Orbital approach to the electronic structure of solids,

ORB ITAL APPROACH TO THE ELECTRON IC

STRUCTURE OF SOLIDS

This page intentionally left blank

Orbital Approach

to the Electronic

Structure of Solids

ENRIC CANADELL

Institut de Ciencia de Materials de Barcelona (CSIC) `

MARIE-LIESSE DOUBLET

CNRS – University of Montpellier

CHRISTOPHE IUNG

University of Montpellier

1

Great Clarendon Street, Oxford ox

3 2 6DP

Oxford University Press is a department of the University of Oxford.

It furthers the University’s objective of excellence in research, scholarship,

and education by publishing worldwide in

Oxford New York

Auckland Cape Town Dar es Salaam Hong Kong Karachi

Kuala Lumpur Madrid Melbourne Mexico City Nairobi

New Delhi Shanghai Taipei Toronto

With offices in

Argentina Austria Brazil Chile Czech Republic France Greece

Guatemala Hungary Italy Japan Poland Portugal Singapore

South Korea Switzerland Thailand Turkey Ukraine Vietnam

Oxford is a registered trade mark of Oxford University Press

in the UK and in certain other countries

Published in the United States

by Oxford University Press Inc., New York

c E. Canadell, M.-L. Doublet & C. Iung 2012

The moral rights of the authors have been asserted

Database right Oxford University Press (maker)

First published 2012

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,

without the prior permission in writing of Oxford University Press,

or as expressly permitted by law, or under terms agreed with the appropriate

reprographics rights organization. Enquiries concerning reproduction

outside the scope of the above should be sent to the Rights Department,

Oxford University Press, at the address above

You must not circulate this book in any other binding or cover

and you must impose the same condition on any acquirer

British Library Cataloguing in Publication Data

Data available

Library of Congress Cataloging in Publication Data

Data available

Typeset by SPI Publisher Services, Pondicherry, India

Printed and bound by

CPI Group (UK) Ltd, Croydon, CR0 4YY

ISBN 978–0–19–953493–7

10 9 8 7 6 5 4 3 2 1

Preface

Understanding the electronic structure of the materials on which he/she is

working may not be an essential need for an experimental scientist but certainly

can make his/her everyday work easier and more intellectually pleasing. The

electronic structure is the most obvious and useful link between the structure

and properties of any solid. Thus, understanding how the electronic structure

of a given material can be assembled (and thus how it can be altered) from

that of the chemically significant building blocks from which it is made up is a

simple yet very suggestive approach to the main goal of any materials science

researcher: the design and preparation of materials with controlled properties.

Whether the new materials suggested in this way can be actually prepared or

not is something that depends, among other things, on the preparative skills

and art of the scientist. This is why knowledge of the electronic structure

may not be essential. However, it can make the quest much more rational and

straightforward, or it can direct the attention to something which otherwise

could seem bizarre.

The impressive increase in computing power and the development of highly

performing simulation codes for solids in recent years has provided chemists,

physicists, and materials science researchers with very efficient tools to access

the details of the electronic structure of practically any periodic solid. However,

this does not necessarily mean that we can understand the electronic structure

of any solid in a precise yet simple way. Indeed this is what is needed to

truly master the link between the structure and properties of the solids of

interest. The development of efficient computational and conceptual tools is

the only way towards a fruitful interaction between theoretical and experimen￾tal approaches with the intention of developing a sound understanding in this

field. Materials science being an essentially interdisciplinary field, the training

of scientists in the area is very much dependent on the physical or chemical

orientation of their curriculum. Nevertheless, understanding the structure–

properties correlation needs both physical and chemical concepts, which are

usually taught using quite different languages. The reason why the writing

of this book has been undertaken is the observation that, to the best of our

knowledge, none of the materials science books available at present extensively

use a blend of band theory, the appropriate physical approach to the under￾standing of the structure and properties of many solids, and orbital interaction

arguments, which is a transparent and chemically very insightful concept.

We believe that this kind of interdisciplinary approach may be extremely

enlightening.

There is certainly nothing novel in saying that knowledge of electronic

structure is one of the more effective ways of making significant advances

in materials science. J. Goodenough was among the first to systematically use

vi Preface

concepts of electronic structure closely linked to structural details in looking

for trends and predicting what materials could exhibit a certain physical prop￾erty. This work had, and still has, a lasting influence on materials science.

