(a) Electron density

Atom positions

a

− -π α

EF -κF

+κF

Wavevector κ

π − α

Electron density

Atom positions

λc = π/κF

Energy E(κ)

(b)

-π − α

2Δ -κF

+κF

Wavevector κ

π − α

Figure 10.12. Schematic representation of the mechanism that leads to the formation of a chargedensity wave. The left parts show the electron density and the atomic lattice, and the right parts show the band structures. Reproduced with permission of the American Institute of Physics from [24].

169

CURRENT Ic (microamps)

10.4. Metal trichalcogenides

200

100

ET

0 0

0.2

0.4

0.6

0.8

1

ELECTRIC FIELD (volts/centimeter) Figure 10.13. The current as a function of applied voltage for NbSe3. In particular, the non-vanishing threshold voltage is observed. Reproduced with permission of the American Institute of Physics from [24].

conductivity. Here, we mention that the existence of solitons is also assumed for the CDW materials. Therefore, they shall be discussed brieﬂy here. In order to understand what a soliton is, we return to Section 2.1. In Figure 2.1 we have shown a linear chain of identical atoms. Let us assume that we have numbered the atoms, n ¼ . . . ; 2; 1; 0; 1; 2; . . . ; and that we have introduced a coordinate u for each atom that describes the position of that atom relative to the position in the upper panel of Figure 2.1. In that panel we have un ¼ 0,

(10.1)

whereas for the atoms in the lower panel we have un ¼ ð1Þn u0

(10.2)

u0 is a constant. We now introduce the so-called order parameter xn ¼ ð1Þn un ,

(10.3)

which is constant for the regular structures of Figure 2.1. It can easily be recognized that the two structures with the same constant value of jxn j but differing in the sign of xn are equivalent. It is therefore possible that domain walls across which xn changes from one of the two values to the other may exist and, moreover, be highly mobile. Mathematically, the order parameter may have the form n n 0 xn ¼ u0 tanh , (10.4) L where n0 is the position and L the width of the domain wall. This domain wall is the so-called soliton. When L is not very small (i.e., 51), the soliton is so delocalized that varying its centre, n0 ; continuously leads to only small changes in the total energy of the system. Then, the soliton can move rather freely up and down the chain and eventually carry charge and/or spin. Through such a large mobility of the solitons, a large conductivity can result.

170

Chapter 10. Crystalline Chain Compounds

As mentioned, we shall return to this below when we discuss trans polyacetylene,but here we mention that for the CDW materials the crucial issue is the deﬁnition of an order parameter that may take more values that all lead to equivalent structures. Local changes from one value to another, like that of equation (10.4), can then exist. The occurrence of CDW requires a uniform lattice without impurities. Any irregularity may disturb the CDWs, i.e., pin it. Thus, having impurities or defects in the otherwise regular chain structure, the CDW may get pinned to those and the conductivity is reduced dramatically. There exists many different types of materials that show CDW characteristics (see e.g., Ref. [29]), but here we shall just brieﬂy discuss the structures of the metal trichalcogenides. They have all that in common that they consist of parallel chains, with each chain containing triangular X3 units separated by M atoms so that trigonal prismatic columns occur (see Figure 10.14). However, different materials have different structures of the chains and, in some cases, even more different structures of the chains occur in the same material. Since the band-structure crossings at the Fermi level are crucial for the CDW properties, each of those chains may have different CDW properties, leading to a quite complicated behaviour of the total conductance. These issues have been discussed in detail by Wilson [30]. Here, we shall brieﬂy illustrate the complicated structures through some selected examples. These are shown in Figure 10.15 for three different materials, i.e., ZrSe3, TaSe3, and NbSe3. Of these, NbSe3 may be the most popular member of this class of materials. The different types of chains have been given different pattern coding, showing that the ﬁrst material has one type of chains, the second one has two different types of chains, whereas the last one has three types of chains.

Se Nb

Figure 10.14. Schematic representation of the structure of NbSe3. Reproduced with permission of the American Institute of Physics from [24].

171

10.5. Metal tetrachalcogenides

ZrSe3

x

x

x

x

a θ

a

x

β

c

c

x

x

x

2.42 2.87 286

TaSe3 9

2.73

3.2

2. 3.3

3.7

3.7

x

2.91

6

2.7

x

x

x0

5

2.7

3

x

2.7

2.9

2.49

2.73

3

x

x

1

x

4

x

15

3.8 2. 37

3.7

π

0

3.6

x

21

3.

3

3 3.3

3.7

15 3.6 7

x

0

x

3.3

c

a

1

0

4.02

x 2π−

x

5 3.5 93 3.20 .33

7 3.4

NbSe3

2.7 2.98 5 2.58 2.6

3.74

3.07

3.83

2.34

0 2.8 2.72

186

x

x

Figure 10.15. Schematic representation of the structure of ZrSe3, TaSe3, and NbSe3. The presentation shows a top view of the chains, and different types of chains have been given different patterns. Reproduced with permission of the American Physical Society from [30].

The prototype of these materials is NbSe3 that, accordingly, has been the subject of many studies. These include the experimental study of Haifeng and Dianlin [25] who used tunnelling spectroscopy as a function of temperature to study the formation of the gap at the Fermi energy due to the CDW. Their results are reproduced in Figure 10.16, where the occurrence of a gap of 100–150 meV at temperatures below 250 K can be identiﬁed.

10.5.

Metal tetrachalcogenides

Some metal tetrachalcogenides also form quasi-one-dimensional materials. These include, e.g., TaTe4, NbTe4, (NbSe4)3I, and (TaSe4)2I. Figure 10.17 shows the

172

dl/dv (a.u.)

Chapter 10. Crystalline Chain Compounds

257K 247K 195K 152K 124K 92K

214K 170K 135K 108K 77.3K

−200

−150

−100

−50

0

50

100

150

200

Voltage (mV) Figure 10.16. Tunnelling spectra for NbSe3 at different temperatures between 77 and 260 K. Reproduced with permission of the American Physical Society from [25].

1/2 1/2 1/4,3/4 1/2 1/2 Ta TaTe4

Te

Figure 10.17. Structure of TaTe4 (and NbTe4) perpendicular to the chain direction. For (NbSe4)3I, the I atoms are situated between the chains. Reproduced with permission of the American Physical Society from [31].

structure of those that indeed resembles those of the previous section, except for the fact that the triangular X3 units here are replaced by square X4 units and, in some cases, counterions also occur. Some of these materials, for instance NbTe4, exhibits a CDW, as the materials of the previous section, whereas others do not [31]. Coluzza et al. [31] performed

10.6. Metal oxides: spin-chain and spin-ladder compounds

173

high-resolution photoemission experiments on these materials in order to study in detail the spectra closest to the Fermi energy. They found clear signals of the quasione-dimensional character of the materials, but no indications of Luttinger-liquid behaviour, although, in principle, any quasi-one-dimensional metallic system could posses Luttinger-liquid properties.

10.6.

Metal oxides: spin-chain and spin-ladder compounds

Transition metal oxides form a class of materials that possesses many unusual properties, often ascribed to many-body, or correlation, effects. One of the most well-known examples of these materials are the high-temperature superconductors for which quasi-two-dimensional layers of Cu and O atoms are an essential building block. Other Cu-O materials contain quasi-one-dimensional chains that shall be discussed here. One of the most intensively studied members of this class of materials is GeCuO3 whose structure is shown in Figure 10.18. It is seen that the structure consists of vertex-sharing CuO6 octahedra with Ge acting as spacers that keep the chains formed by the linked octahedra well separated. Hase et al. [33] observed that when the temperature was lowered below 14 K, the magnetic susceptibility abruptly decreases. They interpreted this as a phase transition to a state with an antiferromagnetic ordering of the spins of neighbouring octahedra. Accordingly, GeCuO3 is a spin-12 chain and at the lowest temperatures these spins are ordered, whereas they are disordered at higher temperatures. The transition is hence a spin-Peierls transition, i.e., a symmetry-lowering where not the structure but the spin density obtains a lower symmetry. Only partly recognizable in Figure 10.18 is the fact that the oxygen atoms are not equivalent. Instead, the CuO6 are stretched such that the two apical oxygen atoms

Figure 10.18. Structure of GeCuO3. Reproduced with permission of the American Physical Society from [32].

174

Chapter 10. Crystalline Chain Compounds

GeCuO3

LAPW

TB

4 2 EF

Energy (eV)

0 −2 −4 −6 −8 −10 Γ

X S

Y r

Z

UR

T Z

Γ

X S

Y r

Z

UR

T Z

Figure 10.19. Band structures of GeCuO3 as obtained with (left panel) density-functional calculations and (right panel) a simple parameterized tight-binding model. Reproduced with permission of the American Physical Society from [32].

have a signiﬁcantly larger distance to the copper atoms (2.77 A˚) than the other four oxygen atoms (1.94 A˚). Only the latter are relevant for the orbitals closest to the Fermi level, resulting ultimately in a spin-12 linear-chain CuO2 system. Mattheiss [32] performed density-functional calculations on GeCuO3 without taking a spin polarization into account (i.e., corresponding to the high-temperature phase). His calculated band structures are reproduced in Figure 10.19. The occurrence of two ﬂat bands that are crossing the Fermi level is readily recognized. Analysing these orbitals in detail he found that they are formed by one d orbital on the copper atoms and two p functions on the oxygen atoms, cf. Figure 10.20. Wu et al. [34] extended the study of Mattheiss and showed that correlation effects indeed could open up a gap at the Fermi level when allowing for a spin polarization. In that case, each of the narrow bands which (Figure 10.19) cross the Fermi level is split into two, one for each spin direction, with one of them being fully occupied and the other being completely empty. The spin-Peierls transition has two components, i.e., the occurrence of an (antiferromagnetic) ordering of the spins of the repeated units along the chains and a lowering of the symmetry (a doubling of the length of the repeated units). In order to see this experimentally, one may use, e.g., neutron-diffraction experiments, as done, for instance, by Nishi [35]. If the length of the repeated unit is doubled, extra diffraction peaks will show up at half the spacing in reciprocal space. The spectra in this part of the reciprocal space at different temperatures are reproduced in Figure 10.21. At the lowest temperature, 2.7 K, a clearly identiﬁable peak is seen

175

10.6. Metal oxides: spin-chain and spin-ladder compounds

GeCuO3 kz = (π/c)

kz = 0 16

Distance along [100] (a.u.)

14 12 10 8 6 4 2 0 0

2

4

6

8

10

0

2

4

6

8

10

Distance along [320] (a.u.) Figure 10.20. Charge density of the orbitals closest to the Fermi level for GeCuO3. Reproduced with permission of the American Physical Society from [32].

1000 T = 2.7 K T = 7.3 K T = 20 K

CuGeo3 (0,1,0.5)

900

Intensity/2 min

800 700 600 500 400 300 200

48

48.5

49 49.5 Omega (deg)

50

Figure 10.21. The intensity of the satellite peak at the superlattice position for different temperatures for GeCuO3. Reproduced with permission of the Institute of Physics from [35].

176

Chapter 10. Crystalline Chain Compounds

1.8 b c

1.4 1.2

0.5

1

0.4

0.8

0.3

dχb/dT

Molar susceptibility (memu/mole)

1.6

0.6

0.2

0.4

0.1

0.2

0 13

14

15

16

0 0

10

20

30

40

50

60

70

80

Temperature (K) Figure 10.22. Molecular magnetic susceptibility of GeCuO3 along two crystalline axis (c being along the chains, b perpendicular to those) as a function of temperature. The inset shows the derivatives whereby the phase transition becomes clearly recognizable. Reproduced with permission of the American Physical Society from [36].

which is not found at the higher temperatures, conﬁrming the proposed existence of a spin-Peierls transition. A similar study was undertaken by Pouget et al. [36] who also measured the magnetic susceptibility as a function of temperature. Their results are reproduced in Figures 10.22 and 10.23. Also these show clearly the existence of the spin-Peierls transition. Also other experimental techniques have been used in exploring the phase transition. These include Cu NMR [37], electron diffraction [38], and neutron scattering at different pressures [39]. For the metal trichalcogenides we have mentioned above the possible existence of the so-called solitons. Their occurrence was related to the existence of two energetically degenerate but structurally different structures. Also for GeCuO3 the occurrence of solitons has been considered [40]. Thus, starting with the hightemperature phase where all CuO6 octahedra are equivalent, two equivalent structures can be formed when passing to the low-temperature phase, differing only in the way the pairs (dimers) are formed. Phase boundaries between these two structures may then occur, and these are the solitons. Kiryukhin et al. [40] studied experimentally the existence of those and found that they were unusually wide. Thus, the parameter L above [equation (10.4)] was found to be well above the length of 10 octahedra. We have so far discussed only GeCuO3, but other related compounds also exist. Sr2CuO3 and Ca2CuO3 (see for instance Ref. [41]) and SrCuO2 [42] belong to that category. In Figure 10.24 we show the structures of Sr2CuO3 and SrCuO2. It is seen

10.6. Metal oxides: spin-chain and spin-ladder compounds

177

600 CuGeO3

Neutron intensity (counts / 25 s)

500

400

300 500 400

200

300 200 100

100

Q = (q,6q,1/2) 0 0.48 0.49 0.50 0.51 0.52 q (r.l.u.)

0

0

5

10 Temperature (K)

15

20

Figure 10.23. Temperature dependence of the intensity of the satellite peak in the neutron-scattering experiments. The inset shows the peak. The sharp increase for temperatures dropping below 14 K is clearly seen. Reproduced with permission of the American Physical Society from [36].

Figure 10.24. Structure of Sr2CuO3 and SrCuO2. The chains are running along the a direction. Reproduced with permission of the American Physical Society from [42].

178

Chapter 10. Crystalline Chain Compounds

Figure 10.25. Structure of SiCuO3. Reproduced with permission of the American Physical Society from [43].

that the structures are different from those of GeCuO3 but also that the important motif, i.e., Cu atoms that are fourfold coordinated by O atoms, is found for all the materials. SiCuO3 and CaCu2O3 are other members, again with their own special properties (partly due to somewhat different structures; cf. Figures 10.25 and 10.26 and Refs. [43,44]), but still sharing many of the general properties of these spinchain compounds, as is also the case for LiCu2O2 [45]. Although we have concentrated on spin properties, other properties have also been studied for those materials. Signals that could indicate the existence of Luttinger-liquid behaviours, which we have discussed several times, have also been explored for the materials of this section. We mention here the work of Kim et al. [46] who performed angle-resolved photoelectron spectroscopy experiments on SrCuO2 and found indications of spinon and holon excitations, as would be the case

10.6. Metal oxides: spin-chain and spin-ladder compounds

179

Figure 10.26. Structure of CaCu2O3. The corner-shared CuO2 zigzag chains running along the b axis are alternatively tilted by roughly 28:6 : Reproduced with permission of the American Physical Society from [44].

Figure 10.27. Spin-ladder compounds. In (B) and (C) the black dots are Cu atoms, and oxygen atoms are placed at the intersections of the solid lines. Reproduced with permission of the American Physical Society from [48,49].

for a Luttinger liquid. To our knowledge, however, this has not been conﬁrmed by later studies. On the other hand, SrCu2O3 is one example of the so-called spin-ladder compounds, with (VO)2P2O7 and Sr2Cu3O5 being other examples; see, e.g., Figure 10.27. Actually, GeCuO3 may be considered an example of the simplest possible case, i.e., a linear chain of spins, where there is coupling between essentially only nearest neighbours. If one imagines putting two such chains of spins next to each other, so that the spins of the two chains interact both inside of the individual chains (or legs, as they are called) and with the spins of the nearest atoms from the next chain, one has a twoleg spin-ladder compound. Of course, this picture is a strong simpliﬁcation, but as we saw for GeCuO3 the orbitals of one single band may be those that describe the spin

180

Chapter 10. Crystalline Chain Compounds

properties of the system of interest and one may, therefore, simplify the description of the properties of the system of interest through a model where only one orbital per unit is of importance. Thus, when passing to SrCu2O3, Sr2Cu3O5, and (VO)2P2O7 one may employ similar simpliﬁcations and have, then, two- or three-leg ladder compounds. As indicated in Figure 10.27, the effective interactions between the different spins can be estimated and, subsequently, one may study the ground state and excitations with this simpliﬁed model; see, e.g., [47]. The theoretical analysis of such spin systems may be based on models related to the Heisenberg model. Then the total energy is written in terms of interactions between spins localized to individual atoms, X ~^m S ~^n , H^ ¼ J m;n S (10.5) m;n

~p is the spinon the pth atom, and the strengths of the interactions are where S described through the parameters J that depend on the types of the two interacting atoms as well as on their interatomic distance. For Jo0; structures with antiparallel spins are energetically preferred, which in some cases causes frustration (e.g., for three atoms it is not possible to construct a structure where all nearestneighbour spins are anti-parallel), which ultimately may result in interesting spin structures. A further discussion of these issues is, however, beyond the scope of this presentation. Also oxides based on vanadium have interesting spin properties. Of those, a0 NaV2O5 is probably the most well-known example, whose structure is shown in Figure 10.28. This has also been discussed in the context of spin-Peierls transitions [50,51] with, however, not a half-ﬁlled but a quarterly ﬁlled band [52] leading to important modiﬁcations in the properties. Among those modiﬁcations is the fact that the V atoms would then become inequivalent, although this is still a matter of debate [53]. As a further modiﬁcation we add that one may replace Na by Mg [54] arriving at the structure of Figure 10.29. Before closing this section we also mention brieﬂy the study by Nakamura et al. [55] on BaVS3 whose structure is shown in Figure 10.30. They found that this material possesses a number of phase transitions: below 240 K the chains change from linear to zigzag structures, below 70 K the material changes from metallic to semiconducting, and below 35 K it changes from paramagnetic to antiferromagnetic. The authors performed several different types of experiments and suggested that a Jahn–Teller distortion is responsible for the transition at 240 K, whereas a Luttinger-liquid behaviour was held responsible for the vanishing signals at the Fermi level in angle-resolved photo-electron spectroscopy experiment below 70 K. Thus, also for this material many-body, or correlation, effects are thought to be important although in this case the existence of a Luttinger liquid (which could be a further consequence of strong correlation effects in quasi-one-dimensional systems) has not been observed. Finally, Kuntscher et al. [56] studied experimentally the quasi-one-dimensional conducting chain compound SrNbO3:41 : They found an extremely small energy gap at the Fermi level (only some few milli electron volts) and suggested that a complicated interplay between phonons, electrons, and the high polarizability of the three-

10.7. Incommensurate elemental crystals

181

Figure 10.28. Structure of a0 V2O5. The ﬁlled circles represent Naþ ions, and the white and shaded pyramids represent two kinds of VO5 pyramids. Finally, A and B show the V4þ O5 and V5þ O5 chains. Reproduced with permission of the Japanese Physical Society from [50].

dimensional medium all are needed in order to explain the experimental observation. And Yonemitsu and Bishop [57] studied theoretically the optical excitation spectra of CuO chains that are parts of some of the high-temperature superconductors.

