Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UF00090206/00001
## Material Information- Title:
- Growth kinetics of faceted solid-liquid interfaces
- Creator:
- Peteves, Stathis D., 1957- (
*Thesis advisor*) Abbaschian, Gholamreza J. (*Thesis advisor*) DeHoff, Robert T. (*Reviewer*) Reed-Hill, Robert E. (*Reviewer*) Narayanan, Ranganathan (*Reviewer*) Anderson, Timothy J. (*Reviewer*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1986
- Copyright Date:
- 1986
- Language:
- English
- Physical Description:
- xxi, 340 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Atoms ( jstor )
Crystal growth ( jstor ) Crystals ( jstor ) Free energy ( jstor ) Kinetics ( jstor ) Liquids ( jstor ) Nucleation ( jstor ) Solids ( jstor ) Solutes ( jstor ) Supercooling ( jstor ) Crystal growth ( lcsh ) Dissertations, Academic -- Materials Science and Engineering -- UF Materials Science and Engineering thesis Ph.D - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Abstract:
- A novel method based on thermoelectric principles was developed to monitor in-situ the interfacial conditions during unconstrained crystal growth of Ga crystals from the melt and to measure the solid-liquid (S/L) interface temperature directly and accurately. The technique was also shown to be capable of detecting the emergence of dislocation(s) at the crystallization front, as well as the interfacial instability and breakdown. The dislocation-free and dislocation-assisted growth kinetics of (111) and (001) interfaces of high purity Ga, and In-doped Ga, as a function of the interface supercooling (AT) were studied. The growth rates cover the range of 10-3 to 2 x 104 m/s at interface supercoolings from 0.2 to 4.60C, corresponding to bulk supercoolings of about 0.2 to 53C. The dislocation-free growth rates were found to be a function of exp(-1/AT) and proportional to the interfacial area at small super- coolings. The dislocation-assisted growth rates are proportional to AT2tanh(l/AT), or approximately to ATn at low supercoolings, with n around 1.7 and 1.9 for the two interfaces, respectively. The classical two-dimensional nucleation and spiral growth theories inadequately describe the results quantitatively. This is because of assumptions treating the interfacial atomic migration by bulk diffusion and the step edge energy as independent of supercooling. A lateral growth model removing these assumptions is given which describes the growth kinetics over the whole experimental range. Furthermore, the results show that the faceted interfaces become "kinetically rough" as the supercooling exceeds a critical limit, beyond which the step edge free energy becomes negligible. The faceted-nonfaceted transition temperature depends on the orientation and perfection of the interface. Above the roughening supercooling, dislocations do not affect the growth rate, and the rate becomes linearly dependent on the supercooling. The In-doped Ga experiments show the effects of impurities and microsegregation on the growth kinetics, whose magnitude is also dependent on whether the growth direction is parallel or anti-parallel to the gravity vector. The latter is attributed to the effects of different connective modes, thermal versus solutal, on the solute rich layer ahead of the interface.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 1986.
- Bibliography:
- Bibliography: leaves 318-339.
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- Stathis D. Peteves.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 030249169 ( AlephBibNum )
16243779 ( OCLC ) AEP2535 ( NOTIS )
## UFDC Membership |

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GROWTH KINETICS OF FACETED SOLID-LIQUID INTERFACES By STATHIS D. PETEVES A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1986 To the antecedents of phase changes: Leucippus, Democritus, Epicurus and, the other Greek Atomists, who first realized that a material persists through a succession of transformations (e.g. freezing-melting-evaporation-condensation). ACKNOWLEDGEMENTS The assumption of the last stage of my graduate education at the University of Florida has been due to people, aside from books and good working habits. It is important that I acknowledge all those individ- uals who have made my stay here both enjoyable and very rewarding in many ways. Professor Reza Abbaschian sets an example of hard work and devotion to research, which is followed by the entire metals processing group. Although occasionally, in his dealings with other people, the academic fairness is overcome by his strong and genuine concern for the research goals, I certainly believe that I could not have asked more of a thesis advisor. I learned many things through his stimulation of my thinking and developed my own ideas through his strong encouragement to do so. His constant support and guidance and his unlimited accessibility have been much appreciated. I am grateful to him for making this research possible and for passing his enthusiasm for substantive and interesting results to me. At the same time, he encouraged me to pursue any side interests in the field of crystal growth, which turned out to be a very exciting and "lovable" field. Finally, I thank him for his understand- ing and his tolerance of my character and habits during "irregular" moments of my life. Professors Robert Reed-Hill and Robert DeHoff have contributed to my education at UF in the courses I have taken from them and discussions of my class work and research. Their reviews of this manuscript and iv their insight to several parts of it was greatly appreciated. Professor Ranganathan Narayanan has been very helpful with his expertise in fluid flow; his suggestions and review of this work is very much acknowledged. I thank Professor Tim Anderson for many helpful comments and for critic- ally reviewing this manuscript. My thanks are also extended to Profes- sor Robert Gould for his acceptance when asked to review this work, for his advice, and for his continuous support. Julio Alvarez deserves special thanks. We came to the University at the same time, started this project, and helped each other in closing many of the "holes" in the crystal growth of gallium story. He intro- duced me to the world of minicomputers and turned my dislike for them into a fruitful working tool. He did the work on the thermoelectric effects across the solid-liquid interface. His collaboration with me in the laboratory is often missed. The financial support of this work, provided by the National Science Foundation (Grant DMR-82-02724), is gratefully acknowledged. I am also grateful to several colleagues and friends for their moral support. I thank Robert Schmees and Steve Abeln for making me feel like an old friend during my first two semesters here. Both hard- core metallurgists helped me extend my interest in phase transforma- tions; I shared many happy moments with them and nights of Mexican dinners and "mini skirt contests" at the Purple Porpoise. With Robert, I also shared an apartment; I thank him for putting up with me during my qualifying exams period, teaching me the equilibrium of life and making the sigma phase an unforgettable topic. Joselito Sarreal, from whom I inherited the ability to shoot pictures and make slides, taught me to stop worrying and enjoy the mid-day recess; his help, particularly in my last year, is very much acknowledged. Tong Cheg Wang helped with the heat transfer numerical calculations and did most of the program writ- ing. From Dr. Richard Olesinski I learned surface thermodynamics and to argue about international politics. Lynda Johnson saved me time during the last semester by executing several programs for the heat transfer calculations and corrected parts of the manuscript. I would also like to thank Joe Patchett, with whom I shared many afternoons of soccer, and Sally Elder, who has been a constant source of kindness, and all the other members of the metals processing group for their help. I have had the pleasure of sharing apartments with George Blumberg, Robert Schmees, Susan Rosenfeld, Diana Buntin, and Bob Spalina, and I am grateful to-them for putting up with my late night working habits, my frequent bad temper, and my persistence on watching "Wild World of Animals" and "David Letterman." I am very thankful to my friends, Dr. Yannis Vassatis, Dr. Horace Whiteworth, and others for their continuous support and encouragement throughout my graduate work. I would also like to thank several people for their scientific advice when asked to discuss questions with me; Professors F. Rhines (I was very fortunate to meet him and to have taken a course from him), A. Ubbelohde, G. Lesoult, A. Bonnissent, and Drs. N. Eustathopoulos (for his valuable discussions on interfacial energy), G. Gilmer, M. Aziz, and B. Boettinger. Sheri Taylor typed most of my papers, letters, did me many favors, and kept things running smoothly within the group. I also thank the typist of this manuscript, Mary Raimondi. My very special thanks to Stephanie Gould for being the most im- portant reason that the last two years in my life have been so happy. I am so grateful to her for her continuous support and understanding and particularly for forcing me to remain "human" these final months. I also especially thank my parents and my sister for 29 and 25 years, respectively, of love, support, encouragement, and confidence in me. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ................................................... iv LIST OF TABLES .................. ................................... xii LIST OF FIGURES ............................. ........................xiii ABSTRACT ................................ ... ....................... xxi CHAPTER I INTRODUCTION ................................ ....................... 1 CHAPTER II THEORETICAL AND EXPERIMENTAL BACKGROUND ........................... 6 The Solid/Liquid (S/L) Interface ................................... 6 Nature of the Interface ............................. ............. 6 Interfacial Features .................... ......................... 8 Thermodynamics of S/L Interfaces .............................. .. 10 Models of the S/L Interface .............. ....................... 14 Diffuse interface model ....................... ..... .......... 14 The "a" factor model: roughness of the interface .............. 22 Other models ................................................... 25 Experimental evidence regarding the nature of the S/L interface 30 Interfacial Roughening ................................. ............ 36 Equilibrium (Thermal) Roughening .................. ................. 36 Equilibrium Crystal Shape (ESC) .................................. 46 Kinetic Roughening .............................................. 48 Interfacial Growth Kinetics ........................................ 53 Lateral Growth Kinetics (LG) ................... ................. .. 53 Interfacial steps and step lateral spreading rate (u ) ......... 54 Interfacial atom migration ...................... .............. 57 Two-dimensional nucleation assisted growth (2DNG) .............. 58 Two-dimensional nucleation ........................ ........... 59 Mononuclear growth (MNG) ..................................... 62 viii Polynuclear growth (PNG) ..................................... 64 Screw dislocation-assisted growth (SDG) ........................ 68 Lateral growth kinetics at high supercoolings ................... 72 Continuous Growth (CG) ........................................... 73 Growth Kinetics of Kinetically Roughened Interfaces .............. 78 Growth Kinetics of Doped Materials ............................... 83 Transport Phenomena During Crystal Growth .......................... 87 Heat Transfer at the S/L Interface ............................... 88 Morphological Stability of the Interface ......................... 93 Absolute stability theory during rapid solification ............ 98 Effects of interfacial kinetics ................................ 99 Stability of undercooled pure melt ............................. 100 Experiments on stability ....................................... 101 Segregation .................. .................................. 102 Partition coefficients ......................................... 102 Solute redistribution during growth ............................ 10. Convection ..................................................... 106 Experimental S/L Growth Kinetics ................................... 112 Shortcomings of Experimental Studies ............................. 112 Interfacial Supercooling Measurements ............................ 113 CHAPTER III EXPERIMENTAL APPARATUS AND PROCEDURES .............................. 117 Experimental Set-Up ..................... ........................... 117 Sample Preparation .................... ............................. 120 Interfacial Supercooling Measurements .............................. 125 Thermoelectric (Seebeck) Technique ............................... 125 Determination of the Interface Supercooling ...................... 129 Growth Rates Measurements .......................................... 134 Experimental Procedure for the Doped Ga ............................. 10 CHAPTER IV RESULTS ............................................................ 146 (111) Interface .......................... .......................... 146 Dislocation-Free Growth Kinetics ................................. 150 MNG region .............................. ....................... 155 PNG region ....................................... .............. 156 Dislocation-Assisted Growth Kinetics ............................. 159 Growth at High Supercoolings, TRG Region ......................... 161 (001) Interface .................................................... 164 Dislocation-Free Growth Kinetics ................................. 166 MNG region ..................................................... 166 PNG region ....................................... .......... 172 Dislocation-Assisted Growth Kinetics ............................. 173 Growth at High Supercoolings, TRG Region ......................... 174 In-Doped (111) Ga Interface ........................................ 175 Ga-.01 wt% In .................................................... 175 Ga-.12 wt% In .................................................... 187 CHAPTER V DISCUSSION ......................................................... 194 Pure Ga Growth Kinetics ............................................ 194 Interfacial Kinetics Versus Bulk Kinetics ........................ 194 Evaluation of the Experimental Method ............................ 197 Comparison with the Theoretical Growth Models at Low Supercoolings 203 2DNG kinetics .................................................. 204 SDG kinetics ................................................... 209 Generalized Lateral Growth Model .................................. 213 Interfacial Diffusivity .......................................... 218 Step Edge Free Energy ............................................ 220 Kinetic Roughening ....................... ......................... 230 Disagreement Between Existing Models for High Supercoolings Growth Kinetics and the Present Results ............................ 235 Results of Previous Investigations ................................. 242 In-Doped Ga Growth Kinetics ........................................ 246 Solute Effects on 2DNG Kinetics ................................. 246 Segregation/Convection Effects .................................. 249 CHAPTER VI CONCLUSIONS AND SUMMARY ............................................ 258 APPENDICES I GALLIUM ........................................................ 263 II Ga-In SYSTEM ................................................... 278 III HEAT TRANSFER AT THE S/L INTERFACE ............................ 280 IV INTERFACIAL STABILITY ANALYSIS ................................. 299 V PRINTOUTS OF COMPUTER PROGRAMS ................................ 305 VI SUPERSATURATION AND SUPERCOOLING ............................... 316 REFERENCES ......................................................... 318 BIOGRAPHICAL SKETCH ................................................ 340 LIST OF TABLES Page TABLE 1 Mass Spectrographic Analysis of Ga (99.9999%) ........... 122 TABLE 2 Mass Spectrographic Analysis of Ga (99.99999%) .......... 123 TABLE 3 Seebeck Coefficient and Offset Thermal EMF of the (111) and (001) S/L Ga Interface .............................. 131 TABLE 4 Typical Growth Rate Measurements for the (111) Interface. 137 TABLE 5 Analysis of In-Doped Ga Samples ......................... 141 TABLE 6 Seebeck Coefficients of S/L In-Doped (111) Ga Interfaces 142 TABLE 7 Experimental Growth Rate Equations ...................... 176 TABLE 8 Experimental and Theoretical Values of 2DNG Parameters .. 205 TABLE 9 Experimental and Theoretical Values of SDG Parameters ... 210 TABLE 10 Growth Rate Parameters of General 2DNG Rate Equation .... 213 TABLE 11 Calculated Values of g .................................. 238 TABLE 12 Solutal and Thermal Density Gradients ................... 252 TABLE A-i Physical Properties of Gallium .......................... 265 TABLE A-2 Metastable and High Pressure Forms of Ga ................ 267 TABLE A-3 Crystallographic Data of Gallium (a-Ga) ................. 271 TABLE A-4 Thermal Property Values Used in Heat Transfer Calculations ............................................ 289 LIST OF FIGURES Page Figure 1 Interfacial Features. a) Crystal surface of a sharp interface; b) Schematic cross-sectional view of a diffuse interface. After Ref. (17) ................... 9 Figure 2 Variation of the free energy G at Tm across the solid/liquid interface, showing the origin of asz. After Ref. (22) ........................................ 13 Figure 3 Diffuse interface model. After Ref. (6). a) The sur- face free energy of an interface as a function of its position. A and B correspond to maxima and minima con- figuration; b) The order parameter u as a function of the relative coordinate x of the center of the inter- facial profile, i.e. the Oth lattice plane is at -x .... 16 Figure 4 Graph showing the regions of continuous (B) and lateral (A) growth mechanisms as a function of the parameters P and y, according to Temkin's model.7 ................ 21 Figure 5 Computer drawings of crystal surfaces (S/V interface, Kossel crystal, SOS model) by the MC method at the indicated values of KT/0. After Ref. (112) ............ 42 Figure 6 Kinetic Roughening. After Ref. (117). a) MC inter- face drawings after deposition of .4 of a monolayer on a (001) face with KT/4 = .25 in both cases, but differ- ent driving forces (Ap). b) Normalized growth rates of three different FCC faces as a function of Au, showing the transition in the kinetics at large supersaturations 50 Figure 7 Schematic drawings showing the interfacial processes for the lateral growth mechanisms a) Mononuclear. b) Poly- nuclear. c) Spiral growth. (Note the negative curva- ture of the clusters and/or islands is just a drawing artifact.) .............................. ............... 63 Figure 8 Free energy of an atom near the S/L interface. QL and Qs are the activation energies for movement in the liquid and the solid, respectively. Qi is the energy required to transfer an atom from the liquid to the solid across the S/L interface ........................ 74 xiii Figure 9 Interfacial growth kinetics and theoretical growth rate equations .............................................. 79 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 Transition from lateral to continuous growth according to the diffuse interface theory;25 no is the melt viscosity at Tm ........................................ 81 Heat and mass transport effects at the S/L interface. a) Temperature profile with distance from the S/L interface during growth from the melt and from solution. b) Concentration profile with distance from the interface during solution growth .................................. 90 Bulk growth kinetics of Ni in undercooled melt. After Ref. (201) ............................................. 92 Solute redistribution as a function of distance solid- ified during unidirectional solidification with no con- vection ................................................ 105 Crystal growth configurations. a) Upward growth with negative GL. b) Downward growth with positive GL. In both cases the density of the solute is higher than the density of the solvent ................................. 109 Experimental set-up .................................... 118 Gallium monocrystal, X 20 .............................. 124 Thermoelectric circuits. a) Seebeck open circuit, b) Seebeck open circuit with two S/L interfaces ........... 126 The Seebeck emf as a function of temperature for the (111) S/L interface .................................... 132 Seebeck emf of an (001) S/L Ga interface compared with the bulk temperature ................................... 133 Seebeck emf as recorded during unconstrained growth of a Ga S/L (111) interface compared with the bulk supercool- ing; the abrupt peaks (D) show the emergence of disloca- tions at the interface, as well as the interactive effects of interfacial kinetics and heat transfer ...... 135 Experimental vs. calculated values of the resistance change per unit solidified length along the [111] orientation vs. temperature ............................ 139 Seebeck emf vs. bulk temperature as affected by dis- location(s) and interfacial breakdown, recording during growth of In-doped Ga .................................. 144 xiv Figure 23 Figure 24 Figure 25 Figure 26 Figure 27 Figure 28 Figure 29 Figure 30 Figure 31 Figure 32 Figure 33 Figure 34 Figure 35 Dislocation-free and Dislocation-assisted growth rates of the (111) interface as a function of the interface supercooling; dashed curves represent the 2DNG and SDG rate equations as given in Table 7 ..................... 149 Growth rates of the (111) interface as a function of the interfacial and the bulk supercooling .............. 151 The logarithm of the (111) growth rates plotted as a function of the logarithm of the interfacial and bulk supercoolings; the line represents the SDG rate equation given in Table 7 ........................................ 152 The logarithm of the (111) growth rates versus the reciprocal of the interfacial supercooling; A is the S/L interfacial area ........................................ 153 Dislocation-free (111) low growth rates versus the inter- facial supercooling for 4 samples, two of each with the same capillary tube cross-section diameter .............. 157 The logarithm of the MNG (111) growth rates normalized for the S/L interfacial area plotted versus the recip- rocal of the interface supercooling ..................... Polynuclear (111) growth rates versus the reciprocal of the interface supercooling; solid line represents the PNG rate equation, as given in Table 7 .................. Dislocation-assisted (111) growth rates versus the inter- face supercooling; line represents the SDG rate equation, as given in Table 7 ..................................... Dislocation-free and Dislocation-assisted growth rates of the (001) interface as a function of the interface supercooling; dashed curves represent the 2DNG and SDG rate equations, as given in Table 7 ..................... The logarithm of the (001) growth rates versus the log- arithm of the interface supercooling; dashed line rep- resents the SDG rate equation, as given in Table 7 ...... Growth rates of the (001) and (111) interfaces as a function of the interfacial supercooling ................ The logarithm of the (001) growth rates versus the reciprocal of the interface supercooling ................ The logarithm of dislocation-free (001) growth rates versus the reciprocal of the interface supercooling for 10 samples; lines A and B represent the PNG rate equa- tions, as given in Table 7 .............................. 158 160 162 165 167 168 169 170 Figure 36 Figure 37 Figure 38 Figure 39 Figure 40 Figure 41 Figure 42 Figure 43 Figure 44 Figure 45 Figure 46 The logarithm of the (001) low growth rates (MNG) nor- malized for the S/L interfacial area plotted versus the reciprocal of the interface supercooling ................ Growth rates as a function of distance solidified of Ga-.01 wt% In at different bulk supercoolings; (t ) indicates interfacial breakdown ......................... Photographs of the growth front of Ga doped with .01 wt% In showing the entrapped In rich bands (lighter region) X 40 ............................................ Initial (111) growth rates of Ga-.01 wt% In as a func- tion of the interface supercooling; ('. ---) effect of distance solidified on the growth rate, and (--) growth rate of pure Ga ......................................... Effect of distance solidified on the growth rate of Ga-.01 wt% In grown in the direction parallel to the gravity vector (a,b), and comparison with that grown in the antiparallel direction (a) .......................... Initial (111) growth rates of Ga-.01 wt% In grown in the direction parallel to the gravity vector; ('--C-) effect of distance solidified on the growth rate, and (--) growth rate of pure Ga .................................. Comparison between the growth rates of Ga-.01 wt% In in the direction parallel ( 0) and antiparallel ( 0 ) to the gravity vector as a function of the interface super- cooling; line represents the growth rate of pure Ga ..... Growth Growth Growth darker behavior of Ga-.12 wt% In (111) interface; a) rates as a function of distance solidified, b) front of Ga-.12 wt% In, X 40; solid shows as regions .......................................... 188 Initial (111) growth rates of Ga-.12 wt% In as a function of the interface supercooling; (*-0-") effect of distance solidified on the growth rate, and (- ) growth rate of pure Ga ................................................. 189 Initial (111) growth rates of Ga-.01 wt% In ( 0 ) and Ga-.12 wt% In ( < ) as a function of the interface supercooling; line represents the growth rate of pure Ga ...................................................... 191 Initial (111) growth rates of Ga-.12 wt% In growth in the direction parallel to the gravity vector as a function of the interface supercooling; (*D0-") effect of distance solidified, and (--) growth rate of pure Ga ........... 192 Figure 47 Figure 48 Figure 49 Figure 50 Figure 51 Figure 52 Figure 53 Figure 54 Figure 55 Figure 56 Figure 57 Figure 58 Initial (111) growth rates of Ga-.01 wt% In ( [ O ) and Ga-.12 wt% In ( X 0 ) grown in the direction parallel ( X 0 ) and antiparallel ( 0 0 ) to the gravity vector; continuous line represents the growth rate of pure (111) Ga interface .................. The logarithm of the (111) rates versus the reciprocal of the interfacial (open symbols) and bulk supercooling (closed symbols) for two samples sizes .................. Absolute thermoelectric power of solid along the three principle Ga crystal axes and, liquid Ga as a function of temperature .......................................... 193 196 199 Comparison between optical and "resistance" growth rates; the latter were determined simultaneously by two inde- pendent ways (see programs #2, 3 in Appendix IV) ........ 202 Comparison between the (111) experimental growth rates and calculated, via the General 2DNG rate equation, as a function of the supercooling ......................... 214 Comparison of the (001) experimental growth rates and those calculated, using the General 2DNG rate equation, growth rates as a function of the supercooling; note that the PNG calcu lated rates were not formulated so as to include the two observed experimental PNG kinetics ...... 215 The step edge free energy as a function of the inter- facial supercooling. a) oe (AT) for steps on the (001) interface. b) oe (AT) for steps on the (111) interface .222 The (111) and (001) growth rates as a function of the interfacial supercooling. The dashed lines are calcu- lated in accord with the general 2DNG rate equation "cor- rected" for Di and supercooling dependent oe ............ 226 Comparison between the (111) dislocation-assisted growth rates and the SDG Model calculations shown as dashed lines ................................................... 227 Experimental (001) dislocation-assisted growth rates as compared to the SDG Model calculated rates (dashed lines) as a function of the interface supercooling ............. 229 The (111) growth rates versus the interface supercooling compared to those determined from CS on the solid/vapor interface (Ref. (117)) .................................. The (111) growth rates versus the interface supercooling compared to the combined mode of 2DNG and SDG growth rates (dashed line) at high supercoolings .............. 232 234 xvii Figure 59 Figure 60 Figure 61 Figure A-i Figure A-2 Figure A-3 Figure A-4 Figure A-5 Figure A-6 Figure A-7 Figure A-8 Figure A-9 Figure A-10 Figure A-11 Comparison between the (001) growth curves and those predicted by the diffuse interface model.6 .............. 236 Normalized (111) growth rates as a function of the nor- malized supercooling for interface supercoolings larger than 3.5C; continuous line represents the universal dendritic law growth rate equation.336 .................. 243 Density gradients as a function of growth rate .......... 253 The gallium structure (four unit cells) projected on the (010) plane; triple lines indicate the covalent (Ga2) bond .................................................... 272 The gallium structure projected on the (100) plane; double lines indicate the short covalentt) bond distance dl. Dashed lines outline the unit cell ................. 273 The gallium structure projected on the (001) plane; double lines indicate the covalent bond and dashed lines outline the unit cell ................................... 274 Ga-In phase diagram ..................................... 279 Geometry of the interfacial region of the heat transfer analysis; Lf is the heat of fusion ...................... 282 Temperature correction 6T for the (111) interface as a function of Vri for different heat-transfer conditions, Uiri; --- Analytical calculations (KL = Ks = K), -- Numerical calculations .................................. 290 Temperature correction 6T for the (001) interface as a function of Vri for different values of Uiri; --- Anal- ytical, -- Numerical calculations ...................... 291 Temperature distribution across the S/L (111) and (001) interfaces as a function of the interfacial radius; --- Analytical model calculations, -- Numerical calcula- tions ................................................... 292 Ratio of the Temperature correction at any point of the interface to that at the edge as a function of r' for different values of Uiri/Ks ............................. 294 Comparison between the (111) Experimental results ( O ) and the Model (--- Analytical, -- Numerical) calcula- tions, at low growth rates (V < .2 cm/s) ................ 295 Comparison between the (111) Experimental results (0,0) and the Model (--- Analytical, -- Numerical) calcula- tions as a function of Vri for given growth conditions .. 296 xviii Figure A-12 Figure A-13 Comparison between the (001) Experimental results ( O ) and the Model (--- Analytical, -- Numerical) calcula- tions as a function of Vri for given growth conditions .. 298 The critical wavelength Xcr at the onset of the insta- bility as a function of growth rate; hatched area indi- cates the possible combination of wavelengths and growth rates that might lead to unstable growth front for the given sample size (i.d. = .028 cm) ...................... 303 Figure A-14 The stability term R(w) as a function of the perturba- tion wavelength and growth rate ......................... 304 xix Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy GROWTH KINETICS OF FACETED SOLID-LIQUID INTERFACES By STATHIS D. PETEVES December 1986 Chairman: Dr. Gholamreza Abbaschian Major Department: Materials Science and Engineering A novel method based on thermoelectric principles was developed to monitor in-situ the interfacial conditions during unconstrained crystal growth of Ga crystals from the melt and to measure the solid-liquid (S/L) interface temperature directly and accurately. The technique was also shown to be capable of detecting the emergence of dislocation(s) at the crystallization front, as well as the interfacial instability and breakdown. The dislocation-free and dislocation-assisted growth kinetics of (111) and (001) interfaces of high purity Ga, and In-doped Ga, as a function of the interface supercooling (AT) were studied. The growth rates cover the range of 10-3 to 2 x 104 m/s at interface supercoolings from 0.2 to 4.60C, corresponding to bulk supercoolings of about 0.2 to 53C. The dislocation-free growth rates were found to be a function xx of exp(-1/AT) and proportional to the interfacial area at small super- coolings. The dislocation-assisted growth rates are proportional to AT2tanh(l/AT), or approximately to ATn at low supercoolings, with n around 1.7 and 1.9 for the two interfaces, respectively. The classical two-dimensional nucleation and spiral growth theories inadequately des- cribe the results quantitatively. This is because of assumptions treat- ing the interfacial atomic migration by bulk diffusion and the step edge energy as independent of supercooling. A lateral growth model removing these assumptions is given which describes the growth kinetics over the whole experimental range. Furthermore, the results show that the fac- eted interfaces become kineticallyy rough" as the supercooling exceeds a critical limit, beyond which the step edge free energy becomes negli- gible. The faceted-nonfaceted transition temperature depends on the orientation and perfection of the interface. Above the roughening supercooling, dislocations do not affect the growth rate, and the rate becomes linearly dependent on the supercooling. The In-doped Ga experiments show the effects of impurities and microsegregation on the growth kinetics, whose magnitude is also depend- ent on whether the growth direction is parallel or antiparallel to the gravity vector. The latter is attributed to the effects of different connective modes, thermal versus solutal, on the solute rich layer ahead of the interface. CHAPTER I INTRODUCTION Melt growth is the field of crystal growth science and technology of "controlling" the complex process which is concerned with the forma- tion of crystals via solidification. Melt growth has been the subject of absorbing interest for many years, but much of the recent scientific and technical development in the field has been stimulated by the in- creasing commercial importance of the process in the semiconductors in- dustry. The interest has been mainly in the area of the growth of crys- tals with a high degree of physical and chemical perfection. Although the technological need for crystal growth offered a host of challenging problems with great practical importance, it sidetracked an area of re- search related to the fundamentals of crystal growth. The end result is likely obvious from the common statement that "crystal growth processes remain largely more of an art rather than a science." The lack of in- depth understanding of crystal growth processes is also due, in part, to the lack of sensors to monitor the actual processes that take place at the S/L interface. Indeed, it is the "conditions" which prevail on and near the crystal/liquid interface during growth that govern the forma- tion of dislocations and chemical inhomogeneities of the product crys- tal. Therefore, a fundamental understanding of the melt growth process requires a broad knowledge of the solid-liquid (S/L) interface and its energetic and dynamics; such an understanding would, in turn, result in many practical benefits. 2 Crystal growth involves two sets of processes; one on the atomic scale and the other on the macroscopic scale. The first one deals with the attachment of atoms to the interface and the second with the trans- port of heat and mass to or from the growth front. Information regard- ing the interfacial atomistic process, both from a theoretical and tech- nical point of view, can be obtained from the interfacial growth kinet- ics. Growth kinetics, in turn, express the mathematical relationship between the growth rate (V) and the thermodynamic driving force, as re- lated to the supercooling (AT) or supersaturation (AC), the analytical form of which portrays a particular growth mechanism related to the nature of the interface. The main emphasis of this dissertation is to study the atomistic processes occurring in the S/L interfacial region where the atoms or molecules from the liquid assume the ordered structure of the crystal, and to evaluate the effects of different factors, such as the structure and nature of the interface, the driving force, and the crystal orienta- tion, physical defects, and impurities on the growth behavior and kin- etics. Another aim of the work is to obtain accurate and reliable growth kinetics that would a) allow further insight to the growth mech- anisms and their dependence on the above mentioned factors and b) pro- vide accurate data against which the existing growth models can be test- ed. In this respect, the growth behavior at increased departures from equilibrium and any possible transitions in the kinetics is of prime interest. A reliable kinetics determination, however, cannot be made without the precise determination of the interface temperature and rate. This 3 investigation plans to overcome the inherent difficulty of measuring the actual S/L interface by using a recently developed technique during a conjunct study about thermoelectric effects across the S/L interfaces.' As shown later, this technique will also provide the means of a sensi- tive and continuous way of in-situ monitoring of the local interfacial conditions. The growth rates will also be measured directly and corre- lated with the interfacial supercoolings for a wide range of supercool- ings and growth conditions, well suited to describe the earlier men- tioned effects on the growth processes. High purity gallium, and gallium doped with known amounts of In were used in this study because, a) it is facet forming material and has a low melting temperature, b) it is theoretically important because it belongs to a special class of substances which are believed to offer the most fruitful area of S/L interfacial kinetics research, and c) of prac- tical importance in the crystal growth community. Furthermore, detailed and reliable growth rate measurements at low rates are already available for Ga;2 the latter study is among the very few conclusive kinetics studies for melt growth which provides a basis of comparison and a chal- lenge to the present study for continuation of the much needed remaining work at high growth rates. The remainder of this introduction will briefly describe the fol- lowing chapters of this thesis. Chapter II is a critical overview of the theoretical and experimental aspects of crystal growth from the melt. This subject demands an unusually broad background since it is a truly interdisciplinary one in the sense that contributions come from many scientific fields. The various sections in the chapter were arranged so that they follow a hierarchal scheme based on a conceptual view of approaching this subject. The chapter starts with a broad dis- cussion of the S/L interfacial nature and its morphology and the models associated with it, together with their assumptions, predictions, and limitations. The concept of equilibrium and dynamic roughening of interfaces are presented next, which is followed by theories of growth mechanisms for both pure and doped materials. Finally, transport phe- nomena during crystal growth and the experimental approaches for deter- mination of S/L interfacial growth kinetics are presented. In Chapter III the experimental set-up and procedure are presented. The experimental technique for measuring the growth rate and interface supercooling is also discussed in detail. In Chapter IV the experimental results are presented in three sec- tions; the first two sections are for two interfaces of the pure mater- ial, while the third one covers the growth kinetics and behavior of the doped material. Also, in this chapter the growth data are analyzed and, whenever deemed necessary, a brief association with the theoretical models is made. In Chapter V the experimental results are compared with existing theoretical growth models, emphasizing the quantitative approach rather than the qualitative observations. The discrepancies between the two are pointed out and reasons for this are suggested based on the concepts discussed earlier. The classical growth kinetics model for faceted interfaces is also modified, relying mainly upon a realistic description of the S/L interface. Finally, the effects of segregation and fluid flow on the growth kinetics of the doped material are interpreted. 