Local Geometry of the Fermi Surface: And High-Frequency Phenomena in Metals

257 Pages·2012·26.281 MB·English

by  Natalya A. Zimbovskaya

Local Geometry of the Fermi Surface
Springer-Science+ Business Media, LLC
Nataliya A. Zimbovskaya
Local Geometry of the
Fermi Surface
And High-Frequency Phenomena in Metals
With 37 lliustrations
'SPringer
Nataliya A. Zimbovskaya
Department ofPhysics, J419
City College of New York
138 Street and Convent Avenue
New York, NY 10031
USA
Library of Congress Cataloging-in-Publication Data
Zimbovskaya, Natallya A.
Local geometry of the Fermi surface: and high-frequency phenomena in metals I
Natallya A. Zimbovskaya.
p. cm.
1ncludes bibliographical references and index.
ISBN 978-1-4612-6557-3 ISBN 978-1-4613-0193-6 (eBook)
DOI 10.1007/978-1-4613-0193-6
1. Fermi surfaces. 2. Fermi liquid theory. 1. Title.
OCI76.8.F4 Z56 2001
530.4'1-<121 00-061862
Printed on acid-free paper.
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Preface
The electronic properties of normal metals have been under active study for several
decades. In the 1950s and 1960s most investigations sought to describe the Fermi
surfaces of metals. These studies were based on experimental data obtained as a re
sult of numerous observations of particular phenomena responsive to the structure
of the electronic spectra of metals and thus to band-structure calculations [lJ. The
high-frequency properties of metals were also actively studied. These investiga
tions were initiated by the development of the theory of the anomalous skin effect
[2J. Later, significant achievements were reached in studies of the high-frequency
properties of metals in the presence of an applied magnetic field. Cyclotron res
onance in a parallel magnetic field [3], electromagnetic waves in metals [4]-[6J,
size effects [7J, [8], Doppler shifted cyclotron resonance, and dopplerons [9], [10]
were predicted in theoretical studies and repeatedly observed in experiments. This
offered new scope for analysis of the properties of the electron system of metals.
The main results of theoretical and experimental studies of the electronic charac
teristics of metals performed during this period are expounded in several books
and review articles (see, e.g., [11]-[14], [16]).
Great progress was also achieved in studies of the interaction between an elec
tron system and the ultrasonic waves propagating in metals. At low temperatures
(T < 10 K) the electrons produce a strong effect on the dispersion and attenuation
of ultrasound waves. Studies in this field were first stimulated by the work of Pip
pard [15J who proposed using magnetoacoustic effects to determine the shape of
the Fermi surfaces. Magnetoacoustic oscillations became one of the most impor
tant tools to study the geometry of the Fermi surfaces of various metals [16]-[19J.
One more effect promoted further development of investigations of the effects
arising when an ultrasound wave propagates in metal: the discovery of giant quan
tum oscillations of the ultrasound attenuation rate in the presence of a quantizing
magnetic field [20].
vi Preface
In the 1970s significant progress was achieved in investigations of the possible
manifestations of the electron-electron interaction in observables. It follows from
the theory of an electron Fermi liquid [4], [21] that a response of the electron
system of a metal to high-frequency and spatially inhomogeneous disturbance can
exhibit some qualitative difference compared to the response of the noncorrelated
electron gas under the same conditions.
An experimental observation of the collective excitations, which arise due to the
electron-electron correlation (so-called Fermi liquid spin and cyclotron waves) in
alkali metals [22]-[24] and which gave information about the parameters of the
Fermi-liquid interaction in sodium and potassium, was an important success in
this field. It generated further development of the phenomenological theory of
the electron Fermi liquid. The theory was extended to be applicable to research
on a broad class of phenomena arising in the presence of a strong (quantizing)
magnetic field. These new methods were first applied to study the spectra of Fermi
liquid quantum waves in metals [25], [26]. A systematic exposition of the theory
of electron Fermi liquid in a quantizing magnetic field was first presented in [27],
[28J.
At the same time investigations proceeded to develop a microscopic theory
of correlated electron systems [29]-[31]. One of the main purposes of these
treatments was to derive the basic equations of the phenomenological theory by
means of the microscopic approach. Such a justification of the basic points of the
phenomenological theory of electron liquid was achieved.
Those effects which are determined by the fundamental geometric characteris
tics of Fermi surfaces (connectivity, presence or absence of open orbits, and so on)
were studied rather well by the mid-1970s. The major manifestations of electron
correlation in the electronic plasma of metals were also investigated in detail for
simple metals whose Fermi surfaces can be supposed to be spherical. At the same
time, new problems and objects to be studied were put forward. Both theorists and
experimentalists started to do research on the electronic properties of thin metallic
films. Semiclassical and quantum size effects [13], [32], [33], collective excitations
[27], [34]-[37], and transport [38] in these metallic films were analyzed in detail.
