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- Herausgeber: Wiley-VCH GmbH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Solid State Physics
Enables readers to easily understand the basics of solid state physics
Solid State Physics is a successful short textbook that gives a clear and concise introduction to its subject. The presentation is suitable for students who are exposed to this topic for the first time. Each chapter starts with basic principles and gently progresses to more advanced concepts, using easy-to-follow explanations and keeping mathematical formalism to a minimum.
This new edition is thoroughly revised, with easier-to-understand descriptions of metallic and covalent bonding, a straightforward proof of Bloch’s theorem, a simpler approach to the nearly free electron model, and enhanced pedagogical features, such as more than 100 discussion questions, 70 problems – including problems to train the students’ skills to find computational solutions – and multiple-choice questions at the end of each chapter, with solutions in the book for self-training.
Solid State Physics introduces the readers to:
- Crystal structures and underlying bonding mechanisms
- The mechanical and vibrational properties of solids
- Electronic properties in both a classical and a quantum mechanical picture, with a treatment of the electronic phenomena in metals, semiconductors and insulators
- More advanced subjects, such as magnetism, superconductivity and phenomena emerging for nano-scaled solids
For bachelor’s students in physics, materials sciences, engineering sciences, and chemistry, Solid State Physics serves as an introductory textbook, with many helpful supplementary learning resources included throughout the text and available online, to aid in reader comprehension.
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Veröffentlichungsjahr: 2022
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Table of Contents
Cover
Title Page
Copyright
Preface to the First Edition
Preface to the Second Edition
Preface to the Third Edition
Physical Constants and Energy Equivalents
1 Crystal Structures
1.1 General Description of Crystal Structures
1.2 Some Important Crystal Structures
1.3 Crystal Structure Determination
1.4 Further Reading
1.5 Discussion and Problems
Notes
2 Bonding in Solids
2.1 Attractive and Repulsive Forces
2.2 Ionic Bonding
2.3 Covalent Bonding
2.4 Metallic Bonding
2.5 Hydrogen Bonding
2.6 Van der Waals Bonding
2.7 Further Reading
2.8 Discussion and Problems
Note
3 Mechanical Properties
3.1 Elastic Deformation
3.2 Plastic Deformation
3.3 Fracture
3.4 Further Reading
3.5 Discussion and Problems
Note
4 Thermal Properties of the Lattice
4.1 Lattice Vibrations
4.2 Heat Capacity of the Lattice
4.3 Thermal Conductivity
4.4 Thermal Expansion
4.5 Allotropic Phase Transitions and Melting
References
4.6 Further Reading
4.7 Discussion and Problems
Notes
5 Electronic Properties of Metals: Classical Approach
5.1 Basic Assumptions of the Drude Model
5.2 Results from the Drude Model
5.3 Shortcomings of the Drude Model
5.4 Further Reading
5.5 Discussion and Problems
Notes
6 Electronic Properties of Solids: Quantum Mechanical Approach
6.1 The Idea of Energy Bands
6.2 The Free Electron Model
6.3 The General Form of the Electronic States
6.4 Nearly‐Free Electron Model: Band Formation
6.5 Tight‐binding Model
6.6 Energy Bands in Real Solids
6.7 Transport Properties
6.8 Brief Review of Some Key Ideas
References
6.9 Further Reading
6.10 Discussion and Problems
Notes
7 Semiconductors
7.1 Intrinsic Semiconductors
7.2 Doped Semiconductors
7.3 Conductivity of Semiconductors
7.4 Semiconductor Devices
7.5 Further Reading
7.6 Discussion and Problems
Note
8 Magnetism
8.1 Macroscopic Description
8.2 Quantum‐Mechanical Description of Magnetism
8.3 Paramagnetism and Diamagnetism in Atoms
8.4 Weak Magnetism in Solids
8.5 Magnetic Ordering
Reference
8.6 Further Reading
8.7 Discussion and Problems
Notes
9 Dielectrics
9.1 Macroscopic Description
9.2 Microscopic Polarization
9.3 The Local Field
9.4 Frequency Dependence of the Dielectric Constant
9.5 Other Effects
9.6 Further Reading
9.7 Discussion and Problems
Notes
10 Superconductivity
10.1 Basic Experimental Facts
10.2 Some Theoretical Aspects
10.3 Experimental Detection of the Gap
10.4 Coherence of the Superconducting State
10.5 Type‐I and Type‐II Superconductors
10.6 High‐Temperature Superconductivity
10.7 Concluding Remarks
References
10.8 Further Reading
10.9 Discussion and Problems
Notes
11 Finite Solids and Nanostructures
11.1 Quantum Confinement
11.2 Surfaces and Interfaces
11.3 Magnetism on the Nanoscale
11.4 Further Reading
11.5 Discussion and Problems
Appendix A
A.1 Explicit Forms of Vector Operations
A.2 Differential Form of the Maxwell Equations
A.3 Maxwell Equations in Matter
Note
Appendix B
B.1 Solutions to Basic Concepts Questions
Index
End User License Agreement
List of Tables
Chapter 4
Table 4.1 Comparison between vibrational frequencies estimated from Young's ...
Table 4.2 Molar heat capacities of different solids at the boiling point of ...
Table 4.3 Debye temperatures and frequencies for selected materials.
Table 4.4 Thermal conductivity
for some metals and insulators at room temp...
Table 4.5 Coefficient of thermal expansion
at room temperature.
Chapter 5
Table 5.1 Numbers
of conduction electrons per atom, calculated conduction ...
Table 5.2 Observed values of the plasma energy
together with the values ca...
Chapter 7
Table 7.1 Gap sizes for common semiconductors (above the horizontal line) an...
Table 7.2 Intrinsic carrier densities for Si and GaAs.
Table 7.3 Effective masses for some semiconductors.
