Physics of Crystal Growth (Collection Alea-Saclay: Monographs and Texts in Statistical Physics)

Physics of Crystal Growth

This text introduces the physical principles of how and why crystals grow. The first three chapters recall the fundamental properties of crystal surfaces at equilibrium. The next six chapters describe simple models and basic concepts of crystal growth including diffusion, thermal smoothing of a surface, and applications to semiconductors. Following chapters examine more complex topics such as kinetic roughness, growth instabilities, and elastic effects. A brief closing chapter looks back at the crucial contributions of crystal growth in electronics during this century. The book focuses on growth using molecular beam epitaxy. Throughout, the emphasis is on the role played by modern statistical physics. Informative appendices, interesting exercises and an extensive bibliography reinforce the text. This book will be of interest to graduate students and researchers in statistical physics, materials science, surface physics and solid state physics. It will also be suitable for use as a coursebook at beginning graduate level.

Collection Alea-Saclay: Monographs and Texts in Statistical Physics

General editor: Claude Godreche

C. Godreche (ed.): Solids far from equilibrium P. Peretto: An introduction to the modeling of neural networks C. Godreche and P. Manneville (eds.): Hydrodynamics and nonlinear instabilities A. Pimpinelli and J. Villain: Physics of crystal growth D. H. Rothman and S. Zaleski: Lattice-gas cellular automata B. Chopard and M. Droz: Cellular automata modeling of physical systems

Physics of Crystal Growth Alberto Pimpinelli Universite Blaise Pascal - Clermont-Ferrand II Jacques Villain Centre d'Etudes Nucleaires de Grenoble

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521551984 © Cambridge University Press 1998 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1998 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Pimpinelli, Alberto. Physics of crystal growth /Alberto Pimpinelli, Jacques Villain. p. cm. — (Collection Alea-Saclay: 4) Includes bibliographical references and index. ISBN 0 521 55198 6. - ISBN 0 521 55855 7 (pbk.) 1. Solids-Surfaces. 2. Crystal growth. 3. Statistical physics. I. Villain, Jacques. II. Title. III. Series. QC176.8.S8P56 1997 548'.5-dc21 96-51772 CIP ISBN-13 978-0-521-55198-4 hardback ISBN-10 0-521-55198-6 hardback ISBN-13 978-0-521-55855-6 paperback ISBN-10 0-521-55855-7 paperback Transferred to digital printing 2007

Contents

Preface List of symbols

xiii xv

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Morphology of a crystal surface A high-symmetry surface observed with a microscope In situ microscopy and diffraction Step free energy and thermal roughness of a surface The roughening transition Smooth and rough surfaces The SOS model and other models Roughening transition of a vicinal crystal surface The roughening transition: a very weak transition Conclusion

1 2 7 7 10 12 15 16 17 21

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Surface free energy, step free energy, and chemical potential Surface tension Small surface fluctuations and surface rigidity Singularities of the surface tension Chemical potential Step line tension and step chemical potential Step line tension close to a high-symmetry direction at low T Thermal fluctuations at the midpoint of a line of tension y The case of a compressible solid body Step-step interactions

23 23 26 27 28 33 35 37 38 39

3 3.1 3.2 3.3

The equilibrium crystal shape Equilibrium shape: general principles The Wulff construction The Legendre transformation

43 43 46 47

vn

viii 3.4 3.5 3.6 3.7 3.8 3.9

Contents

3.10

Legendre transform and crystal shape Surface shape near a facet The double tangent construction The shape of a two-dimensional crystal Adsorption of impurities and the equilibrium shape Pedal, taupins and Legendre transform Conclusion

48 50 51 53 55 56 57

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Growth and dissolution crystal shapes: Frank's model

Frank's model Kinematic Wulff 's construction Stability of self-similar shapes Frank's theorem Examples Experiments Roughening and faceting Conclusion

60 61 62 64 66 66 69 69 69

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Crystal growth: the abc

5.10

The different types of heterogrowth Wetting Commensurate and incommensurate growth Effects of the elasticity of the solid Nucleation, steps, dislocations, Frank-Read sources Supersaturation Kinetic roughening during growth at small supersaturation Evaporation rate, saturating vapour pressure, cohesive energy, and sticking coefficient Growth from the vapour Segregation, interdiffusion, buffer layers

6 6.1 6.2 6.3 6.4 6.5 6.6 6.7

General equations The quasi-static approximation The case without evaporation: validity of the BCF model The Schwoebel effect Advacancies and evaporation Validity of the BCF model in the case of evaporation Step bunching and macrosteps

5.9

7 7.1 7.2

Growth and evaporation of a stepped surface

Diffusion

Mass diffusion and tracer diffusion Conservation law and current densitv

70 70 73 73 75 75 78 79 81 83 84 88 89 90 92 94 97 98 100 111 112 114

Contents

ix

7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10

Vacancies and interstitial defects in a bulk solid Surface diffusion Diffusion under the effect of a chemical potential gradient Diffusion of radioactive tracers Activation energy Surface melting Calculation of the diffusion constant Diffusion of big adsorbed clusters

114 116 117 119 119 123 124 124

8 8.1 8.2 8.3 8.4 8.5 8.6 8.7

Thermal smoothing of a surface General features The three ways to transport matter Smoothing of a surface above its roughening transition Thermal smoothing due to diffusion in the bulk solid Smoothing below the roughening transition Smoothing of a macroscopic profile below TR Grooves parallel to a high-symmetry orientation

130 131 132 133 136 137 140 142

9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9

Silicon and other semiconducting materials The crystal structure The (001) face of semiconductors Surface reconstruction Anisotropy of surface diffusion and sticking at steps Crystalline growth vs. amorphous growth The binary compounds AB Step and kink structure The (111) face of semiconductors Orders of magnitude

144 144 146 147 148 148 149 150 150 151

10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10

Growth instabilities of a planar front Diffusion limited aggregation: shape instabilities Linear stability analysis: the Bales-Zangwill instability Stabilizing effects: line or surface tension Stability of a regular array of straight steps: the general case The case of MBE growth without evaporation Beyond the linear stability analysis: cellular instabilities The Mullins-Sekerka instability Growth instabilities in metallurgy Dendrites Conclusion

156 156 158 162 164 167 168 170 174 176 177

11 11.1

Nucleation and the adatom diffusion length The definition of the diffusion length

181 182

x

Contents

11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 11.12

The nucleation process Adatom lifetime and adatom density Adatom-adatom and adatom-island collisions The case f = 1 Numerical simulations and controversies Experiments The case f =2 Generalization Diffraction oscillations in MBE Critical nucleus size and numerical simulations Surfactants

183 185 185 186 187 188 190 194 195 196 197

12 12.1 12.2 12.3 12.4 12.5 12.6 12.7

Growth roughness at long lengthscales in the linear approximation What is a rough surface? Random fluctuations and healing mechanisms A subject of fundamental, rather than technological, interest The linear approximation (Edwards & Wilkinson 1982) Lower and upper critical dimensions Correlation length Scaling behaviour and exponents

201 201 202 204 205 206 207 208

13 13.1 13.2 13.3 13.4 13.5 13.6 13.7

The Kardar-Parisi-Zhang equation The most general growth equation Relevant and irrelevant terms in (13.1): the KPZ equation Upper critical dimension and exponents Behaviour of X near solid-fluid equilibrium A relation between the exponents of the KPZ model Numerical values of the coefficients X and v The KPZ model without fluctuations (Sf = 0)

