Growth Kinetics of Chemical Compound Layers 9781904602842, 9781898326342

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GROWTH KINETICS OF CHEMICAL COMPOUND LAYERS

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GROWTH KINETICS OF CHEMICAL COMPOUND LAYERS

V.I. Dybkov

Department of Physical Chemistry of Inorganic Materials Institute for Problems of Materials Science National Academy of Sciences of Ukraine, Kyiv 252180, Ukraine

CAMBRIDGE INTERNATIONAL SCIENCE PUBLISHING iii

Published by Cambridge International Science Publishing 7 Meadow Walk, Great Abington, Cambridge CB1 6AZ, UK http://www.cisp-publishing.com Published 2004

© V I Dybkov © Cambridge International Science Publishing

Conditions of sale All rights reserved. No part of this publication may be reproduced or trans mitted in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permission in writing from the publisher

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library

ISBN 1 898326 347

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SUMMARY The monograph deals with a physicochemical approach to the problem of solid state growth of chemical compound layers in binary heterogeneous systems formed by two solids as well as a solid with a liquid or a gas. It is explained why the number of compound layers growing at the interface between initial phases is usually much less than the number of chemical compounds on the phase diagram of a given binary system. Many experimentally observed kinetic dependences of the layer thickness (or mass) upon the time (linear, parabolic, linear-parabolic, asymptotic, paralinear, etc.) were obtained from a single theoretical view-point based on two almost obvious postulates. The nature of kinetic instability of compound layers resulting in their gradual degradation with passing time is discussed . A comparative analysis of the growth rate of the same compound layer in various reaction couples consisting of elements A and B and their other chemical compounds is presented. The effect of dissolution in the solid-liquid systems and of evaporation in the solid-gas systems on the layer-growth rate was taken into account. The reasons for the great difference in values of reaction- and selfdiffusion coefficients of the components of a chemical compound are analysed. Comparison of the consequences following from physicochemical and purely diffusional approaches is given to show that the latter is one of the limiting cases of the former. Theoretical conclusions are illustrated by the available experimental data on the formation of intermetallics, silicides, oxides, salts and other chemical compounds. The book only contains the material which cannot be found in other similar books and is addressed to scientific workers, students and postgraduates (chemists, physisists, materials scientists, metallurgists, etc.) involved into the study of solid state processes and their practical applications.

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PREFACE The main ideas, concepts and equations which form the basis of this book have been first presented in a series of papers published in Zhurnal Fizicheskoi Khimii, Poroshkovaya Metallurgiya, Journal of the Less-Common Metals, Journal of Materials Science and Journal of Physics and Chemistry of Solids. Then, a more detailed consideration of the process of reaction diffusion has been given in my monograph “Kinetics of Solid State Chemical Reactions: Growth of Chemical Compound Layers in Binary Heterogeneous Systems” (Naukova Dumka Publishers, 1992, in Russian). Since that time, I have felt an incessant interest of colleagues, who are involved into the investigation of layer formation at phase interfaces, to theoretical results obtained in my previous works. Therefore, I decided to write a new, more extended version of the book. I am grateful to Mr.Victor Riecansky, Cambridge International Science Publishing, who kindly agreed to publish it in English. This monograph deals with a physicochemical theory of the formation of chemical compound layers. The consideration is based mainly upon my own results. This does not of course mean that I ignore the theoretical works of other investigators. Simply, my views regarding the reaction diffusion differ, in some cases diametrically, from the views of many researchers, including the so-called commonly accepted views. To avoid compilativity, I preferred to give my interpretation of both theoretical and experimental data on solid state chemical kinetics supposing that those investigators, who cannot agree with it, will have the reasons to write their own books and thus will be satisfied. The book is addressed, in the first place, to the actively working researchers, students and post-graduates. As this part of scientific community has not got sufficient time to read lengthy books, I tried to be concise. Nevertheless, I explained, as far as I could, the main results in word, not relying only upon the clarity of the language of mathematical formulae, their number in the book being probably somewhat greater than in other books written for chemists. As all the formulae are obtained from very simple assumptions, I have no doubt that the reader will be able to derive, analyse and apply them without any diffuculties, in spite of their seemingly complex look.