10.7.

Incommensurate elemental crystals

Barium and rubidium are most often considered typical examples of simple metals that are, accordingly, easy to understand. Nevertheless, these materials have surprising properties and have chain-like structures. Under pressure Ba as well as Rb undergo a variety of phase transitions and for both it has been found that for pressures of the order of 20 GPa new structures occur in which chains are embedded inside a host [58–61]. Figure 10.31 shows the structure of this crystalline phase for Ba [58]. It is seen to consist of an ordered host (the dark atoms in the ﬁgure) that contains channels inside which a guest (the lighter atoms in the ﬁgure) can sit. The powder diffraction x-ray experiments of Nelmes et al. [58] showed that the guest chains have a structure that is incommensurate with that of the host. Although the precise reason behind this is not known it does suggest that the interactions between guest and host are weak or, at least, unstructured, so that the chains can be considered as rather

182

Chapter 10. Crystalline Chain Compounds

Figure 10.29. Structure of Mg2O5. Reproduced with permission of the Institute of Physics from [54].

independent entities, maybe stabilized by an unstructured, homogeneous potential from the host. A similar suggestion can be made for the crystalline phase of Rb that is shown in Figure 10.32 [59]. The structure of this was determined by Schwarz et al. [59] using

10.8. CH3BiI2

183

Figure 10.30. Structure of BaVS3. Reproduced with permission of the American Physical Society from [55].

Figure 10.31. Structure of the crystalline phase IVa of Ba. The dark and the light atoms mark the host and the guest structures, and the right panel shows two different guest structures. Reproduced with permission of the American Physical Society from [58].

synchrotron x-ray diffraction, and although the structure is not identical to that mentioned above for Ba, it does share the property of consisting of a host with channels inside which linear chains are placed. Moreover, the structure of the chains was found to be essentially incommensurate with that of the host. Theoretical calculations have been performed on the incommensurate Ba structure in order to understand its origin [60]. Reed and Ackland [60] were able to reproduce the occurrence of a structure with at least a very low symmetry, if not incommensurate, at certain pressures and suggested a complex interplay between free-electron behaviour and s ! d electron transfers as being at the root of the origin of this phenomenon.

10.8.

CH3BiI2

We discuss CH3BiI2 as a single, out of many, example of an organometallic, quasione-dimensional material. This material was synthesized and characterized by Wang et al. [62] and the structure is shown in Figure 10.33. It consists of parallel

184

Chapter 10. Crystalline Chain Compounds

Figure 10.32. Structure of the crystalline phase IV of Rb. Reproduced with permission of the American Physical Society from [59].

C(1) I(2)

I(2)' C(1) Bi(1) I(2) I(3)

I(1) Bi(1) Bi(2) I(4)

I(1)

I(1)' I(3)

I(3)' Bi(2)

C(2) I(4)'

c

I(4) C(2)

b a

b

Figure 10.33. The structure of CH3BiI2. Reproduced with permission of the American Chemical Society from [62].

chains, with each chain having a backbone of Bi atoms, each linked with the next Bi atom via two I atoms. Finally, a CH3 group is attached to each Bi atom. Wang et al. [62] also performed electronic-structure calculations on the resulting material, using the fairly simple extended-Hu¨ckel approach. The results of their calculations gave that this material is clearly quasi-one-dimensional: the electronic interactions perpendicular to the chain direction are essentially vanishing. Moreover, it was found to be an insulator with a relatively large gap.

10.9.

Pt(CN)4-based chain materials

The square planar [Pt(CN)4] (tetracyanoplatinate) ion is an important building block for a whole class of chain materials [63]. As shown in Figure 10.34, orbital overlapping between the Pt d z2 orbitals lead to signiﬁcant covalent interactions along the stacking direction.

185

10.10. Conclusions

CN

CN pt CN

CN

pt

CN

CN

CN CN

CN pt

CN

CN

pt

CN

CN

CN CN

CN pt

CN

CN

Figure 10.34. The structure of stacked square-planar [Pt(CN)4] units. Reproduced with permission of the American Physical Society from [63].

The precursor of these materials, K2[Pt(CN)4]3H2O, is an insulator, but in the presence of other substances, e.g., halogens, it may change into a conductor. This change may be accompanied by a signiﬁcant reduction in the Pt–Pt interatomic distances. Thus, for K2[Pt(CN)4]3H2O this distance is 3.48 A˚, but for K2[Pt(CN)]Br0:3 3H2O it is reduced to 2.89 A˚. Scrocco [63] demonstrated that through variation of the water content, the properties of these materials could be modiﬁed. Scrocco studied materials K2[Pt(CN)4]xH2O, where x is of the order of 0.3, i.e., markedly lower than the value for the precursor, 3. For this lower value the conduction is signiﬁcantly improved, but a clear explanation for this is still lacking. Among the proposals is one according to which the Pt–Pt interatomic distances change when changing x: According to other proposals, the [Pt(CN)4] squares (cf. Figure 10.34) rotate relative to each other about the stacking direction, thereby modifying the overlaps between the orbital of neighbouring units. Although we here do not offer any ﬁnal explanation for the experimental ﬁndings, we emphasize that these materials constitute an interesting example of how composition can be used in modifying the materials properties.

10.10.

Conclusions

In contrast to the previous section, where we studied mixed systems that, however, were non-crystalline, we have in this section studied crystalline systems of which

186

Chapter 10. Crystalline Chain Compounds

most consisted of more than one type of atoms. Therefore, the properties of these materials are markedly inﬂuenced by the fact that the systems are true threedimensional ones, which is a clear difference from many of the systems we have discussed earlier in this presentation. Nevertheless, for many of the systems one may assume that the three-dimensional structure ﬁrst of all provides a frame for stabilizing chains, whose intrinsic properties are largely determined by the intra-chain interactions and only to a smaller extent by the interactions between the individual chains and their surroundings. Our ﬁrst example, CaNiN, clearly demonstrated this aspect. Theoretical studies could be used to decompose the interactions into those due to charge transfers between the NiN chains and the Ca counterions and those from interactions between the NiN chains. Although these interactions clearly were non-negligible it was also clear that the electronic properties were primarily due to the individual (and charged) NiN chains. Similar results were found for (SN)x with, however, one very important difference. For the isolated chain a single band would cross the Fermi level and, accordingly, making the structure unstable with respect to a structural relaxation that would open up a gap at the Fermi level. The interactions between the chains, although weak, were sufﬁcient to suppress this distortion so that, instead, the material was superconducting at very low temperatures. Thus, exactly this highly interesting property of the material is critically dictated by the three dimensionality of the system. For the metal containing MX2 chains (that often are charged) we found also that effects due to three dimensionality were important. Thus, although relatively unstructured, the Coulomb potential from the charges was clearly dictated by the complete three-dimensional structure, whereas the electronic properties were determined by the intra-chain interactions. The metal trichalcogenides and metal tetrachalcogenides comprised an interesting class of materials with the existence of CDW as the most signiﬁcant property that could be ascribed to the quasi-one-dimensionality of these systems. The occurrence of CDW could be identiﬁed in a number of experimental studies and was responsible for many unusual properties. Although for most of the systems considered, we have focused on electronic and structural properties and largely ignored magnetic or spin-dependent properties, there exists an important class of materials, most notably transition-metal oxides, for which the spin properties clearly possess quasi-one-dimensionality. We have brieﬂy discussed those materials. Even though these materials are truly three-dimensional ones, we emphasize that when focusing exactly on spin properties only some of the atoms contribute to those and these atoms are arranged in a chain-like or ladder-like arrangement; in the latter case with different number of legs. Therefore, a theoretical description of the spin properties could be provided by looking at quasi-one-dimensional systems. We have also brieﬂy discussed two elements, Ba and Rb, for which experimental studies at high pressures (of the order of 20 GPa) have shown the existence of surprising crystal structures with chains embedded into a host where, moreover, the chain structures were incommensurable with the host structures.

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Most organic substances are non-metallic. On the other hand, the richness of synthetic organic chemistry suggests that systems that unite carbon-based parts and metallic parts may be stable and may allow for the synthesis of materials with speciﬁc properties, including some with chain-like structures, that could be metallic. Therefore, we have discussed brieﬂy two examples of quasi-one-dimensional materials where organic groups were used in stabilizing them. It turned out, however, that the metallic properties were only marginally recovered and, ultimately quasi-one-dimensional materials solely based on carbon atoms have been those that have allowed for the most successful development in the construction of quasione-dimensional, polymeric, metallic systems. We shall discuss these systems in a later chapter (Chapter 12).

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Chapter 11

Mixed-Valence MX Chain Compounds and Related Systems

The MX chain compounds that we shall discuss in the present chapter are closely related to the systems we have discussed in the previous chapter. Nevertheless, we shall devote a special section to these materials for one reason: the materials became the centre of interest at the end of the 1980s and the beginning of the 1990s as a class of materials that had many properties similar to those of the conjugated polymers (synthetic metals) that we shall discuss further later in this presentation (Chapter 12). Thus, the MX chain compounds form, for our purpose, a bridge between inorganic crystalline materials and organic polymeric materials. And therefore they are discussed in a separate chapter.

11.1.

The MX chain compounds

The MX chain compounds are crystalline materials consisting of linear chains with alternating M and X atoms. M is a metal like Ni, Pt, or Pd, whereas X is a halogen like Cl, Br, or I. Most often, the metal atoms have further (groups of) atoms attached to them, leading to a coordination larger than two. One example of such a system is shown in Figure 11.1 for which M equals Pt and X equals Br [1]. It is seen that each Pt atom is surrounded not only by the two Br nearest neighbours along the chain but also by two further Br atoms and two NH3 groups, giving a total coordination for the Pt atoms of six. Another example, ½NiðchxnÞ2 BrBr2 with ðchxnÞ ¼ 1R; 2R – cyclohexanediamine, is shown in Figure 11.2 [2]. In this case, M equals Ni and X equals Br, whereas additional Br atoms are placed between the chains, ultimately leading to a situation where the chains are charged and surrounded by counterions. A similar situation is encountered for ½PtðenÞ2 ½PtðenÞ2 X2 ðClO4 Þ4 with (en) being ethylenediamine. Here the chains are also charged. AuX2 (dibenzylﬁde) with X being Cl or Br also belongs to these materials [3,4]. If the X atoms are placed symmetrically between the neighbouring M atoms (cf. Figure 11.3), one arrives at a structure like M3þ X M3þ X M3þ X M3þ X ,

(11.1)

whereas when the X atoms are shifted alternatively in one or the other direction along the chain, another structure is obtained, Mð3qÞþ X Mð3þqÞþ X Mð3qÞþ X Mð3þqÞþ X . (11.2) 191

192

Chapter 11. Mixed-Valence MX Chain Compounds and Related Systems

Figure 11.1. Structure of a single chain of ½PtðNH3 Þ2 Br2 ½PtðNH3 Þ2 Br4 : Reproduced with permission of the American Physical Society from [1].

The different materials differ in the size of the displacement of the halogen atoms and, consequently, in the value of q: For M ¼ Ni; the displacement and q are small, whereas they are larger for M ¼ Pt and Pd. For later purpose we notice that the two structures Mð3qÞþ X Mð3þqÞþ X Mð3qÞþ X Mð3þqÞþ X , X Mð3þqÞþ X Mð3qÞþ X Mð3þqÞþ X Mð3qÞþ ð11:3Þ are energetically degenerate. Therefore, as we shall see, domain walls (solitons) on individual chains between the two structures may exist. In two papers, Gammel et al. [5,6] presented a thorough theoretical study of the ground-state and excitation properties of these materials. We shall here brieﬂy outline their approach and results. The starting point was to assume that each atom, M or X, along the chain contributes with only one orbital to the bands closest to the Fermi energy. We shall brieﬂy discuss this assumption below. Assuming that the chain is lying along the

193

11.1. The MX chain compounds

N1

Br2

Br1

(a) Br1 C10

C1 C2 N2

C9

N1 Ni

C3

N2

C4

N3 CS

C8

C7 C6 Br2 (b)

Figure 11.2. (a) The structure of ½NiðchxnÞ2 BrBr2 : (b) The building block of the MX chain compound. Reproduced with permission of the American Physical Society from [2].

Figure 11.3. Structure of the backbone of an M–X chain (upper part) without or (lower part) with an alternation along the backbone. Black and white circles represent M and X atoms, respectively.

194

Chapter 11. Mixed-Valence MX Chain Compounds and Related Systems

z axis, this orbital is the valence d z2 orbital for the M atom and the valence pz orbital for the halogen atom. Having two orbitals per MX unit, the resulting model is correspondingly a two-band model. Moreover, they allowed the atoms to be displaced along the z axis away from the position where all M–X interatomic distances have the same length. The displacement for the lth atom is denoted yl with l being even (odd) for the metal (halogen) atoms. Thus, for the two structures of equation (11.3) we have y2n ¼ 0 y2nþ1 ¼ ð1Þn u0

ð11:4Þ

where u0 is a constant, and where the two structures of equation (11.3) have the same value of ju0 j but the opposite sign of u0 : Gammel et al. [5,6] noticed that a critical parameter was the difference in the onsite energies for the metal and the halogen orbitals, 2e0 ; compared with the average hopping integral between the neighbouring metal and halogen orbitals, t0 ; and depending on the value of e0 =2t0 the ground state has different structures. Thereby the difference between the Ni-based and the Pt- or Pd-based chain compounds could be rationalized. Gammel et al. [5,6] also included many-body terms like those of the Hubbard model that we have discussed in Section 3.6 as well as additional terms that should assure that the proper ground-state structure was obtained. These details are, however, not relevant to the present discussion. Through proper choice of the values of the parameters, Gammel et al. [5,6] could identify different types of ground states. For some of those the lowest total energy was found for a system containing a charge-density wave as that of equation (11.2), whereas for others the metal atoms but not the halogen atoms would be displaced. Still others were characterized by a spin-density wave, i.e., whereas the former two solutions can be considered Peierls transitions, the latter is a spin-Peierls transition. Finally, yet other parameter values led to other structures with an even lower symmetry. Whereas the ﬁrst solutions correspond to physical realizations for speciﬁc MX chain compounds, this is not really the case for the latter. More interesting is to study the response of the system to extra charge or to excitation. This was also studied by Gammel et al. [5,6]. The fact that the two structures of equation (11.3) are energetically degenerate leads to the possibility of the formation of solitons or polarons. A soliton can in this case be considered a domain wall between two segments of the (approximately) inﬁnite chain across which the structure changes from one of the two forms of equation (11.3) to the other form. The domain wall may be more or less wide. On the other hand, a polaron is a spatially conﬁned region where the structure is perturbed, e.g., from that of the one form of equation (11.3) towards the other form and back again. Thus, the existence of polarons does not require the existence of two energetically degenerate structures, whereas that of solitons does. Often all these structural defects lead to extra states in the gaps. Studying those, Gammel et al. [5,6] found the results of Figures 11.4, 11.5, and 11.6. Figure 11.4 shows that the occurrence of solitons or polarons indeed leads to extra gap states that ultimately can be used experimentally in identifying these

11.1. The MX chain compounds

195

Figure 11.4. The energy levels for different MX chains with M being Pt and X being (a) Cl, (b) Br, or (c) I. U marks the neutral, undistorted chain, P polarons, B bipolarons, and K solitons. Moreover, the upper indices indicate whether the chain is neutral (0), having one extra electron ðÞ; or having one less electron (+). The energy zero is placed at the top of the occupied bands for the neutral, undistorted chain. Reproduced with permission of the American Physical Society from [5].

196

Chapter 11. Mixed-Valence MX Chain Compounds and Related Systems

Figure 11.5. Excess charge (solid curves) and spin (dashed curves) density distribution of some of the structural distortions of Figure 11.4 for MX chains with M being Pt and X being Cl, except for (c) where X equals I. Reproduced with permission of the American Physical Society from [5].

structural distortions. This has been done, e.g., by Sakai et al. [7] and by Okamoto et al. [8] on a charged PtCl compound, thereby obtaining support for their existence. In Figure 11.5 we show the calculated extra charge and spin density due to some of the defects of Figure 11.4. It is seen that the extra charge and spin is localized to the region of the lattice distortion. Parameters that quantify the charge-density and spin-density waves are shown in Figure 11.6. For the neutral, undistorted chain,

11.1. The MX chain compounds

197

Figure 11.6. Parameters quantifying the occurrence of charge-density (solid curves) or spin-density waves (dashed and dot-dashed curves; here the dashed curves are for the X atoms, the dot-dashed curves for the M atoms) as functions of site index for the same systems as in Figure 11.5. The parameters are the order parameters that are constant for the regular, periodic structure, but non-constant for structures containing local distortions. The width of the distortions can be extracted from the ﬁgures. Reproduced with permission of the American Physical Society from [5].

there is a vanishing spin-density wave, whereas a charge-density wave is uniform throughout the whole chain. Introducing the local distortions as in Figure 11.5 leads to either local variations [Figure 11.6(a) and (b)] or shifts [Figure 11.6(c)] in the charge-and/or spin-density wave, as one would expect. We ﬁnally address the question as to whether the model used by Gammel et al. [5,6] is realistic. Alouani et al. [1] have performed parameter-free density-functional

198

Chapter 11. Mixed-Valence MX Chain Compounds and Related Systems

Figure 11.7. The band structures for the system of Figure 11.1. The dots mark results from densityfunctional calculations, whereas the full curves are results from a tight-binding ﬁt. Finally, the Fermi energy is set equal to 0. Reproduced with permission of the American Physical Society from [1].

