Solid State Physics E-Book

J. R. Hook

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: The Manchester Physics Series
• Sprache: Englisch

Seitenzahl: 805

Veröffentlichungsjahr: 2013

Beschreibung

This Second Edition is aimed at students taking a firstcourse in this subject, although it will also be of interest toprofessional physicists and electronic engineers requiring a graspof the fundamentals of this important area of physics. Basicconcepts are introduced in an easily accessible context: forexample, wave propagation in crystals is introduced using one-andtwo-dimensional geometries. Only when these basic ideas arefamiliar are generalisations to three dimensions and the elegantframework of the reciprocal lattice made. Extensively rewritten,the Second Edition now includes new and expanded coverage ofsemiconductor devices, the quantum Hall effect, quasicrystals, hightemperature superconductors and techniques for the study of thesurfaces of solids. A chapter on dielectrics and ferroelectrics hasalso been added. Solid State Physics, Second Edition features:
* A carefully written and structured text to help students fullyunderstand this exciting subject.
* A flow diagram allowing topics to be studied in differentorders or omitted altogether.
* Optional "starred" and highlighted sections containing moreadvanced and specialised material for the more ambitiousreader.
* Carefully selected problems at the end of each chapter designedto assist learning. Solutions are provided at the end of thebook.

Leseprobe

Contents

Editor’s preface to the Manchester Physics Series

Foreword

Author’s preface to second edition

Flow Diagram

CHAPTER 1 CRYSTAL STRUCTURE

1.1 Introduction

1.2 Elementary Crystallography

1.3 Typical Crystal Structures

1.4 X-ray Crystallography

1.5 Quasi-crystals

1.6 Interatomic Forces

CHAPTER 2 CRYSTAL DYNAMICS

2.1 Introduction

2.2 SoundWaves

2.3 Lattice Vibrations of One-dimensional Crystals

2.4 Lattice Vibrations of Three-dimensional Crystals

2.5 Phonons

2.6 Heat Capacity from Lattice Vibrations

2.7 Anharmonic Effects

2.8 Thermal Conduction by Phonons

CHAPTER 3 FREE ELECTRONS IN METALS

3.1 Introduction

3.2 The Free Electron Model

3.3 Transport Properties of the Conduction Electrons

CHAPTER 4 THE EFFECT OF THE PERIODIC LATTICE POTENTIAL—ENERGY BANDS

4.1 Nearly Free Electron Theory

4.2 Classification of Crystalline Solids into Metals, Insulators and Semiconductors

4.3 The Tight Binding Approach

4.4 Band Structure Effective Masses

CHAPTER 5 SEMICONDUCTORS

5.1 Introduction

5.2 Holes

5.3 Methods of Providing Electrons and Holes

5.4 Absorption of Electromagnetic Radiation

5.5 Transport Properties

5.6 Non-equilibrium Carrier Densities

CHAPTER 6 SEMICONDUCTOR DEVICES

6.1 Introduction

6.2 The p–n Junction with Zero Applied Bias

6.3 The p–n Junction with an Applied Bias

6.4 Other Devices Based on the p-n Junction

6.5 Metal–oxide–semiconductor technology and the MOSFET

6.6 Molecular Beam Epitaxy and Semiconductor Heterojunctions

CHAPTER 7 DIAMAGNETISM AND PARAMAGNETISM 198

7.1 Introduction

7.2 Paramagnetism

7.3 Diamagnetism

CHAPTER 8 MAGNETIC ORDER

8.1 Introduction

8.2 The Exchange Interaction

8.3 Ferromagnetism

8.4 The Néel Model of Antiferromagnetism

8.5 Spin Waves

8.6 Other Types of Magnetic Order

8.7 Ferromagnetic Domains

CHAPTER 9 ELECTRIC PROPERTIES OF INSULATORS

9.1 Dielectrics

9.2 Pyroelectric Materials

9.3 Piezoelectricity

CHAPTER 10 SUPERCONDUCTIVITY

10.1 Introduction

10.2 Magnetic Properties of Superconductors

10.3 The London Equation

10.4 The Theory of Superconductivity

10.5 Macroscopic Quantum Phenomena

10.6 High-temperature Superconductors

CHAPTER 11 WAVES IN CRYSTALS

11.1 Introduction

11.2 Elastic Scattering of Waves by a Crystal

11.3 Wavelike Normal Modes—Bloch’s Theorem

11.4 Normal Modes and the Reciprocal Lattice

CHAPTER 12 SCATTERING OF NEUTRONS AND ELECTRONS FROM SOLIDS

12.1 Introduction

12.2 Comparison of X-rays, Neutrons and Electrons

12.3 Neutron Scattering Techniques

12.4 Determination of Phonon Spectra

12.5 Magnetic Scattering

12.6 Electron Scattering

CHAPTER 13 REAL METALS

13.1 Introduction

13.2 Fermi Surfaces

13.3 Electron Dynamics in a Three-dimensional Metal

13.4 Experimental Determination of the Fermi Surface

13.5 Why do Electrons Behave Independently?

13.6 Electromagnetic Waves in Metals

CHAPTER 14 LOW-DIMENSIONAL SYSTEMS

14.1 Introduction

14.2 The Two-dimensional Electron Gas

14.3 The Quantum Hall Effect

14.4 Resonant Tunnelling Devices

APPENDIX A Coupled probability amplitudes

APPENDIX B Electric and magnetic fields inside materials

APPENDIX C Quantum mechanics of an electron in a magnetic field

APPENDIX D The exchange energy

Bibliography

Solutions to problems

Index

Table of Constants

The Manchester Physics Series

General Editors

D. J. SANDIFORD:   F. MANDL:   A. C. PHILLIPS

Department of Physics and Astronomy, Faculty of Science, University of Manchester

Properties of Matter:

B. H. Flowers and E. Mendoza

Statistical Physics:

Second Edition

F. Mandl

Electromagnetism:

Second Edition

I. S. Grant and W. R. Phillips

Statistics:

R. J. Barlow

Solid State Physics:

Second Edition

J. R. Hook and H. E. Hall

Quantum Mechanics:

F. Mandl

Particle Physics:

Second Edition

B. R. Martin and G. Shaw

The Physics of Stars:

Second Edition

A. C. Phillips

Computing for Scientists:

R. J. Barlow and A. R. Barnett

Nuclear Physics:

J. S. Lilley

Copyright © 1974 by John Wiley & Sons, Ltd, The Atrium, Southern Gate,

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First edition 1974

Second edition 1991

Reprinted June and November 1994, July 1995, August 1996, February 1997, February 1998 and

February 1999

Reprinted with corrections December 2000, November 2001, April 2003, August 2004, April 2005,

October 2006, December 2007, October 2008, February 2010

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Library of Congress Cataloging-in-Publication Data

Hook, J. R. (John R.)