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The experimental data have been invoked in the minimal necessary amount in order to illustrate the theoretical conclusions and consequences. This seems to be justified as each experimentalist knows the state of affairs with experiment in the field better than I do. For students and postgraduates, the excess amount of experimental data, often contradictory, seems to be even undesirable, giving an impression of the lack of any order in science. Those readers, who wish to have more details, are referred to original works indicated in the list of references. The consideration is devoted to the case where the layers of initial substances and the layers of growing compounds are parallel-plane. Probably, this does not mean that the results obtained are of no interest to researchers working, for example, in the field of sintering or reaction kinetics in powder mixtures. The processes analysed in the book take place also during the interaction of powders but under more complicated conditions as the influence of the surface curvature and the extent of closeness of reacting particles becomes important or even decisive in determining the kinetics of layer formation. For the works in this field, it is traditional to try to obtain a formal kinetic description of the reaction rate, the main aim being establishing the time dependence of the extent of transformation of the reactants into the final product. This question is not considered in the book at all as it has been analysed in detail in many other books and it is very difficult to add something new to the well-known data. It is not accidental that the consideration bears a polemic character. It was not my intention simply to give a number of ready mathematical formulae for the experimentalists to treat their data and to obtain some constants, although such a work is clearly also necessary and unavoidable. It seemed to be much more important to show that, firstly, not all has yet been done in the field of theory of solid state reaction kinetics and, secondly, some of widespread views should be modified or even rejected as contradicting not only with the available experimental data but with the common sense as well. The results of this work may in turn seem to be questionable. I would be grateful to the readers, who could draw my attention to the facts necessitating the reconsideration of one or the other of theoretical consequences and conclusions following from the proposed approach. I express my deep gratitude to colleagues from the Department of Physical Chemistry of Inorganic Materials, who were first listeners of my works. This department has been organised and headed for more than thirty years by Professor V.N. Yeremenko, now unfortunately late. His contribution to my growth as a researcher in the field of layer growth is very significant. I acknowledge this contribution with sincere gratitude. My thanks are also due to Professor V.V. Skorokhod for the help with publication of the book ‘Kinetics of Solid State Chemical Reactions’ and many papers. I greatly appreciate the friendly support of my work by Professors F.J.J.van Loo, F.M. d’Heurle, J. Philibert and P. Gas. Stimulating discussions with them in Paris, Aussois and Orsay in 1993 and in Marseille during my stay as a Visiting Professor at Faculte des Sciences St Jerome in 1996 were vii

of special value to me. Helpful discussions with Professors L.N.Larikov, B.Pieraggi, V.O.Lavrenko, G.Blaise, V.P.Kazimirov, V.M.Danilenko, O.I.Raichenko, A.M.Gusak and V.R.Sidorko are also acknowledged. I would like to express my gratitude to the researchers, who sent me the reprints of their works. Each of these works has been used, to a greater or lesser extent, during preparation of this book. I hope that the authors will be tolerant in those cases where my and their own interpretations of the results obtained do not coincide. I am also indebted to the publishers for the permission to reproduce some figures.