211 211 213 215 217 218 219 219

14 14.1 14.2 14.3 14.4 14.5 14.6 14.7

Growth without evaporation Where X is shown to vanish in the KPZ equation Diffusion bias and the Eaglesham-Gilmer instability A theorist's problem: the case X = v = 0 Calculation of the roughness exponents Numerical simulations The Montreal model Conclusion

221 221 222 224 225 227 227 228

15 15.1 15.2

Elastic interactions between defects on a crystal surface Introduction Elastic interaction between two adatoms at a distance r

230 231 232

Contents

xi

15.3 15.4 15.5 15.6 15.7

Interaction between two parallel rows of adatoms Interaction between two semi-infinite adsorbed layers Steps on a clean surface More general formulae for elastic interactions Instability of a constrained adsorbate

234 236 238 241 243

16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10 16.11

General equations of an elastic solid Memento of elasticity in a bulk solid Elasticity with an interface The isotropic solid Homogeneous solid under uniform hydrostatic pressure Free energy The equilibrium free energy as a surface integral Solid adsorbate in epitaxy with a semi-infinite crystal The Grinfeld instability Dynamics of the Grinfeld instability Surface stress and surface tension: Shuttleworth relation Force dipoles, adatoms and steps

249 249 252 255 256 257 259 261 264 268 268 270

17 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8

Technology, crystal growth and surface science Introduction The first half of the twentieth century: the age of the radio The third quarter of the twentieth century: the age of transistors The last quarter of the twentieth century: the age of chips MOSFETS and memories From electronics to optics Semiconductor lasers Quantum wells

277 277 278 278 281 282 284 285 287

Appendix A - From the discrete Gaussian model to the two-dimensional Coulomb gas

289

Appendix B - The renormalization group applied to the two-dimensional Coulomb gas

293

Appendix C - Entropic interaction between steps or other linear defects

296

Appendix D - WulfTs theorem finally proved

300

Appendix E - Proof of Frank's theorem

304

Appendix F - Step flow with a Schwoebel effect

309

Appendix G - Dispersion relations for the fluctuations of a train of steps

312

xii

Contents Appendix H - Adatom diffusion length /,• and nucleation

316

Appendix I - The Edwards-Wilkinson model

319

Appendix J - Calculation of the coefficients of (13.1) for a stepped surface

322

Appendix K - Molecular beam epitaxy, the KPZ model, the Edwards-Wilkinson model, and similar models

324

Appendix L - Renormalization of the KPZ model

326

Appendix M - Elasticity in a discrete lattice

332

Appendix N - Linear response of a semi-infinite elastic, homogeneous medium

335

Appendix O - Elastic dipoles in the z direction

342

Appendix P - Elastic constants of a cubic crystal

345

References Index

347 374

Preface

In writing a preface, an author is faced with the question: what is this book of mine? Of course, in the end only the reader will decide what it really is. The scope of this preface, as of all prefaces, is to say what it was intended to be. This book tries to offer a reasonably complete description of the physical phenomena which make solid materials grow in a certain way, homogeneous or not, rough or smooth. These phenomena belong to chemistry, quantum physics, mechanics, statistical mechanics. However, chemistry, mechanics and quantum physics are essentially the same during growth as they are at equilibrium. The statistical aspects are quite different. For this reason, the authors have insisted on statistical mechanics. Another reason to emphasize the statistical mechanical concepts is that they will probably survive. The concepts developed many years ago by Frank, or more recently by Kardar, Parisi and Zhang are still valid while, for instance, quantum mechanical calculations of the relevant energy parameters will certainly evolve a lot in the next few years. We have not considered it useful to devote too many pages to them, but we have tried to present the frame in which the data can be inserted, as soon as they are known. However, although emphasis is on statistical mechanics, other aspects are not ignored, even though they may have been treated somewhat superficially. The reader will find more detailed information in an extensive bibliography, where all titles are given in extenso, thus making its use much easier. This book is mainly devoted to growth, and therefore to non-equilibrium processes. Nevertheless, we have tried to make it self-contained and to incorporate some elements of equilibrium surface physics, for instance the roughening transition and the equilibrium shape. The reader eager to know more will again find the necessary references in the bibliography. xin

xiv

Preface

The authors are theorists and their book is mainly devoted to theory. Few details are given on experimental methods, but many experimental pictures (mostly from scanning tunneling microscopy) show how real materials do behave. In this domain, too, an abundant bibliography is available. Although the responsability for all which is written here-good or badis completely ours, we owe a lot to all those who contributed to our understanding of the subject. We wish to thank J.M. Bermond, H. Bonzel, J.-P. Bucher, J. Chevrier, G. Comsa, J. Ernst, J. Frenken, M. Hanbiicken, J.C. Heyraud, K. Kern, R. Kern, M. Lagally, J. Lapujoulade, J.J. Metois, B. Mutafschiev and E. Williams, whose experimental works revealed to us all the beauty of Surface Physics and Crystal Growth-and often contributed to the iconographic asset of the book. We are also very grateful to D. Wolf, Ph. Nozieres, R. Kern again, J. Krug, P. Jensen, J. Langer, C. Misbah, L. Sander, D. Vvedensky and A. Zangwill, who shared with us some of their secrets. Special thanks are due to P. Politi and M. Schroeder, whose untiring reading of preliminary versions has been a source of most valuable suggestions and improvements. A good share of the chapters on elasticity has much profited from the competence of C. Duport, who corrected all our formulae, and even explained some of them to us! We thank and beg pardon to all who are omitted here either for space or memory limitations. A final thank is due to the people in Cambridge University Press, and most of all to R. Neal, for waiting patiently for the completion of this work.

List of symbols

D L A : Diffusion limited aggregation (section 11.2) M B E : Molecular beam epitaxy M L : Monolayer R H E E D : Reflection high-energy electron diffraction a: lattice constant or atomic distance (usually taken equal to 1 in this book). Az, Aay, A: kinetic coefficients in sections 13.1 and 13.2 A(t): amplitude appearing in section 13.7 B: a thermodynamic coefficient in (2.38) B^: kinetic coefficient in chapter 13 C: a constant (section 15.7) d: dimension of the space (usually 3) df: surface dimension d—1 df = D/D', d" = D/D": see (6.20) duc: upper critical dimension (section 12.4) dfc: lower critical dimension (section 12.4) df: fractal dimension of a fractal terrace (section 11.2) D: Fick diffusion constant (sections 7.1, 8.4) Ds, D: surface Fick diffusion constant D*: tracer diffusion constant (section 7.1) Di nt ; diffusion constant of interstitials DVac-' diffusion constant of vacancies in section 7.3. D°s: see eq. (7.13) Do: see eq. (11.28) D2: diffusion constant of dimers at a surface (sections 7.10 and 11.8) D1, D": step kinetic coefficients (section 6.4) Ds (D when no ambiguity is possible): surface diffusion constant of adatoms e ; basis of Napierian logarithms E: Young modulus Eim. binding energy of a dimer (section 11.8) xv

xvi

List of symbols

-^ inR . \ / n2 This is formula (1.12), but with a well-defined value of the prefactor, in which we have explicitated the lattice constant a (usually taken equal to 1 in this book). ii) The specific heat does not diverge, and all its derivatives with respect to T are continuous, in contrast with usual transitions (e.g. at the Curie temperature of a ferromagnet). The specific heat has therefore no observable singularity. It has, however a singularity called an essential singularity-a seemingly strange name (coined by mathematicians, probably inspired by philosophic language) for an invisible singularity!