Solid state physics/J. R. Hook, H. E. Hall.—2nd ed.

p. cm.—(The Manchester physics series)

Rev. ed. of: Solid state physics/H. E. Hall, 1st ed. 1974.

Includes bibliographical references and index.

ISBN 0 471 92804 6 (cloth)—ISBN 0 471 92805 4 (paper)

1. Solid state physics. I. Hall, H. E. (Henry Edgar), 1928–

II. Hall, H. E. (Henry Edgar), 1928– Solid state physics.

III. Title. IV. Series.

QC176.H66 1991

530.481—dc20

90–20571

CIP

British Library Cataloguing in Publication Data

Hook, J. R.

Solid state physics.

1. Solids. Structure & physics properties

I. Title II. Hall, H. E. (Henry Edgar) III. Series

530.41

ISBN 978 0 471 92804 1 (hbk)

ISBN 978-0471-92805-8 (pbk)

To my family in partial compensation for taking so much ofmy time away from them

Editors’ preface to the Manchester Physics Series

The Manchester Physics Series is a series of textbooks at first degree level. It grew out of our experience at the Department of Physics and Astronomy at Manchester University, widely shared elsewhere, that many textbooks contain much more material than can be accommodated in a typical undergraduate course; and that this material is only rarely so arranged as to allow the definition of a shorter self-contained course. In planning these books we have had two objectives. One was to produce short books: so that lecturers should find them attractive for undergraduate courses; so that students should not be frightened off by their encyclopaedic size or their price. To achieve this, we have been very selective in the choice of topics, with the emphasis on the basic physics together with some instructive, stimulating and useful applications. Our second objective was to produce books which allow courses of different lengths and difficulty to be selected, with emphasis on different applications. To achieve such flexibility we have encouraged authors to use flow diagrams showing the logical connections between different chapters and to put some topics in starred sections. These cover more advanced and alternative material which is not required for the understanding of latter parts of each volume.

Although these books were conceived as a series, each of them is self-contained and can be used independently of the others. Several of them are suitable for wider use in other sciences. Each Author's Preface gives details about the level, prerequisites, etc., of his volume.

The Manchester Physics Series has been very successful with total sales of more than a quarter of a million copies. We are extremely grateful to the many students and colleagues, at Manchester and elsewhere, for helpful criticisms and stimulating comments. Our particular thanks go to the authors for all the work they have done, for the many new ideas they have contributed, and for discussing patiently, and often accepting, the suggestions of the editors.

Finally, we would like to thank our publishers, John Wiley & Sons Ltd, for their enthusiastic and continued commitment to the Manchester Physics Series.

D. J. Sandiford

F. Mandl

A. C. Phillips

February 1997

Foreword

The story of the creation was told in 200 words. Look it up if you don't believe me.—Edgar Wallace

When the time came to consider a second edition of Solid State Physics I felt that I had already said what I had to say on the subject in the first edition. I also felt that the book was rather too idiosyncratic for many students. For these reasons I thought it would be better if the revision and updating were undertaken by another hand, and the editors shared this view.

We therefore approached Dr John Hook, a friend and colleague for many years, and I think the result justifies the decision. The new edition is, in my opinion, a substantial improvement on the old one, but it would not have occurred to me to write it like that.

September 1990

Henry Hall

Author’s preface to second edition

I accepted the invitation of the editors of the Manchester Physics Series to write a second edition of Solid State Physics for two main reasons. Firstly I felt that, although the approach adopted in the first edition had much to commend it, some re-ordering and simplification of the material was required to make the book more accessible to undergraduate students. Secondly there was a need to take account of some of the important developments that have occurred in solid state physics since 1973.

To achieve re-ordering and simplification it has been necessary to rewrite most of the first edition. A major change has been to introduce the idea of mobile electron states in solids through the free electron theory of metals rather than through the formation of energy bands by overlap of atomic states on neighbouring atoms. The latter approach was used in the first edition because it could be applied first to the dilute electron gas in semiconductors where an independent particle model might be expected to work. Although this was appealing to the experienced physicist, it proved difficult to the undergraduate student, who was forced to assimilate too many new ideas at the beginning. One feature of the first edition that I have retained is to delay for as long as possible a formal discussion of the reciprocal lattice and Brillouin zones in a three-dimensional crystal. Although these concepts provide an elegant general framework for describing many of the properties of crystalline solids, they are, like Maxwell’s equations in electromagnetism, more likely to be appreciated by students if they have met some of the ideas earlier in a simpler context. The use of the formal framework is avoided in the early chapters by using one- and two-dimensional geometries whenever necessary.

To take account of recent developments the amount of material on semiconductor physics and devices has been substantially increased, a chapter has been added on the two-dimensional electron gas and quantum Hall effect, and sections on quasi-crystals, high-Tc superconductors and the use of electrons to probe surfaces have been included. A chapter on the electrical properties of insulators has also been added.

I have tried to conform to the aim of the Manchester Physics Series by producing a book of reasonable length (and thus cost), from which it is possible to define self-contained undergraduate courses of different length and difficulty. The problem with solid state physics in this context is that it contains many diverse topics so that many quite different courses are possible. I have had to be very selective therefore in my choice of subjects, which has been strongly influenced by the third year undergraduate solid state physics courses at Manchester. We currently have a basic course of 20 lectures, which is given at two levels; the courses cover material from Chapters 1–5 of this book and the higher level course also incorporates appropriate sections of Chapters 11–13. A further course of 20 lectures on selected topics in solid state physics currently covers magnetism, superconductivity and ferroelectricity (Chapters 7–10). The flow diagram inside the front cover can be used as an aid to the design of courses based on this book.