V.I.Dybkov

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INTRODUCTORY REMARKS The improvement of existing materials as well as the development of new materials is often based on the use of a chemical reaction in which a solid reacts with another solid, a liquid or a gas to form a solid product (an intermetallic, an oxide, a salt, etc.) at the interface between initial substances. Therefore, solid-state formation of chemical compound layers is of interest not only to chemists (researchers and technologists) but also to metal physicists, materials scientists, metallurgists, specialists in the field of corrosion, protective coatings, welding, soldering and microelectronics. A number of theoretical and experimental works were devoted to the investigation of solid-state growth kinetics of compound layers in binary heterogeneous systems. The results so far obtained were summarised in the books by V.I.Arkharov,1 V.Z.Bugakov,2 O.Kubaschewski and B.E.Hopkins,3 W.Seith,4 K.Hauffe, 5 U.R.Evans, 6 B.Ya.Pines, 7 I.M.Frantsevich et al., 8 P.Kofstad,9 Yu.D.Tret’yakov, 10 Ya.E.Geguzin,11 K.P.Gurov et at.12 and others. Various aspects of layer formation were analysed in comprehensive papers by U.Gösele and K.N.Tu, 13 R.A.Rapp,14 F.M.d’Heurle and P.Gas, 15 G.Ottaviani, 16 U.Gösele, 17 F.M.d’Heurle, 18,19 J.Philibert,20,21 F.J.J.van Loo,22 E.G.Colgan, 23 P.Gas,24 H.Schmalzried 25 and many other investigators. From an analysis of the literature data, it can easily be seen that the views of researchers of different specializations regarding the mechanism of reaction diffusion resulting in the occurrence of compound layers at interfaces differ, sometimes very considerably. Meanwhile, the main features of the process of growth of the layers are the same, irrespective of the fact whether these compounds are oxides onto the metal surface, or intermetallics during welding the dissimilar metals, or silicides in making the very-large-scale-integrated circuits for microelectronics. In spite of their variety, theoretical approaches of different authors to the consideration of the problem of solid-state heterogeneous kinetics can distinctly be divided into two groups. The first group takes account of both the step of diffusional transport of reacting particles (atoms, ions or radicals) through the bulk of a growing layer to the reaction site (a phase interface) and the step of subsequent chemical transformations with the participation of these diffusing particles and the surface atoms (ions) of the other component (or ‘molecules’ of the other chemical compound of a binary multiphase system). This is the physicochemical approach, ix

the main concepts and consequences of which were given in the most consistent form in works by V.I.Arkharov. 1,26,27 Historically, it dates from the early twenties. Namely, in 1924 U.R.Evans (see Ref.6) proposed an equation showing the comparative influence of chemical and physical phenomena on the growth rate of a chemical compound layer. Unfortunately, its importance for understanding the essence of the process of reaction diffusion was not estimated properly. In the majority of subsequent works, the step of chemical transformations was simply ignored as such, i.e. no distinction was made between the formation of a solid solution and a chemical compound (a phase of constant composition). As a result, during a long period of time the diffusional theory in its different modifications was dominating. This is the second group of approaches which originate from the ideas expressed in the most consistent form by C.Wagner in the thirties (see Ref.5,9,28). Based upon the Fick diffusion laws, Wagner ’s theory helped the chemists to reveal the main features of the kinetics of solid state heterogeneous reactions which have little in common with the kinetics of homogeneous chemical reactions taking place in solutions or gas mixtures. In particular, from a diffusional view-point, C.Wagner was able to theoretically derive the parabolic law of growth of a chemical compound layer, established experimentally by G.Tammann in studying the interaction of metals with halogens (see, for example, Ref.10). It was the great success of the diffusional theory since in the framework of purely chemical considerations such a dependence of the layer thickness on the time could not be explained. Further development of the diffusional approach is due to the works by Th.Heumann,29 G.V.Kidson,30 Ya.E.Geguzin,11 K.P.Gurov et al.,12 B.Schröder and V.Leute,31 A.T.Fromhold and N.Sato, 32 D.S.Williams et al.,33 G.-X.Li and G.W.Powell,34 M.Danielewski35 and other researchers. With time, it became, however, clear that in the case of formation of chemical compounds, no improvements of the diffusional approach can lead to satisfactory agreement of the theory with the available experimental data, even qualitative. For example, in many systems, thin compound layers (some tens to some hundreds nanometers thick) are known to grow linearly with passing time.6,8,9,36-44 The diffusional approach does not admit the existence of such a kinetic dependence. Besides, from an analysis of the experimental data of F.J.J.van Loo,45 K.N.Tu et al.,46 J.M.Poate and T.C.Tisone,47 W.K.Chu et al.,48 G.J.van Gurp and C.Langereis,49 S.S.Lau et al.,50 G.J.van Gurp et al.,51 G.Ottaviani and M.Costato,52 P.T.Vianco et al., 53 B.Y.Tsaur et al.,54 D.M.Scott and M.-A.Nicolet, 55 J.E.E.Baglin et al., 56 K.N.Tu et al,57,58 T.G.Finstad, 59 F.M.d’Heurle et al., 60 G.Majni et al., 61,62 M.Natan and S.W.Duncan, 63 F.M.d’Heurle and C.S.Petersson, 64 Z.Marinkovic and V.Simic, 65,66 B.Coulman and H.Chen, 67 M.V.Belous et al., 68 A.A.Naem, 69 G.E.White and H.Chen, 70 O.Thomas et al., 71 L.Zhang and D.G.Ivey, 72 Yu.N.Makogon,73 A.K.Pant et al.,74 K.Radermacher et al.,75 A.Thevand et al.,76 E.G.Colgan et al., 77 X.-A.Zhao et al.,78,79 R.J.Tarento and G.Blaise,80 S.B.Jung et al.,81 H.T.G.Hentzell et al.,82 S.U.Campisano et al.,83 E.G.Colgan,84 ˆ

