Table Of Content61 Springer Series in Solid-State Sciences
Edited by Peter Fulde
Springer Series in Solid-State Sciences
Editors: M. Cardona P. Fulde H.-J. Queisser
40 Semiconductor Physics 51 Phonon Scattering in Condensed Matter
An Introduction By K Seeger Editors: W. Eisenmenger, K LaBmann,
41 The LMTO Method and S. D6ttinger
Muffin-Tin Orbitals and Electronic 52 Superconductivity in Magnetic and Exotic
Structure Materials
By H. L. Skriver Editors: T. Matsubara and A. Kotani
42 Crystal Optics with Spatial Dispersion,
53 Two-Dimensional Systems,
and Excitons
Heterostructures, and Superlattices
By V. M. Agranovich and V. L. Ginzburg
Editors: G. Bauer, R Kuchar,
43 Resonant Nonlinear Interactions of and H. Heinrich
Light with Matter
54 Magnetic Excitations and Fluctuations
By V. S. Butylkin, A. E. Kaplan,
Editors: S. Lovesey, U. Balucani, R Borsa,
Yu. G. Khronopulo, and E. 1. Yakubovich
and V. Tognetti
44 Elastic Media with Microstructure II
Three-Dimensional Models 55 The Theory of Magnetism II.
By 1. A. Kunin Thermodynamics and Statistical Mechanics
By D. C. Mattis
45 Electronic Properties of Doped
Semiconductors 56 Spin Fluctuations in Itinerant Electron
By B. I. Shklovskii and A. L. Efros Magnetism By T. Moriya
46 Topological Disorder in Condensed Matter 57 Polycrystalline Semiconductors,
Editors: R Yonezawa and T. Ninomiya Physical Properties and Applications
47 Statics and Dynamics of Nonlinear Editor: G. Harbeke
Systems
58 The Recursion Method and
Editors: G. Benedek, H. Bilz,
Its Applications
and R. Zeyher
Editors: D. Pettifor and D. Weaire
48 Magnetic Phase Transitions
Editors: M. Ausloos and R. 1. Elliott 59 Dynamical Processes and
49 Organic Molecular Aggregates, Electronic Ordering on Solid Surfaces
Editors: A. Yoshimori and M. Tsukada
Excitation and Interaction Processes
Editors: P. Reineker, H. Haken, 61 Localization, Interaction, and
and H. C. Wolf Transport Phenomena
50 Multiple Diffraction of X-Rays in Crystals Editors: B. Kramer, G. Bergmann,
By Shih-Lin Chang and Y. Bruynseraede
Volume 1-39 are listed on the back inside cover
Localization,
Interaction, and
Transport Phenomena
Proceedings of the International Conference,
August 23-28, 1984
Braunschweig, Fed. Rep. of Germany
Editors: B. Kramer, G. Bergmann, and
Y Bruynseraede
With 125 Figures
Springer- ¥e rlag
Berlin Heidelberg New York Tokyo
Professor Dr. Bernhard Kramer
Physikalisch-Technische Bundesanstalt Braunschweig, Postfach 3345
D-3300 Braunschweig, Fed. Rep. of Germany
Professor Dr. Gerd Bergmann
Institut flir Festkorperforschung, Kernforschungsanlage lillich, Postfach 1913
D-5170 lillich, Fed. Rep. of Germany
Professor Dr. Yvan Bruynseraede
Laboratorium voor Vaste Stof-Fysika en Magnetisme, Celestijnenlaan 200 D
B-3030 Leuven, Belgium
Series Editors:
