Mass Transport in Solids
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Mass Transport in Solids Edited by
F. Beniere University of Rennes I Rennes, France
and
Springer Science+ Business Media, LLC
Proce~dings of a NATO Advanced Study Institute on Mass Transport in
Solids, held June 28-July 11, 1981, in Lannion, France
Library of Congress Cataloging in Publication Data
NATO Advanced Study Institute on Mass Transport in Solids (1981:
Lannion, France) Mass transport in solids. (NATO advanced science
institutes series. Series B, Physics; v. 97) "Published in
cooperation with NATO Scientific Affairs Division." "Proceedings of
a NATO Advanced Study Institute on Mass Transport in
Solids, held June 28-July 11, 1981, in Lannion, France"-T.p. verso.
Includes bibliographical references and index. 1.
Diffusion-Congresses. 2. Solids-Congresses. 3. Mass
transfer-Con
gresses.I.1Beniere, F.ll. Catlow, C. R. A. (Charles Richard
Arthur), 1947- . Ill. North Atlantic Treaty Organization.
Scientific Affairs Division. IV. Title. V. Series. QC176.8.D5N37
1981 530.4'1 83-8142 ISBN 978-1-4899-2259-5 ISBN 978-1-4899-2257-1
(eBook) DOI 10.1007/978-1-4899-2257-1
© 1983 Springer Science+Business Media New York Originally
published by Plenum Press, New York in 1983 Softcover reprint of
the hardcover 1st edition 1983
All rights reserved. No part of this book may be reproduced, stored
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ACKNOWLEDGEMENTS
We would like first to thank NATO for their support of the ASI on
which this book is based.
In addition we would like to thank the following organisations for
financial support: the British Council, Centre National d'Etudes
des Telecommunications, Universite de Rennes, University College
London, L'Institut Universitaire de Technologie de Lannion, Commis
sariat de L'Energie Atomique, Centre National de la Recherche
Scientifique.
Finally we would like to thank Mrs. Nina Paterson, Ms. Rosemary
Rosier and Mffie H. Halopeau for their efficiency in preparing the
manuscript.
v
PREFACE
Atomic transport in solids is a field of growing importance in
solid state physics and chemistry, and one which, moreover, has
important implications in several areas of materials science. This
growth is due first to an increase in the understanding of the
fund amentals of transport processes in solids. Of equal
importance, however, have been the improvements in the last decade
in the experi mental techniques available for the investigation of
transport phenomena. The advances in technique have stimulated
studies of a wider range of materials; and expansion of the field
has been strong ly encouraged by the increasing range of applied
areas where transport processes play an essential role. For
example, mass transport phenomena play a critical role in the
technology of fabrication of components in the electronics
industry. Transport processes are involved both during the
fabrication and operation of devices and with the growing trend to
miniaturisation there are increasing demands on accurate control of
diffusion processes.
The present book (which is based on a NATO sponsored Advanced Study
Institute held in 1981 at Lannion, France) aims to present a
general survey of the subject, highlighting those areas where work
has been especially active in recent years. Thus following
introductory accounts in chapters (I) and (2) of the basic
theoretical and experi mental aspects of transport in solids, the
book continues with a detailed account by Lidiard in chapter (3) of
important recent theo retical advances in diffusion theory -
principally the development of a kinetic theory of transport
processes in solids. In chapter (4) Jacobs then surveys the state
of present understanding of the con ductivities of strongly ionic
solids, mainly the halides of the alkali and alkaline earth metals
- systems for which there probably exists the most detailed and
accurate transport data.
The use of sophisticated techniques has, as remarked, played a
notable role in recent advances in our understanding of transport
processes. Among these, the contribution of computer simulation
methods deserves emphasis. Probably the most successful to date
have been the 'static' simulation methods discussed by Mackrodt in
chapter (5). These yield values of formation and migration
energies
v~
PREFACE
of the defects which control transport in solids, which have proved
of considerable use in analysing and interpreting experimental
data. Of potentially greater power are the dynamical simulation
techniques described by Jacucci in chapter (6), although to date
their applica tion to solids has been limited. In chapter (7),
Wolf presents a general survey of the theory of correladon effects
in atomic trans port; in particular, a recent theoretical
development - the encounter model - is discussed in detail.
Advances in experimental techniques include the application of NMR
methods to the elucidation of ion migration mechanisms; this topic
is also discussed in chapter (7). Lechner in chapter (8) then
presents a detailed account of the use of inelastic neutron
scattering techniques which are becoming of in creasing importance
in studies of transport in solids with more mobile atoms. Chapter
(9) describes a r~cent theory of diffusion in a temperature
gradient
The book then continues with a survey of mass transport in
different classes of material: metals are discussed by Brebec in
chapter (10); molecular solids by Chadwick in chapter (11); Pfister
in chapter (12) discusses transport in semiconductors and Faivre
des cribes amorphous materials. The next three chapters are
devoted to oxide materials owing to the importance and diversity of
these sys tems. Wuensch in chapter (14) describes diffusion in
relatively simple binary oxides, while S~rensen (chapter 15) and
Catlow (chap ter 16) discuss the complexities which arise due to
disorder induced by deviation from stoichiometry which occurs in a
large number of transition metal, lanthanide and actinide oxides.
The simpler prob lems posed by the ionic halides are reviewed by
Jacobs in ·chapter (4).
Most of the discussion in these chapters relates to bulk trans
port, i.e. transport through a crystal (or region of amorphous
material). However, in manypractical situations transport is con
trolled by non-bulk mechanisms: grain boundary diffusion, pipe dif
fusion down dislocations or surface transport. Thus, chapter (17)
by Heyne concentrates on grain boundary effects, while Tasker
considers surface properties in chapter (18).
The last three chapters of the book consider applications. Three
topics are discussed. Corish and Atkinson in chapter (19) consider
corrosion - in particular, the extent to which knowledge of
fundament al transport properties of oxide and sulphide films
assists our under standing of corrosion processes. Vedrine in
chapter (20) describes the importance of mass-transport in the
operation of heterogeneous catalysts, while in the final chapter
(21) Steele reviews the topical field of battery materials and
'superionic' conduction. Again, emphasis is given to the role of
knowledge at a fundamental level in understanding problems of
applied importance.
Finally, in order to show the diversity of contemporary
research
PREFACE ix
~n this field, we have collected in the Appendix,abstracts
submitted by participants of the NATO ASI.
The book aims therefore to lead the reader through from the fund
mentals to the applied areas of this field. We also hope that the
book shows how the field interacts with many of the most important
modern physical techniques employed in this exciting and expanding
subject.
F. B~ni~re C.R.A. Catlow
CONTENTS
CHAPTER (1): Introduction to Mass Transport in Solids . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 C.R.A.
Catlow
CHAPTER (2): Les Techniques de la Diffusion.............. 21 F.
Beniere
CHAPTER (3): The Kinetics of Atomic Transport
CHAPTER (4):
Ionic Conductivity P.W.M. Jacobs
81
CHAPTER (5): Theory of Defect Calculations for Ionic and Semi-Ionic
Materials •••••••••••••••••••• 107 W .C. Mackrodt
CHAPTER (6): Computer Experiments on Point Defects and Diffusion •
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
131 G. Jacucci
CHAPTER (7): Theory of Correlation Effects in Diffusion • • • • • •
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • 149 D.
Wolf
CHAPTER (8): Neutron Scattering Studies of Diffusion in Solids . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
169 R.E. Lechner
CHAPTER (9): Diffusion in a Temperature Gradient ••••••••• 227 M.J.
Gillan
CHAPTER (10): La Diffusion dans les Metaux •••.••••.••••••• 251 G.
Brebec
xi
CHAPTER ( 12):
321
CHAPTER (13): Diffusion dans les Solides Amorphes ••..•••. 333 G.
Faivre
CHAPTER (14): Diffusion in Stoichiometric Close-Packed Oxides • • •
. . . • • . . • . • . . • . . . . . • . • • . . . . • . • • • • • •
353 B.J. Wuensch
CHAPTER (15): Highly Defective Oxides ..•...•..••.••.....• 377 O.T.
Sl,lirensen
CHAPTER (16): Non-Stoichiometry and Disorder in Oxides •.. 405
C.R.A. Catlow
CHAPTER (17): Interfacial Effects in Mass Transport in Ionic Solids
. . . . . . . . . • . . . . . . • . . . . • • • . . • • 425 L.
Heyne
CHAPTER (18): The Surface Properties of Ionic Materials.. 457 P.W.
Tasker
CHAPTER (19): Corrosion . • . • • . . • . . . • . . . • • • . • . .
. . . • . • • . . • . . 477 A. Atkinson and J. Corish
CHAPTER (20): Mass Transport in Heterogeneous Catalysis . . • . . .
. . . . . • . . . • . . • . • . . . . . . . . . . . . . 505 J.C.
Vedrine
CHAPTER (21): Electrochemical Applications of Super ionic
Conductors . . . . . . . . . . . . . • . . . . . . . . 53 7 B.C.H.
Steele
APPENDIX Short contributions 567
C.R.A. Catlow
Department of Chemistry University College London 20 Gordon Street,
London WClH OAJ
1. Introduction
This chapter aims first to outline the basic features of the theory
of transport in solids and the relationship between macroscopic
transport coefficients and atomistic migration mechanisms. Secondly
we shall provide the necessary background in defect physics, giving
emphasis, however, to areas where there have been notable
theoretical developments in recent years. Our survey will look
forward to the more detailed theoretical surveys of Lidiard
(Chapter 3) and Wolf (Chapter 7), and to the summaries of
experimental work, particularly that of Jacobs (Chapter 4) on ionic
materials. We stress, as in later chapters, interpretation of
experimental data at an atomistic level, which indeed is a major
theme of this book, and we aim to show the importance in transport
studies of the concerted use of several techniques including both
theoretical and experimental methods.
