Diffusion in Solids. Recent Developments

494 Pages·1975·10.152 MB·English

by  A.S. Nowick (Eds.)

MATERIALS SCIENCE AND TECHNOLOGY
EDITORS
ALLEN M. ALPER JOHN L. MARGRAVE A. S. NOWICK
GTE Sylvania Inc. Department of Chemistry Henry Krumb School
Precision Materials Group Rice University of Mines
Chemical & Metallurgical Houston, Texas Columbia University
Division New York, New York
Towanda, Pennsylvania
A. S. Nowick and B. S. Berry, ANELASTIC RELAXATION IN CRYSTALLINE SOLIDS,
1972
E. A. Nesbitt and J. H. Wernick, RARE EARTH PERMANENT MAGNETS, 1973
W. E. Wallace, RARE EARTH INTERMETALLICS, 1973
/. C. Phillips, BONDS AND BANDS IN SEMICONDUCTORS, 1973
H. Schmalzried, SOLID STATE REACTIONS, 1974
/. H. Richardson and R. V. Peterson (editors), SYSTEMATIC MATERIALS ANALYSIS,
VOLUMES I, II, AND III, 1974
A. /. Freeman andJ. B. Darby, Jr. (editors), THE ACTINIDES: ELECTRONIC STRUC­
TURE AND RELATED PROPERTIES, VOLUMES I AND II, 1974
A. S. Nowick and J. J. Burton (editors), DIFFUSION IN SOLIDS: RECENT DEVELOP­
MENTS, 1975
/. W. Matthews (editor), EPITAXIAL GROWTH, PARTS A AND B, 1975
/. M. Blakely (editor), SURFACE PHYSICS OF MATERIALS, VOLUMES I AND II, 1975
G. A. Chadwick and D. A. Smith (editors), GRAIN BOUNDARY STRUCTURE AND
PROPERTIES, 1975
John W. Hostie, HIGH TEMPERATURE VAPORS; SCIENCE AND TECHNOLOGY, 1975
In preparation
John K. Tien and George S. Ansell (editors), ALLOY AND MICROSTRUCTURAL
DESIGN
Lyle Schwartz and J. B. Cohen, DIFFRACTION FROM MATERIALS
DIFFUSION IN SOLIDS
RECENT DEVELOPMENTS
Edited by A. S. NOWICK
Henry Krumb School of Mines
Columbia University
New York, New York
J. J. BURTON
ESSO Research Center
Linden, New Jersey
ACADEMIC PRESS New York San Francisco London 1975
A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT © 1975, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
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United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
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Library of Congress Cataloging in Publication Data
Nowick, A S
Diffusion in solids.
(Materials science and technology series)
Includes bibliographies.
1. Diffusion. 2. Solids. I. Burton, James
Joseph, Date joint author. II. Title.
QC176.8.D5N68 531'.3 74-5704
ISBN 0-12-522660-8
PRINTED IN THE UNITED STATES OF AMERICA
List of Contributors
Numbers in parentheses indicate the pages on which the authors' contributions begin.
G. ALEFELD (231), Physik-Department der Technischen Universitä
München, 8046 Garching, West Germany
T. R. ANTHONY (353), Metallurgy and Ceramics Laboratory, Researc
and Development Center, General Electric Company, Schnectady, Ne
York
CHARLES H. BENNETT (73), IBM Thomas J. Watson Research Center,
Yorktown Heights, New York
WILBUR M. FRANKLIN (1), Physics Department, Kent State University,
Kent, Ohio
W. J. FREDERICKS (381), Chemistry Department, Oregon State
University, Corvallis, Oregon
ROBERT A. HUGGINS (445), Center for Materials Research, Stanford
University, Stanford, California
H. B. HUNTINGTON (303), Department of Physics, Rensselaer Poly­
technic Institute, Troy, New York
N. L. PETERSON (115), Materials Science Division, Argonne National
Laboratory, Argonne, Illinois
D. TURNBULL (171), Division of Engineering and Applied Physics,
Harvard University, Cambridge, Massachusetts
J. VÖLKL (231), Physik-Department der Technischen Universität
München, 8046 Garching, West Germany
. K. WARBURTON (171), Division of Engineering and Applied Physics,
Harvard University, Cambridge, Massachusetts
IX
Preface
The study of diffusion in crystalline solids has become an increasingly
active field since 1945 when a wide variety of radioactive isotopes of the
elements began to be available. Because of this increased activity, our
concepts of diffusion mechanisms have developed rapidly in the last 25 years
or so; simultaneously, our detailed knowledge of the point defects which
are so often responsible for the elementary diffusion step increased. There
have been many fine review articles and several textbooks on diffusion,
which provide an adequate systematic introduction to the subject. Such is
not the objective of the present volume. Rather, it was desired to bring
up to date several of the most active areas of study of the past decade by
presenting reviews written by foremost authorities in these areas. Often the
present status of the subject will be found to be still in a state of flux, but it
is hoped that each of these critical reviews will clarify the field for the
reader and better help him to understand the rapidly evolving literature.