Chapter 9
Table 9.1 Dielectric constants
of selected materials at room temperature....
Table 9.2 Dielectric constants
of selected materials in the electrostatic ...
List of Illustrations
Chapter 1
Figure 1.1 A two‐dimensional Bravais lattice.
Figure 1.2 Illustration of (primitive and nonprimitive) unit cells and of th...
Figure 1.3 A two‐dimensional Bravais lattice with different choices for the ...
Figure 1.4 (a) Simple cubic structure; (b) body‐centered cubic structure; an...
Figure 1.5 Structures of CsCl and NaCl. The spheres are depicted much smalle...
Figure 1.6 Close packing of spheres leading to the hcp and fcc structures.
Figure 1.7 Structures for (a) graphene, (b) graphite, and (c) diamond. Bonds...
Figure 1.8 Construction for the derivation of the Bragg condition. The horiz...
Figure 1.9 Three different lattice planes in the simple cubic structure char...
Figure 1.10 Illustration of X‐ray scattering from a sample. The source and d...
Figure 1.11 Top: A chain with a lattice constant
as well as its reciprocal...
Figure 1.12 Ewald construction for finding the directions in which construct...
Figure 1.13 Two‐dimensional lattices.
Figure 1.14 Left: two‐dimensional NiO crystal; Right: possible choices of th...
Figure 1.15 A two‐dimensional crystal.
Figure 1.16 (a) Two‐dimensional crystal structure of a hexagonal close‐packe...
Figure 1.17 Two‐dimensional Bravais lattices.
Chapter 2
Figure 2.1 (a) Typical interatomic potential
for bonding in solids accordi...
Figure 2.2 (a) Formation of bonding and antibonding energy levels in the
i...
Figure 2.3 Linear combination of orbitals on neighboring atoms. (a) Two s or...
Figure 2.4 The energy changes
and
for the formation of a hydrogen molecu...
Figure 2.5 One‐dimensional chain of ions.
Chapter 3
Figure 3.1 (a) Illustration of stress
and strain
.
and
are the length...
Figure 3.2 Typical stress of a solid as a function of applied strain.
and
Figure 3.3 Young's modulus for different materials. The values are merely a ...
Figure 3.4 Estimate of the yield stress for shearing a solid. (a) Atoms in e...
Figure 3.5 An edge dislocation formed by an extra sheet of atoms in a crysta...
Figure 3.6 (a) Shearing of a solid in the presence of an edge dislocation. T...
Figure 3.7 Calculated local stress field for a crack along the (1,1,1) plane...
Figure 3.8 Exposing a wire to tensile stress. The forces
act on the entire...
Chapter 4
Figure 4.1 (a) One‐dimensional chain with one atom per unit cell.
denotes ...
Figure 4.2 Motion of the atoms in the chain for (a)
and (b)
.
Figure 4.3 Instantaneous positions of atoms in a chain for two different wav...
Figure 4.4 (a) One‐dimensional chain with two atoms per unit cell. (b) Allow...
Figure 4.5 Motion of the atoms for
in the optical branch.
represents an ...
Figure 4.6 Vibrational spectrum for a finite chain of atoms with a length of...
Figure 4.7 (a) Energy level diagram for one harmonic oscillator. (b) Energy ...
Figure 4.8 Phonon dispersion in aluminum and diamond. The dispersion is plot...
Figure 4.9 Obtaining the interatomic force constant from Young's modulus for...
Figure 4.10 Temperature‐dependent heat capacity of diamond. Data from Desnoy...
Figure 4.11 Temperature‐dependent heat capacity according to the Einstein mo...
Figure 4.12 (a) Points of integers
that represent the allowed vibrational ...
Figure 4.13 Temperature‐dependent thermal conductivity of Si. Adapted from G...
Figure 4.14 Classical picture for the thermal expansion of a solid. The inte...
Figure 4.15 Gibbs free energy for two competing phases A and B. At the tempe...
Figure 4.16 (a) Melting temperature as a function of cohesive energy for the...
Figure 4.17 (a) Two‐dimensional square lattice with forces acting only betwe...
Chapter 5
Figure 5.1 Measured and calculated electrical conductivities of metals as a ...
Figure 5.2 (a) Illustration of the Hall effect. (b) Equilibrium between the ...
Chapter 6
Figure 6.1 The formation of energy bands in solids. (a) Bonding and antibond...
Figure 6.2 Band formation in Si. The lower band is completely occupied with ...
Figure 6.3 Electronic states in the free electron model. The increasing ener...
Figure 6.4 (a) Density of states
for a free electron gas. (b) Fermi–Dirac ...
Figure 6.5 Most of the electrons in a metal (roughly those in the dark gray ...
Figure 6.6 Sketch of the electronic and lattice contributions to the heat ca...
Figure 6.7 Screening of a positively charged impurity in a metal. (a) The oc...
Figure 6.8 Potential due to a positive point charge in a metal compared to t...
Figure 6.9 Electronic states in the nearly‐free electron model for a one‐dim...
Figure 6.10 Qualitative explanation for the gap openings at the Brillouin zo...
Figure 6.11 Bands for a one‐dimensional solid calculated within the tight‐bi...
Figure 6.12 (a) Electronic energy bands in Al along the
–X direction. The i...
Figure 6.13 Electronic energy bands for Si and GaAs. These materials have th...
Figure 6.14 Illustration of the difference between metals and semiconductors...
Figure 6.15 Origin of the electronic energy bands for graphene. (a) sp
hybr...
Figure 6.16 Simple picture of conduction in a metal. The circles symbolize f...
Figure 6.17 Temperature‐dependent heat capacities of two solids.
Figure 6.18 Band structure and Brillouin zone of a material.
Chapter 7
Figure 7.1 Charge neutrality and the position of the chemical potential in a...