5 Final comments and conclusions are found in Chapter VI. The Appen- dices contain detailed calculations and background information on the Ga crystal structure, Ga-In system, morphological stability, heat transfer, computer programming, and supercooling/supersaturation relations. CHAPTER II THEORETICAL AND EXPERIMENTAL BACKGROUND The Solid/Liquid (S/L) Interface Nature of the Interface The nature and/or structure of interfaces between the crystalline and fluid phases have been the subject of many studies. When the fluid phase is a vapor, the solid-vapor (S/V) interface can easily be des- cribed by associating it with the crystal surface in vacuum,3'4 which can be studied directly on the microscopic scale by several experimental techniques.s However, this is not the case for the S/L interface, which separates two adjacent condensed phases, making any direct experimental study of its properties very difficult, if not impossible. In contrast with the S/V interface, here the two phases present (S and L) have many properties which are rather similar and the separation between them may not be abrupt. Furthermore, liquid molecules are always present next to the solid and their interactions cannot be neglected, as can be done for vapors. The S/L interface represents a far more peculiar and complex case than the S/V and L/V interfaces; therefore, ideas developed for the latter interfaces do not properly portray the actual structure of the solid/liquid interface. In the following section, the conceptual des- cription of the various types of S/L interfaces will be given, and each type of interface will be briefly related to a particular growth mechan- ism. Two criteria have been used to classify S/L interfaces. The first one, which is mainly an energetic rather than a structural criterion, 6 7 considers the interface as a region with "intermediate" properties of the adjacent phases, rather than as a surface contour which separates the solid and the liquid side on the atomic level. According to this criterion, the interface is either diffuse or sharp.6-10 A diffuse interface, to quote,6 "is one in which the change from one phase to the other is gradual, occurring over several atom planes" (p. 555). In other words, moving from solid to liquid across the interface, one should expect a region of gradual transition from solid-like to liquid- like properties. On the other hand, a sharp interfaces-10 is the one for which the transition is abrupt and takes place within one inter- planar distance. A specific feature related to the interfacial diffuse- ness, concerning the growth mode of the interface, is that in order for the interface to advance uniformly normal to itself (continuously), a critical driving force has to be applied.6 This force is large for a sharp interface, whereas it is practically zero for an "ideally diffuse" interface. The second criterion8-12 assumes a distinct separation between solid and liquid so that the location of the interface on an atomic scale can be clearly defined. In a manner analogous to that for the S/V interface, the properties of the interface are related to the nature of the crystalline substrate and/or macroscopic thermodynamicc) properties via "broken-bonds" models. Based on this criterion, the interface is either smooth (singular, 13 faceted) or rough (non-singular, non- faceted). A smooth interface is one that is flat on a molecular scale, represented by a cusp (pointed minimum) in the surface free energy as a * Sometimes these interfaces are called F- and K-faces, respectively.13 8 function of orientation plot (Wulff's plot"4 or y-plot15). In contrast, a rough interface has several adatoms (or vacancies) on the surface layers and corresponds to a more gradual minimum in the Wulff's plot. Any deviation from the equilibrium shape of the interface will result in a large increase in surface energy only for the smooth type. Thus, on smooth interfaces, many atoms (e.g. a nucleus) have to be added simul- taneously so that the total free energy is decreased, while on rough interfaces single atoms can be added. Another criterion with rather lesser significance than the previous ones is whether or not the interface is perfect or imperfect with re- spect to dislocations or twins.11 In principle this criterion is con- cerned with the presence or absence of permanent steps on the interface. Stepped interfaces, as will become evident later, grow differently than perfect ones. Interfacial Features There are several interfacial features (structural, geometric, or strictly conceptual) to which reference will be made frequently through- out this text. Essentially, these features result primarily from either thermal excitations on the crystal surface or from particular interfa- cial growth processes, as will be discussed later. These features which have been experimentally observed, mainly during vapor deposition and on S/L interfaces after decanting the liquid,16 are shown schematically in Fig. la for an atomically flat interface. (Note that the liquid is omitted in this figure for a better qualitative understanding of the structure.) These are a) atomically flat regions parallel to the top- most complete crystalline layer called terraces or steps; b) the edges Terraces, Steps Edge (ledge) Liquid Figure 1 Interfacial Features. a) Crystal surface of a sharp interface; b) Schematic cross-sectional view of a diffuse interface. After Ref.(17) 10 (or ledges) of these terraces that are characterized by a step height h; c) the kinks, or jogs, which can be either positive or negative; and d) the surface adatoms or vacancies. From energetic considerations, as understood in terms of the number of nearest neighbors, adatoms "prefer" to attach themselves first at kink sites, second at edges, and lastly on the terraces, where it is bonded to only one side. With this line of reasoning, then, atoms coming from the bulk liquid are incorporated only at kinks, and as most crystal growth theories imply,18 growth is strongly controlled by the kink-sites. Although the above mentioned features are understood in the case of an interface between a solid and a vapor where one explicitly can draw a surface contour after deciding which phase a given atom is in, for S/L interfaces there is considerable ambiguity about the location of the interface on an atomic scale. However, the interfacial features (a-c) can still be observed in a diffuse interface, as shown schematically in Fig. lb. Thus, regardless of the nature of the interface, one can refer, for example, to kinks and edges when discussing the atomistics of the growth processes. Thermodynamics of S/L Interfaces Solidification is a first order change, and, as such, there is dis- continuity in the internal energy, enthalpy, and entropy associated with the change of state.19 Furthermore, the transformation is spatially discontinuous, as it begins with nucleation and proceeds with a growth process that takes place in a small portion of the volume occupied by the system, namely, at the interface between the existing nucleus (crys- tal seed or substrate) and the liquid. The equilibrium thermodynamic 11 formulation to interfaces, first introduced by Gibbs20 forms the basis of our understanding of interfaces. The intention here is not to review this long subject, but rather to introduce the concepts previously high- lighted in a simple manner. If the temperature of the interface is exactly equal to the equilibrium temperature, Tm, the interface is at local equilibrium and neither solidification nor melting should take place. Deviations from the local equilibrium will cause the interface to migrate, provided that any increase in the free energy due to the creation of new interfacial area is overcome so that the total free energy of the system is decreased. On the other hand, the existence of the enthalpy change, AH = HL HS, means that removal of a finite amount of heat away from the interface is required for growth to take place. At equilibrium (T = Tm) the Gibbs free energies of the solid and liquid phases are equal, i.e. GL = GS. However, at temperatures less than Tm, only the solid phase is thermodynamically stable since GS < GL. The driving force for crystal growth is therefore the.free energy dif- ference, AGv, between the solid and the supercooled (or supersaturated) liquid. For small supercoolings, AGv can be written as LAT G, LT (1) where L is the heat of fusion per mole and Vm is the solid molar volume. The S/L interfacial energy is likely the most important parameter des- cribing the energetic of the interface, as it controls, among others, the nucleation, growth, and wetting of the solid by the liquid. Accord- ing to the original work of Gibbs, who considered the interface as a physical dividing surface the S/L interfacial free energy is related to 12 the "work done to create unit area of interface." Analytically Oas can be given by Osz = UsT TSs 1 + PVi = Us TSsZ (2) where UsZ is the surface energy per unit area, SsZ is the surface en- tropy per unit area, and the surface volume work, PVi, is assumed to be negligible. A further understanding of the surface energy, as an excess quantity for the total energy of the two phase system (without the interface), can be achieved by considering Fig. 2. Here the balance in free energy across the interface is accomodated by the extra energy of the interface, Ost. The step edge (ledge) free energy is concerned with the effect of a step on the crystal surface of an otherwise flat face. As discussed later, this quantity is a very important parameter related to the exist- ence of a lateral growth mechanism versus a continuous one and the roughening transition. In order to understand the concept of edge free energy, consider the step (see Fig. 1) as a two-dimensional layer that perfectly wets the substrate. In this particular case, the extra inter- facial area created (relative to that without the step) is the periph- ery; the energetic barrier for its formation accounts for the step edge energy. Based on this concept, the step edge free energy is comparable to the interfacial energy and, in some sense, the values of these two parameters are complementary. For example, it has been stated21 that for a given substance and crystal structure, the lower the surface free energy of an interface, the higher the edge free energy of steps on it and vice-versa. However, such a suggestion is contradictory to the traditionally accepted analytical relation given as6 13 HL HS 0 GS GL --TT S m L S / L Figure 2 Variation of the free energy G at T across the solid- liquid interface, showing the origin of a s. After Ref. (22). 14 oe = Os9 h (3) where oe is the edge energy per unit length of the step and h is the step height. However, this relation, as discussed later, has not been supported by experimental results. Models of the S/L Interface As may already be surmised, the most important "property" of the interface in relation to growth kinetics is whether the interface is rough or smooth, sharp or diffuse, etc. This, in turn, will largely determine the behavior of the interface in the presence of the driving force. Before discussing the S/L interface models, one should disting- uish between two interfacial growth mechanisms, i.e. the lateral (step- wise) and the continuous (normal) growth mechanisms. According to the former mechanism, the interface advances layer by layer by the spreading of steps of one (or an integral number of) interplanar distance; thus, an interfacial site advances normal to itself by the step height only when it has been covered by the step. On the other hand, for the con- tinuous growth mechanism, the interface is envisioned to advance normal to itself continuously at all atomic sites. Whether there is a clear cut criterion which relates the nature of the interface with either of the growth mechanisms and how the driving force affects the growth behavior are discussed in the following sec- tions. Diffuse interface model According to the diffuse interface growth theory,6 lateral growth will take over "when any area in the interface can reach a metastable equilibrium configuration in the presence of the driving force, it will 15 remain there until the passage of the steps" (p. 555). Afterwards, ob- viously, the interface has the same free energy as before, since it has advanced by an integral number of interplanar spacings. On the other hand, if the interface cannot reach the metastable state in the presence of the driving force, it will move spontaneously. This model, which involves an analogy to the wall boundary between neighboring domains in ferromagnets,23 assumes that the free energy of the interface is a peri- odic function of its mean position relative to the crystal planes, as shown in Fig. 3a. The maxima correspond to positions between lattice planes. The free energy, F (per unit area), of the interface is given as 00 F = a E {f(un) + Ka-2(un .n+1)2} (4) where a is the interplanar distance and the subscripts n, n + 1, repre- sent lattice planes and K is a constant; u is related to some degree of order, and f(un) is the excess free energy of an intermediate phase characterized by u, formed from the two bulk phases (S and L). The second term represents the so-called gradient energy,24 which favors a gradual change (i.e. the diffuseness) of the parameter un. Leaving aside the analytical details of the model, the solution obtained for the values of u's which minimize F are given as u(z) = tanh (z) (5) na where z is a distance normal to the interface and the quantity w >1 (U cP a) a) ai lu ai 44 Q) U (a '4-4 3 cn A B Position of interface -3 -2 -1 I I I Su(z) 1 ul ___ __ ___ I____hjluuI _L IH -en Figure 3 Diffuse interface model. After Ref. (6). a) The surface free energy of an interface as a function of its position. A and B correspond to maxima and minima configuration; b) The order parameter u as a function of the relative coordinate x of the center of the interfacial profile, i.e. the Oth lattice place is at -x. l v n = (2/a) (K/f)1/2 (6) signifies the thickness of the interface in terms of lattice planes. As expected, the larger diffuseness of the interface, the larger is the co- efficient K characterizing the gradient energy and the smaller the quan- tity fo which relates to the function f(un). The interesting feature of this model is that the surface energy is not constant, but varies peri- odically as a function of the relative coordinate x of the center of the interface where the lattice planes are at z = na -x (see Fig. 3b). Assuming the interface profile to be constant regardless of the value of x we have o(x) = o, + g(x)oo (7) where oo is the minimum value for a, and cog(x) represents the "lattice resistance to motion" and g(x) is the well known diffuseness parameter that for large values of n is given as 2 -4 4 3 2nirx t n g(x) = 2 4 n (1 cos --) exp (- ) (8) a 2 Note that g(x) decreases with the increasing diffuseness n. Its limits are 0 and 1, which represent the cases of an ideally diffuse and sharp interface, respectively. In the presence of a driving force, AGv, if the interface moves by 6x, the change in free energy is given as 6F = (AG + o d(x)) 6x (9) v o dx For the movement to occur, 6F must be negative. The critical driving force is given by -AG = dg(x) Trogmax(10) v dx max a where 2 3 2 n an max 8 exp (- ) (11) max 8 2 Thus, if the driving force is greater than the right hand side of eq. (10), which represents the difference between the maxima and minima in Fig. 3a, the interface can advance continuously. The magnitude of the critical driving force depends on g(x), which is of the order of unity and zero for the extreme cases of sharp and ideally diffuse interfaces, respectively. In between these extremes, i.e. an interface with an intermediate degree of diffuseness, lateral growth should take place at small supercoolings (low driving force) and be continuous at large AT's. Detailed critiques from opponents and proponents of this theory have been reported elsewhere.25-27 A summary is given next by pointing out some of the strong points and the limitations of this theory: 1) The concept of the diffuse interface and the gradient energy term were first introduced for the L/V interface,24 which exhibits a second order transition at the critical temperature, Tc, where the thickness of the interface becomes infinite.28 Since a critical point along the S/L line in a P-T diagram has not been discovered yet, the quantities f(un) and the gradient energy are hard to qualify for the solid-liquid interface. The diffuseness of the interface is determined by a balance between the energy associated with a gradient, e.g. in density, and the energy re- quired to form material of intermediate properties. The concept of the diffuseness was extended to S/L interfaces6 after observing29 that the grain boundary energy (in the cases of Cu, Au, and Ag) is larger than two times the OsZ value. 2) The theory does not provide any analytical 19 form or rule for prediction of the diffuseness of the interface for a given material and crystal direction. However, the model predicts6 that the resistance to motion is greatest for close-packed planes and, thus, their diffuseness will comparatively be quite small. 3) The theory, which has been reformulated for a fluid near its critical point30 (and received experimental support24,31), provides a good description of spinodal decomposition32'33 and glass formation.3 The present author believes that this theory's concept is very rea- sonable about the nature of the S/L interface. Indeed, recent studies, to be discussed next, indirectly support this theory. However, there are several difficulties in "following" the analysis with regard to the motion of the interface, which stem primarily from the fact that it a) does not explicitly consider the effect of the driving force on the dif- fuseness of the interface, and b) conceives the motion of the interface as an advancing averaged profile rather than as a cooperative process on an atomic scale, which is important for smooth interfaces. In a later development7 about the nature of the S/L interface, many aspects of the original diffuse interface theory were reintroduced via the concept of the many-level model." Here the thickness of the inter- face, i.e. its diffuseness, is considered a free parameter that can ad- just itself in order to minimize the free energy of the interface (F); the latter is evaluated by introducing the Bragg-Williams35 approxima- * As contrasted to other models where the transition from solid to liquid is assumed to take place within a fixed and usually small num- ber of layers, e.g. two-level or two-dimensional models. 20 tion,* and depends on two parameters of the model, namely B and y, given as AG v 4W S= and y 4W KT KT here W = Es (Ess + EZg)/2 is the mixing energy, EsZ is the bond energy between unlike molecules and Ess, Ezz are the bond energies between solid-like and liquid-like molecules, respectively; K is the Boltzman's constant. Numerical calculations show that the interface under equilibrium is almost sharp for y > 3 and increases its diffuseness with decreasing y. It can also be shown that the roughness of the interface defined as10i36 U U S = U (12) o where Uo is the surface energy of a flat surface and U that of the act- ual interface. The latter increases with decreasing y, with a sharp rise at y -2.5. This is expected since U is related to the average num- ber of the broken bonds (excess interfacial energy).37 When the interface is undercooled, AGv < 0, the theory shows a pro- nounced feature. The region of positive values of the parameters B and y can be divided into two subregions, as shown in Fig. 4. In region A there are two solutions, each corresponding to a minimum and a maximum of F, respectively, while in region B there are no such solutions. In * The Bragg-Williams or Molecular or Mean Field approximation35 of stat- istical mechanics assumes that some average value E can be taken as the internal energy for all possible interfacial configurations and that this value is the most probable value. Then, the free energy of the interface becomes a solvable quantity. Qualitatively speaking, this approximation assumes a random distribution of atoms in each layer; therefore, clustering of atoms is not treated. 10-1 A 10-2 A 10-3 10-4 10-5 l 0 1 2 3 Y Figure 4 Graph showing the regions of continuous (B) and lateral (A) growth mechanisms as a function of the parameters B and y, according to Temkin's model.7 22 this region, F varies monotonically so that the interface can move con- tinuously. On the other hand, in region A the interface must advance by the lateral growth mechanism. Moreover, depending on the y value, a material might undergo a transition in the growth kinetics at a measur- able supercooling. For example, if y = 2, the transition from region A to region B should take place at an undercooling of about .05 Tm (assum- ing that L/KT, 1, which is the case for the majority of metals). How- ever, to make any predictions, W has to be evaluated; this is a diffi- cult problem since an estimate of the EsZ values requires a knowledge of the interfaciall region" a-priori. It is customarily assumed that Egs = EZ which leads to a relation between W and the heat of fusion, L. But this approximation, the incorrectness of which is discussed elsewhere, leads, for example, to negative values of asZ for pure metals.38 Never- theless, if this assumption is accepted for the moment, it will be shown that Temkin's model stands somehow between those of Cahn's and Jackson's (discussed next). The "a" factor model: roughness of the interface Before discussing the "a" factor theory,8'9 the statistical mechan- ics point of view of the structure of the interface is briefly des- cribed. The interfacial structure is calculated by the use of a parti- tion function for the co-operative phenomena in a two-dimensional lat- tice. Indeed, the change of energy accompanying attachment or detach- ment of a molecule to or from a lattice site on the crystal surface can- not be independent of whether the neighboring sites are occupied or not. A large number of models39 have been developed under the assumptions i) 23 the statistical element is capable of two states only and ii) only interactions between nearest neighbors are important. The "a" factor theory, introduced by Jackson,8 is a simplified approach based on the above mentioned principles for the S/L interface. This model considers an atomically smooth interface on which a certain number of atoms are randomly added, and the associated change in free energy (AG) with this process is estimated. The problem is then to minimize AG. The major simplifications of the model are a) a two-level model interface: as such it classifies the molecules into "solid-like" and "liquid-like" ones, b) it considers only the nearest neighbors, and c) it is based on Bragg-Williams statistics. The main concluding point of the model is that the roughness of the solid-liquid interface can be discriminated according to the value of the familiar "a" factor, defined as a= L (13) KTm where E represents the ratio of the number of bonds parallel to the interface to that in the bulk; its value is always less than one and it is largest for the most close-packed planes, e.g. for the f.c.c. struc- ture (111) = .5, (100) = 1/3, and (110) = 1/6. It should be noted that the a factor is actually the same with y in Temkin's theory. For values of a < 2, the interface should be rough, while the case of a > 2 may be taken to represent a smooth interface. Alternately, for mater- ials with L/KTm < 2, even the most closely packed interface planes should be rough, while for L/KTm > 4 they should be smooth. According to this, most metallic interfaces should be rough in contrast with those of most organic materials which have large L/KTm factors. In between 24 these two extremes (2 and 4) there are several materials of considerable importance in the crystal growth community, such as Ga, Bi, Ge, Si, Sb, and others such as H20. For borderline materials (a = 2), the effect of the supercooling comes into consideration. For these cases, this model qualitatively suggests26,40 that an interface which is smooth at equil- ibrium temperature may roughen at some undercooling. Jackson's theory, because of its simplicity and its somewhat broad success, has been widely reviewed in many publications.25,26,27,'3 The concluding remarks about it are the following: a) In principle, this model is based on the interfacial "roughness" point of view.10'36 As such, it attempts to ascribe the interfacial atoms to the solid or the liquid phase, which, as mentioned elsewhere, is likely to be an unrealistic picture of the S/L interface. Thus, the model excludes a probable "interface phase" that forms between the bulk phases so that its quantitative predictions are solely based on bulk properties (e.g. L). b) The model is essentially an equilibrium one since the effect of the undercooling on the nature of the interface was hardly treated. Hence, it is concluded that a smooth interface will grow laterally, re- gardless of the degree of the supercooling. A possible transition in the nature of the interface with increasing AT is speculated only for materials with a 2. Indeed, it is for these materials that the model actually fails, as will be discussed later. c) The anisotropic behavior of the interfacial properties is lumped in the geometrical factor E, which could be expected to make sense only 25 for flat planes or simple structures, but not for some complex struc- tures. d) In spite of the limitations of this model, the success of its predictions is generally good, particularly for the extreme cases of very smooth and very rough interfaces.26'27'34 Other models The goal of most other theoretical models of the S/L interface is the determination of the structural characteristics of the interface that can then be used for the calculation of thermodynamic properties which are of experimental interest; the majority of these models follow the same approaches that have been applied for modeling bulk liquids. Therefore, these are concerned with spherical (monoatomic) molecules that interact with the (most frequently used) Lennard-Jones, 12-6, potential.42 The L-J potential, which excludes higher than pair contri- bution to the internal energy, is a good representation of rare gasses and its simple form makes it ideal for computer calculations. The model approach can be classified into three groups: a) hard-sphere, b) computer simulations (CS); molecular dynamics (MD), or Monte Carlo (MC), and c) perturbation theories. In the Bernal model (hard-sphere),43 the liquid as a dense random packing of hard spheres is set in contact with a crystal face, usually with hexagonal symmetry (i.e. FCC (111), HCP (0001)). Computer algor- ithms of the Bernal model have been developed4 based on tetrahedral packing where each new sphere is placed in the "pocket" of previously 26 deposited spheres on the crystalline substrate. Under this concept, the model44,'4 shows how the disorder gradually progresses with distance from the interface into the liquid. The beginning of disorder, on the first deposited layer, is accounted by the existence of "channels""4 (p. 6) between atom clusters, whose width does not allow for an atom to be placed in direct contact with the substrate. As the next layer is de- posited, new sites are eventually created that do not continue to follow the crystal lattice periodicity, which, when occupied, lead to disorder. However, the very existence of the formed "channels" is explained by the peculiarity of the hcp or fcc close-packed crystal face that has two interpenetrating sublattices of equal occupation probabilities.4 The density profiles calculated at the interface also show a minimum associ- ated with the existence of poor wetting; on the other hand, perfect wet- ting conditions were found when the atoms were placed in such a way that no octahedral holes were formed.46 Thermodynamic calculations from these models allow for an estimate of the interfacial surface energy (oUs), which are in qualitative agreement with experimental findings. In conclusion, these models give a picture of the structure of the interface which seems reasonable and can calculate asg. However, they neglect the thermal motion of atoms and assume an undisturbed crystal lattice up to the S/L interface, eliminating, therefore, any kind of interfacial roughness. Computer simulation of MC and MD techniques are linked to micro- scopic properties and describe the motion of the molecules. In contrast with the MD technique, which is a deterministic process, the MC tech- nique is probabilistic. Another difference is that time scale is only 27 involved in the MD method, which therefore appears to be better suited to study kinetic parameters (e.g. diffusion coefficients). From the simulations the state parameters such as T, P, kinetic energy, as well as structural interfaciall) parameters, can be obtained. Furthermore, free energy (entropy) differences can be calculated provided that a ref- erence state for the system is predetermined. The limitations of the CS techniques are4 a) a limited size sample (-1000 molecules), as compared to any real system, because of computer time considerations; the small size (and shape) of the system might eliminate phenomena which might have occurred otherwise. b) The high precision and long time required for the equilibriation of the system (for example, the S/L interface is at equilibrium only at T,, so that precise conditions have to be set- up). c) The interfacial free energy cannot be calculated by these tech- niques. MD simulations of a L-J substance have concluded47 for the fcc (100) interface that it is rather diffuse since the density profile nor- mal to the interface oscillates in the liquid side (i.e. structured liquid) over five atomic diameters. Similar conclusions were drawn from another MD48 study where it was shown that, in addition to the density profile, the potential energy profile oscillates and that physical prop- erties such as diffusivity gradually change across the interface from those of the solid to those of the liquid. Note that none of these studies found a density deficit (observed in the hard sphere models) at the interface. However, in an MC simulation49 of the (111) fcc inter- face with a starting configuration as in the Bernal model, a small defi- cit density was observed in addition to the "channeled-like" structure 28 of the first 2-3 interfacial layers. A more precise comparison of the (100) and (111) interfaces concluded50 that the two interfaces behave similarly. Interestingly enough, this study also indicates that the L S transition, from a structural point of view, as examined from mol- ecular trajectory maps parallel to the interface, is rather sharp and occurs within two atomic planes, despite the fact that density oscilla- tions were observed over 4-5 planes. However, these trajectory maps, in terms of characterizing the atoms as liquid- or solid-like, are very subjective and critically depend on the time scale of the experiment;51 an atom that appears solid on a short time could diffuse as liquid on a longer time scale. The perturbation method of the S/L interface52 has not yet been widely used to determine the interfacial free energy or the structure of the liquid next to the solid, but only to determine the density profiles at the interface. The latter results are shown to be in good agreement with those found from the MD simulations, but do not provide any add- itional information. In a study of the (100) and (111) bcc inter- faces,51'53 calculations suggest that the interfacial liquid is "struc- tured," i.e. with a density close to that of the bulk liquid and a solid-like ordering. The interfacial thickness was estimated quite large (10-15 layers) and the observed density profile oscillations were less sharp than those observed47-50 for the fcc interfaces. This was rationalized by the lower order and plane density (area/atom) for the bcc interfaces. Despite the differences in the density profiles among the (100) and (111) interfaces, the interfacial potential energies and S/L surface energies were found to be nearly equal (within 5%).51 The interfacial phenomena were also studied by a surface MD method,4',55 meant to investigate the epitaxial growth from a melt. It was observed that the liquid adjacent to the interface up to 4-5 layers had a "stratified structure" in the direction normal to the interface which "lacked intralayer crystalline order"; intralayer ordering started after the establishment of the three-dimensionally layered interface regions. In contrast with the previously mentioned MD studies, non- equilibrium conditions were also examined by starting with a supercooled melt. For the latter case, the above mentioned phenomena were more pro- nounced and occurred much faster than the equilibrium situation. These results are supported by calculations56 of the equilibrium S/L interface (fcc (001) and (100)) in a lattice-gas model using the cluster variation method. In addition, it was shown that for the nonclose-packed face (110), the S L transition was smoother and the "intermediate" layer observed for the (001) face was not found for the (110) face. However, despite these structural differences, the calculated interfacial ener- gies for these two orientations differed only by a few percent.57 Most of the methods presented here give some information on the structure and properties of the S/L interface, particularly of the liquid adjacent to the crystal. In spite of the fact that these models provide a rather phenomenological description of the interface, their information seems to be useful, considering all the other available techniques for studying S/L interfaces. In this respect, they rather suggest that the interfacial region is likely to be diffuse, particu- larly if one does not think of the solid next to the liquid as a rigid wall. Such a picture of the interface is also suggested from recent 30 experimental works that will be reviewed next. These simulations re- sults then raise questions about the validity of current theories on crystal growth58'59 and nucleation60 which, based on theories discussed earlier, such as the "a" factor theory, assume a clear cut separation between solid and liquid; this hypothesis, however, is significantly different from the cases given earlier. Experimental evidence regarding the nature of the S/L interface Apparently, the large number of models, theories, and simulations involved in predicting the nature of the S/L interface rather illus- trates the lack of an easy means of verifying their conclusions. In- deed, if there was a direct way of observing the interfacial region and studying its properties and structures, then the number of models would most likely reduce drastically. However, in contrast to free surfaces, such as the L/V interface, for which techniques (e.g. low-energy dif- fraction, Auger spectroscopy, and probes like x-rays61) allow direct analysis to be made, no such techniques are available at this time for metallic S/L interfaces. Furthermore, structural information about the interface is even more difficult to obtain, despite the progress in techniques used for other interfaces.62 Therefore, it is not surprising that most existing models claim success by interpreting experimental re- sults such that they coincide with their predictions. Some selected examples, however, will be given for such purposes that one could relate experimental observations with the models; emphasis is given on rather recent published works that provide new information about the interfa- cial region. A detailed discussion about the S/L interfacial energies will also be given. Indirect evidence about the nature of the 31 interface, as obtained from growth kinetics studies, will not be covered here; such detailed information can be found, for example, in several review papers25,26,63 and books.64,65 Interfacial energy measurements for the S/L interface are much more difficult than for the L/V and S/V interfaces.62 For this reason, the experiments often rely upon indirect measurement of this property; in- deed, direct measurements of asz are available only for a very few cases such as Bi,66 water,67 succinonitrile,68 Cd,"69 NaCI and KCl1,70 and several metallic alloys.62 However, even in these systems, excepting Cd, NaCI, and KC1, information regarding the anisotropy of asz is lack- ing.71"76 Nevertheless, most evaluations of the S/L interfacial ener- gies come from indirect methods. In this case, the determinations of as deal basically with the conditions of nucleation or the melting of a solid particle within the liquid. For the former, that is the most widely used technique, Osz is obtained from measured supercooling limits, together with a crystal-melt homogeneous nucleation theory in which asZ appears as a parameter60'77 in the expression 3 M o J = K exp (- ) (14) AT Here J is the nucleation frequency, Ky is a factor rather insensi- tive to small temperature changes, and M is a material constant. On the * Strictly speaking, only these measurements are direct; the rest, still considered direct in the sense that the S/L interface was at least ob- served, deal with measurements of grain boundary grooves or intersec- tion angles (or dihedral angles) between the liquid, crystal, and grain boundary.7174 The level of confidence of these measurements75 and whether or not the shape of the boundaries were of equilibrium or growth form76 remain questionable. 32 other hand, the latter method, i.e. depression of melting point of small particles (spherical with radius r) by AT, is based on the well known Gibbs-Thomson equation78 2o T AT = s m (15) Lr Homogeneous nucleation experiments were performed by subdividing liquid droplets and keeping them apart by thin oxide films, or by sus- pending the particles in a suitable fluid in a dilatometer and measuring the nucleation rates (J) and associated supercoolings (AT).77,79 The determined values were correlated with the latent heat of fusion with the well known known relation77,80* cal ao .45 L (units of g-at). sz g-atom However, more recent experiments have shown that much larger supercool- ings than those observed earlier are possible,81 and the ratio AT/Tm considerably exceeds the value of .2 T, 77,79 which is often taken as the limiting undercooling at which homogeneous nucleation occurs in pure metals. As a consequence, many of the experimentally determined values are in error by as much as a factor of 2. The main criticism of the OsZ values determined from nucleation experiments includes the following: a) the influence of experimental conditions (e.g. droplet size, droplet coating, cooling rates, and initial melt superheat) on the amount of maximum recorded undercooling,8lb b) whether a crystal nucleus (of atomic dimensions, a few hundred atoms)/melt interface can be adequately described with asz of an infinite interface, which is a macroscopic * A slope of .45 has also been proposed80 for the empirical relation of the ratio o s/agb (ogb is the grain boundary surface tension). s2. gb gb 33 quantity,76 c) whether the observed nucleation is truly homogenous or rather if it is taking place on the surface of the droplets,82 d) the assumption that the nucleus has a spherical shape or that asZ is isotropic," and e) the fact that the values obtained represent some average interfacial energy over all orientations. In spite of these limitations, the asz values deduced from nucleation experiments still constitute the major source of S/L interfacial energies; if used with skepticism, they provide a reference for comparison with other inter- facial parameters. Moreover, it should be mentioned that these values have been confirmed in some cases using other techniques or theoretical approaches which have not been reviewed here. However, the theoretical approaches84-87 have also been criticized because they assume complete wetting, atomically smooth interfaces, and that the liquid next the interface retains its bulk character. Experimental attempts to find a critical point between the solid and the liquid by going to extreme temperatures and pressures (high or low) have always resulted in non-zero entropy or volume changes at the limit of the experiment, suggesting that a critical point does not exist. Similar conclusions are drawn from MD studies,88 despite the wide range of T and P accessible to computer simulations. Theoretical studies,89 which disregard lattice defects, also predict that no crit- ical point exists for the S/L transition because the crystalline sym- metry cannot change continuously. In contrast to these results, a critical point was found in the vicinity of the liquidus line of a K-Cs * Note that the temperature coefficient of asZ has also been neglected in most studies. 34 alloy;90 also, a CS of a model for crystal growth from the vapor found that the phase transition proceeds from the fluid phase to a disordered solid and afterwards to the ordered solid.91 Strong molecular ordering of a thin liquid layer next to a growing S/L interface has been suggested92 as an explanation of some phenomena observed during dynamic light scattering experiments at growing S/L interfaces of salol and a nematic liquid crystal.93 In an attempt to rationalize this behavior, it was proposed that only interfaces with high "a" factors can exert an orienting force on the molecules in the interfacial liquid; however, such an idea is not supportive of the ob- servation regarding the water/ice (0001) interface (a = 1.9).94-96 The ice experiments94'95 have shown that a "structure" builds up in the liquid adjacent to the interface (1.4-6 pm thick), when a critical growth velocity (-1.5 pm/s) is exceeded, that has different properties from that of the water (for example, its density was estimated to be only .985 g/cc, as compared to 1 g/cc of the water) and ice, but closer to that of water. Interpreting these results from such models as that of the sharp and rough interface, of nucleation (critical size nuclei) ahead of the interface and of critical-point behavior, as in second- order transition* were ruled out. Similar experiments performed on salol revealed97 that the S/L interface resembles that of the ice/water system, only upon growth along the [010] direction and not along the [100] direction. The "structured" (or density fluctuating) liquid layer * It should be noted they95 determined the critical exponent of the relation between line width and intensity of the scattered light in close agreement with that predicted29'30 for the diffuse liquid-vapor interface at the critical point. 35 was estimated to be in the order of 1 pm. An explanation of why such a layer was not formed for the (100) interface was not given. Still, these results agree in most points with the ones mentioned earlier92 and are indirectly supported by the MD simulations54'56 discussed earlier. However, despite the excellence of these light scattering experiments for the information they provide, there is still some concern regarding the validity of the conclusions which strongly depend on the optics framework. 9 Aside from the computer simulations and the dynamic light-scatter- ing experiments, experimental evidence of a diffuse interface is usually claimed by observing a "break" in the growth kinetics V(AT) curve; this is associated with the transition from lateral to continuous growth kin- etics. As such, these will be discussed in the section regarding kin- etic roughening and growth kinetics at high supercoolings. Confirmation of the "a" factor model has been provided via observa- tions of the growth front (faceted vs. non-faceted morphology) for sev- eral materials.26 Although experimental observations are in accord with the model for large and small "a" materials, there are several materials which facet irrespective of their "a" values. These are Ga,2,63,99 Ge'100o', Bi,63 Si,102 and H20,103 which have L/KTm values between 2 and 4 and P4'04 and Cd69 whose L/KTm values are about 1. Other common fea- tures of these materials are a) complex crystal structures, oriented molecular structure; b) semi-metallic properties; c) some of their interfaces have been found to be non-wetted by their melts; and d) their S/L interfacial energies do not follow the empirical rule of ost .45 L. Hence, these materials belong to a special group and it would be difficult to imagine that simple statistical models could be adequate to describe their interfaces. However, these materials are of great theor- etical importance in the field of crystal growth, as well as of techni- cal importance referring to the electronic materials industry. Next, the effect of temperature and supercooling on the nature of the interface is discussed. Interfacial Roughening For many years, one of the most perplexing problems in the theory of crystal growth has been the question of whether the interface under- goes some kind of smooth to rough transition connected with thermody- namic singularities at a temperature below the melting point of the crystal. This transition is usually called the "roughening transition" and its existence should significantly influence both the kinetics dur- ing growth and the properties of the interface. The transition could also take place under non-equilibrium or growing conditions, called the "kinetic roughening transition," which differs from the above mentioned equilibrium roughening transition. These subjects, together with the topic of the equilibrium shape of crystals, are discussed next. Equilibrium (Thermal) Roughening The concept of the roughening transition, in terms of an order- disorder transition of a smooth surface as the temperature increases was first considered back in 1949-1951.10,36 The problem then was to calcu- late how rough a (S/V) interface of an initially flat crystal face (close-packed, low-index plane) might become as T increases. This was possible after realizing that the Ising model for a ferromagnet could be 37 adapted to the treatment of phase transformations (order-disorder, second-order phase transformation) by recognizing that the equilibrium structure of the interface is mathematically equivalent to the structure of a domain boundary in the Ising model for magnetism. Statistical mechanics,39 as mentioned previously, have long been associated with co-operative phenomena such as phase transition; more- over, in recent years, the important problem of singularities related with them has been a central topic of statistical mechanics. Its appli- cation to a system can be reduced to the problem of calculating the par- tition function of the system. One of the most popular tractable models for applications to phase changes is the Ising or two-dimensional lat- tice gas model.