R?ssler U. Solid State Theory: An Introduction
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Solid State Theory: An Introduction
U. Rössler
Solid State Theory:
An Introduction
Second revised and extended edition
With 123 Figures and 6 Tables
Prof. Dr. h.c. Ulrich Rössler
Universität Regensburg
Inst. Theoretische Physik
Universitätsstr. 31
93040 Regensburg
Germany
ISBN 978-3-540-92761-7 e-ISBN 978-3-540-92762-4
DOI 10.1007/978-3-540-92762-4
Springer Dordrecht Heidelberg London New York
Library of Congress Control Number : 2009929708
c
Springer-Verlag Berlin Heidelberg 2009
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To Erika
Preface to Second Edition
Given the fact that the first edition of this book is almost sold out, the editor
has a sked for a second edition. I readily accepted this opportunity, because –
as many users and I myself had found out in the meantime – there were quite
a lot of typing and other errors in the first edition which had to be eliminated.
But more important, I saw the chance to extend the book by a new chapter
on light–matter interaction, which complements the canon of the book with
respect to both the physics of this interaction and the theoretical methods
used for its description. Moreover, during the four years since the first edition
the whole field has developed as can be seen in the list of references which
now contains about 30 new titles.
Part of the contents of this book has b een presented during a guest lecture
at the National Mechnikov University in Odessa in the spring term of 2007.
I would like to acknowledge the hospitality of the Department of Theoretical
Physics and its head, Prof. V.M. Adamyan, and the contacts with the students
who attended this one as their first lecture in English. Finally, I appreciate
the support by and easy cooperation with the publisher.
Regensburg, Ulrich R¨ossler
June 2009
Preface to First Edition
The history of my involvement in this book project starts more than 30 years
ago. During the years 1969–1972, my thesis advisor, Otfried Madelung, wrote
a series of three textbook s on Solid State Theory, entitled Festk¨orpertheorie
I–III, which app eared in the Springer pap erback series Heidelberger Taschen-
b¨ucher. My fellow graduate students and friends Manfred Lietz, Rolf Sandrock,
Joachim Treusch, and I, were the first to proof-read these books. Better still,
we were given the unique opportunity to provide input based on insights
gained during our studies. In 1978, when Festk¨orpertheorie I–III were partly
rewritten and translated into English, Otfried Madelung again asked his for-
mer disciples, then already established in university positions, for comments
and contributions based o n their respective research and teaching experience
in this field. The result, entitled Introduction to Solid-State Theory, became a
widely used textbook, published in several further editions over the following
years.
Like most textbooks on Solid-State Theory currently used in university
physics courses all over the world, Introduction to Solid-State Theory has
meanwhile become somewhat outdated. Solid State Physics has evolved sig-
nificantly and many topics, which 30 years ago were still the subject of active
research or even beyond its leading edge, have now become part of our stan-
dard knowledge. The idea of accounting for this development in a textbook
has been lingering in the collective mind of the Solid State Physics commu-
nit y for quite a while, but it took an initiative by Springer to concretize the
project. When Springer editors asked Otfried Madelung to rework on Intro-
duction to Solid-State Theory accordingly, he convinced them that it would
make more sense to write a completely new book, proposing me as a potential
author. This is how I got involved.
Due to the formative influence of Otfried Madelung and his approach to
science, my research in Solid State Theory has from the very beginning been
oriented towards experimental work, often directly stimulated by concrete
experimental results. This tendency was solidified during a year as postdoc-
toral researcher with Manuel Ca rdona at Brown University in Providence,
XPreface
RI, USA. Quite commonly, my research projects were initiated by discussions
with researchers renowned for their experimental work and have frequently
been conducted in fruitful cooperations. It is to this continuous contact with
the physics reality that I owe the down-to-ea rth approach which characterizes
my research and teaching and which should also be noticeable in this book.
At the University of Regensburg, where I became professor in 1972, Solid
State Physics has been a strong research field, both in experiment and theory.
Over the years the topics evolved from magnetism, phase transitions, lattice
dynamics, and electronic structure of bulk material to the upcoming fields of
high-T
c
superconductors, correlated electron systems, surface physics, quan-
tum wells, nano-structures, and composite materials. Quite naturally, Solid
State Theory has been a standard part of the physics curriculum in Regens-
burg. It started as a one-semester course with four weekly lectures in the
fourth year of the German diploma curriculum (corresponding to the first
year of the graduate education in the Anglo-American system). Soon it was
supplemented by a second course on special topics, with the purpose of guid-
ing the students into active research fields. For more than 30 years, I taught
these courses on a regular basis, taking turns with my colleagues Joachim
Keller, Uwe Krey, Ulrich Schr¨oder, and Dieter Strauch. The exchange of teach-
ing concepts and problems with these colleagues, and also our joint research
projects considerably enriched my lectures. During the last decade I benefit-
ted much fr om the expertise of my senior coworker Michael Suhrke. Further,
important input came from discussions with many colleagues from all over
the world during conferences, visits, and sabbaticals in different places. My
lecture notes for these courses, accumulated and continuously modified over
the years, constitute the backbone of this book.
