Electrodynamics of solids dressel, gruner

E LECTRODYNAMICS OF S OLIDS
OPTICAL PROPERTIES OF ELECTRONS IN MATTER
The authors of this book present a thorough discussion of the optical properties of
solids, with a focus on electron states and their response to electrodynamic fields. A
review of the fundamental aspects of the propagation of electromagnetic fields, and
their interaction with condensed matter, is given. This is followed by a discussion
of the optical properties of metals, semiconductors, and collective states of solids
such as superconductors.
Theoretical concepts, measurement techniques, and experimental results are
covered in three inter-related sections. Well established, mature fields are dis-
cussed, together with modern topics at the focus of current interest. The substantial
reference list included will also prove to be a valuable resource for those interested
in the electronic properties of solids.
The book is intended for use by advanced undergraduate and graduate students,
and researchers active in the fields of condensed matter physics, materials science,
and optical engineering.
M
ARTIN DRESSEL received his Doctor of Sciences degree in 1989 from the
Universit
¨
at G
¨
ottingen where he subsequently worked as a postdoctoral research
fellow. Since then he has held positions in the University of British Columbia at
Vancouver; the University of California, Los Angeles; the Technische Universit
¨
at,
Darmstadt; and the Center of Electronic Correlations and Magnetism at the Uni-
versit
¨

at Augsburg. Professor Dressel is now Head of the 1. Physikalisches Institut
at the Universit
¨
at Stuttgart.
G
EORGE GR
¨
UNER obtained his Doctor of Sciences degree from the E
¨
otv
¨
os Lorand
University, Budapest, in 1972, and became Head of the Central Research Institute
of Physics in Budapest in 1974. In 1980 he took up the position of Professor of
Physics at the University of California, Los Angeles, and later became Director
of the Solid State Science Center there. Professor Gr
¨
uner has been a distinguished
visiting professor at numerous institutions worldwide and is a consultant for several
international corporations and advisory panels. He is a Guggenheim Fellow and is
also a recipient of the Alexander Humboldt Senior American Scientist Award.
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Electrodynamics of Solids
Optical Properties of Electrons in Matter
Martin Dressel
Stuttgart
and
George Gr
¨
uner

Los Angeles
PUBLISHED BY CAMBRIDGE UNIVERSITY PRESS (VIRTUAL PUBLISHING)
FOR AND ON BEHALF OF THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
The Pitt Building, Trumpington Street, Cambridge CB2 IRP
40 West 20th Street, New York, NY 10011-4211, USA
477 Williamstown Road, Port Melbourne, VIC 3207, Australia

© 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
2 1 Introduction
where i and j refer to the components of the electric field E and displacement
field D; thus
¯
¯
ij
is the so-called dielectric tensor. For homogeneous solids, the
response depends only on r − r

(while time is obviously a continuous variable),
and Eq. (1.0.1) is reduced to
D
i
(r, t) =

¯
¯
ij
(r − r

, t −t

)E
j

(r

, t

) dt

dr

. (1.0.2)
We further assume linear response, thus the displacement vector D is proportional
to the applied electric field E. In the case of an alternating electric field of the form
E(r, t) = E
0
exp
{
i(q · r − ωt)
}
(1.0.3)
the response occurs at the same frequency as the frequency of the applied field with
no higher harmonics. Fourier transform then gives
D
i
(q,ω) =
¯
¯
ij
(q,ω)E
j
(q,ω) (1.0.4)
with the complex dielectric tensor assuming both a wavevector and frequency

dependence. For
¯
¯
ij
(r − r

, t − t

) real, the q and ω dependent dielectric tensor
obeys the following relation:
¯
¯
ij
(r − r

, t −t

) =
¯
¯
∗
ij
(r − r

, t −t

),
where the star (
∗
) refers to the complex conjugate. Only cubic lattices will be

considered throughout most parts of the book, and then ˆ is a scalar, complex
quantity.
Of course, the response could equally well be described in terms of a current at
position r and time t, and thus
J(r, t) =