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B.Blanpain et al., 85 K.Barmak et al., 86 K.P.Rodbell et al.,87 Y.Fujiwara et al., 88 M.Millares et al., 89 and other researchers (see Ref.5,8,9,90-95), it can be concluded that the simultaneous growth of more than two compound layers in any reaction couple of a multiphase binary system is an exception rather than the rule. Contrary to these observations, the diffusional theory starts from the quite opposite point of view (see, for example, Ref.12). Again, it is based upon the assumption of local equilibrium or quasi-equilibrium. It appears, however, to be clear that no local equilibrium can exist in any reaction couple in which the layers of some part of thermodynamically stable compounds are missing. Note that, supposing the existence of local equilibrium and applying the Gibbs phase rule, it is easy to come to the ‘logical’ conclusion that under constant temperature and presure conditions no compound layer can occur between two reactants in a binary system as in this case the largest number of co-existing phases at the zero number of degrees of freedom is two. Such a conclusion is clearly absurd since two phases (initial substances) always exist from the very beginning of the experiment. Therefore, the assumption of the existence of local equilibrium between all the phases involved into the interaction is incompatible with the thesis of simultaneous growth of compound layers. As pointed out by W.Jost,96 the number of growing layers cannot be restricted by the phase rule as in the course of chemical reactions resulting in their formation the system is too far from equilibrium. Thus, the assumption of local equilibrium should be used with great care. Its too straightforward application may be misleading. The unjustified neglect of a chemical interaction step in analysing the process of layer formation appears to be the main source of discrepancies between the theory and experiment. The primary aim of this book is, on the basis of physicochemical views regarding solid state reaction kinetics, to attempt (a) to show the comparative role of diffusion and chemical transformations in the course of growth of a chemical compound layer at the interface between reacting substances; (b) to explain why all the compound layers of a multiphase binary system not only should not but in most reaction couples cannot occur and grow simultaneously; (c) to obtain, in the framework of a single theoretical approach, the main experimentally observed kinetic dependences of the layer thickness upon the time; (d) to persuade the experimentalist not to hesitate to publish the results which seem to be conflicting with existing diffusional views. It may well happen that, from a physicochemical view-point, such results are quite natural and therefore might be expected.