2. Macroscopic transport coefficients
Bulk transport measurements generally refer either to a flux of
matter in a chemical potential gradient or of a flux of charge in
an electrical potential gradient. The former case results in
diffusion; the latter in conductivity. In both cases the flux may
be taken to a first approximation, as linear in the appropriate
gradient. Thus for diffusion, if we consider the simple case in
which there is a gradient in the chemical potential (and hence the
concentration) that is solely in the x direction, we may write down
the fallowing relationship
2 C. R. A. CATLOW
JM=-n (.E..) X kT
(1)
where D is the diffusion coefficient, ~ is the chemical potential;
JM is the rate of particle tr~nsport across unit area, and n is the
~umber of particles per unit volume. On substituting the standard
expression for ~x as a function of n, equation (1) simplifies to
the more familiar form:
}f = -D X
(2)
However, the original equation, representing the basic idea of
macroscopic transport down a potential gradient, which plays the
role of a 'force', is of great value and will be developed and
discussed in greater detail in Chapter ( 3 ) • An analogous
equation cen be written for the rate of charge transport, Jq, in an
electrical field gradient 3$/3x; that is we may ~ite
34> Jq = -a __ x (3)
X 3x
in which a is the electrical conductivity. Equation (3) may,
however, be meaningfully rearranged following the concept, outlined
above, of transport being driven by potential gradients which play
the role of 'forces'. However, in the case of charge transport the
'force' is q(3$/3x), where q is the charge of the migrating
species, Thus we have
Jq = _ ~ (q acpx)· X q dX
(4)
Furthermore, we note that~= q-l Jq. Hence, if we make the
assumption that charge tran~port res~onds to the driving force
q(3$/ax) just as mass transport does to the force (a~/ax) we are
led to the relationship:
a _ nq2 D- kT (5)
which is known as the Nernst-Einstein equation, a more formal
derivation of which is given by Mott and Gurney1•
Transport may be affected by a third type of force, which is
produced by gradients in temperature. The theory of this phenomenon
is developed by Gillan* (Chapter 9). However, our
* A notable contribution to the experimental study of this
phenomenon is also presented in the abstract of Zeqiri,
INTRODUCTION TO MASS TRANSPORT
present discussion continues with the consequences of the
Nernst-Einstein relationship, as it is evident that the validity of
the relation provides a simple, if crude, way of extracting
mechanistic information. Deviation of the ratio cr/D from the
predicted value implies that charge and mass transport are effected
by different mechanisms; alternatively the same mechanism may be
operative, but may effect transport of charge and mass to different
extents. The most obvious examples of the former case are when
diffusion js effected by an ionic exchange mechanism which effects
no charge transport;* an example will be given below. The latter
case is more subtle and is generally described in terms of
'correlation effects', a simple example
3
of which can be given for diffusion effected by the simplest defect
mechanism, i.e. vacancy migration.
Consider a chloride ion vacancy in NaCl. The species behaves as a
charged entity and may migrate by a succession of random jumps,
effecting charge transport. However, Cl- ion transport, as measured
for example by radioactive tracer methods (a discussion of which is
given in Chapter 2 by Beniere) even though it is effected by the
same mechanism is not entirely a random process, since after a
tracer ion has been transferred by a vacancy jump there is a
probability (equal to 1/Ncl where Ncl is the number of nearest
neighbour Cl- ions) that the subsequent jump of the vacancy returns
the tracer ion to its original site; that is after an initial
tracer jump, the subsequent jump is not purely random but is
correlated with the initial jump. The consequence of this
correlation effect in tracer transport compared with the purely
random nature of charge transport is a deviation from the
Nernst-Einstein relationship, which is represented by a correlation
factor f. Thus, assuming conductivity and diffusion are effected by
the same mechanism, f is measured as:
-1 ( ~/D ) nq /kT
f (6)
Further discussions of correlation factors are given in earlier
reviews of Lidiard2 and Ha~en3 and detailed discussions of the
theory and of the techniques for measuring this important quantity
are given in Chapter ( 7) by Wolf. It is clear, however, from the
discussion given above, that measurement of the correlation factor
(which of course necessitates highly accurate experimental data)
provides valuable mechanistic information, and indeed determination
of the correlation factor provides one of the hest ways of
investigating transport at an atomic level.
* Another simple case is the occurrence of diffusion by a neutral
species (e.g. a vacancy pair) which clearly cannot effect charge
transport.
4 C. R. A. CATLOW
The mechanisms which effect ion transport are almost invariably
associated with point defects, a simple example of which - vacancy
migration - has been given above. Non-defect mechanisms have been
proposed in a limited number of cases; an example is the direct
fluoride ion exchange mechanism illustrated in Figure 1. However,
in the remainder of this chapter we shall assume that the magnitude
of D and cr is controlled by mobilities and concentrations, x, of
point defects.* Thus we shall take the result of the application of
random walk theory to a hopping model of diffusion which gives for
the diffusion coefficient
1 2 D= 0 xvr (7)
*
Figure I Direct exchange mechanism proposed for F- ions in alkaline
earth fluorides. Lattice fluoride ions (which are at the corners of
the cubes in the diagram) exchange by migration through
interstitial sites.
By concentration we mean mole fraction, i.eo fraction of lattice
sites occupied by defects.
INTRODUCTION TO MASS TRANSPORT
3. Basic defect physics
The previous section established that,for most crystalline
materials,transport is effected by migration of well defined
defects; and is, therefore, according to equations (5) and (7),
governed by two crucial factors: first, the number of defects
present in the lattice and secondly, their mobility. Both factors
will now be considered.
3.1 Defect mobility
Defect transport has generally been treated in terms of a 'hopping'
mechanism, in which transport is effected by a set of discrete
events, the time scale of which Th, is such that when compared with
the time TR' spent by the migrating particle between the events, we
have in general
(8)
Given the applicability of this description, the rate of particle
transport may then be treated by a method based on Absolute Rate
Theory4. The theory gives the following expressions for the
frequency, v, of defect jumps
v = vo exp(-Eact/kT) (9)
where E t is the potential energy of the 'saddle-point' for the
defect &tgration mechanism - that is the maximum in the
potential energy profile for the migratjon route relative to the
energy of the ground state.* The concept of thermally activated
defect migration mechanisms characterised by well identified
activation energies, is central to our present understanding of
mass transport in solids. Activation energies can readily be
determined from experimental studies since the rate of transport,
according to equation t9), shows 'Arrhenius' behaviour (i.e. a
linear dependence of log(v) on T-1), Moreover, activation energies
are amenable to theoretical calculations by the computer simulation
techniques discussed in Chapter ( 5) by Mackrodt. Indeed the
comparison of experimental and calculated activation energies has
proved to be a most useful way of investigating mechanism.
*
5
Other assumptions enter into the derivation of equation (9); one
important one is that the potential surface around the saddle point
is harmonic.
6 C.R.A.CATLOVV
important condition is that
E >> kT act (10)
i.e. for the hopping description to be valid, the activation energy
must be considerably greater than the average thermal energies of
ions in the crystal. We emphasize this point as for an important
class of materials, namely the superionic conductors (which are
considered in detail in Chapter (2l)),exceptionally low activation
energies may occur, in which case the validity of hopping descrip
tions could possibly become questionable. However, recent workS
suggests that even for these systems the hopping descriptions freq
uently provide a reasonable approximation.
A further point to note is that the treatment of defect transport
given above is applicable to reorientation in addition to bulk
transport processes. The commonest case of the former involves
migration of a defect around an impurity centre to which it is
bound; an example is provided by the migration of vacancies around
divalent impurities in alkali halide crystals, possible mechanisms
for which are illustrated in Figure 2. Defect re orientation has
been widely studied by relaxation techniques, including dielectric
and anelastic relaxation and ionic thermocurrent (ITC) methods.
These clearly establish Arrhenius type behaviour for the
reorientation process. The theory is considered in detail by
L~diard both in the present book (Chapter 3) and elsewhere •
Activation energies for defect re orientation may be obtained, and
one should note that comparisons of calculated and experimental
activation energies have again proved of value in elucidating
mechanism.
Thus to summarise, the rate of defect transport (both bulk
transport and dipolar reorientation) is to a large extent
controlled by the energies of the saddle point configurations of
the migration mechanisms. The remainder of the chapter will be
concerned primarily with the second crucial factor in equation (7),
i.e. the concent~qtion term.
3.2 Point defect concentrations
We distinguish between two different origins for the equilibrium
point defect concentration* in a crystal: the first we shall
identify as thermally produced disorder; the second occurs as a
response to impurities. Both types of defect population may
contribute to transport in a given crystal, their relative
importance depending on temperature and on impurity
concentrations.
* Non-equilibrium concentrations defects may also be produced by
mech- anical processes or radiation damage. The latter is indeed a
major area of defect physics an account of which is given by
Hendersonl4.
INTRODUCTION TO MASS TRANSPORT 7
Figure 2 Reorientation mechanisms for dopant vacancy pairs in
alkali halides; mechanisms are effected by vacancy jumps, i.e.
lattice cations ( 0) and dopant ions (~) jump into vacancy
neighbouring dopant ion. For further discussion see chapter
(3).
A. Thermally generated disorder
The existence of an equilibrium concentration of defects in pure
crystals can best be understood in terms of the concept of 'defect
reactions' of which there are two basic types. The first, known as
Frenkel disorder involves the displacement of atoms (or ions) from
lattice to interstitial sites, which can be represented by the
equation
LAT (1)
~VAC ~(a)
(11)
where lAT indicates a perfect lattice site (whose activity may be
written as unity for low defect conc~ntrations). VAC and !NT
indicate vacancies and interstitials* respectively whose activities
are av and ar. KF is the equilibrium constant for the reaction. In
the subsequent treatment in this section, activities will be
approximated by concentrations, in which case a standard
mass-action treatment based on chemical thermodynamics allows us to
write the following equation for the vacancy and interstitial
concentrations xv and x1
* We should stress that the Frenkel pair (vacancy + interstitial)
indicated on the right hand side of equation (11) refers to an
isolated pair of non-interacting defects, that is not a pair of
defects on neighbouring sites.·
8 C. R. A. CATLOW
(12)
where gF is the standard Gibbs free energy of Frenkel pair
formation; the subgcript p is added to emphasize that the term
refers to measurements at constant pressure. Expressing gF in terms
of its component enthalpy hF and entropy sF, we have P
p p
x xi= exp(//k) exp(-hF/kT) (13) v p p
showing that the defect concentration, as well as the mobility
term, is governed by an exponential term thus giving rise to
Arrhenius behaviour in the concentration in addition to mobility
factors.