Important subjects that are not covered in this book are crystal defects and disordered solids. I would have liked to include a chapter on each of these topics but would have exceeded the length limit set by the publishers and editors had I done so.

Like the first edition, this book presupposes a background knowledge of properties of matter (interatomc potentials and their relation to binding energies and elastic moduli, kinetic theory), quantum mechanics (Schrodinger’s equation and its solution to find energy eigenvalues and eigenfunctions), electricity and magnetism (Maxwell’s equations and some familiarity with electric and magnetic fields in matter) and thermal physics (the Boltzmann factor and the Fermi and Bose distributions). Books in which this background information can be found are listed in the bibliography along with selected general reference books on solid state physics and some books and articles that provide further information on specific topics.

This book includes some more advanced and detailed material, which can be omitted without loss of continuity. Complete sections in this category are identified by starring and parts of sections are printed on a grey background.

The use of bold type for a technical term in the text, normally when the term is first encountered, indicates that a definition or explanation of the term can be found there. Italic type is used for emphasis.

I am very grateful to David Sandiford and Henry Hall for their helpful advice and constructive criticism. I would also like to thank Manchester undergraduate Colin Lally, who read the manuscript from the point of view of a prospective consumer; his reaction reassured me that the level was appropriate. Ian Callaghan’s draughtmanship and photography was invaluable in producing many of the figures, and my son James helped willingly with some of the more mundane manuscript-preparation tasks.

September 1990

JOHN HOOK

FLOW DIAGRAM

Solid lines indicate essential prerequisites.Dashed arrows indicate some assumption of material from a previous chapter.

Cover photography Visible light from a semiconductor laser developed at Philips Opteoelectronic in Eindhoven. (Photography provided by Professor Eoin O’Reilly of Surrey University and reproduced with the kind permission of Philips.)

CHAPTER 1

Crystal structure

Beauty when uncloth’d is clothè d best.—Phineas Fletcher (1582–1650)

1.1 INTRODUCTION

The aim of solid state physics is to explain the properties of solid materials as found on Earth. For almost all purposes the properties are expected to follow from Schrödinger’s equation for a collection of atomic nuclei and electrons interacting with electrostatic forces. The fundamental laws governing the behaviour of solids are therefore known and well tested. It is nowadays only in cosmology, astrophysics and high-energy physics that the fundamental laws are still in doubt.

In this book we shall be concerned almost entirely with crystalline solids, that is solids with an atomic structure based on a regular repeated pattern, a sort of three-dimensional wallpaper. Many important solids are crystalline in this sense, although this is not always manifest in the external form of the material. Because calculations are easier, more progress has been made in understanding the behaviour of crystalline than of non-crystalline materials. Many common solids—for example, glass, plastics, wood, bone—are not so highly ordered on an atomic scale and are therefore non-crystalline. Only recently has progress been made in understanding the behaviour of non-crystalline solids at a fundamental level.†

Even in the restricted field of crystalline solids the most remarkable thing is the great variety of qualitatively different behaviour that occurs. We have insulators, semiconductors, metals and superconductors—all obeying different macroscopic laws: an electric field causes an electric dipole moment in an insulator (Chapter 9), a steady current in a metal or semiconductor (Chapters 3 to 6) and a steadily accelerated current in a superconductor (Chapter 10). Solids may be transparent or opaque, hard or soft, brittle or ductile, magnetic or nonmagnetic.

In this chapter we first introduce in Section 1.2 the basic ideas of crystallography. In Section 1.3 we describe some important crystal structures and in Section 1.4 we explain how x-ray diffraction is used to determine crystal structure. In Section 1.5 we discuss quasi-crystals, ordered solids that challenge much of the conventional wisdom concerning crystalline materials. Section 1.6 contains a qualitative description of the interatomic forces responsible for binding atoms into solids.

1.2 ELEMENTARY CRYSTALLOGRAPHY

A basic knowledge of crystallography is essential for solid state physicists. They must know how to specify completely, concisely and unambiguously any crystal structure and they must be aware of the way that structures can be classified into different types according to the symmetries they possess; we shall see that the symmetry of a crystal can have a profound influence on its properties. Fortunately we will be concerned in this book only with solids with simple structures and we can therefore avoid the sophisticated group theoretical methods required to discuss crystal structures in general.

1.2.1 The crystal lattice

We will use a simple example to illustrate the methods and nomenclature used by crystallographers to describe the structure of crystals. Graphite is a crystalline form of carbon in which hexagonal arrays of atoms are situated on a series of equally spaced parallel planes. The arrangement of the atoms on one such plane is shown in Fig. 1.1(a). We choose graphite for our example because a single two-dimensional plane of atoms in this structure illustrates most of the concepts that we need to explain. Solid state physicists often resort to the device of considering a system of one or two dimensions when confronted with a new problem; the physics is often (but not always) the same as in three dimensions but the mathematics and understanding can be much easier.

To describe the structure of the two-dimensional graphite crystal it is necessary to establish a set of coordinate axes within the crystal. The origin can in principle be anywhere but it is usual to site it upon one of the atoms. Suppose we choose the atom labelled O in Fig. 1.1(a) for the origin. The next step is a very important one; we must proceed to identify all the positions within the crystal that are identical in all respects to the origin. To be identical it is necessary that an observer situated at the position should have exactly the same view in any direction as an observer situated at the origin. Clearly for this to be possible we must imagine that the two-dimensional crystal is infinite in extent. Readers should convince themselves that the atoms at A, B, C, D and E (and eight others in the diagram) are identical to the atom at the origin but that the atoms at F, G and H are not; compare for example the directions of the three nearest neighbours of the atom at O with the directions of the three nearest neighbours of the atom at F. The set of identical points identified in this way is shown in Fig. 1.1(b) and is called the crystal lattice; comparison of Figs. 1.1(a) and (b) illustrates clearly that the lattice is not in general the same as the structure. Readers should convince themselves that, apart from an unimportant shift in position, the lattice is independent of the choice of origin. Having identified the crystal lattice in this way the coordinate axes are simply obtained by joining the lattice point at the origin to two of its neighbours.

