NUCLEATION AND CRYSTAL GROWTH

Department of Materials Engineering The Technion

NUCLEATION AND CRYSTAL GROWTH: 318220 Dr. R. Ghez

Topics: I. Motivation In what sense is crystal growth different from any other first-order phase transformation? II. Diffusional models of crystal growth 1. Diffusion equations and Stefan conditions. 2. Quasi-steady-state models, including surface kinetics. 3. Time-dependent models. 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. Why the standard model is incomplete. 4. Nucleation as a Stefan problem. 5. The critical condition is a saddle point in the space of thermodynamic variables. IV. Homogeneous nucleation of an ensemble of clusters The Volmer-Weber-Farkas-Becker-D¨ oring-Zeldovich theory of homogeneous nucleation. 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.

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. i

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.

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 collection of papers (more recent, mostly) is: A Perspective on Crystal Growth, edited by D.T.J. Hurle (North-Holland, 1992). 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. ii