1.8 The roughening transition: a very weak transition

19

As a matter of fact, the surface specific heat would not be easy to measure, since the surface is but a very small part of a solid. It is easier-at least in some instances-to measure the step line tension y, which also possesses an essential singularity: y ~ exp

TR-T

where T\ is a constant. This theoretical result (Nozieres & Gallet 1987) can be compared with the experimental curve of Fig. 1.7. The weakness of this singularity justifies the title of this section. It is also remarkable that the local aspect of a surface does not change very much when crossing the phase transition (Fig. 1.10). Then, if there is no observable singularity of the specific heat, it is usually hard to measure the step free energy and if the aspect of the surface does not change much, how is it possible to determine the roughening transition temperature TRI It is indeed a difficult problem. The usual method is to determine the temperature at which (1.16) is satisfied. An unfortunate situation for an experimentalist, who has to rely entirely on a theoretical formula! The derivation of formula (1.16) disregards a possibly important feature of solid-fluid interfaces, namely the fact that hills and valleys have not the same energy. Symmetry is known to have a great importance in the theory of phase transition, and one should be particularly careful for a very weak transition as the roughening transition. Indeed, according to Rys (1986) in a model without hill-valley symmetry, the transition is killed and the surface is always smooth. According to Den Nijs (1990) Rys's result is incorrect. Rys considered a special model in which steps are not allowed to meet. He claimed that such a model is never rough if the hill-valley symmetry does not exist. This result is correct although the proof is difficult and only accessible to specialists. Den Nijs claimed that, even in the presence of hill-valley symmetry, Rys's model cannot be rougher than the critical roughness. In other words, at sufficiently high temperature, according to Den Nijs, the correlation function (1.4) has a logarithmic divergence corresponding to (1.12), but the multiplying factor is independent of temperature and equal to the universal value which it has, in well-behaved models, only at the roughening temperature. This value turns out to be 2/TT2, SO that G(R)

^\nR.

Den Nijs concludes plausibly that, since the system can only be critically rough without hill-valley symmetry, it cannot be rough at all with hill-

20

1 Morphology of a crystal surface (a)

o o o o o o o o o

0 0 0 0 0 o o oQ]o

0 0 0 OO o o o oo a o oH-
2.2 Small surface fluctuations and surface rigidity As long as o is not known, formula (2.1) is not very useful. However, one can assume for a(zx,zy) an analytic expansion, whose use is comparable with that of the Landau expansion in the theory of critical phenomena (Landau & Lifshitz 1959a). It will be seen, however, that an analytic expansion does not exist below the roughening transition temperature. It is convenient to transform the surface integral in (2.1) into an integral over Cartesian coordinates x and y. For a closed surface, it is only possible if the surface is cut into a few pieces (e.g. 6) and different projection planes are used for the various pieces (e.g. xy9 yz and zx9 each twice). For a given piece, it is convenient to introduce the function (the projected free energy),

zy) = 6, then £ = Arsh(//2e) is large, and thus y'jle ~ e { /2, £ ~ ln(y/Ne). Hence, f

(

+ kBTy'In

^

and recalling the definition of e, = [Wo + Woyf + kBTy'(\ny' - 1)] .

(2.28a)

ii) If | / | < 6, from (2.27) we have = (Wo - 2kBTe + kBTy—).

(2.28b)

Fig. 2.6. Schematical representation of a step element as a smooth continuous line. One can compare this with the microscopic rough appearance of Fig. 1.6.

2.7 Thermal fluctuations at the midpoint of a line of tension y

37

The approach of the present section differs slightly from that of section 1.3. In section 1.3 we computed the average energy of a step of given length and average direction. Here, we have forced the step to start at a point A and to end in a point 5, and we fixed the length and the direction of the segment AB. In fact, the free energy per unit length turns out to be the same in the two situations if AB is long enough. That is, the free energy depends only weakly on the boundary conditions: we can fix the ends or allow them to fluctuate a little, it does not matter very much. The result is not surprising: it is a consequence of the independence of the free energy of a one-dimensional system with short-range interaction on boundary conditions in the thermodynamic limit. We leave to the reader the task of verifying this property in the framework of the broken bonds model previously studied (see problem 2.2). 2.7 Thermal fluctuations at the midpoint of a line of tension y

Consider a line (Fig. 2.6) of line tension y, whose ends A and B are fixed. Let L be their distance. Define an X axis as passing through these fixed points. Let u(X) the ordinate of the line at the abscissa X. What is (u2(L/2))l The interesting point in asking this question is that this quantity is measurable, when the line is a surface step (Alfonso et al. 1992); y can thence be deduced. Let us define the Fourier components uq of u(X) by

E With this definition, only positive q are to be kept. The boundary conditions u(L) = u(0) = 0 impose q = nm/L, with m an integer. For small deviations, the step free energy (2.21) may be written as SF = #"o + b3?, where b3F represents the contribution of fluctuations, and has the following form, analogous to (1.7):

= \y

fLdXuf

2y Jo ^

rL

= - — / . q c l uqUq> I dX sin(qX) sin(g/X)

where the line stiffness y is: (2.31)

38

2 Surface free energy, step free energy, and chemical potential

The equipartition theorem applied to (2.30) states that (\uq\2)=2kBT/(yq2). Whence: P-32)

It might be tempting at this point to replace the sum by an integral, but then the result would be twice as large as the correct one. The correct way is to use the fact that q = nm/L and X = L/2, so that

where we used formula (O.234.2) in Gradshteyn & Ryzhik (1980). The measurement of {u2(^L)) allows thus to deduce y. Alfonso et al. (1992) found y/kBT « 3 at about 900 °C for steps on Si(lll). The order of magnitude of the line tension y is probably not very different. 2.8 The case of a compressible solid body

We are now going to briefly address the question of elasticity, neglected up to this point. The extension of formula (2.1) to a compressible solid requires the intervention of the deformation (or strain) tensor e = {e}ar We recall that it is defined starting from the displacements ua(r) through the relation eay = (5aw7 + dyua). The free energy is, under hydrostatic pressure, and for small deformations (harmonic approximation),

\ JIJIJId r J2 Qg*«y(r)e (r) + / 3

K

J J

/ or> equivalently, in terms of the graph p = g(f) of a function of the slope of / . In fact, take a function /(x) (Fig. 3.4), and consider the tangent line to its graph in the point (xi,/(xi)). The equation of the tangent is y = m\x + n\9 where m\ = f{x\)9 and n\ = f{x\) — m\x\. We can thus associate to the function /(x) of x, the function g(m) of the slope m of / in x, according to the following prescription: for each x, take

48

3 The equilibrium crystal shape

f(x)

Fig. 3.4. Geometric meaning of the Legendre transformation of a function f(x). the slope m = f\x) of / . Then invert this relation to get x = x(m) (which is a well-defined function if / is convex). Finally, define the function g(m) as g(m) = f(x(m)) - x(m)m = f(x(m)) - x(m)f'(x(m)). (3.7) This works well in thermodynamics because convexity is a necessary condition constraining free energies. It is well-known, however, that statistical mechanics models (e.g. mean field models) often produce non-convex free energies; these are for instance responsible for the van der Waals loops in the equation of state of model gases. In this case, it is an easy exercise to realize that the Legendre transformed function is multivalued (see Fig 3.8 below). A fact that shall not in any case plunge us into despair, for no other reason than that we already know the cure for it: it is the double-tangent construction, which gives the equal-area Maxwell rule. We will see later how this applies to equilibrium shapes. Indeed, all properties of Legendre transformations apply lock, stock and barrel to the relation between the free energy and the equilibrium crystal shape. 3.4 Legendre transform and crystal shape First of all, note that (3.6) justifies in some sense the existence of Wulff's construction: the surface free energy and the crystal shape carry the same content of geometrical information, since they are related by a Legendre transformation. Knowing the shape of the crystal is completely equivalent to knowing the graph of the projected surface free energy.