Pioneered by R. Hoffmann, J. K. Burdett, and M.-H. Whangbo in the 1980s,

the introduction to materials science of the ideas of orbital interaction, which

had been so useful in rationalising the structure and reactivity of molecules,

was a major breakthrough. It soon became clear that the step-by-step building

up of many of the tools used within the context of the band theory of solids,

such as band structure, density of states, Fermi surface, etc., based on orbital

interaction ideas, provided an invaluable yet intuitive and easy-to-handle tool

with which to analyse the results of quantitative calculations or to rationalise

experimental observations. Structural and transport properties, the origin of

different phase transitions and structural modulations, the nature of scanning

tunnelling and atomic force microscopy images of complex materials, etc.

were successfully rationalised on the basis of this type of approach. Very

detailed structural information is encoded within orbital-interaction-type argu￾ments so that through this approach it is relatively easy to link the effect

of possible structural modifications into say the band structure or the Fermi

surface, etc. and, consequently, to anticipate how these changes could alter the

stability, conductivity or related properties of a given structure.

With these developments in mind, around 1990 we thought that it would

be timely to introduce these ideas into the curricula of chemistry, physics or

materials science courses at the postgraduate or final-year undergraduate lev￾els. This idea materialised as a course on the orbital approach to the electronic

structure of solids given at Universite de Paris-Sud Orsay, which was quite ´

successful and was repeated for a number of years. It was also introduced at

other French institutions such as the Ecole Normale Superieure de Cachan, ´

Universite de Montpellier, and Universit ´ e de Pau, as well as in several interna- ´

tional events. Based on this experience a book entitled Description orbitalaire

de la structure electroniques des solides ´ by C. Iung and E. Canadell, covering

the general principles and applications of such approach to one-dimensional

solids was published in French by Ediscience International in 1997. Over

the years many colleagues prompted us to complete this work by writing

a new book fully covering the course, but academic and professional duties

continuously delayed this project. The present book is a natural follow-up of

the initial French publication in which we have generalised the content to cover

two- and three-dimensional solids and added some new material.

The book contains 12 chapters, the first two being a sort of prelude. The first

is a very brief overview of the free electron theory of solids with the purpose

of introducing some very basic physical notions, which we will use throughout

the book. In the second chapter we present a short overview of the basic notions

currently used to understand the electronic structure of molecules, emphasising

the symmetry and orbital interaction arguments. One of the purposes of this

chapter is to show that the molecular orbital theory used for molecules and

the band theory used for periodic solids are really simple variations of the

same idea due to the discrete or periodic nature of the systems. The essential

machinery of the band theory of solids and its orbital interaction analysis is

Preface vii

developed in Chapter 3. Most of the formal tools that will be used throughout

the book are explained there using the simplest periodic system we can think

of: the infinite chain of hydrogen atoms. This keeps the formal developments

simple and allows us to treat the same system in different ways so that the

reader may be aware of different ways to approach a given problem. The fourth

chapter is devoted to the ubiquitous Peierls distortions of solids. This is an

important phenomenon exhibited by many solids and has strong consequences

for transport and other properties. Chapters 5, 7, and 8 are essentially different

applications of the ideas developed in the third and fourth chapters to organic

and inorganic one-dimensional solids. Chapter 6 is a brief introduction to

the handling of symmetry when studying the electronic structure of solids.

The use of symmetry in band theory is an elegant yet not always simple

matter, which cannot be developed at length in a book like the present one.

However, we have discussed some useful and quite basic aspects of symmetry

in this chapter. Up to the end of Chapter 8 the work is restricted to one￾dimensional systems. Chapters 9–11 generalise the approach to two- and three￾dimensional solids. In Chapter 9 the basic theoretical notions are generalised

for systems of any dimensionality and some model systems are considered.

The increase in dimensionality and structural complexity soon leads to the

need to consider many orbitals and several directions of the Brillouin zone.