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Solid-State Growth of a Chemical Compound Layer

Chapter 1

SOLID-STATE GROWTH OF A CHEMICAL COMPOUND LAYER AT THE INTERFACE OF TWO ELEMENTARY SUBSTANCES 1.1 Special features of description of the kinetics of solid-state heterogeneous reactions Initially, main attention will be paid to characteristic features of the solidstate formation of the layers of chemical compounds common for solid– solid, solid–liquid and solid–gas systems, and this will be followed by a description of the effect of dissolution of a solid phase in a liquid in the solid–liquid system and its evaporation in the solid–gas system on the process of growth of a chemical compound layer. Thus, under the conditions of occurrence of a reaction only its product is assumed to be solid. It forms a continuous adherent layer at the interface of initial phases. Analysis of the kinetics of formation of the chemical compound layers in heterogeneous systems starts with the simplest case – the growth of a solid layer between elementary substances A and B which form, according to the equilibrium phase diagram of the A–B binary system, only one chemical compound ApBq, where p and q are positive numbers (Fig.1.1). The substances A and B are considered to be solid at reaction temperature T 1 and mutually insoluble. It is easy to see that the usual concepts and laws of the kinetics of homogeneous reactions cannot be used directly in analyzing the examined heterogeneous process. In fact, problems already arise when using the main concepts of chemical kinetics: the concentration of a given reactant in the system and the rate of a chemical reaction. The definition of the concentration as the amount of a substance in unit volume applicable in the case of a homogeneous reaction, to the entire system consisting of initial substances and reaction products as a whole, in the case of a heterogeneous system is only rational for components A and B within each of its homogeneous parts (phases). In the examined case, it is possible to consider the concentration of component, for example, A, in the initial phases A and B or in a growing layer of the compound Ap Bq but it is irrational to discuss the concentration of substances A, B or A pBq in the entire heterogeneous system consisting of non-mixing phases A, ApB q, and B. This is due to the fact that in contrast to the homogeneous system in which the concentration of a given substance at any moment of time is the same at all points of this system or, at least, 1

Growth Kinetics of Chemical Compound Layers

I cS

c I2

Binary phase diagram A–B

I1

A

ApBq

B

1

2 B

Reaction couple

A

B

A pB q A x

c B(B) CB c B(B) Content of B cB(A pB q )

c B(A) Distance Fig.1.1 Schematic diagram illustrating the growth of the A p B q chemical compound layer at the interface of initial substances A and B.

changes continuously from point to point, in the heterogeneous system the concentration of the components differs at different points and at the phase interfaces its value changes abruptly (see Fig.1.1). Therefore, the distribution of the concentration of components A and B with the distance becomes important in heterogeneous systems. By definition, the chemical compound is an ordered phase with a constant composition. Ordering means that every component of the compound forms its own sublattice in the crystal lattice of the compound in which all sites are occupied by atoms or ions of only this component (see Ref.96). The constancy of the composition is a consequence of the valency rule as, for example, in the case of oxides (Al 2O 3), or of more complicated laws as in the case of intermetalic compounds where compounds of a somewhat strange composition (NiBi 3) form. Although the solid clearly contains no molecules of Al2O3 or NiBi3 as such, the composition of the solid phases 2

Solid-State Growth of a Chemical Compound Layer

Al 2O3 and NiBi3 is on the average described by these chemical formulae. It is clear that the concept of the rate of a chemical reaction, defined for a homogeneous system as the change of the concentration of reacting substances or its products per unit time at a constant reaction volume, 97–99 cannot be used for the examined heterogeneous systems. In this case, the concentration of components A and B in the initial phases and in the growing layer of the ApB q compound with no homogeneity range, remains constant, regardless of the occurrence of the chemical reaction. Therefore, a quantitative characteristic of the rate of a chemical reaction in any heterogeneous system is usually a change in the thickness or mass of the solid layer formed per unit time. Choice of the method of controlling the growth rate of a layer (by its thickness or mass) depends on the efficiency of a chosen method of experimental investigation of the interaction of initial substances. In this book, attention will be paid only to the parallel-plane layers whose thickness is the same over the entire surface of contact of the initial substances. In addition, it is assumed that the length of the layer in the direction normal to the direction of diffusion of components A and B (see Fig.1.1) is considerably greater than its thickness. In this case, the edge effects on the process of layer growth can be neglected. It should be noted that the layers observed in practice seldom have ideal appearance. Firstly, one or both boundaries of a layer with the initial phases may be uneven. As an example, Fig.1.2 shows a micrograph of a layer of the Fe 2Al5 intermetalic compound formed at the interface between iron and aluminium. 100 The specimen was produced by interaction of solid commercially pure iron with liquid aluminium at 700°C followed by their subsequent joint cooling in water. It may be seen that the interface of the Fe2Al 5 layer with aluminium is more or less flat, which cannot be said of its interface with iron. Secondly, the layers formed often contain cracks, pores and other macrodefects. Undoubtedly, this has a considerable and often controlling effect on the kinetics of their growth. Initially, attention will be given to the growth of a layer which is ideal

