Tai ngay!!! Ban co the xoa dong chu O R B I TA L A P P R O A C H T O T H E E L E C T R O N I C S T RU C T U R E O F S O L I D S This page intentionally left blank Orbital Approach to the Electronic Structure of Solids ENRIC CANADELL Institut de Ciència de Materials de Barcelona (CSIC) MARIE-LIESSE DOUBLET CNRS – University of Montpellier CHRISTOPHE IUNG University of Montpellier 1 3 Great Clarendon Street, Oxford ox2 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. 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. 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. 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 Université 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 Supérieure de Cachan, Université de Montpellier, and Université de Pau, as well as in several interna- tional events. Based on this experience a book entitled Description orbitalaire de la structure électroniques des solides by C. 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.