Fig. 1.1 Two-dimensional crystal of carbon atoms in graphite: (a) shows how the atoms are situated at the corners of regular hexagons; (b) shows the crystal lattice obtained by identifying all the atoms in (a) that are in identical positions to that at O. The crystal axes, lattice vectors and conventional unit cell are shown in both figures

The positions of all the lattice points of the two-dimensional graphite crystal are reached by drawing all possible vectors of the form

(1.1)

from the origin, where u and υ take on all possible integer values, positive, negative and zero. That the crystal appears identical when viewed from all the positions given by this equation is an indication that it possesses the important property of translational invariance.

The generalization of the above ideas to a three-dimensional crystal is straightforward. An origin is chosen and all the points within the crystal that are identical to it are identified; this set of points constitutes the three-dimensional crystal lattice. The directions of the crystal coordinate axes are then defined by joining the lattice point at the origin to three of its near neighbours (Fig. 1.2). The choice of neighbours is sometimes obvious but, where this is not the case, convention usually dictates the choice that most clearly reflects the symmetry of the lattice. The distances and directions of the nearest lattice points along the crystallographic x, y and z axes are specified by the three lattice vectors a, b and c. The lattice is completely specified by giving the lengths of a, b and c, and the angles α, β, and γ between them (Fig. 1.2). The positions of all the lattice points are reached by drawing all possible vectors of the form

Fig. 1.2 Crystallographic axes and unit cell for a three-dimensional crystal lattice

(1.2)

from the origin. The ability to express the positions of the points in this way, with a suitable choice of a, b and c, may be taken as a definition of a lattice in crystallography. A crystalline material may be defined as a material that possesses a lattice of this kind; the translational invariance property of the crystal is that it appears identical from all positions of the form of Eq. (1.2). Note that the only effect of a shift in choice of origin on a crystal lattice is a shift in the lattice as a whole by the same amount.

The lattice vectors also define the unit cell of a crystal. This concept is most easily explained by returning to the two-dimensional graphite crystal of Fig. 1.1, for which the unit cell is the parallelogram OACB defined by the vectors a and b. It is so called because stacking such cells together generates the entire crystal lattice, as is indicated by the broken lines in Fig. 1.1(b). The analogous three-dimensional object in Fig. 1.2, defined by lattice vectors a, b and c, is called a parallelopiped and is the unit cell for the three-dimensional lattice. The unit cell obtained from the conventional choice of lattice vectors is known as the conventional unit cell.

The concept of the unit cell as a building block allows us to understand the remarkable similarities between different crystals of the same material. In particular we can explain the law of constancy of angle (first stated by Nicolaus Steno in 1761) that: In all crystals of the same substance the angles between corresponding faces have a constant value. Fig. 1.3 is an illustration from an early book on mineralogy showing how macroscopically plane faces in various orientations can be built up by using cubic unit cells as building blocks. We shall see in Chapter 12 that the surfaces of crystals are not in fact constructed in the manner suggested by this illustration.

The reader will have noticed that the two-dimensional lattice of graphite (Fig. 1.1(b)) possesses symmetry properties other than the translational invariance indicated by Eq. (1.1). The lattice is transformed into itself, for example, by a rotation of 60° about an axis perpendicular to the xy plane through a lattice point; this axis is the crystallographic z axis of graphite, which is therefore a sixfold rotation axis of the lattice. In 1845 Bravais deduced that any three-dimensional lattice of the form of Eq. (1.2) could be classified into one of 14 possible types according to the symmetry that it possessed. The 14 Bravais lattices contain only one-, two-, three-, four- and six-fold rotation axes.

Fig. 1.3 The way in which the stacking of cubic unit cells can produce crystal faces of different orientations (Hauy, Traite de crystallographie)

1.2.2 The basis

Once the crystal lattice has been determined in the way described in the previous section and used to identify suitable coordinate axes and a unit cell, the description of the crystal structure is completed by specifying the contents of the unit cell. This is accomplished by identifying the group of atoms which, when associated with each lattice point, completely generates the structure. This group of atoms is known as the basis of the structure. The basis is specified by giving the position and chemical type of all the atoms within it. We again use the two-dimensional graphite structure of Fig. 1.1(a) to illustrate this procedure. The unit cell OACB of this structure contains the atom F, and readers should convince themselves that a suitable (but not unique) choice of basis for the two-dimensional graphite crystal can be obtained by associating the carbon atoms at O and F with the lattice point at O. This is so because the association of the corresponding pair of atoms with each lattice point (atoms G and A with the lattice point at A, for example) does indeed generate the entire structure. The position of an atom within the cell is most easily described by using the basis vector r, which connects the atom to the origin. Thus the position of the atom at F may be written

This atom is said to be at position (,). Our choice of basis for the two-dimensional graphite crystal can therefore be written concisely as

where the chemical type of the atom (carbon in this case) is specified by giving its chemical symbol. That a basis of two atoms is required to specify completely the two-dimensional graphite structure is an indication that each primitive unit cell contains just two atoms. In a three-dimensional crystal the basis vector of an atom can always be written as

and this atom is therefore said to be at (x, y, z).

Taking the symmetry of the basis as well as that of the lattice into consideration allows any crystal to be sorted into one of 32 possible point symmetry groups (sometimes referred to as the 32 crystal classes) and one of 230 possible space symmetry groups. A knowledge of these classifications is not assumed in this book, and readers requiring an understanding of them are recommended to consult one of the standard texts on crystallography.