Figure 7.2 Transport of charge in an electric field
for a partially filled...
Figure 7.3 (a) Sketch of the valence band and conduction band in the vicinit...
Figure 7.4 The measurement of cyclotron resonance. The electrons (or holes) ...
Figure 7.5 Nonionized dopant atoms in a Si lattice: (a) donor (b) acceptor....
Figure 7.6 Energy levels of dopant atoms. (a) The donor ground state is plac...
Figure 7.7 Electron density and position of the chemical potential for an n‐...
Figure 7.8 The pn junction. (a) Energy levels and carrier densities in separ...
Figure 7.9 Idealized model of the depletion zone solved using the Poisson eq...
Figure 7.10 Definition of the energies in the pn junction. The VBM on the n ...
Figure 7.11 The pn junction as a diode (considering only the electrons, not ...
Figure 7.12 Characteristic
curve for a pn junction operated as diode.
Figure 7.13 Design and working principle of a MOSFET: (a) without applied vo...
Figure 7.14 Generation of an inversion layer in the MOSFET. The positive gat...
Figure 7.15 Optoelectronic devices. (a) A light‐emitting diode works because...
Figure 7.16 Sketch of a silicon solar cell and the electrical contact to an ...
Figure 7.17 Band structure of a semiconductor near the band gap.
Figure 7.18 Valence band and conduction band near the junction between two s...
Chapter 8
Figure 8.1 (a) Precession of an atomic magnetic moment in an external field....
Figure 8.2 Paramagnetic susceptibility of a solid with localized magnetic mo...
Figure 8.3 (a) Density of occupied states for free electrons at
K, split u...
Figure 8.4 Types of magnetic ordering. The arrows denote the direction and s...
Figure 8.5 Temperature‐dependent magnetization of Fe, Co, and Ni below the C...
Figure 8.6 (a) Occupied density of states in a 3d transition metal, separate...
Figure 8.7 (a) Domains of different magnetization in a ferromagnetic solid. ...
Figure 8.8 (a) Magnetic material with a single domain leading to a strong ex...
Figure 8.9 Magnetization of a ferromagnetic sample as a function of external...
Chapter 9
Figure 9.1 A parallel‐plate capacitor. (a) Charges on the plates of the capa...
Figure 9.2 Mechanisms leading to microscopic electric polarization. (a) The ...
Figure 9.3 The local field on microscopic polarizable units. (a) Microscopic...
Figure 9.4 Dielectric function for a damped, driven harmonic oscillator clos...
Figure 9.5 (a) Contributions to the imaginary part of
by transitions betwe...
Figure 9.6
Upper part
: The unit cell of barium titanate
with the charges o...
Figure 9.7 (a) Exposing a piezoelectric material to mechanical stress result...
Chapter 10
Figure 10.1 Typical temperature‐dependent resistivities of a normal metal an...
Figure 10.2 Periodic table of the elements with the superconducting elements...
Figure 10.3 (a) Combined effect of a magnetic field and a finite temperature...
Figure 10.4 The Meissner effect is not merely a consequence of zero resistiv...
Figure 10.5 Illustration of the isotope effect. The graph shows the critical...
Figure 10.6 Exponential damping of an external magnetic field near the surfa...
Figure 10.7 Local deformation of the lattice via the electrostatic interacti...
Figure 10.8 Occupation of single‐electron levels at zero temperature in (a) ...
Figure 10.9 Gap size for a superconductor in the BCS model as a function of ...
Figure 10.10 Tunneling experiment between a superconductor and a normal meta...
Figure 10.11 Qualitative low‐temperature heat capacity of a superconductor i...
Figure 10.12 (a) A superconducting ring enclosing a magnetic flux. The magne...
Figure 10.13 Resistivity
, internal magnetic field
, and magnetization
a...
Figure 10.14 Magnetic flux in a type‐II superconductor. The field penetrates...
Figure 10.15 Increase of the highest critical temperature
of known superco...
Chapter 11
Figure 11.1 (a) A thin metal film on a semiconducting or insulating substrat...
Figure 11.2 (a) Matching of a bulk electronic state (a Bloch wave) to an exp...
Figure 11.3 Illustration of topologically protected metallic states between ...
Appendix B
Figure B.1 Two‐dimensional lattices with unit cell.
Guide
Cover
Table of Contents
Title Page
Copyright
Preface to the First Edition
Preface to the Second Edition
Preface to the Third Edition
Physical Constants and Energy Equivalents
Begin Reading
Appendix A
Appendix B
Index
End User License Agreement
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Solid State Physics
An Introduction
Third Edition
Philip Hofmann
Author
Prof. Philip HofmannAarhus UniversityDepartment of Physics and AstronomyNy Munkegade 1208000 Aarhus CDenmark
Solution manual for instructors available from www.wiley-vch.de/ISBN9783527414109
Cover Image: Band structure of aluminum determined by angle‐resolved photoemission spectroscopy. Data taken from Physical Review B 66, 245422 (2002).
All books published by WILEY‐VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
Library of Congress Card No.: applied for
British Library Cataloguing‐in‐Publication DataA catalogue record for this book is available from the British Library.
Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at <http://dnb.d‐nb.de>.
© 2022 WILEY‐VCH GmbH, Boschstraße 12, 69469 Weinheim, Germany
All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
Print ISBN: 978‐3‐527‐41410‐9ePDF ISBN: 978‐3‐527‐83725‐0ePub ISBN: 978‐3‐527‐83726‐7
Cover Design: FORMGEBER, Mannheim, Germany
Preface to the First Edition
This book emerged from a course on solid state physics for third‐year students of physics and nanoscience, but it should also be useful for students of related fields such as chemistry and engineering. The aim is to provide a bachelor‐level survey over the whole field without going into too much detail. With this in mind, a lot of emphasis is put on a didactic presentation and little on stringent mathematical derivations or completeness. For a more in‐depth treatment, the reader is referred to the many excellent advanced solid state physics books. A few are listed in the Appendix.