* The Ising model is a square two-dimensional array of magnetic atomic dipoles. The dipoles can only point up or down (i.e. an occupied and a vacant site, respectively); the nearest neighbor inter- action energy is zero when parallel and p/2 when antiparallel. Thus, this model restricts atoms to lattice sites and assumes only nearest neighbor interactions with the potential energy being the sum of all such pair interactions. This simple model has been rigorously solved'06 to obtain the partition function and the transition temperature Tc (Curie temperature) for the ferromagnetic phase transition paramagneticc - ferromagnetic). Hoping that this discussion provides a link between the roughening transition and statistical mechanics, the earlier discus- sion about roughening continues. * Strictly speaking, the two models are different, but because of their exact correspondence,105 they are considered similar. Burton et al.10 considered a simple cubic crystal (100) surface with (/2 nearest neighbor interaction energy per atom. Proving that this two level problem corresponds exactly to the Ising model, a phase transition is expected at Tc. This transition then is related to the roughening of the interface ("surface melting") and the temperature at which it takes place is related to the interaction energy as KT exp (- ) = 1, or-- .57 2KT ( where TR is the roughening temperature. For a triangular lattice, e.g. (111) f.c.c. face KTR/p is approximately .91. The authors also consid- ered the transition for higher (than two) level models of the interface using Bethe's approximation. It was shown that, with increasing the number of levels, the calculated TR decreases substantially, but remains practically the same for a larger number of levels. Although this study did not rigorously prove the existence of the roughening transition,i07 it gave a qualitative understanding of the phenomenon and introduced its influence on the growth kinetics and interfacial structure. The latter, because of its importance, motivated in turn a large number of theoret- ical works'08 during the last two decades. This upsurge in interest about interfacial roughening brought new insight in the nature of the transition and proved59'109'110 its existence from a theoretical point of view. In principle, these studies use mathematical transformations to relate approximate models of the interface to other systems, such as * Exact treatments of phase transitions can be discussed only for special systems and two dimensions, as discussed previously. For more than two dimensions, approximate theories have to be considered. Among them are the mean field, Bethe, and low-high temperature expan- sions methods. 39 two-dimensional Coulomb gas, ferroelectrics, and the superfluid state, which are known to have a confirmed transition. As mentioned prev- iously, it is out of the scope of this review to elucidate these studies, detailed discussion about which can be found in several reviews.107,111,112 At the present time, the debate about the roughening transition seems to be its universality class or whether or not the critical behav- ior at the transition depends on the chosen microscopic model. Based on experiments, the physical quantities associated with the phase transi- tion vary in manner IT-TcIP when the critical temperature Tc is ap- proached. The quantities such as p in the above relation that charac- terize the phase transition are called critical exponents. They are inherent to the physical quantities considered and are supposed to take universal values (universality class) irrespective of the materials under consideration. For example, in ferromagnetism, one finds as T Tc (Curie temperature): susceptibility, x a (T Tc)-Y (T > Tc) specific heat, C(T) = (T Tc)-a Another important quantity in the critical region is the correla- tion length, which is the average size of the ordered region at temper- atures close to Tc. In magnetism, the ordered region (i.e. parallel spin region) becomes large at Tc, while in particle systems the size of the clusters of the particles become large at Tc. The correlation length also obeys the relation'05 IT TcI- (T > Tc) T (16)(T < T |Tc TI-V (T < Tc) 40 or, according to a different model, E diverges in the vicinity of TR as113 as1 T T R 1/2 = exp (C/( TR) (T < TR) TR (17) C = m (T > TR) where C is a constant (about 1.5i13 or 2.1114). The above mentioned illustrates that the universality class can be different depending on the model in use. To be more specific, the difference in behavior can be realized by comparing the relations (16) vs. (17); the former, which belongs to the two-dimensional Ising model, indicates that E diverges by a power law, while the latter of the Kosterlitz-Thouless113 theory shows that diverges exponentially. One, however, may wonder what the importance of the correlation length is and how it relates, so to speak, to "simpler" concepts of the interface. In this view, E relates to the interfacial width;59 hence, for temperatures less than the roughening transition, the interfacial width is finite in contrast with the other extreme, i.e. for T's > TR; E also corresponds to the thickness of a step so that the step free energy can then be calculated from E. Indeed, it has been shown that oe is re- lated to the inverse of &.110,115 Thus, these results predict that the step edge free energy approaching TR diverges as T -T o e exp (-C/( ) 1/2) (18) e TR and is zero at temperatures higher than TR.116 Hence, the energetic barrier to form a step on the interface does not exist for T's higher than TR. 41 In summary, the key points of the roughening transition of an interface between a crystal and its fluid phase (liquid or vapor) are the following: a) At T = TR a transition from a smooth to a rough interface takes place for low Miller index orientations. At T < TR the interface is smooth and, therefore, is microscopically flat. The edge free energy of a step on this interface is of a finite value. Growth of such an interface is energetically possible only by the stepwise mode. On the other hand, for T > TR, the interface is rough, so it extends arbitrarily from any reference plane. The step edge energy is zero, so that a large number of steps (i.e. arbitrarily large clusters) is al- ready present on a rough interface. It can thus grow by the continuous mechanism. Pictorial evidence about the roughening transition effects can be considered from the results of an MC simulation117 of the SOS model* (S/V interface), shown in Fig. 5. Also, a transition with in- creasing T from lateral kinetics to continuous kinetics above TR was found for the interfaces both on a SC11 and on an fcc crystal'17 for the SOS model, b) It is claimed that most theoretical points of the transition have been clarified. Based on recent studies, the tempera- ture of the roughening transition is predicted to be higher than that of the BCF model. Furthermore, its universality class is shown to be that of the Kosterlitz-Thouless transition. Accordingly, the step edge free * If, for the ordinary lattice gas model in a SC crystal, it is required that every occupied site be directly above another occupied site, one ends up with the solid-on-solid (SOS) model. This model can also be described as an array of interacting solid columns of varying heights, hr = 0, 1, ..., -; the integer hr represents the number of atoms in each column perpendicular to the interface, which is the height of the column. Neighboring sites interact via a potential V = Klhr-hr'j. If the interaction between nearest neighbor columns is quadratic, one ob- tains the "discrete Gaussian" model. Figure 5 Computer drawings of crystal surfaces (S/V interface, Kossel crystal, SOS model) by the MC method at the indicated values of KT/d. After Ref. (112). 43 energy goes to zero as T TR, vanishing in an exponential manner. These points have been supported and/or confirmed by several MC simula- tions results,19 in particular, for the SOS model. As may already be surmised, the roughening transition is also ex- pected to take place for a S/L interface. Indeed, its concept has been applied, for example, in the "a" factor model;8'9 the "a" factor is in- versely related to the roughening transition temperature TR, assuming that the nearest neighbor interactions (p) are related to the heat of fusion. Such an assumption is true for the S/V interface where only solid-solid interactions are considered (Ess = p, Esv = Evv 0). Then, for the Kossel crystal,120" Lv = 3( where Lv is the heat of evaporation. Unfortunately, however, for the S/L interface all kinds of bonds (Ess, Es9, EZ) are significant enough to be neglected so that one could not assume a model that accounts only vertical or lateral (with respect to the interface plane) bonds. Assumptions such as EZZ = EsZ cannot be justified, either. Several ways have been proposed"21 to calculate Esz. Their accuracy, however, is limited since both Es, and EZZ, to a lesser extent, depend on the actual properties of the interfacial region which, in reality, also varies locally. Nevertheless, such information is likely to be available only from molecular dynamics simulations at the present. Quantitative experimental studies of the roughening transition are rare, and only a few crystals are known to exhibit roughening. Because of the reversible character of the transition, it is necessary to study * As Kossel crystal120 is considered a stacking of molecules in a primi- tive cubic lattice, for which only nearest neighbor interactions are taken into account. 44 a crystal face under growth and equilibrium conditions above and below TR. That means the "a" factor, which is said to be inversely propor- tional to TR, has to change continuously (with respect to the equilib- rium temperature) or that L/KTm has to be varied. For a S/V interface, depending on the vapor pressure, the equilibrium temperature can be above or below TR, so that "a" can vary. The only exception in this case is the He S/L superfluidd) interface, at T < 1.76 K. For this system, by changing the pressure, the "a" factor can be varied over a wide range, in a small experimental range (i.e. .2 K < T < 1.7 K), where equilibrium shapes, as well as growth dynamics, can be quantitatively analyzed.96 For a metallic solid in contact with its pure melt though, this seems to be impossible because only very high pressure will influ- ence the melting temperature. Thus, at Tm a given crystal face is either above or below its TR;122 crystals facet at growth conditions provided that Ti < TR, where Ti is the interface temperature. Thus, the roughening transition of a S/L interface of a metallic system cannot be expected, or experimentally verified. In spite of the fact that most of the restrictions for the S/L interface do not exist for the S/V one, most models predict TR's (for metals) higher than Tm, thus defying experimentation on such interfaces. The majority of the reported experiments are for non-metallic mate- rials such as ice,123 naphthalene,124 C2C16 and NH4C1,125 diphenyl,126 adamantine,127 and silver sulphide;128 in these cases the transition was only detected through a qualitative change in the morphology of the crystal face (i.e. observing the "rounding" of a facet). The likely conclusions from these experiments are that the transition is gradual 45 and that the most close-packed planes roughen the last (i.e. at higher T). Also, it can be concluded that the phenomena are not of universal character (e.g. for diphenyl and ice the most dense plane did not roughen even for T = T,, while for adamantine the most close-packed plane roughened below the bulk melting point) and that the theoretically predicted TR's for S/V interfaces are too high (e.g. for C2C16 the theoretical value of KTR/Lv is 1/16 compared with the theoretical value of 1/8). It was also found that impurities reduce TR.127 The roughening transition for the hcp He crystals has been experi- mentally found for at least three crystal orientations ((0001), (1100), (1101)129,130). Moreover, a recent study130 of the (0001) and (1100) interfaces, is believed to be the first quantitative evidence that couples the transition with both the growth kinetics and the equilibrium shape of the interface. Below TR the growth kinetics were of the lat- eral type; that allowed for a determination of the relationship ce(T). At TR it was shown that oe vanished as T T exp (-C/( R )1/2) TR in accord with the earlier mentioned theories. At T > TR the interfaces advanced by the continuous mechanism. As far as S/L interfaces of pure metallic substances are concerned, the roughening transition is likely non-existent experimentally. A faceted to non-faceted transition, however, has been observed for a metallic solid-solution (other liquid metals or alloys) interface in the Zn-In and Zn-Bi-In systems.'31',32 The transition, which was studied isothermally, took place in the composition range where important 46 changes in Osz occurred. Evidence about roughening also exists for several solvent-solute combinations during solution growth.133 Additional information about the roughening transition concept comes from experimental studies on the equilibrium shape of microscopic crystals. This topic is briefly reviewed in the next section. Equilibrium Crystal Shape (ESC) The dynamic behavior of the roughening transition can also be understood from the picture given from the theory of the evolution of the equilibrium crystal shape (ECS). In principle, the ECS is a geomet- rical expression of interfacial thermodynamics. The dependence of the interfacial free energy (per unit area) on the interfacial orientation n determines r(T,n), where r is the distance from the center of the crys- tal in the direction of n of a crystal in two-phase coexistence.14'15 At T = 0, the crystal is completely faceted.134" As T increases, facets get smaller and each facet disappears at its roughening temperature TR(n). Finally, at high T, the ECS becomes completely rounded, unless, of course, the crystal first melts. As discussed earlier, facets on the ECS are represented with cusps in the Wulff plot, which, in turn, are related to nonzero free energy per unit length necessary to create a step on the facet; 13 the step free energy also vanishes at TR(n), where the corresponding facets disappear. Below TR, facets and curved areas on the crystal meet at edges with or without slope discontinuity (i.e. smooth or sharp); the former corresponds to first-order phase transition and the latter to second-order transitions. The edges are the * It is generally believed that macroscopic crystals at T = 0 are facet- ed; however, this claim that comes only from quantum crystals still remains controversial.134 47 singularities of the free energy r(T,n)136 that determines the ECS phase diagram.137 The shape of the smooth edge varies y = A(x xc)8 + higher-order terms where xc is the edge position; x, y are the edge's curvature coordin- ates. The critical exponent 8 is predicted to be as 8 = 2136 or 9 = 3/2.137,138 The 3/2 exponent is characteristic of a universality class'39,140 and it is therefore independent of temperature and facet orientation as long as T < TR. Indeed, the 3/2 value has been reported from experimental studies on small equilibrium crystals (Xe on Cu sub- strate141 and Pb on graphite134). For the equilibrium crystal of Pb grown on a graphite substrate, direct measurements of the exponent 6 via SEM yielded a value of 8 = 1.60, in the range of temperatures from 200- 3000C, in close agreement with the Pokrovsky-Talapov transition139 and smaller than the prediction of the mean-field theory.137 Sharp edges have also been seen in some experiments, as in the case of Au,142,143 but they have received less theoretical attention. At the roughening transition, the crystal curvature is predicted to jump from a finite universal value for T = TR+ to zero for T = TR-,130,138144 as contrasted to the prediction of continuously vanish- ing curvature.136 Similarly, the facet size should decrease with T and vanish as T TR-, like exp (-C/V(TR T)),113 as opposed to the behav- ior as (TR T)1/2.136 The jump in the crystal curvature has been ex- actly related59 to the superfluid jump of the Kosterlitz-Thouless trans- ition in the two-dimensional Coulomb gas.113,130'134'141 In addition, the facet size of Ag2S crystals128 was found (qualitatively) to de- crease, approaching TR, in an exponential manner. 48 Although the recent theoretical predictions seem to be consistent with the experimental results, the difficulty of achieving an ECS on a practical time scale imposes severe limitations on the materials and temperatures that can be investigated. The only ideal system to study these phenomena is the 4He (see an earlier discussion), for which sev- eral transitions have already been discovered in the hcp phase. Whether the superfluid 4He liquid resembles a common metallic liquid and how the quantum processes affect the interface still remain unanswered. Kinetic Roughening In the last decade or so, MC simulations of SOS kinetic model" of (001) S/V interface of a Kossel crystal have revealed117,145',46 a very interesting new concept, the "kinetic roughening" of the interface; in distinction with the equilibrium roughening caused by thermal fluctua- tions, the kinetic roughening is due to the effect of the driving force on the interface during growth. The simulations show that when a crys- tal face is growing at a temperature below TR (T < TR) under a driving force AG less than a critical value AGc, it is smooth on an atomic scale and it advances according to a lateral growth mechanism. However, if the crystal face is growing at T < TR, but at a driving force such that AG > AGc, it will be rough on an atomic scale and a continuous growth * This is an extension of the SOS model for (S/V) growth kinetics studies. Atoms are assumed to arrive at the interface with an extern- ally imposed rate K+. The evaporation rate K-, on the other hand, is a function of the number of nearest neighbors, i.e. fn,m' which is the fraction of surface atoms in the n/th layer with m lateral neighbors. The net growth rate is then the difference between condensation and evaporation rates in all layers. Unless some specific assumptions are made concerning K-, and/or about fnm, the system cannot be solved. Indeed, all the existing kinetic SOS models essentially differ only in the above mentioned assumptions. (See, for example, references 117 and 119.) 49 mechanism will be operative. The transition in the interface morphology and growth kinetics as a function of the driving force is known as kin- etic roughening. Computer drawings of the above mentioned simulations, shown in Figs. 6a and 6b, show the kinetic roughening phenomenon. It can be seen that at a low driving force the growth kinetics are non-linear, as contrasted with the high driving force region where the kinetics are linear. These correspond respectively to lateral and continuous growth kinetics, as discussed in detail later. It is believed that the high driving force results in a relatively high condensation rate with re- spect to the evaporation rate. In addition, the probability of an atom arriving on an adjacent site of an adatom and thus stabilizing it, is overwhelming that of the adatom evaporation. These result in smaller and more numerous clusters, as contrasted to the low driving force case where the clusters are large and few in number. As far as the author knows, an experimental verification of kinetic roughening for a S/L interface in a quantitative way is non-existent. There are a few studies which identify the transition with morphological changes occurring at the interface with increasing supercooling.133 Such conclusions are of limited qualitative character and under certain circumstances could also be erroneous, because 1) there may be a clear- cut distinction between equilibrium and growth forms of the interface,12 2) even when the growth is stopped, the relaxation time for equilibrium may be quite long130 for macroscopic dimensions, and 3) a "round" part of a macroscopically faceted interface does not necessarily have to be rough on an atomic scale. Such microscopic detailed information can be gained only from the standpoint of interfacial kinetics, which also a) A--D (100) 0 1*1 1 0 3 4 5 6 7 A/kT 20 Figure 6 Kinetic Roughening. After Ref. (117). a) MC interface drawings after deposition of .4 of a monolayer on a (001) face with KT/4 = .25 in both cases, but different driving forces (A. b) Normalized growth rates of three different FCC faces as a function of Al, showing the transition in the kinetics at large supersaturations. B f o (100) Figure 6 Kinetic Roughening. After Ref. (117). a) MC interface drawings after deposition of .4 of a monolayer on a (001) face with KT/p = .25 in both cases, but different driving forces (AH). b) Normalized growth rates of three different FCC faces as a function of Aj, showing the transition in the kinetics at large supersaturations. 51 allow for a reliable determination of critical parameters linked to the transition. There are a few growth kinetics studies which provide a clue regarding the transition from lateral to continuous growth; these will be reviewed next rather extensively due to the importance of the kinetic roughening in this study. A faceted (spiky) to non-faceted (smooth spherulitic) transition was observed for three high melting entropy (L/KTm 6-7) organic sub- stances, salol, thymol, and O-terphenyl. 47 The transition that took place at bulk supercoolings ranging from 30-50C for these materials was shown to be of reversible character; it also occurred at temperatures below the temperature of maximum growth rate. An attempt to rational- ize the behavior of all three materials in accord with the predictions of the MC simulation results"17 was not successful. The difference in the transition temperatures (20, 13, and -10C for the 0-terphenyl, salol, and thymol, respectively) were attributed to the dissimilar crys- tal structures and bonding. Morphological changes corresponding to changes from faceted to non- faceted growth form together with growth kinetics have been reported148 for the transformation I-III in cyclohexanol with increasing supercool- ing. The morphological transition was associated with the change in growth kinetics, as indicated by a non-linear to linear transition of the logarithm of the growth rates, normalized by the reverse reaction term [1 exp(- AG,/KT)], as a function of 1/T (i.e. log(V/1- exp(- AGv/KT)) vs. 1/T plot); the linear kinetics (continuous growth) * This feature will be further explained in the continuous growth sec- tion. 52 took place at supercoolings larger than that for the morphological change and also larger than the supercooling for the maximum growth rate. The change in the kinetics was found to be in close agreement with Cahn's theory.25 It should be noted that the low supercoolings data, which presumably represented the lateral growth regime, were not quantitatively analyzed; also, the "a" factor of cyclohexanol lies in the range of 1.9-3.7, depending on the E value. It was also sug- gested149 that normalization of the growth rates by the melt viscosity at high AT's might mask the kinetics transition. The morphological transition for melt growth has also been ob- served133 for the (111) interface of biphenyl at a AT about .03C; the "a" factor of this interface was calculated to be about 2.9. For growth from the solution, the transition has been observed at minute supercool- ing for facets of tetraoxane crystals with an "a" factor in the order of 2. 150 Based on kinetic measurements, it was initially suggested that P4 undergoes a transition from faceted to non-faceted growth at supercool- ings between 1-9C.Isl However, this was not confirmed by a later study by the same authors, who reported that P4 grew with faceted dendritic form at high supercoolings."4 In conclusion, a complete picture of the kinetic roughening phe- nomenon has not been experimentally obtained for any metallic S/L interface. It seems that for growth from the melt because of the lim- ited experimental range of supercoolings at which a change in the growth morphology and kinetics can be accurately recorded, only materials with 53 an "a" factor close to the theoretical borderline of 2 are suitable for testing. Even in such cases the transition cannot be substantiated and quantified in the absence of detailed and reliable growth kinetics anal- ysis. Interfacial Growth Kinetics Lateral Growth Kinetics (LG) It is generally accepted that lateral growth prevails when the interface is smooth or relatively sharp; this in turn implies the fol- lowing necessary conditions for lateral growth: 1) the interfacial temperature Ti is less than TR and 2) the driving force for growth is less than a critical value necessary for the dynamic roughening transi- tion, and/or the diffuseness of the interface. The problem of growth on an atomically flat interface was first considered by Gibbs,20 who suggested that there could be difficulty in the formation of a new layer (i.e. to advance by an interplanar or an interatomic distance) on such an interface. When a smooth interface is subjected to a finite driving force (i.e. a supercooling AT), the liquid atoms, being in a metastable condition, would prefer to attach them- selves on the crystal face and become part of the solid. However, by doing so as single atoms, the free energy of the system is still not de- creased because of the excess surface energy term associated with the unsatisfied lateral bonds. Thus, an individual atom, being weakly bound on the surface and having more liquid than solid neighbors, is likely to "melt" back. However, if it meant to stay solid, it would create a more favorable situation for the next arriving atom, which would rather take 54 the site adjacent to the first atom rather than an isolated site. From this simplified atomistic picture, it is obvious that atoms not only prefer to "group" upon arrival, but also choose such sites on the sur- face as to lower the total free energy. These sites are the ones next to the edges of the already existing clusters of atoms. The edges of these interfacial steps (ledges) are indeed the only energetically favorable growth sites, so that steps are necessary for growth to pro- ceed (stepwise growth). The interface then advances normal to itself by a step height by the lateral spreading of these steps until a complete coverage of the surface area is achieved. Although another step might simultaneously spread on top of an incomplete layer, it is understood that the mean position of the interface advances one layer at a time (layer by layer growth). Steps on an otherwise smooth interface can be created either by a two-dimensional nucleation process or by dislocations whose Burgers vec- tors intersect the interfaces; the growth mechanisms associated with each are, respectively, the two-dimensional nucleation-assisted and screw dislocation-assisted, which are discussed next. Prior to this, however, we will review the atomistic processes occurring at the edge of steps and their energetic, since these processes are rather independent from the source of the steps. Interfacial steps and step lateral spreading rate (us) In both lateral growth mechanisms the actual growth occurs at ledges of steps, which, like the crystal surface, can be rough or smooth; a rough step, for example, can be conceived as a heavily kinked step. For S/V interfaces it has been shown107'112 that the roughness of 55 the steps is higher than that of their bonding surfaces and it decreases with increasing height; moreover, MC simulations find that steps roughen before the surface roughening temperature TR. On the other hand, for a diffuse interface, the step is assumed6 to lose its identity when the radius of the two-dimensional critical nucleus, rc, becomes larger than the width of the step defined as w = h/(g)1/2 (19) Note that the width of the step is thought to be the extent of its pro- file parallel to the crystal plane; hence, the higher the value of w, the rougher the step is and vice versa. Interestingly enough, even for relatively sharp interfaces, i.e. when g ~ .2-.3, the step is predicted to be quite rough. Based on this brief discussion, the edge of the steps is always assumed to be rough. Atoms or molecules arrive at the edge of the steps via a diffusive jump across the cluster/liquid interface. Diffusion towards the kink sites can occur either directly from the liquid or vapor (bulk diffu- sion) or via a "surface diffusion" process from an adjacent cluster, or simultaneously through both. For the case of S/L interfaces, however, it is assumed that growth of the steps is via bulk diffusion only.152 Furthermore, anisotropic effects (i.e. the edge orientation) are ex- cluded. The growth rate of a straight step is derived as152" 3DLAT AT S= K D T- (20) e hRTT E T m For detailed derivation, see further discussion in the continuous growth section. 56 where D is the liquid self-diffusion coefficient and R is the gas con- stant. Cahn et al.25 have corrected eq. (20) by introducing the phenom- enological parameter B and the g factor as -1/2 DLAT e = 5(2 + g-1/2) DLAT (21) e hRTT Here B corrects for orientation and structural factors; it principally relates the liquid self-diffusion coefficient to interfacial transport, which will be considered next. B is expected to be larger than 1 for symmetrical molecules (i.e. molecularly simple liquids for which "the molecules are either single atoms or delineate a figure with a regular polyhedral shape"''5) and less or equal to 1 for asymmetric molecules. In spite of these corrections, the concluding remark from eqs. (20) and (21) is that ue increases proportionally with the supercooling at the interface. When the step is treated as curved, then the edge velocity is de- rived as17 = Ue (1 rc/r) (22) where r is the radius of curvature. In accord with eq. (22), the edge of a step with the curvature of the critical nucleus is likely to remain immobile since u = 0. If one accounts for surface diffusion, ue is given according to the more refined treatment of BCF10 as Ue = 2axsV exp (- W/KT) (23) where a is the supersaturation, xs is the mean diffusion length, v is the atomic frequency (v 1013 sec-l), and W is the evaporation energy. For parallel steps separated by a distance yo, the edge velocity is derived as 57 Ue = 2oxsv exp (- W/KT) tanh (yo/2xs) (24) which reduces to (23) when yo becomes relatively large. Interfacial atom migration The previously given analytical expression (eq. (20)) for the edge velocity can be written more accurately as ue = c AGvexp(- AGi/KT) (25) where c is a constant and AGi is the activation energy required to transfer an atom across the cluster/L interface. This term is custom- arily assumed 54 to be equal to the activation energy for liquid self- diffusion, so that ue in turn is proportional to the melt diffusivity or viscosity (see eq. (20)). Before examining this assumption, let it be supposed that the transfer of an atom from the liquid to the edge of the step takes place in the following two processes: 1) the molecule "breaks away" from its liquid-like neighbors and reorients itself to an energetically favorable position and 2) the molecule attaches itself to the solid. Assuming that the second process is controlled by the number of available growth sites and the amount of the driving force at the interface, it is ex- pected that AGi to be related to the first process. As such, the inter- facial atomic migration depends on a) the nature of the interfacial region, or, alternatively, whether the liquid surrounding the cluster or steps retains its bulk properties; b) how "bonded" or "structured" the liquid of the interfacial region is; c) the location within the interfacial region where the atom migration is taking place; and d) the molecular structure of the liquid itself. Thus, the combination and the magnitude of these effects would determine the interfaciall 58 diffusivity," Di. Alternatively, suggesting that Di = D, one explicitly assumes that the transition from the liquid to the solid is a sharp one and that the interfacial liquid has similar properties to those of the bulk. Although this assumption might be true in certain cases,25,153 its validity has been questioned25,153'155 for the case of diffuse interface, clustered, and molecularly complex liquids. These views have been supported by recent experimental works92'95',56 and previously dis- cussed MD simulations of the S/L interface,5s0,s53,-s6 which indicate that a liquid layer, with distinct properties compared to those of the bulk liquid and solid, exists next to the interface. Within this layer then the atomic migration is described by a diffusion coefficient Di that has been found to be up to six orders of magnitude smaller92'95 than the thermal diffusivity of the bulk liquid; if this is the case, the transport kinetics at the cluster/L interface should be much slower than eq. (20) indicates. Moreover, if the interfacial atom migration is 3-6 orders of magnitude slower than in the bulk liquid, one should also have to question whether atoms reach the edge of the step as well by surface diffusion. As mentioned earlier, these factors are neglected in the determination of ue. Finally, it should be noted that AGi also enters the calculations of the two-dimensional nucleation rate via the arrival rate of atoms (Ri) at the cluster, which is discussed next. Two-dimensional nucleation-assisted growth (2DNG) As indicated earlier, steps at the smooth interface can be created by a two-dimensional nucleation (2DN) process, analogous to the three- dimensional nucleation process. The main difference between the two is that for 2DN there is always a substrate, i.e. the crystal surface, 59 where the nucleus forms. The growth mechanism by 2DN, conceived a long time ago;157 can be described in terms of the random nucleation of two- dimensional clusters of atoms that expand laterally or merge with one another to form complete layers. In certain limiting cases, the growth rate for the 2DNG mechanism is predominantly determined by the two- dimensional nucleation rate, J, whereas in other cases the rate is determined by the cluster lateral spreading velocity (step velocity), ue as well as the nucleation rate. These two groups of 2DNG theories are discussed next, succeeding a presentation of the two-dimensional nuclea- tion theory. Two-dimensional nucleation. The prevailing two-dimensional nucle- ation theory is based on fundamental ideas formulated several decades ago.158-161 These classical treatment, which dealt with nucleation from the vapor phase, and the basic assumptions were later followed in the development of a 2DN theory in condensed systems. The classical theory assumes that clusters, including critical nuc- lei, have an equilibrium distribution in the supercooled liquid or that the growth of super-critical nuclei is slow compared with the rate of formation of critical size clusters. It also assumes, as the three- dimensional nucleation theory, single atom addition and removal from the cluster, as well as the kinetic concept of the critical size nuc- leus.162" The expression -for the nucleation rate is given as 1 1 * The validity of these assumptions has been the subject of great con- troversy and continues to be so. For detailed discussion, see, for example, ref. 162. 60 where w? is the rate at which individual atoms are added to the critical cluster (equal to the product of arrival rate, Ri, and the surface area of the cluster, S), ni is the equilibrium concentration of critical nuc- lei with i" number of atoms, and Z is the Zeldovich non-equilibrium fac- tor which corrects for the depletion of the critical nuclei when nuclea- tion and growth proceed. Z has a typical value of about 10-2,163 and is given as tG i 1/2 1 Z = ( ) z= TaTKT where AGI is the free energy of formation of the critical cluster. For the growth of clusters in the liquid, it is assumed that the clusters fluctuate in size by single atom increments so that the edge of the cluster is rough. The arrival rate Ri is then defined as described pre- viously for the growth of a step. Finally, the concentration of the critical nuclei is given as * SAG. n. = n exp (- )K 1 KT where n is the atom concentration. For a disk-like nucleus of height h, the work needed to form it is given as 2 AG = e (26) h AG v where oe is the step edge free energy per unit length of the step. For small supercoolings at which the work of forming a critical two-dimen- sional nucleus far exceeds the thermal energy (KT), the nucleation rate per unit area can be approximately written, as derived by Hillig,164 in the form of N LAT 1/2 30D AG J = ( -) exp (- ) (27) V RTT 2a KT m m o where N is Avogadro's number and ao is the atomic radius. This expres- sion, that confirmed an earlier derivation,165 is the most widely accepted for growth from the melt. The main feature of eq. (27) is that J remains practically equal to zero for up to a critical value of super- cooling. However, for supercoolings larger than that, J increases very fast with AT, as expected from its exponential form. Relation (27) can be rewritten in an abbreviated form as jT 1/2 AG AG J KD( )12 exp (- --) Kn exp (- K-) (28) where Ko is a material constant and Kn is assumed to be constant within the usually involved small range of supercooling. Although theoretical estimates of Kn are generally uncertain because of several assumptions, its value is commonly indicated in the range of 10212.163 The very large values of Kn, and the fact that it is essentially insensitive to small changes of temperature, have made it quite difficult to check any refinements of the theory. Indeed, such approaches to the nucleation problem that account for irregular shape clusters166 and anisotropy effects167 lead to same qualitative conclusions as expressed by eq. (28). Also, a recent comparison of an atomistic nucleation theory from the vapor145 with the classical theory leads to the same conclusion. In contrast, the nucleation rate is very sensitive to the exponential term, therefore to the step edge free energy and the supercooling at the clus- ter/liquid (C/L) interface. The nature of the interface affects J in two ways. First, in the exponential term, AG", through its dependence upon oe and in the pre-exponential term through the energetic barrier 62 for atomic transport across the C/L interface. The assumptions of the classical theory are simple in both cases, since oe is taken as con- stant, regardless of the degree of the supercooling, and the transport of atoms from the liquid to the cluster is described via the liquid self-diffusion coefficient. These assumptions are not correct when the interface is diffuse6 and at large supercoolings.32 These aspects will be discussed in more detail in a later chapter. Mononuclear growth (MNG). As was mentioned earlier, two-dimen- sional nucleation and growth (2DNG) theories are divided into two regions according to the relative time between nucleation and layer com- pletion (cluster spreading). The first of these is when a single crit- ical nucleus spreads over the entire interface before the next nuclea- tion event takes place (see Fig. 7a). Alternatively, this is correct when the nucleation rate compared with the cluster spreading rate is such that 1/JA > I/ue or for a circular nucleus A < (ue/J)2/3 (29) where A, 1 are the area and the largest diameter of the interface, re- spectively. If inequality (29) is satisfied, each nucleus then results in a growth normal to the interface by an amount equal to the step (nucleus) height, h. Thus, the net crystal growth rate for this class- ical mononuclear (and monolayer) mechanism (MNG) is given as164,168 V = hAJ (30) In this region, the growth rate is predicted to be proportional to the interfacial area (i.e. crystal facet size). The practical limitations of this model, as well as the experimental evidence of its existence, will be given later. 63 a) A MNG AT, k A h i 2DNG b) PNG Figure 7 Schematic drawings showing the interfacial processes for c) AT >0 Figure 7 Schematic drawings showing the interfacial processes for the lateral growth mechanisms a) Mononuclear. b) Poly- nuclear. c) Spiral growth. (Note the negative curvature of the clusters and/or islands is just a drawing artifact.) Polynuclear growth (PNG). At supercoolings larger than those of the MNG region, condition (29) is not fulfilled and the growth kinetics are described by the so called polynuclear (PNG) model.* According to this model, a large number of two-dimensional clusters nucleate at ran- dom positions at the interface before the layer is completed, or on the top of already growing two-dimensional islands, resulting in a hill- and valley-like interface, as shown in Fig. 7b. Assuming that the clusters are circular and that ue is independent of the two-dimensional cluster size, anisotropy effects, and proximity of neighboring clusters, the one layer version of this model was analytically solved.169 This was poss- ible by considering that for a circular nucleus the time, T, needed for it to cover the interface is equal to the mean time between the genesis of two nuclei (i.e. the second one on top of the first), or otherwise given by SJ(u t)2 dt = 1 (31) Integration of this expression and use of the relation V = h/T yields the steady state growth rate (for the polynuclear-monolayer model) given as V = h (rJu2/3)1/3 (32) This solution has been shown by several approximate solutions164,168'170 and simulations168,171,172 to represent well the more complete picture of multilevel growth by which several layers grow concurrently through * It should be mentioned that the use of the term "polynuclear growth" in this study should not be confused with the usually referred unreal- istic model,18 which considers completion of a layer just by deposi- tion of critical two-dimensional nuclei. nucleation and spreading on top of lower incomplete layers. The more general and accurate growth rate equation in this region is given by V= ch (JUe2)1/3 (33) where the constant c falls between 1-1.4. It is interesting that eq. (32), being an approximation to the asymptotic multilevel growth rate, has been shown to be very close to the exact value of steady state con- ditions that are achieved after deposition of 3-4 layers.173 It was also suggested from these studies that for irregularly shaped nuclei the transient period is shorter than for the circular ones. Nevertheless, the growth rate is well described by eq. (33). The effect of the nucleus shape upon the growth rate has been con- sidered in a few MC simulation experiments for the V/Kossel crystal interface. 