1.2.3 Crystal planes and directions

Within a crystal lattice it is possible to identify sets of equally spaced parallel planes. Two examples of sets of lattice planes for a two-dimensional lattice are illustrated in Fig. 1.5. The density of lattice points on each plane of a set is the same and all the lattice points are contained on each set of planes. Planes of lattice points play an important role in the physics of the diffraction of waves by crystals, and it is necessary therefore to have a method of identifying the different sets. Miller indices are used for this purpose. These are derived from the intercepts made on the crystal axes by the plane that is nearest to the origin (but not the one that actually passes through the origin). Thus in Fig. 1.5(b) the nearest plane to the origin has intercepts a/3 and b/2, and this set of planes is therefore referred to by the Miller indices (3 2); note that it is the reciprocal of the intercept that determines the Miller index so that a large index indicates a small intercept. Fig. 1.5(a) illustrates a special case in which one intercept is infinite so that the corresponding Miller index is zero; thus the planes (1 0) are parallel to the y axis.

Fig. 1.5 (a) The (1 0) set of planes in a two-dimensional lattice, (b) The (3 2) set of planes in a two-dimensional lattice

For a set of lattice planes in a three-dimensional lattice the plane nearest the origin will have intercepts a/h, b/k and c/l and the set is referred to by Miller indices (h k l). Some three-dimensional examples are illustrated in Fig. 1.6. The planes (1 0 0) are parallel to both the y and z axes, and hence to the yz plane. Negative intercepts are indicated by a bar over the corresponding index, as in (1 1) and (2 0). The () set of planes is however identical to the (h k l) set. Note that if a, b and c define a primitive unit cell, then the Miller indices do not have a common factor. To see why this is, consider the special case of a set of planes with Miller indices (6 4) on Fig. 1.5(b); such a set would be parallel to the (3 2) planes shown but would have half the spacing. The (6 4) planes would therefore have lattice points only on alternate planes.

For crystals of high symmetry certain sets of planes may be related by symmetry and thus be equivalent from an atomic point of view. Thus for crystals of cubic symmetry, in which the unit cell sides a, b and c are equal in magnitude and mutually perpendicular, the three sets of planes (1 0 0), (0 1 0) and (0 0 1) are related by symmetry; they are said to belong to the form {1 0 0} where the curly brackets mean all planes equivalent by symmetry to the given plane.

Fig. 1.6 Some crystal planes inscribed in a unit cell, with their Miller indices

1.3 TYPICAL CRYSTAL STRUCTURES

1.3.1 Cubic and hexagonal close-packed structures

The crystal structure adopted by a particular material depends on the nature of the forces between the atoms within it. In some solids, particularly the inertgas solids and many metals, the forces are such that, to a good approximation, the atoms look like attracting hard spheres. For minimum energy in such cases it is necessary that the spheres should be packed as closely as possible. In two dimensions this principle leads to the close-packed layer structure shown in Fig. 1.7; this is a two-dimensional crystal in which the centres of the spheres lie on a triangular lattice like that of Fig. 1.4(d). The close packing can be extended to three dimensions if a second close-packed layer is placed over the first such that the spheres in the second layer are centred over interstices in the first. Let us suppose, for example, that the second layer occupies the positions marked B in Fig. 1.7. In this way each sphere in the second layer will touch three spheres in the first and the packing will be as close as possible. Inspection of Fig. 1.7 shows that such packing may be continued in various ways, for a third layer can occupy either positions C or A; both of these sets of positions mark interstices in the second layer.

A very common stacking sequence is ABCABC. . ., which gives a structure known as cubic close-packed (ccp) or face-centred cubic (fcc). The cubic unit cell of this structure is shown in Fig. 1.8; there are atoms at the corners of the cell and at the centres of each face. To make clear the relation to Fig. 1.7 a close-packed layer of atoms is shaded in Fig. 1.8; it is a (1 1 1) plane, normal to a body diagonal of the cube. It follows from symmetry that all planes of the form {1 1 1} are close-packed planes. In the fcc structure the environment of every atom is identical so that the crystal lattice corresponds to the atomic structure in this case. The rhombohedral primitive unit cell of the lattice is shown in Fig. 1.8(b). The conventional non-primitive choice of unit cell is however the cubic unit cell of Fig. 1.8(a) because it more obviously shows the full cubic symmetry. The conventional cell has a volume four times that of the primitive cell and thus contains four lattice points. Not surprisingly the lattice of the fcc structure is denoted an fcc lattice in the Bravais classification. Examples of elements that crystallize into the fcc structure are aluminium, calcium, nickel, copper, silver, gold, lead, neon, argon, krypton and xenon.

Fig. 1.7 A close-packed layer of spheres occupying positions A. The adjacent layers can occupy positions B or C

Fig. 1.8 The cubic close-packed (ccp) or face-centred cubic (fcc) structure

The environment of an atom in the fcc structure is best visualized by looking at the atomic coordination polyhedron. This is the figure formed from planes which are the perpendicular bisectors of lines joining an atom to its neighbours. Suppose you built a model of the structure out of Plasticine spheres and then compressed it; if you then picked it apart you would find that the spheres had deformed into coordination polyhedra.†

Fig. 1.9 Coordination polyhedron of the ccp structure: the rhombic dodecahedron. The origin of the crystallographic axes has been shifted so that there is an atom at the centre of the unit cell. The bonds from this atom to its nearest neighbours are shown

The coordination polyhedron thus represents the ‘sphere of influence’ of an atom. The coordination polyhedron of the fcc structure is shown in Fig. 1.9, together with the positions of the nearest neighbour atoms. Fig. 1.9 shows a cubic unit cell with the origin shifted from that of Fig. 1.8 by half a cell side along a cube axis, the [1 0 0] direction. This polyhedron is called a rhombic dodecahedron; it has 12 faces corresponding to contact with the 12 nearest neighbours; hence each atom is said to have a coordination number of 12. On this polyhedron the symmetries characteristic of the cubic structure (and of the cube itself) may be identified. The rhombic dodecahedron has four three-fold axes of symmetry through opposite pairs of corners A ([1 1 1] directions); it looks the same after rotation by 120° about any of these axes. There are also three four-fold axes through opposite pairs of corners B ([1 0 0] directions) and six two-fold axes through the centres of opposite faces ([1 1 0] directions).

