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© Martin Dressel and George Grüner 2002
This edition © Martin Dressel and George Grüner 2003
First published in printed format 2002
A catalogue record for the original printed book is available
from the British Library and from the Library of Congress
Original ISBN 0 521 59253 4 hardback
Original ISBN 0 521 59726 9 paperback
ISBN 0 511 01439 2 virtual (netLibrary Edition)
Contents
Preface page xi
1 Introduction 1
PART ONE
: CONCEPTS AND PROPERTIES 7
Introductory remarks 7
General books and monographs 8
2 The interaction of radiation with matter 9
2.1 Maxwell’s equations for time-varying fields 9
2.1.1 Solution of Maxwell’s equations in a vacuum 10
2.1.2 Wave equations in free space 13
2.2 Propagation of electromagnetic waves in the medium 15
2.2.1 Definitions of material parameters 15
2.2.2 Maxwell’s equations in the presence of matter 17
2.2.3 Wave equations in the medium 19
2.3 Optical constants 21
2.3.1 Refractive index 21
2.3.2 Impedance 28
2.4 Changes of electromagnetic radiation at the interface 31
2.4.1 Fresnel’s formulas for reflection and transmission 31
2.4.2 Reflectivity and transmissivity by normal incidence 34
2.4.3 Reflectivity and transmissivity for oblique incidence 38
2.4.4 Surface impedance 42
2.4.5 Relationship between the surface impedance and the reflectivity 44
References 45
Further reading 46
3 General properties of the optical constants 47
3.1 Longitudinal and transverse responses 47
v
vi Contents
3.1.1 General considerations 47
3.1.2 Material parameters 49
3.1.3 Response to longitudinal fields 52
3.1.4 Response to transverse fields 55
3.1.5 The anisotropic medium: dielectric tensor 55
3.2 Kramers–Kronig relations and sum rules 56
3.2.1 Kramers–Kronig relations 57
3.2.2 Sum rules 65
References 69
Further reading 70
4 The medium: correlation and response functions 71
4.1 Current–current correlation functions and conductivity 72
4.1.1 Transverse conductivity: the response to the vector potential 73
4.1.2 Longitudinal conductivity: the response to the scalar field 78
4.2 The semiclassical approach 79
4.3 Response function formalism and conductivity 81
4.3.1 Longitudinal response: the Lindhard function 81
4.3.2 Response function for the transverse conductivity 87
References 91
Further reading 91
5 Metals 92
5.1 The Drude and the Sommerfeld models 93
5.1.1 The relaxation time approximation 93
5.1.2 Optical properties of the Drude model 95
5.1.3 Derivation of the Drude expression from the Kubo formula 105
5.2 Boltzmann’s transport theory 106
5.2.1 Liouville’s theorem and the Boltzmann equation 107
5.2.2 The q = 0 limit 110
5.2.3 Small q limit 110
5.2.4 The Chambers formula 112
5.2.5 Anomalous skin effect 113
5.3 Transverse response for arbitrary q values 115
5.4 Longitudinal response 120
5.4.1 Thomas–Fermi approximation: the static limit for q < k
F
120
5.4.2 Solution of the Boltzmann equation: the small q limit 122
5.4.3 Response functions for arbitrary q values 123
5.4.4 Single-particle and collective excitations 130
5.5 Summary of the ω dependent and q dependent response 132
References 133
Further reading 134
Contents vii
6 Semiconductors 136
6.1 The Lorentz model 137
6.1.1 Electronic transitions 137
6.1.2 Optical properties of the Lorentz model 141
6.2 Direct transitions 148
6.2.1 General considerations on energy bands 148
6.2.2 Transition rate and energy absorption for direct transitions 150
6.3 Band structure effects and van Hove singularities 153
6.3.1 The dielectric constant below the bandgap 154
6.3.2 Absorption near to the band edge 155
6.4 Indirect and forbidden transitions 159
6.4.1 Indirect transitions 159
6.4.2 Forbidden transitions 162
6.5 Excitons and impurity states 163
6.5.1 Excitons 163
6.5.2 Impurity states in semiconductors 165
6.6 The response for large ω and large q 169
References 171
Further reading 171
7 Broken symmetry states of metals 173
7.1 Superconducting and density wave states 173
7.2 The response of the condensates 179
7.2.1 London equations 180
7.2.2 Equation of motion for incommensurate density waves 181
7.3 Coherence factors and transition probabilities 182
7.3.1 Coherence factors 182
7.3.2 Transition probabilities 184
7.4 The electrodynamics of the superconducting state 186
7.4.1 Clean and dirty limit superconductors, and the spectral weight 187
7.4.2 The electrodynamics for q = 0 188