Figure 11.8. Variation in the total energy for the system of Figure 11.1 as a function of a parameter that describes the position of the Br atoms relative to the positions in the middle between the neighbouring Pt atoms both without (dashed curve) and with (full curve) the inclusion of the NH3 groups. Reproduced with permission of the American Physical Society from [1].

calculations on ½PtðNH3 Þ2 Br2 ½PtðNH3 Þ2 Br4 : For this system they found that indeed the band structures closest to the Fermi level resemble those of the model, i.e., the bands are formed essentially by Pt d z2 and Br pz functions; cf. Figure 11.7. A surprising result was that the inclusion of the (closed-shell) NH3 (ammonia) molecules was important. Ignoring them, extra orbitals centred on the Pt atoms showed up around the Fermi level and, more importantly, the proper charge-density wave could not be obtained, cf. Figure 11.8.

11.2. The MMX chain compounds

11.2.

199

The MMX chain compounds

As an extension of the systems of the previous section, Kimura et al. [9] synthesized a chain compound, ðNH3 Þ4 ½Pt2 XðpopÞ4 with (pop) being P2 O5 H2 2 and X being Cl, Br, or I. The resulting structure is shown in Figure 11.9. The fact that here two metal atoms are placed between the halogen atoms give rise to more variations in the possible valence pattern of the metal atoms, like Pt2þ Pt2þ X Pt3þ Pt3þ X Pt2þ Pt3þ X Pt2þ Pt3þ X Pt2:5þ Pt2:5þ X Pt2:5þ Pt2:5þ X

ð11:5Þ

Using NMR spectroscopy, Kimura et al. [9] could identify signals related to the occurrence of Pt3þ and Pt2þ ; thus ruling out the third pattern above. A further

Figure 11.9. Structure of the MMX chain compound (NH3 Þ4 ½Pt2 XðpopÞ4 ] with (pop) being P2 O5 H2 2 and X being Cl. The positions of the Cl atoms are disordered. Reproduced from [9].

200

Chapter 11. Mixed-Valence MX Chain Compounds and Related Systems

analysis gave that the ﬁrst pattern is the one that is observed in accordance with x-ray diffraction studies [10]. Nevertheless, the richness in the possible valence patterns is most likely matched by a similar richness in the possible excitations. However, to the best of our knowledge this has not been studied in further detail.

11.3.

Magnus’ green salt

As the last example of a crystalline compound with a dominating quasi-one-dimensionality we shall, once more, discuss a material based on Pt atoms, this time the so-called Magnus’ green salt, PtðNH3 Þ4 PtCl4 : Its structure, cf. Figure 11.10, consists of chains with alternating PtðNH3 Þ4 and PtCl4 units. The Pt atoms form the backbone of the linear chain, and as for the other systems of this section one may formally ascribe the Pt atoms a valence that alternate along the chain. Magnus’ green salt has been known for almost 200 years (see, e.g., Ref. [11]) and over the years a number of related compounds have been produced upon substituting Pt with Pd and/or some of the sidegroups through other ones. Also mixed compounds like PtðNH3 Þ4 PdCl4 and PdðNH3 Þ4 PtCl4 have been produced [12]. Lately the interest in these materials has grown, partly due to the prospects of varying the (semiconducting) properties of those materials in a controlled way, and partly due to their solubility properties [13–15].

NH3 H3N

Pt

H3N Cl

Pt

Cl H3N

Pt

H3N Cl

Pt

Cl H3N

Pt

H3N Cl Cl

NH3 Cl

2-

Cl NH3

2+

NH3 Cl

2-

Cl NH3

2+

NH3 Cl

Pt

2+

2-

Cl

Figure 11.10. Structure of Magnus’ green salt, PtðNH3 Þ4 PtCl4 : Reproduced with permission of the Royal Society of Chemistry from [13].

11.4. Conclusions

11.4.

201

Conclusions

Pt and the other elements of its group in the periodic table (i.e., Pd and Ni) have been the central elements in this section. The materials were crystalline and highly anisotropic so that the quasi-one-dimensionality could be clearly identiﬁed in their properties. The interesting properties were related to being able to modulate the charge or spin distribution along the backbone of the chains, i.e., of the metal atoms. This option was made possible by including further atoms or groups of atoms in the structure. A structural modulation like the small displacements of the halogen atoms in the MX and the MMX chain compounds could be used in obtaining a strong modulation in the electronic properties, i.e., the electrons respond strongly on the relatively small perturbations of the structure. This means, on the other hand, that manipulating the modulation may lead to strong responses. These responses are, e.g., the occurrence of solitons or polarons. Here, we add that the generation of a soliton requires a much larger structural perturbation than that of a polaron: in the former case, half-part of an essentially inﬁnite chain has to be perturbed, whereas in the latter case the perturbation is conﬁned to a ﬁnite region. Independent of whether solitons or polarons are generated, extra electronic orbitals localized to the region of the structural distortion and with energies in the gap between occupied and unoccupied orbitals are generated. When the distortion is not very localized, it can move up and down the chain with only small energy costs, ultimately being able to carry extra charge or spin from electrons occupying the defect-induced gap states. Accordingly, these materials can conduct. This type of charge-transport processes is not unique to the present type of materials. Actually, it is closely related to the Grotthus mechanism [16] which since the ﬁrst part of the 19th century has been used in explaining the high electrical conductivity in hydrogen-bonded systems with H2 O being the most prominent example (it may be added here that the conductivity requires the addition of charge carriers, e.g., ions). For this presentation it is more important to notice that exactly the same mechanism have been used in explaining the conduction properties of the conjugated polymers (also called synthetic metals) that we shall treat in detail in the next chapter. The interest in the MX chain compounds during the late 1980s and early 1990s evolved actually from the research activity in the conjugated polymers and many of the concepts from the latter were modiﬁed and adapted to the MX chain compounds. The MX chain compounds (and later the MMX chain compounds) have provided an interesting additional class of materials which, compared with the conjugated polymers, had the advantages of being crystalline and ordered. Moreover, they offer additional possibilities of ﬁne tuning the materials properties. As we have discussed, through variation of the metals, the halogens, and side groups a variety of properties can be obtained. Finally, it is interesting to notice that the last example of the present section, i.e., Magnus’ green salt, has become of interest as another material with materials properties like those of the conjugated polymers.

202

Chapter 11. Mixed-Valence MX Chain Compounds and Related Systems

References [1] M.E. Alouani, R.C. Albers, J.M. Wills, and M. Springborg, Phys. Rev. Lett. 69, 3104 (1992). [2] H. Okamoto, Y. Shimada, Y. Oka, A. Chainani, T. Takahashi, H. Kitagawa, T. Mitani, K. Toriumi, K. Inoue, T. Manabe, and M. Yamashita, Phys. Rev. B 54, 8438 (1996). [3] H. Tanino, K. Syassen, and K. Takahashi, Phys. Rev. B 39, 3125 (1989). [4] H. Tanino, M. Holtz, M. Hanﬂand, K. Syassen, and K. Takahashi, Phys. Rev. B 39, 9992 (1989). [5] J.T. Gammel, A. Saxena, I. Batistic´, A.R. Bishop, and S.R. Phillpot, Phys. Rev. B 45, 6408 (1992). [6] S.M. Weber-Milbrodt, J.T. Gammel, A.R. Bishop, and E.Y. Loh, Jr., Phys. Rev. B 45, 6435 (1992). [7] M. Sakai, N. Kuroda, and Y. Nishina, Phys. Rev. B 40, 3066 (1989). [8] H. Okamoto, T. Mitani, K. Toriumi, and M. Yamashita, Phys. Rev. Lett. 69, 2248 (1992). [9] N. Kimura, H. Ohki, R. Ikeda, and M. Yamashita, Chem. Phys. Lett. 220, 40 (1994). [10] S. Jin, T. Ito, K. Toriumi, and M. Yamashita, Acta Crystallogr. C 45, 1415 (1989). [11] M. Atoji, J.W. Richardson, and R.E. Rundle, J. Am. Chem. Soc. 79, 3017 (1957). [12] M.E. Cradwick, D. Hall, and R.K. Phillips, Acta Crystallogr. B 27, 480 (1971). [13] J. Bremi, W. Caseri, and P. Smith, J. Mat. Chem. 11, 2593 (2001). [14] M. Fontana, H. Chanzy, W.R. Caseri, P. Smith, A.P.H.J. Schenning, E.W. Meijer, and F. Gro¨hn, Chem. Mat. 14, 1730 (2002). [15] W.R. Caseri, H.D. Chanzy, K. Feldman, M. Fontana, P. Smith, T.A. Tervoort, J.G.P. Goossens, E.W. Meijer, A.P.H.J. Schenning, I.P. Dolbnya, M.G. Debije, M.P. de Haas, J.M. Warman, A.M. van de Craats, R.H. Friend, H. Sirringhaus, and N. Stutzmann, Adv. Mat. 15, 125 (2003). [16] C.J.T. Grotthus, Ann. Chim. Phys. 58, 54 (1806).

Chapter 12

Synthetic Metals: Conjugated Polymers

Organic materials are most often insulating and only rarely discussed in the connection with metallic behaviour. Nevertheless, one class of organic materials has turned out to possess conductivities that match those of the best crystalline metals like copper. This class of material, the conjugated polymers, has for that reason been coined ‘synthetic metals’. The presently large research activity in the conjugated polymers was initiated by three works [1–3] where it was reported that the electrical conductivity of polyacetylene, ðCHÞx ; could be increased by many orders of magnitude when being doped by other materials like AsF5 or iodine. The three persons, Heeger, MacDiarmid, and Shirakawa, who were considered the driving forces behind this study were ultimately honoured for this work with the Nobel Prize in Chemistry for the year 2000 [4–6] (see also Refs. [8,7]). In the meantime it has become possible to vary the electrical conductivity of polyacetylene and related conjugated polymers over very many orders of magnitude (see Figure 12.1) simply through variation of the polymer and/or dopant. It is seen that the conductivity ranges from that of insulators, via that of semiconductors, to that of metals. Although the initial interest in those materials was related to the metal-like conductivity, the current research focuses more on exploiting the semiconducting properties of the conjugated polymers. Nevertheless, we shall include these

Conjugated polymers

Semiconductors

Insulators s/m

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Metals 102

104

106

108

C op pe Si Iro r lv n er

lic on G er m an iu m

Si

G la ss

D ia m on d

Q ua rtz

Conductivity

Figure 12.1. The conductivity of conjugated polymers compared to those of others, well-known materials. Reproduced with permission of the Royal Swedish Academy of Sciences from [7]. 203

204

Chapter 12. Synthetic Metals: Conjugated Polymers

materials in our presentation of metallic chains, since many of the concepts we have discussed earlier are closely related to those of the conjugated polymers.

12.1.

The prototype: polyacetylene

We shall start out with studying the material that has become the prototype of the conjugated polymers/synthetic metals, i.e., polyacetylene. Figure 12.2 shows the experimental results of Chiang et al. [2] on the increase in conductivity upon doping this material, i.e., one of the work that was honoured with the Nobel Prize in Chemistry in 2000. In order to be able to understand the experimental ﬁnding we have to ﬁrst discuss the fundamental structural and electronic properties of polyacetylene ðCHÞx :

Figure 12.2. The conductivity of polyacetylene doped with arsene pentaﬂuoride, i.e., ½ðCHÞðAsF5 Þy x ; as a function of y: The inset shows the structure of two isomers of polyacetylene. Reproduced with permission of the American Physical Society from [2].

12.1. The prototype: polyacetylene

205

Figure 12.3. The structure of polyacetylene. (a), (b) The trans forms, and (c)–(e) the cis forms. Moreover, the C–C bond lengths alternate in (b), (d), (e) and not in (a), (c). Black and white circles represent carbon and hydrogen atoms, respectively. Reproduced from [9].

Polyacetylene may exist in two high-symmetry forms, trans and cis polyacetylene, both shown in Figure 12.3. The backbone is formed by threefold coordinated carbon atoms with one hydrogen atom bonded to each carbon atom, too. Four of the ﬁve valence electrons per CH unit participate in strong s bonds between the nearest neighbours. These s bonds are formed by sp2 hybrids on the carbon atoms and the 1s functions on the hydrogen atoms. The last valence electron per CH unit occupies

206

Chapter 12. Synthetic Metals: Conjugated Polymers

Figure 12.4. Band structures for the ﬁve structures of Figure 12.3. The dashed lines mark the Fermi level. All calculations were performed with two CH units per repeated unit, so that a translational symmetry was used for the trans isomers and a zigzag symmetry for the cis isomers. Reproduced from [9].

a p function on the carbon atom that is perpendicular to the plane of the nuclei. This orbital participates in weaker p bonds between the carbon atoms. Figure 12.4(a) shows the band structures of the trans polyacetylene isomer of Figure 12.3(a) that has no bond-length alternation, i.e., of the undimerized isomer. This structure possesses a zigzag symmetry with one CH unit per repeated unit and, therefore, the band structures in Figure 12.4(a) possess band degeneracies at the zone edge, k ¼ 1: The two bands that meet at k ¼ 1 at the Fermi energy are those formed by the p orbitals, whereas the energetically deeper bands are due to the s orbitals.

12.1. The prototype: polyacetylene

207

As we have discussed in Section 2.1, the total energy of this structure can be lowered by lowering the symmetry, which in this case corresponds to letting the C–C bond lengths alternate, whereby the structure of Figure 12.3(b) is obtained. As Figure 12.4(b) shows, the symmetry lowering is accompanied by the opening up of a gap at the Fermi level. When considering the cis isomer of polyacetylene, the structure without C–C bond-length alternation, Figure 12.3(c), has the smallest repeated unit containing two CH units and, therefore, already this structure has a gap at the Fermi level, cf. Figure 12.4(c). Allowing the C–C bond lengths to alternate, two different structures can be created, cf. Figure 12.3(d) and (e). The bond-length alternation does inﬂuence the band gap at the Fermi level, as shown in Figure 12.4(d) and (e), and from those it is also clear that the two isomers are not equivalent. Despite this difference to the trans isomer, for the cis isomer, it is also known that the structure of the lowest total energy is one with a C–C bond-length alternation, i.e., the cis–trans isomer of Figure 12.3(d), whereas the trans–cis isomer of Figure 12.3(e) may not even exist (see, e.g., Ref. [10]). In Section 2.1 we introduced the model due to Su et al. [11–13] that originally was developed for trans polyacetylene in order to explain the experimental results of Figure 12.2. Su et al. [11–13] realized that the p orbitals are the important ones when describing low-energy excitations or the addition or removal of electrons from the material. Thus, as a simpliﬁcation, one may consider the material as being (quasi-)one-dimensional and having only one electron per (CH) site. Then, the occurrence of the bond-length alternation is a simple consequence of a Peierls distortion [14]. In order to quantify this, Su et al. [11–13] wrote the total energy of a single transpolyacetylene chain according to equation (2.5), i.e., as a sum of one term from the p electrons and a remainder, E tot ¼ E p þ E s .

(12.1)

The p-electron contribution is determined from a single-particle Hamiltonian, X H^ p ¼ ½n c^yn c^n þ tn;nþ1 ð^cynþ1 c^n þ c^yn c^nþ1 Þ, (12.2) n

c^yn

where and c^n are the creation and annihilation operator for the p function of the nth site, and n and tn;nþ1 the on-site energy and hopping integral, respectively. When passing from the regular structure of Figure 12.5(a) without a C–C bond-length alternation to one of the two equivalent structures of Figure 12.5(b) and (c) with a C–C bond-length alternation, the nth CH unit is displaced a little, approximately parallel to the chain axis. Denoting this displacement un ; we have un ¼ ð1Þn u0 .

(12.3)

For the more complex structures of Figure 12.5, which we shall discuss further below, un has a more complicated dependence on n: Su et al. assumed that the hopping integrals depend linearly on un ; tn;nþ1 ¼ t0 aðun1 un Þ.

(12.4)

208

Chapter 12. Synthetic Metals: Conjugated Polymers

Figure 12.5. Different structures of trans polyacetylene. (a) The regular structure without a C–C bondlength alternation. (b) and (c) The two regular structures with a bond-length alternation. (d) Represents a structure with a sharp soliton and (e) one with a polaron. Reproduced from [9].

Moreover, the on-site energies are independent of n; n ¼ 0 .

(12.5)

Finally, E s ; the part of the total energy that is not contained in H^ p ; is written to lowest order in the fun g as a harmonic function, KX Es ¼ ðunþ1 un Þ2 . (12.6) 2 n Using parameter-free density-functional calculations these assumptions can be studied in more detail [15]. This leads to results like those of Figure 12.6. Here, the total energy is obtained from parameter-free density-functional calculations for different bond-length alternations, whereas the p term is calculated from the band structures, and the s term is the remainder. First of all, it is seen that the total energy is lowered upon the bond-length alternation, which is the most important ﬁnding. Second, it is also seen that the separation of the total energy into a p term that is a decreasing function of increasing bond-length alternation and a s term that favours the undimerized structure, is supported by the calculations.

12.1. The prototype: polyacetylene

209

Figure 12.6. Variation in the total energy when passing from the regular structure of Figure 12.5(b), via that of Figure 12.5(a), to that of Figure 12.5(c). The total energy E tot has been split into an electronic part, E p ; and a remainder, E s : Reproduced from [9].