17 Square-like172 and irregular nuclei result in higher growth rates. This increase in the growth rate can be understood in terms of a larger cluster periphery, which, in turn, should (statistic- ally) have a larger number of kink sites than the highly regular cluster shapes assumed in the theory. This situation would cause a higher atom deposition to evaporation flux ratio. Furthermore, surface diffusion during vapor growth was found to cause a large increase in the growth rate.174 As indicated earlier, eqs. (32) and (33) were derived under the assumption that the nucleus radius increases linearly with time. Al- though this assumption does not really affect the physics of the model, it plays an important role in the kinetics because it determines the 1/3 exponent in the rate equations. For example, assuming that the cluster radius grows as r(t) tl/2 (i.e. the cluster area increases linearly 66 with time) as in a diffusion field, the growth rate equation is derived as175,176 V z c'h (JUe2)1/2 (34) where c' is a constant close to unity. Indeed, growth data (S/V) of a MC simulation study were represented by this model.176 Alternatively, if the growth of the cluster is assumed to be such that its radius in- creases with time as r(t) t + t1/2 (i.e. a combined case of the above mentioned submodels), it can be shown that the growth rate takes the form of V = c"h (JUe2)2/5 (35) where c" is a constant. Therefore, according to these expressions, the power in the growth rate equation varies from 1/3 to 1/2.177 A faceted interface that is dislocation free grows by any of the two previously discussed 2DN growth mechanisms. At low supercoolings the kinetics are of the MNG mode, while at higher supercoolings the interface advances in accord with PNG kinetics. The predicted growth rate equations (eqs. (30) and (32)) can be rewritten with the aid of eqs. (27), (26), and (20) as 2 AT 1/2 e (36) (MNG) V = K A ( AT) exp (- ae (36) 1 T TAT MC AT 5/6 _e_ (PNG) V= K (i-) exp (- 3T) (37) 2 T 3TAT Here, KI, K2, and M are material and physical constants whose analytical expressions will be given in detail in the Discussion chapter. The growth rates as indicated by eqs. (36) and (37) are strongly dependent upon the exponential terms, and therefore upon the step edge free energy 67 and the interfacial supercooling. Although the pre-exponential terms of the rate equations, strictly speaking, are functions of AT and T, prac- tically they are constant within the usually limited range of supercool- ings for 2DNG. The distinct features associated with 2DNG kinetics are the following: a) A finite supercooling is necessary for a measurable growth rate (-10-3 um/s); this is related to the threshold supercooling for 2DN, mentioned earlier, and it is governed by oe in the exponential term. The smaller ce is, the smaller the supercooling at which the interfacial growth is detectable. b) Only the MNG kinetics are depend- ent on the S/L interfacial area. c) Since the pre-exponential terms are relatively temperature independent, both MNG and PNG kinetics should fall into straight lines in a log(V) vs. 1/AT plot. d) From the slope of the log(V) vs. 1/AT curve (i.e. Moe2/T), the step edge free energy can be calculated,63177-181 provided that the experimental data have been measured accurately. oe can then be used to estimate the diffuse- ness parameter "g" via the proposed relation6 oe = osz h (g)1/2 (38) e) Furthermore, in the semilogarithmic plot of the growth data, the ratio of the slopes for the MNG and PNG regimes should be 3, according to the classical theory; however, as discussed earlier, this ratio can actually range from 2 to 3 depending on the details of the cluster spreading process. Detailed 2DNG kinetics studies are very rare, in particular for the MNG region, which has been found experimentally only for Ga2 and Ag.182 The major difficulties encountered with such studies are 1) the necess- ity of a perfect interface; 2) the commonly involved minute growth 68 rates; 3) the required close control of the interfacial supercooling and, therefore, its accurate determination; and 4) the problems associ- ated with analyzing the growth data analysis when the experimental range of AT's is small or it falls close to the intersection of the two MNG and PNG kinetic regimes for a given sample size. Nevertheless, there are a couple of experimental studies which rather accurately have veri- fied the 2DN assisted growth for faceted metallic interfaces.2,63,99,182 Screw dislocation-assisted growth (SDG) Most often crystal interfaces contain lattice defects such as screw dislocations and these can have a tremendous effect on the growth kinet- ics. The importance of dislocations in crystal growth was first pro- posed by Frank,183 who indicated that they could enhance the growth rate of singular faces by many orders of magnitude relative to the 2DNG rates. For the past thirty years since then, researchers have observed spirals caused by growth dislocations on a large variety of metallic and non-metallic crystals grown from the vapor and solutions,16 and on a smaller number grown from the melt.'84 When a dislocation intersects the interface, it gives rise to a step initiating at the intersection, provided that the dislocation has a Burgers vector (t) with a component normal to the interface.185 Since the step is anchored, it will rotate around the dislocation and wind up actually in a spiral (see Fig. 7c). The edges of this spiral now pro- vide a continuous source of growth sites. After a transient period, the spiral is assumed to reach a steady state, becoming isotropic, or, in terms of continuous mechanics, an archimedian spiral. This further means that the spiral becomes completely rounded since anisotropy of the 69 kinetics and of the step edge energy are not taken into account. How- ever, it has been suggested119 that on S/V interfaces sharply polygoni- zed spirals may occur at low temperatures or for high "a" factor mater- ials. Nonrounded spirals have been observed during growth of several materials,186''87 as well as on Ga monocrystals during the present study. Most theoretical aspects of the spiral growth mechanism were first investigated by BCF in their classical paper,10 which presented a revo- lutionary breakthrough in the field of crystal growth. Interestingly enough, although their theory assumes the existence of dislocations in the crystal, it does not depend critically on their concentration. The actual growth rate depends on the average distance (yo) between the arms of the spiral steps far from the dislocation core. This was evaluated to be equal to 4nrc; later, a more rigorous treatment estimated it as 19rc.188 The curvature of the step at the dislocation core, where it is pinned, is assumed to be equal to the critical two-dimensional nucleus radius rc. On the other hand, for polygonized spirals, the width of the spiral steps is estimated186 to be in the range of 5rc to 9rc. According to the continuum approximation, the spiral winds up with a constant angular velocity w. Thus, for each turn, the step advances Yo in a time yo/ue = 2nr/. Then the normal growth rate V is given aso1 V = bw/27 = byo/ue (39) where b is the step height (Burgers vector normal component). According to the BCF notation, from eq. (24) where yo = 4rrc 47Ye/KTo (here Ye is the step edge energy per molecule), one gets the BCF law V = f-v exp (- W/KT) (02/01) tanh (ol/o) (40) where 2nYeb S= x and f is a constant. 1 KTx s BCF also considered the case when more than one dislocation merges at the interface. For instance, for a group of S dislocations, each at a distance smaller than 2nrc from each other, arranged in a line of length L, eq. (40) holds with a new yo = Yo/S when L < 4Arc and yo 2L/S when L > 47rc. Nevertheless, the growth rate V can never surpass the rate for one dislocation, regardless of the number and kind of dis- locations involved. For growth from the melt, the rate equation for the screw disloca- tion growth (SDG) mechanism has been derived as152,189 DL AT2 V = (41) 41rT RTo V m sZ m Canh et al.25 have modified eq. (41) for diffuse interfaces with a multiplicity factor B/g. The physical reason for this parabolic law is that both the density of spiral steps and their velocity increases pro- portionally with AT. Models for the kinetics of nonrounded spirals also predict a parabolic relationship between V and AT.190 However, another model that accounts for the interaction between the thermal field of the dislocation helices has shown that a power less than two can be found in the kinetic law V(AT).191 The influence of the stress field in the vicinity of the disloca- tion has shown to be significant on the shape of growth and dissolution (melting) of spirals in several cases.192 It can be shown'88 that the effect of the stress field extends to a distance rs from the core of the dislocation given as 2 b c 1/2 s 2- where p is the shear modulus. Nevertheless, corrections due to the stress field are usually neglected since most of the time rs < Yo. In conclusion, dislocations have a major effect on the kinetics of growth by enhancing the growth rates of an otherwise faceted perfect interface, as it has been shown experimentally for several materi- als.2,25,26,34,63 Predictions from the classical SDG theory describe the phenomena well enough, as long as spiral growth is the dominant pro- cess.145 As far as growth from the melt is concerned, most experimental results are not in agreement with the commonly referred parabolic growth law, eq. (41); indeed, the majority of the S/L SDG kinetics found in the literature are expressed as V ATm with m < 2. In contrast with the perfect (and faceted) interface, a dislocated interface is mobile at all supercoolings. Moreover, the SDG rates are expected to be several orders of magnitudes higher than the respective 2DNG rates, regardless of the growth orientation. Like the 2DNG kin- etics, the dislocation-assisted rates can fall on two kinetic regimes according to the BCF theory. This can be understood by considering the limits of SDG rate equation, eq. (40), with respect to the supersatura- tion a. It is realized that when a < a0, i.e. low supersaturation, then one has the parabolic law V 02 and for o D o0 the linear law V o For the parabolic law case, yo is much greater than xs and the reverse is true for the linear law. In between these two extreme cases, i.e. at intermediate supersaturations, the growth rates are expected to fall in a kinetics mode faster than linear but slower than parabolic; such a mode could be, for example, a power law, V = ATn, with n such that 1 < n < 2. For growth from the melt, the BCF rate equation can be rewritten as V = N AT2 tanh (P/AT) (42) where N and P are constants. Equation (42) reduces to a parabolic or to a linear growth when the ratio P/AT is far less or greater, respective- ly, than one. Lateral growth kinetics at high supercoolings According to the classical LG theory, the step edge free energy is assumed to be constant with respect to supercooling, regardless of poss- ible kinetics roughening effects on the interfacial structure at high AT's. Based on a constant oe value, the only change in the 2DNG growth kinetics with AT is expected when the exponent AG*/3KT (see eq. (37)) is close to unity. In this range, the rate is nearly linear (-ATn, n = 5/6). An extrapolation to zero growth rates from this range intersects the AT axis to the right of the threshold supercooling for 2DN growth. For SDG kinetics, based on the parabolic law (eq. (40)), no changes in the kinetics are expected at high AT's. However, the BCF law (eq. * For detailed relations between supersaturation and supercooling see Appendix VI. (39)), as discussed later, for large supercoolings reduces to an equa- tion in the form V = A' AT B' (43) where A' and B' are constants. Note: if eq. (43) is extrapolated to V = 0, it does not go through the origin, but intersects the AT axis at a positive value. It should be mentioned that none of the above discussed transitions has ever been found experimentally for growth from a metallic melt. The parabolic to linear transition in the BCF law has been verified through several studies of solution growth.181,193 Continuous Growth (CG) The model of continuous growth, being among the earliest ideas of growth kinetics, is largely due to Wilson194 and Frenkels95 (W-F). It assumes that the interface is "ideally rough" so that all interfacial sites are equivalent and probable growth sites. The net growth rate then is supposed to be the difference between the solidifying and melt- ing rates of the atoms at the interface. Assuming also that the atom motion is a thermally activated process with activation energies as shown in Fig. 8, and from the reaction rate theory, the growth rate is given as154,196 Q. i LAT V = V exp (- -) [1 exp (- )] (44) o KT KT T m where Vo is the equilibrium atom arrival rate and Qi is the activation energy for the interfacial transport. As mentioned earlier, for practi- cal reasons, Qi is equated to the activation energy for self-diffusion in the liquid, QL, and Vo = avi where a is the jump distance interlayerr spacing/interatomic distance) and vi is the atomic vibration frequency. 74 S/L S L Qi QL Scc L _______ X - Figure 8 Free energy of an atom near the S/L interface. QL and Q are the activation energies for movement in the liquid and the solid, respectively. Qi is the energy required to transfer an atom from the liquid to the solid across the S/L interface. 75 Hence, aviexp (- Qi/KT) = D/a where D is the self-diffusion coefficient in the liquid. A similar expression can be derived based on the melt viscosity, n, by the use of the Stokes-Einstein relationship aDn = KT. Therefore, eq. (44) can be rewritten as LAT V = F(T) [1 exp (- )] (45) KT T m where F(T) in its more refined form is given as197 F(T) Da f - X2 n in which f is a factor (5 1) that accounts for the fact that not all available sites at the interface are growth sites and A is the mean dif- fusional jump distance. Note that if A =a, then F(T) = Df/a. Further- more, for small supercoolings, where LAT/KTmT < 1, eq. (45) can be re- written as (in molar quantities)25 DL V = AT = KAT (46) aRTT c m which is the common linear growth law for continuous kinetics. For most metals the kinetic coefficient Kc is of the order of several cm/seco'C, resulting in very high growth rates at small supercoolings. Because of this, CG kinetics studies for metallic metals usually cover a small range of interfacial supercoolings close to Tm; in view of this, most of the time linear and continuous kinetics are used interchangeably in the literature. However, this is true only for small supercoolings, since for large supercoolings the temperature dependence of the melt diffusiv- ity has to be taken into account. Accordingly, the growth rate as a function of AT is expected to increase at small AT's and then decrease at high AT's. On the other hand, a plot of the logarithm of LAT V/[l exp (- )T kT T m as a function of 1/T should result in a straight line, from the slope of which the activation energy for interfacial migration can be obtained. Indeed, such behavior has been verified experimentally25,26,63,198 in a variety of glass-forming materials and other high viscosity melts. An alternative to eqs. (45) and (46) was proposed by suggesting that the arrival rate at the interface for simple melts might not be thermally activated;199,200 the kinetic coefficient Kc then was assumed to depend on the speed of sound in the melt. This treatment was in good agreement with the growth data for Ni,201 but not with the data of glass-forming materials. Another approach suggested that the growth rate is given as202 a 3KT 1/2 LAT V = KT)1/2 f [1 exp (- )]T A m KTT m 1/2 where the atom arrival rate is replaced by (3KT/m) which is the thermal velocity of an atom. This equation was in good agreement with recent MD results on the crystallization of a Lennard-Jones liquid.202,203 Other approaches for continuous growth are mostly based on the kin- etic SOS model for a Kossel crystal in contact with the vapor.117,145 As mentioned elsewhere, the basic difference among these models is the assumption concerning clustering (i.e. number of nearest neighbors), which strongly effects the evaporation rate and, therefore, the net growth rate.204 In addition, these MC simulations only provide informa- tion about the relative rates in terms of the arrival rate of atoms. For vapor growth, the latter is easily calculated from gas kinetics. 77 For melt growth, however, the arrival rate strongly depends on the structure of the liquid at the interface, which is not known in detail. Therefore, these models cannot treat the S/L continuous growth kinetics properly. Some general features revealed from these models are dis- cussed next to complete this review. All MC calculations for rough interfaces indicate linear growth kinetics. The calculated growth rates are smaller than those of the W-F law, eq. (44). This is understood since the latter assumes f = 1. Interestingly enough, the simulations show that some growth anisotropy exists even for rough interfaces. For example, for growth of Si from the melt, MC simulations predicted205 that there is a slight difference in growth rates for the rough (100) and (110) interfaces. The observed anisotropy is rather weak as compared to that for smooth interfaces, but it is still predicted to be inversely proportional to the fraction of nearest neighbors of an atom at the interface (5 factor). Nevertheless, true experimental evidence regarding orientation dependent continuous growth is lacking. If there is such a dependence, the corresponding form of the linear law would then be V = Kc(n) AT (47) This is illustrated by examining the prefactor of AT in eq. (46). Note that the only orientation dependent parameter is (a), so that the growth rate has to be normalized by the interplanar spacing first to further check for any anisotropy effect. If there is any anisotropy, it could only relate to the diffusion coefficient D, otherwise Di to be correct, and, therefore, to the liquid structure within the interfacial region. At present, the author does not know of any studies that show such 78 anisotropy. In contrast, it is predicted'17 that there is no growth rate difference between dislocation-free and dislocated rough inter- faces. This is because a spiral step created by dislocation(s) will hardly alter the already existing numerous kink sites on the rough interface. A summary of the interfacial growth kinetics together with the theoretical growth rate equations is given in Fig. 9. Next, the growth mode for kinetically rough interfaces is discussed. Growth Kinetics of Kinetically Roughened Interfaces As discussed earlier, an interface that advances by any of the lat- eral growth mechanisms is expected to become rough at increased super- coolings. Evidently, the growth kinetics should also change from the faceted to non-faceted type at supercoolings larger than that marking the interfacial transition. In accord with the author's view regarding the kinetic roughening transition, the following qualitative features for the associated kinet- ics could be pointed out: a) Since the interface is rough at driving forces larger than a critical one, its growth kinetics are expected to resemble those of the intrinsically rough interfaces. Thus, the growth rate is expected to be unimpeded, nearly isotropic, and proportional to the driving force. Moreover, the presence of dislocations at the inter- face should not affect the kinetics, b) It is clear that the faceted interface gradually roughens with increasing AT over a relatively wide range of supercoolings. The transition in the kinetics should also be a gradual one. c) In the transitional region the growth rates should be faster than those predicted from the lateral, but slower than the |

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328 203. J. Q. Broughton and G. H. Gilmer, J. Chem. Phys., 79 (1983) 5119. 204. F. Rosenberger, in: Interfacial Aspects of Phase Transformations, B. Mutaftschiev, ed. (D. Reidel, Dordrect, Netherlands, 1982), p. 315. 205. K. A. Jackson, private communication. 206. J. D. E. McIntyre and W. F. Peck, Jr., J. Electrochem. Soc., 123 (1976) 1800. 207. J. M. Cases and B. Mutaftschiev, Surf. Sci., 9 (1960 57. 208. V. V. Voronkov, Sov. Phys.-Cryst., 19 (1974) 296. 209. G. J. Abbaschian and R. Mehrabian, J. Cryst. Growth, 43 (1978) 433. 210. E. A. Brener and D. E. Temkin, Sov. Phys.-Cryst., 30 (1985) 140. 211. M. J. Aziz, Appl. Phys. Lett., 43 (1983) 552. 212. R. Kern, in: Growth of Crystals, Vol. 8, N. N. Sheftal, ed. (Con sultants Bureau, New York, 1969), p. 3. 213. R. Boistelle, in: Industrial Crystallization, J. W. Mullin, ed. (Plenum Press, New York, 1976), p. 203. 214. B. Simon and R. Boistelle, J. Cryst. Growth, 52 (1981) 779. 215. N. V. Stoichev, G. A. Alftintsev, and D. E. Ovsienko, Sov. Phys.- Cryst., 20 (1976) 504. 216. V. T. Borisov and Y. E. Matveev, Sov. Phys.-Cryst., 14 (1970) 765; 16 (1971) 207. 217. M. J. Aziz, in Undercooled Alloy Phases, E. W. Collings and C. C. Koch, eds. (TMS-AIME Spring 1986 Meeting, New Orleans, LA), to be published. 218. H. Beneking, W. Vits, in: Proc. 2nd Int. Symp. on GaAs, Inst. Phys. Soc. Conf. Ser. No. 7, p. 96. 219. See for example: a) F. Rosenberger, Fundamentals of Crystal Growth I (Springer-Verlag, Berlin, 1979); b) R. L. Parker, Sol. State Phys., 25 (1970) 151. 220. S. D. Peteves and G. J. Abbaschian, in: Undercooled Alloy Phases, E. W. Collings and C. C. Koch, eds., TMS-AIME Symp. Proc., 1986 Spring Meeting (New Orleans, LA), to be published. 221. P. Bennema, Phys. Stat. Sol., 17 (1966) 555. Stability Term, R(ia); C/cm Figure A-14 The stability term R(w) as a function of the perturbation wavelength and the growth rate. 304 221 76C. However, as shown earlier, the (111) kinetics deviate from those of the 2DNG theory at much lower supercoolings. The same arguement also holds for the (001) kinetics. It is believed that the deviation from the classical rate law is due to a reduction in the 2D nucleation barrier as the driving force for growth (i.e., the supercooling) increases, thus implying that the step edge energy oe is decreasing with supercooling. As understood, the be havior of oe is closely related to the concept of kinetic roughening, which is expected to prevail at high supercoolings. The variation of the step edge free energy with the supercooling can be determined directly from the experimental results and the general equation for two-dimensional nucleation growth (see eq. (69)). To achieve this, the mobility of the steps, which governs the pre-exponen tial term in the 2DNG rate equation (eq. (37)) was assumed to have about the same order of magnitude of mobility at the lower end of the PNG regime. The oe values of the best fit are shown in Fig. 53 as a func tion of the interface supercooling. It can be seen that the step edge energy is approximately constant up to supercoolings of about 3C. At higher supercoolings it starts decreasing, first gradually and later rapidly with AT. The functional form of oe(AT) was found to be best expressed by an exponential relation given as e = U ~ exp[-2.69 (ATr AT)]} (90) where o is a constant equal to 20.3 ergs/cm^ (i.e. the step edge energy near the 2DN threshold supercooling) and ATR is the supercooling at Comparison of the (001) experimental growth rates and those calculated using the general 2DNG rate equation, as a function of the supercooling; note that the PNG calculated rates were not formulated so as to include the two observed experimental PNG kinetics. Figure 52 215 137 Table 4. Typical Growth Rate Measurements for the (111) Interface. Lot: B, Sample: 1, Tm = 29.74C, S z (29C) = 1.84 Mv/C .28 mv, AR/Ae (27C) = 1.17pQ/pm, ATb = 8.22C, I = At = 1.4 sec. Eoff = 5 x 10-_i A, Distance Time, AU V, optical V, resistance Results Solidified Optical, MV pm/ s pm/ s pm sec 1750 2.13 821.6 1750 2.18 802.7 1750 7.09 837 1750 2.15 815 1750 7.021 829 3500 1.95 1790 Dislocations 1750 15.6 1842 Dislocations 1750 15.84 1871 Dislocations 1750 6.85 809 1750 2.14 819 155 where A is the S/L interface area. The reasoning behind the general 2DNG equation, eq. (69), as well as the magnitude of the parameters K-^, K2, and B and their dependence upon the growth variables will be given in the following chapter. As indicated above, the dislocation-free (111) data could be divided into three regions, as shown in Fig. 26. The first two regions, I (MNG) and II (PNG), will be discussed in detail below; the third region, III (TRC), which covers growth rates higher than about 1500 pm/s and interface supercoolings larger than 3.5C, will be discussed in a later section. The cut-off points for each region are established by realizing a systematic deviation of data points from carefully deter mined regression lines representing the kinetics for the region adjacent to them. The regression lines were initially determined from data points well beyond or above (with respect to AT) the cut-off points. Subsequently, transitional data points would be included in the regres sion analysis only if their deviation from the former line was small enough so that it did not significantly affect the parameters of the rate equation. Furthermore, the population of the data points and the number of samples used, quantitatively ensures the justification for assigning a borderline supercooling between each region. MNG region Region I (MNG), ranges from 1.5 to about 1.9C supercoolings and for growth rates up to about 1 pm/s. The growth rates in this region depend on the size of the capillary tube cross section. For each 34 alloy;90 also, a CS of a model for crystal growth from the vapor found that the phase transition proceeds from the fluid phase to a disordered solid and afterwards to the ordered solid.91 Strong molecular ordering of a thin liquid layer next to a growing S/L interface has been suggested92 as an explanation of some phenomena observed during dynamic light scattering experiments at growing S/L interfaces of salol and a nematic liquid crystal.93 In an attempt to rationalize this behavior, it was proposed that only interfaces with high "a" factors can exert an orienting force on the molecules in the interfacial liquid; however, such an idea is not supportive of the ob servation regarding the water/ice (0001) interface (a ~ 1.9).94 96 The ice experiments9495 have shown that a "structure" builds up in the liquid adjacent to the interface (1.4-6 pm thick), when a critical growth velocity (~1.5 pm/s) is exceeded, that has different properties from that of the water (for example, its density was estimated to be only .985 g/cc, as compared to 1 g/cc of the water) and ice, but closer to that of water. Interpreting these results from such models as that of the sharp and rough interface, of nucleation (critical size nuclei) ahead of the interface and of critical-point behavior, as in second- order transition" were ruled out. Similar experiments performed on salol revealed97 that the S/L interface resembles that of the ice/water system, only upon growth along the [010] direction and not along the [100] direction. The "structured" (or density fluctuating) liquid layer * It should be noted they95 determined the critical exponent of the relation between line width and intensity of the scattered light in close agreement with that predicted2930 for the diffuse liquid-vapor interface at the critical point. 129 For example, in the case of Ga (orthorhombic structure, see more in Appendix I), the tensor is actually a diagonal matrix; the elements along the diagonal represent the Seebeck coefficient for the three principal axes of the Ga crystal. Furthermore, eqs. (64) -(67) are, strictly speaking, valid only if the circuit conductors are structurally and chemically homogeneous.320 Strong textures and intense segregation in the conductors result in spurious emf's caused by secondary effects such as Bennedick and Volta effects.321 Nevertheless, with a suitable experimental arrangement and instrumentation, the Seebeck voltage can be utilized to determine the interfacial temperature, as discussed next. Determination of the Interface Supercooling Prior to making the kinetics measurements, the following parameters for each sample were determined: a) the melting point, Tm. This temp erature was used to double check and recalibrate, if necessary, the thermocouples. The thermocouple output would give the bulk supercooling of the liquid, b) The values of the "offset emf", EQff* According to the previous discussion about the thermoelectric technique, when the two S/L interfaces are at the same temperature, the recording emf (see eq. (66)) should be zero. However, in practice this is rarely the case be cause of the several other junctions involved in the circuitry and the possible minute temperature differences between them. For example, a constant temperature difference of .01C between the W-Cu junctions would result in an offset emf of the order .02 qV. Other causes result ing in a non-zero are inhomogeneities in the Cu-leads and the junc tion between Cu leads and the instrument's cables, and offset potentials of the recording instruments. For each sample, the value of 159 is independent of the sample size. These are in qualitative agreement with the theoretical predictions; as indicated in the discussion of 2DN growth models, the mononuclear gradually changes to the polynuclear growth mechanism above a certain supercooling. The growth rates are still exponential functions of (-1/AT), and fall into a line in the log (V) vs. l/AT plot of Fig. 29. The equation of the regression line for data points up to interfacial supercoolings of 3.51C and growth rates of 1455 pm/s is given as312 log V = 5.98 10.42/AT with a coefficient of determination of .992 and a coefficient of corre lation of .996. The growth rate equation for the PNG region is thus given as V = 9.56 x 105 exp(- 23.995/AT) (71) where V is the growth rate in pm/s. Dislocation-Assisted Growth Kinetics The dislocation-assisted (111) data seem also to be divided into two growth regions, as shown in Fig. 25. The data fall on a straight line up to interfacial supercoolings of about 2C; this region is called the SDG region. At larger supercoolings than this, the data points deviate from the line for the lower growth rates and approach the (high supercoolings) dislocation-free growth rates. The growth kinetics in the first region are determined for super coolings up to about 2C and corresponding growth rates of 2100 pm/s. The dislocation-assisted growth kinetics, like the dislocation-free kinetics, can be represented by an equation in the form of AT AT c (72) c 30 experimental works that will be reviewed next. These simulations re sults then raise questions about the validity of current theories on crystal growth5859 and nucleation60 which, based on theories discussed earlier, such as the "a" factor theory, assume a clear cut separation between solid and liquid; this hypothesis, however, is significantly different from the cases given earlier. Experimental evidence regarding the nature of the S/L interface Apparently, the large number of models, theories, and simulations involved in predicting the nature of the S/L interface rather illus trates the lack of an easy means of verifying their conclusions. In deed, if there was a direct way of observing the interfacial region and studying its properties and structures, then the number of models would most likely reduce drastically. However, in contrast to free surfaces, such as the L/V interface, for which techniques (e.g. low-energy dif fraction, Auger spectroscopy, and probes like x-rays61) allow direct analysis to be made, no such techniques are available at this time for metallic S/L interfaces. Furthermore, structural information about the interface is even more difficult to obtain, despite the progress in techniques used for other interfaces.62 Therefore, it is not surprising that most existing models claim success by interpreting experimental re sults such that they coincide with their predictions. Some selected examples, however, will be given for such purposes that one could relate experimental observations with the models; emphasis is given on rather recent published works that provide new information about the interfa cial region. A detailed discussion about the S/L interfacial energies will also be given. Indirect evidence about the nature of the 60 where w'? is the rate at which individual atoms are added to the critical cluster (equal to the product of arrival rate, R^, and the surface area of the cluster, S), nÂ£ is the equilibrium concentration of critical nuc lei with i" number of atoms, and Z is the Zeldovich non-equilibrium fac tor which corrects for the depletion of the critical nuclei when nuclea- tion and growth proceed. Z has a typical value of about 10^, 163 and is given as Z AG? () V4ttKT; 1/2 1 i where AG^ is the free energy of formation of the critical cluster. For the growth of clusters in the liquid, it is assumed that the clusters fluctuate in size by single atom increments so that the edge of the cluster is rough. The arrival rate R is then defined as described pre viously for the growth of a step. Finally, the concentration of the critical nuclei is given as n^ = n exp (- AG. i KT ) where n is the atom concentration. For a disk-like nucleus of height h, the work needed to form it is given as TTO AG = - h AG (26) v where ae is the step edge free energy per unit length of the step. For small supercoolings at which the work of forming a critical two-dimen sional nucleus far exceeds the thermal energy (KT), the nucleation rate per unit area can be approximately written, as derived by Hillig,164 in the form of GROWTH KINETICS OF FACETED SOLID-LIQUID INTERFACES By STATHIS D. PETEVES 1 A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA My very special thanks to Stephanie Gould for being the most im portant reason that the last two years in my life have been so happy. I am so grateful to her for her continuous support and understanding and particularly for forcing me to remain "human" these final months. I also especially thank my parents and my sister for 29 and 25 years, respectively, of love, support, encouragement, and confidence in me. Vll Table 8. Experimental and Theoretical Values of 2DNG Parameters. Growth Mode Interface K1 (pm sec) * K2 pm/sec o e ergs/cm^ Slope ratio MNG/PNG in log(V) vs. 1/AT Exp. Theor. Exp. Theor. Exp. o . si Exp Theor. MNG (111) 2.25x1010 4.8xl0U 20.3xf(AT) 67-40 2.45 3 or 2.5 or 2 PNG 5.2xl07 1.6xl08 MNG (001) 6.08xl010 5.9xl01* 11.7xf'( AT) 2.6 PNG 6.3xl07 2.5xl07 1.8x108 205 7 considers the interface as a region with "intermediate" properties of the adjacent phases, rather than as a surface contour which separates the solid and the liquid side on the atomic level. According to this criterion, the interface is either diffuse or sharp.6-10 A diffuse interface, to quote,6 "is one in which the change from one phase to the other is gradual, occurring over several atom planes" (p. 555). In other words, moving from solid to liquid across the interface, one should expect a region of gradual transition from solid-like to liquid like properties. On the other hand, a sharp interface8-10 is the one for which the transition is abrupt and takes place within one inter- planar distance. A specific feature related to the interfacial diffuse ness, concerning the growth mode of the interface, is that in order for the interface to advance uniformly normal to itself (continuously), a critical driving force has to be applied.6 This force is large for a sharp interface, whereas it is practically zero for an "ideally diffuse" interface. The second criterion8-12 assumes a distinct separation between solid and liquid so that the location of the interface on an atomic scale can be clearly defined. In a manner analogous to that for the S/V interface, the properties of the interface are related to the nature of the crystalline substrate and/or macroscopic (thermodynamic) properties via "broken-bonds" models. Based on this criterion, the interface is either smooth (singular,''13 faceted) or rough (non-singular,'' non- faceted). A smooth interface is one that is flat on a molecular scale, represented by a cusp (pointed minimum) in the surface free energy as a * Sometimes these interfaces are called F- and K-faces, respectively.13 67 and the interfacial supercooling. Although the pre-exponential terms of the rate equations, strictly speaking, are functions of AT and T, prac tically they are constant within the usually limited range of supercool ings for 2DNG. The distinct features associated with 2DNG kinetics are the following: a) A finite supercooling is necessary for a measurable growth rate (~10~3 pm/s); this is related to the threshold supercooling for 2DN, mentioned earlier, and it is governed by oe in the exponential term. The smaller oe is, the smaller the supercooling at which the interfacial growth is detectable. b) Only the MNG kinetics are depend ent on the S/L interfacial area, c) Since the pre-exponential terms are relatively temperature independent, both MNG and PNG kinetics should fall into straight lines in a log(V) vs. 1/AT plot, d) From the slope of the log(V) vs. l/AT curve (i.e. Moe^/T), the step edge free energy can be calculated,63177"181 provided that the experimental data have been measured accurately. oe can then be used to estimate the diffuse ness parameter "g" via the proposed relation6 e = si h (g)1/2 (38) e) Furthermore, in the semilogarithmic plot of the growth data, the ratio of the slopes for the MNG and PNG regimes should be 3, according to the classical theory; however, as discussed earlier, this ratio can actually range from 2 to 3 depending on the details of the cluster spreading process. Detailed 2DNG kinetics studies are very rare, in particular for the MNG region, which has been found experimentally only for Ga2 and Ag.182 The major difficulties encountered with such studies are 1) the necess ity of a perfect interface; 2) the commonly involved minute growth 2 M \ E ZL 1 > CD O _J 0 -1 -2 3 .4 .5 .6 1/AT, *C_1 Figure 41 Initial (111) growth rates of Ga-.Ol wt% In grown in the direction parallel to the gravity vector; () effect of distance solidified on the growth rate, and ( ) growth rate of pure Ga. 185 206 (37) these constants are given as = chKQ" Kg~ D where c is a con- 1 /0 stant of the order of unity. KQ = (N/Vm)(L/RTm) 3P/2a and Kg is given as 3p)/2a L/RTm; 0 is assumed to be one in the calculations. The results reveal about four orders of magnitude difference be tween the experimental and calculated terms for the MNG and about one order of magnitude difference for the PNG kinetics. The experimental values of and Ko were determined from careful linear regression anal ysis of the growth data for both growth rates and interfaces, as pre sented earlier. Although there is some uncertainty in the theoretical calculations concerning the constants involved, it is believed that the lack of coincidence between the experimental and theoretical values lies on the use of the liquid self-diffusion coefficient D (1.6 x lO"-1 cm~/s329) in the calculations for the migration of the atoms across the interface. For the PNG region, the two values are quite close, consid ering the uncertainty in calculating the pre-exponential terfo of the nucleation rate equation. However, this discrepancy between the experi mental and calculated PNG values can still be reconciled, as explained later. The step edge free energy (oe) was also calculated from the expo nential term Mo exp ( AT mo V T , e m riu 6Xp hkT AT i.e. from the slope of the log(V) vs. 1/AT line for the data in the MNG regime." The values of og per unit length of steps on the (111) and 1/2 * Properly, oe should be computed using the slope of logfV/AT ) vs. l/AT. However, since the AT range is small (from 1.5 to 1.9C and from .6 to .8C), the pre-exponential factor is large, the incor poration of the AT factor has a negligible effect on og. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS iv LIST OF TABLES xii LIST OF FIGURES xiii ABSTRACT xxi CHAPTER I INTRODUCTION 1 CHAPTER II THEORETICAL AND EXPERIMENTAL BACKGROUND 6 The Solid/Liquid (S/L) Interface 6 Nature of the Interface 6 Interfacial Features 8 Thermodynamics of S/L Interfaces 10 Models of the S/L Interface 14 Diffuse interface model 14 The "a" factor model: roughness of the interface 22 Other models 25 Experimental evidence regarding the nature of the S/L interface 30 Interfacial Roughening 36 Equilibrium (Thermal) Roughening 36 Equilibrium Crystal Shape (ESC) 46 Kinetic Roughening 48 Interfacial Growth Kinetics 53 Lateral Growth Kinetics (LG) 53 Interfacial steps and step lateral spreading rate (u ) 54 Interfacial atom migration 57 Two-dimensional nucleation assisted growth (2DNG) 58 Two-dimensional nucleation 59 Mononuclear growth (MNG) 62 viii 220 for Ga). In this case, the region next to the clusters of the top layer is more "liquid-like" than that in the first layer next to the crystal. In the PNG region, because of dimensional arguments, nucleation events are predominant at the top layers while layer spreading is con trolled by the lower layers. Also keeping in mind that in the case of the PNG process could be written as the product of (D^nUC'*) (D^r)^^, a value of about 10^ 10^ cm^/s can be estimated. The above mentioned argument explains the discrepancy between the pre-exponential theoretical and experimental terms at low supercoolings. It does not explain, though, the observed transitional kinetics for both interfaces occurring at high supercoolings, as shown in Figs. 23 and 31, where the dashed lines represent the calculated rates in accord with the 2DNG models corrected for D^. The reasoning for the observed devia tion in the growth kinetics of high supercoolings is discussed next. Step Edge Free Energy As discussed earlier, the classical 2DNG theory assumes that the step edge free energy is independent of the interfacial supercooling. Based on this assumption, the only deviation in the growth kinetics one could expect at high supercoolings is when the free energy for the form ation of a critical nucleus AG" equals the thermal energy KT (in other words, when the exponential term in eqs. (37) and (69) diminishes). According to the experimental values of ae, the (ill) interface should deviate from the 2DNG rate equation at supercoolings in the order of * Note that this criterion, AG" = KT, has recently been identified with the onset of the kinetic roughening, incorrectly, as discussed later. 50 40 w e 30 > 20 10 0 1. (Ill) 0 V//g V//-g x V//g mo <Â¡> Ax o o m x 50 1. 75 2. 00 2. 25 AT. C 2. 50 2. 75 Figure 47 Inicial (111) growth rates of Ga-.01 wt% In ( O ) and Ga-.12 wt% In ( X O ) grown in the direction parallel ( X ) and antiparallel ( O O ) to the gravity vector; continuous line represents the growth rate of pure (111) Ga interface. 