To follow this text, a basic university‐level physics course is required as well as some working knowledge of chemistry, quantum mechanics, and statistical physics. A course in classical electrodynamics is of advantage but not strictly necessary.
Some remarks on how to use this book: Every chapter is accompanied by a set of “discussion” questions and problems. The intention of the questions is to give the student a tool for testing his/her understanding of the subject. Some of the questions can only be answered with knowledge of later chapters. These are marked by an asterisk. Some of the problems are more of a challenge in that they are more difficult mathematically or conceptually or both. These problems are also marked by an asterisk. Not all the information necessary for solving the problems is given here. For standard data, for example, the density of gold or the atomic weight of copper, the reader is referred to the excellent resources available on the World Wide Web.
Finally, I would like to thank the people who have helped me with many discussions and suggestions. In particular, I would like to mention my colleagues Arne Nylandsted Larsen, Ivan Steensgaard, Maria Fuglsang Jensen, Justin Wells, and many others involved in teaching the course in Aarhus.
Preface to the Second Edition
The second edition of this book is slightly enlarged in some subject areas and improved throughout. The enlargement comprises subjects that turned out to be too essential to be missing, even in a basic introduction such as this one. One example is the tight‐binding model for electronic states in solids, which is now added in its simplest form. Other enlargements reflect recent developments in the field that should at least be mentioned in the text and explained on a very basic level, such as graphene and topological insulators.
I decided to support the first edition by online material for subjects that were either crucial for the understanding of this text but not familiar to all readers, or not central enough to be included in the book but still of interest. This turned out to be a good concept, and the new edition is therefore supported by an extended number of such notes; they are referred to in the text. The notes can be found on my website www.philiphofmann.net.
The didactic presentation has been improved, based on the experience of many people with the first edition. The most severe changes have been made in the chapter on magnetism but minor adjustments have been made throughout the book. In these changes, didactic presentation was given a higher priority than elegance or conformity to standard notation, for example, in the figures on Pauli paramagnetism or band ferromagnetism.
Every chapter now contains a “Further Reading” section at the end. Since these sections are supposed to be independent of each other, you will find that the same books are mentioned several times.
I thank the many students and instructors who participated in the last few years' Solid State Physics course at Aarhus University, as well as many colleagues for their criticism and suggestions. Special thanks go to NL architects for permitting me to use the flipper‐bridge picture in Figure 11.3, to Justin Wells for suggesting the analogy to the topological insulators, to James Kermode for Figure 3.7, and to Arne Nylandsted Larsen and Antonija Grubišić Čabo for advice on the sections on solar cells and magnetism, respectively.
Preface to the Third Edition
The third edition of this book introduces numerous improvements throughout the text, in particular in the description of covalent bonding in Chapter 2 and in the discussion of the Bloch theorem and the nearly‐free electron model in Chapter 6.
The most significant changes are related to the problem sections in each chapter. In addition to the “traditional” type of problems that require an analytical solution, I have now included a number of problems that need to be solved numerically. Their complexity varies from plotting a simple function to calculating the carrier densities in a semiconductor. By introducing this new type of problems I hope to strengthen the students' computational skills, to overcome the restriction of being able to calculate solely what can be approximated using a simple model, and to impart upon students the capability to “play” with model parameters in order to explore what might happen in different physical situations. Moreover, exercises such as 4.4 and 6.10 are intended to help students understand the way phonon dispersions or electronic states are plotted as one‐dimensional cuts through a multi‐dimensional Brillouin zone. For instructors, Python scripts for individual problems are provided as part of the instructor resources that are available from the publisher.
Another major change in the problem sections is the addition of a “basic concepts” section in addition to the “discussion questions” and the more complex “problems” from the second edition. Many (but not all) of the new “basic concepts” questions are of the multiple‐choice type and the solutions to all of them are given in Appendix B. As in the first two editions, the “discussion questions” can serve as an inspiration to think about the central new concepts of each chapter or for discussing them in class, whereas the “problems” serve for a more in‐depth exploration of the subjects. As in the previous editions, problems marked by an asterisk * are particularly challenging. The “basic concepts” section can be used in self‐studies to test one's understanding of the most important ideas. Most of the questions do not require any calculations but they still go beyond a simple repetition of the chapter's content and involve some thinking. The number of “basic concepts” questions in a given chapter depends on the number and complexity of new concepts introduced in this chapter. Chapter 1, for instance, introduces difficult and very important ideas such as the reciprocal lattice, and therefore it contains a large number of “basic concepts” questions. Chapter 3, on the other hand, is conceptually less difficult and contains only a few of them. Many more of this type of questions along with their solutions can be found on my website at www.philiphofmann.net.
The multiple‐choice questions have only one correct answer, or, if several correct answers exist, there is an explicit option to choose this, e. g., “C. Both A. and B. are correct.” In some cases, there is an overlap between a “basic concepts” question testing a conceptual understanding of a subject and a “problem” with a more in‐depth treatment of the same question.
My thanks go to the 2021 class of the Statistical Physics and Solid State Physics course at Aarhus University for testing much of the new content, as well as to the teaching instructors Paulina Majchrzak, Alfred Jones, Michael Iversen and Nikolaj Rønne. I also thank Davide Curcio for introducing me to a new set of advanced writing tools and Charlotte E. Sanders for many helpful comments on the manuscript.