Professor Dr. Manuel Cardona . Professor Dr. Peter Fulde
Professor Dr. Hans-Joachim Queisser
Max-Planck-Institut flir Festkorperforschung, Heisenbergstrasse 1
D-7000 Stuttgart 80, Fed. Rep. of Germany
Organizing Committee
B. Kramer (Chairman) (Physikalisch-Technische Bundesanstalt Braunschweig, RR.G.)
G. Bergmann (Kernforschungsanlage Jti1ich, RR.G.)
Y. Bruynseraede (Universiteit Leuven, Belgium)
L. Van Gerven (Universiteit Leuven, Belgium)
W. Gey (Technische Universitiit Braunschweig, RR.G.)
Advisory Committee
R. Huguenin, Switzerland L. P. Gor'kov, USSR W. Weller, G.D.R.
M. Pepper, U.K. A. A. Abrikosov, USSR M. Ya. Az bel', Israel
P. Chaudhari, USA A. G. Aronov, USSR G. Deutscher, Israel
P. W. Anderson, USA A. A. Rogachev, USSR S. Kobayashi, Japan
E. Abrahams, USA R. Taylor, Canada S. Hikami, Japan
A. W. Overhauser, USA W. GOtze, RR.G. P. Wyder, The Netherlands
Secretary Elisabeth Meyer (Physikalisch·Technische Bundesanstalt Braunschweig, RR.G.)
Sponsors
The conference was organized under the auspices of the Deutsche Physikalische Gesellschaft
Financial support was obtained from: Volksbank Braunschweig
Physikalisch-Technische Bundesanstalt Braunschweig Firma Richard Borek Braunschweig
Stadt Braunschweig IBM Deutschland
ISBN-13:978-3-642-82518-7 e-ISBN-13:978-3-642-82516-3
DOl: 10.1007/978-3-642-82516-3
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
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by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copy
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Wort", Munich.
© Springer-Verlag Berlin Heidelberg 1985 .
Softcover reprint of the hardcover 1st edition 1985
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of a specific statement, that such names are exempt from the relevant protective laws and regulations
and therefore free for general use.
2153/3130-543210
Preface LITPIM
When we first had the idea of organizing the International Conference on
Localization, Interaction, and Transport Phenomena in Impure Metals we
expected to bring together at most a hundred physicists. The fact that more
than a hundred and fifty participated clearly shows that the topic of the
meeting was of great interest to an important fraction of the solid state
physics community. In fact, remembering that the localization problem is
already a quarter of a century old, it is quite amazing to see how, during the
last five years, new and very successful theoretical models emerged which
were confirmed by sometimes ingenious experiments. The number of groups
involved in the study of localization or related problems in the transport
properties of matter even seems to be increasing.
The main purpose of this conference was to review the present status of
activities in the localization field and hopefully to stimulate new ideas. A
study of the Conference Proceedings ascertains that we were successful in
reaching these two goals. Moreover, the presence of the authors of the about
ninety contributed papers published in the supplement volume assured the
very lively atmosphere which characterizes successful conferences. We think
that this was the most important ingredient for achieving the second goal
in particular.
We thank our sponsors for their support, which was given unreluctantly
and generously. Especially, we gratefully acknowledge the hospitality of the
PTB and the city of Braunschweig during the time of the meeting. With
out this help ,the organization of the conference would hardly have been
possible in the form which we all have experienced. We also thank the nu
merous people who have contributed by helping during the preparation of
the conference and in the daily affairs, especially Wolfgang Woger, Lud
wig Schweitzer, and Robert Johnston. In particular, we would also like to
thank Mr. Wolfgang Pogrzeba, who was responsible for the technical ar
rangements, and our conference secretary, Mrs. Elisabeth Meyer. She was
extremely efficient in organizing all the details of LITPIM, preparing the
Conference Program, the Abstract Booklet, and the Proceedings. It would
have been very hard, if not impossible, to run the meeting without her
continuous and extremely reliable work.