3.4 Legendre transform and crystal shape

49

This has the interesting practical consequence that it is possible to measure (up to a proportionality factor) the surface free energy of a crystal by observing its equilibrium shape; Fig. 2.2 was obtained this way. Clearly, the difficult part is just obtaining the equilibrium shape. From (3.6) we immediately prove -— = —x ,

-— = —y

CZX

CZy

(3.8)

and therefore dx ozx

dy CZy

If we explicitate again the scale factor, we recover equation (3.1). Formula (3.6) gives thus a solution of (3.1). We already stressed that (3.1) is not applicable when the surface free energy-and thus the function (/>-has singularities. We saw in section 2.3 that such singularities exist: they are commonly angular points for (j) as a function of the surface slope, or cusps in the polar plot of a as a function of the crystalline directions n. It is easy to see that these singularities correspond to facets in the equilibrium shape: a proof based on Legendre transforms can be found in problem 3.1. Here we prefer to give a physical argument. Suppose that the crystal surface has a singular orientation in a neighbourhood of the point P. This point must be surrounded by steps. Let R be the radius of the smallest of such steps. Along it, as well as inside it, the chemical potential S/J, is given by (2.12). It is thus equal to \/R times a positive constant. In other words, 1/R has a macroscopic value proportional to Sfi, and thus, according to (2.9), proportional to the reciprocal of the crystal radius. This reasoning is only approximate, since it neglects the effect of step-step interactions which may modify the formula (2.12). We leave to the reader the task of investigating this problem. Thus, the chemical potential on a facet is not fixed by formula (2.9). The latter is in fact not defined on a facet, since (j)(zx,zy) is not differentiate at that point, according to (2.6). If the facet is at equilibrium, the chemical potential is uniform all over it, and equal to the value it takes at the facet edge, where (2.9) can be given a meaning. Therefore the chemical potential on a facet is not fixed only by the orientation of the facet itself, as (2.9) seems to imply; it also depends on the size and the shape of the facet. Our definition of the equilibrium shape in terms of Wulff's construction is also valid for facets. As we have just discussed, the chemical potential on a facet at equilibrium is fixed by its value at the facet edge. The Wulff construction guarantees that the chemical potential is constant on the

50

3 The equilibrium crystal shape

O*

L

Fig. 3.5. Wulff 's construction for a crystal with rough and non-rough orientations. The facet AB joins smoothly to the neighbouring rounded parts. facet edge (which is necessary for the facet to be at equilibrium) and that it has the required value. For instance, in the-unrealistic-case of a step line tension y independent of the step orientation, the facet must be circular. Its radius R fixes the chemical potential over the facet according to the Gibbs-Thomson formula /n = /LLO + y/R (chapter 2). Wulff's construction now warrants that the chemical potential takes precisely this value near the facet-as well as everywhere else on the crystal surface. For the sake of visualization, let us take a two-dimensional crystal (Fig. 3.5). The point H defined by (3.5) spans a curve (Z) which has an angular point at K — (O,ffo). When H tends to K from the right (or the left, respectively) side, the point of intersection of (A) and (Ao), normal to, respectively, OH and OK, tends to the point A (resp. B). Considering the angular point of (S) as the limit of an arc of very large curvature, one easily realizes that the curve (S) resulting from Wulff's construction (i.e. the crystal surface) contains the segment AB. We recall that cusps in polar plots of a are called roughness singularities because a crystal can only possess facets with a given orientation if the temperature is lower than the roughening temperature relative to that orientation. 3.5 Surface shape near a facet Two cases are possible: the facet may join smoothly with the neighbouring surface, or with a sharp edge. Thus, the equilibrium shape may possess

3.6 The double tangent construction

51

or not slope discontinuities. We will first consider the case of a smooth junction. The problem is: how does the slope tend to zero? Let us take a cylindrical symmetry, and choose the xy plane parallel to the facet. The lines z = constant are parallel to the y axis. While the size (in the x direction) of the facet depends on the step line tension (see problem 3.1), the shape of the rounded part near a facet depends crucially on the step-step interactions. Several mechanisms, as we saw in section 2.9, predict interactions decaying as I / / 2 = (z') 2, where t is the step spacing, and zf is the step density; the projected surface free energy 0 is then, as in (2.38): ct>{zf) = y \ z f \ + B \ z f \ \

(3.10)

whose Legendre transform is easily obtained (see problem 3.2):

-2/(27B)kx-y)K =l 0,

l

x

+ y)5

x>y -y<x\ and cpi are determined by boundary conditions at steps. In the absence of the Schwoebel effect, (6.2) yields = po-f

= (po

or (po = q>\ exp(oci?/2) + (p2 exp(a 2 ?/2) = cpi exp(—ai?/2) + (p2 exp(—a2?/2) It follows s i n h («?

where we used ai — a2 = ^v2 + 4. The velocity must be found self-consistently from matter conservation at the step:

108

6 Growth and evaporation of a stepped surface

The result reads

sinh2(f)

tanh(-\/t; 2 + 4 ) -

(2)

Recalling that v = KTVV it is now an easy matter to verify that equation (6.9), giving the step velocity in the quasi-static approximation, is recovered when v < 1. This means v < \/DS/TV . The step flow velocity must be negligible with respect to the 'adatom velocity' y/Ds/Tv for the approximation to be valid. In the absence of evaporation, TV —> oo, (2) reduces to v = ivf-an equality which is a special case of (6.10) for £ = £' in this limit-to first order in v//Ds. Thus, the condition of validity of the quasi-static approximation is F/Ds < 1 for pure growth. This condition is always satisfied in MBE. 6.2. Derive the step flow velocity in the absence of evaporation. Solution FF

F/

p({) = P o + _ coth (vt/2D) - —

r

F/

r

exp (-v

2vsirih(vt/2D) and

6.3. Write (6.19) for D = D"', t much larger than the atomic distance, and K(

< 1.

Solution - Using the lengths d! = D/A! and d" = D/A!', equation (6.19a) reads vr =

KD(PO-FTV)X

2 - (1 - Kd") exp (-*/') - (1 + ted") exp (K/') X

(1 + Kd") (1 + Kd!) exp (K/') - (1 - Kd") (1 - Kd!) exp ( - K / ' ) '

The assumption D = D/r implies that A" is infinite, and d" vanishes. Expanding to first order in K/ we get 9 _

(\

_

jr/'^

_

h

-U v/f\

4- ^ / ^

1

2

^r = (Po - i 7 ^) (1 + Kd') (1 + K/') - (1 - Kd ) (1 - Kf) and

"'^^•-""•ii^Tri

Problems In the case of growth, the product PQK2 may be neglected, but not which is equal to 1 according to (6.6). One thus finds V

109 DTVK2

~ 2{d! + /') '

Equation (6.19b) reads: v" =

KD

(p0 -

FTV)

x

2 - (1 - Kd') exp(-K/") - (1 + Kd') exp(K/") X

(1 + Kd")(\ + K

Making again d" = 0, and expanding the exponentials, yields

The step velocity v is then in pure growth (1/Ty = K = 0)

Note that, if / r = / " = /, v becomes

F/ which expresses the fact that the step velocity is equal to the flux of atoms on each terrace. 6.4. Write and solve the equations of evolution of a stepped surface during growth with desorption, when all terrace widths are much smaller than xs = 1/K. Discuss the stabilizing or destabilizing action of the Schwoebel effect. Solution - We only discuss the case of a total Schwoebel effect. During growth with evaporation, the positions xn of the steps obey the equations xn = (F -

PO/TV)

(x n+ i - xn) .