We have already noted that the thermodynamic parameters used in
equation (12) are constant pressure terms. 7However, we should note
a relationship recently derived by Gillan who showed that
gp = fv (14)
where f is the free energy of defect formation at constant volume.v
This enables us to write
F F sv -u
xvxi = exp(k) exp(k~) (15)
F F h 1 . d . where uv and s are t e constant vo ume energtes an
entroptes. We draw attention to equation (15) as it is of
importance in the interpretation of thermodynamic parameters
derived from transport measurements (discussed in greater detail by
Jacobs in Chaper 4 ). In addition we should note that constant
volume terms are much more amenable to calculation (using the
techniques discussed by Mackrodt in Chapter ( 5)) than are constant
pressure parameters; the use of constant volume terms in
expressions such as equation (15) allows therefore, in principle,
direct calculation of defect concentrations.
Before leaving this brief survey of basic defect thermodynamics,
two further points should be noted. The first concerns the
magnitude of the differences between the constant pressure and
volume terms, which are related by the following expressions
h Tf3p
Sp (17)
~ s +-- v v KT p
INTRODUCTION TO MASS TRANSPORT 9
where Sp is the expansivity of the solid and Kr the isothermal
compressibility; Vp is the volume of defect formation, the theory
and calculations of which have been successfully treated by
Lidiard8. The magnitude of the differences between the constant
volume and pressure terms can be quite considerable at higher
temperatures, as shown in Table 1, which presents calculated values
for the cation Frenkel energy in AgCl obtained in a successful
theoretical study of ion transport in this material by Cat low,
Corish and Jacobs 9.
Table 1 Energies and Enthalpies of
Frenkel pair formation in AgCl
T/K u (T)/eV v hp(T)/eV
300 1.37 1.48 400 1.33 1.49 500 1.28 1.56 600 1.22 1.64 700 1.14 1.
78
The second point to which we draw attention concerns the
temperature dependence of the defect parameters. Equations (16) and
(17) clearly demonstrate an explicit dependence of h and s on
temperature. The constant volume terms u and s alsg, howe~er, vary
with temperature - an effect which appeXrs to ge attributable
largely to the temperature dependence of the lattice ~arameter.
Thus the theoretical study of AgCl referred to above , calculated
the variation of u , for the cation Frenkel pair, using the quasi
harmonic approxima¥ion in which temperature effects are described
entirely in terms of the dependence on temperature of the lattice
parameter. The results summarised in Table (1) clearly demonstrate
that the temperature dependence of u becomes appreciable close to
the melting point. Such factors ~e believe should be included in
all accurate analyses of transport data - a point which will be
discussed in greater detail by Jacobs in Chapter ( 4 ) •
An entirely parallel treatment may be applied to the second major
class of disorder, namely the generation of Schottky defects, in
which vacancies are created by the displacement of lattice atoms to
the surface. As normally used, the term refers to ionic crystals in
which the requirement of site conservation leads to the necessity
of creating oppositely charged defects in concentrations inversely
proportional to their charges. Thus in 1:1 crystals such as NaCl,
equal concentrations of Na+ and Cl- vacancies must be present in
the pure crystal. Thus for such crystals the Schottky disorder may
be represented by the expression
10
C.R.A.CATLOVV
(17)
+ where x and x are the cation and anion vacancy concentration
respectively, Xnd gS is the Gibbs free energy of Schottky pair
formation. For cryEtals, such as metals or rare gases, that are
constructed from neutral atoms, the vacancy equilibrium can
evidently be written in terms of only one type of species as
discussed by Chadwick and Brebec in Chapters (II) and (10);
otherwise the basic defect thermodynamics of vacancy disorder in
these crystals resembles that of Schottky disorder in ionic
systems.
The discussion given above concerning the temperature dependence of
defect parameters and the relationship between constant pressure
and constant volume parameters is evidently equally applicable to
vacancy as to interstitial disorder. The same type of thermodynamic
approach will also prove of value in discussing certain features of
the behaviour of impurity induced defects that are discussed
below.
B. Impurity induced defects
The simplest way of understanding the presence of impurity induced
defect populations is via the concept of charge compensation. Let
us consider an ionic crystal, e.g. NaCl, which contains a small
concentration of divalent impurity ions, e.g. Mg2+, ca2+, which
enter as substitutionals on the cation sub lattice. These impurity
ions will have an effective charge (i.e. charge relative to that o~
the perfect lattice ion) of +1. Preservation of electroneutrality
requires the creation of oppos itely charged defects. The nature
of this charge compensating defect population depends on the
intrinsic disorder of the crystal. Thus for a rock salt structured
crystal, vacancy will d~minate over interstitial compensation if
the condition, g8 < g is satisfied, i.e. if Schottky disorder is
predominanf, wh~ch is the case in NaCl. *Divalent substitutionals
will therefore be compensated by cation vacancies.
The presence of an impurity induced defect population clearly will
enhance the rate of ion transport, particularly at low temperatures
where the concentration of thermally generated defects, governed by
expressions such as equation (12), becomes small. Indeed for
certain classes of crystal, e.g. the ceramic oxides, MgO and Al
2o3, transport is probably almost always
* Similar, but slightly more complex expressions, hold for other
crystals, e.g. those of the fluorite structure, where it can be
shown that anion interstitials will compensate impurities with a
positive effective charge provided that g~ < g~/2.
INTRODUCTION TO MASS TRANSPORT 11
effected by charge compensating defects owing to the high formation
energies of the intrinsic defects - a topic discussed in further
detail by Wuensch in Chapter (14). However, in those crystals
where, at higher temperatures, intrinsic disorder is appreciable,
the effect of impurities is seen most obviously from the Arrhenius
plots for conductivity or diffusion; these generally show a dis
tinct 'knee' at the temperature at which the impurity induced
defect population is being replaced by the intrinsic disorder as
the dominant defect population. An example is shown in Figure
3.
2
-5
1·5 2·0 2·5 3·0 3·5lC 103
T"I(K"I)
Figure 3 Arrhenius plot for conductivity in KCl (see chapter 4).
Note that, following the Nernst-Einstein relationship, log(crT)
rather than log(cr) is plotted against T-1.
How may we interpret the energies deduced from the slopes of these
Arrhenius plots? At high temperatures, the Arrhenius energy* is
clearly a sum of formation and activation terms, since the rate of
ion transport is the product of a population and mobility term (see
equation 7), both of which have an exponential dependence on an
energy term as shown by equations (9) and (12). In the dopant
induced or 'extrinsic' region, the defect population is fixed. It
might be expected
* The term 'Arrhenius energy' which is an experimentally determined
quantity, should be distinguished from 'activation energy' which
refers to a specific migration mechanism, and does not include
contributions from defect formation terms.
12 C. R. A. CA TLOW
therefore that the Arrhenius energy would simply equal the defect
activation energy. Although this simple prediction holds in some
cases, in general it is an over-simplification owing to the
interaction between defects, which has a major effect on ion
transport as discussed in the next section.
3.3 Short and long range defect interactions
In ionic materials, interactions between defects are predominantly
Coulombic in origin, although recent theoretical worklO,ll has
suggested that elastic forces may make an important contribution to
the interaction energy; elastic forces provide, of course, the only
source of interaction in non-polar solids such as the rare gas
crystals. Short-range defect inter actions are generally described
in terms of distinct defect clusters, most commonly containing an
impurity ion and its charge compensating defect. An example is
shown in Figure 4 for the case of Ca doped NaCl. A second example
is illustrated in 2 Figure 5 where we show a simple pair cluster
comprising a Ca + substitutional and an oxygen vacancy in calcia
stabilised zirconia - an important solid electrolyte material in
which high concentrations of Ca2+ substitutional ions are
compensated by oxygen vacancies in the fluorite structured ZrOz
host lattice. (Note that zro2 is only stable with the fluorite
structure when doped with low valence ions.)
Divalent \cl- dopant
CLUSTER IN DOPED NaCI
Figure 4 Dopant vacancy cluster in ca2+ doped NaCL @represents the
ion; the vacancy is situated on the nearest neighbour site.
orientation of this cluster was illustrated in Figure (2)
dopant Re-
Figure 5 2+ Dopant vacancy cluster in Ca doped Zro2
If the dopant ion itself is immobile, clustering will in turn
immobilise the charge compensating defects, The effect on transport
properties can be simply treated by extending the mass action
formalism to include defect aggregation reactions, Thus the
formation of the substitutional-vacancy cluster in NaCl shown in
Figure 5 may be represented by the reaction
(SUBS) + (VAC) (x ) (x+)
c
where SUBS and VAC represent free (i.e. unassociated)
substitutionals and vacancies;--the concentrations of the free
defects and the complex are represented by the symbols underneath
the equation. The reaction above leads to the mass action
equation
X
\ .. K = exp ( -g c /kT) (18) X X C p
s v in which gc is the free energy of cluster formation. The most
obvious efFect of such equilibria is to add an additional term to
the measured Arrhenius energy for the transport coefficients.
Indeed, detailed analyses of the variation with temperature and
with dopant concentration of conductivity and diffusion co
efficients allow the energies of clustering as well as those of
formation and activation to be deduced. The analysis is
13
14 C. R. A. CA TLOW
normally achieved by solving the complex set of mass-action
equations representing defect formation and clustering to which are
added constraints representing the invariance of the total dopant
concentration and the electroneutrality condition. This allows
transport coefficients to be calculated as a function of the
thermodynamic parameters; the latter parameters may then be
adjusted by a least squares fitting procedure in order to reproduce
the experimental data. Further discussion of this important
procedure is given by Jacobs in Chapter (4).
The role of defect clustering in limiting ion transport is of
partiaular importance in superionic materials, where considerable
effort is devoted to achieving the maximum possible conductivities.