7.4.3 Optical properties of the superconducting state:
the Mattis–Bardeen formalism 190
7.5 The electrodynamics of density waves 196
7.5.1 The optical properties of charge density waves: the Lee–Rice–
Anderson formalism 197
7.5.2 Spin density waves 198
7.5.3 Clean and dirty density waves and the spectral weight 199
References 202
Further reading 203
viii Contents
PART TWO: METHODS 205
Introductory remarks 205
General and monographs 206
8 Techniques: general considerations 207
8.1 Energy scales 207
8.2 Response to be explored 208
8.3 Sources 210
8.4 Detectors 212
8.5 Overview of relevant techniques 214
References 215
Further reading 216
9 Propagation and scattering of electromagnetic waves 217
9.1 Propagation of electromagnetic radiation 218
9.1.1 Circuit representation 218
9.1.2 Electromagnetic waves 221
9.1.3 Transmission line structures 223
9.2 Scattering at boundaries 230
9.2.1 Single bounce 231
9.2.2 Two interfaces 233
9.3 Resonant structures 234
9.3.1 Circuit representation 236
9.3.2 Resonant structure characteristics 238
9.3.3 Perturbation of resonant structures 241
References 243
Further reading 243
10 Spectroscopic principles 245
10.1 Frequency domain spectroscopy 246
10.1.1 Analysis 246
10.1.2 Methods 247
10.2 Time domain spectroscopy 250
10.2.1 Analysis 251
10.2.2 Methods 253
10.3 Fourier transform spectroscopy 258
10.3.1 Analysis 260
10.3.2 Methods 264
References 267
Further reading 267
Contents ix
11 Measurement configurations 269
11.1 Single-path methods 270
11.1.1 Radio frequency methods 271
11.1.2 Methods using transmission lines and waveguides 273
11.1.3 Free space: optical methods 275
11.1.4 Ellipsometry 278
11.2 Interferometric techniques 281
11.2.1 Radio frequency bridge methods 281
11.2.2 Transmission line bridge methods 282
11.2.3 Mach–Zehnder interferometer 285
11.3 Resonant techniques 286
11.3.1 Resonant circuits of discrete elements 288
11.3.2 Microstrip and stripline resonators 288
11.3.3 Enclosed cavities 290
11.3.4 Open resonators 291
References 295
Further reading 297
PART THREE: EXPERIMENTS 299
Introductory remarks 299
General books and monographs 300
12 Metals 301
12.1 Simple metals 301
12.1.1 Comparison with the Drude–Sommerfeld model 302
12.1.2 The anomalous skin effect 312
12.1.3 Band structure and anisotropy effects 316
12.2 Effects of interactions and disorder 319
12.2.1 Impurity effects 319
12.2.2 Electron–phonon and electron–electron interactions 321
12.2.3 Strongly disordered metals 329
References 336
Further reading 337
13 Semiconductors 339
13.1 Band semiconductors 339
13.1.1 Single-particle direct transitions 340
13.1.2 Forbidden and indirect transitions 353
13.1.3 Excitons 354
13.2 Effects of interactions and disorder 357
13.2.1 Optical response of impurity states of semiconductors 357
x Contents
13.2.2 Electron–phonon and electron–electron interactions 361
13.2.3 Amorphous semiconductors 366
References 368
Further reading 370
14 Broken symmetry states of metals 371
14.1 Superconductors 371
14.1.1 BCS superconductors 372
14.1.2 Non-BCS superconductors 382
14.2 Density waves 387
14.2.1 The collective mode 387
14.2.2 Single-particle excitations 393
14.2.3 Frequency and electric field dependent transport 394
References 395
Further reading 396
PART FOUR: APPENDICES 397
Appendix A Fourier and Laplace transformations 399
Appendix B Medium of finite thickness 406
Appendix C k · p perturbation theory 421
Appendix D Sum rules 423
Appendix E Non-local response 429
Appendix F Dielectric response in reduced dimensions 445
Appendix G Important constants and units 461
Index 467
Preface
This book has its origins in a set of lecture notes, assembled at UCLA for a graduate
course on the optical studies of solids. In preparing the course it soon became
apparent that a modern, up to date summary of the field is not available. More than
a quarter of a century has elapsed since the book by Wooten: Optical Properties of
Solids – and also several monographs – appeared in print. The progress in optical
studies of materials, in methodology, experiments and theory has been substantial,
and optical studies (often in combination with other methods) have made definite
contributions to and their marks in several areas of solid state physics. There
appeared to be a clear need for a summary of the state of affairs – even if with
a somewhat limited scope.