The first two chapters deal with aspects of the theory of diffusion. The
elementary diffusion jump has continued to fascinate the theoretician as
one of the simplest examples of a chemical rate process, and yet it has
turned out to be an elusive one. In Chapter 1, the problem is attacked
from several different viewpoints, for example, harmonic and anharmonic
theories, and the use of quantum mechanical concepts. In Chapter 2, a
relatively new approach is presented; detailed and exact information about
hypothetical materials is calculated using high-speed computers.
Almost all the experimental chapters deal with new tools or approaches
which supplement conventional diffusion measurements. Examples are
isotope effects, vacancy wind effects, ionic conductivity, and electromigration.
Some of the topics have become especially active in the last few years
because of their great technological potential, for example, electromigration
and very rapid ionic transport. Others, such as fast diffusion in metals, and
the unique aspects of diffusion of the lightest element, hydrogen, are topics
of enormous intrinsic interest, although not without technological impor­
tance too. Finally, Chapter 8 shows how diffusion in alkali halides, long
XI
Xll PREFACE
thought to be the simplest of systems, is really extremely complex, to an
extent that is only just beginning to be understood.
While these topics are not the only active ones in the field of diffusion,
in order to limit the size of this book, it was necessary to restrict our
coverage. Nevertheless, we do feel that those areas presented here represent
a set of topics of prime importance at this stage in the development of the
field of diffusion.
I
Classical and Quantum Theory of Diffusion in Solids
WILBUR M. FRANKLIN
PHYSICS DEPARTMENT
KENT STATE UNIVERSITY
KENT, OHIO
I. Introduction 1
II. Jump Frequency 5
A. Introduction 5
B. Classical Harmonic Approximation 6
C. Model with Anharmonicity and Deformed Force Constants 10
III. Lattice Vibration Theory 14
A. Introduction 14
B. Harmonic Theory 17
C. Anharmonic Theory 24
IV. Defect Modes 26
A. A descriptive Model for Diffusion in Terms of Localized, Resonance,
and Band Modes 27
B. Inhomogeneity Matrix 33
C. Linear Chain with Impurity Adjacent to a Vacancy 36
D. Green's Function in Terms of Defect Lattice Eigenfunctions 40
E. Displacement and Momentum Correlation Functions 41
F. Internal Degrees of Freedom 42
G. Defect Calculations Related to Diffusion 44
V. Temperature Dependence—Classical Case 50
VI. Mass Dependence—Classical Case 56
VII. Quantum and Anharmonic Effects 58
VIII. Lattice Vibration Theory and Diffusion Experiments 69
References 70
I. Introduction
The seemingly simple process of diffusion in solids is, when analyzed
in detail, one of the most complex theoretical problems in solid state
physics and materials science. As a simple example, consider the Arrhenius
1
2 WILBUR M. FRANKLIN
plot for the diffusivity, D = D exp(-AH/kT), which shows an exponential
0
temperature dependence and an activation enthalpy AH and a preexponential
diffusion constant D, which are constant with respect to the temperature.
0
This empirical result is surprising in view of the anharmonic terms, including
many-body forces, that come into play with the very large saddle-point
strains inherent in the migration event. In the classical work of Wert (1950)
on the diffusion of carbon in a iron (which is reproduced in Fig. 1), he
said, after finding that AH was constant over a large temperature range,
"This fact is perhaps the most surprising result of the analysis ...."J
Temperature (°C)
-40 -10 100 200 400 600 800
1 1 1 1 1
- / -
/*
r -
/D
- -
s /
- -\
A/
1 1 1 1 1 1 1 1 1 1 1 1 1 1
4.4 3.8 3.0 2.2 1.4 0.6 0
1000 /T
Fig. 1. The diffusion coefficient for C in a iron, taken from Wert (1950). This plot
shows that Arrhenius-type behavior is followed by D vs T~l in this system for more than
14 orders of magnitude.
There are, in addition to the linearity of the Arrhenius plot, many other
features of diffusion, both overt and obstruse, which challenge theoretical
understanding or explanation. The contributions to the understanding of
migration events by theorists such as Wigner (1933), Hirschfelder and
Wigner (1939), Bardeen and Herring (1952), Eyring (1935), Montroll and
Potts (1955), and Bak and Prigogine (1959) indicate the complexity to the
problems involved as well as the intrigue of attempting to formulate viable
% While more recent data for the diffusion of C in Fe show a definite departure from the
Arrhenius relation, other systems (e.g. N in Fe) do obey the relation very well (Lord and
Beshers, 1966).
1. CLASSICAL AND QUANTUM THEORY OF DIFFUSION 3
theoretical paradigms and constructs for a difficult problem. However,
despite the work of many capable theorists, a good attempt has not yet
been made to find the localized and resonance modes of the equilibrium
configuration! (EC) or the saddle-point configuration (SPC) of even the
most commonly studied diffusion defect—an impurity adjacent to a
vacancy—in a three-dimensional model. There are numerous other examples
of complex problems in atomic migration theory which have not yet been
solved or are only partially solved. Some of these will be stated or
inferred in this chapter which deals, principally, with the theory of the
basic migration event in the solid state.