One .way of enhancing conductivity is by increasing the level of
impurity ions and hence of charge compensating defects. The effect
of injecting these additional defects may, however, be reduced if
the defects are trapped by the impurity ions. Thus in the
superionic oxygen conducting material Ca/Zro2 discussed above, it
is now clear that the magnitude of the conductivity is, to a
considerable extent, controlled by the strength of the
dopant-defect interactions in clusters of the type shown in Figure
s; although we should note that at higher dopant concentrations
clustering is unquestionably more complex than implied by simple
aggregates of the type shown in the figure. Indeed it is now clear
that in heavily defective materials - either heavily doped or
non-stoichiometric solids* - clustering may become exceedingly
complex - a topic which we raise in Chapter 16.
The treatment of short-range interactions given above strongly
resembles that given to ionic interactionsin electrolyte solutions.
This resemblance is extended to the treatment of long-range
interactions, which being largely Coulombic in origin, are most
simply described in terms of the Debye-Huckel theory; this
considers each ion as being surrounded by 'clouds' or 'ionic
atmospheres' of oppositely charged ions, whose effect is
incorporated into the mass action treatment of defect equilibria by
activity coefficients, f., so that, for example, the Frenkel mass
action equation (equation 12) is now written as
where the activity coefficients f. are given by 1
*
The resemblance between heavily doped and non-stoichiometric solids
is discussed further in Chapters (15) and (16).
INTRODUCTION TO MASS TRANSPORT
1 5
(20)
where q. is the effective change of the ith defect, e: is the
static dielectfic constant and R is the distance of closest
approach of defects; x-1 is the Debye-Huckel screening length for
which we have the relationship
4 7r E q~ x. X2= i 1 1
ve: kT (21)
where xiis the mole fraction of the ith defect and v is the volume
of the unit cell. The Debye-Huckel theory is strictly only
applicable to systems with very low defect concentrations (<
lo-3 molar). It has, however, been successfully applied to more
concentrated systems, and indeed it has been found that more
sophisticated theories often reproduce the results of the simple
Debye-Huckel approach.
This now completes our account of basic diffusion and defect
theory. The remainder of our discussion is concerned with a number
of more special topics, to which it is nevertheless necessary to
draw attention in the introductory chapter. These concern first the
migration of 'minority' defects species, secondly the mechanisms of
impurity transport and thirdly the occurrence of non-bulk transport
mechanisms.
3.4 Minority defect transport
We are concerned here with the migration of defects which are not
produced by the dominant intrinsic disorder reaction in the
crystal. A good example is cation diffusion in the fluorite
structured compounds where the intrinsic disorder is invariably of
the anion Frenkel type. Cation diffusion is effected by the low
levels of cation vacancies, which in turn are controlled in such
systems by a coupling of the Schottky and Frenkel disorder
reactions; and the influence of impurities is understood via the
effect on the Schottky equilibrium of the perturbation of the
Frenkel disorder reaction.
The interest and complexity of such systems is well illustrated by
the non-stoichiometric uo2+ phase, which shows both oxygen excess
(x > 0) and oxygen der1~ient (x < 0) composition regions. The
former contains an excess of oxygen interstitials and the latter of
oxygen vacancies. Thus in the oxygen excess regions in accordance
with the mass-action equation for Frenkel disorder, the oxygen
vacancy concentration is suppressed; maintenance of the Schottky
equilibrium then requires
16 C. R. A. CA TLOW
enhancement of the cation vacancy population. Thus if cation
diffusion takes place by a vacancy mechanism, diffusion w~.ll be
enhanced in the oxygen excess region of the non-stoichiometric
phase. Similar arguments show that the cation vacancy concentration
and hence the cation diffusion coefficients are reduced in the
vacancy excess, oxygen deficient regions. Reference (12) shows how
these changes in the vacancy concentration can be discussed in
terms of the variation in the effective formation energy of the
defect.
These predictions are borne out experimentally as illustrated by
the measured variation in the Arrhenius energy for cation diffusion
shown in Figure 6. Reference (12) discusses in detail the
consequences of variations in the Arrhenius energy. The results
illustrate the dramatic effects which variation in the chemical
composition of a system may have on diffusion rates, and the way in
which this may be understood by application of mass-action
theory.
1·9
2
ratio of oxygen to metal
Figure 6 Variation in Arrhenius energy for cation diffusion in
uo2+x• For further discussion of theoretical and experimental
aspect~, we refer to reference (12).
INTRODUCTION TO MASS TRANSPORT 1 7
3.5 Impurity transport
Although in some materials, impurities are immobile, impurity
migration is often important. Impurity diffusion may, for example,
be effected by normal lattice defect mechanisms; thus vacancies can
effect impurity transport in the same manner as for lattice atom
transport. One special type of mechanism to which we wish to draw
attention involves the migration of dopant-vacancy clusters without
dissociation. Thus the cluster shown in Figure 4 may migrate by a
two-step mechanism involving a jump of the dopant into the vacancy
followed by a jump of the vacancy around the dopant the types of
mechanism illustrated in Figure 2. It will be seen that although
the individual steps effect only reorientation of the complex, the
successive operation of the two jumps results in bulk migration of
the impurity ion. Similar but more complex processes are involved
in the more exotic example provided by the fission gas transport in
uo2• Calculations13 suggest that Xe, produced as a fission product,
occupies a complex compr1s1ng one cation and two anion vacancies -
see Figure 7 - which may migrate by a non-dissociative mechanism
involving interchange of the gas atom with a neighbouring cation
vacancy. Fission gas migration is, we should note, of major
importance in controlling the behavior of uo2 fuels during
operation of fission reactors.
Figure 7
[±] cation vacancy
Xe atom occupying trivacancy (cation+ two anion vacancies) in uo2
•
18 C. R. A. CATLOW
3.6 Non-bulk migration mechanisms
The account presented in this chapter has essentially concerned
transport through the bulk of a single crystal. However, in many
practical situations transport may be effected by non-bulk
processes which give rise to higher mobilities than are found for
bulk migration. We draw attention to two related types of
mechanism. The first which may be important in studying
polycrystalline materials is known as grain-boundary diffusion, and
involves transport of atoms along the interfaces between grains - a
process which commonly occurs more rapidly than bulk diffusion and
which is discussed in Chapter (17) by Heyne. The second related
mechanism is surface diffusion which may be import ant in porous
materials; atom transport occurs along the surface, in some cases
by the agency of surface defects. A discussion of surface structure
and properties is given by Tasker in Chapter (18).
In general, far less is known about non-bulk than bulk trans port
processes. Their importance, however, should be stressed,
particularly in materials such as ceramic oxides, where bulk trans
port is slow. Examples will be given in the discussions presented
in Chapters (14) and (19).
Summary and Conclusions
The discussion in this chapter has aimed to provide a basis for the
subsequent detailed discussion of the theoretical and experimental
study of mass transport in solids. Certain points raised in our
discussion need, however, special emphasis. The first concerns the
importance of a detailed understanding of transport mechanisms at
an atomic level if reliable predictions of th~ effects of
temperature and dopants are to be made. The second concerns the
temperature dependence of basic defect thermodynamic parameters,
particularly at temperaturesclose to the melting point, which may,
we believe, have an important consequence for the analysis of
transport data. Thirdly, we should stress the role of defect
interactions in limiting ion transport in doped systems. Finally,
we repeat the important possible role of grain boundary and surface
diffusion in many ceramic materials. All these points will be
referred to in greater detail in the chapters which follow.
INTRODUCTION TO MASS TRANSPORT 19
References
I. Mott, N.F. and Gurney, R. in 'Electronic Processes 1n Ionic
Crystals'. O.U.P. I957.
2. Lidiard, A.B. in 'Handbuch der Physik'. (Ed. S. Flugge), Vol. 20
(Springer Verlag, Berlin), I957.
3. Haven, Y. in 'Fast Ion Transport in Solids. (Ed. W. van Gool),
p.35, North Holland, I972.
4. Vineyard, G. J. Phys. Chern. Solids, l• I57 (I957).
5. Catlow, C.R.A. Solid State Ionics - in press.
6. Lidiard, A.B. 'International Centre for Theoretical Physics'.
Report IC/8I/I9 (I98I).
7. Gillan, M.J. Phil. Mag. A43, 30I (I98I).
8. Lidiard, A.B. Phil Mag. A43, 29I (I98I).
9. Catlow, C.R.A., Corish, J. and Jacobs, P.W.M. J. Phys. C.~, 3433
(I979).
IO. Catlow, C.R.A., Corrish, J., Jacobs, P.W.M. and Quigley, J., J.
Phys. Chern. Solids~. 23I (I980).
II. Catlow, C.R.A., Corish, J. and Jacobs, P.W.M. Phys. Rev. B. -
1n press.
I2. Cat low, C.R.A. Proc. Roy. Soc. A353, 533 (I 977) .
I3. Catlow, C.R.A. Proc. Roy. Soc. A364 ,_ 473 (I 978).
I4. Henderson, B., in 'Defects in Crystalline Solids' . Edward
Arnold, I972.
CHAPTER (2): LES TECHNIQUES DE LA DIFFUSION*
F. Beniere
UNIVERSITE DE RENNES I - 35042 RENNES-Beaulieu (France)
Pour des raisons de commodite d'usinage, les echantillons utili
ses pour la diffusion, aussi bien dans les laboratoires de
recherche que dans 1' industrie, presentent generalement une face
principale plane a partir de laquelle s'effectue la diffusion. On
choisit comme axe x'x la direction normale a ce plan d' abscisse x
= 0. Dans ce cas tres frequent de la diffusion plane, 1' equation
de Fick prend la forme simple :
ac at 2._ (D _££)
ax ax
La resolution de cette equation differentielle aux derivees par
tielles a partir des conditions initiales et limites conduit au
pro fil de diffusion theorique C(x,t) dont la forme n'est
generalement pas analytique.