193 Polynuclear growth (PNG) 64 Screw dislocation-assisted growth (SDG) 68 Lateral growth kinetics at high supercoolings 72 Continuous Growth (CG) 73 Growth Kinetics of Kinetically Roughened Interfaces 78 Growth Kinetics of Doped Materials 83 Transport Phenomena During Crystal Growth 87 Heat Transfer at the S/L Interface 88 Morphological Stability of the Interface 93 Absolute stability theory during rapid solification 98 Effects of interfacial kinetics 99 Stability of undercooled pure melt 100 Experiments on stability 101 Segregation 102 Partition coefficients 102 Solute redistribution during growth 104 Convection 106 Experimental S/L Growth Kinetics 112 Shortcomings of Experimental Studies 112 Interfacial Supercooling Measurements 113 CHAPTER III EXPERIMENTAL APPARATUS AND PROCEDURES 117 Experimental Set-Up 117 Sample Preparation 120 Interfacial Supercooling Measurements 125 Thermoelectric (Seebeck) Technique 125 Determination of the Interface Supercooling 129 Growth Rates Measurements 134 Experimental Procedure for the Doped Ga 140 CHAPTER IV RESULTS 146 (111) Interface 146 Dislocation-Free Growth Kinetics 150 MNG region 155 PNG region 156 ix 244 optically, whereas the interfacial supercooling was assumed to be equal to the bulk supercooling at low growth rates. At higher rates, the interface temperature was determined by a thermocouple. They reported that the growth of perfect crystals is characterized by 2DNG kinetics and that of the imperfect crystals by the SDG mechanism. Although their results are in qualitative agreement with the present ones, a quantita tive comparison between the two is not possible. This is because their growth rates extend in a small range (2-4 pm/s for the dislocation-free and 10-280 pm/s for the dislocated interfaces), and the data points are plotted in small linear graphs. Semiquantitatively, however, their re sults are in fair agreement with those of the present study. Another difficulty in comparing their data to the present ones is that their growth rate equations, given by the authors in references (104) and (215), are not consistent with each other. They relate the discrepan cies to the differences between the experimental conditions of each experiment. Kinetics of solidification of dislocation-free and dislocated single crystals of Ga, grown in glass capillaries, were also studied by Abbaschian et al.2>" The growth rates (up to 2500 pm/s) were measured optically and the interface supercoolings were determined from a heat transfer model. Their (111) results are in good agreement with those of the present experiments, up to growth rates of about 500-600 pm/s. Above these rates, the present results show slightly higher growth rates (about +7% in the range of 600 pm/s only). This is believed to be due to the limited accuracy of the heat transfer model used to determine the interface supercooling at high growth rates (see more about it in their Figure 26 The logarithm of the (111) growth rates versus the reciprocal of the interfacial supercooling; A is the S/L iterfacial area. 153 144 Figure 22 Seebeck emf compared with the bulk temperature as affected by dislocation(s) and interfacial breakdown, recorded during growth of In-doped Ga. 87 and AT 2-2.5C), it was thought that the interface lost its faceted character. Additions of Ag, Cu63 caused a sharp increase in the growth veloc ity of the pure Ga and a replacement of the two-dimensional nucleation by the dislocation growth mechanism. The source of the dislocations was attributed to impurity segregation and separation of second phases (CuGa2, for example). Whether the adsorption of the impurity on different crystal facets changes, resulting in habit modifications, is not clearly understood as yet. During growth of Si217 and GaAs,218 such effects have been ob served. Although the role of impurities is quite important during growth of facet forming materials, there have been very few studies de voted to this field of research and the essential features of growth in the presence of impurities are not very well understood. Theoretical interpretations are not yet possible, but based on experimental results some interpretations allow for guidelines regarding the possible solute effects on the growth kinetics. However, aside from the technical point of view, the role of impurities is worth further investigation for the better comprehension of the crystal growth mechanisms, and, most import antly, of the S/L interface. Transport Phenomena During Crystal Growth Growth of a solid from the liquid phase involves two sets of pro cesses; one on the atomic scale and the other on the macroscopic scale. The first is associated with the interfacial atomistic processes. The second involves the transport of matter (solute, impurities) and latent 329 222. See for example reviews in Refs. (25) and (26); for recent example see: a) K. F. Kobayashi, M. I. Kumikawa, and P. H. Shingu, J. Cryst. Growth, 67 (1984) 85. 223. G. A. Colligan and B. J. Bayles, Acta Met., 10 (1962) 895. 224. J. J. Kramer and W. A. Tiller, J. Chem. Phys., 42 (1965) 257. 225. D. A. Rigney and J. M. Blakely, Acta Met., 14 (1966) 1375. 226. G. T. Orrok, Ph.D. Thesis, Harvard University, 1958. 227. A. Rosenberg and W. C. Winegard, Acta Met., 21 (1954) 342. 228. G. J. Abbaschian and M. E. Eslamloo, J. Cryst. Growth, 28 (1975) 372. 229. G. L. F. Powell, G. A. Colligan, V. A. Surprenant, and V. Urquhart, Met. Trans., A8 (1977) 971. 230. V. V. Nikonova and D. E. Temkin, in: Growth and Imperfections of Metallic Crystals, D. E. Ovsienko, ed. (Consultants Bureau, New York, 1967), p. 43. 231. J. J. Favier and M. Turpin, presentation at I.C.C.G.5, Boston, MA, July 17-22, 1977. 232. R. T. Delves, in: Crystal Growth, Vol. 1, B. R. Pamplim, ed., (Pergamon Press, Oxford, 1974), p. 40. 233. R. F. Sekerka, in: Crystal Growth: An Introduction, P. Hartman, ed. (North-Holland, Amsterdam, 1973), p. 403. 234. D. J. Wollkind, in: Preparation and Properties of Solid State Materials, Vol. 4, W. R. Wilcox, ed. (M. Dekker, New York, 1979), p. 111. 235. J. S. Langer, Rev. Mod. Phys., 52 (1980) 1. 236. A. A. Chernov, Sov. Phys.-Cryst., 16 (1972) 734. 237. W. Rutter and B. Chalmers, Can. J. Phys., 35 (1953) 15. 238. W. A. Tiller, K. A. Jackson, J. W. Rutter, and B. Chalmers, Acta Met., 1 (1953) 428. 239. W. W. Mullins and R. F. Sekerka, J. Appl. Phys., 34 (1963) 323. 240. W. W. Mullins and R. F. Sekerka, J. Appl. Phys., 35 (1964) 444. 241. V. V. Voronkov, Sov. Phys. Solid State, 6 (1965) 2378. i 9 Terraces, Steps b) Liquid Solid Figure 1 Interfacial Features. a) Crystal surface of a sharp interface; b) Schematic cross-sectional view of a diffuse interface. After Ref.(17) 63 20NG Figure 7 Schematic drawings showing the interfacial processes for the lateral growth mechanisms a) Mononuclear. b) Poly nuclear. c) Spiral growth. (Note the negative curvature of the clusters and/or islands is just a drawing artifact.) CHAPTER IV RESULTS The experimental investigation of the high purity Ga interfacial kinetics covers a range of 10^ to 2 x 10^ pm/s growth rates and inter facial supercoolings up to about 4.6C, corresponding to bulk supercool ings up to 53C. In addition, the kinetics have been determined as a function of crystal perfection (dislocation-free versus dislocation- assisted interface) and crystal orientation ([111] and [001]). On the other hand, the In-doped Ga kinetics study covers a range of 10^ to 45 pm/s growth rates and interfacial supercoolings up to about 2.5C. For the doped material the kinetics have also been determined for two ini tial compositions, Ga .01 wt% In and Ga .12 wt% In, for dislocation- free interfaces of the (ill) type. Furthermore, the growth rates have been measured as a function of solidified length and growth direction with respect to the gravity force. In this chapter the growth kinetics results are presented and, whenever it is obvious, they are qualitatively related to the earlier discussed growth theories. (ill) Interface When the bulk supercooling'' was less than about 1.5C, the undis turbed (111) S/L interface was practically stationary in contact with * It should be noted that for a motionless interface, the bulk and interface supercoolings are the same. 146 95 GL > Thus, for solidification into a supercooled liquid (G^ < 0), the consti tutional supercooling criterion always predicts instability. Also, since the instability is predicted to be proportional to the growth rate, the interface is expected to be unstable during rapid solidifica tion of alloys. The second theoretical approach on the morphological stability (MS) of the interface is based on the dynamics of the entire process.239-242 In this approach, small perturbation, which can be a temperature, con centration, or shape fluctuation at the S/L interface, is imposed on the system. When the mathematical equations are linearized with respect to perturbation, in order to make the problem solvable, the time dependence of the amplitude of the perturbation is calculated under given growth conditions. If the perturbation grows, the interface is unstable, while if it decays, the interface is stable. The morphological instability problem is then solved by taking into account CS, surface tension (os) and transport of heat from the interface through both the liquid and the solid. Assuming constant velocity during unidirectional solidification of a dilute binary alloy in the z-direction, the perturbation of the interface is given as z = 6 exp (at + i(u)xx + w^y)) where 6 is the perturbation amplitude and oox y are its spatial frequen cies. The interface is unstable if the real part of a is positive for any perturbation (the imaginary part of a has been shown rigorously to vanish243 at the stability/instability demarcation (a = 0)). The value of a for local equilibrium conditions and isotropic S/L interface is given as24 0 > 2 44 2 4 s Table 7. Experimental Growth Rate Equations; V in pm/sec and A in pm . Interface Growth Mechanism Supercooling Range, C Growth Kinetics (111) Dislocation-free, 2DNG MNG 1.5 1.9 ,, ,r.9 e 58.76^ V = 1.7x10 A exp ( T) PNG 2 3.5 11 nr / 23 9 \ V = 9.5 x 10 exp (- T) Dislocation-assisted, SDG .2.- 2 V = 700 AT1,7 Kinetic Roughening > 3.5 V a AT (001) Dislocation-free, 2DNG MNG .6 .8 V = 2.95x 109 A exp ("^^) PNG .8 1.45 (A) V = 6xl05 exp (_^j) (B) V 2.4x10 exp (- Dislocation-assisted, SDG .2 1 1 93 V = 1640 AT Kinetic Roughening > 1.5 . V a AT 176 21 Figure 4 Graph showing the regions of continuous (B) and lateral (A) growth mechanisms as a function of the parameters B and Y, according to Temkin's model.7 80 60 40 20 0 Vrj x 103, cm2/#ec -6 Temperature correction, ST, for the (111) interface as a function of Vr. for different heat-transfer conditions, U.r.; Analytical calculations (K =K =K), Numerical calculations. 1 1 L s 290 297 0C) the above mentioned assumption will introduce some error. Indeed, if one recalculates the case of Ihr^ = .02K using the thermal parameters of the fluid for 0C, it is found that these conditions correspond to U.r. = .0193K. The latter value, as understood from Fig. A-ll, would re- 11 suit in slightly higher <5T values than the previous ones. Finally, Fig. A-12 shows the comparison between experimental and calculated results for the (001) interface. The agreement between the two is quite satisfact ory, as shown in Fig. A-12; similar to the (111) interface, the experimental results for the (001) interface are slightly lower than the numerical at Vr^ values larger than .01 cm^/s. The present numerical calculations have shown to be in excellent agreement with the experimental results, determined directly via the Seebeck technique. The above conclusion assures the reliability of the heat transfer model if no assumptions are made regarding the conduction of heat to the solid and the liquid phase. The assumption that the liquid and the solid have the same thermal properties (as made in the earlier calculations2181) will introduce errors, particularly for the (001) interface at high growth rates. The numerical results can be used to estimate the thermal gradients at the interface and the interfacial temperature, whenever the Seebeck technique is not feasible. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. / Gholamreza J. Abbaschian, Chairman Professor of Materials Science and Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Robert T. DeHo^i^; Co-Chairman Professor of Materials Science and Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Robert E. Reed-Hill Professor Emeritus of Materials Science and Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. C^ . Ranganathan Narayanan Associate Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Associate Professor of Chemical Engineering This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1986 ikjhjjt Q Dean, College of Engineering Dean, Graduate School 198 As discussed earlier, the direct measurement of the interfacial supercooling via the Seebeck principle requires the knowledge of the Seebeck coefficient of the S/L interface, SgÂ£, as a function of tempera ture and orientation of the solid. SsÂ£ was measured directly in this study (see also Ref. (311)); some of the determined values for the (111) and (001) interfaces were given previously in Table 3. Note that these values agree well with those calculated from the absolute Seebeck coef ficients of solid and liquid Ga, according to the following relation ship1 (111) [111] S (T) = S (T) S(T) = 1.86 mV/C at T = 29C si s l (001) [001] S (T) = s (T) S(T) = 2.2 mV/C at T = 29C sic s l where Sg and S^ are the absolute Seebeck coefficients of the Ga. Sg was determined as a function of crystal orientation and temperature, from the following equation gfhkl](T) = S[001](T) 2 + S[010](T) 2 + S[100](T) 2 (86) s s Is 2 s 3 where cp^> The temperature coefficients of the Seebeck coefficients were311 .0107 and .012 pV/(C)2 (negative) for the (111) and (001) interfaces, respec tively. Figure 49 shows the Seebeck coefficient of the liquid and solid along the principal axes as a function of temperature. According to the theoretical background of the Seebeck technique, the recorded Seebeck emf, Es, is related to the supercooling of the mov ing interface (II) as 320 36. W. K. Burton and N. Cabrera, Disc. Faraday Soc., 5 (1949) 33. 37. A. Ookawa, in: Crystal Growth and Characterization, R. Ueda and J. B. Mullin, eds. (North-Holland, Amsterdam, 1975), p. 5. 38. D. Nason and W. A. Tiller, J. Cryst. Growth, 10 (1971) 117. 39. K. Huang, Statistical Mechanics (J. Wiley, New York, 1963). 40. J. R. O'Connor, J. Electrochem. Soc., 110 (1963) 338. 41. J. D. Ayers, R. J. Schaeffer, and M. E. Glicksman, J. Cryst. Growth, 37 (1977) 64. 42. J. E. Lennard-Jones and A. F. Devonshire, Proc. Roy. Soc., A169 (1938) 317. 43. J. D. Bernal, Proc. Roy. Soc., A280 (1954) 299. 44. A. Bonissent, in: Modern Theory of Crystal Growth, A. S. Chernov and H. Muller-Krumbhaar, eds. (Springer-Verlag, Berlin, 1983), p. 1. 45. A. Bonissent and B. Mutaftschiev, Phil. Mag., 35 (1977) 65. 46. F. Spaepen, Acta Met., 23 (1975) 731. 47. A. J. C. Ladd and L. V. Woodcock, Chem. Phys. Let., 51 (1977) 155; J. Phys. C., 11 (1978) 3565. 48. S. Toxvaerd and E. Praestgaard, J. Chem. Phys., 11 (1977) 5291. 49. A. Bonissent, E. Gauthier, and J. L. Finney, Phil. Mag., B39 (1979) 49. ^ 50. J. W. Broughton, A. Bonissent, and F. F. Abraham, J. Chem. Phys., 74 (1981) 4029. 51. D. W. Oxtoby and A. D. J. Haymet, J. Chem. Phys., 76 (1982) 12. 52. F. F. Abraham and Y. Singh, J. Chem. Phys., 67 (1977) 2384. 53. A. D. J. Haymet and D. W. Oxtoby, J. Chem. Phys., 74 (1981) 2559. 54. U. Landman, C. L. Cleveland, C. S. Brown, and R. N. Barnett, in: Nonlinear Phenomena at Phase Transitions and Instabilities, T. Riste, ed. (Plenum, New York, 1981), p. 379. 55. C. L. Cleveland, U. Landman, and R. N. Barnett, Phys. Rev. Lett., 49 (1982) 790. 56. J. W. Cahn and R. Kikuchi, Phys. Rev., B31 (1985) 4300. 92 x / / / / _ / / / / 1 1 / 1 / r .*-i 1 1 - ** | (175 C)2 j /* i l_ 1 1 1 o 100 200 300 400 500 600 AT^ x 10'2, ro2 Figure 12 Bulk growth kinetics of Ni in undercooled melt. After Ref. (201). 700 64 Polynuclear growth (PNG). At supercoolings larger than those of the MNG region, condition (29) is not fulfilled and the growth kinetics are described by the so called polynuclear (PNG) model." According to this model, a large number of two-dimensional clusters nucleate at ran dom positions at the interface before the layer is completed, or on the top of already growing two-dimensional islands, resulting in a hill- and valley-like interface, as shown in Fig. 7b. Assuming that the clusters are circular and that ue is independent of the two-dimensional cluster size, anisotropy effects, and proximity of neighboring clusters, the one layer version of this model was analytically solved.169 This was poss ible by considering that for a circular nucleus the time, t, needed for it to cover the interface is equal to the mean time between the genesis of two nuclei (i.e. the second one on top of the first), or otherwise given by irj J(uet)2 dt = 1 (31) Integration of this expression and use of the relation V = h/x yields the steady state growth rate (for the polynuclear-monolayer model) given as V = h (irJ\Jg/3)^/^ (32) This solution has been shown by several approximate solutions164168170 and simulations168171172 to represent well the more complete picture of multilevel growth by which several layers grow concurrently through * It should be mentioned that the use of the term "polynuclear growth" in this study should not be confused with the usually referred unreal istic model,18 which considers completion of a layer just by deposi tion of critical two-dimensional nuclei. Growth Rate/Interfacial Area,(/i.m*s) Figure 28 The logarithm of the MNG (111) growth rates normalized for the S/L interfacial area plotted versus the reciprocal of the interface supercooling. 158 301 For small AT or high Ti mt MA > 0. However, for practical purposes, as for the Ga growth kinetics experimental range, it can be assumed that the growth rate is only a function of supercooling. Accordingly, then .n-1 i B'w B'v cm N MT Mo AT exP(_ AT) (n + AT) (sec. Therefore, eq. (A34) can be rewritten as [-KLGL(L b + 2 K a The stability/instability demarcation can be obtained by letting o -* 0, provided that the thermal steady state approximation (to V/2kt ) is valid. The largest meaningful perturbation wavelength A = 2 it/to < d where d^ is the interfacial area diameter. Since, in the present ex periment, d = .014 cm, to then is given as to > 224.3 cmL Since for Ga kt = .1376 and = .1862 cm^/s the condition to > V/2kt holds, even for velocities up to about 60 cm/s. Thus, for growth rates involved in this investigation (V < 2 cm/s), it can be safely assumed that to > V/2k Based on this assumption, then a., as, and a = to (see eq. (54) of the text and analytical forms of coefficients therein). Therefore, eq. (A35) can be written as to{(-K G K G )m m 2KT Tu } 0 = g-g-JL T m (A36) hyMrj-, + 2Kt0 or that the interface is unstable when -LG, K G n -ti 5-* > T rul2 2K m (A37) Based on the morphological stability criterion, as expressed in eq. (A37), the conditions under which the planar (111) interface may become 112 For the Al-Cu experiments,:98 the 5 values (200-400 pm) appeared to be insensitive to the growth rates, but sensitive to the gradient and initial composition (for fixed CQ, t as G^l and for given G^, * as C0f). The Mn-Bi (Bi rich) system was studied solidifying both upwards and downward; the former resulting in solutal and the latter in thermal convection. As expected, a higher degree of convection was observed for the solutally unstable configuration. The determined ('250 pm) values were found to increase with concentration (note that here k >1) and slightly with the growth rate. However, the important effect of the liquid gradient was overlooked in this study. Experimental S/L Growth Kinetics Shortcomings of Experimental Studies Despite the numerous experimental studies reported over the past years, little conclusive information is available regarding crystal growth kinetics from the melt. To a large extent this is a consequence of the fact that experiments for melt growth kinetics, particularly for metals, are difficult. The difficulties associated with S/L interfacial kinetics are: high melting temperatures, opacity, impurities, sample perfection, i.e. the structural and chemical homogeneitv of the sample, and, most importantly, the determination of the actual temperature at the interface. The latter because of its importance will be discussed separately next. There are also several shortcomings in interpreting growth kinetics results. This is because in most studies a) the S/L interfacial kinetics are "confused" with the bulk kinetics, b) the kin etics measurements are not carried out over a wide enough supercooling 245 discussion2 and in Appendix III). The authors have cautioned the use of heat transfer calculations at high rates, since the calculations become very sensitive to the errors involved in the thermo-physical property values and assumptions of the calculations. Based on the numerical cal culations and the present direct measurements, the analytical solution underestimates the supercooling at rapid rates, as discussed in Appendix III. For the (001) interface there is a difference, up to about 15%, between the growth rate parameters for the PNG and SDG kinetics. Abbaschian et al.2 have also showed that their dislocation-free and assisted data for both interfaces approach each other at high supercool ings and that for the (001) interface they meet at about 1.6C interface supercooling. They also reported that, for dislocation-assisted growth at minute supercoolings (<0.1C), the growth kinetics could be inter preted by a linear relationship in the form of V = K AT. At larger than about 0.05C AT, the kinetics of the dislocated interface followed a ATn relationship with n around 1.7. The linear growth can be explained, assuming that growth was due to S dislocations (of the same sign) arranged in an array of length L so that L > 2urc > L/S. Accordingly, the rate is then proportional to ue S/L (see earlier discussion in Chap ter II) and, therefore, linearly dependent on AT. In order for the lin ear law to extend to AT's as low as .005C, a possible combination of parameter values required is for example L = 200 pm and S =50, which imply that xg 4 pm and ue about 1.6 cm/s (based on the 40 pm/s,0C2 experimental kinetic coefficient), which appear to be quite reasonable. Borisov et al.216,337 reported solidification data of Ga thin layers with growth rates up to 200 cm/s and corresponding supercoolings 134 detects the emergence of dislocations at the interface. This unique capability of the technique is illustrated in Fig. 20 where the Seebeck emf generated across the S/L interface of a (111) sample together with the bulk supercooling (emf of thermocouple II) are shown. The Seebeck emf changes proportionally to the interface temperature, which is in turn related to the bulk supercooling, heat transfer conditions, and the growth kinetics313 (also see the previous discussion on transport phe nomena at the interface). The abrupt peaks in the steady Seebeck emf indicate the emergence of screw dislocation(s) at the interface; when a dislocation intersects the faceted interface, the growth rate, is dras tically altered, which changes the interface supercooling and, there fore, the Seebeck emf. Growth Rates Measurements To measure the growth rate, the interface was initially positioned outside the observation bath II by keeping the heater 2 on, while the water temperature was set at the desired level of bulk supercooling. After the temperature had reached the steady state, heater II was turned off, allowing the interface to enter the bath and to grow into the supercooled liquid inside the observation bath. The growth rate was then measured via the optical microscope and/or by the resistance change of the sample, as described below. For growth rates in the range of 10'^ to 1.5 x 1(P pm/s, the interface velocity was measured directly by observing the motion of the trace of the interface on the capillary glass wall via the graduated optical microscope (20-40x) and timing it by a stop watch. Rate measurements were made only when the growth was 23 the statistical element is capable of two states only and ii) only interactions between nearest neighbors are important. The "a" factor theory, introduced by Jackson,8 is a simplified approach based on the above mentioned principles for the S/L interface. This model considers an atomically smooth interface on which a certain number of atoms are randomly added, and the associated change in free energy (AG) with this process is estimated. The problem is then to minimize AG. The major simplifications of the model are a) a two-level model interface: as such it classifies the molecules into "solid-like" and "liquid-like" ones, b) it considers only the nearest neighbors, and c) it is based on Bragg-Williams statistics. The main concluding point of the model is that the roughness of the solid-liquid interface can be discriminated according to the value of the familiar "a" factor, defined as where Â£ represents the ratio of the number of bonds parallel to the interface to that in the bulk; its value is always less than one and it is largest for the most close-packed planes, e.g. for the f.c.c. struc ture Â£ (111) = .5, Â£ (100) = 1/3, and E, (110) = 1/6. It should be noted that the a factor is actually the same with y in Temkin's theory. For values of a < 2, the interface should be rough, while the case of a > 2 may be taken to represent a smooth interface. Alternately, for mater ials with L/KTm < 2, even the most closely packed interface planes should be rough, while for L/KTm > 4 they should be smooth. According to this, most metallic interfaces should be rough in contrast with those of most organic materials which have large L/KTm factors. In between 187 interface that advances downward are higher by as much as 15% in the range of 30 urn/s growth rates. The rate equations for the two growth regions are determined to be as MNG: V = 14.6 x 107 A exp(-54.8/AT) PNG: V = 2.217 x 105 exp(-22.019/AT) (82) (83) where V is in pm/s. Ga .12 wt% In Increasing the In content to .12 wt% In had a similar effect on the growth kinetics, as compared with the .01 wt% In alloy, but several dis tinguishing features were observed. The addition of .12 wt% In to Ga caused very frequent interruptions in growth and multifacet formation at the growth front. Nevertheless, the growth behavior was similar to that of .01 wt% In; the growth rates at a constant bulk temperature were still a function of the distance solidified. Here, following the inter facial breakdown, there was always entrapped liquid, which sometimes ex tended as much as .5 cm along the capillary. The size of the entrapped liquid decreased with increasing supercooling, as before. The frequency of the interfacial breakdown was more for the .12 wt% In-doped Ga, as shown in Fig. 43. The growth rates as a function of the interface supercooling for the .12 wt% In-doped (111) Ga interface are shown in Fig. 44 on a log(V) vs. 1/AT plot. It appeared that, over the range of supercoolings indi cated, the interface retained its faceted character, and the growth 246 up to 35C. The growth direction, and the perfection of the crystals, were not specified. In their view, the lateral type of growth for Ga, reported in the above mentioned investigations,2 99104215 was caused by impurities. They claim that the observed linear rate equation (V = 5.3 AT cm/s) agrees satisfactorily with their own theory of normal growth mechanisms. Based on their explanation, Ga should not behave as a facet forming material and should grow continuously at any supercool ing. Their inaccuarate results have been attributed to the rather heur istic experimental conditions used,3383 which produced very strained crystals. Finally, Gutzow and Pancheva338^ have reported that solidifi cation of Ga single crystals, growth by the capillary technique, was of the faceted type.. In-Doped Ga Growth Kinetics The kinetic results presented earlier indicate that the growth rate of the (111) interface of Ga doped with In up to .12 wt% depends on the supercooling, In content, distance solidified, and growth direction with respect to the gravity vector. Furthermore, it is shown that the faceted interface breaks down as the growth process proceeds; the fre quency of which depends on the In content and the supercooling. The effects of each parameter on the growth kinetics are discussed in the following sections. Solute Effects on the 2DNG Kinetics The inital growth rates of the Ga-.012 wt% In samples versus the interfacial supercooling, as shown in Figs. 39 and 41, compared with the 2DNG kinetics of pure Ga indicate that at growth rates less than .5 REFERENCES 1. J. Alvarez, S. D. Peteves, and G. J. Abbaschian, in: Thin Films and Interfaces II, J. E. E. Baglin, D. R. Campbell, and W. K. Chu, eds. (Elsevier, New York, 1984), p. 345. 2. G. J. Abbaschian and S. F. Ravitz, J. Cryst. Growth, 44 (1978) 453. 3. E. A. Flood, The Solid-Gas Interface, Vol. 1 (Marcel-Dekker, New York, 1967). 4. A. Bonissent, in: Interfacial Aspects of Phase Transformations, B. Mutaftschiev, ed. (D. Reidel Publ. Co., Dordrect, Netherlands, 1982), p. 143. 5. See for example: H. Bethge, Interfacial Aspects of Phase Trans formations, B. Mutaftschiev, ed. (D. Reidel Publ. Co., Dordrect, Netherlands, 1982), p. 669. 6. J. W. Cahn, Acta Met., 8 (1960) 554. 7. D. E. Temkin, in: Crystallisation Processes, N. N. Sirota, F. K. Gorskii, and V. M. Varikash, eds. (Consultants Bureau, New York, 1966), p. 15. 8. K. A. Jackson, in: Liquid Metals and Solidification, (ASM, Cleveland, OH, 1958), p. 174. 9. K. A. Jackson, in: Growth and Perfection of Crystals, R. H. Doremus, B. W. Roberts, and D. Turnbull, eds. (J. Wiley, New York, 1958), p. 319. 10. W. K. Burton, N. Cabrera, and F. C. Frank, Phil. Trans. R. Soc. London, A243 (1951) 299. 11. F. C. Frank, in: Growth and Perfection of Crystals, R. H. Doremus, B. W. Roberts, and D. Turnbull, eds. (J. Wiley, New York, 1958), p. 304. 12. C. Herring, in: Structure and Properties of Solid Surfaces, R. Gomer and F. S. Smith, eds. (Univ. of Chicago Press, Chicago, IL, 1953), p. 1. 13. P. Hartman and W. G. Perdok, Acta Cryst., 8 (1955) 49; 521. 14. G. Wulff, Z. Krist., 34 (1901) 449. 15. C. Herring, Phys. Rev., 82 (1951) 87. 318 268 beyond the scope of this study. Briefly reviewed below are some of the aforementioned studies which might provide some insight into its proper ties and behavior related to this work. A contradictory aspect of the solid to liquid transition of Ga is the premelting phenomenon.365 For example, the abrupt rise of the spe cific heat (c ) near T^366 of Ga thin samples was explained by a surface melting model.367 The thickness of the premelted surface liquid layer was estimated to be in the range of 10-80 nm, an extremely large value, and to be dependent upon the differences in the interfacial surface energies (a -a -o^)368 The temperature of the premelting transition and the crystal orientation of the samples were not specified. Although the model explained the observed rise in the c^, the magnitude of its parameters and their physical relevance cannot be explained. Premelting phenomena in Ga, very close to T were also associated with premonitory effects observed in a thermoelectric study of Ga single crystals.318 However, the premonitory emf anomaly is believed to be within the ex perimental error range of the investigation.369 In contrast to these studies, premelting effects could not be detected up to temperatures of 10 ^C close to the melting temperature,370 during an anomalous x-ray transmission study of Ga perfect crystals (<010>). Interfacial free energies for the liquid/vapor and solid/vapor Ga interfaces have been measured by several investigators. The former ranges from 700-900 ergs/cm^,371"373 while the latter is about 780-850 2 ergs/cm 374 3 7 5 Based on these values and those for the S/L interface 2 (40-70 ergs/cm ), it can be shown that the Young's condition for perfect wetting (o > o + o ) is most likely satisfied. However, this in & sv sÂ£ Z.v 44 a crystal face under growth and equilibrium conditions above and below Tp. That means the "a" factor, which is said to be inversely propor tional to Tr, has to change continuously (with respect to the equilib rium temperature) or that L/KTm has to be varied. For a S/V interface, depending on the vapor pressure, the equilibrium temperature can be above or below T^, so that "a" can vary. The only exception in this case is the ^He S/L (superfluid) interface, at T < 1.76 K. For this system, by changing the pressure, the "a" factor can be varied over a wide range, in a small experimental range (i.e. .2 K < T < 1.7 K), where equilibrium shapes, as well as growth dynamics, can be quantitatively analyzed.96 For a metallic solid in contact with its pure melt though, this seems to be impossible because only very high pressure will influ ence the melting temperature. Thus, at Tm a given crystal face is either above or below its T^;122 crystals facet at growth conditions provided that T^ < Tr, where T is the interface temperature. Thus, the roughening transition of a S/L interface of a metallic system cannot be expected, or experimentally verified. In spite of the fact that most of the restrictions for the S/L interface do not exist for the S/V one, most models predict T^'s (for metals) higher than Tm, thus defying experimentation on such interfaces. The majority of the reported experiments are for non-metallic mate rials such as ice,123 naphthalene,121* C2Clg and NH^Cl,125 diphenyl,126 adamantine,127 and silver sulphide;128 in these cases the transition was only detected through a qualitative change in the morphology of the crystal face (i.e. observing the "rounding" of a facet). The likely conclusions from these experiments are that the transition is gradual 99 accordingly.245 The second restriction regarding the analytical form of absolute stability is that the net heat flow must be into the solid. KtCt K G L L + S S ^ r. \ (i.e. > o;. 2K The third one is that to use this criterion the conditions must be such that the regime of interest is far from the MSC regime. Effect of interfacial kinetics The effect of interfacial kinetics on morphological stability has been treated by several researchers256-260 by incorporating non-equilib rium (kinetic) effects at the appropriate interfacial boundary conditions of the heat-flow and diffusion equations. These treatments include growth rate and kinetics dependent interfacial supercooling and partition ratios.261-263'' (This subject will be treated in more detail in Appendix III.) Briefly, the analysis indicates that, for small supercoolings (i.e. V = f(AT) and k is given by the phase diagram), the numerator of eq. (54) remains unchanged, but the denominator is increased by an extra kinetic term253 _ 3f mt a(AT) For slow kinetics (small p^), this term leads to a reduction of the vel ocity at which the perturbation grows; in other words, a larger value of concentration, as compared to the case of local equilibrium, is needed for instability at fixed V. For fast kinetics (p^, > 5 cm/sC), on the other hand, not only the stability/instability demarcation, but also the magnitude of a (eq. (54)) are unaffected by the growth kinetics. * Convection effects on k leading to longitudinal and lateral instabil ities have also been incorporated in the stability analysis during unidirectional solidification.2 612 6 3 249 face.209 The part of the interface with the lower In concentration will move ahead of the rest since it is more supercooled; however, when the top of the protuberance reaches an adjacent region of liquid with low In concentration, it will then spread across, entrapping the solute rich strip. At high supercoolings (>5C), the inner surfaces of these bands developed several protuberance-like cells, indicating that the growth inside these bands is rather purely diffusive because of the high In content. Next, the effect of the interfacial segregation process on the growth kinetics is discussed in relation to the fluid flow effects associated with the growth direction with respect to the gravity vector. Segregation/Convection Effects As shown earlier, the initial growth rates for the parallel to gravity growth direction were higher than those for the antiparallel at a given interface supercooling. A possible cause for this finding could be due to differences in the nature and magnitude of convection in the liquid. The convection could be caused either by density inversion of the liquid and/or by the contact forces between the glass wall and the melt. When a Ga-In alloy is solidified upwards, a solute boundary layer is built up ahead of the S/L interface where the solute concentration decreases exponentially with the distance from the interface. Hence, the density of the liquid is higher at the interface and decreases with the distance from the S/L interface, x'. Thus, based on the previous discussion about convection in Chapter II, the composition gradient does not result in a solutally driven convection. On the other hand, the negative temperature gradient (see Appendix III) acts in the opposite 61 J N_ fLAT_vl/2 3Â£D A V ^RTT 7 2a p v KT (27) mm o where N is Avogadro's number and aQ is the atomic radius. This expres sion, that confirmed an earlier derivation,165 is the most widely accepted for growth from the melt. The main feature of eq. (27) is that J remains practically equal to zero for up to a critical value of super cooling. However, for supercoolings larger than that, J increases very fast with AT, as expected from its exponential form. Relation (27) can be rewritten in an abbreviated form as T v AT,1/2 AG , AG , J = KqD() exp (- jp^-) ~ Kn exp (- ^-) (28) where KQ is a material constant and is assumed to be constant within the usually involved small range of supercooling. Although theoretical estimates of are generally uncertain because of several assumptions, its value is commonly indicated in the range of 10^1-2.163 The very large values of Kp, and the fact that it is essentially insensitive to small changes of temperature, have made it quite difficult to check any refinements of the theory. Indeed, such approaches to the nucleation problem that account for irregular shape clusters166 and anisotropy effects167 lead to same qualitative conclusions as expressed by eq. (28). Also, a recent comparison of an atomistic nucleation theory from the vapor145 with the classical theory leads to the same conclusion. In contrast, the nucleation rate is very sensitive to the exponential term, therefore to the step edge free energy and the supercooling at the clus ter/liquid (C/L) interface. The nature of the interface affects J in two ways. First, in the exponential term, AG", through its dependence upon oe and in the pre-exponential term through the energetic barrier 230 Kinetic Roughening When a smooth interface is growing at a temperature below T^, but at a driving force which is larger than a certain value, it will become rough and the non-linear V(AT) relation (i.e. lateral growth mechanism) will be replaced by a linear one. This phenomenon, as discussed earl ier, is known as kinetic roughening. In comparison with the thermal roughening transition, little attention has been paid to the character ization of the former. From a theoretical point of view, the transition could be best related to the conditions where the necessity for interfacial steps (in order for the smooth interface to grow) ceases to exist and, therefore, the two-dimensional nucleation barrier diminishes and dislocations have no effect on the growth rate. The nucleation barrier is meaningless either when thermal fluctuations result in a vast number of critical nuclei or when the step edge free energy becomes zero. The former im plies a critical supercooling AT^ at which the free energy for forming a critical 2D nucleus AG becomes equal to the thermal energy KT; on the other hand, the latter indicates a supercooling dependent step edge energy. From the experimental point of view AT^ is relatively small only when og is small to begin with. For example, assuming that the (111) interface kinetically roughens at AT = 4C by satisfying the con dition AG = KT, oe is estimated to be about 5.32 ergs/cm^; this is about 25% of the measured oe via the MNG rate equation. It is possible to rationalize a smaller oe value assuming that the 2D clusters are an isotropic. For example, if it is assumed that the nuclei are parallelo gram-like (instead of circular) with the two sides r^ and r2 such that 286 ve W CUir1/KL) J0(Yt) O (A27) are satisfied. By assigning the roots of the above equations to be y sn and the temperature distribution in the solid and liquid region have the following forms 00 7 71/7* 9 = E A J (y r') exp{[(Vr./2< ) ((Vr./2< ) + y ) ]z } (A28) s n o sn r is is sn s n=l and 00 9i = Bn Jo(YÂ£nr,) exp{[(Vr.ps/2KLpL) ((Vr.p^K^)2 + y2^)1 2]z^} n=l (eq. (A29)) Values of y and y that satisfy the above equations can be found in S jG table form,407 whereas the values of the coefficients An and Bn can be obtained from the remaining two boundary conditions, equations (A9) and (A10). Truncating equations (A28) and (A29) at Nth terms and inserting them into equations (A9) and (A10), yields N E n=l J (y r') o sn N Z n=l B n J (y0 r') o in (A30) Vp Lr. s 1 T T, m b Vr iN2 N Vr. K E A J (y r) [r-1 (() + y ) s n o sn 2k v2k sn n=l s s 2 ,1/2. (A31) N Vr.p Vr.p n ,~ kt e b j (y r') [T-i-s ((^-^)2 + y2 )1/2] L n o in 2kt pT v2ktp yn n1 Li i_i Li Since the above equations must be satisfied for all values of r1, one can randomly chooses N values of r' between 0 and 1 and insert them into equations (A30) and (A31); this results into 2N equations with 2N un knowns and unique solutions of A^ and Bn of any order are thus assured. By inserting the solutions back into equations (A28) and (A29), the axial 272 a 'y' coordinates O o -5 Figure A-1 The Gallium structure( four unit cells ) projected on the (010) plane; triple lines indicate the covalent( Ga9 ) bond. 321 57. R. Kikuchi and J. W. Cahn, Acta Met., 27 (1979) 1337. 58. K. A. Jackson, J. Cryst. Growth, 24/25 (1974) 130. 59. S. T. Chui and J. D. Weeks, Phys. Rev., B14 (1976) 4978. 60. D. Turnbull, J. Appl. Phys., 21 (1950) 1022. 61. B. C. Lu and S. A. Rice, J. Chem. Phys., 68 (1978) 5558. 62. N. Eustathopoulos and J. C. Joud, in: Current Topics in Materials Science, Vol. 4, E. Kaldis, ed. (North-Holland, Amsterdam, 1980), p. 281. 63. D. E. Ovsienko and G. A. Alfintsev, in: Crystals, Vol. 2, H. C. Freyhardt, ed. (Springer-Verlag, Berlin, 1980), p. 119. 64. A. A. Chernov, Modern Crystallography III; Crystal Growth (Springer-Verlag, Berlin, 1984). 65. R. Strickland-Constable, Kinetics and Mechanisms of Crystallization (Academic, London, 1968). 66. M. E. Glicksman and C. L. Void, Acta Met., 17 (1969) 1. 67. S. R. Coriell, S. C. Hardy, and R. F. Sekerka, J. Cryst. Growth, 11 (1971) 53; S. C. Hardy, Phil. Mag., 35 (1977) 471. 68. R. J. Scaeffer, M. E. Glicksman, and J. D. Ayers, Phil. Mag., 32 (1975) 725. 69. B. Mutaftschiev and J. Zell, Surf. Sci., 12 (1968) 317. 70. G. Grange, R. Landers, and B. Mutaftschiev, J. Cryst. Growth, 49 (1980) 343. 71. G. F. Bolling and W. A. Tiller, J. Appl. Phys., 31 (1960) 1345. 72. E. Arbel and J. W. Cahn, Surf. Sci., 66 (1977) 14. 73. W. A. Miller and G. A. Chadwick, The Solidification of Metals, Publ. 110 (Iron and Steel Institute, London, 1968). 74. C. S. Smith, Trans. AIME, 175 (1948) 15. 75. J. D. Ayers and R. J. Schaefer, Scripta Met., 139 (1969) 225. 76. N. Eustathopoulos, Int. Metals Rev., 28 (1983) 189. 77. D. Turnbull, J. Appl. Phys., 20 (1949) 817; J. Chem. Phys., 20 (1952) 411. 