ˆσ(r, r

, t, t

)E(r

, t

) dt

dr

(1.0.5)
leading to a complex conductivity tensor ˆσ(q,ω)in response to a sinusoidal time-
varying electric field. The two response functions are related by
ˆ(q,ω) = 1 +
4πi
ω
ˆσ(q,ω) ; (1.0.6)
this follows from Maxwell’s equations.
Except for a few cases we also assume that there is a local relationship between
the electric field E(r, t) and D(r, t) and also j(r, t), and while these quantities may
display well defined spatial dependence, their spatial variation is identical; with
J(r)

E(r)
=ˆσ and
D(r)
E(r)
=ˆ (1.0.7)
two spatially independent quantities. This then means that the Fourier transforms
1 Introduction 3
of ˆ and ˆσ do not have q = 0 components. There are a few notable exceptions
when some important length scales of the problem, such as the mean free path 
in metals or the coherence length ξ
0
in superconductors, are large and exceed the
length scales set by the boundary problem at hand. The above limitations then
reduce
ˆσ(ω) = σ
1
(ω) + iσ
2
(ω) and ˆ(ω) = 
1
(ω) + i
2
(ω) (1.0.8)
to scalar and q independent quantities, with the relationship between ˆ and ˆσ as
given before. We will also limit ourselves to non-magnetic materials, and will
assume that the magnetic permeability µ
1
= 1 with the imaginary part µ
2
= 0.

We will also make use of what is called the semiclassical approximation. The
interaction of charge e
i
with the radiation field is described as the Hamiltonian
H =
1
2m

i

p
i
−
e
i
c
A(r
i
)

2
, (1.0.9)
and while the electronic states will be described by appropriate first and second
quantization, the vector potential A will be assumed to represent a classical field.
We will also assume the so-called Coulomb gauge, by imposing a condition
∇·A = 0 ; (1.0.10)
this then implies that A has only transverse components, perpendicular to the
wavevector q.
Of course one cannot do justice to all the various interesting effects which arise
in the different forms of condensed matter – certain selections have to be made,

this being influenced by our prejudices. We cover what could loosely be called the
electrodynamics of electron states in solids. As the subject of what can be termed
electrodynamics is in fact the response of charges to electromagnetic fields, the
above statement needs clarification. Throughout the book our main concern will
be the optical properties of electrons in solids, and a short guide of the various
states which may arise is in order.
In the absence of interaction with the underlying lattice, and also without
electron–electron or electron–phonon interactions, we have a collection of free
electrons obeying – at temperatures of interest – Fermi statistics, and this type
of electron liquid is called a Fermi liquid. Interactions between electrons then lead
to an interacting Fermi liquid, with the interactions leading to the renormalization
of the quasi-particles, leaving, however, their character unchanged. Under certain
circumstances, notably when the electron system is driven close to an instability, or
when the electronic structure is highly anisotropic, this renormalized Fermi-liquid
picture is not valid, and other types of quantum liquids are recovered. The – not
too appealing – notion of non-Fermi liquids is usually adopted when deviations
4 1 Introduction
from a Fermi liquid are found. In strictly one dimension (for example) the nature
of the quantum liquid, called the Luttinger liquid, with all of its implications,
is well known. Electron–phonon interactions also lead to a renormalized Fermi
liquid.
If the interactions between the electrons or the electron–phonon interactions are
of sufficient strength, or if the electronic structure is anisotropic, phase transitions
to what can be termed electronic solids occur. As is usual for phase transitions,
the ordered state has a broken symmetry, hence the name broken symmetry states
of metals. For these states, which are called charge or spin density wave states,
translational symmetry is broken and the electronic charge or spin density assumes
a periodic variation – much like the periodic arrangement of atoms in a crystal.
The superconducting state has a different, so-called broken gauge symmetry. Not
surprisingly for these states, single-particle excitations have a gap – called the