Apres la diffusion effectuee a la temperature T pendant le temps t,
les atomes qui ont penetre dans le solide sont distribues selon le
profil C(x,t)· La figure 1 ci-apres montre un profil experimental
correspondant a la diffusion a partir d'une couche infiniment mince
(dans deux echantillons symetriques), done au profil
theorique
x2 C = Cs exp (- --) 4Dt (1)
Le premier paragraphe de ce chapitre decrit les methodes experl
mentales de determination du profil de diffusion des a tomes. Par
comparaison aux profils theoriques caracteristiques, on deduit le
mecanisme de diffusion et la valeur de D. Les autres methodes di
rectes et indirectes de mesure de D sont indiquees dans les deux
pa ragraphes suivants : "Methodes electriques" et "Methodes
dynamiques'~
21
_x X
Fig. I. Profil d'autodiffusion C(x t) =f(x) et LnC=</l(x2 ) de
l'ion chlorure dans un monocristal de chlorure de potassium
(ref.l)
DETERMINATION DES PROFILS DE DIFFUSION
La preparation du couple de diffusion est importante. Un exempre
ideal est fourni par la technologie des semi-conducteurs : on
forme, par epitaxie sur un substrat de silicium pur, une couche de
silicium dope. L'impurete diffuse (heterodiffusion) a partir de
conditions initiales bien definies dans le uleme reseau cristallin.
On forme
LES TECHNIQUES DE LA DIFFUSION 23
d 1 autres couples d 1heterodiffusion solide A I solide B selon
diffe rents procedes metallurgiques (soudure, compression,
colaminage .•• ), electrochimiques (galvanoplastie, oxydation
anodique, ••. ) ou physi ques (evaporation sur filament,
sublimation sous vide, pulverisation cathodique, ••. ) • La
diffusion gaz/ solide (voire liquide/ solide) est plus facile a
realiser. Il est commode de fixer la valeur de la pres sion du gaz
afin de se placer dans les conditions de la diffusion a partir d 1
une concentration constante. Celle-ci peut etre avantageu sement
la pression de vapeur de l 1 espece diffusante a 1 1 etat solide
dont il suffit alors de contr6ler la temperature. Dans le cas de l
1 autodiffusion, toutes ces methodes de preparation sont en
principe applicables a condition de remplacer 1 1 espece diffusante
B par un melange d I isotopes de A - dont 1 I un, A'*, de
preference radioactif de composition differente de celle
constitutive du solide. En fait, on se place le plus souvent dans
le cas de la diffusion a partir d 1 une couche infiniment mince de
A* deposee sur la face plane de A.
Un autre aspect, tout aussi important que la preparation des
echantillons, est le traitement thermique. Il faut maintenir cons
tante la temperature T' souvent elevee (jusqu I a pres de 3 000 K
dans les oxydes et metaux refractaires), pendant une duree t qui
doit parfois atteindre plusieurs dizaines de jours. En effet, D
depend exponentiellement de T selon la loi d 1 Arrhenius
w D = D0 exp ( - kT ) (2)
Il faut simultanement veiller aux conditions de proprete afin d I
eviter la diffuSiOn d I impureteS inVOlOntaires qui pourraient mo
difier totalement les proprietes de transport de 1 1 element
etudie.
I. Sectionnement des Echantillons apres Diffusion
La zone de diffusion (approximativement 4 Vi5t) do it etre decou
pee en tranches paralleles d1 epaisseur ~x, si possible constante.
La methode de sectionnement a employer depend de l 1 epaisseur ~X
desiree qui, elle-meme, depend de la valeur de D. Les indications
du tableau ci-apres donnent la gamme des valeurs de D correspondant
a l 1 equa tion : 13 ~x = 4VDf (soit : C = 10-2 Cs a la 13eme
section) pour la gamme des durees extremes 10 3 < t < 10 6
secondes. Au-dessous de 103 secondes, les corrections de montee et
descente en temperature de viennent difficiles. Les durees
superieures a 106 secondes sont aussi a eviter car elles
immobilisent le materiel trop longtemps.
Usinage. Le principe est de sectionner 1 I echantillon a 1 f aide d
1 un tour ou d 1 une fraiseuse parallelement au front initial de
dif fusion en recherchant la plus grande precision. Le reperage du
plan origine se fait a l 1 aide de fils repere. Le reglage du
parallelisme entre le plan des reperes et le plan de coupe est
effectue par un microscope. Les copeaux sont recueillis a chaque
passe. L 1 epaisseur enlevee est determinee par pesee. Cette
methode permet d 1 obtenir des coupes de quelques microns.
24 F. BENIERE
Tableau 1. Gammes des valeurs des coefficients de diffusion
accessibles par les differentes techniques.
Methode /1x D (cm2• sec- 1)
Usinage, abrasion 5 ]J.m 10-12 - 10-9
Microtome 1-100 ]J.m 10-13 - 10-6
Chimique 10 ]J.m 10-11 - 10-8
Electrochimique 50 nm 10-16 -ro-13
Micro sonde ionique 1-10 nm 10-19 - 10-14
Microsonde electronique 10-12 - 10-8
Resonance magnetique 10-16 - 10-5
Diffraction neutronique 10-7 - 10-5
Abrasion. Certaines substances ne se pretent pas a 1 'usinage
(verres, ceramiques, •.• ) ou les echantillons sont trop petits
pour etre pris dans un mandrin. On peut utiliser une surface
abrasive (papier emeri) et recuperer la substance abrasee dont 1'
epaisseur se chiffre encore en microns.
Microtome. Parmi les differents appareils de sectionnement me
canique, le microtome est de loin le plus precis. Cette fois, c'est
l'echantillon qui est mobile et l'outil de coupe qui est fixe (fig.
2). Ce dernier est un solide rasoir en acier special, carbure de
tungstene ou diamant pour les substances les plus dures. Meme le
silicium2 a pu etre debite en sections d'un micron
d'epaisseur.
Fig. 2
[c] ua
Fig. 3. Profils de diffusion d'impuretes metalliques obtenus par
abrasion chimique dans le silicium. (D' apres BLONDIAUX, ref.
3).
25
L'echantillon est monte sur une rotule orientable. Le reglage du
pa rallelisme entre la surface de diffusion et le fil du rasoir
est effectue au moyen, soit d'une lunette autocollimatrice, soit
d'un jeu de miroirs. La substance arasee apres chaque passage est
re cueillie directement dans les coupelles ou les tubes des
compteurs de radioactivite. C'est par cette methode que sont
obtenues les va leurs les plus precises des coefficients de
diffusion qui servent de references ou d'etalonnage aux autres
methodes. L'epaisseur des coupes est egale a un ou plusieurs
microns. Pour des zones plus epaisses, un nombre important de
coupes peut etre collecte dans chaque coupelle.
26 F. BENIERE
I' Ga atoms.cm3
micron
Fig. 4
Profils de diffusion du gallium obtenus par ana lyse par
activation et par sectionnement elec trochimique dans le si
licium dope a differen tes teneurs Na et Nd d' accepteurs et
donneurs. (D'apres CORISH et al., ref. 4).
Sectionnement chimique. Pour les substances les plus recalci
trantes au sectionnement mecanique, telles que les verres ou les
semi-conducteurs, on emploie parfois un reactif chimique (melange
d'acides) qui dissout l'echantillon. L'epaisseur dissoute est pro
portionnelle au temps d'irnmersion. L1 epaisseur optimale est
environ dix microns. L' inconvenient majeur de la methode est son
manque de regularite dans la decoupe. C'est pourtant la seule
technique appli cable au cas de substances tres dures dans
lesquelles la zone de diffusion est tres profonde (Fig. 3).
Sectionnement electrochimique. Dans cette methode, reservee a
l'etude des semi-conducteurs, l'echantillon est serre dans une
pince de tantale plongee dans une cuve a electrolyse ou elle joue
le role de pole positif (anode). La cathode est en platine
(degagement d'hy drogene). Un electrolyte du type
N-methylacetamide est rendu conduc teur par addition de KN03. Une
alimentation delivre un courant cons tant. La formation d'oxygene
a l'anode oxyde instantanement l'echan tillon qui st= recouvre
d'une pellicule dont l'epaisseur depend du
LES TECHNIQUES DE LA DIFFUSION 27
temps d'electrolyse. L'echantillon est ensuite plonge dans une
solu tion f~uorhydrique pour dissoudre la couche d'oxyde et le
cycle est repete pour decouper toute la zone interessante. Ce
precede est uti lise pour la diffusion dans silicium, arseniure de
gallium, etc ••• Chaque oxydation retire une couche d'epaisseur de
l'ordre de 50 nm, ce qui convient bien a la microelectronique au
les profils de dis tribution des impuretes interessent des zones
de l'ordre du micron (fig. 4).
Erosion ionique. Cette technique a apporte, depuis sa relative
ment recente mise en oeuvre, un progres decisif dans les etudes des
faibles valeurs des coefficients d'heterodiffusion, et parfois meme
d'autodiffusion. La region a analyser est soumise a un bombardement
d'ions positifs acceleres. La matiere - atomes du reseau et impure
tes - est pulverisee sous 1' impact du faisceau. Quelques-uns des
atomes ejectes se trouvent ionises positivement. Ces derniers sont
acceleres par un champ electrique, puis devies par un champ magne
tique qui les separe selon leur nombre de masse suivant les princi
pes de la spectrometrie de masse. Les avantages essentiels de cette
methode sont sa rapidite et surtout sa haute resolution spatiale
puisqu'elle permet des mesures de la concentration par pas de quel
ques nanometres seulement.