8 function of orientation plot (Wulff's plot14 or y-plot15). In contrast, a rough interface has several adatoms (or vacancies) on the surface layers and corresponds to a more gradual minimum in the Wulff's plot. Any deviation from the equilibrium shape of the interface will result in a large increase in surface energy only for the smooth type. Thus, on smooth interfaces, many atoms (e.g. a nucleus) have to be added simul taneously so that the total free energy is decreased, while on rough interfaces single atoms can be added. Another criterion with rather lesser significance than the previous ones is whether or not the interface is perfect or imperfect with re spect to dislocations or twins.11 In principle this criterion is con cerned with the presence or absence of permanent steps on the interface. Stepped interfaces, as will become evident later, grow differently than perfect ones. Interfacial Features There are several interfacial features (structural, geometric, or strictly conceptual) to which reference will be made frequently through out this text. Essentially, these features result primarily from either thermal excitations on the crystal surface or from particular interfa cial growth processes, as will be discussed later. These features which have been experimentally observed, mainly during vapor deposition and on S/L interfaces after decanting the liquid,16 are shown schematically in Fig. la for an atomically flat interface. (Note that the liquid is omitted in this figure for a better qualitative understanding of the structure.) These are a) atomically flat regions parallel to the top most complete crystalline layer called terraces or steps; b) the edges 115 coefficients of the solid and the liquid, the probe material, and the thermal field within the sample, as well as on the experimental details. This method has been further used308 to determine the growth rate during constrained growth of Sn, Bi, and Sn-Pb. Another method of determining the interface temperature relies upon mathematical analysis of heat flow conditions at the moving S/L boundary during unconstrained growth into a supercooled melt.2>178 *181 For these cases, the bulk and interfacial supercoolings are related via a tempera ture correction as AT = ATb hcV, ATb > AT > 0 where hc is the parameter representing the interfacial heat transfer coefficient which depends on the experimental design and the physical and thermal properties, such as latent heat, thermal conductivities, and densities of the materials involved. There are a few developed heat transfer models2>178181 that allow for calculation of hc and, there fore, of AT if ATb and V are measured (see detailed discussion in Appen dix III). Besides the complex mathematics of these models and the de pendence of their accuracy on thermal property data, their major draw back lies in the lack of verifying their validity as long as the inter face temperature is not measured directly. Furthermore, at fast growth rates and for rapid interfacial kinetics, the problem of calculating the interface temperature is very complex.309 Therefore, it seems rather difficult, if not impossible, to obtain accurate kinetic data as long as the interface supercooling has to be determined indirectly. Because of the above mentioned limitations of the previous tech niques, a novel technique for directly and accurately determining the 260 4) A quantitative justification of the experimental results for both 2DNG and SDG mechanisms is possible by removing the existing assumptions which treat the interfacial atomic migration as the liquid bulk diffusion process and the step edge energy as independent of the supercooling. 5) Based on the above mentioned, the diffusion coefficient within the interfacial layer, D^, was found to be up to about 3-4 orders of magnitude smaller than the liquid Ga self-diffusion coefficients (10-^ cm"/s). 6) At higher supercoolings, the results show that the faceted interfaces gradually become kinetically rough as the supercooling increases. The step edge free energy, which as indicated should be treated as a function of the supercooling, was shown to diverge exponen tially with the supercooling at the faceted-nonfaceted transition. The roughening supercooling was found to be smaller for the faster growing (001) interface. 7) A lateral growth model, which includes the interfacial diffusiv- ity and supercooling dependent step edge free energy, was found to des cribe well the growth kinetics of both interfaces up to the supercool ings marking the kinetic roughening transition. 8) At supercoolings higher than that of the transition disloca tions, which at lower supercoolings enhance the growth rate of the per fect interface by several orders of magnitude, do not affect the growth rate. Furthermore, beyond the transition the growth rates are linearly 36 difficult to imagine that simple statistical models could be adequate to describe their interfaces. However, these materials are of great theor etical importance in the field of crystal growth, as well as of techni cal importance referring to the electronic materials industry. Next, the effect of temperature and supercooling on the nature of the interface is discussed. Interfacial Roughening For many years, one of the most perplexing problems in the theory of crystal growth has been the question of whether the interface under goes some kind of smooth to rough transition connected with thermody namic singularities at a temperature below the melting point of the crystal. This transition is usually called the "roughening transition" and its existence should significantly influence both the kinetics dur ing growth and the properties of the interface. The transition could also take place under non-equilibrium or growing conditions, called the "kinetic roughening transition," which differs from the above mentioned equilibrium roughening transition. These subjects, together with the topic of the equilibrium shape of crystals, are discussed next. Equilibrium (Thermal) Roughening The concept of the roughening transition, in terms of an order- disorder transition of a smooth surface as the temperature increases was first considered back in 1949-1951. 10 536 The problem then was to calcu late how rough a (S/V) interface of an initially flat crystal face (close-packed, low-index plane) might become as T increases. This was possible after realizing that the Ising model for a ferromagnet could be 262 14) During growth of the In-doped Ga in the direction parallel to the gravity vector, gravitationally induced convection takes place be cause of the solutal gradient. 62 for atomic transport across the C/L interface. The assumptions of the classical theory are simple in both cases, since ae is taken as con stant, regardless of the degree of the supercooling, and the transport of atoms from the liquid to the cluster is described via the liquid self-diffusion coefficient. These assumptions are not correct when the interface is diffuse6 and at large supercoolings.32 These aspects will be discussed in more detail in a later chapter. Mononuclear growth (MNG). As was mentioned earlier, two-dimen sional nucleation and growth (2DNG) theories are divided into two regions according to the relative time between nucleation and layer com pletion (cluster spreading). The first of these is when a single crit ical nucleus spreads over the entire interface before the next nuclea tion event takes place (see Fig. 7a). Alternatively, this is correct when the nucleation rate compared with the cluster spreading rate is such that 1/JA > Â£/ue or for a circular nucleus A < (ue/J)^^ (29) where A, l are the area and the largest diameter of the interface, re spectively. If inequality (29) is satisfied, each nucleus then results in a growth normal to the interface by an amount equal to the step (nucleus) height, h. Thus, the net crystal growth rate for this class ical mononuclear (and monolayer) mechanism (MNG) is given as161*168 V = hAJ (30) In this region, the growth rate is predicted to be proportional to the interfacial area (i.e. crystal facet size). The practical limitations of this model, as well as the experimental evidence of its existence, will be given later. 6TCr')/ 5TCr=l) Figure A-9 Ratio of the Temperature correction(6T) to that at tlie edge as a function of r' of U r./K . i i s at any point of the interface (r'=r/r^) for different values 294 223 which ae vanishes; ATR was determined as 4.75C. Similarly, near this supercooling, it was found that oe approaches zero as oe exp(- .736/(T TR)1/2) where Tr = Tm ATR. It should be indicated that attempts to quantify oe(AT) as a power law resulted in poor regression analysis coefficients. Furthermore, for the best fit power law behavior, ATR was found to be in the order of 3.5C. a value which is much smaller than that of the (111) experimental results. Similarly for the (001) interface, the relation between the step edge free energy and the supercooling, shown in Fig. 53, was determined to be given as ae = a {1 exp[-6.36(ATR AT)]} (91) where a is equal to 11.7 ergs/cm2 and ATR is found to be equal to 2.28C. Interestingly enough, such a rapid divergence of the step energy upon approaching the roughening transition temperature is predicted by theoretical studies, as discussed earlier in Chapter 2. The exponential divergence of the step edge energy with the supercooling seems quite reasonable, since, for example in the Kosterlitz-Thouless113 model, oe vanishes at TR as exp(C| (T Tr)/Tr|^2). However, this theoretical behavior is related to the thermal roughening of the interface and, therefore, to the temperature dependence of oe rather than the driving force. Similar behavior has been supported recently by a growth kin etics study of the (0001)-face of ^He,130 where the values of oe were deduced from a 2DNG rate equation as a function of the equilibrium temperature of the S/L ^He interface. 24 these two extremes (2 and 4) there are several materials of considerable importance in the crystal growth community, such as Ga, Bi, Ge, Si, Sb, and others such as H2O. For borderline materials (a = 2), the effect of the supercooling comes into consideration. For these cases, this model qualitatively suggests2640 that an interface which is smooth at equil ibrium temperature may roughen at some undercooling. Jackson's theory, because of its simplicity and its somewhat broad success, has been widely reviewed in many publications.25262734 The concluding remarks about it are the following: a) In principle, this model is based on the interfacial "roughness" point of view.1036 As such, it attempts to ascribe the interfacial atoms to the solid or the liquid phase, which, as mentioned elsewhere, is likely to be an unrealistic picture of the S/L interface. Thus, the model excludes a probable "interface phase" that forms between the bulk phases so that its quantitative predictions are solely based on bulk properties (e.g. L). b) The model is essentially an equilibrium one since the effect of the undercooling on the nature of the interface was hardly treated. Hence, it is concluded that a smooth interface will grow laterally, re gardless of the degree of the supercooling. A possible transition in the nature of the interface with increasing AT is speculated only for materials with a ~ 2. Indeed, it is for these materials that the model actually fails, as will be discussed later. c) The anisotropic behavior of the interfacial properties is lumped in the geometrical factor Â£, which could be expected to make sense only 107 the detailed effects and nature of convection in these processes are not fully understood yet, and their understanding is likely to be limited at this point. In this section, a qualitative review of some convective phenomena during unidirectional solidification of a dilute alloy is given, in order to provide some background for the discussion related to the growth kinetics of the In-doped Ga. For complete information re garding this subject, the reader is referred to review papers288-290 and books.2193291292 During crystal growth of multicomponent systems, temperature and compositional gradients needed to drive heat and mass flows. However, these gradients induce variations in the properties of the liquid from which the crystal grows. The most important property that changes is the density. In a gravitational field, a density gradient will always result in fluid motion293 when the gradient is not aligned parallel to the gravity force. This type of flow is called natural or free convec tion; it is driven by body (buoyancy) forces (e.g. gravitational, elec tric, magnetic fields) and/or surface tension as contrasted with forced convection that arises from surface" (contact) forces. Density gradients in a fluid can be due to existing thermal gradi ents, since density increases as temperature decreases (thermal expan sion). The resulting convection is termed thermal convection. However, during growth of a multicomponent system, a density variation can be caused by compositional differences due to, for example, the interfacial Surface tension should not be confused with surface forces that re quire direct contact between matter elements. An example of surface force is the frictional force exerted from the rotating crystal on the melt during pulling. 75 Hence, av^exp (- Q^/KT) = D/a where D is the self-diffusion coefficient in the liquid. A similar expression can be derived based on the melt viscosity, n> by the use of the Stokes-Einstein relationship aDn = KT. Therefore, eq. (44) can be rewritten as V = F(T) [1 exp (- H^)] (45) m where F(T) in its more refined form is given as197 F(T) f a I A2 n in which f is a factor (< 1) that accounts for the fact that not all available sites at the interface are growth sites and A is the mean dif- fusional jump distance. Note that if A =a, then F(T) = Df/a. Further more, for small supercoolings, where LAT/KTmT 1, eq. (45) can be re written as (in molar quantities)25 V = DL aRTT AT = K AT c (46) m which is the common linear growth law for continuous kinetics. For most metals the kinetic coefficient Kc is of the order of several cm/sec,0C, resulting in very high growth rates at small supercoolings. Because of this, CG kinetics studies for metallic metals usually cover a small range of interfacial supercoolings close to Tm; in view of this, most of the time linear and continuous kinetics are used interchangeably in the literature. However, this is true only for small supercoolings, since for large supercoolings the temperature dependence of the melt diffusiv- ity has to be taken into account. Accordingly, the growth rate as a function of AT is expected to increase at small AT's and then decrease at high AT's. On the other hand, a plot of the logarithm of 156 capillary size, the data points fall on a straight line with a negative slope, indicating that the growth rates are exponential functions of (1/AT). As discussed earlier, the mononuclear growth mechanism is likely to be observed at AT's just larger than a critical threshold supercooling, with growth rates that are exponential functions of (l/AT) and proportional to the interface area (see eq. (40)). The pre dictions of the MNG model are satisfied for the low growth rates data (< 1 pm/s), as shown in Fig. 27 in a log (V) vs. l/AT plot. Note that the data fall on two parallel lines, each corresponding to different samples with the same capillary tube inside diameter, D. The growth rate in this region is also proportional to the S/L interfacial area. The pro portionality of the growth rates upon the S/L interfacial area is better illustrated in Fig. 28, where a plot of the quantity log (V/D^) as a function of l/AT, for all samples, results in a straight line. The equation for this line, as determined by regression, is given by log ~= 17.132 D 25.517 AT with a coefficient of determination .9991 and of correlation .9995. Thus, the growth rate equation for the MNG region is determined as V = 1.731 x 109 A exp (- 58.759/AT) (70) V is the growth rate in pm/s and A is the S/L interfacial area in pm^. PNG Region The second region, called PNG, covers the supercoolings range from about 2 to 3.5C and growth rates in the range of 1 1.5 x 10^ pm/s. The data points here, as shown in Fig. 26, still fall into a line, but with a smaller slope than that of Region I. Moreover, the growth rate 324 118. G. H. Gilmer and P. Bennema, J. Appl. Phys., 43 (1972) 1347. 119. H. Muller-Krumbhaar, in: Current Topics in Materials Science, Vol. 1, E. Kaldis, ed. (North-Holland, Amsterdam, 1978), p. 1. 120. W. Kossel, Nachr. Ges. Wiss. Gottingen, Math-Physik, K1 (1927) 135. 121. See for example: L. Coudurier, N. Eustathopoulos, P. Desre, and A. Passerone, Acta Met., 26 (1978) 465. 122. K. A. Jackson, in: Progress in Solid State Chemistry, Vol. 4, H. Reiss, ed. (Pergamon, London, 1967), p. 53. 123. D. Nenov and V. Stoyanova, J. Cryst. Growth, 46 (1979) 779. 124. A. Pavlovska and D. Nenov, J. Cryst. Growth, 12 (1972) 9. 125. K. A. Jackson and C. E. Miller, J. Cryst. Growth, 40 (1977) 169. 126. A. Pavlovska and D. Nenov, J. Cryst. Growth, 8 (1971) 209. 127. A. Pavlovska, J. Cryst. Growth, 46 (1979) 551. 128. T. Ohachi and I. Taniguchi, J. Cryst. Growth, 65 (1983) 84. 129. D. Balibar and B. Castaing, J. Physique Lett., 41 (1980) 329. 130. P. E. Wolf, F. Gallet, S. Balibar, E. Rolley, and P. Nozieres, J. Physique, 46 (1985) 1987; also, see references herein. 131. A. Passerone and N. Eustathopoulos, J. Cryst. Growth, 49 (1980) 757. 132. A. Passerone, R. Sangiorgi, and N. Eustathopoulos, Scripta Met., 14 (1980) 1089. 133. See for example: H. J. Human, J. P. Van der Eerden, L. A. M. J. Jetten, J. G. M. Oderkerken, J. Cryst. Growth, 51 (1981) 589. 134. C. Rottman, M. Wortis, J. C. Heyraud, and J. J. Metois, Phys. Rev. Lett., 52 (1984) 1009 and references therein. 135. L. D. Landau, Collected Papers, D. Terhaar, ed. (Gordon-Breach, New York, 1965), p. 540. 136. A. F. Andreev, Sov. Phys. JETP, 53 (1981) 1063. 137. C. Rottman, M. Wortis, Phys. Rev., B29 (1984) 328. 138. C. Jayaprakash, W. F. Saam, and S. Teitel, Phys. Rev. Lett., 50(1983) 2017. 257 corresponding distance (~7000 pm) under the diffusive conditions only. Similarly, for 6 = 50 pm, x is calculated as about 44 pm. Therefore, depending on the conditions, it is uncertain whether the solute tempera ture correction should be evaluated from the transient or steady state solute profiles. It seems that detailed quantitative work would obviously require a compositional analysis along the solidified length; although the latter does not assure the former (see, for example, the conclusions of refer ences (297) and (298), which are just qualitative), it still provides valuable information concerning the convection process. ACKNOWLEDGEMENTS The assumption of the last stage of my graduate education at the University of Florida has been due to people, aside from books and good working habits. It is important that I acknowledge all those individ uals who have made my stay here both enjoyable and very rewarding in many ways. Professor Reza Abbaschian sets an example of hard work and devotion to research, which is followed by the entire metals processing group. Although occasionally, in his dealings with other people, the academic fairness is overcome by his strong and genuine concern for the research goals, I certainly believe that I could not have asked more of a thesis advisor. I learned many things through his stimulation of my thinking and developed my own ideas through his strong encouragement to do so. His constant support and guidance and his unlimited accessibility have been much appreciated. I am grateful to him for making this research possible and for passing his enthusiasm for substantive and interesting results to me. At the same time, he encouraged me to pursue any side interests in the field of crystal growth, which turned out to be a very exciting and "lovable" field. Finally, I thank him for his understand ing and his tolerance of my character and habits during "irregular" moments of my life. Professors Robert Reed-Hill and Robert DeHoff have contributed to my education at UF in the courses I have taken from them and discussions of my class work and research. Their reviews of this manuscript and IV 212 for using the interfacial diffusivity instead of bulk diffusion will be given later. As realized in this section, the pioneering work of BCF has been borrowed in applying theory to practice. Their model, which is still considered among the most elegant, is hard to apply since most of its parameters are not known but have to be estimated, particularly for melt growth. Moreover, one might legitimately question whether such a theory, that strongly depends on surface diffusion, could relate to the growth of a S/L interface. However, as discussed earlier, only this model could explain the observed non-parabolic growth laws. If the experimental results are "forced" to follow a parabolic law, the rate equations for the two interfaces can be represented (V in pni/s) as'' (111): V = 730 AT2 (.2 < AT < 1.9) (111): V = 1703 AT2 (.2 < AT < 1.1) with coefficients of correlation of .87 and .9, respectively. Next, the experimental growth rate coefficients are compared with the calculated ones from the parabolic law, eq. (41), of the SDG model. Substituting oSÂ£ in eq. (41) with oe, as determined from the 2DNG kinetics and assum ing that the Burgers vector b of the dislocations is equal to the step heights used earlier, the kinetic coefficients of the (111) and (001) interfaces is calculated as 72 and 124 (Mm/s*C2), respectively. By comparing these values with those of the experimentally determined rate equation, it is realized that the latter are greater by about a factor * Note that the kinetic coefficient here is larger than that given in Table 7 because the largest population of data points is for AT's less than 1C supercooling. Figure A-12 Comparison between the (001) Experimental results ( O ) and the Model ( Analytical, Numerical) calcula tions as a function of Vr^ for given growth conditions .. 298 Figure A-13 The critical wavelength Acr at the onset of the insta bility as a function of growth rate; hatched area indi cates the possible combination of wavelengths and growth rates that might lead to unstable growth front for the given sample size (i.d. = .028 cm) 303 Figure A-14 The stability term R(to) as a function of the perturba tion wavelength and growth rate 304 xix Thermoelectric Power Figure 49 T,K Absolute thermoelectric power of so]id along the three principle Ga crystal axes and, liquid Ga as a function of temperature. '661 336 354. 0. Hunderi and R. Ryberg, J. Phys., F4 ( 1974) 2096. 355. F. Greuter and P. . Oelhapen, Z. Phyzik, B34 (1979) 123. 356. D. I. Page, D. H. , Saunderson , and C. G. Windsor, J. Phys., C6 (1973) 212. 357. M. I. Barker, i M. W. Johnson, N. H. March, and D. I. Page, in: Properties of Liquid Metals: Proceedings, S. Takeuchi, ed. (Halsted Press, New York, 1973), p. 99. 358. A. Defrain, J. Chim. Phys., 74 (1977) 851. 359. L. Bosio, R. Cortes, and A. Defrain, J. Chim. Phvs., 70 (1973) 357. 360. R. D. Heyding, W. Keeney, and S. L. Segel, J. Phys. Chem. Solids, 34 (1973) 133. 361. A. J. Mackintosh, K. N. Ishihara, and P. H. Shingu, Scripta Met., 17 (1983) 1441. 362. J. D. Stroud and M. J. Stott, J. Phys., F5 (1975) 1667. 363. L. Bosio, A. Defrain, and I. Epelboin, C. R. Acad. Sci., 268 (1969) 1344. 364. A. Jayaraman, W. Klement, R. Newton, and G. J. Kennedy, J. Phys. Chem. Solids, 24 (1963) 7. 365. A. R. Ubbelohde, Melting and Crystal Structure (Clarendon Press, Oxford, 1965). 366. G. Fritsch, R. Lachner, H. Diletti, and E. Liischer, Phil. Mag., A46 (1982) 829. 367. J. K. Kristensen, R. M. J. Cotterill, Phil. Mag., 36 (1977) 347. 368. G. Fritsch and E. Liischer, Phil. Mag., A48 ( 1983) 21. 369. A. R. Ubbelohde, private communication. 370. H. Wenzl ad G. Mair, Z. Physik, B21 (1975) 95. 371. U. Konig and W. Keck, J. Less-Common Met., 90 (1983) 299. 372. A. R. Miedema and F. J. A. den Broeder, Z. , Metallk., 70 (1979) 14 373. G. J. Abbaschian, J. Less-Common Met., 40 (1975) 329. 374. W. R. Tyson and W. A, . Miller, Surf. Sci., 62 (1977) 267. CHAPTER I INTRODUCTION Melt growth is the field of crystal growth science and technology of "controlling" the complex process which is concerned with the forma tion of crystals via solidification. Melt growth has been the subject of absorbing interest for many years, but much of the recent scientific and technical development in the field has been stimulated by the in creasing commercial importance of the process in the semiconductors in dustry. The interest has been mainly in the area of the growth of crys tals with a high degree of physical and chemical perfection. Although the technological need for crystal growth offered a host of challenging problems with great practical importance, it sidetracked an area of re search related to the fundamentals of crystal growth. The end result is likely obvious from the common statement that "crystal growth processes remain largely more of an art rather than a science." The lack of in- depth understanding of crystal growth processes is also due, in part, to the lack of sensors to monitor the actual processes that take place at the S/L interface. Indeed, it is the "conditions" which prevail on and near the crystal/liquid interface during growth that govern the forma tion of dislocations and chemical inhomogeneities of the product crys tal. Therefore, a fundamental understanding of the melt growth process requires a broad knowledge of the solid-liquid (S/L) interface and its energetics and dynamics; such an understanding would, in turn, result in many practical benefits. 1 Figure 34 The logarithm of the (001) growth rates versus the reciprocal of the interface supercooling. 169 18 -AG v c dx 'max a o max (10) where 2 3 it n ( tt n N exp ( ^ ) (11) ^max 8 Thus, if the driving force is greater than the right hand side of eq. (10), which represents the difference between the maxima and minima in Fig. 3a, the interface can advance continuously. The magnitude of the critical driving force depends on g(x), which is of the order of unity and zero for the extreme cases of sharp and ideally diffuse interfaces, respectively. In between these extremes, i.e. an interface with an intermediate degree of diffuseness, lateral growth should take place at small supercoolings (low driving force) and be continuous at large AT's. Detailed critiques from opponents and proponents of this theory have been reported elsewhere.25-27 A summary is given next by pointing out some of the strong points and the limitations of this theory: 1) The concept of the diffuse interface and the gradient energy term were first introduced for the L/V interface,24 which exhibits a second order transition at the critical temperature, Tc, where the thickness of the interface becomes infinite.28 Since a critical point along the S/L line in a P-T diagram has not been discovered yet, the quantities f(un) and the gradient energy are hard to qualify for the solid-liquid interface. The diffuseness of the interface is determined by a balance between the energy associated with a gradient, e.g. in density, and the energy re quired to form material of intermediate properties. The concept of the diffuseness was extended to S/L interfaces6 after observing29 that the grain boundary energy (in the cases of Cu, Au, and Ag) is larger than two times the oSÂ£ value. 2) The theory does not provide any analytical 29 The interfacial phenomena were also studied by a surface MD method,5455 meant to investigate the epitaxial growth from a melt. It was observed that the liquid adjacent to the interface up to 4-5 layers had a "stratified structure" in the direction normal to the interface which "lacked intralayer crystalline order"; intralayer ordering started after the establishment of the three-dimensionally layered interface regions. In contrast with the previously mentioned MD studies, non equilibrium conditions were also examined by starting with a supercooled melt. For the latter case, the above mentioned phenomena were more pro nounced and occurred much faster than the equilibrium situation. These results are supported by calculations56 of the equilibrium S/L interface (fee (001) and (100)) in a lattice-gas model using the cluster variation method. In addition, it was shown that for the nonclose-packed face (110), the S + L transition was smoother and the "intermediate" layer observed for the (001) face was not found for the (110) face. However, despite these structural differences, the calculated interfacial ener gies for these two orientations differed only by a few percent.57 Most of the methods presented here give some information on the structure and properties of the S/L interface, particularly of the liquid adjacent to the crystal. In spite of the fact that these models provide a rather phenomenological description of the interface, their information seems to be useful, considering all the other available techniques for studying S/L interfaces. In this respect, they rather suggest that the interfacial region is likely to be diffuse, particu larly if one does not think of the solid next to the liquid as a rigid wall.48 Such a picture of the interface is also suggested from recent 211 Based on this model, ATC is given as 4tt o V T AT = _e ? m c 2x L s Assuming oe to be independent of AT (the general case where oe is a function of AT is discussed later) and neglecting the temperature de pendence of xs within the temperature interval under consideration, ATC should then be constant. From the experimental values of ATC, the mean diffusion length, xs, is estimated to be about 430 for both inter faces. The latter value of xg indicates1933 an activation energy for atomic migration in the order of 3 Kcal/mole, as compared to that one for liquid self-diffusion of about 1.85 Kcal/mole. One the other hand, if multiple dislocations are considered," i.e. a number of S disloca tions, the earlier calculated value of xs reduces to xs/S. In utilizing the other constant Kp, it is uncertain about estimating some parameters from the BCF theory. For melt growth the term before the tanh term (see eqs. (40) and (88)) could be approximated with the term L D x AT E s 2trr T c Accounting for the experimental value of AT_, the value of K_ is calcu- lated to be 73000 and 57000 qm/s'0C, using D = 10^ cm^/s. These values are about two orders of magnitude smaller than the experimental values of given in Table 9. It should be noted that by replacing D with D^, the interfacial diffusivity, in the order of 10"^ cm^/s would bring the calculated values in agreement with the experimental ones. The reason * This considers the case of an array, L length, of S dislocations that satisfies the condition 2irrc > L; the latter implies that L < 850 . 127 dependence of the Seebeck emf generated across the S/L interface upon the interface temperature and crystal orientation, as well as the dopant concentration. Thermoelectricity, in principle, is concerned with the generation of electromotive forces by thermal means in a circuit of conductors.314 Since its discovery, the thermoelectric phenomenon has extensively been used to measure temperatures. The first discovered thermoelectric phe nomenon is the Seebeck effect, upon which the method of determining AT in this study is entirely based. For the Seebeck effect, one generally envisages an open circuit, shown in Fig. 17a, constructed out of conduc tors A and B with their junctions 1 and 2 held at temperatures Tj and T2* The thermoelectric emf, Es, developed by this couple is given by T 2 (S, S_) dT (64) T A B where S^ and Sg are the absolute thermopowers (the rate of change of the thermoelectric voltage with respect to temperature) or Seebeck coeffic ients of metals A and B. The thermoelectric power of the couple is de fined as 3 143 1 5 S._(T) = S (T) S(T) = lim (AE /AT) (65) AB A B m s This relation permits the determination of the Seebeck coefficient of the junction if the absolute thermoelectric powers of the components are known. The Seebeck effect can be used to measure the S/L interfacial temp erature by an arrangement that is shown in Fig. 17b. The thermoelectric loop in Fig. 17b is identical to that of Fig. 17a, except that conduc tors A and B are replaced with a solid and liquid metal; similarly, the 180 The initial growth rates versus the reciprocal of the interface supercooling are shown on a semi-log scale in Fig. 39 as solid symbols. The open symbols connected to them by the dotted curves show the effect of distance solidified on the growth rate at a given ATb. The line in Fig. 39 represent the growth rate equations of the pure (111) Ga interface, as given in Table 6. The interface supercooling was calculated based on the bulk supercooling and liquidus temperature, Tj of the Ga-.Ol wt% In as AT = ATb <5T <5TS = TL Tb <5T <5TS (79) where <5T is the heat transfer correction (see calculations and discus sion in Appendix III) and <5TS is the temperature correction because of the solute build up given as <5TS = -m (CA CG) As defined earlier, m is the liquidus slope, is the instantaneous interface composition, and CQ is the bulk composition. As discussed in the "Segregation" section, as the doped interface grows, it rejects sol ute ahead of the interface (for k < 1). Hence, the interfacial composi tion is higher than the initial bulk composition CQ. <5TS is zero at the onset of growth when = CQ and since m < 0, it increases with dis tance solidified (see eq. (62) where = C). Therefore, <5TS, by it self, should result in a decreased interface supercooling since the interface equilibrium temperature is lower than that of the interface at 200 (87) where is the interface temperature. Note that eq. (87), as indi cated, is an approximation since it is assumed that Ss^(Tm) = SsÂ¡i(T); its use introduces an error in the AT estimate that is small at low AT's, for example, about .008C for AT = 2C, but it increases at higher AT's. Therefore, the temperature coefficients of the Seebeck coeffi cients, dSSj^/dT, were taken into account so that the interfacial super cooling was calculated from the following relation dS (S,(T) + -r T ) AT + AE = 0. sit m dT m dT The emf output of the sample was measured with an accuracy of .005 pV, which corresponds to an accuracy of about .003C in supercooling. It should be noted that this accuracy is one order of magnitude better than the constancy of the bulk temperature, which was within .025C at any set temperature. The growth rate measurements, as indicated in the previous chapter, cover a range of seven orders of magnitude, i.e. from about 10"^ to 10^ pm/s. Such a broad range of measurements assures not only a complete picture of the growth behavior of the interfaces, but also eliminates any possible misinterpretation of the growth kinetics. In addition, as mentioned previously, for several of the used samples, the rate measure ments extended over 6-7 orders of magnitude while using the same experi mental techniques, thus defying any questions regarding the "uniformity" of samples and experimental procedures. The high growth rates (V > .15 cm/s) were determined by the square- wave current technique described earlier. The values of the current I were chosen so that the potential drop was stable enough to be resolved Growth Rate 1.000 .010 .001 . 550 .575 .600 .625 .1 .650 Figure 27 Dislocation-free (111) low growth rates versus the interfacial super cooling for 4 samples, two of each with the same capillary tube cross- section diameter. 157 11 formulation to interfaces, first introduced by Gibbs20 forms the basis of our understanding of interfaces. The intention here is not to review this long subject, but rather to introduce the concepts previously high lighted in a simple manner. If the temperature of the interface is exactly equal to the equilibrium temperature, Tm, the interface is at local equilibrium and neither solidification nor melting should take place. Deviations from the local equilibrium will cause the interface to migrate, provided that any increase in the free energy due to the creation of new interfacial area is overcome so that the total free energy of the system is decreased. On the other hand, the existence of the enthalpy change, AH = H^ Hg, means that removal of a finite amount of heat away from the interface is required for growth to take place. At equilibrium (T = Tm) the Gibbs free energies of the solid and liquid phases are equal, i.e. G^ = Gg. However, at temperatures less than Tm, only the solid phase is thermodynamically stable since Gg < G^. The driving force for crystal growth is therefore the.free energy dif ference, AGV, between the solid and the supercooled (or supersaturated) liquid. For small supercoolings, AGV can be written as AC - AGV V T mm (1) where L is the heat of fusion per mole and Vm is the solid molar volume. The S/L interfacial energy is likely the most important parameter des cribing the energetics of the interface, as it controls, among others, the nucleation, growth, and wetting of the solid by the liquid. Accord ing to the original work of Gibbs, who considered the interface as a physical dividing surface the S/L interfacial free energy is related to 276 packing considerations, [100] direction; this is understood based on the explanation that slip along the [100] would disrupt the strong covalent bonds. Twinning occurs readily in Ga with compression along [100], by transforming the a and b axes of the matrix to the b and a axes of the twin; it is believed to be associated with the rotation of the Ga^ mole cules about the c axis.393 Based on the results of the deformation study,390 a standard (001) stereographic projection for Ga was pre pared.3914 Upon melting, the low symmetry Ga structure changes into a state with 9-10 nearest neighbors, about 2.8 apart from each other395, arranged in somewhat loose close packing.396 This pronounced change in short range order upon the melting point is reflected in the anomalous density increase of 3.2%, whereas most metals show a density decrease of 2-6%. As far as the structural form of the interfaces under consideration is concerned, the (001) appears to be very flat, because the centers of its atoms lie on the plane. There are two Ga atoms per unit face area (axb). If the crystal is sliced along this plane, it is realized that each atom is missing four nearest neighbor bonds, two of the d^ type and two of the d^ type. On the other hand, the (ill) plane, in contrast with the (001), is not flat, but appears to have a zig-zag like struc ture composed of flat stripes which are part of the (211) plane. On the plane each Ga atom has three neighbors of the d^, d^, and d^ kind. In the calculation of the geometric factor Â£ (one of the parameters included in the Jackson's model "a" factor), which is the ratio of the 4 arranged so that they follow a hierarchal scheme based on a conceptual view of approaching this subject. The chapter starts with a broad dis cussion of the S/L interfacial nature and its morphology and the models associated with it, together with their assumptions, predictions, and limitations. The concept of equilibrium and dynamic roughening of interfaces are presented next, which is followed by theories of growth mechanisms for both pure and doped materials. Finally, transport phe nomena during crystal growth and the experimental approaches for deter mination of S/L interfacial growth kinetics are presented. In Chapter III the experimental set-up and procedure are presented. The experimental technique for measuring the growth rate and interface supercooling is also discussed in detail. In Chapter IV the experimental results are presented in three sec tions; the first two sections are for two interfaces of the pure mater ial, while the third one covers the growth kinetics and behavior of the doped material. Also, in this chapter the growth data are analyzed and, whenever deemed necessary, a brief association with the theoretical models is made. In Chapter V the experimental results are compared with existing theoretical growth models, emphasizing the quantitative approach rather than the qualitative observations. The discrepancies between the two are pointed out and reasons for this are suggested based on the concepts discussed earlier. The classical growth kinetics model for faceted interfaces is also modified, relying mainly upon a realistic description of the S/L interface. Finally, the effects of segregation and fluid flow on the growth kinetics of the doped material are interpreted. To the antecedents of phase changes: Leucippus, Democritus Epicurus and, the other Greek Atomists, who first realized that material persists through a succession of transformations (e.g freezing-melting-evaporation-condensation). Bulk Supercooling Al^, 135 Figure 20 Seebeck emf as recorded during unconstrained growth of a Ga S/L (111) interface compared with the bulk supercooling; the abrupt peaks (D) show the emergence of dislocations at the interface, as well as the interactive effects of interfacial kinetics and heat transfer. Seebeck eml 173 and .992 for line B. It should be realized that, unlike the MNG region, the difference between the two rate equations is not due to the effect of the interfacial area; for example, each line comes from samples of different sizes (e.g. for line B max i.d. = .0595 and min i.d. = .024 cm) and samples with a given size fall on both lines. Figure 35 also designates the lot of Ga from which the samples were prepared. As can be seen, for samples prepared from the same Ga lot, the data points fell on either of the two curves. This indicates that the difference between the two lines is not due to possible differences in the residual impur ities in the as received Ga lots. Futhermore, for a sample, which gave data points belonging to curve A, was melted in the capillary tube and reseeded, the data points shifted to curve B. Interestingly enough, all samples with the fastest kinetics (line A) also had higher Seebeck coef ficients (about 2.5 mV/C, see Table 3) than that for the perfectly ori ented (001) interface (2.2 uV/C), and their interface trace on the glass wall was inclined with respect to the capillary axis by 4-10. Needless to say, the difference between B and A lines is not due to the inclination, since the actual growth rates of the latter have been cor rected to account for the normal growth rates. Dislocation-Assisted Growth Kinetics The qualitative similarities between the growth behavior of the (111) and (001) interfaces also hold for the dislocation-assisted growth. The growth rates of the dislocated (001) interface, as shown in Fig. 31, can be fit into the general SDG rate equation, eq. (72), given earlier. The correlation parameters IC, ATC will be presented in the 14 e = as h (3) where oe is the edge energy per unit length of the step and h is the step height. However, this relation, as discussed later, has not been supported by experimental results. Models of the S/L Interface As may already be surmised, the most important "property" of the interface in relation to growth kinetics is whether the interface is rough or smooth, sharp or diffuse, etc. This, in turn, will largely determine the behavior of the interface in the presence of the driving force. Before discussing the S/L interface models, one should disting uish between two interfacial growth mechanisms, i.e. the lateral (step wise) and the continuous (normal) growth mechanisms. According to the former mechanism, the interface advances layer by layer by the spreading of steps of one (or an integral number of) interplanar distance; thus, an interfacial site advances normal to itself by the step height only when it has been covered by the step. On the other hand, for the con tinuous growth mechanism, the interface is envisioned to advance normal to itself continuously at all atomic sites. Whether there is a clear cut criterion which relates the nature of the interface with either of the growth mechanisms and how the driving force affects the growth behavior are discussed in the following sec tions . Diffuse interface model According to the diffuse interface growth theory,6 lateral growth will take over "when any area in the interface can reach a metastable equilibrium configuration in the presence of the driving force, it will 113 range, c) the perfection (dislocation-free vs. dislocation-assisted) or morphology of the interface are not reported, and d) certain critical data such as L, aSÂ£, q(T), crystal structure along the growth direction, etc. were unavailable. The methods of determination of the growth rate during crystal growth are: 1) optical measurements via a microscope by directly ob serving and timing the motion of the interface; 2) resistometric,228 which utilizes the resistance change across the sample during growth; 3) photocells, where the passage of time of the growth front for a certain length is determined with the aid of two or more photocells;63 A) high speed photography of the advancing interface and subsequent frame by frame analysis; and 5) conductance, which is related to the thickness of a molten layer so that the growth velocity can be calculated from the current transient.300 During constrained growth experiments, the steady state growth rate is usually assumed to be equal to the rate with which the thermal zone moves along the sample. In the present study, methods (1) and (2) were utilized, as it will be further discussed later. Interfacial Supercooling Measurements Several methods of direct or indirect determination of the S/L interface temperature have been attempted in the past. The most com monly used direct method consists of embedding a thermocouple probe in the crystal or the melt.100 However, the presence of the thermocouple not only disturbs the thermal and solutal fields at the interface, but it also affects the actual growth process; in several cases it has been reported63 that thermocouples were used to intentionally introduce dis locations . Moreover, in controlled solidification experiments, this Surface free energy 16 Figure 3 Diffuse interface model. After Ref. (6). a) The surface free energy of an interface as a function of its position. A and B correspond to maxima and minima configuration; b) The order parameter u as a function of the relative coordinate x of the center of the interfacial profile, i.e. the Oth lattice place is at -x. Growth Rate, f.imlt (111) 2 10 1.5 x 10 10* 5 x 10' Dislocation Assisted .oo 0 nTTTTXTrt^-QP ,L Q CD-' O O OO o 8 o O ccg o o o o a Dislocation Free / _oP t||iiiiiimjiAi7mroTTirrniDO^^^ 2 3 AT, C Figure 23 Dislocation-free and Dislocation-assisted growth rates of the (111) interface as a function of the interface supercooling; dashed curves represent the 2DNC and SDG rate equations as given in Table 7. 