Fig.1.2 Microstructure of the interface between commercial purity iron and aluminium. The Al+intermetallic compound eutectic is distributed at the grain boundaries of the aluminium solid solution; magnification ×200.

3

Growth Kinetics of Chemical Compound Layers

both in the chemical (compound with a constant composition) and the physical sense (ordered structure, and no macrodefects). 1.2 Reaction diffusion The process of formation of a chemical compound in the form of a continuous solid layer between initial substances is often termed the reaction diffusion. This term reflects the most important feature of the layer formation mechanism, namely that the layer growth is due to continuous alternation of two consecutive steps: a) Diffusion of atoms (ions) of reacting substances across its bulk in opposite directions; b) Subsequent chemical transformations taking place at the layer interfaces with the participation of diffusing atoms of one of the components and the surface atoms of another component. It should be stressed that the term ‘diffusion growth’ reflects only one aspect of the layer growth mechanism – atomic diffusion. The differences in terminology are not so harmless as may seem at first sight. The concept ‘chemical transformations’ or ‘chemical reaction’ in the examined case unites the following processes: 1. Transition of the atoms (ions) of a given kind through the interface from one phase to another. This is ‘external diffusion’, according to the terminology proposed by B.Ya. Pines.7 2. Redistribution of the electronic density of atomic orbitals resulting in the formation of molecules, ions, radicals or other stable groupings of atoms included in the growing layer. 3. Rearrangement of the lattice of the initial phase into the lattice of a chemical compound formed. It should be noted that some elementary act of ‘external’ diffusion also occurs in homogeneous reactions taking place in solutions or gases. Indeed, in order to be combined into a molecule, the reacting particles must move (diffuse) to each other. The second of these processes in a liquidphase or gas homogeneous system results in the formation of an individual molecule which can move relatively freely in the reaction bulk. In the examined solid-state heterogeneous system, the ‘molecule’ formed is rigidly fixed in the crystal lattice of a chemical compound together with a number of other ‘molecules’, thus lost their individuality. What is possible in this case is only the substitution of atoms of any of the ‘molecules’ comprising the layer, for equivalent atoms, not disturbing the general balance of atoms in the entire system. In the general case, a layer of the ApB q compound grows as a result of diffusion of the B atoms to interface 1 (see Fig.1.1) where a chemical reaction then takes place in accordance with the equation qBdif + pAsurf = Ap Bq

(1.1)

and also as a result of diffusion of the A atoms to interface 2 followed by 4

Solid-State Growth of a Chemical Compound Layer

the reaction (1.2)