Our intention was to summarize those aspects of the optical studies which have
by now earned their well deserved place in various fields of condensed matter
physics, and, at the same time, to bring forth those areas of research which are
at the focus of current attention, where unresolved issues abound. Prepared by
experimentalists, the rigors of formalism are avoided. Instead, the aim was to
reflect upon the fact that the subject matter is much like other fields of solid state
physics where progress is made by consulting both theory and experiment, and
invariably by choosing the technique which is most appropriate.
‘A treatise expounds, a textbook explains’, said John Ziman, and by this yard-
stick the reader holds in her or his hands a combination of both. In writing the book,
we have in mind a graduate student as the most likely audience, and also those not
necessarily choosing this particular branch of science but working in related fields.
A number of references are quoted throughout the book, these should be consulted
for a more thorough or rigorous discussion, for deeper insight or more exhaustive
experimental results.
There are limits of what can be covered: choices have to be made. The book
focuses on ‘mainstream’ optics, and on subjects which form part of what could be
termed as one of the main themes of solid state physics: the electrodynamics or
xi
xii Preface
(to choose a more conventional term) the optical properties of electrons in matter.
While we believe this aspect of optical studies will flourish in future years, it is also
evolving both as far as the techniques and subject matter are concerned. Near-field
optical spectroscopy, and optical methods with femtosecond resolution are just two
emerging fields, not discussed here; there is no mention of the optical properties
of nanostructures, and biological materials – just to pick a few examples of current
and future interest.
Writing a book is not much different from raising a child. The project is
abandoned with frustration several times along the way, only to be resumed again
and again, in the hope that the effort of this (often thankless) enterprise is, finally,
not in vain. Only time will tell whether this is indeed the case.
Acknowledgements
Feedback from many people was essential in our attempts to improve, correct,
and clarify this book, for this we are grateful to the students who took the course.
Wolfgang Strohmaier prepared the figures. The Alexander von Humboldt and the
Guggenheim Foundations have provided generous support; without such support
the book could not have been completed.
Finally we thank those who shared our lives while this task was being completed,
Annette, Dani, Dora, and Maria.
1
Introduction
Ever since Euclid, the interaction of light with matter has aroused interest – at least
among poets, painters, and physicists. This interest stems not so much from our
curiosity about materials themselves, but rather to applications, should it be the
exploration of distant stars, the burning of ships of ill intent, or the discovery of
new paint pigments.
It was only with the advent of solid state physics about a century ago that this
interaction was used to explore the properties of materials in depth. As in the field
of atomic physics, in a short period of time optics has advanced to become a major
tool of condensed matter physics in achieving this goal, with distinct advantages
– and some disadvantages as well – when compared with other experimental tools.
The focus of this book is on optical spectroscopy, defined here as the information
gained from the absorption, reflection, or transmission of electromagnetic radia-
tion, including models which account for, or interpret, the experimental results.
Together with other spectroscopic tools, notably photoelectron and electron energy
loss spectroscopy, and Raman together with Brillouin scattering, optics primarily
measures charge excitations, and, because of the speed of light exceeding sub-
stantially the velocities of various excitations in solids, explores in most cases
the q = 0 limit. While this is a disadvantage, it is amply compensated for
by the enormous spectral range which can be explored; this range extends from
well below to well above the energies of various single-particle and collective
excitations.
The interaction of radiation with matter is way too complex to be covered by
a single book; so certain limitations have to be made. The response of a solid at
position r and time t to an electric field E(r
, t
) at position r
and time t
can be
written as
D
i
(r, t) =
¯
¯
ij
(r, r
, t, t
)E
j
(r
, t
) dt
dr
(1.0.1)
1
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