The analysis of the results (or the qualitative building up of the electronic

structure) may become too cumbersome, so that a simpler analytical tool

must be devised. The simpler and more useful tool devised for this purpose

is the density of states (DOS). The object of Chapter 10 is to present several

ways to analyse this useful construct from the viewpoint of orbital interaction

analysis using real examples. Chapter 11 deals with low-dimensional solids

and the analysis of the Fermi surface, an extremely useful concept which,

when appropriately decoded, contains much information about the transport

and structural properties of metallic systems. In this chapter we will show that

the essential aspects of the Fermi surface of a given metal may be obtained in

a relatively simple way using the orbital interaction approach. The procedure

will be illustrated by considering several classes of low-dimensional materials,

which have given rise to considerable debate in the literature. Most of the

present book uses a one-electron view of the electronic structure of solids.

Although this is a perfectly legitimate option for a very wide range of materials

and for the purposes of this book, it must be clearly stated that an explicit

consideration of electronic repulsion is indispensable to understand certain

classes of solids such as systems exhibiting magnetic properties. Discussion

of this problem at a level consistent with the detailed approach of this book

would have markedly increased its length and has not been considered realistic.

However, we have included a final chapter in which the essentials of how the

inclusion of electronic repulsion can modify the conclusions of a one-electron

approach are outlined.

This is essentially a teaching book and consequently we have included a

series of exercises so that readers may check their progress from time to time.

Exercises that do not need to be considered on a first reading are marked with

an asterisk. Answers to the exercises are provided, although sometimes they are

viii Preface

deliberately only sketched. Since this is not a research book we have not made

any attempt to present a detailed list of references. We generally mention some

books or publications that may be helpful for readers interested in expanding

their coverage of the subject. For the real examples discussed in the text we

always make reference to the original publications reporting the structure of the

system. In that way, readers interested in carrying out actual calculations for the

system can prepare their inputs. In general we also provide reference to one or

two papers in which the electronic structure is discussed. Because of the nature

of the book we have always chosen those with a strong pedagogic orientation.

We apologise for not mentioning the many excellent papers available for most

of the systems considered.

This book would have been very different (and certainly less satisfying)

without the input of the many students who attended our lectures. We are

deeply indebted to them; their comments and questions have provided the

impetus for the continuous polishing and revising of many aspects of this

book. In addition we have benefited from the comments of many friends and

colleagues who have read parts of the book, both the French and English

versions. This book also owes much to the many discussions that took place

before the actual writing with T. R. Hughbanks (Texas A & M University),

M.-H. Whangbo (North Carolina State University), and the late J. K. Burdett

(University of Chicago), and to Y. Jean (Palaiseau), and F. Volatron (Orsay) for

pushing us to write the initial French version. We thank A. Garc´ıa for his help

in implementing the tight-binding programs and F. Boyrie for his invaluable

help in the LaTeX compilations. We also thank C. Raynaud and E. Clot for

useful discussions about the methodological part of the book. We are grateful

to Dunod Editions for permission to use material from the French edition in the ´

present work. We warmly thank Sonke Adlung, our editor at Oxford University

Press, and his team (Lynsey Livingston, April Warman, and Clare Charles) for

their continuous support, help and infinite patience with three authors who

were continuously delaying the writing of the book. Last, but not least, we

deeply thank our families for patiently enduring the writing of this book.