The thermodynamic description of rate processes was developed initially
by Eyring (1935) and was followed by Wert (1950), who corrected the
description of the activated state to include the proper number of degrees of
freedom. Wert and Zener (1949) and Zener (1951) have given a theoretical
basis for the preexponential part of the diffusivity, D . Basic background
0
material and contact with experimental aspects of diffusion can be found in
the articles by LeClaire (1949, 1953) and Lazarus (1960) and in the Pocono
Manor Symposium (1952). The phonon theory of point defects, which
considers the 3N frequencies of the ΛΓ-body system, was formulated initially
by Huntington et al (1955) who stated that the theory could be extended to
diffusion. Application to the migration process followed in the work by
Vineyard (1957) and, in a different approach, by Rice (1958), Manley and
Rice (1960), and Lawson et al. (1960). Vineyard, in addition to applying the
many-body theory to the jump frequency, introduced the mass dependence,
noting that the simple m~1/2 mass dependence expected for free particle
motion or for the motion of an atom in a simple harmonic lattice does not
hold exactly in a solid. Later, the essential equivalence of the harmonic
theories of Vineyard (1957) and Rice (1958) was shown by Franklin (1967)
and by Glyde (1967). Rice and Frisch (1960) developed an anharmonic
model of the migration process in one dimension. An alternative develop­
ment of the theory of migration has been provided by Flynn (1968, 1972)
and by Feit (1971, 1972).
As the elementary jump process was placed on a reasonably firm
theoretical basis utilizing reaction rate theory, more attention was paid to
fundamental and complex questions such as the role played by correlations,
anharmonicity, many-body effects in the sense of more than two-body
interactions, quantum effects, and refinements of the isotope effect. The first
work on anharmonic effects in rate processes, which was done by Slater
ί The abbreviations EC (equilibrium configuration) and SPC (saddle-point configuration)
are used throughout this article. The SPC is often considered to be an equilibrium state
of the system but we used SPC, nontheless, simply to denote the different atomic
arrangement from that of the EC.
4 WILBUR M. FRANKLIN
(1959), was followed by that of Rice and Frisch (1960), Franklin (1967,
1969), and Flynn (1971b). The thermalization process of an activated species,
which involves anharmonic effects, was studied by Geszti (1967) based on the
original work on anharmonicity in solids by Bak and Prigogine (1959).
Wigner (1933) and Hirschfelder and Wigner (1939) studied the quantum
effects of tunneling on small mass diffusion. Quantum statistics were studied
by LeClaire (1966) and Franklin (1969) and tunneling effects were included,
in addition, by Weiner and Partom (1969, 1970), Flynn (1971a) and
Ebisuzaki et a\. (1967a,b). The zero-point energy, another observable
quantum effect which is a part of quantum statistics, was discussed by
Flynn and Stoneham (1970). The zero-point energy in quantum crystals was
discussed by Varma (1971). The many-body terms of the generalized
anharmonic terms (three- and higher body interactions) were developed by
Burton (1969). The effects of correlations, studied initially by Bardeen and
Herring (1952) and Compaan and Haven (1956, 1958), were developed
further (Manning, 1968). The effects of di-, tri-, and multivacancy diffusion
mechanisms on the Arrhenius plot were discussed by Seeger and Mehrer
(1968, 1969a,b, 1970), Burton (1970), Burton and Jura (1967), Burton and
Lazarus (1970), and Rothman and Peterson (1969). The effects of local
modes were studied first by Montroll and Potts (1955) and then by Schottky
(1965), with reference to the Soret effect.
Most of the topics cited above have become specialized areas of study
so that significant contributions to the theory of diffusion now often require
a detailed theoretical knowledge of contemporary statistical or quantum
mechanics, lattice vibration theory with defects, pseudopotential theory, etc.
Consequently, it is too imposing a task to treat all the significant areas of
diffusion theory adequately and critically in a short article. Therefore,
certain areas, which are outlined below, have been chosen for emphasis since
the remaining areas cited have been treated elsewhere or, on the other
hand, do not fit comfortably within the framework of the topics selected
here, which are couched in the theory of lattice vibrations.
In Section II, the Vineyard theory is developed for the atomic jump
frequency, utilizing the harmonic approximation for the associated lattice
vibrations. Then, anharmonic terms and expansions of deformed saddle-
point force constants are used in one-dimensional models to obtain the
saddle-point frequencies and anharmonic corrections using known
quantities. Section III begins with a brief introduction giving some of
the limitations and domain of applicability of phonon theory. Then, the
harmonic theory is presented, followed by the addition of anharmonic terms.
Lattice vibration theory with the presence of atomic defects is utilized from
the outset; the development is given in second quantized form and the
impingement on diffusion theory is assessed from a basic standpoint