single-particle gap – a form of generalized rigidity. As expected for a phase
transition, there are collective modes associated with the broken symmetry state
which – as it turns out – couple directly to electromagnetic fields. In addition, for
these states the order parameter is complex, with the phase directly related to the
current and density fluctuations of the collective modes.
Disorder leads to a different type of breakdown of the Fermi liquid. With
increasing disorder a transition to a non-conducting state where electron states are
localized may occur. Such a transition, driven by an external parameter (ideally
at T = 0 where only quantum fluctuations occur) and not by the temperature, is
called a quantum phase transition, with the behavior near to the critical disorder
described – in analogy to thermal phase transitions – by various critical exponents.
This transition and the character of the insulating, electron glass state depend on
whether electron–electron interactions are important or not. In the latter case we
have a Fermi glass, and the former can be called a Coulomb glass, the two cases
being distinguished by temperature and frequency dependent excitations governed
by different exponents, reflecting the presence or absence of Coulomb gaps.
A different set of states and properties arises when the underlying periodic lattice
leads to full and empty bands, thus to semiconducting or insulating behavior.
In this case, the essential features of the band structure can be tested by optical
experiments. States beyond the single-electron picture, such as excitons, and also
impurity states are essential features here. All this follows from the fundamental
assumption about lattice periodicity and the validity of Bloch’s theorem. When this
is not relevant, as is the case for amorphous semiconductors, localized states with
a certain amount of short range order are responsible for the optical properties.
The response of these states to an electromagnetic field leads to dissipation,
and this is related to the fluctuations which arise in the absence of driving
fields. The relevant fluctuations are expressed in terms of the current–current
1 Introduction 5
or density–density correlation functions, related to the response through the cel-
ebrated fluctuation-dissipation theorem. The correlation functions in question can

be derived using an appropriate Hamiltonian which accounts for the essential fea-
tures of the particular electron state in question. These correlations reflect and the
dissipation occurs through the elementary excitations. Single-particle excitations,
the excitation of the individual quasi-particles, may be the source of the dissipation,
together with the collective modes which involve the cooperative motion of the
entire system governed by the global interaction between the particles. Electron–
hole excitations in a metal are examples of the former, plasmons and the response
of the broken symmetry ground state are examples of the latter. As a rule, these
excitations are described in the momentum space by assuming extended states and
excitations with well defined momenta. Such excitations may still exist in the case
of a collection of localized states; here, however, the excitations do not have well
defined momenta and thus restrictions associated with momentum conservation do
not apply.
Other subjects, interesting in their own right, such as optical phonons, di-
electrics, color centers (to name just a few) are neglected; and we do not discuss
charge excitations in insulators – vast subjects with interesting properties. Also
we do not discuss the important topic of magneto-optics or magneto-transport
phenomena, which occur when both electric and magnetic fields are applied.
The organization of the book is as follows: underlying theory, techniques, and
experimental results are discussed as three, inter-relating parts of the same en-
deavor. In Part 1 we start with the necessary preliminaries: Maxwell’s equations
and the definition of the optical constants. This is followed by the summary of
the propagation of light in the medium, and then by the discussion of phenomena
which occur at an interface; this finally brings us to the optical parameters which
are measured by experiment. The three remaining chapters of Part 1 deal with the
optical properties of metals, semiconductors, and the so-called broken symmetry
states of metals. Only simple metals and semiconductors are dealt with here, and
only the conventional broken symmetry states (such as BCS superconductors) will
be covered in the so-called weak coupling limit. In these three chapters three
different effects are dominant: dynamics of quasi-free electrons, absorption due

to interband processes, and collective phenomena.
In Part 2 the experimental techniques are summarized, with an attempt to bring
out common features of the methods which have been applied at vastly different
spectral ranges. Here important similarities exist, but there are some important
differences as well. There are three spectroscopic principles of how the response
in a wide frequency range can be obtained: measurements can be performed in the
frequency domain, the time domain, or by Fourier transform technique. There are
also different ways in which the radiation can interact with the material studied:
6 1 Introduction
simply transmission or reflection, or changes in a resonance structure, can be
utilized.
In Part 3 experimental results are summarized and the connection between the-
ory and experiment is established. We first discuss simple scenarios where the
often drastic simplifications underlying the theories are, in the light of experiments,
justified. This is followed by the discussion of modern topics, much in the limelight
at present. Here also some hand-waving arguments are used to expound on the
underlying concepts which (as a rule) by no means constitute closed chapters of
condensed matter physics.
Part one
Concepts and properties
Introductory remarks
In this part we develop the formalisms which describe the interaction of light (and
sometimes also of a test charge) with the electronic states of solids. We follow
usual conventions, and the transverse and longitudinal responses are treated hand in
hand. Throughout the book we use simplifying assumptions: we treat only homo-
geneous media, also with cubic symmetry, and assume that linear response theory
is valid. In discussing various models of the electron states we limit ourselves to
local response theory – except in the case of metals where non-local effects are
also introduced. Only simple metals and semiconductors are treated; and we offer
the simple description of (weak coupling) broken symmetry – superconducting