For comparison we show in Figure 12.7 the variation in the total energy for the cis isomers as a function of a bond-length-alternation parameters. For trans polyacetylene, the total energy is a symmetric function of this parameter, simply reﬂecting the fact that the two structures of Figure 12.5(b) and (c) are equivalent. But a similar symmetry does not exist for the cis isomers, and, as Figure 12.7 shows, actually only one isomer, the cis–trans isomer of Figure 12.3(d), is (meta-)stable. Ultimately, Su et al. [11–13] were interested in explaining the large conductivity of doped polyacetylene. They have studied a single chain with one extra charge and suggested that when introducing a soliton, n n 0 un ¼ ð1Þn u0 tanh , (12.7) L extra states would occur in the gap around the Fermi level. Although it did cost energy to create the soliton, it was energetically favourable to populate or depopulate the gap states, so that when being charged or excited, the soliton-containing chain was the energetically most favourable structure. In equation (12.7), n0 is the centre of the soliton, and L represents the width of the soliton. Su et al. [11–13] predicted that L ’ 7; which is considerably larger than the value L ’ 0 that we have assumed in our pictorial description of the soliton in Figure 12.5(d). Since tanhðxÞ ! 1 for x ! 1; a soliton is a domain wall between the two structures of Figure 12.5(b) and (c). And since the two structures of Figure 12.5(b) and (c) are energetically degenerate, the soliton can move along the chain essentially without energy costs. For most other conjugated polymers, the two structures that differ only in the bond-length alternation are not energetically degenerate, as we have seen for cis

210

Chapter 12. Synthetic Metals: Conjugated Polymers

0.08 0.06

Total energy per unit cell (eV)

0.06

0.04

4 3

0.02

0.04

2

0.00

1 −0.02

0.02 −0.04

0.00

−0.06 −0.03 0.00

0.03

0.06

0.09

O

−0.02

S −0.04

−0.03

0.00 0.03 0.06 Dimerization d (angstrom)

0.09

Figure 12.7. As Figure 12.6, but for the cis isomers. Only the total energy is shown. Reproduced with permission of the American Physical Society from [10].

polyacetylene. Therefore, the arguments leading to the conclusion that solitons can exist do no longer hold and, instead, it has been proposed [16] that polarons can be formed. Figure 12.5(e) shows a polaron for trans polyacetylene, and it is seen that it can be considered a spatially conﬁned structural distortion where the structure changes locally from that of the lowest total energy. It may be modelled through un ¼ ð1Þn u0 tanh

n n n n 1 2 tanh , L L

(12.8)

i.e., as a pair of two solitons that are so close that the gap states that are localized to the region of the soliton are interacting. In equation (12.8) n1 and n2 are the positions of the two solitons. Alternatively, one may consider the polaron as being a local distortion in the otherwise regular structure. An important prediction of the proposal of Su et al. [11–13] is that a soliton in trans polyacetylene leads to a state exactly at the mid-gap position. This prediction can be tested experimentally, either on the doped material or on the excited material. Figure 12.8 shows optical spectra from doped polyacetylene for various degrees of doping [17], and it is readily seen that when increasing the dopant concentration an extra feature in the gap grows simultaneously. Figure 12.9 shows the optical spectra for photo-excited trans polyacetylene [18], and here also the existence of gap states is noticed.

12.1. The prototype: polyacetylene

211

Figure 12.8. Doping-induced optical absorption for trans polyacetylene as a function of dopant concentration. Reproduced with permission of the American Physical Society from [17].

Figure 12.9. Photo-induced optical absorption for trans polyacetylene. Reproduced with permission of the American Physical Society from [18].

212

Chapter 12. Synthetic Metals: Conjugated Polymers

Before discussing these results in more detail, we add for the sake of completeness that the existence of defects like those of Figure 12.5(d) was proposed many years ago by Pople and Walmsley [19]. Moreover, Nechtschein [20] reported experimental results on the spin properties of such chains that nowadays are interpreted as showing the existence of solitons. However, in those early works there was no attempt to relate these defects to charge-transport processes. Obviously, the proposal of Su et al., that solitons exist, is a constructive and healthy prediction (see, e.g., Ref. [13]), but there are quantitative problems, i.e., the gap states in the spectra of Figures 12.8 and 12.9 do not appear at the exact mid-gap positions. In order to explain the deviations, one may invoke a number of extra features that were excluded in the original model of Su et al. These include electronic interactions beyond the nearest neighbours as well as interactions between different chains, but it turns out that their effects are much too small to explain the experimental deviations. Instead, it has been proposed that effects beyond those of a single-particle (Hu¨ckel-like) model are responsible, with the Hubbard model and its extensions that we have discussed in Section 3.6 offering a simple framework for the inclusion of these many-body effects. Su et al. [11–13] have suggested that the average hopping integral t0 has the value 2.5 eV, and it has been noticed (see, e.g., Ref. [21]) that the Hubbard on-site energy U of equation (3.43) should be of the order of U ’ 3:5 t0 ; i.e., very far from negligible, in order to explain the optical properties of trans polyacetylene. Electronic-structure calculations can be used in studying the electronic properties of the conjugated polymers. In Figure 12.10 we show experimental UPS spectra for trans polyacetylene [22] together with the calculated density of states. When comparing with the band structures of Figure 12.4(b) we recognize a sharp feature some 3–5 eV below the Fermi level, which is due to the fairly ﬂat s bands in that region. Moreover, the energetically lowest features in the calculated density of states are also recovered in the experimental results. Finally, the features that are due to the p bands can also be recognized. Having established the performance of the density-functional calculations (which has been the matter of some debate; see, e.g., Ref. [23]), we can use the theoretical calculations in studying the response of the system to modiﬁcations in the electron distribution. With the concept of constrained density-functional calculations [24], one can calculate the total energy as a function of the number of electrons at the different sites. The results can be sought mapped on those of model calculations with some parameters whose values are optimized so that the density-functional results are reproduced accurately. Both for the trans and two cis isomers of Figure 12.3(b) and (d) this was done [25] and related to the generalized p electron model, " Hp ¼

X

n c^yn;s c^n;s

n;s

"

þ

X

# tn;m ð^cyn;s c^m;s þ c^ym;s c^n;s Þ

nom;s

X n

U n r^ n;" r^ n;# þ

X n

# V n;nþ1 r^ n r^ nþ1 ,

ð12:9Þ

213

12.1. The prototype: polyacetylene

125 eV Intensity (arb.units)

90 eV 65 eV 50 eV 40.8 eV 27 eV

−25

−20

−15 −10 Energy (eV)

−5

0

Figure 12.10. Calculated density of states (lower curve) together with experimental UPS spectra recorded at different photon energies for trans polyacetylene. The vertical dashed line marks the top of the occupied orbitals in the calculations. Reproduced from [9].

with r^ n;s ¼ c^yn;s c^n;s

(12.10)

(s being " or #) and r^ n ¼ r^ n;" þ r^ n;# .

(12.11)

Compared with equation (12.2), the Hamiltonian of equation (12.9) includes hopping integrals beyond those of nearest neighbours and many-body interactions are also included. Setting most of the parameter values equal to 0 except for n ¼ 0 , tn;m ¼ t0;mn tmn ; U n ¼ U 0 U, V n;nþ1 ¼ V 0;1 V ,

m ¼ 2; 1; 1; 2 ð12:12Þ

where t1 can take two values since the C–C bond lengths alternate, the values of Table 12.1 were obtained [25]. In the table, the nearest-neighbour hopping integrals t1 have been written in the form of equation (12.4).

214

Chapter 12. Synthetic Metals: Conjugated Polymers

Table 12.1. Parameter values for the model Hamiltonian of equation (12.9) for trans, cis–trans, and trans–cis polyacetylene as obtained with constrained density-functional calculations. Parameter t1 t2 U V t0

(eV) (eV) (eV) (eV) (eV) ( a (eV/A)

trans

cis–trans

trans–cis

3.35, 2.88 0.25 10.7 0.3 3.12 2.98

3.44, 2.59 0.16 10.5 0.0 3.02 5.39

3.14, 2.86 0.18 9.9 1.0 3.00 2.64

For trans polyacetylene the values are in good agreement with those that have been estimated from experimental information, and for the cis isomers we see that the different geometrical structure indeed does lead to modiﬁcations in the values. In total, it is clear that many-body effects are non-negligible. The model of Su et al. [11–13] gives a good qualitative description of the properties of the materials at hand, but for a quantitative description, one needs to include the many-body effects. So far we have discussed the properties of a single polyacetylene chain ðCHÞx : However, the large, measured conductivity is a macroscopic conductivity, i.e., we have to take the complete morphology of the experimental samples into account when trying to explain this. It has turned out that it is not easy to understand the experimental results completely. Most samples of polyacetylene have been produced using the synthesis techniques put forward by Shirakawa and Naarmann (see, e.g., Ref. [26]). The resulting ﬁlms consist of randomly oriented ﬁbrils with a ( (see, e.g., Figure 12.11) [27]. Inside each ﬁbril the ﬁnite diameter of roughly 200 A ðCHÞn chains are roughly parallel and n is of the order of 102 : Thus, one critical issue is how charge is transferred from one chain to another inside a single ﬁbril and, subsequently, how it is transferred between the ﬁbrils. Theoretical studies on coupled, parallel chains [28–31] have shown that there is a certain probability that a soliton can be transferred from one chain to another, making charge transfer within a single ﬁbril possible through such a mechanism. Moreover, it has also been shown theoretically [32–34] that when the doping concentration is sufﬁciently high, the solitons start interacting with each other, and the structure of the chain changes and the material becomes metallic. It is, however, not yet fully clear at which dopant concentration that occurs. Finally, experimental results have shown that not only the chains themselves but also the dopants, that are placed between the chains, are active in the charge-transport processes [35–38].

12.2.

Other carbon-based conjugated polymers

Polyacetylene is an organic material, based solely on carbon and hydrogen. Therefore, one may exploit all the tools and possibilities of organic synthesis in trying to produce other, related materials, and to ﬁne-tune the materials properties for applications: we shall discuss brieﬂy these possibilities in Section 12.5.

12.2. Other carbon-based conjugated polymers

215

Figure 12.11. Electron micrographs of polyacetylene. The upper panel shows the as-grown morphology, and the lower one shows the morphology upon modest stretching. Reproduced with permission from [27].

Before discussing other conjugated polymers it is important to identify those properties of polyacetylene that make this material interesting for applications. First, it has to have a backbone consisting at least mainly, if not exclusively, of sp2 bonded or sp-bonded carbon atoms so that p orbitals are found closest to the Fermi level. Second, the p electrons are relatively loosely bound to the backbone and can fairly easily be polarized, for instance through structural defects like solitons or polarons. Third, a charge-transport process via the structural defects can take place.

216

Chapter 12. Synthetic Metals: Conjugated Polymers

Figure 12.12. Structure of a linear chain of carbon atoms (top) without or (bottom) with a C–C bondlength alternation.

Fourth, the structure of the lowest total energy for the neutral system is one in which the C–C bond lengths are non-uniform. Even simpler than polyacetylene is a linear chain of carbon atoms, polyyne, which is shown in Figure 12.12 and which we have discussed earlier in Section 7.3. Electronic-structure calculations [39,40] show indeed that the lowest total energy is for the structure with a C–C bond-length alternation, that p bands are those closest to the Fermi level (see Figure 7.12), and that various structural defects exist for the charged chain. Thus, in principle this material does fulﬁll the criteria above. However, it turns out that it is very difﬁcult to synthesize this material. Long oligomers tend to form a zigzag-like structure and, ultimately, to interact with each other so that a kind of graphene-like structure results [41–43]. Actually, electronic-structure calculations [44,45] have shown that the linear carbon chains Cn only for the smallest values of n (up to, say, n 20), correspond to the most stable structure, whereas for larger values of n other structures involving, e.g., sp2 -bonded carbon atoms are stabler. Thus, the only way to make long linear carbon chains stable may be to insert them inside some channels, for instance inside carbon nanotubes [46]. The polydiacetylenes (see Figure 12.13) constitute another, interesting class of materials. Upon electromagnetic radiation the monomers in crystalline diacetylene can polymerize whereby the crystalline structure largely remains unaltered. Thus, in contrast to polyacetylene, the polydiacetylenes, (C4 R0 R00 Þx ; are crystalline materials. Moreover, by choosing large, bulky sidegroups R0 and R00 the individual chains can be kept so far from each other that they do not interact, and a pure quasi-onedimensional material is the result (see, e.g., Refs. [47–49]). It turns out, however, that optical excitations in polydiacetylenes result in excitons and not in structural distortions like polarons (see, e.g., Refs. [48,50,51]). Therefore, this class of material is clearly different from materials like polyacetylene. Staying with materials that are based solely on carbon and hydrogen atoms, the ﬁrst material that both exists and truly belongs to the class of conjugated polymers/ synthetic metals (which is the subject of the present section), is poly-para-phenylene (PPP, Figure 12.14) consisting of linked C6 H4 (phenyl) rings. For the PPP polymer of Figure 12.14, p orbitals are indeed found closest to the Fermi level, and, moreover, upon excitation or charge addition polarons are formed. Therefore, this (existing) material does indeed belong to the class of synthetic metals. One problem may be that steric effects between neighbouring rings make these non-planar, so that a strict separation into p and s orbitals is no longer possible, although the

12.2. Other carbon-based conjugated polymers

217

Figure 12.13. Structure of polydiacetylene. The upper panel shows the structure of the monomers in the crystal, and the lower panels show the two different possible structures of the polymer. Here, the black circles represent carbon atoms, and the white ones some more or less bulky sidegroups.

Figure 12.14. Structure of (top) poly-para-phenylene, (middle) poly-para-phenylene–vinylene, and (bottom) polyacene. The black circles represent carbon atoms, and the white ones some more or less bulky sidegroups or just hydrogen atoms.

218

Chapter 12. Synthetic Metals: Conjugated Polymers

deviation from planarity is so small that it still is a good approximation to separate the orbitals into s and p [52]. On the other hand, the more the ring-torsion angles deviate from 0 ; the smaller is the overlap between the p functions of neighbouring rings, and the conduction will be reduced. On the other hand, these angles constitute another degree of freedom that may be of relevance when adding charge to the chain: in that case local structural defects where these angles deviate from that of the surrounding chain may exist and transport charge just like the polarons [53]. Moreover, one may inﬂuence the p electrons by substituting the hydrogen atoms with larger sidegroups that enhance the steric repulsions between the rings and, accordingly, increase the ring-torsion angles [54]. Poly-para-phenylene–vinylene, PPV, of Figure 12.14 is another, intensively studied conjugated polymer/synthetic metal. It may be considered an intermediate between poly-para-phenylene of Figure 12.14 and polyacetylene, and it shares many of the properties with those two. Despite its large relevance for the ﬁeld of synthetic metals, it will not be discussed further in this chapter. As the last pure-carbon polymer of this section we mention polyacene, also shown in Figure 12.14. It is seen to be related to poly-para-phenylene but instead of containing linked rings, it contains fused rings. Moreover, it does resemble a ﬁnite stripe of a graphene sheet with the dangling bonds being saturated by hydrogen atoms. It has the advantage that it has a very rigid backbone, assuring that the s=p separation remains valid and that the p electrons are delocalized over the complete system. We shall, however, discuss a related polymer below and, therefore, not consider it further here.

12.3.

Incorporating heteroatoms

By inserting non-carbon atoms into the backbone of the polymer new possibilities for controlled variation of the materials properties arise. There are then an overwhelmingly large class of polymers which will be discussed brieﬂy through some selected examples. A simple possibility is to substitute a CH group by a single N atom, whereby the number of valence electrons is unaltered. As one example we consider the trans isomer of polycarbonitrile ðCHNÞx that can be considered related to trans polyacetylene, but having every second CH unit replaced by a N atom. Polycarbonitrile was ﬁrst synthesized by Wo¨hrle [55], even before the present intense research activity in the conjugated polymers/synthetic metals started. For the structure without a bond-length alternation, the symmetry is lower than that of the equivalent structure for trans polyacetylene, i.e., the zigzag symmetry of the latter is lacking and ‘only’ the translational symmetry remains. Therefore, a Peierls distortion that lowers the symmetry and, simultaneously, opens up a gap at the Fermi level may not exist. Actually, as seen in Figure 12.15(a) even the chain without a bond-length alternation has a gap at the Fermi energy [56]. Nevertheless, upon letting the C–N bond lengths alternate the total energy can be lowered, cf. Figure 12.16, and, moreover, the band gap at the Fermi level is increased, as shown in Figure 12.15 [56]. A difference to trans polyacetylene is that the orbitals in

12.3. Incorporating heteroatoms

219

Figure 12.15. The band structures for polycarbonitrile ðCHNÞx ; (a) without and (b) with a bond-length alternation. Reproduced from [9].

Figure 12.16. As Figure 12.6, but for polycarbonitrile, ðCHNÞx : Reproduced from [9].

general appear at lower energies, including those around the Fermi level, and that a ﬂat, occupied s band (due to nitrogen lone-pair orbitals) shows up just below the Fermi level. But this polymer fulﬁlls the criteria mentioned in the beginning of the preceding section although it has received only little attention in this respect. By substituting every second H atom of trans polyacetylene by a cyano (CN) group, one arrives at the polymers of Figure 12.17. Once again, the lack of the

220

Chapter 12. Synthetic Metals: Conjugated Polymers

Figure 12.17. Different structures of polyacetylene with every second H atom replaced by a cyano group. Black circles, white circles, and black squares represent carbon, hydrogen, and nitrogen atoms, respectively. Reproduced from [9].

Figure 12.18. Variation in the total energy for ðCHCCNÞx as a function of the bond-length alternation for different structures of Figure 12.17. The structures I and III of the ﬁgure corresponds to vanishing displacement. Reproduced from [9].

higher (zigzag) symmetry suggests that already the structure without a C–C bondlength alternation has a non-vanishing gap at the Fermi energy, but, as for polycarbonitrile, one cannot exclude the existence of such a bond-length alternation. An interesting aspect is that the polymer also may take a fused-ring form like polyacene of Figure 12.14 with every second CH group substituted by a N atom. And in the case that a bond-length alternation exists, there are even two different possible structures, i.e. structures IV and V of Figure 12.17. Electronic-structure calculations have shown [57] that the polymer prefers a structure with a bond-length alternation, cf. Figure 12.18, and also that a structure consisting of fused rings has a lower total energy than the polyacetylene-like structure, cf. Figure 12.19. In that case, structure IV is the one of the lowest total energy, but also structure V corresponds to that of a local total-energy minimum. Finally, as Figure 12.20 shows, all structures have p bands closest to the Fermi energy, and as for polycarbonitrile, the bands are in general at deeper energies than those of polyacetylene.

12.3. Incorporating heteroatoms

221

Figure 12.19. Variation in the total energy for ðCHCCNÞx as a function of a parameter describing the transition between structure I (N displacement ¼ 0) and structure III (N displacement ¼ 1) of Figure 12.17. Reproduced from [9].