For further development of these studies it appeared to be necessary to perform
an in-depth study of the process of electron scattering from a surface of the film
[39]-[43]. The influence of the surface layer of the metal on the electron wave
functions had also been studied in detail [44]. Other important fields of research,
concerning the electron properties of metals in the 1970s, include studies of mag
netic breakdown, nonlinear effects in metals, and some others. Further treatment
of these leads was continued for the next decade. However, major attention moved
to the field of investigations of the electron properties of low-dimensional conduc
tors. The phenomenon of high-temperature superconductivity was also intensively
treated.
The subject of the present book is closely connected with the principal trends
of studies in the field of the electron theory of metal developed in the 1970s and
up to the present. The main subject-matter of the book is the local geometry of
Fermi surfaces and its influence on high-frequency phenomena in metals and other
Preface vii
materials of a metal-like type of conductivity. The author's interest in this subject
was first stimulated by the results of several experiments carried out in the 1960s
[45]-[49] which revealed cyclotron resonance in a magnetic field directed along
a normal to the surface of a metallic sample. In these experiments, the cyclotron
resonance in a normal magnetic field was observed in cadmium [45]-[47], zinc
[48], and potassium [49]. A detailed theoretical analysis showed that this resonance
arose due to a local flattening of the Fermi surfaces of these metals in the vicinities
of some points. This conclusion was a starting point for further research whose
results are presented here. These results were published in part in several papers
[50]-[75].
The first chapter of this book is of an introductory character. It covers material
which can be found in most of the books on the theory of metals. However, a brief
summary of some basic concepts of the theory of metals seems to be of importance
for a better comprehension of the major portion of the book. Here the concept of
the electronic liquid of a metal is introduced; the definition of the Fermi surface is
given and some geometric features of the Fermi surfaces are described. These are
points of flattening and parabolic points of the Fermi surface where its Gaussian
curvature becomes zero, as well as lines of zero or extremely large curvature which
can be found on the Fermi surfaces of some metals. The influence of such points
or lines upon the observables comes from the change in the electron density of
states in their neighborhood. The manifestations of this influence are studied in
detail in the main body of the book. Besides, the first chapter contains a brief
description of several phenomena in metals which can be affected due to the local
geometry of the Fermi surfaces, notably: the skin effect, the cyclotron resonance,
the magnetoacoustic oscillations of the attenuation rate, and the velocity shift of
the ultrasound waves propagating in metals.
It is known that the phenomenological theory of the Fermi liquid in metals is well
developed within the framework of an isotropic model of a metal where the Fermi
surface is supposed to be a sphere. However, real metals, as a rule, have complicated
shape anisotropic Fermi surfaces. This gives rise to significant difficulties in studies
of the Fermi-liquid effects. The point of Chapter 2 of this book is to propose some
asymptotic expansions of the kernel of the Fermi-liquid interaction (Landau F
function) to accommodate the theory of the electron liquid to a more systematic
study of the Fermi-liquid effects in real metals. These asymptotic expansions are
based on the symmetry properties of the Fermi surfaces. Thus Chapter 2 contains
a body of mathematical techniques which permit us to study the response of an
electron liquid of real metals to external disturbances.
In subsequent chapters these techniques are applied to analyze concrete prob
lems. Chapter 3 deals with the anomalous skin effect in metals whose Fermi
surfaces have local anomalies of their curvature. The analysis carried out in this
chapter leads to the conclusion that in the presence of points or lines of zero, or
anomalously large curvature on a Fermi surface, the magnitude and frequency de
pendence of the surface impedance of a semi-infinite metal can be significantly
changed. It is shown that under certain conditions a new kind of weakly damping
electromagnetic wave can propagate in a metal.
viii Preface
The content of Chapter 4 is a systematic theory of a cyclotron resonance in
metals, in a geometry where an external magnetic field is directed along a normal to
the surface of the metal. Unlike the well-known Azbel-Kaner cyclotron resonance
[3] this effect can exhibit itself only in the presence of locally flattened or nearly
cylindrical segments on the effective parts of the Fermi surfaces. When the Fermi
surface of a considered metal everywhere has a finite and nonzero curvature, the
resonance feature corresponding to the cyclotron resonance is smeared out until it
is scarcely detectable. The proposed theory of the cyclotron resonance in metals in
a normal magnetic field gives good agreement with the results of the experiments
of [47]-[49].