Defining £n = xn+i — xn, and letting F — poAu = K one obtains Sn=K (/ n+ i — /n). The Fourier transform Xk of tn satisfies Xk=K

(eik - 1 ) 4 ,

or Afc = K (cos k — 1 + i sin k) Ik • The regular array of steps is stable if all eigenvalues K(cosk — 1 + isin/c) have a non-positive real part. Therefore it is stable if K > 0, (growth) and unstable if K < 0 (evaporation). This result has been obtained in section 6.7.

110

6 Growth and evaporation of a stepped surface

6.5. Discuss the evolution of a stepped profile with an initially random distribution of step widths during MBE growth with Schwoebel effect. Show that the width of the distribution decreases approximately as the inverse fourth root of the deposited coverage. Solution - See Gossmann et al. (1990a,b). 6.6. Discuss the diffusion and incorporation of impurities during the MBE growth of a stepped surface. Consider the following cases, a) Impurities are incorporated with probability 1 when they meet a step, b) Impurities can jump over a step (case of 'surfactants'). What are the modifications with respect to section 6.1? Solution - See Andrieu et al. (1989).

7 Diffusion

meme equation, Celle de Laplace, se rencontre dans la theorie de I'attraction newtonienne,

The same equation-the Laplace equation, appears in the theory of gravitation, in that of

dans Celle dU mOUVement

motion in liquids, in that of the

des liquides, dans celle du potentiel electrique, dans celle du magnetisme, dans celle de la propagation de la chaleur et bien d'autres encore.

electric potential, in that of magnetism, in that of heat propagation as well as in many other theories.

Henri Poincare, (La valeur de la Science)

In the bulk or on a surface, the theory of diffusion is full of unexpected traps. The most obvious one is that the diffusion constant which appears in Fick's law is not the same which describes the random walk of an isolated atom. The latter is called tracer diffusion constant. This means that you need a tracer, e.g. a radioactive tracer, to measure it. In the case of diffusion on a clean surface (self-diffusion) there is no equivalent of Fick's law. One can however define the response to a chemical potential gradient, which turns out to be the quantity measured by many macroscopic techniques... This introduces a 'surface mass diffusion constant' Ds which, on a high-symmetry or a vicinal surface, is a linear combination of the diffusion constants of adatoms and advacancies. At ordinary temperatures, the surfaces of usual metals do not evolve spontaneously because Ds is very low. On the contrary, diffusion is already quite important on these surfaces in MBE growth, because the adatom diffusion constant is much higher.

Ill

112

7 Diffusion 7.1 Mass diffusion and tracer diffusion

In this introductory section, certain general notions concerning diffusion in a d-dimensional space will be recalled. Most of them apply to surface diffusion if d is set equal to 2. More details will be found in textbooks (Adda & Philibert 1966, Philibert 1990) and review articles (Bonzel 1975, 1990, Gomer 1990). It is convenient to distinguish two diffusion processes, to which correspond two diffusion constants (or 'coefficients'). The first process is collective mass diffusion: if the density p(r, t) of some material in a homogeneous medium is not uniform, the motion of atoms tends to equalize it. The process is described by Fick's law, in which appears what we will call the Fick diffusion constant D:

The second process is the individual random walk (Fig. 7.1) of a particular atom. A usual and often good approximation in (and on) a solid at low enough temperature is to assume that the atom moves by uncorrelated jumps. If the atom jumps by a vector an at its n-th step, the mean square path after m steps is 2\

= n(a

(7.2)

n=\

oooooooo yOOOOOO

oooooooo

Fig. 7.1. Random walk of an interstitial impurity in a crystal. The left hand side represents a general random walk. It is often assumed that all jumps are of a single atomic distance as shown on the right hand side.

7.1 Mass diffusion and tracer diffusion

113

The second factor is independent of time and the first one is proportional to the time t. If the position of the atom of interest at time t is r(t), one can write r{tf + t)-x{tf)]2)=2dD*t

(7.3)

where D* is called tracer diffusion constant, because this kind of diffusion is only observable if a few atoms are marked. This is generally achieved by observing the diffusion of a radioactive isotope or tracer. One can be interested in the diffusion of a material in another one (heterodiffusion) or in the diffusion in a pure material (self-diffusion). When one deals with self-diffusion in a solid, one has generally in mind tracer diffusion. There is a special, important case where D is equal to D*. This happens for dilute objects (let us call them 'impurities') which do not interact. The local impurity concentration can be written as p(r, t) = S/P/(M)> where Pj(v,t) is the probability that the 7-th impurity is at r at time t. This probability, after all, is a density, and therefore should satisfy (7.1). The sum of these probabilities also satisfies (7.1) with the same diffusion constant D. On the other hand, solving (7.1) with the initial condition p(r,0) = 5(0), one finds Pj(T,t)

= (4Dt)~d/2Qxp

(-r2/4Dt)

Hence

([x{tf + t ) - v{t!)f) = I ddrp(r, t)r2 = 2dDt. Comparison with (7.3) yields D = D*. It is not quite obvious that this argument is incorrect when the concentration is not weak. This is the issue of problem 7.2. It is illuminating to consider the following examples, which show that D* and D are generally different. 1) Consider a lattice gas in which the only possible process is the exchange of 2 atoms. In this perhaps academic case, D = 0 and D* ^ 0. 2) In a fluid at its critical point, D = 0 because a density fluctuation of weak amplitude and long wavelength feels no restoring force. On the other hand, D* has no reason to vanish. Other examples will follow in the next sections. The effect of interactions between particles on the difference between D and D* has been investigated by Ferrando & Scalas (1993) and in textbooks (Philibert 1990).

114

7 Diffusion 7.2 Conservation law and current density

It follows from (7.1) that the total number of particles, fddrp(r,t), integrated over the whole sample, is conserved, i.e. independent of time. If the integration is on a finite volume V, the time derivative is equal to the flux of the current density j through its boundary surface S, namely

ddrp(r, t) = -J dd"h n • j(r, t) = - J ddr divj(r, t). Since this should be true for any volume, one deduces the continuity (or conservation) equation 0.

(7.4)

i(rj) = -DVp(rj).

