Dynamics of Heavy Electrons

252 Pages·2000·12.63 MB·English

by  Y. Kuramoto

THE
INTERNATIONAL SERIES
OF
MONOGRAPHS ON PHYSICS
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J. BIRMAN S.F. EDWARDS R. FRIEND
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INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS
105. Y. Kuramoto, Y. Kitaoka: Dynamics of heavy electrons
104. D. Bardin, G. Passarino: The Standard Model in the making
103. G. C. Branco, L. Lavoura, J. P. Silva: CP violation
102. T. C. Choy: Effective medium theory
101. H. Araki: Mathematical theory of quantum fields
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91. R. A. Bertlmann: Anomalies in quantum field theory
90. P. K. Gosh: Ion traps
89. E. Simanek: Inhomogeneous superconductors
88. S. L. Adler: Quaternionic quantum mechanics and quantum fields
87. P. S. Joshi: Global aspects in gravitation and cosmology
86. E. R. Pike, S. Sarkar: The quantum theory of radiation
84. V. Z. Kresin, H. Morawitz, S. A. Wolf: Mechanisms of conventional and
high T superconductivity
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83. P. G. de Gennes, J. Prost: The physics of liquid crystals
82. B. H. Bransden, M. R. C. McDowell: Charge exchange and the theory of
ion-atom collision
81. J. Jensen, A. R. Mackintosh: Rare earth magnetism
80. R. Gastmans, T. T. Wu: The ubiquitous photon
79. P. Luchini, H. Motz: Undulators and free-electron lasers
78. P. Weinberger: Electron scattering theory
76. H. Aoki, H. Kamimura: The physics of interacting electrons in disordered systems
75. J. D. Lawson: The physics of charged particle beams
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51. C. M011er: The theory of relativity
46. H. E. Stanley: Introduction of phase transitions and critical phenomena
32. A. Abragam: Principles of nuclear magnetism
27. P. A. M. Dirac: Principles of quantum mechanics
23. R. E. Peierls: Quantum theory of solids
Dynamics of Heavy Electrons
Y. KURAMOTO
Department of Physics
Tohoku University
Y. KITAOKA
Department of Physical Science
Osaka University
CLARENDON PRESS OXFORD
2000
OXFORD
UNIVERSITY PRESS
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Library of Congress Cataloging in Publication Data
Kuramoto, Y. (Yoshio)
Dynamics of heavy electrons / Y. Kuramoto, Y. Kitaoka.
(The international series of monographs on physics; 105)
1. Mesons. 2. Energy-band theory of solids. I. Kitaoka, Y. (Yoshio)
II. Title. III. Series: International series of monographs on physics ; v. 105.
QC793.5.M42K87 2000 530.4'l-dc21 99-42821
ISBN 019 851767 X
Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India
Printed in Great Britan
on acid-free paper by
TJ International Ltd., Padstow
PREFACE
The present book has two-fold purposes. First, the book is intended to be a monograph on
heavy electrons, which have been the focus of very active experimental and theoretical
studies in the last two decades. Heavy electrons are found in a number of lanthanide and
actinide compounds, and are characterized by a large (comparable to the mass of a muon)
effective mass. The heavy electrons exhibit rich phenomena, such as smooth crossover to
local-moment behaviour with increasing temperature, unconventional superconductivity,
weak antiferromagnetism and pseudo-metamagnetism. Although some mysteries still
remain, the authors feel that, overall, a reasonably coherent understanding of heavy
electrons is available as a whole. Therefore, a survey of the properties of heavy electrons
from a global and unified point of view is in order.
The physics of heavy electrons is one of the typical examples where close inter-
action between theory and experiment has revealed systematically the origin of the
seemingly incredible behaviour of electrons. One of the present authors (Kuramoto) has
been engaged in formulating a theoretical scheme which is now applied widely to mag-
netic impurities and heavy electrons and, among other things, has derived the dynamical
response functions. The other author (Kitaoka) has been involved in NMR experiments
on heavy-electron systems and has observed the formation of metallic, insulating, super-
conducting, and weakly magnetic phases of heavy electrons through the NMR. Thus,
the authors think it appropriate for them to write a book which focuses on the dynamical
aspects of heavy electrons. In view of the innumerable and diverse activities in the field,
it is difficult to treat every aspect in detail, although the important thermodynamic and
transport results have been covered reasonably well.