Physical Constants and Energy Equivalents
Planck constant
J s
eV s
Boltzmann constant
J K
eV K
Proton charge
C
Bohr radius
m
Bohr magneton
J T
Avogadro constant
mol
Speed of light
m s
Rest mass of the electron
1Crystal Structures
Our general objective in this book is to understand the macroscopic properties of solids on a microscopic level. In view of the many particles in solids, coming up with any microscopic description appears to be a daunting task. It is clearly impossible to solve the equations of motion (classical or quantum‐mechanical) of the particles. Fortunately, it turns out that solids are often crystalline, with the atoms arranged on a regular lattice, and this symmetry permits us to solve microscopic models despite the vast number of particles involved. In a way, this situation is similar to atomic physics where the key to a quantum‐mechanical description is the spherical symmetry of the atom. We will often imagine a macroscopic solid as one single crystal, a perfect lattice of atoms without any defects whatsoever. While it may seem that such perfect crystals are not particularly relevant for real materials, this is in fact not the case. Many solids are actually composed of small crystalline grains. Such solids are called polycrystalline, in contrast to a macroscopic single crystal, but the number of atoms within a perfect crystalline environment in them is still very large compared to the number of atoms on the grain boundary. For instance, for a grain size on the order of atomic distances, only about 0.1% of all atoms are at the grain boundaries.
There are, however, also solids that are not crystalline. These are called amorphous. The amorphous state is characterized by the absence of any long‐range order. There may exist, however, a degree of short‐range order between the atoms.
This chapter is divided into three parts. In the first part, we define some basic mathematical concepts needed to describe crystals. We keep things simple and mostly use two‐dimensional examples to illustrate the ideas. In the second part, we discuss common crystal structures. For the moment, we will not ask why the atoms bind together in the way they do – this topic will be discussed in Chapter 2. Finally, we delve into a more detailed discussion of X‐ray diffraction, the experimental technique that can be used to determine the microscopic structure of crystals. X‐ray diffraction is used not only in solid state physics but also for a wide range of problems in nanotechnology and structural biology.
1.1 General Description of Crystal Structures
Our description of crystals starts with the mathematical definition of the lattice. A lattice is a set of regularly spaced points with positions defined as multiples of generating vectors. In two dimensions, a lattice can be defined as all the points that can be reached by the vectors , created from two non‐collinear vectors and as
where and are integers. In three dimensions, the corresponding definition is
Such a lattice of points is also called a Bravais lattice. The number of possible Bravais lattices with different symmetries is limited to 5 in two dimensions and to 14 in three dimensions. An example of a two‐dimensional Bravais lattice is given in Figure 1.1. The lengths of the vectors and are often called the lattice constants.
Having defined the Bravais lattice, we move on to the definition of the primitive unit cell. By this we denote any volume of space that, when translated through all the vectors of the Bravais lattice, will fill space without overlap and without leaving any voids. The primitive unit cell of a lattice contains only one lattice point. It is also possible to define nonprimitive unit cells containing several lattice points. These fill space without leaving voids when translated through a subset of the Bravais lattice vectors. Possible choices of a unit cell for a two‐dimensional rectangular Bravais lattice are illustrated in Figure 1.2. It is evident from the figure that a nonprimitive unit cell has to be translated by a multiple of one (or two) lattice vectors to fill space without voids and overlap. A special choice of the primitive unit cell is the Wigner–Seitz cell, which is also shown in Figure 1.2. It is the region of space that is closer to one given lattice point than to any other.
The last definition we need in order to describe an actual crystal is that of a basis. The basis describes the items we “put” on the lattice points, that is, the building blocks for the real crystal. The basis can consist of one or several atoms, or even of complex molecules as in the case of protein crystals. Different cases are illustrated in Figure 1.3.
Figure 1.1 A two‐dimensional Bravais lattice.
Figure 1.2 Illustration of (primitive and nonprimitive) unit cells and of the Wigner–Seitz cell for a rectangular two‐dimensional lattice.
Figure 1.3 A two‐dimensional Bravais lattice with different choices for the basis.
Finally, we add a remark about symmetry. So far, we discussed only translational symmetry. However, a real crystal may also exhibit point symmetry. Compare the structures in the middle and the bottom of Figure 1.3. The former structure possesses a number of symmetry elements that are missing in the latter – for example, mirror lines, a rotational axis, and inversion symmetry. The knowledge of such symmetries can be very useful for the description of crystal properties.
1.2 Some Important Crystal Structures
After this rather formal treatment, we look at a number of common crystal structures for different types of solids, such as metals, ionic solids, or covalently bonded solids. In Chapter 2, we will take a closer look at the details of the bonding in these types of solids.
Figure 1.4 (a) Simple cubic structure; (b) body‐centered cubic structure; and (c) face‐centered cubic structure. Note that the spheres are depicted much smaller than in the situation of most dense packing and not all of the spheres on the faces of the cube are shown in (c).
1.2.1 Cubic Structures
We begin with one of the simplest crystal structures possible, the simple cubic structure shown in Figure 1.4a. This structure is not very common among elemental solids, but it is an important starting point for understanding many other structures. Only one chemical element (polonium) is found to crystallize in the simple cubic structure. The structure is unfavorable because of its openness – there are many voids, if we think of the atoms as solid spheres in contact with each other. In metals, which are the most common elemental solids, directional bonding is not important, and a close packing of the atoms is usually favored. We will learn more about this in the next chapter. For covalent solids, on the other hand, directional bonding is important, but six bonds extending from the same atom in an octahedral configuration is highly uncommon in elemental solids.
The packing density of the cubic structure is improved in the body‐centered cubic (bcc) and face‐centered cubic (fcc) structures that are also depicted in Figure 1.4. In fact, the fcc structure has the highest possible packing density for identical spheres, as we shall see later. These two structures are very common – 17 elements crystallize in the bcc structure and 24 elements in the fcc structure. Note that the simple cubic structure ist the only one for which the cube is identical with the Bravais lattice. While the cube is also a unit cell for the bcc and fcc lattices, ist it not the primitive unit cell in these cases. Still, both structures are Bravais lattices with a basis containing one atom, but the vectors spanning these Bravais lattices are not the edges of the cube.