(7.5)

Comparison with (7.1) yields

In the case of a surface, generally considered in the following chapters, j is the surface current density, defined by the fact that the net flux through a line element on the surface in time St is n • jdA5t, where d/ is the length of the line element and n the unit normal vector. 7.3 Vacancies and interstitial defects in a bulk solid Diffusion in solids takes place mainly through the motion of defects, especially vacancies and interstitial atoms (Fig. 7.2). It is possible to define diffusion constants of these defects. This is not quite obvious, since they have a finite lifetime. However, the defect densities satisfy equations analogous to (7.1) for times shorter than the lifetime, and the mean-square displacements satisfy equations analogous to (7.3) for times shorter than the lifetime. Since the defect concentration is usually low in a solid, the Fick and tracer diffusion constants of a given defect are the same. Let DmX and D vac be the diffusion constants of interstitials and vacancies. If there are very few vacancies, or if they move very slowly, then diffusion is mainly due to interstitials. The density of the crystal (assumed to be pure) is in appropriate units p(r, t) = 1 + Pint(r91). Inserting this relation into (7.1), one finds pmi = Z>V2pint- This relation implies that pint satisfies a Fick equation with a Fick diffusion constant Dint = D. Thus, the Fick diffusion constant D is equal to the diffusion constant Dint of interstitials. However, as mentioned in section 7.1, when dealing with self-diffusion in solids, we are generally more interested in the tracer diffusion constant D*. It is not equal to D[nU but to DmX times the average interstitial concentration pmi. Before proving this, note that formula (7.3), which

7.3 Vacancies and interstitial defects in a bulk solid

3

115

OO OOO O

OOO O (X) O O

ooo>C oooo OO0 OOOOO oooooooo

Fig. 7.2. Three self-diffusion processes in a crystal, represented as two-dimensional for graphical convenience, a) Exchange of atoms at normal sites, b) Diffusion of interstitial defects, c) Vacancy diffusion. Process (a) only contributes to tracer diffusion, not to mass diffusion. defines D*? is only valid for times so long that each atom has accomplished several jumps as an interstitial. This implies that any given interstitial atom exchanges rapidly its position with an ordinary atom. It will now be proved that D* = AntPint for interstitial-dominated selfdiffusion. Indeed, the average displacement (7.3) is equal to the average displacement \Srfnt(t)\ of an interstitial, multiplied by the number of interstitials Npmu divided by the number of atoms N. Since ( 0: ~pn(kx,ky,t)=

-DYk2pn(kx,ky,t) +DY [pn+i(kx,ky, t) + pn-i(kX9ky, t) - 2pn(kx,ky, t)}

and for n=0: po(kx,ky, t) = -Dsk2po(kx,ky,

t) +1>; [pi(fcx,fey, t) - po(kx,ky, t)] .

Let now pn(kx,ky,t) = K(kx,ky)Rn(t), and /c2 = /c2 +/c2. The preceding equations become, respectively: Rn(t) = -Dvk2Rn(t)

+ Dv [Rn+1(t) + Rn^(t) - 2Rn(t)] ,

Ao(t) = -Dsk2Ro(t) + DC [ ^ I ( 0 - i^o(0] • Let us forget for the moment the term proportional to k2. We can cast both equations into one, if we note that they describe a one-dimensional random walk between layers, which cannot cross the boundary. In other words, the layer n = 0 acts as a mirror, reflecting back the approaching walker. The classical method for solving this kind of problems is the method of images (Sommerfeld 1964): one solves the random walk problem without the mirrorthat is, the boundary-adding in its place an initial condition Rn(t = 0) = (>n,o + (>n,-i- Therefore, the net particle current crossing the boundary at n = 0 is always nil. The equations read now: Rn(t) = Dv [Rn+1(t) + Rn-!(t) - 2Rn(t)] , Rn(0) = Sn,0 + V i • These are easily solved using once more the Fourier transforms R(q,t)\ k , t) = -2D V (1 - cos q)R(q, t),

128

7 Diffusion

and by employing known properties of the modified Bessel function of imaginary argument:

1 f71 cosq cos(nq)dq , in(z) = - / e n Jo

It is now easy to show that the solution reads

x[ln(2Df(n)t)+In+l(2D\n)t)] where we defined D(n) = D'(n) = Dv for n ^ 0, and D(0) = Ds> D'(0) = D'v. 7.6. Calculate the chemical potential as a function of the density for a noninteracting lattice gas, using the formulae \x = (dF/dN)TV and F = U — TS. Rederive the result by using the grand-canonical distribution. Solution - In the absence of interactions, (7 = 0 and F = —TS = —/ For N particles on a lattice of JV sites,

The Stirling formula yields

(

Ing = jrinjr

- NlnN - (jr - N)ln(JT - N)

^

^

^

^

^

= -Jf [plnp + (1 - p)ln(l - p ) ]

Hence F = kBTJT [p\np + (1 - p)\n(l - p)] and

»=-^{d-f)

=kBT[lnp-ln(l-p)]=kBTln-?-

or

and

These formulae may be obtained much more easily by the grand-canonical distribution. Indeed the probability of having HR atoms at site R is given by

Problems

129

Const x Y[RQxp(PfiRnR), where fiR is the local chemical potential at site R. Since YIR can be either 0 or 1, it follows p(R) = < nR >=

8 Thermal smoothing of a surface

/ was perpetually asking not for mathematical equations, but for physical circumstances of what they were trying to work out. R. Feynman (Surely you are joking, Mr. Feynman)

It is fortunate that things do not reach their equilibrium shape studied in chapter 3. It would be too bad, if Donatello's or Rodin's statues were to behave as hedge-hogs! However, they would do so if they were heated at high enough temperatures. Annealing is indeed one of the methods used in laboratories in order to get smooth crystal surfaces. During the annealing process, atoms diffuse until they have enough neighbours, i.e. enough satisfied chemical bonds, to be happy. Rather than considering the detailed atomic structure, it is generally preferable to introduce the chemical potential. Atoms go to the places where the chemical potential is low. Since the chemical potential is related to the shape through the Gibbs-Thomson formula, surfaces become smooth. At very long distances, surface diffusion is less efficient than evaporation followed by condensation. In this chapter, we address the case where there is no net evaporation. This situation is easily obtained by putting the sample into a closed container. The distances on which thermal equilibrium can be reached are: • A few microns at temperatures of order 2 T M / 3 . • A few nanometres at temperatures of order T M /3.

130

8.1 General features

131

Fig. 8.0. Ancient works of art do not possess their equilibrium shape, which is not always the case with modern ones!

8.1 General features

In this chapter, we examine the possibility of reducing the roughness of a surface by annealing, i.e. by moderate but long heating. Surface annealing is often used after MBE growth or after cleaning by an ion beam. In addition, it is a method of measurement of the diffusion constant (Bonzel 1975). It will be assumed that there is globally no growth and no evaporation, or in other words, that the chemical potential inside the solid and inside the vapour is the same. This situation is easily obtained by putting the crystal into a closed container. However, if the surface has not its equilibrium shape, the chemical potential is not uniform on the surface as seen in chapter 2. The variations of the chemical potential imply, according to (7.11) a flux of adatoms and advacancies from places where the chemical potential is high to where it is low. For instance, if there are islands and lakes (Fig. 8.1), matter will flow from islands to lakes. This flow generally takes place through surface diffusion and the current density is then given by (7.5). However, surface

8 Thermal smoothing of a surface

132

(a)

(b)

Fig. 8.1. The density of adatoms (and their chemical potential) is higher near a terrace (a or c) and weaker near a hole (b or d). Pictures (c) and (d) correspond to a continuous description, applicable to large terraces and holes.

transport is not always the most relevant process as will be seen in the next section. 8.2 The three ways to transport matter