149 LIST OF TABLES Page TABLE 1 Mass Spectrographic Analysis of Ga (99.9999%) 122 TABLE 2 Mass Spectrographic Analysis of Ga (99.99999%) 123 TABLE 3 Seebeck Coefficient and Offset Thermal EMF of the (111) and (001) S/L Ga Interface 131 TABLE 4 Typical Growth Rate Measurements for the (ill) Interface. 137 TABLE 5 Analysis of In-Doped Ga Samples 141 TABLE 6 Seebeck Coefficients of S/L In-Doped (111) Ga Interfaces 142 TABLE 7 Experimental Growth Rate Equations 176 TABLE 8 Experimental and Theoretical Values of 2DNG Parameters .. 205 TABLE 9 Experimental and Theoretical Values of SDG Parameters ... 210 TABLE 10 Growth Rate Parameters of General 2DNG Rate Equation .... 213 TABLE 11 Calculated Values of g 238 TABLE 12 Solutal and Thermal Density Gradients 252 TABLE A-l Physical Properties of Gallium 265 TABLE A-2 Metastable and High Pressure Forms of Ga 267 TABLE A-3 Crystallographic Data of Gallium (a-Ga) 271 TABLE A-4 Thermal Property Values Used in Heat Transfer Calculations 289 Xll 136 steady and the interface consisted of a single facet of the orientation under consideration. Whenever the trace of the interface was not normal to the tube axis, the measured rates were corrected by the cosine of the angle between the interface's normal and the capillary axis. For each bulk supercooling, at least six rate measurements were made; the stan dard deviation from the mean accounted up to about 3%. A typical set of rate measurements for a sample along the (ill) interface is given in Table A. For growth rates in the range of 500 1.5 x 10"^ pm/s, the inter face velocity was determined from the resistivity change of the sample as a function of time, in addition to the above mentioned optical tech nique. For rates higher than 1.5 x 10-^ pm/s, the growth rates were determined only by the resistance change technique, since the accuracy of the optical measurements was limited at high growth rates. It should be noted that, although no optical growth rates measurements were taken at faster rates than 1.5 x 10^ pm/s, the interface behavior and shape were directly observed (7-20X) and correlated with the rate measure ments. For the resistivity technique, a square wave current at a speci fied periodicity (ranging from 100-500 milliseconds) alternating between less than a picoamp (<1 x 10^ A) and a few milliamps (3 5 x 10~^ A) was passed through the sample. This technique, which was fully control led from the microcomputer (see computer programs //2-//A in Appendix V), made it possible to alternatively measure the interface supercooling and the growth rate. During the picoamps cycle, the Seebeck emf was re corded, which, in turn, yielded the AT values, while during the milli amps cycle the potential drop across the sample was measured; the latter 225 K A (AT)1/2 exp[-B(f(AT))2/AT] V = mTPx 5 TR (92) {1 + K2 (AT)i7~ A 7 exp[-B(f(AT)) /AT]} 7 where the analytical forms of and K9 given earlier still hold, but are corrected for instead of D. B is defined as ir(o)2h Vm Tm/L K T and f(AT) is equal to o_(AT)/o as expressed in eqs. (90) or (91) de- pending on the interface under consideration. The calculated growth rates using the parameters given in Table 10 and eqs. (90) and (91) are presented in Fig. 54. As can be seen, the calculated rates agree very well with the experimental dislocation-free results over about the en tire experimental range for the (ill) interface, but up to supercoolings of about 2.1C for the (001) interface. The (001) growth kinetics be yond this supercooling will be described later. Similarly, for the dislocation-assisted growth kinetics accounting for a supercooling dependent step edge energy, the SDG rate equation (88) could be rewritten as V = K AT d f(AT) tanh(- AT f(AT) AT ) (93) where is equal to K^/ATc, and the analytical forms of f(AT), Kp, and ATC were given previously. From the curve-fitting parameters for the (111) dislocation-assisted growth data, shown in Fig. 55, ATC is evalu ated to be in the order of 3 and 10 at AT's higher than about 3C super cooling. Accordingly, xsS = 250 or 80 (or about 185 and 60 assuming that the spiral steps are polygonized instead of circular); a possible combination of parameters for ATC = 10 is S = 20 xs = 4 . The latter value of xs is close to that calculated based on the assumption that xs = a exp(L/4 K T) where a and L are the interatomic distance and heat of 203 faster rates, the Peltier heating is proportionally much smaller. Simi larly, the maximum Joule heating is estimated to be less than Qj = 6.1 x 10_t cal/s*cm, which is again negligible compared to Qs. For higher growth rates Qs increases much faster and, therefore, the Peltier and Joule effects are still negligible for the current densities used in this study. Comparison with the Theoretical Growth Models at Low Supercoolings At low supercoolings, the faceted Ga (ill) and (001) interfaces grow by two-dimensional nucleation-assisted or screw dislocation- assisted lateral growth mechanisms, as indicated earlier. From a theor etical point of view, the experimental data are of particular interest, especially considering the lack of reliable kinetics studies for growth from a metallic melt, because they provide accurate results against which the existing theoretical growth models can be tested and compared. Prior to comparing the Ga results with the predictions of the classical models, the "a" factor as proposed by Jackson to predict the growth be havior of the two interfaces are considered first. According to Jackson,8 if a, defined as a = LÂ£/KTm, is greater than two, the interface should be smooth, while for values of a < 2 the interface should be rough and normal growth should prevail. The value of L/KTm for Ga is about 2.2. By taking into account the bond strength of first and second neighbors, as discussed in detail in Appendix I, E, is calculated to be about 0.3 for the (001) face and 0.5 for the (111) face. The a parameters become = 0.7 and = 1-1 using the E, factors cited above; therefore, normal growth should be expected for 182 the initiation of growth. Consequently, as the interface supercooling is reduced, the growth rate decreases with distance, as shown in Fig. 37. This simple explanation for the observed behavior of the data points in the log V(x) vs. 1/AT plot, Fig. 39, disregards (for the moment) the effect of solute on the kinetics and the fact that 6T decreases as the growth rate decreases; however, a detailed discussion of these effects, together with a more detailed explanation of the growth rate-supercooling relation will be given in the Discussion chapter. As depicted in Fig. 39, the initial growth rates are exponential functions of 1/AT, and similar to those of the pure material; there is a limiting supercooling of about 1.5C for a measurable growth rate, and the results seem to fall into two regions, i.e. the mononuclear and the polynuclear. The rate equations for the initial growth the two regions were estimated as approximately MNG: V = 9.96 x 106 A exp(-50.23/AT) PNG: V = 1.88 x 105 exp(-21.8/AT) (80) (81) where V is given in pm/s and A is the interfacial area in pm~; the coef ficients of correlation of the two rate equations are .999 and .994, re spectively. Also note that the regression analysis for the PNG region extends up to supercoolings of about 2.55C. In the MNG region, the growth rates are slightly lower than those of the pure Ga. On the other hand, the rate equation, eq. (80), has a slope almost the same as that 242 deviation from the classical laws takes place at lower growth rates, it is believed that the kinetics transition, for both interfaces, is not due to the interfacial breakdown. However, since the analysis indicated that a possible breakdown of the interface might have occurred at rates in the order of .8 cm/s, it is appropriate as a last check to compare the growth data with that of the dendritic growth theory. Figure 60 shows the (ill) growth data as plotted in a normalized growth velocity (Vn) vs. the normalized supercooling (ATn) plot. Vn is defined as336 = V_ = V_ Tm sZ n 2a o 2a A'C P where V is the actual growth rate, a is the thermal diffusivity, Cp is the specific heat, and A is the unit supercooling defined as L/Cp. The normalized supercooling is given as where AT^ is the bulk supercooling. The physical parameters used for the growth parameters are given in Appendix I. The experimental points as shown in Fig. 60 fall into a single line with a slope of about 1.45, as compared with that of 2.65 for the uni versal dendritic growth law rate equation336 (continuous line in Fig. 60). Furthermore, at normalized supercoolings larger than .2, it is pre dicted that the power of the growth law should increase from 2.65. Results of Previous Investigations Alfintsev et al. 104,215 first studied the growth of single crystals of Ga placed between two glass plates. The growth rates were measured 31 interface, as obtained from growth kinetics studies, will not be covered here; such detailed information can be found, for example, in several review papers252663 and books.6465 Interfacial energy measurements for the S/L interface are much more difficult than for the L/V and S/V interfaces.62 For this reason, the experiments often rely upon indirect measurement of this property; in deed, direct measurements of ogÂ£ are available only for a very few cases such as Bi,66 water,67 succinonitrile,68 Cd/'69 NaCl and KC1,"70 and several metallic alloys.62 However, even in these systems, excepting Cd, NaCl, and KC1, information regarding the anisotropy of oSÂ£ is lack ing.71-76 Nevertheless, most evaluations of the S/L interfacial ener gies come from indirect methods. In this case, the determinations of oSÂ£ deal basically with the conditions of nucleation or the melting of a solid particle within the liquid. For the former, that is the most widely used technique, ogÂ£ is obtained from measured supercooling limits, together with a crystal-melt homogeneous nucleation theory in which oSÂ£ appears as a parameter6077 in the expression M o3 J = K exp ( ) (14) AT Here J is the nucleation frequency, Kv is a factor rather insensi tive to small temperature changes, and M is a material constant. On the * Strictly speaking, only these measurements are direct; the rest, still considered direct in the sense that the S/L interface was at least ob served, deal with measurements of grain boundary grooves or intersec tion angles (or dihedral angles) between the liquid, crystal, and grain boundary.71-74 The level of confidence of these measurements75 and whether or not the shape of the boundaries were of equilibrium or growth form76 remain questionable. 17 n = (2/a) (K/f )l/2 (6) o signifies the thickness of the interface in terms of lattice planes. As expected, the larger diffuseness of the interface, the larger is the co efficient K characterizing the gradient energy and the smaller the quan tity fQ which relates to the function fi^). The interesting feature of this model is that the surface energy is not constant, but varies peri odically as a function of the relative coordinate x of the center of the interface where the lattice planes are at z = na -x (see Fig. 3b). Assuming the interface profile to be constant regardless of the value of x we have o(x) = oQ + g(x)aQ (7) where oQ is the minimum value for o, and oQg(x) represents the "lattice resistance to motion" and g(x) is the well known diffuseness parameter that for large values of n is given as g(x) = 2 \4n2 (1 cos ^^) exp (- ^r^) (8) 3. Z Note that g(x) decreases with the increasing diffuseness n. Its limits are 0 and 1, which represent the cases of an ideally diffuse and sharp interface, respectively. In the presence of a driving force, AGV, if the interface moves by 6x, the change in free energy is given as 6F = (AG + a dg^x)) 6x (9) v o dx For the movement to occur, <5F must be negative. The critical driving force is given by 209 1.85. The difference of the ue's most likely comes from dissimilarities in the elementary molecular rearrangement at the edge of the steps. Possibly, for certain orientations close to the [001] direction, the Ga2 molecule becomes the growth unit instead of single atoms. Accordingly, the ratio of the local advancement would be equal to the ratio of the covalent bond length to the atomic radius. This ratio is about 2.4/1.3 = 1.84, which is equal to the ratio of the step edge velocities. This explanation for the observed differences in the kinetics along the (001) interface is further justified by the fact that the samples with faster growth rates had a few degrees misorientation with the tube axis. Al though this rationale is satisfactory for the PNG region, it seems to break down at higher growth rates (>1000 qm/s) where the rates appear to become the same. However, as will be discussed later, the nature of the interface changes as the interface supercooling increases because of the kinetic roughening. SPG kinetics. When dislocation(s) intersect the faceted interface, the kinetic characteristics are entirely different than those of the 2DNG mechanism. The dislocated interfaces are mobile at all supercoolings and their growth rates, at a given AT, are several orders of magnitude higher than those of the dislocation-free interfaces. For example, at 1.5 and 2C supercoolings the SDG rates of the (111) interface are higher than the 2DNG rates by six and three orders of magnitude, respectively. Although the growth rate equations given in Table 7 indicate a nearly parabolic relationship between the rate and the interfacial supercooling for both Al In summary, the key points of the roughening transition of an interface between a crystal and its fluid phase (liquid or vapor) are the following: a) At T = Tp a transition from a smooth to a rough interface takes place for low Miller index orientations. At T < T-^ the interface is smooth and, therefore, is microscopically flat. The edge free energy of a step on this interface is of a finite value. Growth of such an interface is energetically possible only by the stepwise mode. On the other hand, for T > Tr, the interface is rough, so it extends arbitrarily from any reference plane. The step edge energy is zero, so that a large number of steps (i.e. arbitrarily large clusters) is al ready present on a rough interface. It can thus grow by the continuous mechanism. Pictorial evidence about the roughening transition effects can be considered from the results of an MC simulation117 of the SOS model'' (S/V interface), shown in Fig. 5. Also, a transition with in creasing T from lateral kinetics to continuous kinetics above T^ was found for the interfaces both on a SC118 and on an fee crystal117 for the SOS model. b) It is claimed that most theoretical points of the transition have been clarified. Based on recent studies, the tempera ture of the roughening transition is predicted to be higher than that of the BCF model. Furthermore, its universality class is shown to be that of the Kosterlitz-Thouless transition. Accordingly, the step edge free * If, for the ordinary lattice gas model in a SC crystal, it is required that every occupied site be directly above another occupied site, one ends up with the solid-on-solid (SOS) model. This model can also be described as an array of interacting solid columns of varying heights, hr = 0, 1, ..., ; the integer hr represents the number of atoms in each column perpendicular to the interface, which is the height of the column. Neighboring sites interact via a potential V = K|hrhr1|. If the interaction between nearest neighbor columns is quadratic, one ob tains the "discrete Gaussian" model. Dislocation-Assisted Growth Kinetics 159 Growth at High Supercoolings, TRG Region 161 (001) Interface 164 Dislocation-Free Growth Kinetics 166 MNG region 166 PNG region 172 Dislocation-Assisted Growth Kinetics 173 Growth at High Supercoolings, TRG Region 174 In-Doped (111) Ga Interface 175 Ga-.01 wt% In 175 Ga-.12 wt% In 187 CHAPTER V DISCUSSION 194 Pure Ga Growth Kinetics 194 Interfacial Kinetics Versus Bulk Kinetics 194 Evaluation of the Experimental Method 197 Comparison with the Theoretical Growth Models at Low Supercoolings 203 2DNG kinetics 204 SDG kinetics 209 Generalized Lateral Growth Model 213 Interfacial Diffusivity 218 Step Edge Free Energy 220 Kinetic Roughening 230 Disagreement Between Existing Models for High Supercoolings Growth Kinetics and the Present Results 235 Results of Previous Investigations 242 In-Doped Ga Growth Kinetics 246 Solute Effects on 2DNG Kinetics 246 Segregation/Convection Effects 249 CHAPTER VI CONCLUSIONS AND SUMMARY 258 x 5 Final comments and conclusions are found in Chapter VI. The Appen dices contain detailed calculations and background information on the Ga crystal structure, Ga-In system, morphological stability, heat transfer, computer programming, and supercooling/supersaturation relations. 266 High purity Ga supercools very easily and can frequently be held for a long time at a temperature of 0C without solidifying. By divid ing Ga into small droplets, it has been possible to supercool the liquid by more than 150C.330>352353 The marked tendency of Ga to supercool has been discussed as a result of the suggested persistence of the Ga^ molecules in the liquid state.354 The latter, however, is contrasted by other works which believe that the covalent binding is destroyed upon melting, resulting in more metallic-like properties for the liquid.355 Amorphous Ga has also been prepared by vapor deposition onto He-cooled substrates;356 however, calorimetric and DTA measurements on single droplets down to 150K have shown no signs of any glass transition.357 When solid Ga, or even ice, comes in contact with the supercooled liquid, crystallization takes place rapidly. In this manner several grams of Ga can be converted to nicely defined orthorhombic crystals. This was routinely done in this study, where it was also realized that, by increasing the supercooling, the geometry of the crystal changed from trapezoid to pyramid (also, see Ref. 322c). Some metastable phases at atmospheric pressure are obtained from supercooled Ga or by solid-solid phase transitions.358-362 Two phases are formed only at high pressures.363364 The most important of these phases and some of their physical properties are listed in Table A-2. It should be noted that only normal Ga (a or I) expands upon solidifica tion . There are several studies on Ga and its physical properties, mainly because of its peculiar character and of its growing importance, partic ularly in the electronics industry. However, review of all of them is 13 Figure 2 Variation of the free energy G at liquid interface, showing the orig Ref. (22). T .m m across the solid- of o . After sÂ£ CHAPTER V DISCUSSION Pure Ga Growth Kinetics Interfacial Kinetics Versus Bulk Kinetics Since the beginning of this century, when the first crystal growth mechanism was proposed,194 the importance of the interfacial supercool ing has been realized as all theoretical treatments of growth deal with the form of the relation between the growth velocity and the super cooling at the interface. The interfacial supercooling is also import ant in describing various aspects of solidification of undercooled melts such as, for example, morphological stability, microsegration, and growth anisotropy. Notwithstanding, most experimental investigations, as discussed earlier, disregard the essential role of the interfacial supercooling and deal with bulk kinetics. According to the present experimental results, as shown in Fig. 24, the (111) Ga growth kinetics as a function of the bulk supercooling can be represented by V = kb ATb2 where kb is a constant dependent upon the supercoolings range; for example, in the range of 30-500 nm/s, kb is determined as 22 (um/sC). In view of this parabolic relationship, and in the absence of inter facial supercooling measurements, not only the true growth modes of the interface would be hidden, but also a false agreement between theoret ical growth laws and the experiment could be readily concluded. This is 194 314 Program #5 continued CALL LEQT2F (A, M, N IA, B ,IDGT,WKAREA,IER) WRITE(6,70)(B(I),1=1,6) WRITE(6,80)(B(I),1=7,12) 1 FORMAT(5X,15) 10 FORMAT(2X,6F10.6) 70 FORMAT(5X,'COEFFICIENT A(N) ',/(10X.E13.6)) 80 FORMAT(5X,'COEFFICIENT B(N) ',/(10X.E13.6)) 1=1 C XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX C X SUBROUTINE TEMP USE EQS.(28) AND (29) TO CALCULATE THE X C X TEMPERATURE DISTRIBUTION IN BOTH SOLID AND LIQUID REGION X c xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx CALL TEMP(B,I,TS,Z,TB,ASRI) 1=2 CALL TEMP(B,I,TL,Z,TB,ASRI) WRITE(6,90)((TS(I,J),1=1,10),J=1,100) WRITE(6,100)((TL(I,J),1=1,10),J=1,100) 1100 CONTINUE IF(IDX.LT.ICASE)GO TO 999 1000 CONTINUE 90 FORMAT(5X,'TEMPERATURE DISTRIBUTION',/7X,'SOLID REGION',//(5X,10 IE.12.5)) 100 FORMAT(/7X,'LIQUID REGION',//(5X,10E12.5)) STOP END 46 changes in os occurred. Evidence about roughening also exists for several solvent-solute combinations during solution growth.133 Additional information about the roughening transition concept comes from experimental studies on the equilibrium shape of microscopic crystals. This topic is briefly reviewed in the next section. Equilibrium Crystal Shape (ESC) The dynamic behavior of the roughening transition can also be understood from the picture given from the theory of the evolution of the equilibrium crystal shape (ECS). In principle, the ECS is a geomet rical expression of interfacial thermodynamics. The dependence of the interfacial free energy (per unit area) on the interfacial orientation n determines r(T,n), where r is the distance from the center of the crys tal in the direction of of a crystal in two-phase coexistence.1415 At T = 0, the crystal is completely faceted. 134" As T increases, facets get smaller and each facet disappears at its roughening temperature Tj^(). Finally, at high T, the ECS becomes completely rounded, unless, of course, the crystal first melts. As discussed earlier, facets on the ECS are represented with cusps in the Wulff plot, which, in turn, are related to nonzero free energy per unit length necessary to create a step on the facet;135 the step free energy also vanishes at Tj^(n), where the corresponding facets disappear. Below TR facets and curved areas on the crystal meet at edges with or without slope discontinuity (i.e. smooth or sharp); the former corresponds to first-order phase transition and the latter to second-order transitions. The edges are the * It is generally believed that macroscopic crystals at T = 0 are facet ed; however, this claim that comes only from quantum crystals still remains controversial.134 289 Table A-4. Thermal Property Values Used in Heat Transfer Calculations Ref. // Liquid Ga thermal conductivity, cal/seccm*C 00 o ii 348 density, g/cc PL = 6.09 346 thermal diffusivity, cm^/s kl = .1376 415 heat of fusion, cal/g L = 19.15 347 Solid Ga thermal conductivity, cal/seccm*C (111) K = .0978 348 (001) KS s = .0382 348 density, g/cc ps = 5.9 345 thermal diffusivity, cm^/s Coolant a) Water (111) (001) < s K s = .1864 = .0728 416 thermal conductivity, cal/seccm*C Kb = .00145 417 viscosity, poise nb = .0089 417 specific heat, cal/g*C C P = .998 417 b) Water-Ethylene glycol solution thermal conductivity, cal/seccm*C 30%, Kb = .0012 418 40%, Kb = .00108 418 specific heat, cal/g*C 30%, C = .9 418 40%, cp p = .84 418 viscosity, poise 30%, nh = .019 417 40%, nb = .025 417 Capillary Tube thermal conductivity, cal/sec*cm*C k 8 = .0025 417 270 The orthorhombic structure was verified by Bradley,380 who showed that all three axes were different in length and gave more precise values of the atomic position parameters. Subsequent redeterminations of the lat tice constants and of the positional parameters by the use of more ex tensive diffraction studies were made.381-384 The results of the above mentioned works are summarized in Table A-3. It is realized from the lattice constants that the a and b axes are very nearly equal, with the c axis longer than the a and b axes. Not only is the cell almost tetragonal, but c/a is nearly ./3, so that it is also pseudohexagonal. Furthermore, the inequality c > a holds throughout the temperature range,383 in agreement with the thermal ex pansion coefficients reported385 as being in the ratio of 1:0.7:1.9 for the c:a:b axes. The Ga atoms form a network of regural hexagons parallel to the (010) plane at heights x = 0 and x = 1/2 and are distorted in the plane. In addition, the pseudohexagonality of the structure is revealed by an other set of atomic hexagons which are perpendicular to the a axis being buckled normal to their planes; the structure can be thought of as con sisting of a stacking of these distorted hexagonal close packed layers, as shown in Figs. A-l, A-2, and A-3, which show projections of the structure on the (100), (010), and (001) planes, respectively. A Ga atom has seven nearest neighbors, with the shortest Ga-Ga bond being considered as covalent or that the closest pair of atoms form Ga^ molecules.386 This would reduce the structure alternatively to four * In some of the recent work on Ga, the new setting Cmca has been used; however, in this work the old designation is used whenever reference is made to the crystal structure. APPENDIX V PRINTOUTS OF COMPUTER PROGRAMS 305 69 kinetics and of the step edge energy are not taken into account. How ever, it has been suggested119 that on S/V interfaces sharply polygoni- zed spirals may occur at low temperatures or for high "a" factor mater ials. Nonrounded spirals have been observed during growth of several materials,186187 as well as on Ga monocrystals during the present study. Most theoretical aspects of the spiral growth mechanism were first investigated by BCF in their classical paper,10 which presented a revo lutionary breakthrough in the field of crystal growth. Interestingly enough, although their theory assumes the existence of dislocations in the crystal, it does not depend critically on their concentration. The actual growth rate depends on the average distance (yQ) between the arms of the spiral steps far from the dislocation core. This was evaluated to be equal to 4irrc; later, a more rigorous treatment estimated it as 19rc.188 The curvature of the step at the dislocation core, where it is pinned, is assumed to be equal to the critical two-dimensional nucleus radius rc. On the other hand, for polygonized spirals, the width of the spiral steps is estimated186 to be in the range of 5rc to 9rc. According to the continuum approximation, the spiral winds up with a constant angular velocity to. Thus, for each turn, the step advances yQ in a time yQ/ue = 2tt/oj. Then the normal growth rate V is given as10 V = bw/2n = byQ/ue (39) where b is the step height (Burgers vector normal component). According to the BCF notation, from eq. (24) where yQ = 4Trrc ~ 4iTYe/KTa (here ye is the step edge energy per molecule), one gets the BCF law V = fv exp (- W/KT) (o^/o^) tanh (o^/a) (40) 311 Program #4 5 REM THIS PROGRAM READS THE SEEBECK EMF OR THE POTEN TIhL ACROSS THE SAMPLE (DEPENDING WHETHER OR NOT CUR RENT PASSES THROUGH THE SAMPLE) WITH THE KEITHLEY IS 1 -NANOL'QLTMETER <20 mV RANGE). 10 DIM AS<20), BS<20),A<1000),C<1000) 15 ZS = CHRS <26):BS = "R2X" 20 PRINT "TAKE DATA(1) OR SAVE DATA < 2 ) ";: INPUT K 30 IF K = 1 THEN GOTO 70 40 IF K = 2 THEN GOTO 130 50 GOTO 20 65 REM FOLLOWING THE DAT* ARE RETREIVED FROM THE K-18 1 70 PR# 3 SO IN# 3 90 PRINT "RA" 100 PRINT "WTX" ; 2% ;AS 105 PRINT "LF1" 110 PRINT "RDE" ; ZS ; : INPUT "";AS 112 REM NEXT THE DATA ARE PRINTED ON THE APPLE lie SC REEN 113 PRINT UT" 115 PR# 0 117 IN# 0 130 NUM = NUM + 1 140 PRINT NUM.A* 145 A(NUM) = UAL ( MIDS (AS,5,15)) 150 OA = PEEK ( 1 2 S 7) 160 IF OA > 127 THEN GOTO 20 170 GOTO 70 176 REM NEXT THE DATA ARE PRINTED 180 PR# 1 135 FOR I = 1 TO NUM 187 C
190 PRINT CHRS (9) ;"SON" ;(I) ,A
195 NEXT I200 PR# 0 205 NUM = 0 210 GOTO 20 131 Table 3. Seebeck Coefficients (Ss^) of the S/L Ga Interfaces and Offset Thermal emf's for Several of the Used Samples. Sample Interface Ssi, MV/C Eoff, MV A-1 (111) 1.822 .086 B-l (111) 1.84 .286 B-2 (111) 1.901 .63 B-3 (111) 1.906 .17 Cl -1 (111) 1.89 .35 C-2 (111) 1.8805 -.035 D-l (111) 1.78 -.2 D-2 (111) 1.792 .712 E-l (111) 1.874 -.208 F-3 (111) 1.909 .413 G-l (111) 1.886 .52 H-2 (001) 2.107 .932 K-l (001) 2.218 .15 D-3 (001) 2.187 .299 L-l (001) 2.22 -.071 M-2 (001) 2.171 .43 N-l (001) 2.3 .121 K-2 (001) 2.43 .592 C-3 (001) 2.45 -.43 L-2 (001) 2.47 .632 140 All measurements would stop when the interface had reached the top of the observation bath. Then, the interface was melted back all the way out of the observation bath and the procedure was repeated at a dif ferent bulk supercooling. Experimental Procedure for the Doped Ga The Ga-In alloys were prepared by mixing high purity Ga (99.99999% Ga) and In (99.999% In);" a desired amount of In in the form of grind ings, weighed to four decimal places, was added to the as received poly ethylene bag that contained the 25g Ga ingot. After the bag was re sealed, the ingot was melted by the heating lamp, as described earlier; liquid Ga at Tm can dissolve up to 30 wt% of In. (The Ga-In system is described in more detail in Appendix II.) Consequently, a capillary was filled with the doped liquid with a procedure similar to that of the pure Ga. The capillary was seeded for the (111) interface. The sample was initially solidified rapidly, at a rate of about .5-1 cm/s in order to prevent macrosegregation across the sample. The two ends of the sample were then melted and connected to the electrical circuit, as des cribed earlier. The unused portion of the alloy was solidified and was used for chemical analysis. The analysis of the alloys as well as the intended compositions are given in Table 5. The preliminary procedure before the growth kinetics measurements was the same as that of the pure Ga. The experimentally determined Seebeck coefficients for the two compositions used are given in Table 6. Note that because of the effect of In on the Seebeck coefficient, these * As indicated by the supplier, AESAR Johnson Mathey, Inc., N.J. 334 319. J. E. Nye, Physical Properties of Crystals, Their Representation by Tensors and Matrices (Oxford University Press, London, 1976), p. 215. 320. C. A. Domenicali, Phys. Rev., 92 (1953) 293. 321. J. Tauc, Photo and Thermoelectric Effects in Semiconductors (Pergamon, New York, 1962), p. 179. 322. For the liquid Ga: a) M. C. Bellissent-Funel, R. Bellissent, andG. Taurand, J. Phys. F: Metal Phys., 11 (1981) 139. b) M. Pokorny and H. U. Astrdm, J. Phys. F: Metal Phys., 6 (1976) 559. For single crystals of Ga: c) M. Olsen-Bar and R. W. Powell, Proc. Roy. Soc. (London), A209 (1951) 542. d) M. Yaqub and J. F. Cochran, Phys. Rev. 137 (1965) A1182; 140 (1965) A2174. For supercooled liquid and single crystals: e) R. W. Powell, Proc. Roy Soc. (London), A209 (1951) 525. 323. D. R. Hamilton and R. G. Seidensticker, J. Appl. Phys., 31 (1960) 1165; 34 (1963) 1450. 324. H. N. Fletcher, J. Cryst. Growth, 35 (1976) 39. 325. J. M. R. Cotterill, J. Cryst. Growth, 40 (1980) 582. 326. K. Kamada, Progr. Cryst. Growth Charact., 3 (1981) 309. 327. J. R. Owen and E. A. D. White, J. Cryst. Growth, 42 (1977) 449. 328. P. Rudolfh, P. Filie, Ch. Genzel, and T. Boeck, Crystal Res. and Technol., 19 (1984) 1073. 329. S. Larsson, L. Broman, C. Raxbergh, and A. Lodding, Z. Naturforsch, 25a (1970) 1472. 330. V. P. Skripov. G. T. Butorin, and V. P. Koverda, Fiz. Met. Metal loved., 31 (1971) 790. 331. Y. Miyazawa and G. M. Pound, J. Cryst. Growth, 23 (1974) 45. 332. a) M. Volmer and O. Z. Schmidt, Z. Phys. Chem. (Leipzig), 85 (1937) 467. b) M. P. Dokhov, S. N. Zadumkin, and A. A. Karashaev, Russ. J. Phys. Chem., 45 (1971) 1061. c) H. Wenzl, A. Fattah, D. Gustin, M. Mihelcic, and W. Uelhoff, J. Cryst. Growth, 43 (1978) 607. 333. a) S. S. Dshandzhgava, E. F. Sidokhin, 0. V. Utenkova, and G. V. Shcheberdinskii, Metallofizika, 4 (1982) 118. b) A. C. Carter and C. G. Wilson, Brit. J. Appl. Phys., 1 (1968) 515. 334. M. Elwenspoek and J. P. van der Eerden, J. Phys. A, to be pub lished; see also: V. V. Podolinski, J. Cryst. Growth, 46 (1979) 511. 77 For melt growth, however, the arrival rate strongly depends on the structure of the liquid at the interface, which is not known in detail. Therefore, these models cannot treat the S/L continuous growth kinetics properly. Some general features revealed from these models are dis cussed next to complete this review. All MC calculations for rough interfaces indicate linear growth kinetics. The calculated growth rates are smaller than those of the W-F law, eq. (44). This is understood since the latter assumes f = 1. Interestingly enough, the simulations show that some growth anisotropy exists even for rough interfaces. For example, for growth of Si from the melt, MC simulations predicted205 that there is a slight difference in growth rates for the rough (100) and (110) interfaces. The observed anisotropy is rather weak as compared to that for smooth interfaces, but it is still predicted to be inversely proportional to the fraction of nearest neighbors of an atom at the interface (Â£ factor). Nevertheless, true experimental evidence regarding orientation dependent continuous growth is lacking. If there is such a dependence, the corresponding form of the linear law would then be V = Kc(n) AT (47) This is illustrated by examining the prefactor of AT in eq. (46). Note that the only orientation dependent parameter is (a), so that the growth rate has to be normalized by the interplanar spacing first to further check for any anisotropy effect. If there is any anisotropy, it could only relate to the diffusion coefficient D, otherwise to be correct, and, therefore, to the liquid structure within the interfacial region. At present, the author does not know of any studies that show such 147 the supercooled liquid; for example, no motion was detected at 40X mag nification (e.g. a movement of the interface by a distance of about 5-10 pm) when the interface was held at 1.5C below the melting point for about 72 hours. On the other hand, the motionless interface would immediately start to move rapidly when the capillary tube was bent or twisted; frequently during this action several other facets moving at different rates would also form at the interface. Some of the facets (the faster moving ones) would eventually grow out of the interface, leaving only {111} interface(s). When more than one {111} facets were left, they would move one at a time for several seconds; if only one (ill) facet was left, the interface would move in a steady state until it would become stationary again. On many occasions, a similar sudden motion of the stationary (ill) interface was also observed after chang ing the water bath temperature abruptly, e.g. from 1.4C supercooling to 0.5C or after suddenly changing the water flow rate, which, in turn, caused strong vibrations of the glass capillary tube. At supercoolings larger than about 1.5C, the undisturbed (111) interface moved parallel to itself at a constant rate that was strongly dependent on the bulk supercooling. Moreover, similar to the growth behavior at lower supercoolings, disturbing the crystal by mechanical or thermal means caused the interface motion to increase abruptly and other facets (mostly {111} and {001}) to appear at the interface. The inter face moved at the increased rate for a few seconds after which the rate abruptly dropped to its previous undisturbed value. As indicated by the work of Pennington et al." and Abbaschian and Ravitz,2 and as it will become apparent later, the growth of the disturbed interface corresponds 56 where D is the liquid self-diffusion coefficient and R stant. Cahn et al.25 have corrected eq. (20) by introduc enological parameter 3 and the g factor as 1/2, DLAT ug = 3(2 + g ) hRTT is the gas con ing the phenom- (21) m Here 3 corrects for orientation and structural factors; it principally relates the liquid self-diffusion coefficient to interfacial transport, which will be considered next. 3 is expected to be larger than 1 for symmetrical molecules (i.e. molecularly simple liquids for which "the molecules are either single atoms or delineate a figure with a regular polyhedral shape"153) and less or equal to 1 for asymmetric molecules. In spite of these corrections, the concluding remark from eqs. (20) and (21) is that u0 increases proportionally with the supercooling at the interface. When the step is treated as curved, then the edge velocity is de rived as17 u = ue (1 rc/r) (22) where r is the radius of curvature. In accord with eq. (22), the edge of a step with the curvature of the critical nucleus is likely to remain immobile since u = 0. If one accounts for surface diffusion, ue is given according to the more refined treatment of BCF10 as ue = 2oxsv exp (- W/KT) (23) where o is the supersaturation, xg is the mean diffusion length, v is the atomic frequency (v 10^ sec ^), and W is the evaporation energy. For parallel steps separated by a distance yQ, the edge velocity is derived as 74 S/L Figure 8 Free energy of an atom near the S/L interface. and Q are the activation energies for movement in the liquid and the solid, respectively. is the energy required to transfer an atom from the liquid to the solid across the S/L interface. 275 such molecules per unit cell, sitting symmetrically on the a-c (010) plane at angles of about 17 to the (001] direction; the spacing of their planes being b/2. From a geometrical point of view, the pseudo- hexagonality of the structure along the c axis is also revealed as a packing of these molecules into an FCC structure which has been pulled out in the two directions to accommodate the elongated shape of these molecules. These strongest bonds join the rumpled hexagonal layers of Ga atoms with the bonds in the layers being considerably weaker in an order of approximately 1/3. In particular, the assigned bond numbers are 1.21, 0.43, 0.38, and 0.31 to the four respective kinds of short distance.387 It has been indicated that the tendency of Ga to form dia tomic molecules may explain its low melting point; presumably it could melt into diatomic molecules.388 The complexity of the structure as regards axial ratios, nearest neighbors, and covalency effects has been theoretically tested from various points of view,389 and it was indeed found that the observed qualitative features of the structure "make sense." The covalency was evaluated to amount to about 27 Kcal/g-atom. The speculated existence of covalent bonding in the crystal is also intensified by the plastic deformation behavior and anisotropic mechan ical properties of Ga single crystals.390-392 Slip in Ga at room temp erature is confined to these systems: (001) [010], (102) [010], and (011) [Oil].390 Slip takes most easily on a (001) plane because it is among the very few "flat" low indices planes in the unit cell with a high density of atoms. However, as a slip direction, the [010] is always observed57 on the expense of the equally probable, from atom 51 allow for a reliable determination of critical parameters linked to the transition. There are a few growth kinetics studies which provide a clue regarding the transition from lateral to continuous growth; these will be reviewed next rather extensively due to the importance of the kinetic roughening in this study. A faceted (spiky) to non-faceted (smooth spherulitic) transition was observed for three high melting entropy (L/KTm ~ 6-7) organic sub stances, salol, thymol, and O-terphenyl.1*7 The transition that took place at bulk supercoolings ranging from 30-50C for these materials was shown to be of reversible character; it also occurred at temperatures below the temperature of maximum growth rate/' An attempt to rational ize the behavior of all three materials in accord with the predictions of the MC simulation results117 was not successful. The difference in the transition temperatures (20, 13, and -10C for the O-terphenyl, salol, and thymol, respectively) were attributed to the dissimilar crys tal structures and bonding. Morphological changes corresponding to changes from faceted to non- faceted growth form together with growth kinetics have been reported11*8 for the transformation I-III in cyclohexanol with increasing supercool ing. The morphological transition was associated with the change in growth kinetics, as indicated by a non-linear to linear transition of the logarithm of the growth rates, normalized by the reverse reaction term [1 exp(- AGV/KT)], as a function of 1/T (i.e. log(V/l- exp(- AGV/KT)) vs. 1/T plot); the linear kinetics (continuous growth) * This feature will be further explained in the continuous growth sec tion. of exp(-l/AT) and proportional to the interfacial area at small super coolings. The dislocation-assisted growth rates are proportional to AT2tanh(l/AT), or approximately to ATn at low supercoolings, with n around 1.7 and 1.9 for the two interfaces, respectively. The classical two-dimensional nucleation and spiral growth theories inadequately des cribe the results quantitatively. This is because of assumptions treat ing the interfacial atomic migration by bulk diffusion and the step edge energy as independent of supercooling. A lateral growth model removing these assumptions is given which describes the growth kinetics over the whole experimental range. Furthermore, the results show that the fac eted interfaces become "kinetically rough" as the supercooling exceeds a critical limit, beyond which the step edge free energy becomes negli gible. The faceted-nonfaceted transition temperature depends on the orientation and perfection of the interface. Above the roughening supercooling, dislocations do not affect the growth rate, and the rate becomes linearly dependent on the supercooling. The In-doped Ga experiments show the effects of impurities and microsegregation on the growth kinetics, whose magnitude is also depend ent on whether the growth direction is parallel or antiparallel to the gravity vector. The latter is attributed to the effects of different connective modes, thermal versus solutal, on the solute rich layer ahead of the interface. xxi 319 16. L. E. Murr, Interfacial Phenomena in Metals and Alloys (Addison- Wesley, Reading, MA, 1975), p. 165. 