Cubic structures with a more complex basis than a single atom are also important. Figure 1.5 shows the structures of the ionic crystals CsCl and NaCl, which are both cubic with a basis containing two atoms. For CsCl, the structure can be thought of as two simple cubic structures stacked into each other. For NaCl, it consists of two fcc lattices stacked into each other. Which structure is preferred for such ionic crystals depends on the relative size of the positive and negative ions.
Figure 1.5 Structures of CsCl and NaCl. The spheres are depicted much smaller than in the situation of dense packing, but the relative size of the different ions in each structure is correct.
1.2.2 Close‐Packed Structures
Many metals prefer structural arrangements where the atoms are packed as closely as possible. In two dimensions, the closest possible packing of atoms (i.e. spheres) is the hexagonal structure shown on the left‐hand side of Figure 1.6. To build a three‐dimensional close‐packed structure, one adds a second layer as in the middle of Figure 1.6. Now there are two possibilities, however, for adding a third layer. We can either put the atoms in the “holes” just on top of the first‐layer atoms, or we can put them into the other type of “holes.” The result are two different crystal structures. The first has an ABABAB… layer stacking sequence, the second an ABCABCABC… layer stacking sequence. Both have exactly the same packing density with the spheres filling about 74% of the total volume. The former structure is called the hexagonal close‐packed structure (hcp), and the latter turns out to be the fcc structure we already know. An alternative sketch of the hcp structure is shown in Figure 1.16b. The fcc and hcp structures are very common in elemental metals, 36 chemical elements crystallizing in hcp and 24 in fcc lattices. These structures also maximize the number of nearest neighbors for a given atom, the so‐called coordination number. For both the fcc and the hcp lattices, the coordination number is 12.
Figure 1.6 Close packing of spheres leading to the hcp and fcc structures.
It is as yet an unresolved question why not all metals crystallize in the fcc or hcp structures, if coordination is indeed so important. Whereas a prediction of the actual structure for a given element is not possible on the basis of simple arguments, we can identify some factors that play a role. For example, structures that are not optimally packed, such as the bcc structure, have a lower coordination number, but they bring the second‐nearest neighbors much closer to a given atom than in the close‐packed structures. Another important consideration is that the bonding situation is often not quite so simple, particularly in transition metals. In these, bonding is not only achieved through the delocalized s and p valence electrons as in simple metals, but also by the more localized d electrons. Bonding through the latter results in a much more directional character so that not only the close packing of the atoms is important.
The structures of many ionic solids can also be viewed as “close‐packed” in some sense. One can derive these structures by treating the ions as hard spheres that have to be packed as closely to each other as possible.
1.2.3 Structures of Covalently Bonded Solids
In covalent structures, the valence electrons of the atoms are not completely delocalized but shared between neighboring atoms, and bond lengths and directions are far more important than the packing density. Prominent examples are graphene, graphite, and diamond as displayed in Figure 1.7. Graphene is a single sheet of carbon atoms in a honeycomb lattice structure. It is a truly two‐dimensional solid with a number of remarkable properties – so remarkable, in fact, that their discovery has lead to the 2010 Nobel prize in physics being awarded to A. Geim and K. Novoselov. The carbon atoms in graphene are connected through a network of sp hybrid bonds enclosing angles of 120. The parent material of graphene is graphite, which consists of a stack of graphene sheets that are weakly bonded to each other. In fact, graphene can be isolated from graphite by peeling off flakes with a piece of scotch tape. In diamond, the carbon atoms form sp‐type bonds and each atom has four nearest neighbors in a tetrahedral configuration. Interestingly, the diamond structure can also be described as an fcc Bravais lattice with a basis of two atoms.
Figure 1.7 Structures for (a) graphene, (b) graphite, and (c) diamond. Bonds from sp and sp orbitals are displayed as solid lines.
The diamond structure is also found for Si and Ge. Many other isoelectronic materials (i.e. materials with the same total number of valence electrons), such as SiC, GaAs, or InP, also crystallize in a diamond‐like structure but with each element on a different fcc sublattice.
1.3 Crystal Structure Determination
After having described different crystal structures, the question is of course how to determine these structures in the first place. By far the most important technique for this is X‐ray diffraction. In fact, the importance of this technique extends far beyond solid state physics, as it has become an essential tool for fields such as structural biology as well. In biology, the idea is that you can derive the structure of a given protein by trying to crystallize it and then use the powerful methodology of X‐ray diffraction to determine its structure. In addition, we will also use X‐ray diffraction as a motivation to extend our formal description of structures.
1.3.1 X‐Ray Diffraction
X‐rays interact rather weakly with matter. A description of X‐ray diffraction can therefore be restricted to single scattering, meaning that we limit our analysis to the case that X‐rays incident upon a crystal sample get scattered not more than once (most are not scattered at all). This is called the kinematic approximation; it greatly simplifies matters and is used throughout the treatment in this book. Furthermore, we will assume that the X‐ray source and detector are placed very far away from the sample so that the incoming and outgoing waves can be treated as plane waves. X‐ray diffraction of crystals was discovered and described by M. von Laue in 1912. Also in 1912, W. L. Bragg came up with an alternative description that is considerably simpler and will serve as a starting point for our analysis.
1.3.1.1 Bragg Theory
Bragg treated the problem as the reflection of the incident X‐rays at flat crystal planes. These planes could, for example, be the close‐packed planes making up fcc and hcp crystals, or they could be alternating Cs and Cl planes making up the CsCl structure. At first glance, the physical justification for this picture seems somewhat dubious, because the crystal planes appear certainly not “flat” for X‐rays with wavelengths on the order of atomic spacing. Nevertheless, the description proved highly successful, and we shall later see that it is actually a special case of the more complex Laue description of X‐ray diffraction.