A particularly interesting conjugated polymer with nitrogen atoms in the backbone is polyaniline plus its modiﬁcations, shown in Figure 12.21. As seen in the ﬁgure, all materials consist of phenylene rings that are linked via N bridges. However, there exist different modiﬁcations, depending on the number of H atoms bonded to the N atoms, cf. Figure 12.21, and it is possible to switch continuously between the different types, i.e., leucoemeraldine with y ¼ 1; emeraldine with y ¼ 0:5; and pernigraniline with y ¼ 0 (see, e.g., Ref. [5]). Moreover, due to steric effects, the phenylene rings are not lying in the same plane and, therefore, the existence of ring-torsion defects has been proposed [58,59]. These are localized regions where the torsional angle between neighbouring rings deviates from an otherwise constant value. Exactly this system was the one which was studied by Phillips and coworkers [60,61] and whose work was brieﬂy mentioned in Section 9.1. These authors showed that randomly distributed ring-torsion defects would not necessarily lead to a localization of all electronic orbitals, in contrast to the case for other types of disorder in quasi-one-dimensional systems (see Section 9.1). There are many other ways of arriving at new conjugated polymers, either by modifying the sidegroups or inserting (hetero-)cycles into the backbone. We shall discuss here just two other modiﬁcations, of which the ﬁrst is chosen because the sidegroups are more electronegative than what usually is used and, therefore, one may arrive at materials with strongly modiﬁed properties. The second example is chosen because it represents a class of important materials in the context of conjugated polymers. Gould et al. [62] synthesized polyacetylene analogues where all hydrogen atoms were substituted by halogen atoms, thereby obtaining ðCFÞx ; ðCClÞx ; and ðCFCClÞx : Whether these systems still can be considered ‘normal’ synthetic metals was studied with density-functional calculations [63]. First, the structures of the trans isomers

222

Chapter 12. Synthetic Metals: Conjugated Polymers

Figure 12.20. Band structures for the different structures of Figure 12.17 as well as for an intermediate structure between those of structures I and III of Figure 12.17 (labelled I–III). The horizontal dashed lines mark the Fermi level. Reproduced from [9].

without and with a C–C bond-length alternation were determined, and the resulting structural parameters are reproduced in Table 12.2. It is seen that the presence of the halogen atoms leads to an opening up of the C–C–C bond angle, and, in addition for F but not for Cl, the lowest total energy is found for a structure with a bond-length alternation. The reason for this difference can be found in Figure 12.22 where we show the band structures for the different trans isomers without a bondlength alternation. For the F-substituted material additional bands show up, but the bands closest to the Fermi level are still of p symmetry. On the other hand, for the Cl-substituted material, the extra bands appear so high in energy that they also cross the Fermi level. Also for the mixed material, ðCFCClÞx ; the band structures suggest that this system is different from those of the ‘normal’ synthetic metals.

12.3. Incorporating heteroatoms

223

Figure 12.21. Structure of the three forms polyaniline. The uppermost structure is the generalized composition of polyanilines, indicating the reduced and the oxidized repeat units. The second, third, and fourth form have y ¼ 1; 0:5; and 0, respectively, and are often called leucoemeraldine, emeraldine, and pernigraniline, respectively. Finally, carbon atoms are represented by black spheres, hydrogen atoms by white spheres, and nitrogen atoms by black squares.

Table 12.2. The optimized structural parameters (bond lengths in a.u., bond angles in deg.) for the trans form of ðCXÞx .

C–X C–C (single) C–C (double) C–C (non-altern.) C–C–C

X¼H

X¼F

X ¼ Cl

2.064 2.636 2.520 2.576 124

2.652 2.653 2.509 2.581 130

3.281

2.578 133

Note: In the order of their appearance the three values for the C–C bond lengths correspond to the lengths of the long bonds and the short bonds for the structure with a bond-length alternation and to the lengths for the structure without a such alternation, respectively.

Thus only upon F-substitution one arrives at a polymer that may possess the ‘standard’ properties of the synthetic metals. The incorporation of (hetero-)cycles is a very common way of modifying the properties of the materials. Then, structures like those of Figure 12.23 are among the most popular ones. To these belongs polythiophene, ðC4 H2 SÞx : As seen in the ﬁgure, the two structures differing in the bond-length-alternation pattern are not

224

Chapter 12. Synthetic Metals: Conjugated Polymers

Figure 12.22. Band structures for the trans forms of (a) ðCHÞx ; (b) ðCFÞx ; (c) ðCClÞx ; and (d) ðCFCClÞx without C–C bond-length alternations. In all cases, one unit cell contains two C atoms, and the horizontal dashed lines represent the Fermi level. Reproduced from [9].

Figure 12.23. The (a) aromatic and (b) quinoid structure of polythiophene. (c) A polaron in polythiophene. Closed and open circles represent sulphur and carbon atoms, respectively. Reproduced with permission of the Royal Society of Chemistry from [64].

equivalent, which is the case for most conjugated polymers. By replacing the S atoms with O atoms or NH groups, one arrives at polyfuran and polypyrrole, respectively, and for all, the structure of the lowest total energy is the aromatic one of Figure 12.23(a), whereas the quinoid structure of Figure 12.23(b) at most is metastable with a higher total energy. Therefore, not solitons, but polarons [i.e.,

12.3. Incorporating heteroatoms

225

Figure 12.24. Band structures for (a) the aromatic and (b) the quinoid structure of polythiophene. The dashed lines mark the Fermi level, and k ¼ 0 and 1 represent the centre and the edge of the ﬁrst Brillouin zone, respectively. Reproduced with permission of the Royal Society of Chemistry from [64].

local changes in the bond-length alternation, cf. Figure 12.23(c)] are created upon doping or excitation. The band structures for the two regular isomers of polythiophene are shown in Figure 12.24. They show, what often (but not always) is the case for these polymers, i.e., that the (stabler) aromatic isomer has the largest band gap at the Fermi level and that the materials have the characteristic features of the synthetic metals. It can be shown [64–67] that as a consequence of this, a polaron will lead to two states in the gap that most likely are not placed symmetrically in the gap. This last ﬁnding is in contrast to the predictions of the model of Su et al. for polyacetylene [11–13], but is a direct consequence of the presence of the hetero-atoms; i.e., not (necessarily) due to many-body effects. Figure 12.25 shows some further examples of how the polymers can be modiﬁed. By replacing some of the hydrogen atoms in polythiophene with larger alkyl groups, steric effects may make the polymer non-planar, leading to new properties. Alternatively, by increasing the number of cycles in the repeated unit, as for poly(isothianaphthene), it can be shown that the band gap of the aromatic isomer decreases and, ultimately when more cycles are added, one arrives at a situation where the quinoid structure is the one of the lowest total energy and largest gap at the Fermi energy [68]. Finally, more complicated units can also form the backbone of the conjugated polymers/synthetic metals, like in poly(ethylenedioxythiophene).

226

Chapter 12. Synthetic Metals: Conjugated Polymers

Figure 12.25. Structure of various conjugated polymers, i.e., from the top to the bottom: poly(3alkylthiophene) with the alkyl group being hexyl, poly(ethylenedioxythiophene), and poly(isothianaphthene). Black circles, white circles, black squares, black triangles, and stars represent carbon atoms, hydrogen atoms, sulphur atoms, oxygen atoms, and CH2 or CH3 groups, respectively.

12.4.

Incorporating metal atoms

When discussing organic polymers in the context of ‘metallic chains/chains of metals’, we have to mention brieﬂy the efforts to incorporate metal atoms into the backbone of the conjugated polymers. It turns out that the efforts in this direction are very limited.

12.5. Applications

227

Figure 12.26. Structure of metal-containing polyynes. The black squares represent PðC4 H9 Þ3 groups, whereas the black circles, the white circles, and the black stars represent carbon, hydrogen, and metal (Pt or Pd) atoms, respectively.

As the only example we mention the work of Friend et al. [69–71] who studied the Pt- and Pd-containing polymers shown in Figure 12.26. In this case, the metal atoms are a central part of the polymer, and it is not possible for charge to be transported along the chain without passing through the metal atoms. Therefore, an important issue that has to be addressed is, whether the polymers of Figure 12.26 can be interpreted as being ﬁnite, organic molecules, separated from each other by the metal atoms or, alternatively, a long chain where the metal atoms are a part of the chain. Theoretical calculations [72,73] have shown that the second interpretation is the correct one. Thus, the metal atoms contribute to the orbitals, also for those that lie closest to the Fermi level, so that a charge-transfer process through the metal atoms is possible. On the other hand, the presence of the metal atoms makes the existence of solitons or polarons less likely, although not impossible.

12.5.

Applications

The fundamental properties, the processing, and the applications of conjugated polymers has been an active ﬁeld of research since the second half part of the 1970s. Consequently, the ﬁeld has reached some degree of maturity and some technological applications have begun to emerge. Although this presentation is mainly devoted to the fundamental properties of quasi-one-dimensional materials, we shall brieﬂy describe some of the applications of the conducting polymers/synthetic metals. For a more thorough description the reader is referred to, e.g., [74–77]. As the name ‘synthetic metals’ suggests, and as outlined in the beginning of this section, the initial research activity was devoted ﬁrst to increase the doping-induced electrical conductivity of these materials and second to exploit this phenomenon. In parallel with that, a very important research activity has focused on the processing of the conjugated materials and their stability against any kind of degradation, including that due to various types of atmospheres. However, as Figure 12.1 shows, these materials do not only possess metallic regimes (depending on the dopant concentration), but can also be made semiconducting when the dopant concentrations are lower. And currently it seems that it is rather the semiconducting than the metallic properties of these systems that have the best prospects for commercial applications.

228

Chapter 12. Synthetic Metals: Conjugated Polymers A cross section of polymer light-emitting diode Glass

Epoxy AI

Ca (electrode) electron ejector Conducting and transparent polymer (electrode)

Conjugated semi conductive polymers

ITO (Indium Tin Oxide) (conducting and transparent) Glass Light

Figure 12.27. A schematic representation of a polymer-based light-emitting diode. Reproduced with permission of the Royal Swedish Academy of Sciences from [7].

The mechanical properties of the conjugated polymers are as those of normal plastics, i.e., they can be bent without breaking. This is in marked contrast to those of more conventional semiconductors, say silicon, and therefore, for special purposes where mechanically ﬂexible semiconductors are important, the conjugated polymers are believed to be important. It has, however, also been realized that the conjugated polymers never will be able to compete with the silicon-based technology. One of the ﬁrst technologically relevant studies of the conjugated polymers was that of Burroughes et al. [78]. They studied the light-emitting properties of conjugated polymers. A schematic representation of their experimental set-up is shown in Figure 12.27. The idea is fairly simple: through an external applied voltage electrons and holes are injected into the material. These will propagate towards each other until they meet, recombine, and submit the energy released through the recombination as light. In their seminal study, Burroughes et al. [78] used PPV (poly-para-phenylene–vinylene) as the conducting polymer, but in the meanwhile many other polymers have been used. In order to make this work a number of problems has to be solved. These include the condition that the Fermi level of the metals should match the top of the valence bands and the bottom of the conduction bands of the conjugated polymer so that electrons and holes indeed are injected into the polymer material. Therefore, bandstructure engineering are very useful when ‘suitable’ conjugated polymers with ‘good’ values for the energies of the top of the valence bands and the bottom of the conduction bands shall be identiﬁed and, subsequently, synthesized. Equivalently, one also has to identify those metals that can be used for hole and electron injection and here it has turned out that these metals have to have fairly low values of the work function (like Al and Ca). Moreover, through proper choice of the band gap between valence and conduction bands of the conjugated polymers, the energy (colour) of the emitted light can be varied, so that in order to have light-emitting displays with a whole spectrum of colours, one needs to use more different polymers. Here, band-structure engineering plays an important role. Another important aspect is that the electron and hole injection shall be efﬁcient, meaning that the potential barrier between the metal and the polymer has to be low. Ultimately controlling and optimizing this requires identifying metal–polymer

12.5. Applications

229

systems where the bonding between the metal and polymer is as ‘covalent’ as possible. Many studies have therefore been devoted to understanding the metal–polymer interfaces (see, e.g., Refs. [79–82]). Another semiconductor device where conjugated polymers can play a role is a photovoltaic cell. This works in principle as reversing the process of Figure 12.27, i.e., through exposure to electromagnetic radiation, electrons and holes are being generated. These move in opposite directions inside the cell and lead to an electric current that can be used for applications. It is, in principle, a relatively simple step to go from a diode to a transistor. Thus, it may not be surprising that transistors based on the conjugated polymers have also been produced. Moreover, through reversible doping of the conjugated polymers it is possible to use these materials for charge storage, i.e., as batteries. This has been demonstrated, e.g., by Curtis et al. [83] for poly(nonylbithiazole) but many other examples exist. As the last example of applications we mention non-linear optics. Any material that is exposed to an external electromagnetic ﬁeld will respond to this. One may quantify the response through either the dipole moment or the total energy. For the former one obtains X X X mi ¼ mð0Þ aij E j þ bijk E j E k þ gijkl E j E k E l þ , (12.13) i þ j

jk

jkl

where mi is the ith component of the dipole moment, E i that of the electromagnetic ﬁeld, mð0Þ i the static dipole moment, a the polarizability, and b; g; the ﬁrst, second, hyperpolarizabilities. For free electrons, only a will be non-zero, whereas for strongly bonded electrons, all polarizabilities and hyperpolarizabilities will be very small. The intermediate characteristics of the p electrons of the conjugated polymers make both the polarizabilities and hyperpolarizabilities large compared with the values of other materials. Another advantage for technological applications is that they have short response times. The interest in these properties stems from the fact that when the external ﬁelds are dynamic with frequencies o2 ; o3 ; for E j ; E k ; ; the responses also will be frequency dependent, aij ¼ aij ðo1 ; o2 Þ, bijk ¼ bijk ðo1 ; o2 ; o3 Þ, gijkl ¼ gijkl ðo1 ; o2 ; o3 ; o4 Þ,

ð12:14Þ

etc., with o1 being the frequency of the response. For any of those, the sum of all the arguments has to vanish; e.g., for the second hyperpolarizability g; o1 o2 o3 o4 ¼ 0.

(12.15)

Thus, responses with other frequencies than those of the external ﬁelds may be generated. Non-vanishing b and g lead to effects like DC-Kerr effect, second- and third-harmonic generation, etc., which makes it possible to manipulate light with light.

230

Chapter 12. Synthetic Metals: Conjugated Polymers

The very large values of the second hyperpolarizability g have made the research in understanding, varying, and exploiting the conjugated polymers in the context of non-linear optics a very active research ﬁeld both for basic science and for applied science (see, e.g., Refs. [74–77,84–90]).

12.6.

Conclusions

The ﬁeld of the conjugated polymers/synthetic metals has, since the second half part of the 1970s, been a very active research ﬁeld, where the ﬁrst industrial applications have also emerged. This has become possible because these materials unite two properties in a way that is unmatched by any other class of materials, i.e., the mechanical properties of conventional plastics and the electronic properties of semiconductors or metals. Moreover, being based on carbon, all the possibilities of organic chemistry allow for controlled synthesis of a very large variety of related materials that nevertheless may differ in important details like stability, solubility, band-gap (relevant, e.g., for colour), bonding to other materials (e.g., metals in semiconductor devices), Fermi energy (relevant for hole- and electron-injection properties), and processability. The prototype of all these materials is polyacetylene with the trans isomer being the stabler one that has been at the focus of much research activity. Compared to the other, well-known isomer, i.e., cis polyacetylene, trans polyacetylene has one property that is not recovered for almost any other member of this class of materials, i.e., it has two energetically degenerate structures, when being neutral. The two structures differ in the C–C bond-length alternation pattern. Owing to this degeneracy, the existence of solitons, i.e., structural defects involving variations in the C–C bond-length pattern, is possible, whereas for most other conjugated polymers, polarons exist. The polarons are also related to local variations in the C–C bond-length pattern. Electron–phonon couplings lead to the existence of extra gap states, when solitons or polarons are created, and through motion of the solitons or polarons, extra charge (or spin) can be transported through the individual chains. Thus, not only electrons, but also phonons are responsible for the conduction properties. This is in contrast to the situation for many other metallic systems, but is very similar to what we have discussed for the MX chains in the preceding section as well as for the MX3 chains discussed in Section 10.4. We have discussed the properties of polyacetylene in some detail and, subsequently, we have described some few (out of the very many) related materials obtained by introducing hetero-atoms into the backbone (including metal atoms), introducing cycles (also ones that contains hetero-atoms) into the backbone, or replacing (some of) the hydrogen atoms with other atoms or larger sidegroups. In some cases, steric effects would force neighbouring cycles to be non-planar, leading to the extra possibility that polarons involving local variations in the dihedral angle between the cycles could exist and, as for the other type of polarons involving bondlength variations, be mobile and carry charge through the chains. By describing a few of the industrial applications of these materials we could demonstrate why it is relevant to be able to vary the electronic (and other) properties of the conjugated polymers in a controlled way.

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We have also discussed another class of materials, the polydiacetylenes, that at ﬁrst seems to be related to the other conjugated polymers and, in addition, to have the property of being crystalline. However, for those the structural defects do not play any signiﬁcant role, whereas excitons do. Although many (but not all) of the current applications of the conjugated polymers are in the area of semiconductors, their unusual metallic properties are still being exploited. Therefore, they do belong to a discussion of ‘metallic chains/chains of metals’. Moreover, there are still many open questions, also from a basicresearch point of view, including understanding the interface between the metals and the conjugated polymers in devices, and understanding the macroscopic conduction, including the processes between the ﬁnite chains. Finally, we have often mentioned ‘doped polymers’. It should be added that ‘doping’ here involves adding other types of atoms in amounts that correspond to percent, i.e., many orders of magnitude larger than what is usually meant when discussing doped, crystalline semiconductors. Therefore, it may be more correct to describe the doped, conjugated polymers as new materials with different compositions than those of the undoped polymers and where, even, interchain interactions are increased, making the material less quasi-one- and more truly three-dimensional.

