Mass Transport in Solids

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 in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher
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.
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 mechanical 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
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.
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).
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.
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
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
•
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).
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
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
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.
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 Low density Polyethylene Containing an Antioxidant, J. Appl. Phys. 52:5916 (1981).
15. D.R. FIGUEROA, A.V. CHADWICK et J.H. STRANGE, J. Phys. C : Solid State Phys., 11:55 (1978).
16. D. WOLF, Correlation Effects, Dans ce livre. 17. K.D. BECKER, Nuclear Spin Relaxation and Correlated Diffusion of
Impurities, Journal de Physique, C6:249 (1980).
*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

Mass Transport in Solids

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