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Boyan Mutaftschiev

The Atomistic Nature of Crystal Growth

With 98 Figures

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Prof. Boyan Mutaftschiev Laboratoire de Mineralogie-Cristallographie Universites de Paris VI et Paris VII 4, Place Jussieu 75252 1'ans France

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Mutaftschiev, Boyan, 1932-The atomistic nature of crystal growth / Boyan Mutaftschiev.

p. cm. -- (Springer series in materials science, ISSN 0933-033X ; v. 43) Includes bibliographical references and index. ISBN 978-3-642-08577-2 ISBN 978-3-662-04591-6 (eBook) DOI 10.1007/978-3-662-04591-6

l. Crystal growth. I. Title. If. Series.

QD921 .M86200l 548' .5--dc2l

ISSN 0933-o33x ISBN 978-3-642-08577-2

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To my teacher Rostislav Kaischew

To Lea, for her constant encouragement throughout the writing of this book

Preface

Throughout my long involvement in research on crystal growth and nucle­ation, I have witnessed the exponential expansion of this field. It was during my first year of college in 1949 that F. C. Frank presented his, for the time, surprising ideas about the role of screw dislocations in the growth kinetics at a Discussion of the Faraday Society. They were immediately appreciated and propagated by my later teacher, R. Kaischew, who has been, along with W. Kossel and I. N. Stranski, one of the founders of the modern atomistic theories of crystal growth. Only four years later , we were able to observe in situ and film the spreading of spiral fronts during the electrocrystallization of silver.

For several more years, crystal growth remained an academic branch of science. At that time it was difficult to present papers on the topic as part of the programs of conferences on solid state physics or crystallography. It was the demand for semiconductor materials with specific qualities that boosted research and increased interest from a wide, interdisciplinary community in the problems of nucleation, bulk growth of large crystals and, more recently, in the growth of objects with nanometric dimensions.

The almost explosive development of this field resulted, however, in the establishment of schools with quite different backgrounds and theoretical ap­proaches, unlike in classical disciplines, the slow development of which pro­vided the basis for a large consensus on the major issues (a typical example is the nucleation theory, which is periodically "revised" while its sister theories treating chemical kinetics have been much less subject to controversies).

The title of this book already reveals my intention, not to present a com­pilation of all existing theoretical approaches to the considered phenomena but to focus on their microscopic, atomistic interpretation. This intention has both an ambition and a limitation. The ambition is to go as far as possible in the understanding of nucleation, crystal growth and thin film formation through the link with other phenomena, the atomistic approach to which is evident; the limitation requires leaving aside numerous topics related to the transport of heat or matter and treating crystals as continua. However, the reader will encounter some of them (Wulff theorem, BCF theory, morpholog­ical stability, etc.), as they may be considered as milestones in the evolution

VIII Preface

of the field. Also, in these cases, an effort is made to suggest their microscopic interpretation in parallel.

Many persons have contributed to the ripening of the ideas presented in this book, beginning with the great I. N. Stranski and his close associate, R. Kaischew. It is a pleasure to acknowledge fruitful discussions over many years, with R. Kern, J. M. Cases, A. Bonissent, C. Chapon, J. J. Metois and C. R. Henry. X. Duval and A. Thorny introduced me to the world of two­dimensional phases, A. Baronnet shared with me his great enthusiasm for dislocations in crystals. My interest in the atomistic interpretation of equilib­rium shapes and their modification by foreign adsorption has been stimulated by the contacts I had with B. Honigmann and R. Lacmann, while H. Reiss, J. L. Katz, F. F. Abraham and the late K. Nishioka contributed to my under­standing of nucleation with their critiques and suggestions. I am indebted to Robert F. Sekerka for the revision of parts of the manuscript, and to Terry Joseph for her improvement of the English.

The work on the manuscript was completed during my time as visiting professor at the Satellite Venture Business Laboratory of the Tokushima Uni­versity in Japan. I am most grateful to Professor Shiro Sakai for his hospitality during my tenure of this position.

Years ago, my professor, Rostislav Kaischew, told me that writing a textbook should be the swansong of a scientist. Perhaps it is because he never wrote one that this year he celebrated his 93nd birthday. It is to him that this book is dedicated. I feel that I have inherited from him the way of looking at the phenomena of nature and his way of teaching. Thus, I hope that in this book I have succeeded in bridging the gap between the ideas of the "classics" , such as Volmer, Stranski and Kaischew, whom I met in the time when few were interested in crystal growth, and the ideas of the scientists of today's generation.