2. Mesure de la Concentration
Comptage de radioaativite. La methode, de loin la plus utilisee
pour determiner les profils de concentration, consiste a marquer 1
'espece qui diffuse par des a tomes radioactifs. Ce fut longtemps
le seul moyen de mesurer les coefficients d'autodiffusion. Elle fut
mise en oeuvre des 1920 5 , dans le cas de 1 'autodiffusion du
plomb dans des sels de plomb, a 1 'aide d 'un isotope radioactif du
plomb descendant du thorium. I1 fallut cependant attendre les
annees 50 pour que les premiers reacteurs nucleaires puissent
produire des isotopes radioactifs de tres nombreux elements. Des
lors, l'etude des phenomenes de transport de matiere se developpa
rapidement. Les coefficients d' autodiffusion purent etre mesures
dans presque taus les corps solides, de meme que les coefficients
d'heterodiffusion dans une foule de substances. Les resultats
individuels sont regu lierement regroupes dans des articles de
revue (par exemple dans le chapitre de ce livre pour la diffusion
dans les metaux) et des compilations frequemment mises a jour6 •
Apres que la zone de diffu sion soit debitee en une vingtaine de
sections par l'un des prece des ci-dessus decrits, chacune est
introduite dans un compteur de radioactivite alpha, beta ou gamma
selon la nature du rayonnement de l'isotope radioactif qui joue le
role de traceur (d'ou le nom de radiotraceur). Les compteurs
modernes sont equipes de passeurs au tomatiques qui permettent le
comptage sequentiel nuit et jour de plusieurs dizaines ou centaines
de sections. Les donnees sont trai-
28 F. BENIERE
1.17 •• ,._0 1.04
X 0.91 tr c 0 0.78 z 0 0.65 r- 0 <l: 0.52 0:: lJ..
w 0.39 .J 0 0.26 :;
0.13
0.00
1.17
0.91 + N._
Ul 0.78 z 0 0.65 r- 0 <l: 0.52 0:: lJ..
w 0.39 .J 0 0.26 ::2
0.13
o.oo 0 45 90 135 180 225 270 315 360 405
DISTANCE (MICRONS)
Fig. 5. Diffusion simultanee de ca++ et Sr++ dans NaC~. L'emission
beta de Ca-45 est mesuree par un spectro metre a scintillation
liquide et l'emission gamma de Sr-85 par un analyseur multicanal a
scintillation so lide. (D'apres MACHIDA et FREDERICKS, ref.
8).
tees par ordinateur et les coefficients de diffusion sont extraits
des profils experimentaux C = f (x). Au lieu de marquer 1' espece
qui diffuse, il revient au meme, dans quelques cas tres favorables,
de laisser diffuser l'impurete non marquee puis de proceder a son
ac tivation apres diffusion en irradiant les echantillons par des
neu
trons dans un reacteur nucleaire. Il faut que la matrice s'active
le moins possible pendant le meme temps. Le cas ideal est la diffu
sion dans le silicium7 • Cette methode d'analyse des impuretes
dans
LES TECHNIQUES DE LA DIFFUSION
PHOSPHORUS (ARBITRARY UNITS) X
( C5 = 1021 cm-3 ; soo•c, 30mn l
K'COUNTS • • 197 ) ... ~ Au(n,'lf .\l :•#'
0.1 0.5 DEPTH (Jim)
Fig. 6
Profils du phosphore et de l'or apres diffusion du phosphore dans
le si licium dope a l'or (phe nomene de piegeage). L'e mission
gamma de Au-197 est mesuree par un ana lyseur nulticanal a· scin
tillation solide en meme temps que le rayonnement de freinage de 1'
emission beta de P-32. (D'apres LECROSNIER et al., ref. 9).
le phosphure d'indium InP par exemple est inapplicable avec les
neu trons. Par centre, elle est possible par irradiation par des
parti cules chargees telles que des protons de 10 MeV fournis par
cyclo tron.
Il est aussi possible de suivre la diffusion simultanee de deux
impuretes qui interagissent pour modifier leurs vitesses
respectives de diffusion avec un isotope emetteur-beta et un
isotope emetteur gamma en utilisant des appareils de comptage
differents (fig. 5). On peut suivre aussi la diffusion simultanee
de plusieurs impuretes lors qu'il existe des isotopes
emetteurs-gamma que l'on peut discriminer par leur difference d'
energie avec le meme compteur. Il est encore possible, avec le meme
compteur gamma, de determiner les profils de plusieurs
radioelements a la fois, meme si l'un d'eux est emetteur beta, en
utilisant le rayonnement de freinage. L'analyse multicanale de
!'ensemble du rayonnement gamma permet de distinguer les pies des
radioelements emetteurs gamma qui se superposent au rayonnement
con tinu de freinage des rayons beta (fig. 6). La difference entre
les periodes de decroissance radioactive est aussi mise a profit
pour se parer des radioelements, en particulier dans la mesure des
effets isotopiques. Dans ce cas, les coefficients de diffusion ne
different que de quelques pour cent et il faut done une tres grande
precision pour separer les profils de concentration. C'estpourquoi
on utilise a la fois la difference d I energie et la difference de
periode. Si les repartitions des concentrations respectives C et C'
des deux isotopes obeissent a !'equation (1), l'effet isotopique
(D-D')/D est obtenu directement a partir de la droite Ln(C' /C)=
f(x 2 ) comme le montre la figure 7 :
30
F. BENIERE
Fig. 7
Effet isotopique d'autodiffu sion de l'ion sodium dans des
monocristaux de chlorure de sodium. Les isotopes Na-22 et Na-24
sont utilises et leurs rayonnements gamma sont sepa res a la fois
grace a leur difference de periode (respec tivement IS heures et
2,6 ans) et d'energie (respectivement 0,511 et 2,71 MeV). La pente
des droites obtenues en por tant Ln(Czz/C 24 ) vs (x2 /4 D22 t)
donne directement l'effet iso topique (Dzz - D24 ) /Dz 4 qui est
de 1' ordre de 3 %. (D' a pres ROTHMANN et al., ref. 10).
Speatrometrie de masse des ions secondaires ou microsonde ioni
que. Dans la technique ou la zone de diffusion est progressivement
pulverisee sous l'impact d'un faisceau d'ions primaires acceleres,
une certaine pro.portion des atomes de la matrice et des impuretes
sont expulses sous la forme d'ions positifs. Ces ions secondaires
sont analyses par un spectrometre de masse a tres haute
resolution
LES TECHNIQUES DE LA DIFFUSION 31
•
depth (A)
Fig. 8. Profil de diffusion du gallium dans le silicium obtenu par
microsonde ionique. La courbe en pointilles est le profil
theorique. (D 1 apres HARIDOSS et al., ref. 11).
en fonction du temps d 1 erosion. La profondeur du cratere d 1
erosion etant directement proportionnelle au temps d 1 erosion, l 1
enregistre ment du rapport des signaux impurete I matrice donne
directement le profil de concentration. La vitesse de la
pulverisation est de 11or dre de 20 nm.mn-i. L1 allure du profil
suffit souvent pour determiner D. Il est, en plus, parfois
necessaire de connaitre les concentra tions en valeur absolue, ce
que la microsonde ionique ne permet pas. Il faut alors proceder a
un etalonnage par rapport a 1 1 une des me thodes de mesure
absolue des concentrations7 • L1 avantage essentiel reside dans sa
resolution spatiale. On peut obtenir aisement avec precision tout
un profil de concentration dans une epaisseur infe rieure a 100 nm
(fig. 8). La sensibilite depend fortement des ele ments et peut se
trouver reduite s 1 il y a interference de masses. Par exemple,
dans le cas du gallium dans le silicium, elle est de 1 1 ordre de
10-6 en fraction molaire. Le cratere d 1 erosion est forme par le
balayage du faisceau d 1 ions primaires dont la densite est de l 1
ordre de 400 uA.cm-2 et dont le diametre est d 1 environ 10
microns. Le cratere ainsi forme est un carr€ de 200 microns de
cote.
On comprend que cette methode se soit rapidement etendue a rexa
men des profils d 1 impuretes donneurs et accepteurs dans les semi
conducteurs. Signalons enfin qu 1 elle est aussi applicable a l 1
auto-
32 F. BENIERE
diffusion dans la mesure ou l'on dispose d'un isotope stable comme
pour l'oxygene 12 •
Microsonde eZectronique. Cette technique permet d'etudier les
couples d'heterodiffusion A/B. Les echantillons sont scies
perpendi culairement a l'interface apres diffusion. Un faisceau
d'electrons d' environ un micron de diametre balaie cette coupe. Le
spectre de rayons X, emis par le bombardement d'electrons, est
analyse en ener gie ou longueur d'onde. Le rapport des intensites
des raies caracte ristiques des elements A et B donne le rapport
des concentrations A/B, a une constante pres. Quoique la
sensibilite soit de 1 'ordre de ]Q-3 (tres inferieure aux methodes
radioactives), la microsonde est particulierement bien adaptee a 1'
etude de 1 1 heterodiffusion metal A I metal B. Dans le cas
frequent ou la diffusion s'accompagne d 1 un effet Kirkendall, la
position du front initial de diffusion est reperee a l 1 aide de
petits fils de tungstene places a 1 1 inter face avant la soudure
du couple A/B.
Retrodiffusion Rutherford. L I echantillon a analyser est bom
barde par un faisceau d 1 ions legers monoenergetiques ( hydro gene
, helium, etc., d 1 energie I a 2 MeV). Le faisceau incident est
nor mal a la cible. Une partie du rayonnement est diffuse. Soit
E0
4 ~-He+ b!i:4J 1.8 Mev Pt
1200.&.
--------..,\', Sj-edge
\i + \ \..._ ____ _
(a)
pt-edge
fl \ il l
fl I _.!I
1 '. \ f i I ~
ENERGY (MeV)
Fig. 9
Spectre d 1 energie des parti cules alpha retrodiffusees par une
couche de platine deposee sur du silicium (fi gure a). Une couche
de si lice est ajoutee sur la fi gure b.
avant diffusion ; apdos diffusion.
La formation de Pt Si est mise en mise en evidence sur la courbe b
apres dif fusion. (D' apres JOUBERT et al., ref. 13).