The coordination polyhedra of the fcc structure all stack together in the same orientation in such a way as to fill the whole of space. The polyhedron therefore constitutes an alternative choice of primitive unit cell for the crystal. This type of unit cell is known as a Wigner-Seitz cell after those who first used it for a quantum mechanical problem. A Wigner-Seitz cell is defined for a general lattice as the smallest polyhedron bounded by planes that are the perpendicular bisectors of vectors joining one lattice point to the others; Fig. 1.10 illustrates this method of construction of the Wigner-Seitz cell for a two-dimensional lattice. It follows from the definition that the interior of the cell is the locus of points which are nearer to the given lattice point than any other. Only for the special case that each atom represents a lattice point is the Wigner-Seitz cell also the atomic coordination polyhedron.

Fig. 1.10 The Wigner-Seitz cell (broken lines) of a two-dimensional lattice obtained by drawing the perpendicular bisectors (full lines) of the lines joining a lattice point to its neighbours

Another common stacking sequence of the close-packed layers in Fig. 1.7 is ABABAB. . . ; this gives the hexagonal close-packed (hcp) structure illustrated in Fig. 1.11(a). All the A-plane atoms in this structure have an identical environment and can therefore be taken as lattice points. The environment of the B-plane atoms is different to that of the A-plane atoms, so that the B-plane atoms do not then lie on lattice points. The resulting lattice is denoted a hexagonal lattice in the Bravais classification. The conventional choice of crystallographic axes for the lattice is shown in Fig. 1.11(a) and the resulting primitive unit cell is identified by the thicker lines. Fig. 1.11(b) is a simpler two-dimensional way of depicting this three-dimensional unit cell. It shows a plan view as seen along the z axis; the z coordinate of the atom in the cell is indicated as a fraction of the side c of the unit cell. The unit cell contains a basis of an A atom at (0, 0, 0) and a B atom at (,,). The close-packed A planes in the hcp structure are the (0 0 1) planes, with the B planes sandwiched half-way between.

Since the environments of the A and B atoms are different, their coordination polyhedra have the same shape but differ in orientation. Because there are two atoms in each primitive unit cell, the Wigner-Seitz unit cell of the hcp lattice is double the volume of each coordination polyhedron and does not bear a simple relation to it.

Fig. 1.11

Other more complicated stacking sequences of close-packed layers are sometimes found, ABACABAC. . . for example, particularly in the rare-earth metals. These more exotic possibilities will not concern us in this book.

1.3.2 The body-centred cubic structure

A cubic structure only slightly less close-packed than fcc is the body-centred cubic (bcc) structure, three cubic unit cells of which are shown in Fig. 1.12(a). The environments of all the atoms are identical in the bcc structure so that the lattice is the same as the structure. The non-primitive cubic unit cell is the conventional choice for this lattice and contains two lattice points; the lattice vectors of the primitive cell are shown in Fig. 1.12(a). The eight hexagonal faces of the coordination polyhedron shown in Fig. 1.12(b) represent the ‘contact’ of an atom with its eight nearest neighbours; the coordination number of the bec structure is thus 8. This is smaller than the coordination number (12) of the fcc and hcp structures, but the existence of the six square faces of the coordination polyhedron indicates that an atom in the bcc crystal has six second nearest neighbours not much further away than the first nearest neighbours. The metallic elements lithium, sodium, potassium, chromium, barium and tungsten crystallize into the bcc structure.

Fig. 1.12

Fig. 1.13 Unit cell of the NaCl structure, with alternate planes of Na+ and Cl− ions perpendicular to the [1 1 1] direction indicated by shading

1.3.3 Structures of ionic solids

We now progress to consider examples of the simplest type of crystals containing atoms of more then one element, namely ionic solids. Since ions of opposite charge approximate to attracting hard spheres, the crystal structure of these is also often dominated by close-packing considerations. The electrons in the negatively charged anion are in general less tightly bound than those in the positively charged cation and the anion is therefore normally larger. In crystals containing equal numbers of positive and negative ions such as sodium chloride, NaCl, the structure is then likely to be determined by the number of larger anions that will pack tightly around the cation. For NaCl this is only six, and this leads to the three-dimensional chessboard structure shown in Fig. 1.13. All the Na+ ions have identical environments within the crystal so that these can be taken to represent the crystal lattice, which is therefore an example of an fcc Bravais lattice. The Cl− ions also lie on an fcc lattice displaced by half a unit cell in the [1 0 0] direction.

It can also be seen from Fig. 1.13 that planes of the form {1 1 1} of the Na+ lattice contain all the Na+ ions and no Cl− ions; the Cl− ions are contained on parallel planes mid-way between the (1 1 1) planes; this fact was crucial in the elucidation of the structure of NaCl which was the first to be determined by x-ray diffraction (see section 1.4). When this structure was first discovered some chemists were horrified to find that it contained no identifiable NaCl molecule. We now know that the absence of an identifiable molecule is very general in inorganic crystals and we have become used to the idea of a crystal as a single giant molecule.

Fig. 1.14 Primitive cubic unit cell of CsCl shown in plan view. The basis is Cs+ (0,0,0) and Cl− (, , )

The caesium chloride, CsCl, structure of Fig. 1.14 is an alternative structure for an ionic solid containing equal numbers of anions and cations; the cubic unit cell has Cs+ ions at the corners and a Cl− ion at the centre. Each ion in the structure has a coordination number of 8 so that this is the structure likely to be adopted when just eight anions will pack tightly around each cation. All cations (or alternatively all anions) in this structure have an identical environment so that their positions form a crystal lattice. In this case the conventional cubic unit cell is primitive and is designated as simple cubic in the Bravais classification.

1.3.4 The diamond and zincblende structures

A very important structure in solid state physics is that adopted by carbon atoms in diamond. In this structure each carbon atom is covalently bonded to four nearest neighbours arranged at the corners of a regular tetrahedron as in Fig. 1.15. Fig. 1.15 shows that only half of the atoms have identical environments and these lie on an fcc Bravais lattice. The other atoms form an fcc lattice displaced by one-quarter of a unit cell in the [1 1 1] direction. The two types of atom differ only in the orientation of the bonds to the nearest neighbours. The small coordination number (4) of the diamond structure indicates that it is very far from being a close-packed structure and that the interatomic forces are very different in nature to those in most metallic, ionic and ‘inert-gas’ solids. Two other elements from group IV of the periodic table, the semiconducting elements silicon and germanium, crystallize into the diamond structure and this explains its importance in solid state physics.