Clearly, the book follows a well-defined tradition. Target readers are those
students in physics or material science who are interested in understanding the
theoretical approach to Solid State Physics, while maintaining contact to the
experimental facts. The contents are essentially comparable to those o f other
textbooks on the same subject, but emphasis is put on new aspects of the field
that have resulted from more recent research. Extensive references to related
literature in the form of textbo oks, topical series, data collections, and selected
origina l papers ar e provided to establish the connection with the sources of this
subject and with active research fields. Each chapter contains a selection of
problems and solutions, wh ich are meant to help the read er gain practice with
the concepts and the physics explai n ed in the text. Since the number o f pages
is restricted, this book cannot claim completeness. Nevertheless, wherever
possible, reference is g iven to those important topics that could be covered
here only briefly or not at all. In short, this book is intended as an introduction
to Solid State Theory, given from the perspective of more than 30 years of
learning, teaching, an d research in this field.
It is a pleasure to thank all those who contributed in one way or the other to
this project. I have already mentioned some friends and colleagues and would
like to extend my acknowledgments to the students who attended my courses
Preface XI
and enriched them by their constructive feedback. Th is applies especially to
my diploma and Ph.D. students, who contributed ideas during many hours of
discussion about their research projects. A highly visible contribution to this
book came from Ingeburg Zirkl who prepared all the figures. A critical reading
of parts of the manuscript by my friends and colleagues Joachim Keller and
Dieter Strauch, and by my son Thomas has led to considerable improvements
of the contents and the text. Finally, I express my gratitude to Springer, in
particular to Dr. Claus Ascheron and Dr. Angela Lahee, not only for their
expert help and advice but also for their patience in waiting for the final
version of this book.
Regensburg, Ulrich R¨ossler
April 2004
Contents
List of Symbols .............................................. XVII
1 Introduction ............................................. 1
1.1 Aims and Outline ..................................... 2
1.2 The Structure of Solids ................................. 4
1.3 Pair-Distribution Function and Structure Factor ........... 9
Problems ................................................. 12
2 The Solid as a Many-Particle Problem ................... 15
2.1 The Hamiltonian of the Solid ........................... 16
2.2 Separating the Motion of Electrons and Ions .............. 20
2.3 Thermal Expectation Values ............................ 22
2.4 Theory o f Linear Response ............................. 24
2.5 Kubo’s Formulas: Resp onse Functions .................... 28
2.6 Properties of Response Functions ........................ 32
Problems ................................................. 36
3 Lattice Dynamics: Phonons .............................. 37
3.1 Harmonic Approximation ............................... 39
3.2 Normal Coordinates ................................... 42
3.3 Phonons and Occupation Number Representation ......... 44
3.4 Acoustic Phonons ..................................... 48
3.5 Optical Phonons ...................................... 55
3.6 Examples: Phonon Dispersion Curves .................... 60
3.7 The Crystal Lattice at Finite Temperature ............... 68
Problems ................................................. 73
4 The Free Electron Gas ................................... 75
4.1 Free Electrons Without Interaction ...................... 76
4.2 Free Electrons in a Magnetic Field ....................... 84
4.3 Occupation Number Representation for Electrons .......... 93
XIV Contents
4.4 Hartree–Fock Approximation ........................... 97
4.5 The Dielectric Function ................................ 102
4.6 Discussion of the Dielectric Function ..................... 107
4.7 Electronic Correlation .................................. 110
Problems ................................................. 115
5 Electrons in a Periodic Potential ......................... 119
5.1 Density Functional Theory ............................. 120
5.2 Bloch Electrons and Band Structure ..................... 127
5.3 Almost Free Electrons and Pseudo-Potentials ............. 129
5.4 LCA O and Tight-Binding Approximation ................. 139
5.5 Effective-Mass Approximation ........................... 147
5.6 Subbands in Semiconductor Quantum Structures .......... 152
Problems ................................................. 157
6 Spin Waves: M agnons ................................... 161
6.1 Preliminaries.......................................... 162
6.2 The Heisenberg Hamiltonian ............................ 166
6.3 Spin Waves in Ferromagnets ............................ 170
6.4 Spin Waves in Anti-Ferromagnets........................ 175
6.5 Molecular Field Approximation ......................... 179
6.6 Itinerant Electron Magnetism ........................... 185
Problems ................................................. 191
7 Correlated Electrons ..................................... 195
7.1 Retarded Green Function for Electrons ................... 196
7.2 The Hubbard Model ................................... 201
7.3 Fermi Liquids ......................................... 208
7.4 Luttinger Liquids ...................................... 213
7.5 Heavy Fermion Systems ................................ 217
7.6 Fractional Quant um Hall States ......................... 222
Problems ................................................. 228
8 Electron–Phonon Interaction ............................ 231
8.1 Preliminaries.......................................... 232
8.2 Coupling Mechanisms .................................. 235
8.3 Scattering Processes: Lifetime, Relaxation ................ 240
8.4 The Fr¨ohlich Polaron .................................. 247
8.5 Effective Electron–Electron Interaction ................... 250
8.6 Coop er Pairs and the Gap .............................. 255
Problems ................................................. 263