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[48] H. Tanaka, M. Inoue, and E. Hanamura, Solid State Commun. 63, 103 (1987). [49] K. Seki, I. Morisada, K. Edamatsu, H. Tanaka, H. Yanagi, T. Yokoyama, and T. Ohta, Phys. Scr. 41, 172 (1990). [50] A.R. Bishop, Solid State Commun. 33, 955 (1980). [51] G. Dellepiano, C. Cuniberti, D. Comoretto, G. Lanzani, G.F. Musso, P. Piaggio, R. Tubino, A. Borghesi, C. Dell’Erba, G. Garbarino, and L. Moramarco, Phys. Rev. B 45, 6802 (1992). [52] B. Champagne, D.H. Mosley, J.G. Fripiat, and J.-M. Andre´, Phys. Rev. B 54, 2381 (1996). [53] C. Quattrocchi, D.A. dos Santos, and J.L. Bre´das, Solid State Commun. 85, 203 (1993). [54] K. Tanaka, S. Yamanaka, K. Ueda, S. Takeda, and T. Yamabe, Synth. Met. 20, 333 (1987). [55] D. Wo¨hrle, Adv. Polym. Sci. 10, 35 (1972). [56] M. Springborg, Ber. Bunsenges. Phys. Chem. 95, 1238 (1991). [57] M. Springborg, Chem. Phys. Lett. 339, 389 (2001). [58] J.M. Ginder, A.J. Epstein, and A.G. MacDiarmid, Solid State Commun. 72, 987 (1989). [59] J.M. Ginder and A.J. Epstein, Phys. Rev. B 41, 10674 (1990). [60] H.-L. Wu and P. Phillips, Phys. Rev. Lett. 66, 1366 (1991). [61] H.-L. Wu, W. Goff, and P. Phillips, Phys. Rev. B 45, 1623 (1992). [62] G.L. Gould, V. Eswera, R.M. Trifu, and D.G. Castner, J. Am. Chem. Soc. 121, 3781 (1999). [63] M. Springborg, J. Am. Chem. Soc. 121, 11211 (1999). [64] A.Md. Asaduzzaman, K. Schmidt-D’Alosio, Y. Dong, and M. Springborg, Phys. Chem. Chem. Phys. 7, 2714 (2005). [65] M. Springborg, Solid State Commun. 89, 665 (1994). [66] M. Springborg, Z. Phys. B 95, 363 (1994). [67] M. Springborg, Int. J. Quant. Chem. 58, 717 (1996). [68] Y.S. Lee and M. Kertesz, Int. J. Quant. Chem. Symp. 21, 163 (1987). [69] O. Lhost, J.M. Toussaint, J.L. Bredas, H.F. Wittmann, K. Fuhrmann, R.H. Friend, M.S. Khan, and J. Lewis, Synth. Met. 55–57, 4525 (1993). [70] A. Ko¨hler, H.F. Wittmann, R.H. Friend, M.S. Khan, and J. Lewis, Synth. Met. 67, 245 (1994). [71] H.F. Wittmann, R.H. Friend, M.S. Khan, and J. Lewis, J. Chem. Phys. 101, 2693 (1994). [72] G. Frapper and M. Kertesz, Inorg. Chem. 32, 732 (1993). [73] M. Springborg and R.C. Albers, Phys. Rev. B 53, 10626 (1996). [74] T.A. Skotheim, R.L. Elsenbaumer, and J.R. Reynolds (eds.), Handbook of Conducting Polymers (Marcel Dekker, New York, 1998). [75] J. Messier, F. Kajzar, P. Prasad, and D. Ulrich (eds.), Nonlinear Optical Effects in Organic Polymers (Kluwer, Dordrecht, 1989). [76] J.L. Bre´das and R.R. Chance (eds.), Conjugated Polymeric Materials: Opportunities in Electronics, and Molecular Electronics (Kluwer, Dordrecht, 1990). [77] J. Messier, F. Kajzar, and P. Prasad (eds.), Organic Molecules for Nonlinear Optics and Photonics (Kluwer, Dordrecht, 1991).

234

Chapter 12. Synthetic Metals: Conjugated Polymers

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Chapter 13

Charge-Transfer Salts

In the preceding section we discussed organic quasi-one-dimensional materials that, upon doping, possessed an electrical conductivity comparable with that of the best, conventional, crystalline materials like copper. The delocalization of the p electrons along the chains was to a large extent responsible for the conductivity as well as for many other interesting properties of those materials, partly through coupling to structural degrees of freedom (i.e., phonons). In the language of Chapter 2, the hopping, or transfer, integrals t between the repeated units were fairly large. Although inter-electronic interactions, quantiﬁed through the Hubbard parameter U; were non-negligible, a single-particle description most often was sufﬁcient for obtaining a qualitatively correct description of the materials properties. However, many-body effects had to be included in order to arrive at a quantitatively correct understanding. As we saw in Chapter 3, many-body effects, when dominating, may lead to markedly different behaviours. The signiﬁcance of many-body effects can be quantiﬁed through the quantity U=t: Therefore, increasing the importance of many-body effects can be obtained by either letting U become larger and/or letting t become smaller. The charge-transfer salts constitute a class of crystalline materials containing two types of (organic) molecules that form chain-like structures and that are essentially ﬂat. The chains often consist (but not always; see later) of only one type of molecules that are stacked on top of each other, so that a small orbital overlap between the neighbouring molecules result. This leads to hopping integrals t of the order of some few tenths of an electron volt, i.e., one order of magnitude smaller than those of the conjugated polymers. The molecules are not neutral, but a charge transfer between the two types of molecules exist. Although the Hubbard U parameter is also small (o1 eV), U=t is sufﬁciently large to make many-body effects very important. The interchain interactions are also small, but not so small that they can be neglected. This means that Peierls transitions that would change metallic chains into semiconducting ones compete with other types of transitions, like the creation of spin-density waves or of superconducting states, but all the transitions may also be suppressed by the interchain interactions. It is the purpose of this section to discuss some of the properties of those materials, through some few selected examples. A much more detailed overview of the ﬁeld can be found, e.g., in the older review of Je´rome and Schulz [1], whereas

235

236

Chapter 13. Charge-Transfer Salts

shorter, more recent discussions can be found in the brief discussions by Bourbonnais and Je´rome [2], Giamarchi [3], and Carlson and Williams [4].

13.1.

General properties

The ﬁrst class of charge-transfer salts that we shall discuss as its most well-known member TTF–TCNQ, i.e., tetrathiofulvalene–tetracyanoquinodimethane. TCNQ is a large planar molecule with a closed electronic shell that, however, easily accepts an extra electron, whereby it becomes a chemically stable open-shell anion. If TCNQ is put into an environment where electrons can be obtained from an electron-donating partner, one electron will be donated to each ðTCNQÞ2 pair, which will then act as a closed-shell ion. In a charge-transfer crystal, the TCNQ molecules will stack on top of each other. Since the highest occupied molecular orbital of the TCNQ molecule is half ﬁlled, a transfer of electrons through the stack is potentially possible. On the other hand, TTF is a planar molecule that easily can donate electrons to other species (like TCNQ). Thus, TTF and TCNQ together will form an ionic salt. TTF may be substituted by other molecules which also can donate electrons. TCNQ as well as some of these organic cations are shown in Figure 13.1. Some of the other cations are closely related to TTF, whereas others are quite different, although they all are largely planar. For example, TSF (tetraselenafulvalene) is obtained from TTF by substituting the S atoms by Se atoms, whereas the hydrogen atoms have been replaced by methyl groups in TMTSF (tetramethyltetraselenafulvalene). N

N O O

TCNQ

O

O

O

O

O

O

O S

O

O

N

S O

O

O N

C

O

O

O S

S

Se

Se

O

O O

Perylene

O

O

O Se

CH3 O

Qn

O

O

O

O

Se

Se O O

O

O

O + O O N

TSF

Se

O O

TTF

O

O Se

Se

Se

Se

TMTSF

H O

NMP

O

O

O

N

O

O

O

O + O O N O O

O

O

CH3

O O

CH2

O

HMTSF

O Se

Se

Figure 13.1. The structure of different organic anions and anions. Reproduced with permission of Taylor & Francis from [1].

237

13.1. General properties

Finally, the ﬁgure includes also some organic cations (Qn and NMP) that are parts of radical anion salts. In the TTF–TCNQ crystal, the planar molecules crystallize in segregated, parallel stacks of either TTF or TCNQ molecules, as shown in Figure 13.2 [5]. This kind of structure is the most common one observed in the charge-transfer salts. The molecules are stacked directly on top of each other, although slightly tilted. Since the individual molecules are essentially planar, one may separate the orbitals into s and p orbitals, of which the former are mainly conﬁned to the plane of the

z

C(3) S(1) C(1)

C(7) S(2)

C(9)

C(8)

C(2) N(1) C(4)

C(6) C(5) N(2)

r

S1(2) S(2) N(2) N1

x Figure 13.2. The crystal structure of TTF–TCNQ. The upper panel shows a top view along the stacks of the molecules, and the lower panel shows a side view of the stacks. Reproduced with permission of The International Union of Crystallography from [5].

238

Chapter 13. Charge-Transfer Salts

molecule, whereas the latter extends in the direction perpendicular to this plane. Thus, for the structure of Figure 13.2 mainly the p orbitals of the neighbouring molecules will overlap and interact. Figure 13.3 shows the crystal structure of HMTSF–TCNQ [6]. The structure resembles that of TTF–TCNQ with some smaller, although important, differences. y

C3 Se 3.1 C 0

x

C8

1

C2

C5

40

3.

C4

N

C7 C6

N

N

Se

Se N

Se N

N

N N

Se

Se Se

Se Se

Se

N N

N N

Se

N Se

Se

z N

N

x Figure 13.3. As Figure 13.2, but for HMTSF–TCNQ. Reproduced with permission of The Royal Society of Chemistry from [6].

239

13.1. General properties

Thus, the molecules of the individual stacks are not completely parallel, but their tilt angles alternate between two values. Moreover, the molecules of the two types of stacks, i.e., of HMTSF and of TCNQ, are not so far apart that they do not interact. ( is shown, which has a value that is In the ﬁgure the shortest Se–N distance of 3:10 A ( indicating that considerably smaller than the sum of the van der Waals radii, 3:50 A; electronic interactions between the molecules of the different stacks will also exist. For ðTTFÞBr0:79 the two stacks are incommensurate, cf. Figure 13.4 [7]. This can be directly related to the non-stoichiometry of the material. Another difference to the two other examples we have discussed is that the molecules of the individual stacks are not tilted relative to each other, and that there are two different stacking orientations. Finally, Figure 13.5 [8] shows the crystal structure of ðTMTSFÞ2 PF6 : This is an example of the second class (besides TTF–TCNQ) of charge-transfer salts that we shall discuss in some detail in the following two sections. Compared to TTF– TCNQ, the two types of ions in ðTMTSFÞ2 PF6 occur in the relative ratio of 2:1 and not 1:1. Moreover, the molecules in TTF–TCNQ, through the tilting, are stacked not directly on top of each other but slightly displaced in the same direction. On the other hand, a related, but different, displacement pattern is observed for ðTMTSFÞ2 PF6 : For this, the displacement alternate between two directions, leading to a zigzag arrangement of the TMTSF molecules. Furthermore, there is an

C

3.57Å 4.54Å

Figure 13.4. As Figure 13.2, but for ðTTFÞBr0:79 : Reproduced with permission from [7].

240

Chapter 13. Charge-Transfer Salts

f 2

0 3879 (1)

1

3233 (7)

2

12

1 3934 (1)

11

A

3959 (2) 11 f

11

11 B

12 366

4067 (2) A

2 1 363

2

3983 (2)

4044 (2) 12 0'

4133 (2) 11

3927 (2) 11

12

1

3874 (2) C

2

Figure 13.5. As Figure 13.2, but for ðTMTSFÞ2 PF6 : Reproduced with permission of The International Union of Crystallography from [8].

alternation in the distance between the molecules of the TMTSF stacks, i.e., the structure shows some kind of dimerization. The stacking of the molecules in the charge-transfer salts leads to an orbital overlap along the stacking direction, which ultimately results in band structures with non-vanishing band widths. Moreover, the transfer of electrons makes the upper-most bands only partially ﬁlled, which means that the Fermi level cuts through one or more bands. The band widths along the stacking direction lie in the range of 0.1–0.5 eV. Moreover, as we have seen, there are also non-negligible interactions between the different chains, leading to band dispersions perpendicular to the stacking directions. These band widths are typically two orders of magnitude smaller than those parallel to the stacking direction. Thus, the band structures are dominated by the interactions along the stacking direction, slightly modulated by the interactions perpendicular to this direction. When neglecting the latter, and assuming that single-particle effects are dominating, we have a situation as that discussed in Chapter 2. Then, for very low temperatures a Peierls distortion should open up a gap at the Fermi level, and the material should become semiconducting.

241

13.2. The TTF-TCNQ family

10−2

TTF_TCNQ (TMTSF)2 X TSF_TCNQ

Resistivity (Ω .cm)

10−3

HMTSF _TCNQ X = PF6

10−4

Qn (TCNO)2 R/100

10−5

X = ClO4

1

10

102

103

Temperature (K) Figure 13.6. The temperature dependence of the resistivity of some representative organic conductors. Reproduced with permission of Taylor & Francis from [1].

In Figure 13.6 we show the temperature dependence of the resistivity of various organic conductors [1]. It is seen that upon reducing the temperature, several of the materials show an abrupt increase in the resistance, which can be ascribed to the above-mentioned Peierls distortion. This occurs at 53, 29, and 24 K for TTF–TCNQ, TSF–TCNQ, and HMTSF–TCNQ, respectively. For ðTMTSFÞ2 PF6 a transition at 12 K to a spin-density wave (i.e., not to a charge-density wave, as in the case of a Peierls distortion) occurs. The properties of the charge-transfer salts seem at ﬁrst hand to be understandable within a single-particle model. However, as we shall discuss below, there are cases where the simplest single-particle models fail and more advanced models have to be applied. Thus, the fact that the chains are interacting makes the systems different from being purely one dimensional. Moreover, the fairly small hopping integrals make even small Hubbard parameters important, so that many-body descriptions in some cases are necessary. These issues shall now be discussed for two types of charge-transfer salts.

13.2.

The TTF–TCNQ family

The more recent research activity in the charge-transfer salts started with the discoveries of the TCNQ and TTF molecules in the years 1960–1970 [9–11]. The structure of the TTF–TCNQ crystal is shown in Figure 13.2, and in the preceding section we discussed its properties in some detail. Through electron transfer from

242

Chapter 13. Charge-Transfer Salts

TTF to TCNQ both molecules become partially charged, and through the overlaps of the p orbitals of the individual molecules in the same stack, a situation with partially ﬁlled bands results. For a real, one-dimensional material at zero temperature, a symmetry lowering (a Peierls distortion) would open up a gap at the Fermi level, changing the material from being a metal to being a semiconductor. However, temperature, interactions between the different chains (stacks), many-body effects, or spin interactions may lead to other effects. By studying the resistance as a function of temperature and pressure [12] some information on the different structures and phase transitions can be obtained. This is shown in Figure 13.7, and by combining with information on transport properties, a phase diagram like that of Figure 13.8 can be constructed [12]. The main features are the following [1]: 1. At ambient pressure there are three-phase transitions at T H ¼ 54 K; T I ¼ 49 K; and T L ¼ 38 K: The ﬁrst one is due to a Peierls transition of the TCNQ chains, whereas the other two are driven by the ordering of the TTF chains. 2. For T H 4T4T L TTF–TCNQ remains a fairly good conductor, since the TTF chains are metallic, and only below T L a Peierls transition to a semiconductor takes place. 3. With the c axis being the stacking direction, the structure shows a 2a 3:4b c superstructure for T H 4T4T I : For T I 4T4T L the periodicity in a varies 1

0.8

d(logR) /dT (1/K)

1 bar 0.6

4 kbar

15 kbar 0.4

8 kbar

13 kbar

0.2

0 30

40

50

60

T (K) Figure 13.7. The derivative of the logarithm of the resistance with respect to temperature as a function of temperature for TTF–TCNQ at different pressures. Reproduced with permission of Plenum Press from [12].

243

13.2. The TTF-TCNQ family

80 70

Metal

Tc (K)

60 50

Commensurability domain

40

2 kf = b*/3

30 20

Semiconductor

10

0

5

10

15 20 P(kbar)

25

30

35

Figure 13.8. Experimentally determined phase diagram for TTF–TCNQ. Reproduced with permission of Plenum Press from [12].

continuously, and for T L 4T the structure possesses a 4a 3:4b c superstructure. Accordingly, interchain interactions are important. 4. When applying pressure, the superstructure periodicity in a and b varies. 5. The phase transitions are extremely sensitive to defects. 6. There are no indications of the occurrence of any magnetic ordering in TTF– TCNQ. TTF–TCNQ has been studied with angle-resolved photo-electron spectroscopy (ARPES) together with parameter-free and model calculations [13]. The parameterfree (density-functional) calculations provide a picture of the bonding within a single-particle picture. Figure 13.9 shows the band structures closest to the Fermi level as obtained with this approach, where, moreover, the origin of the various bands has been emphasized. The ﬁgure conﬁrms the picture we have pursued above, i.e., a partial electron transfer leads to two sets of bands crossing the Fermi level, with one set from the TTF chains and another set from the TCNQ chains. In Figure 13.10 the calculated band structures are compared with the results from the ARPES experiments. For the latter, a grey-scale representation is used which is so constructed that it, in an unbiased way, emphasizes the dispersions as do the band structures (for details, see Ref. [13]). It is clear that there are no more than qualitative similarities between the two sets of data, suggesting that a single-particle picture is not adequate. At this point it will be emphasized, that the density-functional band structures may not resemble the experimental excitation spectra (this is a well-known fact although it often is a good approximation to ignore it – as we have done repeatedly during this presentation). As an alternative, Sing et al. [13]

244

Chapter 13. Charge-Transfer Salts

1

TCNQ bands

TTF bands

energy relative toEF (eV)

0

-1

-2

-3

Γ

Z Γ

Z

Figure 13.9. Calculated band structures for the bands closest to the Fermi level for the hightemperature phase of TTF–TCNQ. The size of the symbols represents the charge of each state residing on either the TCNQ (left panel) or the TTF (right panel) molecules. Reproduced with permission of The American Physical Society from [13].