Geometric oscillations of the attenuation rate of the ultrasonic waves, propagat
ing in metals perpendicularly to an applied magnetic field, are very sensitive to the
local geometry of the Fermi surface at the points corresponding to the stationary
points of a cyclotron orbit of an electron. The characteristic features of the geo
metric oscillations of the ultrasonic attenuation, and the velocity shift caused by
the local geometry of the Fermi surface, are analyzed in Chapter 5.
Chapter 6 includes a theory of a special kind of collective excitations, the
so-called Fermi-liquid cyclotron dopplerons. These collective excitations are
doppleron-like extensions of the well-known Fermi-liquid cyclotron waves pre
dicted by v.P. SHin [4]. Cyclotron dopplerons can propagate in metals only under
the condition that their Fermi surfaces have a paraboloid-like shape. Thus these
collective excitations can be referred to as a manifestation of the Fermi-liquid
interaction among the electrons in metals with nonspherical Fermi surfaces.
The presence of points and lines of zero curvature affects the density of states
of electrons on the Fermi surface and, consequently, the magnitude and shape
of the quantum oscillations of the observables in a strong magnetic field. Some
manifestations of this influence in metals are analyzed in Chapter 7.
The topics covered by the final chapter are related to those of the preceding
chapters. This chapter deals with the possible manifestations of the local geome
try of the Fermi surface in low-dimensional conductors with a metal-like type of
conductivity. The skin effect, the cyclotron resonance, and the quantum oscillatory
phenomena in organic metals are discussed in the first three sections of the chapter.
The last two sections include the theory of the magnetoacoustic response of modu
lated two-dimensional electron systems in a quantum Hall regime near half-filling
of the lowest Landau level. It is shown that as well as in conventional metals, the
local geometry of the Fermi surfaces can strongly influence the response of such
systems to the electromagnetic or ultrasonic disturbance. The theoretical analysis
corroborates the results of recent experiments on the cyclotron resonance in lay
ered conductors and on the velocity shift and attenuation of the surface acoustic
waves propagating in the modulated GaAs/AIGaAs heterostructures.
An acquaintance with this text should prepare the reader for a more detailed
study of particular areas of the theory of metals and other metal-like systems
especially of the "Fermiology" which is not exhausted to the present. It is assumed
that the essentials of electrodynamics, and quantum and statistical mechanics are a
sufficient theoretical background for reading this book. As far as possible the author
Preface ix
has tried to present the necessary calculations thoroughly. Some special details are
considered in the Appendices. The book is intended to appeal to theoretical and
experimental physisists, whose field of research concerns magnetotransport in
metals and low-dimensional systems or the neighboring areas of solid state theory,
and to graduate students.
During the development of the English edition some alterations were introduced
into the text. New material (Chapter 1, and Sections 8.1, 8.2, 8.4, and 8.5) and a
number of new figures have been added, and the list of references has been changed
substantially.
I am pleased to acknowledge the hospitality of the Physics Department of City
College, City University of New York, which made computer and other facilities
available to me during the preparation of the manuscript for the English edition of
this book.
I am sincerely grateful to all my colleagues with whom I collaborated to make
these studies. This collaboration was a honor for me. It is my pleasant duty to
thank Professor J.L. Birman for his proposal to have the book published. I take
this opportunity to express my deep gratitude to my husband, Dr. a.M. Zimbovsky.
Without his enthusiastic help and support this book would never have been written.
New York Nataliya A. Zimbovskaya
Contents
Preface v
1 The Electronic Liquid of Metals 1
1.1 The Quasi Particle Concept ................. 1
1.2 Local Geometry of the Fenni Surface and High-Frequency
Properties of Metals ..................... 4
1.3 Semiclassical Dynamics of Electrons in a Magnetic Field and
Magnetotransport in Metals 11
1.4 Skin Effect . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Cyclotron Resonance ................... 22
1.6 Ultrasonic Attenuation in Metals: Geometric Resonances 25
1.7 Fermi-Liquid Interaction between Quasi Particles .... 27
2 Main Equations of the Theory of the Electron Fermi Liquid of
Anisotropic Metal 31
2.1 Main Relations of the Quantum Theory of an Electronic
Liquid of Metals . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Approximation of the Fenni-Liquid Kernel for a Metal with a
Simply Connected and Everywhere Convex Fenni Surface. . 39
2.3 Approximation of the Fermi-Liquid Functions in Metals with
a Cubic Symmetry of a Crystal Lattice . . . . . . . . . . . 42
2.4 Approximation of the Fenni-Liquid Functions for a Metal
with an Axial-Symmetric Fenni Surface .......... 49