B. Mutaftschiev

Contents

Part I Introduction and Background

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2. Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Some Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 The Principles of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Internal Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Other Thermodynamic Functions; Maxwell's Relations . . . . . . 10 2.5 Work and Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.6 Integrated Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2. 7 Chemical Potentials and Supersaturation . . . . . . . . . . . . . . . . . . 17

2.7.1 Supersaturated Vapor (Ideal Gas) . . . . . . . . . . . . . . . . . . 18 2.7.2 Supersaturated Solution... . . ..... . ....... . ... . .... 18 2.7.3 Undercooled Condensed Phase . . . . . . . . . . . . . . . . . . . . . 19

2.8 The "Surface Phase" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3. Statistical Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1 Partition Functions; General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Partition Function in a Continuum Phase Space . . . . . . . . . . . . 24 3.3 Examples of Simple Partition Functions . . . . . . . . . . . . . . . . . . . 26

3.3.1 Translational Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.2 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.3 Free Rotation................ . . ... ... . . . ... . . .. . . 30

3.4 Canonical Partition Functions of Pure Phases; Free Energy and Chemical Potential . . . . . . . . . . . . . . . . . . . . . . 32 3.4.1 The Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4.2 The "Lattice Gas" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4.3 The Monatomic Einstein Crystal . . . . . . . . . . . . . . . . . . . 33 3.4.4 The One-Dimensional Non-Einstein Crystal. . . . . . . . . . 34

3.5 Partition Function of Associated Vapor. . . . . . . . . . . . . . . . . . . . 38

X Contents

Part II Equilibria

4. Equilibrium Between Large Phases; The Vapor Pressure of Solids........................ . ... . 43 4.1 The Clausius-Clapeyron Equation . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 Statistical-Thermodynamic Treatment . . . . . . . . . . . . . . . . . . . . 45 4.3 Repetitive-Step; Thermodynamic Frequency . . . . . . . . . . . . . . . 45 4.4 Kinetic Treatment of the Equilibrium Crystal-Vapor . . . . . . . . 49 4.5 Equilibrium in the Different Sites on the Crystal Surface . . . . 51 4.6 Adsorbed and Incorporated Molecules;

the Different Types of Crystal Face . . . . . . . . . . . . . . . . . . . . . . . 55

5. The Surface Tension of Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1 The Broken-Bond Approach to the Surface Energy of a Solid:

the Born- Stern Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.2 Surface Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.3 Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.4 Stefan's Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6. Equilibrium Between Large Three-and Two-Dimensional Phases: Adsorption Phenomena . . . . 73 6.1 Partition Function and Chemical Potential

of an Adsorbed Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.1.1 The Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.1.2 The Mean Field Approximation . . . . . . . . . . . . . . . . . . . . 77

6.2 Desorption Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.3 Frumkin- Fowler's Adsorption Isotherm.................... 79 6.4 Multilayer Adsorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.5 Two-Dimensional Phase Transitions; Spreading Pressure. . . . . 88 6.6 Two-Dimensional Versus Three Dimensional Phases;

a Link to Wetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6. 7 Displacement and Mixing

of Condensed Two-Dimensional Phases . . . . . . . . . . . . . . . . . . . . 94

7. Thin Films, Surface Roughening, and Surface Alloys ...... 103 7.1 Chemisorbed Versus Physisorbed Layers; Thin Solid Films .. 103 7.2 Surface Roughness Considered as a Special Case

of Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.3 Adsorption on a Thermally Rough Substrate:

Surface Alloying ................................... . .... 113 7.4 Surface Melting . . ..................................... . 119

Contents XI

8. Equilibrium Between a Small and a Large Phase .......... 123 8.1 The Gibbs Potential of a Small Phase;

the Capillarity Approximation ........................... 123 8.2 The Size-Dependent Chemical Potential .................. . 128

8.2.1 Droplets in a Vapor ... . ........................... 129 8.2.2 Gas Bubbles in a Liquid ........................... 130 8.2.3 Solid Clusters in a Melt ........................... 132

8.3 Statistical Mechanical Treatment of the Free Energy of a Solid Cluster ....... . ............................... 133

8.4 Capillarity Approximation Versus Model Calculations .. .. ... 138

9. Equilibrium Shapes of Crystals ...................... .. ... 147 9.1 Curie- Wulff's Condition and Wulff's Theorem ............ . 148 9.2 Herring's 1-Plot ........................................ 151 9.3 Faceting of a K Face ................................... . 156 9.4 Equilibrium Shape on a Foreign Substrate ................ . 158 9.5 Equilibrium Shape of a Droplet on a Substrate:

Young's Equation ... .. . .......... . .... . .. ..... . ........ 161 9.6 Entropy Effects on Surface Free Energy

and Equilibrium Shape .................................. 163 9.7 Equilibrium of Small Anisotropic Phases;

the Thermodynamic Approach .......................... . 166 9.8 Equilibrium of Small Anisotropic Phases;

the Atomistic Approach ... ...... . ....................... 168 9.9 The Influence of Foreign Adsorption

on the Equilibrium Shape .. . ............................ 17 4 9. 9.1 The Different Types of Crystal Face . . . . . . . . . . . . . . . . 177 9.9.2 The Equilibrium Shape ................ . .......... 179 9.9.3 Faceting .. . ....................... .. ............. 180

Part III Nucleation

10. Homogeneous Nucleation; the Phase Approach ...... . .... 183 10.1 The Classical Nucleation Work. Droplets in a Vapor ... .. .. . 184 10.2 Bubbles in Liquids: Boiling and Cavitation ................ 189

10.2.1 Boiling under Positive External Pressure .. .. ........ 190 10.2.2 Boiling under Negative External Pressure .... . ...... 192