Γ

Z

energy relative to EF (eV)

0.0

a b

c

-0.5

d

d'

-1.0 0.0

0.5

1.0

momentum along b* (Å-1) Figure 13.10. Grey-scale plot of the ARPES dispersions for the high-temperature phase of TTF–TCNQ together with the results from the density-functional calculations from Figure 13.9. Reproduced with permission of The American Physical Society from [13].

studied a Hubbard model and adjusted the parameters in order to reproduce the experimental ARPES data. This resulted in U ¼ 1:96 eV and t ¼ 0:4 eV; and the spectra shown in Figure 13.11. This ﬁgure shows a signiﬁcantly improved agreement between theory and experiment, indicating that many-body effects (as, e.g.,

245

13.3. The TMTSF2-X and ET2-X families

(a)

0

-kF

3kF

kF

(b)

0.0 −0.2

Energy relative to EF

~O(J)

−0.4 "Charge" −0.6 ~O(t)

−π/2

−0.8 0

π/2

Momentum

π

−0.2 0.0

0.2

Energy relative to EF (eV)

"Spin"

0.4

Momentum along b* (Å-1)

Figure 13.11. As Figure 13.10, but compared with the results obtained using a Hubbard model. Reproduced with permission of The American Physical Society from [13].

quantiﬁed through U=t which here equals 4.9, i.e., it is large) are very important. Moreover, the authors could also identify two different types of excitations, i.e., ‘charge’ and ‘spin’ excitations (cf. Figure 13.11), which resembles the behaviour of a Luttinger liquid. TTF–TCNQ is just one member of a whole group of charge-transfer salts, although it is the most prominent one. One may replace TTF with related molecules, like, e.g., TSF (i.e., substituting S by Se), HMTSF, and HMTTF (related to HMTSF, but with S instead of Se) that are shown in Figure 13.1. Thereby, smaller materials-speciﬁc differences are found. For example, for TSF–TCNQ one observes only a single phase transition (at T ¼ 29 K) at ambient pressure [1]. Moreover, one may expect that the fact that Se is a larger atom than S results in a larger overlap of the p orbitals on the TSF molecules than the overlap for the p orbitals on the TTF molecules. For HMTTF–TCNQ, two phase transitions at T H ¼ 48 K and T L ¼ 43 K are observed which are very similar in nature to the same two transitions for TTF–TCNQ [1]. However, in all cases the phase transitions are related to the occurrence of charge-density waves, i.e., to Peierls transitions, and no indications of magnetic structures are observed. This is different than the materials we shall discuss in the next section.

13.3.

The TMTSF2 -X and ET2 -X families

In 1980, Bechgaard et al. [14] have reported that the family of quasi-one-dimensional organic salts ðTMTSFÞ2 X showed a metal–insulator transition at unusually low temperatures. Here, TMTSF is shown in Figure 13.1, whereas X is an inorganic anion like PF6 ; AsF6 ; SbF6 ; TaF6 ; NO3 ; BF4 ; ReO4 ; or ClO4 : These results are reproduced in Figure 13.12. Shortly afterwards it was discovered [15] that one of

246

Chapter 13. Charge-Transfer Salts

DC Resistivity (Ω .cm)

10−3

10−6

10−5

2x10−6

(TMTSF)2 X X = NO3− X = PF6 X = AsF6− X = BF6− 3

10

30 100 Temperature [K]

300

Figure 13.12. The temperature dependence of the resistivity of various ðTMTSFÞ2 X salts. Reproduced with permission from [14].

these compounds, for X ¼ PF6 ; was superconducting when pressure was applied, and ﬁnally it was shown [16] that for X ¼ ClO4 the material became superconducting without external pressure. Subsequently, superconductivity has been observed in other members of this family of charge-transfer salts, too. In Figure 13.5 we have shown the crystal structure of one of these materials. In contrast to the case for TTF–TCNQ and its analogues, for TMTSF2 -X only one stack of molecules is responsible for the conduction, i.e., the stack of the organic TMTSF molecules. Moreover, it turns out that the structures for the different anions are very similar. As seen in Figure 13.12, the salt for X ¼ PF6 possesses a semiconductor-metal phase transition at temperatures somewhere above 10 K. This transition was originally assumed being due to a Peierls transition, but x-ray measurements showed no (or only very weak) signals of a change in the structure [17,18]. Instead, a spindensity wave (magnetic structure) may be formed, which usually is accompanied by essentially no lattice distortion and which marks a clear distinction to the case of the TTF–TCNQ family.

247

13.3. The TMTSF2-X and ET2-X families

12 Conducting phase

10

1.3

Insulating phase (S D W)

6

Tc /k

Tc / K

8

1.2 1.1

4 1.0 0.9 9

2

10

11

12

Superconducting phase 0

5

10

15

20

P/kbar Figure 13.13. The phase diagram of ðTMTSFÞ2 AsF6 : Reproduced with permission of EDP Sciences from [19].

As a single example of a phase diagram for these materials, we show in Figure 13.13 that of ðTMTSFÞ2 AsF6 [19]. Ultimately, by analysing this and many other phase diagrams of a large number of materials of this family of charge-transfer salts, one has arrived at the generic phase diagram shown in Figure 13.14. In contrast to what we found for the TTF–TCNQ family, we observe non-magnetic spin-ordered, antiferromagnetic, Luttinger liquid, and superconducting phases, i.e., we have a much more rich phase diagram. The discovery of superconductivity in organic, quasi-one-dimensional materials has initiated a very large research activity in related materials that still, at the moment of writing, is going on. Theoretical support for pursuing this had been given in 1964 by Little [20], who had suggested that the attraction between two electrons in a molecular wire with strongly polarizable sidegroups could be much stronger than the interactions between the electrons in the Cooper pairs of the superconducting state of conventional superconductors according to the BCS theory [21]. According to Little [20], the increased attraction would suggest transition temperatures well in excess of room temperature (i.e., of the order of 10 000 K). Although it might be questioned whether such high-transition temperatures can be achieved, it was very realistic to hope for non-negligible transition temperatures when focusing on quasi-one-dimensional materials, like the charge-transfer salts. In the meantime, transition temperatures well above 10 K have been achieved (see, e.g., Ref. [4]) for charge-transfer salts of the ðETÞ2 X family. Here, ET stands for bis(ethylenedithio)tetrathiafulvalene and is shown in Figure 13.15, and, as for the ðTMTSFÞ2 X family, X is an inorganic anion; in this case typical examples

248

Chapter 13. Charge-Transfer Salts

Tx

Tρ

LLρ+σ

Temperature (K)

100 LLσ 10 AF

SP 1

SC 10

20

30

40

Pressure (kbar)

(TMTTF)2PF6 (TMTTF)2Br (TMTSF)2PF6 (TMTSF)2ClO4 Figure 13.14. A generic phase diagram of the ðTMTSFÞ2 X salts. LL marks Luttinger-liquid phases that are either insulating (LLs ) or conducting (LLrþs ). SP marks a non-magnetic spin-ordered state, whereas AF represents an antiferromagnetic state. Finally, SC represents a superconduction state, and FL a Fermi-liquid state. Notice that both sulphur- (TMTTF) and selenium-based (TMTSF) salts are represented. Reproduced with permission of Institute of Physics from [2].

Figure 13.15. The structure of ET, bis(ethylenedithio)tetrathiafulvalene. Black circles, white circles, and black squares represent C, H, and S atoms, respectively.

include ReO4 ; I3 ; IBr2 ; CuðNCSÞ2 ; and Cu½NðCNÞ2 Cl: Moreover, more different crystal structures exist for these salts. Nevertheless, we shall not enter into the discussion of these materials further.

13.4.

The TTF–CA family

In the preceding two sections we have studied charge-transfer salts where donor (D) and acceptor (A) molecules form parallel, separate stacks, i.e., –D–D–D–D–D– D– and –A–A–A–A–A–A– : Other types of charge-transfer salts form mixed stacks of the type –D–A–D–A–D–A– : The prototype of those is TTF– CA, where CA equals chloranil (C6 Cl4 O2 ; cf. Figure 13.16). Thus, the charge transfer occurs within the stacks. Moreover, they are all small-gap semiconductors [22]. Some of these mixed-stack materials possess electronic instabilities that lead to structural transitions when exposed to pressure, temperature, or photo-irradiation. A common feature of all these transitions is the loss of a symmetry inversion below a critical temperature T c or above a critical pressure Pc : Accompanying this symmetry lowering is a sharp variation in the charge transfer between donor and

249

13.4. The TTF-CA family

Figure 13.16. The structure of CA, chloranil. Black circles, white circles, and black squares represent C, Cl, and O atoms, respectively.

3.9 3.3

3.7

CB

CB

3.5

E(eV)

3.1

3.3

2.9

2.5

3.1

VB

VB

2.7

2.9 Z

U Y

Γ

X

W T

S

Z

U Y

Γ

X

W T

S

2.7

Figure 13.17. The bands closest to the Fermi level for TTF–CA in the (left panel) high-temperature and (right panel) low-temperature phase. Reproduced with permission of The American Physical Society from [24].

acceptor molecules. Close to the transition point, surprising phenomena are observed, including colour variation, phase coexistence, thermal hysteresis, disorder, and non-linear excitations [22]. Owing to the sharp increase in charge transfer, the transition has been coined ‘neutral–ionic’ phase transition. Le Cointe et al. [23] performed a detailed experimental study of the structure at three different temperatures, i.e., 40, 90, and 300 K, of TTF–CA. The abovementioned phase transition occurs at T c ’ 81 K; so that the lowest and highest temperatures are well separated from the phase-transition temperatures. Later, Oison et al. [24] used this structural information as input for density-functional calculations. From their study it is found (see Figure 13.17) that the phase transition has some effects on the bands closest to the Fermi level, and that the gap indeed is small. It is also clear that the dispersion perpendicular to the stacking direction is very small. In fact, Oison et al. [24] found that the interchain interactions can be neglected. Despite the relatively small differences in the two sets of bands of Figure 13.17, important charge-transfer changes occur. This is seen in Figure 13.18 which indicates that in the low-temperature structure, the weights of the band orbitals on the acceptor molecules are much larger, suggesting that the acceptor molecules have a larger number of electrons in the low-temperature phase. A more detailed analysis of Oison et al. [24] gave that 0:51 0:08 electrons are transferred in the

250

|CAlk|2

Chapter 13. Charge-Transfer Salts

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

Z

U Y

Γ

X

W T

S

Z

U Y

Γ

X

W T

S

0

Figure 13.18. The weights on the acceptor molecules for TTF–CA in the (left panel) high-temperature and (right panel) low-temperature phase for the occupied valence bands of Figure 13.17. Reproduced with permission of The American Physical Society from [24].

high-temperature phase, whereas 0:66 0:06 electrons are transferred in the lowtemperature phase. Experimental values suggest a somewhat larger differences between the two phases, i.e., 0:2 0:1 and 0:7 0:1; respectively [25]. Although the trends agree, some quantitative differences remain. Whether these are due to manybody effects is unclear, and shall not be discussed further here. Instead we notice an interesting feature of the low-temperature phase. Due to the symmetry-lowering, the donor and acceptor molecules have alternating distances, so that they form (Dþ A ) pairs. This means that two regular structures may exist, ðDþ A ÞðDþ A ÞðDþ A ÞðDþ A ÞðDþ A ÞðDþ A ÞðDþ A Þ , ðA Dþ ÞðA Dþ ÞðA Dþ ÞðA Dþ ÞðA Dþ ÞðA Dþ ÞðA Dþ Þ .

ð13:1Þ

Ultimately, domain walls (solitons) like those we have discussed for the conjugated polymers or the MX chains may exist and carry charge, ðDþ A ÞðDþ A ÞðDþ A ÞD0 ðA Dþ ÞðA Dþ ÞðA Dþ Þ , ðA Dþ ÞðA Dþ ÞðA Dþ ÞA0 ðDþ A ÞðDþ A ÞðDþ A Þ ,

ð13:2Þ

or spin, ðDþ A ÞðDþ A ÞðDþ A ÞDþ ðA Dþ ÞðA Dþ ÞðA Dþ Þ , ðA Dþ ÞðA Dþ ÞðA Dþ ÞA ðDþ A ÞðDþ A ÞðDþ A Þ .

ð13:3Þ

Except for pointing out the similarity to solitons which we have discussed for MX3 chains in Section 10.4, for MX chains in Chapter 11, and for polyacetylene in Section 12.1, these will not be discussed further.

13.5.

Conclusions

The charge-transfer salts that we have discussed in this section is the last example of (metallic) quasi-one-dimensional systems that we shall treat. Similar to some, but not all, of the other systems of our discussion, they are crystalline, but the difference to the other materials is ﬁrst of all related to the scale of the interactions. Thus,

References

251

whereas the covalent interactions within the individual molecules are as those of other covalently bonded materials, the interactions within the chain directions are much smaller, typically at least one order or smaller than those of the covalent interactions. This difference has as a further consequence, that many-body effects, that have numerical values as those of other, stronger interacting materials, become important. Ultimately, this makes the occurrence of superconductivity possible. We have discussed three classes of charge-transfer salts. For the ﬁrst two, the donor and acceptor molecules formed separate stacks, whereas mixed stacks were found for the third class. The ﬁrst two classes differed in the ratio between anions and cations and, in addition, in the fact that for the second class conduction takes place solely through one of the two types of chains, i.e., the chains of organic molecules. However, for such materials the quasi-one-dimensionality may be considered more pronounced than for those of the other two classes. The materials of the TTF–TCNQ family were those whose properties mostly resembled those of the other materials of this presentation. We saw cases of Peierlslike transitions, above which good conduction could be achieved. The materials of the TMTSF2 -X and ET2 -X families perhaps had the most interesting phase diagrams. For those materials, a large number of ‘unusual’ phases could be observed, including cases where spin-density waves occurred as well as superconducting phases. It may therefore be not surprising that since the beginning of the 1980s these materials have been at the focus of the most intense research activity of all the charge-transfer salts, where attempts for increasing the transition temperature for superconductivity and for discovering new unusual properties has been the dominating reason. To this end, very many new materials based on other, planar, organic molecules and/or counterions have been examined. Finally, the materials of the TTF–CA family contained mixed donor–acceptor stacks, and possessed an neutral–ionic phase transition, whose fundamental origin still is not completely understood. It was interesting to compare these materials with those we have discussed earlier and to observe that for this class of materials (but not for the other charge-transfer salts of this presentation) structural defects like solitons (and maybe also polarons) may also exist and transport charge or spin through the system.

References D. Je´rome and H.J. Schulz, Adv. Phys. 31, 299 (1982). C. Bourbonnais and D. Je´rome, Phys. World 11, 41 (1998). T. Giamarchi, Chem. Rev. 104, 5037 (2004). D. Carlson and J. Williams, New Sci. 1847, 26 (1992). T.J. Kistenmacher, T.E. Phillips, and D.O. Cowan, Acta Crystallogr. B 30, 763 (1974). [6] T.E. Phillips, T.J. Kistenmacher, A.N. Bloch, and D.O. Cowan, J. Chem. Soc. Chem. Commun. 9, 334 (1976). [7] S.J. La Placa, P.W.R. Corﬁeld, R. Thomas, and B.A. Scott, Solid State Commun. 17, 635 (1977). [1] [2] [3] [4] [5]

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Chapter 13. Charge-Transfer Salts

[8] N. Thorup, G. Ringdorf, H. Soling, and K. Bechgaard, Acta Crystallogr. B 37, 1236 (1981). [9] D.S. Acker, R.J. Harder, W.R. Hertler, W. Mahler, L.R. Melby, R.E. Benson, and W.E. Mochel, J. Am. Chem. Soc. 82, 6408 (1960). [10] L.R. Melby, Can. J. Chem. 43, 1448 (1965). [11] F. Wudl, G.M. Smith, and E.J. Hufnagel, Chem. Commun. 21, 1453 (1970). [12] D. Je´rome and H.J. Schulz, in Extended Linear Chain Compounds, Vol. 2, ed. J.S. Miller (Plenum Press, New York, 1982). [13] M. Sing, U. Schwingenschlo¨gl, R. Claessen, P. Blaha, J.M.P. Carmelo, L.M. Martelo, P.D. Sacramento, M. Dressel, and C.S. Jacobsen, Phys. Rev. B 68, 125111 (2003). [14] K. Bechgaard, C.S. Jacobsen, K. Mortensen, H.J. Pedersen, and N. Thorup, Solid State Commun. 33, 1119 (1980). [15] D. Je´rome, A. Mazaud, M. Ribault, and K. Bechgaard, J. Phys. Lett. Paris 41, L95 (1980). [16] K. Bechgaard, K. Carneiro, M. Olsen, F.B. Rasmussen, and C.S. Jacobsen, Phys. Rev. Lett. 46, 852 (1981). [17] J.P. Pouget, Chem. Scr. 17, 85 (1981). [18] J.P. Pouget, R. Moret, R. Come`s, K. Bechgaard, J.M. Fabre, and L. Giral, Mol. Cryst. Liq. Cryst. 79, 129 (1982). [19] R. Brusetti, M. Ribault, D. Je´rome, and K. Bechgaard, J. Phys. Paris 43, 801 (1982). [20] W.A. Little, Phys. Rev. 134, A1416 (1964). [21] J. Bardeen, L.N. Cooper, and J.R. Schrieffer, Phys. Rev. 108, 1175 (1957). [22] C. Koenig and C. Hoerner, Ck Newsletter August 10th, 58 (1995). [23] M. Le Cointe, M.H. Leme´e-Cailleau, H. Cailleau, B. Toudic, L. Toupet, G. Heger, F. Moussa, P. Schweiss, K.H. Kraft, and N. Karl, Phys. Rev. B 51, 3374 (1995). [24] V. Oison, C. Katan, P. Rabiller, M. Souhassou, and C. Koenig, Phys. Rev. B 67, 035120 (2003). [25] C.S. Jacobsen and J.B. Torrance, J. Chem. Phys. 78, 112 (1983).