Group III–Group V semiconducting compounds (see Chapter 5), such as gallium arsenide, GaAs, and indium antimonide, InSb, crystallize into the closely related zincblende, ZnS, structure. This structure differs from that of diamond only in that one type of carbon atom is replaced by zinc atoms and the other by sulphur atoms.

Fig. 1.15

1.4 X-RAY CRYSTALLOGRAPHY

1.4.1 The Bragg law

The wavelength of x-rays is typically 1 Å, comparable to the interatomic spacings in solids. This means that a crystal behaves as a three-dimensional diffraction grating for x-rays. In an optical diffraction experiment it is possible to deduce the spacing of the lines on the grating from the separation of the diffraction maxima; by measuring the relative intensities of different orders information about the structure of the lines on the grating can be obtained. In an exactly similar way, measurement of the separation of the x-ray diffraction maxima from a crystal allows us to determine the size of the unit cell, and from the intensities of the diffracted beams we obtain information on the arrangement of atoms within the cell.

The general laws of diffraction as formulated by von Laue will be considered in Chapter 11. For the present, the simpler and more physical formulation discovered by Bragg and used by him in his earliest structure determinations will suffice. Bragg derived the condition for constructive interference of the x-rays scattered from a set of parallel lattice planes. Consider x-rays incident at a glancing angle θ on one of the planes of the set as shown in Fig. 1.16(a). The figure illustrates that there will be constructive interference of the waves scattered from the two successive lattice points A and B in the plane if the distances AC and DB are equal. This is the case if the scattered wave makes the same angle θ to the plane as the incident wave; the diffracted wave thus looks as though it has been reflected from the plane. The use of the glancing angle θ rather than the angle of incidence is conventional in x-ray crystallography; the reflection condition implies that the x-ray beam is deflected through an angle 2θ. Note that we consider the scattering associated with lattice points rather than atoms because it is the basis of atoms associated with each lattice point that is the true repeat unit of the crystal; the lattice point is the analogue of the line on an optical diffraction grating and the basis represents the structure of the line.

Coherent scattering from a single plane is not sufficient to obtain a diffraction maxium. It is also necessary that successive planes should scatter in phase. This will be the case if the path difference for scattering off two adjacent planes is an integral number of wavelengths. From Fig. 1.16(b) we see that this is so if

(1.3)

where d is the spacing of the planes and n is an integer. This is Bragg’s law.

The diffracted beams (often referred to as reflections) from any set of lattice planes can only occur at the particular angles predicted by the Bragg law. X-ray crystallographers use the Miller indices of the planes to label the reflections. A beam corresponding to a value of n greater than 1 could be identified by a statement such as ‘the nth-order reflection from the (h k l) planes’. This however is rather cumbersome and such a beam is described instead more concisely as the (nh nk nl) reflection. Thus the third-order reflection from the (1 1 1) planes is described as the (3 3 3) reflection. This notation is justified by rewriting the Bragg law as

which makes nth-order diffraction off (h k l) planes of spacing d look like first-order diffraction off planes of spacing d/n. Planes of this reduced spacing would have Miller indices (nh nk nl).

The details of the structure were then deduced from the differences between the diffraction patterns for NaCl and KCl. The major difference was the absence of the (1 1 1) reflection in KCl compared to a weak but clearly detectable (1 1 1) reflection in NaCl. This arises because the K+ and Cl− ions both have the argon electron shell structure and hence scatter x-rays almost equally whereas the Na+ and Cl− ions have different scattering strengths. The (1 1 1) reflection in NaCl corresponds to one wavelength of path difference between neighbouring (1 1 1) planes, and thus to half a wavelength difference between the alternate planes of Na+ and Cl− ions that make up the crystal structure of Fig. 1.13. The difference in scattering of x-rays by the Na+ and Cl− ions is necessary therefore to prevent elimination of the (1 1 1) reflection by destructive interference. Bragg was able to deduce that the structures of NaCl and KCl corresponded to alternate planes of positive and negative ions perpendicular to the [1 1 1] direction and the structure of Fig. 1.13 followed from this.

1.4.2 Experimental arrangements for x-ray diffraction

Since the pioneering work of Bragg, x-ray diffraction has developed into a routine technique for the determination of crystal structure. In most experiments the x-rays are produced by accelerating electrons through a potential difference of order 30 keV and allowing them to collide with a metal target; the x-ray emission is a mixture of the characteristic lines (K, L, M, etc.) of the metal atoms and a continuous background which varies smoothly with wavelength. By changing the accelerating voltage it is possible to vary the relative amounts in the mixture to obtain either almost monochromatic x-rays or a broadened white spectrum.

If a higher-intensity source of x-rays is required, the intense radiation emitted by the charged particles (usually electrons) in a synchrotron can be used. The particles radiate predominantly in a direction tangential to their path as a result of the acceleration associated with their orbits. The intensity of the synchrotron radiation normally varies smoothly with increasing wavelength above a minimum cut-off value, which depends on the radius of curvature of the path and the energy of the particles. The intensity can be made to peak at a particular wavelength by placing bending magnets at regular intervals along a straight section of the synchrotron. This configuration is known as an undulator and wavelength selection occurs because of constructive interference between the radiation emitted in the vicinity of successive magnets. Examples of situations in which synchrotron radiation is used are for the determination of the structure of very small crystals and of crystals containing biological molecules where the unit cell may contain thousands of atoms. In the latter case it is necessary to measure the intensities of a large number of closely spaced diffracted beams in order to determine the structure.

Many types of x-ray camera have been invented to sort out the reflections from different crystal planes. We shall describe only three very common types of x-ray photograph that are widely used for the simple structures that we study in this book.