Just as human beings do, atoms prefer to travel airborne or by surface transportation (Fig. 8.2). Pursuing this analogy, it may be said that going airborne is faster on long distances, but taking off requires some time (to jump over high energy barriers) so that surface transportation, i.e. surface diffusion, is more appropriate for short trips. These facts will be summarized by precise formulae in the next section. Atoms also have the possibility, as Londoners or New Yorkers do, to travel underground (Fig. 8.2). This process, i.e. volume diffusion, may sometimes be effective at intermediate distances, but will generally be ignored in this book. It is just briefly addressed in section 8.4. The characteristic length r\ beyond which air transportation dominates surface transportation may be roughly evaluated as follows, on a highsymmetry surface. This length is the distance on which atoms can diffuse without evaporating. According to diffusion theory (see formula 7.3) the time t needed to diffuse on a distance r is of order r2/4Ds. The evaporation probability during this time is t/rv « r2/(4DsTv)9 where 1/T,, is the probability per unit time that a given adatom evaporates. Surface transport dominates if this quantity is much smaller than 1. It follows (8.1)

8.3 Smoothing of a surface above its roughening transition

133

Fig. 8.2. Various ways of matter transport which contribute to smoothen a surface. Surface diffusion (a) is generally the dominant process. Transport through the vapour (b) becomes important at high temperatures and long distances. Diffusion through the bulk (c) will generally be neglected here.

This length coincides with the characteristic length xs = 1/K encountered in chapter 6 (equation 6.6). At low temperature, xv is huge, and r\ too, although Ds is rather small. Actually, evaporation is negligible for elements in most cases. It is not so for binary systems. In GaAs, for instance, in usual MBE conditions, arsenic evaporates. 8.3 Smoothing of a surface above its roughening transition

In this section we consider a crystal surface which is approximately planar, but for a small initial perturbation which slowly decays in time. We wish to study this decay. If the average orientation, supposed parallel to the xy plane, is above its roughening temperature, this decay turns out to be described by linear equations which may be solved by a Fourier transformation. Accepting this assumption, which will be checked selfconsistently, it is enough to consider a single Fourier transform. Therefore, we assume a sinusoidal initial profile of the form (Fig. 8.3): 7ZX z\ A, i — \j) — nyyj) tuis

.

^o.zdj

2L It will be found that the equation of motion preserves the sinusoidal form and has the solution (8.2b) z(x, t) = h(t) cos %x 2L The corresponding chemical potential is given, as seen in chapter 2, by

d_ dcp foe

(8.3)

where cp is the projected free energy density per unit area of the xy plane, and z a = dz/dxa. For a weak modulation amplitude h, cp may be

134

8 Thermal smoothing of a surface

(a)

(b)

Fig. 8.3. A method, used in particular by Bonzel et al (1984a,b) to measure the surface diffusion constant of a material, consists of making sinusoidal grooves (a) and measure the time they need to smoothen. When the average orientation of the surface is below its roughening transition, the formation of pseudo-facets (b) is observed. approximated by a quadratic function which will be assumed isotropic for the sake of simplicity: (8.4) It would be straightforward to take the anisotropy of the surface stiffness a into account. Relations (8.3) and (8.4) yield Zyy

(8.5)

The profile (8.3) may represent a single Fourier component of a complex accidental modulation. It may also represent an artificial, macroscopic profile cut in a surface in order to make an appropriate experiment (Bonzel 1975). Evaporation-condensation kinetics At high enough temperatures or long enough distances, evaporation dominates surface diffusion according to (8.1). It will be assumed that transport

8.3 Smoothing of a surface above its roughening transition

135

in the vapour is so fast, with respect to evaporation and adsorption (condensation), that the chemical potential /xo in the vapour is uniform. The chemical potential \i at each point of the surface is given by (8.5). Evaporation dominates adsorption at all places where \i > /io, while the reverse is true where \i < fiQ. According to the axioms of irreversible thermodynamics, the derivative z of the surface height with respect to time will be assumed to be a linear function of the chemical potential difference (// — juo), provided that this difference is small: z = —Const x (fi — /io) •

(8.6)

Insertion of (8.5) into (8.6) yields z = v (zxx + zyy) ,

(8.7)

where v > 0 is a constant. This equation of motion is linear. A consequence of this is that an initially sinusoidal profile keeps its sinusoidal shape. If the initial profile is given by (8.2a), the solution of (8.7) is given by (8.2b). More precisely, as first derived by Mullins (1957, 1959) z(x, y91) = z(x, y, 0) exp {-t/x(L))

(8.8)

where T(L) = Const x L 2 .

(8.9)

At usual temperatures, the vapour at equilibrium with the solid has generally an extremely low density, so that the coefficient v in (8.7) is very small. Surface kinetics are therefore often dominated by surface diffusion, to which we turn now. Surface diffusion kinetics We now address the case where evolution of the surface shape is due to adatom or advacancy diffusion on the surface. It follows from (8.1) that this hypothesis is correct (Mullins 1957, 1959) for nickel at temperatures between 1000 K and 1000 °C with a surface modulation of wavelength L « 10 pm. The first thing to write is a conservation equation which expresses the absence of evaporation: z = -V-js.

(8.10)

The surface current density j s is given by (7.11): h = -PDsVii

(8.11)

where V = (dx,dy). Inserting (8.5) into (8.11) and (8.10), one finds the equation of motion z = -KV2

(V 2 z) ,

(8.12a)

136

8 Thermal smoothing of a surface

where K = pbsa > 0. This equation of motion is still linear, so that an initially sinusoidal profile again remains sinusoidal. This feature has been experimentally observed on the (110) face of nickel (Bonzel et a\. 1984a,b). If z does not depend on y9 (8.12a) reads

*~K0-

< 8I2b >

If the initial shape is given by (8.2), the solution of (8.12) has again the form (8.8), but now (8.9) should be replaced by T(L) = ^KL4 . 7T 4

(8.13)

The experimental determination of the constant yields the surface mass diffusion coefficient Ds = ksTK/a if the surface stiffness a is known. If both evaporation and surface diffusion are taken into account, the general equation of motion obtained combining (8.7) with (8.12a) is z = vV2z - KW2 (V 2 z) .

(8.14)

The Fourier transform of wave vector q, zq, has a relaxation rate 1/Tq=vq2+Kq4 . As said above, v is generally small excepted at very high temperatures, and the second term generally dominates. At high temperature, the first term is important for small wavevectors, i.e. at long distances. 8.4 Thermal smoothing due to diffusion in the bulk solid The previous section has been devoted to the cases when mass transport is due to surface diffusion or to evaporation+condensation. We now investigate the case when transport through the bulk dominates these two processes. In practice, it is generally not so. However, the case of volume diffusion is of interest, not only for the sake of completeness, but also because the equations to be solved are of a very common type in surface physics. In the solid, the equation is the diffusion equation (7.1). As before, the profile is assumed independent of y9 so that this equation reads

Note that, in this section, the matter density p and the current density j are three-dimensional densities, and D is the bulk diffusion constant. As in chapter 6, the quasi-static approximation may be used. The above equation reduces to dx2