Chapter 14

Concluding Remarks

This book is a presentation of the properties of chains, in particular of metallic chains and of chains of metal atoms. Chains may be considered physical realizations of one-dimensional systems that in turn represent idealizations. Our world is threedimensional and even atomically thin wires are still three-dimensional objects. Moreover, no physical object is ever completely isolated from the rest of the universe. Therefore, a repeated issue throughout our presentation has been the question as to whether the systems we are discussing possess the properties that make the one-dimensional systems different from three-dimensional ones. The chains that are produced in mechanically break-junction experiments are ﬁnite and suspended between three-dimensional tips. During the experiment the junction becomes gradually thinner, but even before the thinnest wires are obtained interesting properties emerge. Thus, when the spatial extensions become so small that the electrons feel the conﬁnement, quantum, or ﬁnite size, effects lead to a discretization of the energy levels of the electrons, which shows up both in stability (e.g., through the occurrence of magic sizes for which the systems are particularly stable) and in the quantization of the conductance. Thus, in this case the interesting properties are not related to true one dimensionality but rather to spatial conﬁnement so that the systems are much more extended in one direction than in the other two. On the other hand, many studies were devoted to the optical properties of metal chains deposited on vicinal semiconductor surfaces. These systems clearly demonstrate one of the central issues within the ﬁeld of quasi-one-dimensional systems: to which extent can the properties be ascribed to the internal properties of the chains and, equivalently, to which extent can the surrounding medium be neglected. We discussed several studies where at ﬁrst it was not clear as to whether the substrate was relevant in understanding/describing the optical spectra. Consequently, the observation of Luttinger-liquid behaviour was reported, but later studies showed that (many, if not all of) the observations could be explained as being due to an interplay between the chains and their surroundings (including other chains). The two issues we have just mentioned, i.e., quantization due to spatial conﬁnement and Luttinger-liquid behaviour, are some of the most important aspects in the ﬁeld of quasi-one-dimensional systems. Throughout our presentation we have repeatedly shown examples of this. However, also other aspects of quasi-one-dimensional systems are interesting and have been the driving force for much research activity. When considering chain materials as being materials for which the interactions in one direction are much

253

254

Chapter 14. Concluding Remarks

stronger than in the other two directions, we arrive at a different deﬁnitions and many materials can then also be considered ‘chain materials’. This includes ﬁrst of all crystalline materials with, e.g., sulphur nitride, the MX chains, and the chargetransfer salts being some few examples that we have treated in our presentation. Here, for sulphur nitride and (some of) the charge-transfer salts the interchain interactions are crucial, although relatively weak. Without those the chains would most likely experience a lattice distortion and become semiconducting, whereas this Peierls-like instability could be suppressed by the interchain interactions and, instead, new phases, including superconducting ones, could be observed. ‘Metallic chains’ may be deﬁned as being quasi-one-dimensional materials that possess metallic conductivity, whereby even organic polymers can be included. Such materials were also one of the topics of this presentation. For these, the conjugated polymers (also called synthetic metals), the inherent properties of the individual polymer chains are the dominating ones and the interchain interactions are only of secondary importance. Therefore, modifying the structure or composition of the backbone of the polymers through intelligent organic synthesis offered a way of obtaining a very broad range of controlled variation of the materials properties. ‘Metallic properties’ are often related to the ability to transport charge. In our presentation one charge-transport process was repeatedly mentioned that was strongly related to the materials being quasi-one-dimensional. For example, for many chains, two energetically degenerate, but structurally different structures exist, and domain walls (solitons) that separate the two structures on one chain can be mobile and carry spin or charge along the chain. Polarons were also often relevant for the charge transport. This type of charge-transport process was seen both for the metal-trichalcogenides, the conjugated polymers, the MX chains, and the TTF– CA-like charge-transfer salts. Alternatively, for ﬁnite chains we studied charge transfer within the Landauer conductance formula. This can also be considered as an example of one-dimensional physics. In closing we emphasize that throughout this presentation we have tried to give various examples of the many different types of materials that (to some extent) can be classiﬁed as being ‘metallic chains’ or ‘chains of metals’ when following one of the deﬁnitions above. We have studied many different types of materials and discussed many different experimental and theoretical studies of those. Nevertheless, our presentation is biased. First, the enormous amount of research that has been devoted to such systems makes it impossible to give anything but just a ﬂavour of the complete ﬁeld. Second, in the choice of the systems we have presented, we have tried to give selected examples from many different types of systems, but, nevertheless, most readers will most likely miss her or his favourite examples. Third, the present authors are working on electronic-structure calculations on the properties of materials, including chain compounds and polymers, and this background of ours has for sure made our presentation biased towards giving more emphasis on this type of research and on the systems we have studied ourselves. Fourth, we emphasize that due to the limited space of this presentation no material has been studied in detail which the large research activity in each of them would justify. Fifth, a more fundamental question is related to the deﬁnition of metallic behaviours. Here, we have

Reference

255

used the ‘standard’ deﬁnition, deﬁning metals as being materials with a particularly high electrical conductivity. Probably, the more exact deﬁnitions, relating metallic behaviour to a Berry-phase description of the electronic orbitals (see, e.g., Ref. [1]) is, according to our conviction, beyond the scope of our presentation. We hope, despite these comments, that our presentation will be of interest to the reader. Finally, we, the authors, are very grateful to all the colleagues and friends for the support they have given us over the years. And we are also grateful to the colleagues and the publishers who have given their permission to use their work in this presentation. In particular at the University of Saarland, Germany, we have beneﬁtted from the support of the German Research Council (DFG) through the Sonderforschungsbereich 277.

Reference [1] R. Resta, J. Phys. Condens. Matter 14, R625 (2002).

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Subject Index

C, 119, 121, 124, 126, 147, 150

(ET)2X, 245, 247 (NH3)4[Pt2X(pop)4], 199 (NbSe4)3I, 171, 172 (TMTSF)2PF6, 240, 241 (TTF)Br0.79, 239 (TaSe4)2I, 171 (VO)2P2O7, 179, 180 [Ni(chxn)2Br]Br2, 191, 193 [Pt(NH3)2Br2][Pt(NH3)2Br4], 192, 198 [Pt(en)2][Pt(en)2X2](ClO4)4, 191 Ag, 37, 79–86, 90–93, 95, 100, 137, 139, 151–153 Al, 40, 82, 84, 97–103, 122–124, 126, 131 alloy, 141–143, 150, 155 angle-resolved photoelectron spectroscopy, 70, 75, 84, 243 antiferromagnetic behaviour, 98, 173, 174, 180, 247, 248 armchair nanogtube, 113–116, 119, 145 aromatic structure, 224, 225 As, 123 Au, 37, 45–47, 51, 65, 66, 69, 79–91, 95, 100, 131, 137, 139, 143, 151 AuMg, 143 AuPt, 143 AuZn, 143 Ba, 150, 181, 183, 186 battery, 229 BaVS3, 180, 183 BCS theory, 167, 247 Bi, 107, 123, 141, 151, 153 bipolaron, 195 bis(ethylenedithio)tetrathiafulvalene, 247, 248 Bohr-Sommerfeld quantization, 133 bond-length alternation, 5–8, 30, 31, 80, 84, 98, 109, 141, 154, 206, 207–209, 216, 219, 220, 220–225, 230 bond-order wave, 10 Born-Oppenheimer approximation, 22, 23 break force, 80, 81 break junction, 45, 49, 51, 52, 65, 76, 80, 81, 84, 87–92, 95, 97, 99, 103, 108, 124, 126, 131, 133, 134, 137, 142, 151, 253

C4H2S, 223 Cx(BN)1 x, 143 CA, 249 Ca2CuO3, 176 CaCu2O3, 178, 179 calix[4]hydroquinone, 152, 153 CaNiN, 159–162, 186 carbon, 59, 105–108, 109, 114–119, 121, 126, 141, 143, 144, 148–150, 154, 155, 187, 216, 218, 230 CC1, 221, 224 Cd, 79, 86–87 CDW, 167, 169, 170 CF, 221 CFCC1, 221, 222 CH, 203, 204, 214, 224 CH3BiI2, 183, 184 charge-density wave, 10, 104, 167, 170, 186, 194, 197, 198, 241, 245 charge-transfer salt, 2, 235–237, 239–241, 246, 247–251, 254 chloranil, 248, 249 CHN, 218, 219 cis polyacetylene, 205, 209, 230 cis-trans polyacetylene, 214 cluster, 37, 38, 55, 108, 124, 131, 133, 134 Co, 79, 84, 91, 147, 164 Co3Fe2S5, 165 complex band structures, 11, 13 conductance, 14–19, 39, 41, 43, 45, 49–51, 53, 58, 59, 61, 62, 65, 76, 80, 82–84, 87–93, 95, 99–103, 109, 110, 118–122, 126, 131–139, 143, 170, 253, 254 conductance histogram, 62, 86, 132, 133, 137, 138 conduction, 2, 14, 18, 19, 40, 41, 48, 62, 65, 76, 84, 90, 93, 95, 109, 115, 117, 121, 124, 132, 133, 135, 139, 185, 201, 218, 230, 231, 246, 251 conduction channel, 14, 16–19, 50, 76, 90, 93, 95, 100, 109, 124, 133 conductivity, 201, 203, 204, 209, 214, 227, 235, 254, 255 conjugated polymers, 167, 191, 201, 203, 204, 209, 212, 214–216, 218, 221, 224, 226, 227–231, 235, 250, 254

257

258

Subject Index

constrained density-functional calculations, 212, 214 correlation effects, 2, 21, 25, 26, 173, 174, 183 Cs, 37, 38, 40, 100, 131, 137, 139, 153, 154 CsFeS2, 165 Cu, 79–85, 90, 91, 95, 100, 131, 137, 139, 147, 151 density-functional theory, 25 disorder, 141, 142, 249 domain wall, 169, 192, 194, 209, 250, 254 electron-gas parameter, 38, 62 emeraldine, 221 even-odd oscillations, 18, 87, 100, 109, 135 exchange-correlation, 25, 37, 39, 41 exciton, 216, 231 extended Hubbard model, 29, 30, 212 fcc, 63, 82, 101–103 Fermi liquid, 31, 34, 248 Fermi wave length, 133 Fermi wave vector, 62 Fermi wavenumber, 167 ferromagnetic behaviour, 98 Fibonacci chain, 142 free electron, 85, 86, 131, 183, 229 frustration, 180 fullerene nanotube, 110 Ga, 103 GaSe, 146, 147 Ge, 119–121 GeCuO3, 173–179 gold, 45–48, 50–56, 58–77, 79, 83, 85, 87, 88, 90, 95, 97 graphene sheet, 110–113, 117, 218 Grotthus mechanism, 201 guest-host systems, 141, 151, 155, 181 Hartree-Fock, 23–27, 37, 80, 165 Hartree-Fock-Roothaan, 25–28 Heisenberg model, 180 helical structure, 54, 55, 57, 90, 94, 95, 111, 124, 125, 150, 155 Hg, 79, 86, 87 HMTSF, 238, 239, 245 HMTSF-TCNQ, 238, 241 HMTTF, 245 HMTTF-TCNQ, 245 Hohenberg, Kohn, 25, 27 holon, 32, 33, 178 Hubbard model, 29, 194, 212, 235, 241, 245 hydrocaoncrinite, 155 hyperpolarizability, 229, 230

I, 148, 172 In, 103–106 incommensurate systems, 181, 183, 186, 187, 239 Ir, 79, 82, 87, 88, 91 Jahn-Teller distortion, 180 jellium model, 37, 38, 40–42, 44, 45, 92, 131, 133, 134 K, 37, 131, 137–139, 140, 146, 149, 153, 154 K2[Pt(CN)4], 184, 185 K2[Pt(CN)]Br0.3, 185 K2PtS2, 165, 166 KFeS2, 165 Kohn, Sham, 25, 26, 28, 37 Landauer formula, 14, 16, 19, 62, 76, 254 LDOS, 66 leucoemeraldine, 221 Li, 80, 131, 139, 148 LiCu2O2, 178 light-emitting diode, 228 local density of states, 115 localization, 2, 124, 142, 221 Luttinger liquid, 31–34, 68, 70, 76, 83, 95, 104, 117, 173, 178, 180, 245, 247, 253 magic numbers, 131 magic radii, 43, 44 magnetic susceptibility, 173, 176 magnetism, 10, 31, 38, 80, 84, 90–92, 95, 97, 98, 160, 173, 176, 186, 243, 245–248 Magnus’ green salt, 200, 201 many-body effects, 2, 19, 21, 28, 30, 31, 34, 68, 142, 173, 180, 194, 212, 214, 225, 235, 241, 242, 244, 250, 251 metal oxide, 146, 173, 180, 186 metal tetrachalcogenide, 171, 186 metal trichalcogenide, 168, 170, 176, 186, 254 Mg2O5, 175 MMX, 199, 201 Mo6Se6, 144, 146 molecular dynamics, 47, 54–56, 84, 99, 101 mordenite, 151, 152, 154 multi-shell wire, 55, 90. multi-wall nanotube, 114, 117 MX, 191, 193–196, 201, 230, 250, 254 MX2, 164, 165, 186 MX3, 167, 230, 250 Na, 37–40, 80, 85, 100, 131–133, 135–137, 139, 153, 154 Na2PdS2, 165 Na2PtS2, 165

Subject Index Na3Fe2S4, 165 nanojunction, 52, 53, 56, 57, 64, 132 nanotube, 57–59, 112–114, 116–121, 126, 141, 143–146, 148, 150, 152, 155, 216 Nb, 79, 92, 94, 167 NbSe3, 169–172 NbTe4, 171, 172 neutral-ionic transition, 249, 251 Ni, 79, 80, 82, 90, 91, 146, 191 NiAl, 143 NiN, 159–162, 186 non-linear optics, 229, 230 non-Ohmic behaviour, 88 odd-even oscillations, 18, 87, 100, 106, 109, 135 Ohmic behaviour, 88 optical absorption, 211 optical excitation, 181, 216 order parameter, 169, 197 paramagnetic behaviour, 98, 180 Pb, 87, 102, 107, 119, 121, 122, 126, 151, 153, 155 Pd, 79, 80, 82, 84, 90, 91, 147, 151, 164, 191 Pd(NH3)4PtCl4, 200 PdCl2, 165 Peierls distortion, 4, 8, 19, 29, 34, 74, 107, 154, 163, 167, 173, 174, 176, 180, 194, 218, 235, 242, 245, 246, 251, 254 pernigraniline, 223 photovoltaic cell, 229 platinum, 87 polarizability, 229 polaron, 194, 195, 201, 208, 210, 215, 216, 218, 224, 227, 230, 251, 254 poly(nonylbithiazole), 229 poly-para-phenylene, 216–218 poly-para-phenylene-vinylene, 217, 218, 228 polyacetylene, 3, 167, 170, 203, 204, 207–215, 218, 220, 225, 230, 250 polyaniline, 221 polycarbonitrile, 218–220 polydiacetylene, 216, 217, 231 polythiophene, 223, 225 polyyne, 216, 227 potassium, 148 Pt, 79, 82, 87, 88, 90, 91, 95, 151, 164, 191, 200 Pt(CN)4, 184, 185 Pt(NH3)4PdCl4, 200 Pt(NH3)4PtCl4, 200 PtS2, 166, 167 quantum of conductance, 17, 50, 51 quasi-periodicity, 141, 142 quinoid structure, 224, 225

259

Rb, 139, 153, 154, 181, 182, 184, 186 Rb2PtS2, 165 RbFeS2, 165 reﬂection, 14, 16, 17 relativistic effects, 74, 79, 87, 105, 122, 126, 152 Rh, 82, 84, 91, 92 ring-torsion defects, 221 Ru, 79, 82, 84, 92 rubidium, 181 S, 124–126 Se, 124–126, 154, 155 semiclassical model, 62, 133 semiclassical orbit, 135 semiclassical quantization, 133 Si, 119, 121, 124 SiCuO3, 178 single-wall nanotube, 114, 115, 117, 148 SN, 162–165, 186 sodium, 37 soliton, 167, 169, 176, 192, 194, 195, 201, 208–210, 212, 214, 215, 239, 227, 230, 250, 251, 254 spectral function, 32 spin chain, 173, 178 spin ladder, 173, 179 spin polarization, 39, 40, 80, 160, 174 spin-density wave, 2, 10, 19, 27, 80, 84, 92, 137, 139, 194, 196, 235, 241, 246, 251 spin-orbit couplings, 46, 47, 74, 79, 81, 105, 122, 166, 167 spin-Peierls transition, 19, 173, 174, 176, 180, 194 spinon, 32, 33, 178 Sr2Cu3O5, 179 Sr2CuO3, 176, 177 SrCu2O3, 180 SrCuO2, 176, 177 SrNbO3.41, 180 structural defect, 115, 194, 215, 216, 218, 230, 231, 251 structural distortion, 10, 11, 19, 92, 99, 155, 196, 201, 210, 216 Su et al., 3, 207, 209, 210, 212, 214, 225 superconductivity, 10, 31, 162, 164, 167, 186, 235, 246–248, 251, 254 supershell, 133 synthetic metals, 2, 142, 167, 191, 201, 204, 254 TaSe3, 170, 171 TaTe4; 170, 171 TCNQ, 236–238, 242 tetracyanoquinodimethane, 236, 241 tetramethyltetraselenafulvalene, 236 tetraselenafulvalene, 236

260

Subject Index

tetrathiofulvalene, 236, 241 Thue-Morse chain, 142 Ti, 79, 94, 95, 147 Tl, 103, 105, 107, 153 TMTSF, 236, 248 TMTSF2-X, 246, 247 TMTTF, 248 trans polyacetylene, 167, 170, 206–214, 218, 230 trans-cis polyacetylene, 214 transfer matrix, 12, 14, 16 transition metal, 95, 97 transmission, 11, 14, 16–18, 50, 82, 93, 94, 135 TSF, 236, 245 TSF-TCNQ, 241 TTF, 242, 244, 245, 254, 255

TTF-CA, 248–251 TTF-TCNQ, 236–239, 241–246, 247 van Hove singularity, 115, 144, 161 weird wire, 101–103, 122 WS2, 144, 145 zeolite, 151, 153, 154 zigzag nanogtube, 113–116, 145 Zn, 79, 86, 87, 143, 147 Zr, 79, 95, 167 ZrSe3, 170, 171