Braunschweig, B. Kramer
November 1984 G. Bergmann
Y. Bruynseraede
v
Contents
Localization, Interaction, and Transport Phenomena in Impure
Metals. An Introduction
By B. Kramer, G. Bergmann, and Y. Bruynseraede (With 4 Figures) 1
Part I Invited Papers
Some Unresolved Questions in the Theory of Localization
By P. W. Anderson (With 3 Figures) . . . . . . . . . . . . . . . . . .. 12
Localization and Interaction in Quasi-Two-Dimensional Metallic Films
By Shun-ichi Kobayashi (With 6 Figures) . . . . . . . . . . . . . 18
Weak Localization in Silicon Mosfets: Isotropic and Anisotropic
Two-Dimensional Electron Gases
By D.J. Bishop, D.C. Tsui, and R.C. Dynes (With 9 Figures) ... 31
Transport as a Consequence of Incident Carrier Flux
By R. Landauer (With 3 Figures) .......... . . .. 38
Interaction Effects in Weakly Localized Regime of Metallic Films
By H. Fukuyama (With 9 Figures) . . . . . . . . . . . . . . . .. . 51
The Conductor Insulator Transition in Strongly Disordered Systems
By W. Gotze (With 8 Figures) . . . . . . . . . . . . . . . . . . . . . . 62
Recent Developments in the Metal-Insulator Transition
By G.A. Thomas and M.A. Paalanen (With 10 Figures) ....... 77
The Scaling Theory of Localisation
By A. MacKiI,mon (With 2 Figures) ...... 90
Anderson Transition and Nonlinear a-Model By F. Wegner ...... 99
Localization and Superconductivity
By G. Deutscher, A. Palevski, and R. Rosenbaum (With 8 Figures) . 108
Aperiodic Strl!cture in the Magnetoresistance of Very Narrow Metallic
Rings and Lines. By R.A. Webb, S. Washburn, C.P. Umbach, and
R.B. Laibowitz (With 4 Figures) . . . . . . " . . . . . . . . . . . . 121
Theory of Weak Localization and Superconducting Fluctuations
By S. Maekawa (With 4 Figures) ................. . 130
VII
Preparation and Physics of Novel Microstructures for Localization
Studies
By M.R. Beasley, S. Bending, and J. Graybeal (With 8 Figures) ... 138
Fluctuation and Localization Effects in Quasi-One-Dimensional
Metallic Structures
By D.E. Prober, S. Wind, and P. Santhanam (With 5 Figures) ... 148
Theory of Fluctuation Phenomena in Kinetics
By M. Ya Azbel' .............................. 162
Quasi One-Dimensional Transport in Narrow Silicon Accumulation
Layers. By C.C. Dean and M. Pepper (With 6 Figures) ........ 169
Experiments on Localization and Interaction in a Strong Magnetic
Field. By G. Ebert (With 11 Figures) .............. . . . . 178
Localization and the Integral Quantum Hall Effect
By A.M.M. Pruisken (With 2 Figures) ................. 188
Quantum Hall Effect and Additional Oscillations of Conductivity in
Weak Magnetic Fields
By D.E. Khmel'nitskii (With 2 Figures) ................. 208
Electron-Phonon Scattering in Dirty Metals
By A. Schmid (With 8 Figures) ..................... 212
Electron Scattering Times in Thin Metal Films: Experimental Results
By C. Van Haesendonck, M. Gijs, and Y. Bruynseraede
(With 8 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Inelastic Lifetime of Conduction Electrons as Determined from
Non-Localization Methods
By J.E. MO,?ij and T.M. Klapwijk (With 5 Figures) .......... 233
Comments of Some Aspects of the Localization and Interaction
Problem. By E. Abrahams . . . . . . . . . . . . . . . . . . . . .... 245
Part II Titles of Contributed Papers
Localization in One Dimension . . . . . .252
Weak Localization in Three Dimensions .253
Flux Quantization in Normal Rings .254
Two-Dimensional Systems ..... . .255
Localization, Magnetism and Superconductivity. .256
Metal-Insulator Transition. .257
VIII
Localization and Interaction in the Magnetic Field .259
Inelastic Scattering Times .260
Conference Photograph .261
Index of Contributors . .263
IX
Localization, Interaction, and Transport Phenomena in
Impure Metals. An Introduction
B. Kramer
Physikalisch-Technische Bundesanstalt, Bundesallee 100
0-3300 Braunschweig, Fed. Rep. of Germany
G. Bergmann
Institut fUr Festkorperforschung der KFA JUlich, Postfach 1913
0-5170 JUlich, Fed. Rep. of Germany
Y. Bruynseraede
Laboratorium voor Vaste Stof-Fysika en Magnetisme, Universiteit Leuven
Celestijnenlaan 2000, B-3030 Leuven, Belgium
1. Overview
Transport properties of solids are determined by disorder due to the presen
ce of impurities or imperfections of the lattice, by various interaction ef
fects such as electron-electron, electron-phonon interaction and by spin-de
pendent processes due to spin-orbit and spin-spin coupling. At room tempera
ture the details of these interactions are unimportant. They are usually in
corporated into one of several mean free paths. As the temperature is lowered
these effects become however more and more important. Thus the study of the
transport properties of solids is a typical field of low-temperature physics.