and density wave – states, all more or less finished chapters of condensed matter
physics. Current topics of the electrodynamics of the electron states of solids are
treated together with the experimental state of affairs in Part 3. We make extensive
use of computer generated figures to visualize the results.
After some necessary preliminaries on the propagation and scattering of elec-
tromagnetic radiation, we define the optical constants, including those which are
utilized at the low energy end of the electrodynamic spectrum, and summarize the
so-called Kramers–Kronig relations together with the sum rules. The response to
transverse and longitudinal fields is described in terms of correlation and response
functions. These are then utilized under simplified assumptions such as the Drude
model for metals or simple band-to-band transitions in the case of semiconductors.
8 Part one: Introductory remarks
Broken symmetry states are described in their simple form using second quantized
formalism.
General books and monographs
A.A. Abrikosov, Fundamentals of the Theory of Metals (North-Holland, Amsterdam,
1988)
M. Born and E. Wolf, Principles of Optics, 6th edition (Cambridge University Press,
Cambridge, 1999)
J. Callaway, Quantum Theory of Solids, 2nd edition (Academic Press, New York, 1991)
R.G. Chambers, Electrons in Metals and Semiconductors (Chapman and Hall, London,
1990)
W.A. Harrison, Solid State Theory (McGraw-Hill, New York, 1970)
H. Haug and S.W. Koch, Quantum Theory of the Optical and Electronic Properties of
Semiconductors, 3rd edition (World Scientific, Singapore, 1994)
J. D. Jackson, Classical Electrodynamics, 2nd edition (John Wiley & Sons, New York,
1975)
C. Kittel, Quantum Theory of Solids, 2nd edition (John Wiley & Sons, New York, 1987)
L.D. Landau, E.M. Lifshitz, and L.P. Pitaevskii, Electrodynamics of Continuous Media,
2nd edition (Butterworth-Heinemann, Oxford, 1984)

I.M. Lifshitz, M.Ya. Azbel’, and M.I. Kaganov, Electron Theory of Metals (Consultants
Bureau, New York, 1973)
G.D. Mahan, Many-Particle Physics, 2nd edition (Plenum Press, New York, 1990)
D. Pines, Elementary Excitations in Solids (Addison-Wesley, Reading, MA, 1963)
D. Pines and P. Noizi
`
eres, The Theory of Quantum Liquids, Vol. 1 (Addison-Wesley,
Reading, MA, 1966)
J.R. Schrieffer, Theory of Superconductivity, 3rd edition (W.A. Benjamin, New York,
1983)
F. Stern, Elementary Theory of the Optical Properties of Solids, in: Solid State Physics
15, edited by F. Seitz and D. Turnbull (Academic Press, New York, 1963), p. 299
J. Tauc (ed.), The Optical Properties of Solids, Proceedings of the International School
of Physics ‘Enrico Fermi’ 34 (Academic Press, New York, 1966)
F. Wooten, Optical Properties of Solids (Academic Press, San Diego, CA, 1972)
P.Y. Yu and M. Cardona, Fundamentals of Semiconductors (Springer-Verlag, Berlin,
1996)
J.M. Ziman, Principles of the Theory of Solids, 2nd edition (Cambridge University
Press, London, 1972)
2
The interaction of radiation with matter
Optics, as defined in this book, is concerned with the interaction of electromag-
netic radiation with matter. The theoretical description of the phenomena and the
analysis of the experimental results are based on Maxwell’s equations and on their
solution for time-varying electric and magnetic fields. The optical properties of
solids have been the subject of extensive treatises [Sok67, Ste63, Woo72]; most
of these focus on the parameters which are accessible with conventional optical
methods using light in the infrared, visible, and ultraviolet spectral range. The
approach taken here is more general and includes the discussion of those aspects of
the interaction of electromagnetic waves with matter which are particularly relevant