10.3 Anisotropic Embryos; the Thermodynamic Approach ....... 193 10.4 Anisotropic Embryos; the Atomistic Approach ............. 194 10.5 The Volmer- Weber Treatment of Nucleation Kinetics ....... 196

11. Homogeneous Nucleation; the Chemical Approach ........ 201 11.1 Equilibrium in Associated Vapor ......................... 201 11.2 Frenkel's Size Distribution .. . ........ . . .. .... . . .. ........ 207

XII Contents

11.3 Frenkel's Treatment of Steady State Nucleation Kinetics ..... 210 11.4 The Beeker- Doring Treatment of Steady State

Nucleation Kinetics .. .... ... ..... ................ ... .... 215 11.5 Nucleation Kinetics in Condensed Systems ................. 221 11.6 Cluster Isomers and Equilibrium Shape ...... ... .......... 223

12. Nucleation on a Foreign Substrate ..... ..... ...... . ....... 227 12.1 Nucleation on a Foreign Solid Substrate .......... . ........ 227 12.2 Nucleation on the Interface Between Two Fluids ............ 233 12.3 Two-Dimensional Nucleation .................... . ........ 235 12.4 Role of the Structure in Substrate Nucleation: Epitaxy . . .... 239

12.4.1 Epitaxy by Classical Three-Dimensional Nucleation ... 240 12.4.2 Epitaxy by "Non-Classical" Three-Dimensional

Nucleation ....................................... 241 12.4.3 Epitaxy by Two-Dimensional Nucleation ............ 242

12.5 Nucleation on Foreign Particles .......................... 244 12.5.1 Nucleation on Perfectly Wetted

Spherical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 12.5.2 Nucleation on Better-Than-Perfectly-Wetted

Spherical Particles ................................ 247

13. Some Specific Cases of Nucleation ........................ 249 13.1 Nucleation on Charged Particles .......................... 249 13.2 Solidification of Small Droplets . . .. .......... . ............ 251 13.3 Nucleation in a Small Volume ............ . .. ........ ..... 255 13.4 Nucleation and Ostwald's Rule ............. .... .......... 257 13.5 Nucleation in a Binary Alloy ....... ..... ............ ... .. 262

13.5.1 Three-Dimensional Binary Alloys ................... 262 13.5.2 Two-Dimensional (Surface) Alloys ........ ... .. .. .. . 265

14. Time-Dependent Nucleation Kinetics .. .. ... .............. 267 14.1 The Time Lag in Nucleation .................. ... . . . . .... 267 14.2 Non-Classical Nucleation on a Substrate .. . .......... . ..... 270

Part IV Crystal Growth

15. Elementary Processes on the Surface of a Crystal ......... 279 15.1 Sticking .... ... ............ ... ............... . ......... 281 15.2 Surface Migration .......... . ........................... 283 15.3 Mean Diffusion Length .................................. 285

16. Growth of a "Perfect" K Face ... ..... ... ..... ............ 291 16.1 Kinetics of Growth of a Planar K Face ... . ......... ... .... 292

16.1.1 The Maximum Growth Rate ...................... . 292

Contents XIII

16.1.2 Incomplete Sticking; Kinetic Coefficients ........... . 294 16.1.3 A Simple Case of Diffusion-Controlled Growth ....... 295

16.2 Diffusion Versus Capillarity: Morphological Stability ....... . 297

17. Growth of an F Face of a Perfect Crystal ............ . .... 303 17.1 The Transition from Layer-by-Layer to Continuous Growth . . 303 17.2 A One-Dimensional K Face: the Monomolecular Step ....... 308 17.3 Rate of Propagation of a Single Straight Step .............. 314

17.3.1 Advancement Controlled by Surface Diffusion ........ 314 17.3.2 Advancement Controlled by Volume Diffusion ........ 318

17.4 Rate of Advancement of a Curved Step ... ... .............. 320 17.5 Growth by Two-Dimensional Nucleation;

Mononuclear Growth ................................... 323 17.6 Growth by Two-Dimensional Nucleation;

Polynuclear Growth ... .. .... . ............ . .............. 326

18. Growth of an F Face of an Imperfect Crystal ............ . 331 18.1 Rate of Propagation of a Train of Equidistant Steps ........ 332

18.1.1 Advancement Controlled by Surface Diffusion ........ 332 18.1.2 Advancement Controlled by Volume Diffusion ........ 333

18.2 Growth Spirals ... .................................. ... . 336 18.2.1 Growth Governed by Surface Diffusion .............. 339 18.2.2 Growth Governed by Volume Diffusion . . .... .. ..... . 340

19. Conclusion ................ . ..... .... ..... ... ............ . 341

Appendices . .......................... ....... ................ .. 343 A. Legendre Transformations ....... . ........... . ........... 343 B. Method of Lagrange Multipliers ......... . ...... ..... ..... 344 C. Euler's Theorem ........................................ 344 D. Stirling's Approximation ... ............... . ............. 345 E. Maximum Term Approximation .......... ...... .......... 345 F . Integrals of the Type J0= xn exp( -ax2 ) dx ..... .. .. ... ..... 347

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

Index .......... . ........... ... ................................ 357