The second purpose of the book is to serve as an advanced textbook on the theoretical
and experimental physics of strongly correlated electrons. The necessity to understand
the fascinating experimental results on heavy electrons has stimulated intensive theoret-
ical efforts to go beyond the existing schemes. It has turned out that many established
ideas and techniques are insufficient for understanding heavy electrons. Theoretically, the
extensive use of quantum fluctuation techniques has lowered the superiority of the cele-
brated mean-field theories. On the other hand, extreme experimental conditions, such as
application of strong magnetic fields and pressures at ultralow temperatures, are required.
Thus, heavy-electron systems have been a target of a case study for applying and testing
almost all tools in theoretical and experimental condensed-matter physics. Those gradu-
ate students and researchers who want to work on strongly correlated condensed-matter
systems will find in the book many examples on how the conventional concepts in gen-
eral solid systems work or do not work in the heavy-electron systems. Although we try to
make the book self-contained for those readers who have the knowledge of condensed-
matter physics at the undergraduate level, they may sometimes find the description of
the book too concise. In such a case, we advise consulting any of the available textbooks
on elementary many-body physics.
vi PREFACE
The book tries to build a coherent picture of heavy electrons from a collection of
apparently diverse experimental and theoretical results. However, the authors neither
intend to be exhaustive in citing papers, nor are they excessively on the alert for including
all of the recent developments. At any rate, a monograph cannot compete with a review
in scientific journals for covering the latest findings. Instead, the book tries to provide a
broad outlook which should be especially useful to newcomers to the field of strongly
correlated electrons, experimental or theoretical, and tries to help them to initiate a
new stage of investigation. Thus, the book emphasizes new theoretical methods and
concepts which might be effective also in other related areas. It also emphasizes the
utility and the limitation of the dynamical information brought about by NMR and
neutron scattering. Special attention is paid to similarity with other strongly correlated
systems, e.g. the copper oxide high-temperature superconductors. Since the research
area of heavy electrons is vast, our treatment inevitably reflects our favourite choice
of the topics, and pays unavoidable attention to our own contribution to the field. We
would apologize to those authors whose contributions are not touched upon, or are not
treated properly. Especially, we have omitted many topics on transport, thermodynamic
properties, and dynamical measurements other than the NMR and neutron scattering.
As an existing reference on the Kondo effect and heavy electrons, we mention a book
by A. C. Hewson (Cambridge 1993). The present book puts more emphasis on the peri-
odic systems rather than magnetic impurities, and also on experimental aspects. Hence,
we hope that the present monograph serves to be complementary to Hewson's book. In
order to keep the size of the book reasonable, we omit some interesting theoretical topics
like the Bethe Ansatz and the conformal field theory. These topics are extensively dealt
with in other monographs.
The contents of the book are arranged as follows: Chapter 1 presents first the heavy-
electron phenomena from thermodynamic measurements, and then introduces the two
basic pictures of electrons in solids: itinerant and localized states. These pictures rep-
resent opposite limiting behaviours of electrons and do not reconcile with each other at
the naive level. A brief account of the Fermi-liquid theory is given. Some basic aspects
of NMR and neutron scattering experiments are explained.
Chapter 2 deals with the single-magnetic-impurity problem. The Kondo effect is
discussed from various viewpoints; scaling, 1/n expansion, local Fermi liquid, and so
on. We emphasize an effective Hamiltonian approach which includes the concept of
the renormalization group. The local non-Fermi-liquid state is treated with emphasis on
the unique excitation spectrum. Detailed experimental results are presented that can be
interpreted without taking account of the intersite interactions among f electrons.
In Chapter 3, we turn to the periodic systems where heavy electrons are formed.
Experimental facts on both metallic and insulating systems are presented. Although the
metallic ground state is basically understood by the Fermi-liquid picture, the system
shows strange behaviours in a magnetic field. We present intuitive ideas and a theoret-
ical apparatus for understanding the basic features of heavy electrons. Practical meth-
ods for computing physical quantities such as the density of states and the dynamical
susceptibility are explained.
These two chapters as a whole represent complementary approaches to heavy elec-
trons from the local and itinerant limits. In Chapter 4, we discuss the anomalous
PREFACE vii
magnetism which emerges as a result of a complicated interference between quasi-
particles. The relevant phenomena include the weak antiferromagnetism with tiny
ordered moments, and the orbital ordering accompanying the quadrupole moments.