Enric Canadell,

Marie-Liesse Doublet,

and Christophe Iung

Bellaterra, Montpellier,

February 2011

Contents

1 Elementary introduction to the transport

properties of solids 1

1.1 Free electron model 1

1.1.1 One-dimensional system 2

1.1.2 Generalisation to a three-dimensional system 9

1.2 Conductivity of real solids 10

1.2.1 Factors influencing the conductivity 10

1.2.2 Band structure of real solids 11

1.2.3 Metallic behaviour 11

1.2.4 Semiconducting and insulating behaviour 12

1.2.5 Number of carriers 13

2 Electronic structure of molecules: use of symmetry 14

2.1 Molecular orbital theory 14

2.1.1 Born–Oppenheimer approximation 15

2.1.2 One-electron approximation 15

2.1.3 LCAO approximation 15

2.1.4 Secular equations and secular determinant 16

2.1.5 Basic features of the H¨uckel and extended

H¨uckel methods 17

2.1.6 Symmetry properties of the molecular

orbitals 18

2.2 A short review of the theory of symmetry point groups 19

2.2.1 Different symmetry point groups 19

2.2.2 Classes 21

2.2.3 Basis for an irreducible representation 22

2.3 Application to the study of the π system of regular

cyclobutadiene 25

2.3.1 Decomposition of the (pz) basis 26

2.3.2 Determination of the basis elements for different

irreducible representations 27

2.3.3 Molecular orbital diagram of the π system of

regular cyclobutadiene 30

2.4 Transition metal complexes 30

2.4.1 Ligands and formal oxidation state 31

2.4.2 The ML6 octahedral complex 33

2.4.3 Distortions of a complex 39

x Contents

3 Electronic structure of one-dimensional systems:

basic notions 44

3.1 Bloch and crystal orbitals 45

3.1.1 Bloch orbitals 46

3.1.2 Crystal orbitals 49

3.2 Electronic structure of the model chain Hn 51

3.2.1 Representation of the CO() and CO(X) functions 51

3.2.2 Energy of the crystal orbitals in the H¨uckel

approach 52

3.2.3 Band structure 54

3.2.4 Basis for an energy level E(±k

) 55

3.2.5 Fermi level of the Hn chain 57

3.3 Electronic structure of the dimerised model chain (H2)n 58

3.3.1 Formal determination of the band structure 58

3.3.2 Qualitative determination of the band structure 61

3.4 Comparison of the regular Hn and dimerised (H2)n chains 63

3.4.1 Comparison of the band structures of the regular Hn

chain generated by either a simple or a double unit cell 63

3.4.2 Dimerisation in the Hn chain: notion of distortion

in a periodic system 67

4 First-order Peierls distortions in periodic 1D systems 72

4.1 Analysis of the model system (H0.5+)n 72

4.1.1 Effect of a tetramerisation on the Fermi level 73

4.1.2 Effect of a tetramerisation on the states near the

Fermi level 74

4.1.3 Effect of a tetramerisation on the band structure 76

4.2 Analysis of first-order Peierls distortion in terms of a charge

density wave 77

4.3 Nesting vector 81

4.4 Commensurate and incommensurate distortions 81

4.4.1 Commensurate distortion 81

4.4.2 Incommensurate distortion 83

4.4.3 Comparison 83

4.5 Conclusions 83

5 Application to trans-polyacetylene 85

5.1 Electronic structure of ethylene 86

5.2 Main aspects of the band structure for trans-polyacetylene 87

5.3 Detailed analysis of the band structure of trans-polyacetylene 88

5.4 Determination of the band structure of trans-polyacetylene

using the fragment formalism 89

5.4.1 Calculation of the band structure by means of the

H¨uckel approach 91

5.4.2 Qualitative determination of the band structure 92

5.5 Band gap opening at the Fermi level in trans-polyacetylene 93

Contents xi

6 Handling the symmetry in 1D compounds 96

6.1 Analysis of the An system 96

6.1.1 Analysis of the cyclic An system 96

6.1.2 Analysis of the linear An system 101

6.1.3 Notion of group of a k point 104

6.2 Application to the determination of the band structure for the An

linear system, where A is an atom 104

6.2.1 Group of the different k points 105

6.2.2 Symmetry of the different Bloch orbitals 105

6.2.3 Bands associated with σ-type overlaps 107

6.2.4 Complete band structure 108

6.3 Band structure of the hypothetical (NaCl)n chain 109

6.3.1 Group of the different k points 110

6.3.2 Bands associated with σ-type overlaps 110

6.3.3 Complete band structure 112

6.4 Consequences of the existence of a glide plane 113

6.4.1 Using point group symmetry properties in

trans-polyacetylene 113

