Department of Materials Engineering The Technion

NUCLEATION AND CRYSTAL GROWTH: 318220 Dr. R. Ghez Chapter I. Motivation: Why study the growth of crystals? In what sense is crystal growth different from any other first-order phase transformation? Chapter II. Diffusional models of crystal growth 1. Diffusion equations and Stefan conditions. 2. Quasi-steady-state models, including surface kinetics. 3. Time-dependent models. Appendix A. An “exact” model for steady-state bulk growth Chapter III. Homogeneous nucleation of a single cluster 1. Summary of bulk phase equilibria. 2. The standard thermodynamic model of homogeneous nucleation. Stability and metastability. 3. Thermodynamics of “small” phases. Surfaces cause shifts in phase equilibria: Superheating, undercooling, supersaturation, undersaturation. 4. Nucleation as a Stefan problem. 5. Why the standard model is misleading: The grand potential is favored over Gibbs’s free energy. The critical condition is a ridge in the space of thermodynamic variables. Appendix B. Second derivatives of the free energy surface Appendix C. Kelvin’s formula (case of droplets) Chapter IV. Homogeneous nucleation of an ensemble of clusters The Volmer-Weber-Farkas-Becker-D¨ oring-Zeldovich theory of homogeneous nucleation. Appendix D. Chemical reactions Appendix E. Are clusters different phases or distinct components? And what is a cloud? Chapter V. Heteronucleation, dislocations, and crystal structure 1. The structure of crystal surfaces: Kossel’s “terrace-ledge-kink” model and Frank’s dislocation model. 2. Why crystals grow: The movement of kinks and steps. 3. Heteronucleus dynamics (Volmer-Weber and Stranski-Krastanov theories) vs. dislocation step propagation (BCF theory).

Intention: This course focuses on the scientific aspects of crystal growth. This demands attention to the processes that occur at phase boundaries and to the structure of real crystal surfaces. It is emphasized that crystal growth, by its very nature, is a dynamic process, although equilibrium thermodynamics provides a convenient framework for limiting cases or at specific locations within the system. i

Prerequisites and Requirements: The study of crystal growth is interdisciplinary. The student must thus have attended at least one serious course in diffusion or transport theory, and others on thermodynamics and phase transformations at an advanced level. Therefore, this course is open to graduate students, only. Exercises are required almost every week, and there will be a final examination whose format is yet to be determined. There will also be at least one mid-term test. The final grade will be the arithmetic average of all the exercises, tests, and the final exam.

Selected References 1. Textbooks: 1. 2. 3. 4. 5. 6.

M. Volmer, Kinetik der Phasenbildung (Th. Steinkopff Verlag, 1939). A. Holden and P. Singer, Crystals and Crystal Growing (Doubleday, 1960). The Art and Science of Crystal Growth, edited by J.J. Gilman (Wiley, 1963). Crystal Growth: an introduction, edited by P. Hartman (North-Holland, 1973). Epitaxial Growth, Parts A & B, edited by J.W. Matthews (Academic Press, 1975). Crystal Growth: a tutorial approach, edited by W. Bardsley, D.T.J. Hurle, and J.B. Mullin (North-Holland, 1979). 7. A.A. Chernov, Modern Crystallography, vol. III: Crystal Growth (Springer, 1984). 8. Handbook of Crystal Growth, 6 volumes, edited by D.T.J. Hurle (Elsevier, 1993–94). 9. R. Ghez, Diffusion Phenomena: cases and studies (Springer, 2001), hereafter referred to as DP in text. Click here for an errata list.

2. Key articles: 1. W.K. Burton, N. Cabrera, and F.C. Frank, “The growth of crystals and the equilibrium structure of their surfaces,” Phil. Trans. Roy. Soc. (London) A243, 299 (1951). 2. J.A. Burton, R.C. Prim, and W.P. Slichter, “The distribution of solute in crystals grown from the melt,” J. Chem. Phys. 21, 1987 (1953). 3. D.T.J. Hurle, “Constitutional supercooling during crystal growth from melts,” Sol. State Electr. 3, 37 (1961). 4. A.A. Chernov, “The spiral growth of crystals,” Usp. Fiz. Nauk 73, 277 (1961). [Engl. transl. in Sov. Phys. Usp. 4, 116 (1961).] 5. G.H. Gilmer, R. Ghez, and N. Cabrera, “An analysis of combined surface and volume diffusion processes in crystal growth,” J. Cryst. Growth 8, 79 (1971); ibid. 21, 93 (1974). 6. G.H. Gilmer, “Computer models of crystal growth,” Science 208, 355 (1980). 7. P.M. Petroff, A. Lorke, and A. Imamoglu, “Epitaxially self-assembled quantum dots,” Phys. Today, May 2001, p. 46. 8. A very good annotated collection of articles, some from the distant past, can be found in Crystal Form and Structure (Benchmark papers in geology, Vol. 34), edited by C.J. Schneer (Dowden, Hutchinson, & Ross, 1977). 9. Another fine collection of papers (more recent, mostly) is: A Perspective on Crystal Growth, edited by D.T.J. Hurle (North-Holland, 1992).

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3. Journals: 1. 2. 3. 4. 5. 6. 7.

Journal of Crystal Growth. Crystallography Reports (formerly, Soviet Physics Crystallography). Journal of Applied Physics. Acta Materialia (formerly, Acta Metallurgica). Journal of the Electrochemical Society. Materials Research Bulletin. Journal of Electronic Materials.

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