LES TECHNIQUES DE LA DIFFUSION 33
l 1 energie du rayonnement incident et E(S) 1 1 energie du
rayonnement qui a fait un angle 8 par rapport a la direction
incidente. Une par tie de 1 1 energie cinetique incidente est
perdue au cours du choc avec un noyau de la cible. La difference d
1 energie E(S) -E0 depend -pour un angle donne et pour une masse
donnee de la particule inci dente- de la masse de l 1 atome de la
cible. Plus celle-ci est grande et plus la difference E(S) -E0 est
faible. C1 est ainsi que, si la cible est constituee de deux types
d 1 elements, 1 1 analyse en energie du rayonnement diffuse montre
deux pies E(S) et E1 (8), plus ou moins elargis selon la resolution
en energie du systeme de detection. La deuxieme cause de perte d 1
energie est due au freinage des ions au cours de leur parcours dans
la cible en raison des interactions avec les electrons. C1 est
pourquoi on observe, a laplace d 1 un pic d 1 ener gie E(S), une
bande d 1 energie ~ E(S). Si, par exemp.le, une couche tres mince
de B se trouve implantee dans A a la profondeur x, on ob serve un
pic a une valeur E(S) qui est fonction de x. Cette propriete est
mise a profit pour l 1 etude des profils. Dans le cas de la distri
bution d 1 une espece B dans une matrice A, ranalyse-detaillee du
spec tre energetique permet de determiner le profil de
distribution de B (Fig. 9) puisque le signal E(S) depend de la
position de B dans la matrice A. I1 est evidemnent plus favorable d
I etudier la distribu tion d 1 atomes lourds dans une matrice d 1
atomes legers, comme dans 1 1 exemple de la figure 9 qui concerne
un profil de platine dans le silicium13 • La retrodiffusion des
ions peut etre compliquee par le phenomene de canalisation. Lorsque
le faisceau incident arrive se lon une rangee cristallographique,
les ions soot canalises et pene trent beaucoup plus profondement
que lorsqu 1 ils ricochent sur le me me corps amorphe, ce qui
entraine une diminution du nombre de parti cules retrodiffusees.
Ce phenomene est, selon les cas, soit a eviter (pour retrouver les
memes caracteristiques de diffusion que sur les echantillons
amorphes), soit a rechercher (pour diminuer le rende ment de
retrodiffusion sur les atomes de la matrice pour mieux dece ler
les atomes etrangers). A titre d 1 exemple, la sensibilite d 1 ions
He+ de 1.5 MeV est de IQ13 atomes.cm-2 pour le platine. La resolu
tion en profondeur, en dehors de la resolution en energie du
systeme de detection, depend de l 1 energie des particules
incidentes et de la nature de la cible. Elle est de 10 nm dans 1 1
exemple du platine et de 30 nm pour le silicium.
METHODES ELECTRIQUES
I. Semi-Conducteurs
Les impuretes donneurs et accepteurs d 1 electrons s 1 ionisent a
partir d 1 une certaine temperature en donnant des defauts
electroni ques : trous ou electrons de conduction detectables par
leurs pro prietes electriques: conductivite electrique, effet
Hall, capacite differentielle, D.L.T.S., etc. Dans un echantillon
massif uniforme ment dope a la teneur de N atomes par cm3, la
conductivite et l 1 effet
34
0
-2
• 3 x1o13
F. BENIERE
Fig. 10
Conductivite du silicium dope par deux teneurs Nd de phos phore en
fonction de la tem perature. Les courbes en trait plein
representent les equa tions theoriques dont sont de duites les
teneurs en defauts electroniques. (D'apres CORISH et al., ref.
4).
Hall sont donnes par des relations tres complexes dans les semicon
ducteurs. Toutefois, on peut parfois se contenter des equations
sim plifiees :
cr=Nqll I R = --H Nq
en admettant que chaque impurete donne la charge q de mobilite ll•
ce qui suppose 1' ionisation complete du donneur ou de 1 'accepteur
dans le domaine extrinseque. La mesure de a en fonction de la tem
perature (Fig. 10) permet de preciser 1 'etendue de ce domaine ex
trinseque. La mesure de la conductivite volumique donne la valeur
moyenne de N dans tout 1 'echantillon, tandis que la conductivite
superficielle permet de connaitre N(x=O). Pour examiner les profils
d'impuretes N(x), on mesure la resistance superficielle de l'echan
tillon par la methode des quatre pointes appliquees sur la surface
principale : un courant est injecte entre les deux pointes extremes
et la tension est mesuree entre les deux pointes interieures. Le
rapport donne la resistance superficielle dont on deduit la conduc
tivite superficielle a une constante geometrique connue pres.
On
LES TECHNIQUES DE LA DIFFUSION 35
obtient ainsi la concentration superficielle N(x=O)• Pour obtenir
le profil complet, il faut eliminer une certaine epaisseur, par
exemple par sectionnement electrochimique qui s'applique
particulierementbien aux semi-conducteurs dopes. L'une ou l 1 autre
des methodes electriques donne la valeur de la teneur a la nouvelle
abscisse x, et ainsi de suite. La large gamme des valeurs de D,
donnee sur le tableau 1, re couvre en fait les zones des
sectionnements chimique et electrochimi que.
2. Cristaux Ioniques
Dans les solides fortement ioniques (halogenures alcalins, halo
genures d'alcalino-terreux, halogenures d'argent, etc.), la conduc
tion du courant est due exclusivement au deplacement des ions
entra1- nes par un champ electrique. Les ions qui migrent sont,
soit les in terstitiels (mecanisme de diffusion interstitielle),
soit des ions du reseau qui sautent dans les lacunes en position de
premier voisin (mecanisme de diffusion lacunaire). Dans le premier
cas, la charge electrique deplacee est egale ala charge de l'ion
lui-meme. Dans le second cas du mecanisme lacunaire, au lieu de
considerer les deplace ments reels des ions, il est plus commode
de considerer les deplace ments des lacunes (en sens contraire)
portant des charges fictives egales et opposees a celles des ions.
En conclusion, l 1 ion qui effec tue un saut dans un solide
ionique assure a la fois le transport de matiere et le transport d
1 electricite. Cette remarque est concreti see par la relation de
Nernst-Einstein qui lie intimement les deux phenomenes : transport
de matiere (par le coefficient d 1 autodiffu sion D) et transport
d'electricite (par la conductivite cr) :
..!:!... = Nq2 D kT
(3)
oii N est le nombre de porteurs par unite de volume, q leur charge,
k la constante de Boltzmann et T la temperature absolue. On peut
done deduire, a partir d 1 une simple mesure de resistance electri
que, le coefficient d'autodiffusion Dcr :
kT D,... = cr v Nq2 (4)
beaucoup plus rapidement que par les methodes analytiques. Il est
cependant tres utile de combiner la conduct;ivite au coefficient
d'autodiffusion D, mesure par exemple a l'aide d 1 un isotope
radio actif, valeur alors notee par un asterisque : D~. En effet,
dans le cas du mecanisme lacunaire par exemple, le mouvement des
atomes est correle mais pas celui des lacunes. On a done la
relation :
~ D = f Dcr (5)
qui offre le meilleur moyen, a l'heure actuelle, de mesurer le fac
teur de COrrelation, Ce dernier permettant d I identifier la nature
des defauts responsables de la diffusion ainsi que le mecanisme de
transport.
36 F. BENIERE
L'etendue des valeurs des coefficients d'autodiffusion accessi
bles par la mesure de la conductivite est donnee par le domaine de
mesure des resistances : R = 1- 10 8 ~. ce qui offre de tres larges
possibilites, particulierement vers les grandes valeurs de D.
METHODES DYNAMIQUES
1. Relaxation Dielectrique
Dans les solides non metalliques (ioniques, semi-conducteurs et
organiques), certaines impuretes portant une charge electrique
rela tive par rapport au reseau sont susceptibles de se deplacer,
soit parce qu'elles sont associees a une lacune, soit parce
qu'elles sont liees a une chaine animee de mouvements de torsion.
Ces defauts de reseau constituent des dipoles electriques
(l'exemple historique est !'association formee d'un cation bivalent
et d'une lacune cationique dans le reseau NaG~) qui peuvent
s'orienter dans un champ electrique. La mesure des pertes
dielectriques en courant alternatif permet de connaitre la
concentration de dipoles ainsi que les frequences de reorientation
du dipole. La concentration des defauts de reseau ainsi que les
frequences de saut des atomes etrangers et des lacu nes entrent
explicitement dans !'expression des coefficients d'hete
rodiffusion, d'ou une methode indirecte d'etude de
l'heterodiffusion.
Une variante de la methode consiste a orienter les dipoles dans un
champ electrique a une temperature suffisamment elevee pour qu'ils
soient bien mobiles. La temperature est alors brutalement
abaissee
9
8
7
6
5
4
3
2
!x1o-13A
290 270 250 230 210 190 170 150 130 110 90 TEMPERATURE (K)
Fig. 11
LES TECHNIQUES DE LA DIFFUSION 37
de fa~on a geler tous les mouvements atomiques. Ensuite, un
rechauf fement lent permet aux dipoles de retrouver
progressivement leur orientation aleatoire et le courant de
depolarisation est mesure en fonction de la temperature a l'aide
d'un electrometre tres sensible (Fig. 11). Cette methode, qui a
re~u des noms differents (courant de depolarisation thermiquement
stirnulee, courant thermoionique, etc.), donne des frequences de
saut et done des informations sur les meca nismes
d'heterodiffusion.
2. Resonance Magnetique Nucleaire
La resonance magnetique (RMN) fut utilisee pour deduire des in
formations sur la diffusion a partir de la largeur de raie. Les re
sultats etaient peu precis. Au contraire, au cours des dernieres
an nees, la RMN est devenue une methode tres puissante et elegante
pour obtenir certains coefficients de diffusion en valeur absolue
avec une bonne precision. Ce progres est du a !'amelioration de la
tech nique15 de mesure des temps de relaxation (grace au
developpement des generateUrS a impulsion et deS detecteurs) ainsi
quI a la nou velle theorie de traitement des temps de relaxation.
Cette theorie, due a WOLF 1E, relie directement les temps de
relaxation aux coeffi cients de diffusion.