8.5 Smoothing below the roughening transition

137

The boundary condition at the surface is obtained by equating the expression (8.5) of the chemical potential \i with the expression (7.7a) which states that its deviation R'9 and vice versa. This proves (in this simple case) the statement that the big island grows at the expense of the small one. Ostwald's theory was improved by Lifshitz & Slyozov (1961) and extended by Chakraverty (1967) and Villain (1986). The most recent theoretical developments have been reviewed by Bray (1994). The agreement with experiment is often only qualitative (Ernst et al 1992a, Sholl & Skodje 1995). Ostwald's argument was actually related to a physically different problem: that of a supersaturated solid solution AXB\-X. For large x, the constituent B forms islands, which are of course three-dimensional. The above argument suggests that big islands grow at the expense of smaller ones. The same argument also applies to the evolution of the StranskiKrastanov or Volmer-Weber droplets of Fig. 5.1. The case of the clean

8.5 Smoothing below the roughening transition

139

surface of Fig. 8.4 is slightly different. Indeed, if the size distribution of terraces and lakes are the same, terraces and lakes can both shrink. Mass conservation does not require the increase of bigger terrace sizes. However, small terraces and lakes will disappear first for two reasons: because the quantity of atoms to be evacuated is smaller; and because the excess chemical potential is higher, in absolute value, for small defects. This is a consequence of the Gibbs-Thomson formula seen in chapter 2. Qualitatively, it will be seen that, after a time t, only islands and lakes of size larger than some value JRC(£) survive. One might use a procedure similar to the method used in chapter 6 for straight steps. First deduce the adatom and advacancy densities p(r) and c(r) from the diffusion equation (7.1) by neglecting the motion of steps (quasi-static approximation). Then the current j reaching the steps is deduced from the equation j(r) = —DVp(r) + AVff(r), and the decay rate of terraces and lakes follows. This non-self-consistent procedure is justified because diffusion is much faster than step motion. Here, an equivalent method will be used: the surface current of matter j(r) will be deduced from (7.11). The chemical potential /i(r) is in principle given by Laplace's equation V2ji = 0 between terraces, with boundary conditions at steps given by the Gibbs-Thomson relation. The latter will just be written 5ji = y/R near a terrace of radius R, and dfi = —y/R near a lake of radius R, where S/J, is the chemical potential counted from its equilibrium value. Solving the Laplace equation for all possible geometries and averaging over them would be a very hard task indeed. We shall be contented with a very simplified calculation. The terraces which decay first are the smaller ones as argued above. If a small terrace of radius R is surrounded by bigger ones (or by lakes) at distances of order r, the chemical potential near this terrace is y/R; the chemical potential gradient is of order y/(Rr), the current density from this terrace is of order (}yD/(Rr) according to (7.11) and the total number of atoms — dN/dt emitted by the terrace per unit time is or order PyD/r. With our usual units, N may be identified with nR2. It will be assumed that all lengths are of the same order of magnitude, r « R. The result is that R decays according to the equation d dt where D is given by (7.12). Integration of the previous equation yields R3(t) - R3(0) « fibyt = (Dp + ACT) 1/3

fiyt.

(8.16)

The exponent 1/3 in R(t) ~ t turns out to be the same as obtained for three-dimensional binary alloys by Lifshitz & Slyozov. In the case of

140

8 Thermal smoothing of a surface

the three-dimensional clusters on a surface (Fig. 5.1), Chakraverti (1967) has obtained an exponent 1/4 instead of 1/3. 8.6 Smoothing of a macroscopic profile below TR

We now come back to the case addressed in section 8.3 and represented by Fig. 8.3, but it will now be assumed that the average orientation z = Const is below its roughening transition temperature TR. Experimentally (Bonzel et al. 1984a,b), the profile is seen to take a trapezoidal shape (Fig. 8.3b). The bottom and the top of the profile look like facets. One might a priori think that this quasi-faceting is similar to the formation of facets at equilibrium. However, this dynamical faceting is not really expected, since it has been seen in chapter 4 that dissolution or evaporation shapes (which might be expected at the top of the profile) should be dominated by orientations with fast kinetics, which are generally not faceted. On the other hand, one cannot have too much confidence in the model of chapter 4. As will be seen, the dynamical faceting observed by Bonzel et al is still subject to controversy. Let us first address an easier problem, the case of a 'bi-directional' profile (Fig. 8.5). Such a profile is not easy to work out in practice, and has never been experimentally studied for this reason. The basic point is that the surface is formed by terraces separated by steps, so that the evolution of the surface may be regarded as a motion of steps: where steps retreat (or advance, respectively), the solid evaporates

Fig. 8.5. Thermal smoothing of a surface in which a bi-directional profile has been cut. The initial shape may be for instance z = Const x cos(Qx) cos(Qy). Below the roughening transition, the transient shape before total smoothing should involve facets on three levels. The experiment has never been done because it is difficult to cut such shapes.

8.6 Smoothing of a macroscopic profile below TR

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Fig. 8.6. a) If the grooves of Fig. 8.3a are perfectly parallel to a high-symmetry orientation, the transient shape (before total smoothing) should have sharp singularities at minima and maxima (formula 8.21). b) On the other hand, a miscut (but not too miscut) surface evolves from the sinusoidal shape of Fig. 8.3a to the trapezoidal shape of Fig. 8.3b. This is because the steps have, near the top, a high curvature which generates a high chemical potential and a fast recoil. The high chemical potential is analogous to the high electric potential near a tip in electrostatics.

(or grows, respectively). As seen from the previous section, small terraces and lakes decay faster than bigger ones. These small terraces are precisely those which are at the top and at the bottom of the profile, and one can therefore expect the top and the bottom to flatten. In addition, faceting also occurs at half-height. This is due to the fact that the chemical potential has an abrupt change of sign at this place. Qualitatively, this result is due to the existence in the initial profile of steps with a high curvature. The argument does not apply to the profile of Fig. 8.3b or 8.6a, when all steps are straight. It does apply, however, if the profile is slightly miscut (Fig. 8.6b), as it is always in real experiments (Surnev et al. 1996). In that case, the qualitative experimental results can be reproduced, as well in the case of diffusion kinetics (Villain & Lan<jon 1989, Bonzel & Mullins 1996) as in the case of evaporation (Lan<jon & Villain 1990). If the miscut angle is too large, however, one recovers the linear problem of section 8.3, and the profile remains sinusoidal.

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8 Thermal smoothing of a surface 8.7 Grooves parallel to a high-symmetry orientation

This problem gives an opportunity to perform exciting exercises of nonlinear mechanics (Rettori & Villain 1988, Lan<jon & Villain 1990, Ozdemir & Zangwill 1990). It is still subject to controversy (Spohn 1993, Selke & Bicker 1993, Selke & Duxbury 1994). The steps are parallel straight lines (Fig. 8.6a) which run parallel to the planes z = Const. The latter are below their roughening transition. Until now, such an ideal situation has never been realized, and the surfaces investigated so far were always slightly miscut. An appropriate form of the continuous approximation will be used. It is not obvious that this is correct, but the same results can be obtained if steps are taken explicitly into account (Rettori & Villain 1988, Ozdemir & Zangwill 1990). The first thing to do is to calculate the chemical potential which corresponds to a given surface shape. It is given by (2.9). Taking the y axis parallel to the steps and using the notation dz/dx = z\ (2.9) reads

The free energy density cp(zr) contains a term proportional to the density of steps, which is \z'\. This term has been discussed in section 2.3. There is an additional term due to interactions between steps. As seen in section 2.9, the interaction free energy per unit length between parallel steps at distance t is proportional to /~ 2 for all accepted interaction mechanisms. Therefore, the interaction free energy per surface area is proportional to *f~3 or |z'| 3 . The projected free energy density is now