For a Laue photograph, historically the first type, a single crystal is illuminated with a collimated beam of ‘white’ (i.e. continuous spectrum) x-rays as in Fig. 1.17(a). Each set of crystal planes will satisfy the Bragg condition, Eq. (1.3), for some wavelength (perhaps several wavelengths if the spread in wavelength is large enough for different orders of diffraction to occur) and the resulting diffracted beams generate a pattern of spots on the photographic film as in Fig. 1.17(b). The symmetry of the spot pattern reflects the symmetry of the crystal when viewed along the direction of the incident beam. The deduction of the symmetry of the crystal is one of the main uses of the Laue method; it is often used to determine the orientation of single crystals that do not have well developed external faces.

Fig. 1.17

When a single crystal is exposed to a collimated monochromatic beam of x-rays no diffraction takes place in general because no set of lattice planes is at the correct angle to satisfy Bragg’s law. If the crystal is rotated about a fixed axis perpendicular to the x-ray beam, then the glancing angle θ varies for sets of planes that are not perpendicular to the rotation axis. A set of such planes is likely to satisfy the Bragg condition for some orientation of the crystal. This is the basis of the rotating crystal method; the crystal is typically surrounded by a photographic film in the form of a cylinder parallel to the rotation axis and the resulting pattern of diffraction spots is analysed to obtain the structure.

An alternative method of ensuring that there are sets of lattice planes in the specimen at the correct angles to satisfy Bragg’s law for a monochromatic incident beam of x-rays is to use a sample in the form of many small crystalline grains glued together. If the orientation of the grains is random then, for any set of lattice planes, some of the grains will be oriented at the Bragg angle θ to the incident x-rays. The locus of the beams reflected from the same set of planes in different grains will be a cone of half-angle 2θ with the incident beam as axis as shown in Fig. 1.18(a); intersection of the x-ray cone with the film produces a line on the photograph. A typical example of a powder photograph is shown in Fig. 1.18(b); each line represents diffraction from a different set of lattice planes. The structure is determined from the measured θ values and the relative intensities of the reflections. Another application of the powder method arises because of the very high resolution that can be obtained for the radiation that is almost back-scattered, as is evidenced by the resolution of the Co Kα doublet in Fig. 1.18(b). Eq. (1.3) shows that when θ is close to 90° it is very sensitive to the precise value of d. Very accurate unit cell dimensions can therefore be obtained from almost back-scattered radiation and this provides a valuable method of measuring thermal expansion.

Fig. 1.18 (a) Experimental geometry for a power photograph, (b) A powder photograph of molybdenum taken with Co Kα radiation. The x-rays enter the camera through the hole in the centre of the film and leave between the ends of the film. Note that the Kα1–Kα2 x-ray doublet (wavelengths 1.789 and 1.793 Å) is resolved for the back-scattered radiation near the entrance hole. (Courtesy of H. Lipson)

1.5 QUASI-CRYSTALS

To the crystallographer devoted to the view of ordered structures described earlier in this chapter the x-ray Laue photograph of Fig. 1.19 presents a seemingly insuperable problem. A diffraction pattern of this type was first observed in 1984 from a sample of Al/Mn alloy, cooled so rapidly from the molten state that the first solid structure to form was ‘frozen in’. Similar patterns have subsequently been observed for other materials. The existence of such sharp spots in the pattern indicates a highly ordered atomic arrangement, containing presumably parallel planar structures capable of scattering x-rays coherently. The spot pattern in Fig. 1.19 however has a tenfold symmetry axis, indicating that the atomic structure must possess similar symmetry,† thus contradicting one of the fundamental theorems of crystallography that any lattice of the form of Eq. (1.2) can contain only two-, three-, four- and sixfold symmetry axes. A tenfold axis fails to appear in the list for essentially the same reason as it is impossible to tile a two-dimensional area with tiles shaped like regular decagons, as is apparent from Fig. 1.20. As the materials giving rise to diffraction patterns like that of Fig. 1.19 cannot therefore possess the property of translational invariance that is expected in crystals, they have been designated as quasi-crystals.

When diffraction patterns for different angles of incidence are viewed, quasi-crystals often appear to have the same symmetry as the icosahedron shown in Fig. 1.21(a). This figure has 20 faces each of which is an equilateral triangle. The line AA is one of the six fivefold axes of symmetry of the icosahedron which lead to the tenfold symmetry of the diffraction pattern. Icosahedral arrangements of atoms arise quite naturally in attempts at close packing if the tetrahedral arrangement of atoms of Fig. 1.21(b) rather than the close-packed plane of Fig. 1.7 is taken as the basic building block. The icosahedron is formed by allowing 20 tetrahedra to share a common vertex. In order to achieve this each tetrahedron has to distort slightly; an atom is about 5% further from its neighbours on the surface of the icosahedron than it is from the atom at the centre. It is this feature that prevents the attainment of long-range close packing by continued stacking of tetrahedra outwards from the original shared vertex. All solids discovered prior to 1984 with a local icosahedral arrangement of atoms had relieved the distortion by incorporating additional atoms in such a way as to regain a structure with translational invariance.

Fig. 1.19 Spot diffraction pattern with tenfold symmetry obtained from a rapidly cooled Al(86 at%)–Mn(14 at%) alloy. (Reproduced with permission from D. Schechtman et al., Phys. Rev. Lett. 53, 1951 (1984))

Fig. 1.20 Three regular decagons sharing a vertex, P, demonstrating the impossibility of tiling a two-dimensional area with unit cells of this shape

Fig. 1.21

It is generally believed that an understanding of the structure of quasi-crystals will be obtained by generalizing to three dimensions the two-dimensional tiling pattern invented by Roger Penrose in 1974, which is shown in Fig. 1.22. In contrast to the stacking of identical parallelogram unit cells, which generates a two-dimensional crystal lattice like that of Fig. 1.1(b), Penrose tiling uses the two building blocks of Fig. 1.22(a). Both the basic tiles are rhombuses, like the unit cell of the rhombic lattice in Fig. 1.4(c), but they have values of the angle γ of 144° and 108°. In the pattern (Fig. 1.22(b)) tiles of the former angle occur exactly (1 + )/2 times as often as those of the latter. Despite the lack of translational invariance, the pattern contains regular decagons all with the same orientation and also sets of almost straight lines intersecting at angles of 72° (Fig. 1.22(b)