pAdif + qBsurf = A p Bq

It is obvious that the rates of these reactions differ. In fact, prior to entering reaction (1.1), the B atoms should lose their contact with the main mass of substance B and transfer through the layer ApB q from interface 2 to interface 1. On the contrary, component A enters the reaction (1.1) in the form of particles (atoms or ions) located on the surface of phase A, i.e. particles bonded with the bulk of substance A. The A atoms diffusing through the layer A pBq from interface 1 to interface 2, and the surface B atoms enter into reaction (1.2). Since the reactions (1.1) and (1.2) are in addition separated in space (they take place at different interfaces of the layer), the equality of their rates is an exception rather than the rule. It should be stressed that an initial period of interaction of the elementary substances when there is no layer and, consequently, there is only one common interface at which the substances A and B react directly, is outside the scope of the proposed macroscopic description. The stage of nucleation of a chemical compound between the initial phases may be the subject of examination in the microscopic theory which should indicate, amongst other parameters of the process, also some smallest thickness of the layer starting from which it can be considered that the interaction product, formed at the A–B interface, is a layer of the chemical compound A pB p with its typical physical and chemical properties. However, it can already now be said with confidence that this value is small in comparison with the really measured thicknesses of the layers and, consequently, has no pronounced effect on the shape of the layer thickness–time kinetic dependence. Indeed, experiments carried out using modern investigation methods, including various types of electron microscopy, x-ray diffraction, Rutherford backscattering of light ions, electron probe microanalysis, ion mass spectrometry, etc., showed the layers of chemical compounds several nanometers thick to have all the properties of bulk phases. For example, in the nickel–aluminium reaction couple, R.J. Tarento and G. Blaise 80 were able not only to identify the nickel aluminides NiAl3, Ni 2Al 3, NiAl and Ni 3 Al in layers 5 nm thick, but also to determine the ranges of homogeneity of the aluminides having these ranges (Ni 2Al3 and NiAl). It is interesting to note that the ranges of homogeneity determined by them were in good agreement with the values indicated on the equilibrium phase diagram of the Ni–Al binary system. The same applies to the transition metal–silicon systems which have been studied sufficiently well as the objects important for microelectronics (see Refs.94 and 95). Taking into account that the lattice spacings of chemical compounds are usually of the order of 0.5 nm or greater, one arrives at the conclusion that a layer 5 nm thick can contain at most 10 crystal lattices. Con5

Growth Kinetics of Chemical Compound Layers

sequently, the results of analysis of the nucleation process, carried out by F.M. d‘Heurle18 for the silicides of transition metals appear to be fully realistic. F.M. d‘Heurle evaluated the specific thickness ds (analogue of the critical radius of a nucleus in a homogeneous system, for more detail see Ref.18) for compounds of the Ni–Si system. For Ni2 Si this value was found to be equal to 0.15 nm, i.e. the ‘nucleus’ does not contain even one atomic layer. Although higher values were obtained for other nickel silicides, they never exceeded 1 nm. Therefore, the nucleation process can hardly play any significant role in the formation of the majority of transition metal silicides, except some special cases. 15,18 It is likely that this also applies to other chemical compounds, although a different viewpoint also exists. 101,102 1.3 Growth of the ApBq layer at the expense of diffusion of only component B Let us assume that reaction (1.1) is the only reaction in the A–ApB q –B system, i.e. the diffusivity of component A in the crystal lattice of the compound Ap Bq is negligible in comparison with the diffusivity of component B. The kinetic equation, expressing the growth rate of the layer as a result of diffusion of the B atoms and subsequent reaction (1.1) can be found using the following assumption (or postulates). 103–109 1. The time dt required for increasing the thickness of the layer by dxB1 (from x to x + dxB1, Fig.1.3) is the sum of the time of diffusion of the B atoms through its bulk to the reaction site dt(B) and the time of their dif subsequent chemical interaction with the surface A atoms at interface 1 dt(B) : chem

a Bf

a Bf

dt = dt dif + dtchem .

(1.3)

is directly proportional to 2. The time of diffusion of the B atoms dt(B) dif both the increase of the thickness of the layer dxB1 and its existing thickness x:

a f= dtdif

x

B

k1B1

dx B1.

(1.4)

where k 1B1 is a physical (diffusional) constant, m 2/s. 3. The time of chemical transformations dt(B) is directly proportional chem to the increase of the thickness of the layer dx B1 and does not depend on its total thickness:

af = dtchem B

1 k0 B1

dx B1.

(1.5)

where k 0B1 is a chemical constant, m/s. 6

Solid-State Growth of a Chemical Compound Layer

t = t1

t = t 1 +dt

Fig.1.3 Schematic diagram illustrating the growth of the ApBq layer between initial substances A and B as a result of diffusion of the B atoms and their subsequent chemical interaction with the A surface atoms. n is the inert marker inside the A pB q layer. An increase in layer thickness takes place only at the A – A p B q interface. Not on scale, in fact, d x B1