Alternatively, the anomalous magnetism is interpreted as originating from a competition
between the Kondo screening and the intersite interaction, if one adopts the viewpoint
from high temperatures. Although there is no established understanding of these phe-
nomena, we present a possible view which emphasizes the dual nature of electrons with
strong correlations: the simultaneous presence of itinerant and localized characters.
Chapter 5 deals with the superconductivity of heavy electrons. We begin with an
introductory treatment of anisotropic pairing in general and symmetry classification of
the singlet and triplet pairings. Then, a detailed review of the experimental situation
for representative superconducting systems is presented. We emphasize the interplay
between superconductivity and anomalous magnetism, and subtle coupling to the lattice
degrees of freedom.
In Chapter 6, we compare heavy-electron systems with some cuprates which show
high-temperature superconductivity. The cuprates also have strongly correlated elec-
trons, but the parameters characterizing the electrons are rather different from those for
heavy electrons. Interestingly, some dynamical properties, such as the NMR relaxation
rate below the superconducting transition, show similar temperature dependence if one
scales the temperature by the transition temperature of each system. We discuss sim-
ilarities and differences between these two systems, and pursue a unified picture for
understanding the strongly correlated electrons including the non-Fermi-liquid states.
In the appendices, we summarize the theoretical techniques frequently used in the
many-body theory. They constitute a compact treatise which should be understandable,
independently of the content of the main text.
It took us an unexpectedly long time to complete the writing of this book. During
this period, we have benefited from discussions with our colleagues and friends. We
are grateful to all of them for helping our understanding on the subject, and especially
to Erwin Muller-Hartmann, who introduced one of the authors (Kuramoto) to the early
stage of the field 20 years ago, and to Kunisuke Asayama and Hiroshi Yasuoka, who
introduced the other author (Kitaoka) to the field of NMR in condensed-matter physics.
We acknowledge fruitful cooperation with and valuable suggestions by Jacques Flouquet,
Kenji Ishida, Yusuke Kato, Hiroaki Kusunose, Frank Steglich, Hideki Tou, and Hisatoshi
Yokoyama. Special thanks are due to Frank Steglich, Yusuke Kato, and Stephen Julian,
who read the first manuscript and made a number of constructive remarks to improve
the book.
Sendai Yoshio Kuramoto
Osaka Yoshio Kitaoka
June 1999
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CONTENTS
1 Fundamental properties of electrons in solids 1
1.1 What are heavy electrons? 1
1.2 Itinerant and localized states 3
1.2.1 Formation of energy bands 4
1.2.2 Localized states 8
1.3 Fundamentals of spin dynamics 13
1.3.1 Itinerant magnetic moments 13
1.3.2 Localized magnetic moments 14
1.3.3 Random phase approximation 17
1.3.4 Fermi-liquid theory 18
1.3.5 Parametrization of the dynamical susceptibility 20
1.3.6 Mode-coupling picture of spin fluctuations 23
1.4 Nuclear magnetic resonance 27
1.4.1 Phenomenology 27
1.4.2 Magnetic hyperfine interaction 29
1.4.3 Electric quadrupolar interaction 31
1.4.4 Nuclear spin-lattice relaxation time T\ 32
1.4.5 Nuclear spin-spin relaxation timeT2 35
1.5 Neutron scattering 36
1.5.1 Characteristics and utility 36
1.5.2 Magnetic scattering 37
Bibliography 40
2 Crossover from localized moment to local
fermi or non-fermi liquid 42
2.1 Description of singlet formation 42
2.1.1 Renormalization of the exchange interaction 42
2.1.2 Numerical renormalization group method 47
2.1.3 Local Fermi-liquid theory 48
2.1.4 The 1/n expansion 51
2.1.5 Effects of spin-orbit and CEF splittings 53
2.2 Dynamics of the Kondo impurity 54
2.2.1 Mean-field theory 54
2.2.2 Dynamical susceptibility in the local Fermi liquid 57
2.2.3 Self-consistent theory at finite temperatures 58
2.3 Deviation from the canonical behaviour 64
2.3.1 Non-Fermi-liquid ground state 64
2.3.2 Mapping to one-dimensional models 65
2.3.3 Properties of the non-Fermi-liquid state 68
2.3.4 Pair of local moments with hybridization 69