6.4.2 Complete space group (non-symmorphic) of

trans-polyacetylene 115

6.4.3 Crystal orbitals of trans-polyacetylene by means of the

non-symmorphic space group G = Tn ⊗ C2h ⊗ {E, gσ } 117

6.4.4 Concluding remarks 119

6.5 Work plan for the study of a 1D system 120

7 Application to polyacene 122

7.1 Band structure near the Fermi level 123

7.1.1 Unit cell definition 123

7.1.2 Symmetry analysis of the chain 123

7.1.3 Appropriate fragment orbitals 123

7.1.4 Crystal orbitals at the and X points 124

7.1.5 π-type band structure of polyacene 126

7.2 Distortions in polyacene 128

7.2.1 Disappearance of the σx y symmetry plane 128

7.2.2 Disappearance of the σyz symmetry plane 128

7.3 General remarks concerning Peierls distortions 130

7.3.1 First-order Peierls distortions 130

7.3.2 Second-order Peierls distortions 131

8 Electronic structure of selected inorganic chains 133

8.1 KCP 133

8.1.1 Band structure of the eclipsed chain [Pt(CN)4]

(2−δ)− 134

8.1.2 Band structure of KCP (staggered chain) 139

8.1.3 Conclusions 142

8.2 (ML4L

)n chains 143

8.2.1 Symmetry 143

xii Contents

8.2.2 Choice of the fragment orbitals to generate the

Bloch orbitals 143

8.2.3 Analysis of the Bloch orbitals at the and X points 144

8.2.4 Symmetry of the Bloch orbitals 144

8.2.5 Band structure 145

8.2.6 Study of the (ReCl4N)n chain 147

8.2.7 Electronic structure of the (Pt(NH2Et)4Cl2+)n chain 149

8.3 Suggested studies 153

9 Electronic structure of 2D and 3D systems 157

9.1 Basic concepts 157

9.1.1 Direct and reciprocal lattices 157

9.1.2 Bloch and crystal orbitals 159

9.1.3 Brillouin zone 161

9.1.4 Symmetry and the Brillouin zone 162

9.2 Analysis of the electronic structure of 2D model systems 166

9.2.1 The square lattice 2

∞[Hn] system 166

9.2.2 The square lattice 2

∞ [An] system 169

9.2.3 π-type band structure of hexagonal graphene layers 173

10 Density of states 181

10.1 Calculation and analysis of the density of states 181

10.1.1 Density of states 181

10.1.2 Projected density of states 183

10.1.3 Crystal orbital overlap population 185

10.2 Combined use of DOS and COOP: electronic structure of the

MPS3 layered phases 186

10.3 Step-by-step determination of the density of states: the

(Pt(NH3)4Cl)2+ chain 188

10.4 Density of states and fragment molecular orbital interaction

analysis: application to the [(C5H5)M] chains 193

10.5 Transition metal diborides with the AlB2 structure type:

a 3D case study 196

11 Fermi surface and low-dimensional metals 203

11.1 Notion of Fermi surface 204

11.2 Nesting vector and electronic instabilities in low-dimensional

metals 207

11.3 Monoclinic TaS3 versus NbSe3 210

11.3.1 Crystal structure and electron counting 211

11.3.2 Qualitative band structure 212

11.3.3 Qualitative Fermi surface: differences between

NbSe3 and TaS3 214

11.4 Molybdenum bronzes 215

Contents xiii

11.4.1 Octahedral distortions and t2g level splitting in

MoO6 octahedra 216

11.4.2 MoO5 chain with corner-sharing octahedra: counting of 2p

oxygen antibonding contributions 217

11.4.3 A0.33MoO3 (A = K, Rb, Cs, Tl) 2D red bronzes: metallic or

insulating? 219

11.4.4 A0.3MoO3 (A = K, Rb, Tl) blue bronzes: 2D solids with

pseudo-1D behaviour 224

11.4.5 Looking for 1D systems where there seem to be none: the

concept of hidden nesting 227

11.5 Low-dimensional molecular conductors 232

11.5.1 An archetypal molecular metal: (TMTSF)2PF6 234

11.5.2 Chemically modifying the electronic structure of molecular

conductors 235

11.5.3 Structurally complex materials with simple band structures 238

11.5.4 A case study: 1D vs 2D character of the carriers in some α

phases of BEDT-TTF 242

11.5.5 Electronic structure and folding: how to relate the band

structure and Fermi surface of different salts of

the same family 247

12 Electron repulsion 256

12.1 From the H¨uckel model to the Hubbard model 256

12.1.1 The delocalised picture of H2 256

12.1.2 The localised picture of H2 260

12.1.3 From the molecule to the solid state 266

12.1.4 Application to one-band systems 269

12.2 Mean-field approaches 273

12.2.1 The many-body problem 273

12.2.2 The Hartree–Fock method 274

12.2.3 Density functional theory 281

12.3 Conclusion 287

Solutions for exercises 291

Appendix: Character tables 342

Index 345

This page intentionally left blank