Figure 1.8 shows the geometrical considerations behind the Bragg description. A collimated beam of monochromatic X‐rays hits the crystal. The intensity of diffracted X‐rays is measured in the specular direction. The angles of incidence and emission are 90. The condition for constructive interference is that the path length difference between the X‐rays reflected from one layer and the next layer is an integer multiple of the wavelength . In the figure, this means that , where is the distance between points A and B and is a natural number. On the other hand, we have , which leads us to the Bragg condition
It is obvious that if this condition is fulfilled for one specific layer and the layer below it, then it will also be fulfilled for any number of layers with identical spacing. In fact, the X‐rays penetrate very deeply into the crystal so that thousands of layers contribute to the reflection. This results in very sharp maxima in the diffracted intensity, similar to the situation for an optical grating with many lines. The Bragg condition can obviously only be fulfilled for , putting an upper limit on the wavelength of the X‐rays that can be used for crystal structure determination.
Figure 1.8 Construction for the derivation of the Bragg condition. The horizontal lines represent the crystal lattice planes that are separated by a distance . The heavy lines represent the X‐rays.
1.3.1.2 Lattice Planes and Miller Indices
Obviously, the Bragg condition will be satisfied not only for a special kind of lattice plane in a crystal, such as the hexagonal planes in an hcp crystal, but for all possible parallel planes in a structure. Thus, we need a more precise definition of the term lattice plane. It proves useful to define a lattice plane as a plane containing at least three non‐collinear lattice points of a given Bravais lattice. If it contains three points, it will actually contain infinitely many because of the translational symmetry of the lattice. Examples for lattice planes in a simple cubic structure are shown in Figure 1.9.
Figure 1.9 Three different lattice planes in the simple cubic structure characterized by their Miller indices.
Following this definition, all lattice planes can be characterized by a set of three integers, the so‐called Miller indices. We derive them in three steps:
We find the intercepts of the specific plane at hand with the crystallographic axes in units of the lattice vectors, for example,
for the leftmost plane in
Figure 1.9
.
We take the “reciprocal value” of these three numbers. For our example, this gives
.
We multiply the numbers obtained in this manner with some factor so that we arrive at the smallest set of integers having the same ratio. In the example given, this is not necessary as all number are already integers.
Such a set of three integers can then be used to denote any given lattice plane. Later, we will encounter a different and more elegant definition of the Miller indices.
In practice, the X‐ray diffraction peaks are so sharp that it is difficult to align and move the sample so that the incoming and reflected X‐rays lie in a plane normal to a certain crystal plane. An elegant way to circumvent this problem is to use a powder consisting of very small crystals instead of a large single crystal. This will not only ensure that some of the many crystals are oriented correctly to get constructive interference from a certain set of crystal planes, it will also automatically yield the interference pattern for all possible crystal planes.
1.3.1.3 General Diffraction Theory
The Bragg theory for X‐ray diffraction is useful for extracting the distances between lattice planes in a crystal, but it has its limitations. Most importantly, it does not provide any information on what the lattice actually consists of, that is, the basis. Also, the fact that the X‐rays are described as being reflected by planes is physically somewhat obscure. In the following, we will therefore discuss a more general description of X‐ray diffraction that goes back to M. von Laue.
The physical process leading to X‐ray scattering is that the electromagnetic field of the X‐rays forces the electrons in the material to oscillate with the same frequency as that of the field. The oscillating electrons then emit new X‐rays that give rise to an interference pattern. For the following discussion, however, it is merely important that something scatters the X‐rays, not what it is.
It is highly beneficial to use the complex notation for describing the electromagnetic X‐ray waves. For the electric field, a general plane wave can be written as
The wave vector points in the direction of the wave propagation with a length of , where is the wavelength. The convention is that the physical electric field is obtained as the real part of the complex field and the intensity of the wave is obtained as
Consider now the situation depicted in Figure 1.10. The source of the X‐rays is far away from the sample at the position so that the X‐ray wave at the sample can be described as a plane wave. The electric field at a point in the crystal at time can be written as
Before we proceed, we can drop the absolute amplitude from this expression because we are only interested in relative phase changes. The field at point is then
Figure 1.10 Illustration of X‐ray scattering from a sample. The source and detector for the X‐rays are placed at and , respectively. Both are very far away from the sample.
A small volume element located at will give rise to scattered waves in all directions. The direction we are interested in is that towards the detector, which we assume to be placed at position , in the direction of a second wave vector . We assume that the amplitude of the wave scattered in this direction will be proportional to the incoming field from Eq. (1.7) and to a factor describing the scattering probability and scattering phase. We already know that the scattering of X‐rays proceeds via the electrons in the material, and for our purpose, we can view as the electron concentration in the solid. For the field at the detector, we obtain
Again, we have assumed that the detector is very far away from the sample so that the scattered wave at the detector can be written as a plane wave. Inserting Eq. (1.7) gives the field at the detector as
We drop the first factor that does not contain and will thus not play a role for the interference of X‐rays emitted from different positions in the sample. The total wave field at the detector can finally be calculated by integrating over the entire volume of the crystal. As the detector is far away from the sample, the wave vector is essentially the same for all points in the sample. The result is therefore
In most cases, it will only be possible to measure the intensity of the X‐rays and not the field amplitude. This intensity is given by
where we have introduced the so‐called scattering vector , which is just the difference of the outgoing and incoming wave vectors. Note that although the direction of the wave vector for the scattered waves is different from that of the incoming wave , their lengths are the same because we consider elastic scattering only.