to experiments conducted at lower frequencies, typically in the millimeter wave and
microwave spectral range, but also for radio frequencies.
After introducing Maxwell’s equations, we present the time dependent solution
of the equations leading to wave propagation. In order to describe modifications
of the fields in the presence of matter, the material parameters which characterize
the medium have to be introduced: the conductivity and the dielectric constant.
In the following step, we define the optical constants which characterize the prop-
agation and dissipation of electromagnetic waves in the medium: the refractive
index and the impedance. Next, phenomena which occur at the interface of free
space and matter (or in general between two media with different optical con-
stants) are described. This discussion eventually leads to the introduction of the
optical parameters which are accessible to experiment: the optical reflectivity and
transmission.
2.1 Maxwell’s equations for time-varying fields
To present a common basis of notation and parameter definition, we want to recall
briefly some well known relations from classical electrodynamics. Before we
consider the interaction of light with matter, we assume no matter to be present.
9
10 2 The interaction of radiation with matter
2.1.1 Solution of Maxwell’s equations in a vacuum
The interaction of electromagnetic radiation with matter is fully described by
Maxwell’s equations. In the case of a vacuum the four relevant equations are
∇×E(r, t) +
1
c
∂B(r, t)
∂t
= 0 , (2.1.1a)
∇·B(r, t) = 0 , (2.1.1b)
∇×B(r, t) −

1
c
∂E(r, t)
∂t
=
4π
c
J(r, t), (2.1.1c)
∇·E(r, t) = 4πρ(r, t). (2.1.1d)
E and B are the electric field strength and the magnetic induction, respectively; c =
2.997 924 58 ×10
8
ms
−1
is the velocity of light in free space. The current density
J and the charge density ρ used in this set of equations refer to the total quantities
including both the external and induced currents and charge densities; their various
components will be discussed in Section 2.2. All quantities are assumed to be
spatial, r, and time, t, dependent as indicated by (r, t). To make the equations
more concise, we often do not explicitly include these dependences. Following the
notation of most classical books in this field, the equations are written in Gaussian
units (cgs), where E and B have the same units.
1
The differential equations (2.1.1a) and (2.1.1b) are satisfied by a vector potential
A(r, t) and a scalar potential (r, t) with
B =∇×A (2.1.2)
and
E +
1
c

∂A
∂t
=−∇. (2.1.3)
The first equation expresses the fact that B is an axial vector and can be expressed
as the rotation of a vector field. If the vector potential vanishes (A = 0) or if A is
time independent, the electric field is conservative: the electric field E is given by
the gradient of a potential, as seen in Eq. (2.1.3). Substituting the above expressions
into Amp
`
ere’s law (2.1.1c) and employing the general vector identity
∇×(∇×A) =−∇
2
A +∇(∇·A) (2.1.4)
gives the following equation for the vector potential A:
∇
2
A −
1
c
2
∂
2
A
∂t
2
=−
4π
c
J +
1

c
∇
∂
∂t
+∇(∇·A) ; (2.1.5)
a characteristic wave equation combining the second time derivative and the second
1
For a discussion of the conversion to rational SI units (mks), see for example [Bec64, Jac75]. See also
Tables G.1 and G.3.
2.1 Maxwell’s equations for time-varying fields 11
spatial derivative. As this equation connects the current to the scalar and vector
potentials, we can find a corresponding relationship between the charge density
and the potentials. Substituting Eq. (2.1.3) into Coulomb’s law (2.1.1d) yields
∇·E = 4πρ =−∇
2
 −
1
c
∂
∂t
(∇·A).
Using the Coulomb gauge
2
∇·A = 0 , (2.1.6)
the last term in this equation vanishes; the remaining part yields Poisson’s equation
∇
2
 =−4πρ , (2.1.7)
expressing the fact that the scalar potential (r, t) is solely determined by the
charge distribution ρ(r, t). From Eq. (2.1.1c) and by using the definition of the