A number of "classical" low-temperature effects in metals are a sig
nature of the presence of well defined microscopic processes [1]. The
residuaZ resistance is due to impurity scattering, the Kondo effect to spin
scattering, and the "classical" temperature-dependence of the resistance,
T5, is within Bloch's theory described as an electron-phonon effect. There
are "classical" rules, how to combine various scattering processes, such as
Matthiesen's rule. Many of these effects and "laws" can be derived neglec
ting quantum mechanical interference effects within the frame work of the
relaxation-time approximation. One of the main subjects of the research du
ring the past thirty years has been to include quantum mechanics into a
microscopic transport theory. This goal is not yet reached. However, one
very important step has become more and more transparent during the past
five years: The problem of localization due to disorder has been formulated
in a way allowing systematic theoretical, and what is perhaps more impor
tant, experimental studies. In addition,one may expect fruitful interaction
with the modern technology of very large-scale integration (VLSI).
It is now little more than 25 years ago that P.W. Anderson published
"The Absence of Diffusion in Certain Random Lattices" [2]. He simul
taneously formulated the problem, made the link between localization and
transport, and gave the first quantitative estimate for the critical disor
der for the transition between the diffusive and the non-diffusive regimes.
Localization may be viewed as one of the most fundamental quantum mechani
cal phenomena related to real condensed matter. The most simple model to
discuss is that of a spinless, noninteracting particle moving in a random
potential. As it forms the basis for the far more complicated theory inclu
ding spin and interactions/we want to discuss its qualitativ~ aspects in
some detail in the following sections. More details will be presented by
the various authors in this volume.
2. Definition of Localization
In this section we consider a system at zero temperature. For a classical
particle in a random potential it is comparatively simple to decide whether
its motion is restricted to finite portions of the space (Fig. 1).
v(x)
£2 -----------------------
.%1 X2
~r-~--~r.~-+;-----~--~~X
Fig. 1: Random potential
If the energy of a particle is higher than Eo it can move through the
whole space. On the other hand, if the energy is smaller than Eo, say EI ,
the motion is restricted to finite intervals, (Xl' X2)' (X3' X4), etc. In
this case, one would assign to Eo a mobility edge. For the corresponding
quantum mechanical problem the situation is more complicated for two rea
sons: Firstly, there is the tunnel effect, which allows the particle to
tunnel through the potential wells, thus delocalizing the particle. Second
ly, there might be interference effects within the wave function scattered
by the random potential. These may lead to localization of the particle via
destructive superposition for energies above Eo. Thus, quantum mechanically,
the conditions for localization are not as simple as in classical mechanics.
An example for the first effect, delocalization due to tunneling, are
the Bloch states in perfect crystals, which can be found in every text book
of solid state physics. An example for localization via interference is the
quantum mechanical behaviour of a particle in a one-dimensional random po
tential. In this case only localized states occur, independent of the mag
nitude of the disorder and/or the energy of the particle [3].
We have deliberately used the words "localization", "delocalization",
and "disorder" without further details. Let us now specify what we mean
bv localization. One of the interestinQ asoects of the oroblem
of localization is the fact that it depends strongly on the dimensionality
of the system. In a hypercube of volume Ld all states are d-dimensional pla
ne waves, either standing or propagating, depending upon the boundary condi
tions. These states are extended since the probability to find an electron
at site x in a volume ddx is constant and equal to ddx/Ld.A trivial case of
exponential localization is obtained if a charged nucleus is placed somewhere
within the hypercube. A bound electron will be restricted to a volume around
the nucleus. The corresponding probability density is exponentially decrea
sing far away from the nucleus. If we introduce however a random potential
v(x) into the hypercube the situation is more complicated.
First of all we have to specify the potential. This is usually done by
replacing the system by a statistical ensemble, which is characterized by a
probability distribution. In the case of a Gaussian white noise potential,
v(x), one has
2