Equation 1.11 is our final result. It relates the measured intensity to the electron concentration in the sample. Except for very light elements, most of the electrons are located close to the ion cores and the electron concentration that scatters the X‐rays is essentially identical to the geometrical arrangement of the atomic cores. Hence, Eq. (1.11) can be used for the desired structure determination. To this end, one could try to measure the intensity as a function of scattering vector and to infer the structure from the result. This is a formidable task, however. It is greatly simplified by the fact that the specimen under investigation is a crystal with a periodic lattice. In the following, we introduce the mathematical tools and concepts that are needed to exploit the crystalline structure in the analysis. The most important of these is the so‐called reciprocal lattice.
1.3.1.4 The Reciprocal Lattice
The concept of the reciprocal lattice is fundamental to solid state physics because it permits us to exploit crystal symmetry in the analysis of many problems. Here, we will use it to describe X‐ray diffraction from periodic structures and we will continue to meet it again in the following chapters. Unfortunately, the meaning of the reciprocal lattice turns out to be difficult to grasp. We will start out with a formal definition and provide some of its mathematical properties. We then go on to discuss the meaning of the reciprocal lattice before we come back to X‐ray diffraction. The full importance of the concept will become apparent in the course of this book.
For a given Bravais lattice
we define the reciprocal lattice as the set of vectors for which
where is an integer. Equivalently, we could require that
Note that this equation must hold for any choice of the lattice vector and reciprocal lattice vector . We can write any as the linear combination of three vectors
where , and are integers. The reciprocal lattice is also a Bravais lattice. The vectors , , and spanning the reciprocal lattice can be constructed explicitly from the lattice vectors 1
From this, one can derive the simple but useful property2
which can easily be verified. Equation (1.17) can then be used to verify that the reciprocal lattice vectors defined by Eqs. (1.15) and (1.16) do indeed fulfill the fundamental property of Eq. (1.13) defining the reciprocal lattice (see also Problem 6).
Another way to view the vectors of the reciprocal lattice is as wave vectors that yield plane waves with the periodicity of the Bravais lattice, because
Using the reciprocal lattice, we can finally define the Miller indices in a much simpler way: The Miller indices define a plane that is perpendicular to the reciprocal lattice vector (see Problem 9).
1.3.1.5 The Meaning of the Reciprocal Lattice
We have now defined the reciprocal lattice in a proper way, and we will give some simple examples of its usefulness. The most important feature of the reciprocal lattice is that it facilitates the description of functions with the same periodicity as that of the lattice. To see this, consider a one‐dimensional lattice, a chain of points with a lattice constant (Fig. 1.11). We are interested in a function with the periodicity of the lattice, such as the electron concentration along the chain, . We can write this as a Fourier series of the form
with real coefficients and . The sum starts at , the constant is therefore outside the summation. Using complex coefficients , we can also write this in the more compact form
To ensure that is still a real function, we require that
that is, that the coefficient must be the complex conjugate of the coefficient . This description is more elegant than the one with the sine and cosine functions. How is it related to the reciprocal lattice? In one dimension, the reciprocal lattice of a chain of points with lattice constant is also a chain of points, now with spacing [see Eq. (1.17)]. This means that we can write a general reciprocal lattice “vector” as
where is an integer. Exactly these reciprocal lattice “vectors” appear in Eq. (1.20). In fact, Eq. (1.20) is a sum of functions with a periodicity corresponding to the lattice vector, weighted by the coefficients . Figure 1.11 illustrates these ideas by showing the lattice and reciprocal lattice for such a chain as well as two lattice‐periodic functions, both in real space and as Fourier coefficients on the reciprocal lattice points. The advantage of describing these functions by the coefficients is immediately obvious: Instead of giving for every point in a range of , the Fourier description consists of just three numbers for the upper function and five numbers for the lower function. Actually, these even reduce to two and three numbers, respectively, because of Eq. (1.21).
Figure 1.11 Top: A chain with a lattice constant as well as its reciprocal lattice, a chain with a spacing of . Middle and bottom: Two lattice‐periodic functions in real space as well as their Fourier coefficients. The magnitude of the Fourier coefficients is plotted on the reciprocal lattice vectors they belong to.
The same ideas also work in three dimensions. In fact, one can use a Fourier sum for lattice‐periodic properties that corresponds to Eq. (1.20). For the lattice‐periodic electron concentration , we get
where are the reciprocal lattice vectors.
Thus we have seen that the reciprocal lattice is very useful for describing lattice‐periodic functions. But this is not the whole story: The reciprocal lattice can also simplify the treatment of waves in crystals in a very general sense. Such waves can be X‐rays, elastic lattice distortions, or even electronic wave functions. We will come back to this point at a later stage.
1.3.1.6 X‐Ray Diffraction from Periodic Structures
Turning back to the specific problem of X‐ray diffraction, we can now exploit the fact that the electron concentration is lattice‐periodic by inserting Eq. (1.23) in our expression from Eq. (1.11) for the diffracted intensity. This yields
Let us inspect the integrand. The exponential represents a plane wave with a wave vector . If the crystal is very big, the integration will average over the crests and troughs of this wave and the result of the integration will be very small (or zero for an infinitely large crystal). The only exception to this is the case where
that is, when the difference between incoming and scattered wave vector equals a reciprocal lattice vector. In this case, the exponential in the integral is 1, and the value of the integral is equal to the volume of the crystal. Equation 1.25 is often called the Laue condition. It is central to the description of X‐ray diffraction from crystals in that it describes the condition for the observation of constructive interference.
Looking back at Eq. (1.24), the observation of constructive interference for a chosen scattering geometry (or scattering vector ) clearly corresponds to a particular reciprocal lattice vector . The intensity measured at the detector is proportional to the square of the Fourier coefficient of the electron concentration