A l'equilibre, les moments magnetiques nucleaires (s'ils exis
tent), orientes par un champ magnetique ~ parallele a l'axe z, sont
distribues sur les niveaux d'energie et donnent le moment resultant
Mb parallele a z. Cet equilibre est modifie par absorption
d'energie a la frequence de Larmor w0 • Le temps de relaxation Tt
(spin-reseau) caracterise la cinetique de retour a l'equilibre. Le
temps de relaxa tion spin-spin T2 est obtenu par la methode d'echo
de spin qui con siste, avec une bobine d'axe normal a l'axe Oz, a
mesurer ramplitude de l'echo de spin cree par deux impulsions
radiofrequence separees de L• La premiere impulsion fait basculer
M6 d'un angle de 90° a par tir de l'axe z pour l'amener dans le
plan xy. Une seconde impulsion qui arrive apres l'intervalle de
temps T tourne ce moment magnetique transversal d'un angle de 180°
et un echo de spin est forme apres un nouvel intervalle de temps T
a la suite de la seconde impulsion. Si le champ magnetique Ho n'est
pas homogene- c'est-a-dire s'il existe un gradient de champ- et si
pendant l'intervalle de temps' les ato mes porteurs des spins ont
eu le temps d'effectuer des sauts (pheno mene de diffusion), 1'
echo s 'en trouve attenue. La mesure de cette attenuation de
l'amplitude de l'echo en fonction de T et en fonction de la valeur
du gradient du champ permet de determiner le coefficient de
diffusion.
La mesure du temps de relaxation Tip• le temps de relaxation
spin-reseau en presence d'un champ magnetique radiofrequence reson
nant et le temps de relaxation TID dans le champ rnagn~tique dipo
laire local peuvent etre utilises pour la mesure du coefficient
de
38
F. BENIERE
Fig. I 2
Influence de la temperature sur le coefficient d' auto diffusion
de F- dans Ba F2 determine par resonance ma gnetique nucleaire a
partir de la mesure de T1 (e) , T]p (o) et T]D (o). (D' apres
FIGUEROA et al., ref. 15).
diffusion, plus particulierement quand le transport est lent. Le
temps de relaxation T1p spin-reseau dans le referentiel tournant
peut etre mesure par une impulsion selon Ox, suivie d'une impulsion
selon Oy. Le temps de relaxation dipolaire TJD peut etre mesure a
partir d'une sequence d'impulsionsplus complexe : 90° selon Ox_T_;
45 ° selon Oy _ t _; 45 ° selon Ox. La me sure par echos de spin de
T2
est peu precise et ne permet d'acceder qu'aux valeurs les plus ele
vees des coefficients de diffusion (- IQ- 5 cm2 • sec- 1 comme dans
l'etat liquide). Par centre, elle presente l'avantage d'etre une
mesure di recte de D. Au contraire, la mesure des temps de
relaxation T1, Tip et TJD en fonction de la temperature permet de
couvrir une tres large gamme : 10-16 a I0-6 cm2 • sec-1 (Fig. 12).
Cependant, la determination du coefficient de diffusion est
indirecte, le passage du temps de re laxation aD mettant en jeu la
theorie de WOLF. La methode est evi demment reservee a l'etude de
l'autodiffusion des atomes qui ont un moment magnetique nucleaire.
C'est le cas de l'isotope F-19 qui cons titue 100% du fluor, alors
que son seul radioisotope, de tres courte periode, F-18, se prete
mal aux techniques radioactives.
LES TECHNIQUES DE LA DIFFUSION 39
La RMN peut etre appliquee egalement a la mesure des coeffi cients
d'heterodiffusion de certaines impuretes a condition que les atomes
du reseau possedent un moment quadripolaire electrique. La largeur
de la raie d'absorption est suivie en fonction de la tempe rature
et de la teneur en impuretes. Les sauts atomiques se mani festent
par un elargissement de la raie 17 •
La theorie16 montre que c'est par des sequences de sauts que les
moments magnetiques perdent leur orientation par rapport a leurs
voi sins. Il en resulte, comme dans le cas de la diffusion
atomique oil les sauts d'un traceur sont generalement correles,
l'existence d'un facteur de correlation aussi sur le coefficient de
diffusion DRMN mesure par RMN. Ce facteur de correlation peut etre
obtenu lorsque la comparaison a d 1 autres methodes est
possiblelS-l?•
3. Diffusion des Neutrons
La diffraction quasi-elastique des neutrons connait un essor tout
aussi rapide que la RMN, bien que son domaine soit limite au
transport rapide. C'est le cas des superconducteurs ioniques oil la
mobilite des ions peut devenir trop elevee pour etre mesuree avec
precision par les autres techniques. L' essor de cette methode re
sulte du puissant developpement de la diffraction des neutrons. Le
spectre de diffraction des neutrons montre une raie quasi-elastique
relativement etroite qui se superpose sur une distribution beaucoup
plus large. Le mouvement incoherent, du a l'autodiffusion des
atomes qui changent de sites, se traduit par un elargissement de la
raie etroite tandis que le mouvement local des atomes dans leurs
sites se traduit par la distribution large. En consequence, le
coefficient de diffusion ne peut etre mesure avec une bonne
precision, a partir de cet effet, que lorsque les deux conditions
suivantes sont rem plies
- les atomes doivent presenter une forte section efficace de
diffraction des neutrons ;
- le temps de residence des atomes dans leurs sites doit etre
comparable a la periods du rayonnement neutronique, ce qui
correspond a des valeurs elevees de 0> 10-6 cm2.sec- 1 •
La methode est decrite en detail dans ce livre dans le chapitre de
LECHNER auquel nous adressons le lecteur.
CONCLUSION
On dispose d'un grand nombre de methodes tres variees pour etu
dier le transport atomique. C'est souvent la gamme des valeurs de
Dx qui conditionne le choix de la methode (tableau 1). Les mesures
sont souvent effectuees a temperature variable et les resultats
donnes par une loi d'Arrhenius. Les valeurs du terme preexponentiel
et de
40 F. BENIERE
l'energie d'activation ont ete compilees dans de nombreux articles
de revues 6 pour toutes les classes de sol ides. Le lecteur
trouvera dans ce livre, soit des compilations specifiques, soit des
referen ces bibliographiques pour les metaux, les cristaux
moleculaires, les semi-conducteurs, les cristaux ioniques, les
oxydes et les verres.
REFERENCES
1. M. BENIERE, These, Paris (1970). 2. F. BENIERE, Nouvelle methode
de microanalyse des composants elec
troniques, Rev. Phys. Appl., 12:1805 (1977). 3. G. BLONDIAUX,
These, Orleans (1980). 4. J. CORISH, F. BENIERE, V.K. AGRAWAL, S.
HARIDOSS etC. DEFEUX,
Lattice Defects in Silicon Doped by Neutron Transmutation, J. Appl.
Phys., 50:6838 (1979).
5. G. VAN HEVESY, Handbuch der Physik, Vol. 13 (1928). 6. Y. ADDA
et J. PHILIBERT, La Diffusion dans les solides, Presses
Universitaires de France, Paris (1966) ; F. BENIERE, Physics of
Electrolytes, Vol. I, Hladik ed., Academic Press, New-York (1972) ;
W.J. FREDERICKS, Diffusion in Solids, A.S. Nowick et J.J. Burton
ed., Academic Press, New-York (1975), H.C. CASEY et G.L. PEARSON,
Point Defects in Solids, J.H. Crawford et L. Slifkin ed., Plenum
Press, New-York (1975) - Radiotracer Diffusion Data, in "Handbook
of Chemistry and Physics", C.R.C. Press, Cleveland-zl980).
7. M. GAUNEAU, A. RUPERT, S. HARIDOSS et F. BENIERE, Etude de la
diffusion du gallium dans le silicium par microanalyse ioni que et
activation neutronique, Analusis, 8:142 (1980).
8. H. MACHIDA et W.J. FREDERICKS, Simultaneous Diffusion of Calcium
and Strontium in NaCQ, Single Crystals, J. dePhys., C7:385
(1976).
9. D. LECROSNIER, J. PAUGAM, F. RICHOU, G. PELOUS et F. BENIERE,
In fluence of Phosphorus-induced Point Defects on a Gold-gettering
Mechanism in Silicon, J. Appl. Phys., 51:1036 (1980).
10. S.J. ROTHMAN, N.L. PETERSON, A.L. LASKAR et L.C. ROBINSON, The
Temperature Dependence of the Isotope Effect for the Diffusion of
Na+ in NaCt, J. Phys. Chern. Solids, 33: 1061 (1972).
11. S. HARIDOSS, F. BENIERE, M. GAUNEAU etA. RUPERT, Diffusion of
Gallium in Silicon, J. Appl. Phys., 51:5833 (1980).
12. F. PERINET, S. BARBEZAT etC. MONTY, New Investigation of Oxygen
Self-Diffusion in Cu20, Journal de Physique, C6:315 (1980).
13. P. JOUBERT, P. AUVRAY, A. GUIVARC'H et G. PELOUS, Growth
Platium Silicide Under Protective Layers, Appl. Phys.
Lett.,31:753(1977).
14. D. RONARC'H et S. HARIDOSS, Depolarization-current Study of
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15. D.R. FIGUEROA, A.V. CHADWICK et J.H. STRANGE, J. Phys. C :
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Nuclear Spin Relaxation and Correlated Diffusion of
Impurities, Journal de Physique, C6:249 (1980).
LES TECHNIQUES DE LA DIFFUSION 41
*ABSTRACT
A general survey is presented of the experimental techniques used
in studying atomic transport in solids. A detailed discussion is
given of the methods of determining diffusion profiles using
radioactive tracers; this account includes a description of
section ing procedures and methods for determining tracer
concentrations. Ionic conductivity measurements are then
considered, and the rela tionship between diffusion coefficients
and ionic conductivity is discussed. The chapter concludes with an
account of dynamical methods including relaxation measurements, NMR
and neutron scatter ing.
CHAPTER (3): THE KINETICS OF ATOMIC TRANSPORT IN CRYSTALS
Alan B. Lidiard
1 • INTRODUCTION
The study of atomic migrations in crystalline solids showing small
degrees of disorder is based on several assumptions or principles.
We may list these in order of decreasing generality as
follows.
(1) The motion of atoms (or ions) can be divided into (i) ther mal
vibrations about defined lattice sites and (ii) displacements or
'jumps' from one such site to another, the mean t1me of stay on any
one site being many times the lattice vibration period and the time
of flight between sites.
(2) These jumps are made possible by the presence of small
concentrations of simple defects or imperfections in the lattice
structure, most notably vacancies and interstitial atoms.
(3