vector potential, we obtain (in the case of static fields) a similar relation for the
vector potential
∇
2
A =−
4π
c
J , (2.1.8)
connecting only A(r, t) to the current density J(r, t). Employing the vector identity
∇·(∇×B) = 0, we can combine the derivatives of Eqs (2.1.1c) and (2.1.1d) to
obtain the continuity equation for electric charge
∂ρ
∂t
=−∇·J , (2.1.9)
expressing the fact that the time evolution of the charge at any position is related to
a current at the same location.
Equation (2.1.7) and Eq. (2.1.9) have been obtained by combining Maxwell’s
equations in the absence of matter, without making any assumptions about a par-
ticular time or spatial form of the fields. In the following we consider a harmonic
time and spatial dependence of the fields and waves. A monochromatic plane
2
Another common choice of gauge is the Lorentz convention ∇·A +
1
c
∂
∂t
= 0, which gives symmetric wave
equations for the scalar and vector potentials:
∇
2

 −
1
c
2
∂
2

∂t
2
=−4πρ ,
∇
2
A −
1
c
2
∂
2
A
∂t
2
=−
4π
c
J ;
or – in the case of superconductors – the London gauge, assuming ∇
2
 = 0. For more details on the
properties of various gauges and the selection of an appropriate one, see for example [Jac75, Por92]. Our
choice restricts us to non-relativistic electrodynamics; relativistic effects, however, can safely be neglected

throughout the book.
12 2 The interaction of radiation with matter
electric wave of frequency f = ω/2π traveling in a certain direction (given by
the wavevector q) can then be written as
E(r, t) = E
01
sin{q · r − ωt}+E
02
cos{q · r − ωt}
= E
03
sin{q · r − ωt + ψ} , (2.1.10)
where E
0i
(i = 1, 2, 3) describe the maximum amplitude; but it is more convenient
to write the electric field E(r, t) as a complex quantity
E(r, t) = E
0
exp{i(q · r − ωt)} . (2.1.11)
We should keep in mind, however, that only the real part of the electric field is
a meaningful quantity. We explicitly indicate the complex nature of E and the
possible phase factors only if they are of interest to the discussion. A few notes
are in order: the electric field is a vector, and therefore its direction as well as
its value is of importance. As we shall discuss in Section 3.1 in more detail, we
distinguish longitudinal and transverse components with respect to the direction of
propagation; any transverse field polarization can be obtained as the sum of two
orthogonal transverse components. If the ratio of both is constant, linearly polar-
ized fields result, otherwise an elliptical or circular polarization can be obtained.
Similar considerations to those presented here for the electric field apply for the
magnetic induction B(r, t) and other quantities. The sign in the exponent is chosen

such that the wave travels along the +r direction.
3
We assume that all fields and
sources can be decomposed into a complete continuous set of such plane waves:
E(r, t) =
1
(2π)
4

∞
−∞
E(q,ω)exp{i(q · r − ωt)}dω dq , (2.1.12)
and the four-dimensional Fourier transform of the electric field strength E(r, t) is
E(q,ω) =

∞
−∞
E(r, t) exp{−i(q · r − ωt)}dt dr . (2.1.13)
Analogous equations and transformations apply to the magnetic induction B. Some
basic properties of the Fourier transformation are discussed in Appendix A.1.
Equation (2.1.12), which assumes plane waves, requires that a wavevector q and
a frequency ω can be well defined. The wavelength λ = 2π/|q| should be much
smaller than the relevant dimensions of the problem, and therefore the finite sample
size is neglected; also the period of the radiation should be much shorter than the
typical time scale over which this radiation is applied. We assume, for the moment,
that the spatial dependence of the radiation always remains that of a plane wave.
3
For exp{i(q · r +ωt)} and exp{−i(q · r + ωt)} the wave travels to the left (−r), while exp{i(q ·r − ωt)} and
exp{−i(q · r −ωt)} describe right moving waves (+r). With the convention exp{−iωt} the optical constants
ˆ and

ˆ
N have positive imaginary parts, as we shall find out in Section 2.2. The notation exp{iωt} is also
common and leads to
ˆ
N = n − ik and ˆ = 
1
− i
2
.