17. M. C. Flemmings, Solidification Processing (McGraw-Hill, New York, 1974). 18. M. Ohara and R. C. Reid, Modeling Crystal Growth Rates from Solution (Prentice Hall, Englewood Cliffs, NJ, 1973). 19. J. Christian, The Theory of the Transformations in Metals (Pergamon, New York, 1965). 20. J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1 (Dover, New York, 1961). 21. B. Mutaftschiev, in: Interfacial Aspects of Phase Transformations (D. Reidel Publ. Co., Dordrect, Netherlands, 1982), p. 63. 22. D. A. Porter and K. E. Easterling, Phase Transformations in Metal Alloys (Van Nostrand, UK, 1981), p. 110. 23. D. C. Mattis, The Theory of Magnetism (Harper-Row, New York, 1965); C. Kittel and J. K. Golt, Solid State Phys., 28 (1958) 258. 24. J. W. Cahn and J. E. Hillard, J. Chem. Phys., 28 (1958) 258. 25. J. W. Cahn, W. B. Hillig, and G. W. Sears, Acta Met., 12 (1964) 1421. 26. K. A. Jackson, D. R. Uhlmann, and J. D. Hunt, J. Cryst. Growth, 1 (1967) 1. 27. D. P. Woodruff, The Solid-Liquid Interface (Cambridge Univ., London, 1973). 28. C. Domb and M. S. Green, Phase Transitions and Critical Phenomena (J. Wiley, New York, 1972). 29. J. E. Hillard and J. W. Cahn, Acta Met., 6 (1958) 772. 30. B. Widom, J. Chem. Phys., 43 (1965) 3892. 31. J. Meunier and D. Langevin, J. Physique Lett., 43 (1982) 185. 32. J. W. Cahn and J. E. Hillard, J. Chem. Phys., 31 (1959) 688. 33. J. W. Cahn, J. Chem. Phys., 42 (1965) 93. 34. J. C. Brice, The Growth of Crystals from Liquids (North-Holland, Amsterdam, 1973), p. 117. 35. T. L. Hill, An Introduction to Statistical Thermodynamics (Addison- Wesley, Reading, MA, 1960). 22 this region, F varies monotonically so that the interface can move con tinuously. On the other hand, in region A the interface must advance by the lateral growth mechanism. Moreover, depending on the y value, a material might undergo a transition in the growth kinetics at a measur able supercooling. For example, if y = 2, the transition from region A to region B should take place at an undercooling of about .05 Tm (assum ing that L/KTm ~ 1, which is the case for the majority of metals). How ever, to make any predictions, W has to be evaluated; this is a diffi cult problem since an estimate of the Esj^ values requires a knowledge of the "interfacial region" a-priori. It is customarily assumed that ESÂ£ = EÂ£Â£, which leads to a relation between W and the heat of fusion, L. But this approximation, the incorrectness of which is discussed elsewhere, leads, for example, to negative values of oSÂ£ for pure metals.38 Never theless, if this assumption is accepted for the moment, it will be shown that Temkin's model stands somehow between those of Cahn's and Jackson's (discussed next). The "a" factor model: roughness of the interface Before discussing the "a" factor theory,89 the statistical mechan ics point of view of the structure of the interface is briefly des cribed. The interfacial structure is calculated by the use of a parti tion function for the co-operative phenomena in a two-dimensional lat tice. Indeed, the change of energy accompanying attachment or detach ment of a molecule to or from a lattice site on the crystal surface can not be independent of whether the neighboring sites are occupied or not. A large number of models39 have been developed under the assumptions i) 38 Burton et al.10 considered a simple cubic crystal (100) surface with (p/2 nearest neighbor interaction energy per atom. Proving that this two level problem corresponds exactly to the Ising model, a phase transition is expected at TQ. This transition then is related to the roughening of the interface ("surface melting") and the temperature at which it takes place is related to the interaction energy as KT exp (- ) = V2 1, or = .57 2hTR cp where is the roughening temperature. For a triangular lattice, e.g. (Ill) f.c.c. face KT^/tp is approximately .91. The authors also consid ered the transition for higher (than two) level models of the interface using Bethe's approximation/' It was shown that, with increasing the number of levels, the calculated Tr decreases substantially, but remains practically the same for a larger number of levels. Although this study did not rigorously prove the existence of the roughening transition, L7 it gave a qualitative understanding of the phenomenon and introduced its influence on the growth kinetics and interfacial structure. The latter, because of its importance, motivated in turn a large number of theoret ical works108 during the last two decades. This upsurge in interest about interfacial roughening brought new insight in the nature of the transition and proved5 9 1 0 9 110 its existence from a theoretical point of view. In principle, these studies use mathematical transformations to relate approximate models of the interface to other systems, such as * Exact treatments of phase transitions can be discussed only for special systems and two dimensions, as discussed previously. For more than two dimensions, approximate theories have to be considered. Among them are the mean field, Bethe, and low-high temperature expan sions methods. 72 For the parabolic law case, yQ is much greater than xs and the reverse is true for the linear law. In between these two extreme cases, i.e. at intermediate supersaturations, the growth rates are expected to fall in a kinetics mode faster than linear but slower than parabolic; such a mode could be, for example, a power law, V <* ATn, with n such that 1 < n < 2. For growth from the melt, the BCF rate equation can be rewritten as V = N AT2 tanh (P/AT) (42) where N and P are constants. Equation (42) reduces to a parabolic or to a linear growth when the ratio P/AT is far less or greater, respective ly, than one. Lateral growth kinetics at high supercoolings According to the classical LG theory, the step edge free energy is assumed to be constant with respect to supercooling, regardless of poss ible kinetics roughening effects on the interfacial structure at high AT's. Based on a constant oe value, the only change in the 2DNG growth kinetics with AT is expected when the exponent AG"/3KT (see eq. (37)) is close to unity. In this range, the rate is nearly linear (~ATn, n = 5/6). An extrapolation to zero growth rates from this range intersects the AT axis to the right of the threshold supercooling for 2DN growth. For SDG kinetics, based on the parabolic law (eq. (40)), no changes in the kinetics are expected at high AT's. However, the BCF law (eq. * For detailed relations between supersaturation and supercooling see Appendix VI. 86 The above mentioned conclusions are rather qualitative, at least theoretically, in the sense that they do not provide a background to make any predictions for a given system. The complex and usually con flicting effects of the impurity on nucleation and growth mechanisms and the poorly understood adsorption phenomena (between the solute, the sol vent, and the interfacial sites) make the problem quite difficult to treat analytically. Furthermore, this important and yet complicated problem has not been investigated in detail; the recent theoretical treatments on this subject mainly treat the diffusion controlled to dif fusionless solidification210211 transition at rapid rates rather than the "impure" interfacial kinetics. Although there are several investi gations concerned with the impurity effects in crystal growth from sol ution,212-214 the number of experiments dealing with growth from the melt is very limited. The effects of small additions of Al,209 In,215 Ag and Cu,63 and other impurities (Sn, Zn)216 on the growth kinetics of Ga has been studied. However, only the Ga-Al study209 is complete in the sense that the effect of solute build-up on the growth rate was reported. In all cases but those of Ag and 20 ppm Al addition, it was reported that the effect of solute was to decrease the growth rate progressively as the percent concentration increased. It was also observed63209 that the faceted interface would occasionally break down as a result of excessive interfacial build-up. Growth in the presence of Al209 and In215 occurred by the 2DNG mechanism with the major effect of the addition be lieved to be on ue rather than on the step edge free energy. At higher concentrations and supercoolings209215 (i.e. 1000 ppm Al, .1 wt% In, Figure 9 Interfacial growth kinetics and theoretical growth rate equations 79 Figure 10 Transition from lateral to continuous growth according to the diffuse interface theory;25 nQ is the melt viscosity at Tm 81 Figure 11 Heat and mass transport effects at the S/L interface. a) Temperature profile with distance from the S/L interface during growth from the melt and from solution. b) Concentration profile with distance from the interface during solution growth . . 90 Figure 12 Bulk growth kinetics of Ni in undercooled melt. After Ref. (201) 92 Figure 13 Solute redistribution as a function of distance solid ified during unidirectional solidification with no con vection 105 Figure 14 Crystal growth configurations, a) Upward growth with negative G^. b) Downward growth with positive G^. In both cases the density of the solute is higher than the density of the solvent 109 Figure 15 Experimental set-up 118 Figure 16 Gallium monocrystal, X 20 124 Figure 17 Thermoelectric circuits. a) Seebeck open circuit, b) Seebeck open circuit with two S/L interfaces 126 Figure 18 The Seebeck emf as a function of temperature for the (111) S/L interface 132 Figure 19 Seebeck emf of an (001) S/L Ga interface compared with the bulk temperature 133 Figure 20 Seebeck emf as recorded during unconstrained growth of a Ga S/L (111) interface compared with the bulk supercool ing; the abrupt peaks (D) show the emergence of disloca tions at the interface, as well as the interactive effects of interfacial kinetics and heat transfer 135 Figure 21 Experimental vs. calculated values of the resistance change per unit solidified length along the [111] orientation vs. temperature 139 Figure 22 Seebeck emf vs. bulk temperature as affected by dis locations) and interfacial breakdown, recording during growth of In-doped Ga 144 xiv Figure 32 The logarithm of the (001) growth rates versus the logarithm of the interface supercooling; dashed line represents the SDG rate equation, as given in Table 167 43 energy goes to zero as T - T^, vanishing in an exponential manner. These points have been supported and/or confirmed by several MC simula tions results,119 in particular, for the SOS model. As may already be surmised, the roughening transition is also ex pected to take place for a S/L interface. Indeed, its concept has been applied, for example, in the "a" factor model;8,9 the "a" factor is in versely related to the roughening transition temperature T^, assuming that the nearest neighbor interactions (cp) are related to the heat of fusion. Such an assumption is true for the S/V interface where only solid-solid interactions are considered (Ess = cp, Esv = Ew ~ 0). Then, for the Kossel crystal,120'' Lv ~ 3cp where Lv is the heat of evaporation. Unfortunately, however, for the S/L interface all kinds of bonds (Ess, ESÂ£, E^) are significant enough to be neglected so that one could not assume a model that accounts only vertical or lateral (with respect to the interface plane) bonds. Assumptions such as E^ = ESÂ£ cannot be justified, either. Several ways have been proposed121 to calculate Es^. Their accuracy, however, is limited since both Esj and E^, to a lesser extent, depend on the actual properties of the interfacial region which, in reality, also varies locally. Nevertheless, such information is likely to be available only from molecular dynamics simulations at the present.4 Quantitative experimental studies of the roughening transition are rare, and only a few crystals are known to exhibit roughening. Because of the reversible character of the transition, it is necessary to study * As Kossel crystal120 is considered a stacking of molecules in a primi tive cubic lattice, for which only nearest neighbor interactions are taken into account. 148 to the dislocation-assisted growth/'323-326 whereas that of the undis turbed interface belongs to one of the 2D nucleation and growth mechan isms. The latter is termed as dislocation-free growth in this study, as contrasted with dislocation-assisted growth. The dislocation-free and dislocation-assisted growth rates are plotted on a linear scale versus the interface supercooling in Fig. 23. As can be seen in the supercooling range of about 1.5-3.5C, one clearly distinguishes two growth rates for the same AT; one belonging to the undisturbed samples, the other belonging to the disturbed samples. At lower than 1.5C (AT), the data points belong only to the latter. As indicated earlier, below this supercooling the (111) interface remained practically stationary "indefinitely"; it would advance only when the crystal was disturbed by mechanical or thermal means. The existence of the threshold supercooling and the functional relationship between the growth rates of the undisturbed samples and the interfacial supercool ings, as discussed below, are indicative of 2D nucleation-assisted growth, whereas those of the disturbed samples correspond to disloca tion-assisted growth. At higher than about 3.5C supercoolings, the two growth rates become approximately similar, and it is rather difficult to clearly differentiate them. At these high supercoolings, thermally in duced dislocations also emerge and grow out of the interface very rapidly, sometimes faster than the measurement rate of 48-25 per second. Therefore, the measured rates in this range are sometimes the mixture of This growth mechanism refers only to the classical SDG mechanism and not to any other growth modes proposed for imperfect interfaces323 and/or associated with dislocations in the bulk liquid and solid.324-326 228 fusion, respectively. Note that even the largest value of xg = 81 is still larger than the spiral step spacing which is estimated to be about 280 A at 4C supercooling. The reason why ATC increases from 3 to 10 at high supercoolings could be due either to a different distribution of dislocations (i.e. on S) at higher supercoo1ings or to a 2DN contribu tion to the spiral growth process. On the other hand, is evaluated as 746 Mm/s-C which, together with the above mentioned value of ATC (e.g. ATC = 0), implies that the kinetic coefficient of the step lat eral spreading rate is about .75 cm/s. The latter value in turn indi- cates that is about 3 x 10"' cm/s (or about 10_o cm-/s for ATC = 3), which agrees with the earlier estimates of D^. Extending the calcula tions for the (001) dislocation-assisted growth data shown in Fig. 56, using eqs. (91) and (93), ATC and are evaluated as 9 and 1840, re spectively. These values indicate that xsS = 47 A and D,- =6 x 10 ' o , cm*/s. All of the above mentioned parameters, besides the fact that they are reasonable as far as numerical values are concerned, they are "con sistent" between interfaces and growth mechanisms, and most importantly, they point out consistently that a) the growth rate equations (92) and (93) describe the results well, b) < D, and c) oe is a function of the supercooling. These conclusions will be further strengthened later, where it is shown that several proposed "hypotheses" for explaining the high growth rates kinetics fail to describe the present results. Next, the kinetic roughening of the interface is discussed. classical regime 81 Figure 10 Transition from lateral to continuous to the diffuse interface theory;25 r| viscosity at T . m growth according is the melt 207 (001) interface (see detailed discussion in Appendix I) were calculated to be 59.4 and 44.8 x 10 ergs/cm, respectively; in the calculations h is 2.9 and 3.8 for the (111) and (001) interfaces, respectively, while Vm is equal to 11.8 cm'Vmole. The experimentally found oe values per unit area of the edge of the step (i.e. oe/h) of 20.3 and 11.7 ergs/cm^ are much smaller than the reported oSÂ£ values of 40,330 56,60 and 673 3 1 ergs/cm^ from "homogeneous" nucleation experiments of Ga, and 52s7 from theoretical calculations. It should be noted, as discussed earlier, that the surface energy per unit area of the edge of the step is not necessarily the same as the S/L interfacial energies. Furthermore, the values obtained from homogeneous nucleation experiments have been sub jected to broad criticism, particularly in the existence of poor wetting of the crystal by the melt which has been reported for Ga.332 According to the classical 2DNG models, the ratio of the slopes of the mononuclear and polynuclear kinetics in a log(V) vs. l/AT plot should be either three for the case of uniform 2D cluster spreading, or two for the non-isotropic case for which the cluster area increases lin early with time (see eqs. (33) and (34), respectively). However, for both growth directions this ratio was found to be between these limits, as 2.4 and 2.6 for the (111) and (001) interfaces, respectively, indi cating that the growth of clusters is controlled simultaneously by the attachment kinetics and the arrival flux of the atoms. For such a case, the growth rate is given as V c'h (Jue^)^^. Indeed, the experimental ratios of 2.4 and 2.6 are close to the value of 2.5, which is the pre dicted value for the PNG model (see eq. (35)) discussed earlier. Although the (001) interface growth behavior was similar to that of the (111) interface, the former showed a unique feature that the (001) 120 100 80 (001) O Experimental Analytical Numerical U,r, = .04 K O UD 60 40 20 10 15 20 25 Vrj x 10'? cm2/8ec Figure A-1J Comparison between the (001) Experimental results (o) and the Model ( Analytical, Numerical) calculations as a function of Vr. for given growth conditions. 1 298 106 characteristic distance about D/V. The solute concentrations in the liquid ahead of the interface for small values of k are given as285286 CL = Co{l + exp(- ^-)} (61) Cl = Co{1 k k[1 exP^' exp( ^-) + 1} (62) for the steady and the initial transient regions, respectively. Here x' is the distance from the interface into the liquid and x is the distance from the onset of growth. In both regions the solute profile decays within a distance D/V from the interface. However, since there is usually convection in the melt, as discussed later, the solute transport is purely diffusive only within a distance 6 from the interface; beyond this distance the liquid is mixed by convection flows. Under such con ditions, the distribution coefficient kQ is replaced by an effective distribution coefficient, kgff, defined as287 k k = 2 (63) k + (1 k ) exp(- ) o o D Note that the equilibrium coefficient kQ, is usually used in calculating ^ef f Convection Macroscopic mass and heat transport play a central role in crystal growth processes. Fluid flow is beneficial to crystal growth, by reduc ing the diffusional barriers for interfacial heat and matter transport, provided that the flow is uniform (steady state). However, because of the complex geometries and boundary conditions, as well as the adverse vertical and nonvertical thermal fields encountered in crystal growth, 274 'z' coordinates O .15 .35 .65 .85 Figure A-3 The (Â¡allium structure projected on tne (001) plane: double lines indicate the covalent bond and dashed lines outline the unit cell. Figure 23 Dislocation-free and Dislocation-assisted growth rates of the (111) interface as a function of the interface supercooling; dashed curves represent the 2DNG and SDG rate equations as given in Table 7 1A9 Figure 2 A Growth rates of the (111) interface as a function of the interfacial and the bulk supercooling 151 Figure 25 The logarithm of the (111) growth rates plotted as a function of the logarithm of the interfacial and bulk supercoolings; the line represents the SDG rate equation given in Table 7 152 Figure 26 The logarithm of the (111) growth rates versus the reciprocal of the interfacial supercooling; A is the S/L interfacial area 153 Figure 27 Dislocation-free (111) low growth rates versus the inter facial supercooling for A samples, two of each with the same capillary tube cross-section diameter 157 Figure 28 The logarithm of the MNG (111) growth rates normalized for the S/L interfacial area plotted versus the recip rocal of the interface supercooling 158 Figure 29 Polynuclear (ill) growth rates versus the reciprocal of the interface supercooling; solid line represents the PNG rate equation, as given in Table 7 160 Figure 30 Dislocation-assisted (111) growth rates versus the inter face supercooling; line represents the SDG rate equation, as given in Table 7 162 Figure 31 Dislocation-free and Dislocation-assisted growth rates of the (001) interface as a function of the interface supercooling; dashed curves represent the 2DNG and SDG rate equations, as given in Table 7 165 Figure 32 The logarithm of the (001) growth rates versus the log arithm of the interface supercooling; dashed line rep resents the SDG rate equation, as given in Table 7 167 Figure 33 Growth rates of the (001) and (111) interfaces as a function of the interfacial supercooling 168 Figure 3A The logarithm of the (001) growth rates versus the reciprocal of the interface supercooling 169 Figure 35 The logarithm of dislocation-free (001) growth rates versus the reciprocal of the interface supercooling for 10 samples; lines A and B represent the PNG rate equa tions, as given in Table 7 170 xv 248 assumed that the most preferred sites should be the most energetic ones, i.e. the kinks. At higher concentrations, adsorption should take place at less energetic sites. In any case, ue is expected to decrease either because of the reduction of kinks or because of the decrease in the step spreading process. For example, if a step tries to pass through two ad sorbed In molecules, this would be possible only if their distance d is less than 2rc. If this condition is satisfied, the step can bow out and pass the impurities. However, then its curvature will increase and, therefore, its velocity will decrease (see eq. (22)). Thus, the inter action of the In molecules with the 2DNG processes becomes more pro nounced at higher supercoolings, as shown in these experiments. For example, the decrease in the growth rate at supercoolings 1.8 and 2.3C accounts for up to about 28 and 50%, respectively. Increasing the In concentration to .12 wt% results in a more effective decrease of the growth rates of the (ill) doped Ga interface relative to those of the pure Ga, as shown in Figs. 44 and 45. The 2D nucleation rate further decreases as shown in Fig. 45. At growth rates higher than about 15 pm/s, the rates start deviating in the direction of faster growth rates, i.e. towards the pure Ga and Ga-.012 wt% In growth curves. The reasoning for this behavior is that, at these supercoolings and decreased values of oe, the critical nucleus becomes quite small so that the interface possibly starts roughening. By observing the crystal growth of the In-doped Ga samples, it was revealed that liquid rich bands were entrapped by the growth front, as shown in Fig. 38, which had faceted boundaries. This is believed to happen due to a non-uniform solute distribution across the S/L inter- 302 unstable were determined. The calculations are summarized in Figs. A-13 and A-14; Fig. A-13 is a plot of the growth rate versus the critical wavelength defined as . -K.G. K G . .-1 L L s s1 / 2 cr 4ttK t r m Note that for A > Acr the interface is unstable. On the other hand, Fig. A-14 is a linear plot of the stability R(w), given as 2 -(KtGt -KG) R(u>) = T IV m 2K as a function of the perturbation wavelength, A, and the growth rate. Note that here the interface is stable for conditions such that R(w) < 0. The calculations, as shown in these figures, were performed based on actual experimental data. The thermal fields within the sample were determined with the aid of the heat transfer model, which was discussed earlier in Appendix III. The analysis indicates that the S/L interface should be stable at growth rates up to about .8 cm/s if the perturbation wavelength is equal to the interface diameter. For smaller perturbations, the interface should be stable even at higher growth rates. Seebeck emf, pV Figure 18 The seebeck emf as a function of temperature for the (111) S/L interface. 132 261 dependent on the supercooling, which implies that the growth mode changes from lateral to normal. 9) The growth of the In-doped Ga, similar to the dislocation-free growth of pure Ga, takes place by the two-dimensional nucleation assisted mechanism. 10) The small additions of In reduce the growth rate of Ga but do not effect the growth mode. In as a dopant decreases the step edge free energy, but slows down the transport kinetics and decreases the lateral step spreading rate, particularly at high dopant levels. 11) The growth rate at a given bulk temperature decreases with dis tance solidified because of the solute build-up at the interface. 12) The faceted In-doped interface breaks down as growth proceeds, because of the solute enriched boundary layer at the interface. Upon breakdown, In-rich bands are entrapped by the advancing crystal. The frequency of these breakdowns and size of the bands increases and de creases, respectively, as the In concentration and the supercooling increase. 13) For a given In dopant level and bulk temperature, the growth rates in the direction parallel to the gravity vector were found to be higher than those in the antiparallel direction. Furthermore, in the parallel direction, the growth rate decayed, as a function of distance solidified at a slower rate and the frequency of interfacial breakdowns was less than that for the antiparallel growth direction. These differ ences are explained based on the convection effects in the interfacial solute boundary layer. 174 next chapter. At supercoolings less than 0.8C, the growth rates can also be correlated by a nearly parabolic equation. The rate equation, as evaluated from the regression analysis is given as V = 1640 AT1'93 (79) where V is the growth rate in pm/s. The coefficients of determination and of correlation for this analysis are .988 and .994, respectively. Growth at High Supercoolings, TRG Region The behavior of the (001) high supercoolings data is quite similar to that of the corresponding (ill) data. This is shown in Figs. 31 and 32, where the results indicate that as AT increases, the kinetics devi ate from both dislocation-assisted and dislocation-free kinetic laws shown as dashed curves and continuous lines in these figures. The devi ation from the low supercoolings laws is in the direction of faster rates at a given AT. The two growth modes (SDG and PNG) approach each other as AT increases and finally fall under the same kinetics range for supercoolings higher than about 1.75C. Above this supercooling, the growth rates increase very drastically with AT up to supercoolings of about 2C; still, at higher supercoolings than the latter value of AT, the relationship V(AT) becomes linear, as can be seen in Fig. 31. Al though this feature is also observed in the (ill) V vs. AT linear plot (Fig. 23), it seems more pronounced for the (001) growth data. Further more, the scatter of the (001) data in the range of the fastest growth rates is less than that of the (ill) interface. If a line is drawn from the origin of the V vs. AT plot through the data points with growth rates higher than 14000 pm/s, it results in a rate equation given as 238 Table 11. Calculated Values of g. Equation (38) Equation (94) Equation (49) Interface asi=67 as=40 asi=67 CTsi=40 sr67 asi>=40 (111) .09 .26 .12 .34 .025 .04 (001) .03 .085 .04 . 1 .013 .022 247 pm/s, the solute has no appreciable effect on the rates. At higher rates, however, the addition of the solute decreases the growth rates. Based on the experimental growth rate equations (eqs. (78) and (80)), the pre-exponential term for doped Ga is smaller than that for the MNG rate equation of the pure Ga (eq. (69)) by 1-2 orders of magnitude. In the PNG region, this difference (compare the pre-exponential terms of eqs. (79) and (81) versus eq. (70)) reduces to less than one order of magnitude. The solute effect on the pre-exponential term of the 2DNG rate equations might be due to the interaction of adsorbed solute atoms on the interface with the lateral spreading process, i.e. the edge spreading rate ue and/or the change of the kinetic factor Kn of the 2DN rate equation (eq. (28)). The edge free energy of the steps on the (111) Ga interface de creases slightly, about 3-4% by the addition of In. The In additions decrease ae, and thus the size of the critical nucleus, reducing the activation free energy of the 2DN process. Nevertheless, the overall nucleation rate seems to be decreased, as depicted from the lower growth rates, in comparison to those of the pure Ga in the mononuclear regime. Therefore, only a minute amount of an impurity is necessary to drastic ally decrease Kn, while its effect on ae is still negligible. This is because adsorption takes place mostly at the growth sites of the clus ters and decreases the molecular kinetics across the cluster/bulk inter face. Regarding the solute effect on the step edge velocity, one has to distinguish whether the In molecules adsorb separately on the surface or in the kinks and steps. Although such a distinction is rather imposs ible since the S/L interface cannot be investigated directly, it can be 126 Figure 17 Thermoelectric circuits. a) Seebeck open circuit. b) Seebeck open circuit with two S/L interfaces. 39 two-dimensional Coulomb gas, ferroelectrics, and the superfluid state, which are known to have a confirmed transition. As mentioned prev iously, it is out of the scope of this review to elucidate these studies, detailed discussion about which can be found in several reviews .107>11:L>112 At the present time, the debate about the roughening transition seems to be its universality class or whether or not the critical behav ior at the transition depends on the chosen microscopic model. Based on experiments, the physical quantities associated with the phase transi tion vary in manner |T-Tc|m when the critical temperature Tc is ap proached. The quantities such as p in the above relation that charac terize the phase transition are called critical exponents. They are inherent to the physical quantities considered and are supposed to take universal values (universality class) irrespective of the materials under consideration. For example, in ferromagnetism, one finds as T -* Tc (Curie temperature): susceptibility, x <= (T TC)T specific heat, C(T) <= (T Tc) a (T > Tc) Another important quantity in the critical region is the correla tion length, which is the average size of the ordered region at temper atures close to Tc. In magnetism, the ordered region (i.e. parallel spin region) becomes large at Tc, while in particle systems the size of the clusters of the particles become large at Tc. The correlation length also obeys the relation105 (T > Tc) Â£ { IT TCH |TC T| -v (T < T) (16) 295 133 Figure 19 Seebeck emf of an (001) S/L Ga interface compared with the bulk temperature Critical Wavelength, cm Growth Rate, cm/s Figure A-13 The critical wavelength A at the onset of the instability as a function of growth rate; hatched area indicates the possible combination of wavelengts and growth rates that might lead to an unstable growth front for the given sample size(i.d.=.028cm). 303 183 of the pure Ga. In the PNG region, the rates for the doped material are also slower; however, as the supercooling increases, the rates gradually increase and fall above the extrapolated regression line for these of the lower supercoolings. This behavior is intensified with increasing C^, as shown in Fig. 39, for the rates as a function of the distance solidified. The growth rate of the doped interface when growing parallel to the gravity vector, wersus the distance solidified, is shown in Fig. 40. Note that the rate decreases with distance from the onset of growth until the interface breaks, as for the interface growing upward. How ever, in comparing Figs. 37 and 40, it should be noted that at a given bulk supercooling, the doped interface becomes unstable less frequently when it grows downward; furthermore, for the latter growth direction, the initial growth rates are higher and seem to decrease less drastic ally with distance than those for upwards growth. The growth rates of the interface growing downwards versus the interface supercooling are given in Fig. 41. In this figure, similar to Fig. 39, the effect of the distance solidified in V is also shown by dotted lines. For the parallel g growth direction, the rates also closely follow those of the pure Ga, with the latter ones still being higher. However, in comparison with the rates of the interface moving upward, it is revealed that the latter are smaller at a given interface supercooling, as shown in Fig. 42. At the lower supercoolings, the two rates are comparable. However, as the AT increases, the rates of the 201 for the lowest growth rates (.5 2 x 10^ pni/s) within a time interval of about two seconds. Within the above mentioned range of growth rates, the interface velocities were also measured optically. A comparison be tween the latter rates and those determined via the potential drop using two different potentiometers and data acquisition programs (programs iil- 4, as presented in Appendix IV) is shown in Fig. 50. The agreement be tween the two is very satisfactory considering the fact that the rates are determined by two independent techniques. Finally, it should be noted that the values of I and At (see eq. (68)) were stable within .01% and .02 .5%, respectively. The standard deviation of the AR/A2. values at a given bulk supercooling never exceeded 5% from the mean. On the other hand, the current value had to be kept minimum to avoid any Peltier heating (or cooling), as well as Joule heating at the interface. These effects are, however, negligible for the parameters used in this study. This is due to the fact that the Peltier coeffi cient of the S/L interface is rather small for Ga. The coefficient is defined from the Kelvin relations314 as ^sZ, ^sZ. Hence, for the (ill) interface supercooled by about 3C, Pg^ = 5.38 x 10 ^ V. Based on the current densities used, about 8 A/cm^, the Peltier heat is calculated as Qp = .0043 W/cm^ = .001 cal/s*cm. Taking into account that the heat of fusion for Ga is 119 cal/ cm^, and the lowest growth rate of 500 pm/s, the rate of heat evolution at the interface be cause of solidification is Qs = 6 cal/s'cm^. Therefore, Qp accounts for only about .016% of the heat evolved for the lowest growth rate. For V xlO 2 1. 5 . 5 G 0 1 2 3 4 5 AT, t Figure 55 Comparison between the (111) dislocation-assisted growth rates and the SDG Model calculations shown as dashed lines. 227 71 ,1/2 rs = (7T> or o s l where p is the shear modulus. Nevertheless, corrections due to the stress field are usually neglected since most of the time rs < yQ. In conclusion, dislocations have a major effect on the kinetics of growth by enhancing the growth rates of an otherwise faceted perfect interface, as it has been shown experimentally for several materi als 2252634b3 Predictions from the classical SDG theory describe the phenomena well enough, as long as spiral growth is the dominant pro cess 14 5 As far as growth from the melt is concerned, most experimental results are not in agreement with the commonly referred parabolic growth law, eq. (41); indeed, the majority of the S/L SDG kinetics found in the literature are expressed as V ATm with m < 2. In contrast with the perfect (and faceted) interface, a dislocated interface is mobile at all supercoolings. Moreover, the SDG rates are expected to be several orders of magnitudes higher than the respective 2DNG rates, regardless of the growth orientation. Like the 2DNG kin etics, the dislocation-assisted rates can fall on two kinetic regimes according to the BCF theory. This can be understood by considering the limits of SDG rate equation, eq. (40), with respect to the supersatura tion o. It is realized that when o o^, i.e. low supersaturation, then one has the parabolic law V c and for o o-^ the linear law V o 57 oe = 2oxsv exp (- W/KT) tanh (yQ/2xs) (24) which reduces to (23) when yQ becomes relatively large. Interfacial atom migration The previously given analytical expression (eq. (20)) for the edge velocity can be written more accurately as u0 ~ c AGv-exp(- AG^/KT) (25) where c is a constant and AG^ is the activation energy required to transfer an atom across the cluster/L interface. This term is custom arily assumed154 to be equal to the activation energy for liquid self- diffusion, so that og in turn is proportional to the melt diffusivity or viscosity (see eq. (20)). Before examining this assumption, let it be supposed that the transfer of an atom from the liquid to the edge of the step takes place in the following two processes: 1) the molecule "breaks away" from its liquid-like neighbors and reorients itself to an energetically favorable position and 2) the molecule attaches itself to the solid. Assuming that the second process is controlled by the number of available growth sites and the amount of the driving force at the interface, it is ex pected that AG^ to be related to the first process. As such, the inter facial atomic migration depends on a) the nature of the interfacial region, or, alternatively, whether the liquid surrounding the cluster or steps retains its bulk properties; b) how "bonded" or "structured" the liquid of the interfacial region is; c) the location within the interfacial region where the atom migration is taking place; and d) the molecular structure of the liquid itself. Thus, the combination and the magnitude of these effects would determine the "interfacial Figure A-5 Geometry of the interacial region of the heat transfer analysis; L is the heat of fusion. Constant Flow Rate(& Temperature) Heat Transfer Fluid O*1 l l l < i l ( i l 1 \ l l l l t * l ( l t ill t i l l | < / i l 1 l- r rT~rt tr* 282 195 due to the fact the a (AT)^ growth law could arise in different ways, as indicated earlier. Another misconception in using the bulk kinetics is that the value of the coefficient k^ depends on the heat transfer conditions, sample and interface geometry, and the specifics of the experimental set-up. This complicates interpretation of the results and may explain the con tradictory conclusions reached by various investigators on the growth mode and kinetics, even for the same material (see earlier example on the growth kinetics of Sn). An example of the effect of sample size on bulk kinetics is illustrated in Fig. 48, where the interfacial and bulk kinetics for the Ga (111) interface are given for samples of different S/L interfacial areas but growth under otherwise identical conditions. The heat transfer calculations (see Appendix III) show that the inter face temperature is related to the bulk temperature by an overall heat transfer coefficient hQ. The coefficient is shown to increase approxi mately proportionally to the interfacial area. Therefore, at a given bulk temperature, the sample with the largest diameter exhibits the highest interface temperature, and, consequently, the slowest bulk kin etics. In contrast, the interfacial kinetics (not the growth rate) are the same for both cases in this example. On the other hand, as dis cussed earlier, there are situations where differences in the mechanism of growth of one and the same material really exist because of the un similar conditions in the crystal/melt interface. The problem with using bulk kinetics in this case is that, as can be seen in Fig. 24, it is not sensitive enough to allow for detection of these growth mechan isms. In the following sections, the reliability and accuracy of the 179 Figure 38 Photographs of the growth front of Ga doped with .01 wt% In showing the entrapped In rich bands (lighter regions), X 40. 130 remained constant; values of it for several samples are listed in Table 3. c) The Seebeck coefficient of the S/L interface, SsÂ£. The values of the S/L interface thermoelectric powers, SsÂ£, were determined directly for each sample and were verified by the results of the previous study.311 Direct determination of SsÂ£ was possible because of the faceted character of the involved Ga interfaces. When these interfaces are free of dislocations, they remain practically stationary up to cer tain values of AT (see earlier discussion on LG kinetics). Therefore, within this range of supercoolings, the S/L interface temperature is not affected by the heat of fusion and it is equal to the bulk temperature T^. Based on this, the value of SSÂ£ was determined as follows. Ini tially, the two interfaces were brought just below Tm. Subsequently, interface II was cooled to about 1.4C for the (111) type and about .6C for the (001) interface below Tm and then heated up to its original temperature. During the cooling and heating cycle of the S/L interface, the thermoelectric voltage generated was recorded as a function of temp erature, as shown in Fig. 18 (also see print out of the computer program (//1) involved in Appendix V). The slope of the fitted line is the Ssi value at the mean temperature. The determined Ss2_ values for several samples for the (ill) and (001) interfaces are listed in Table 3. Dur ing growth conditions, since the Seebeck emf changes proportionally to the interface supercooling, if the conditions (growth) at the interface remain then otherwise similar, it also "follows" proportionally the changes in AT^. This is indeed shown in Fig. 19. The Seebeck technique, as mentioned earlier, not only allows for direct and accurate measurement of the interface supercooling, but also 239 AGt = 50KT If, instead of 50KT, 40KT is assumed, then the calculated g values from eq. (94) are in agreement with those calculated via eq. (38). The critical supercooling at which the transition off the lateral growth (i.e. for both 2DNG and SDG mechanisms) occurs, can be used to calculate g from the previously derived equation (eq. 49) as a g V T AT5'" = sl. T m (49) h L Assuming that AT" is the supercooling for which the deviation from the low supercoolings 2DNG rate equations is observed (i.e. AT''(111) = 3.5C and AT"(00l) = 1.5), the latter equation yields estimated g values as .02 to .04, and .01 to .02 for the (ill) and (001) interface, respec tively. These values of g are less than those calculated previously from the step edge free energy and from ATt. It should be mentioned that AT" is assumed to be such that the thickness of the interface be comes equal to the 2D critical nucleus radius. It is interesting to note that, although the theory implies that oe should approach zero in the transitional regime, it does not predict any quantitative decrease in the g parameter (i.e. increased diffuseness) with supercooling. The prediction of the onset for the transition from lateral to continuous growth at a supercooling such that the width of the nucleus exceeds its radius seems to be correct, since it implies that the step will loose its "identity" in the background of the interface following the trans ition. The estimated g values indicate a one to two layer S/L inter face, as expected for a faceted interface. However, these values, as shown previously, are not quantitatively self-consistent with the pro posed tests of the theory. Space Group Lattice Constants in A Positional Parameters Atomic Coordinates in the Unit Cell (8 Atoms) Table A-3. Crystallographic Data of Gallium (a-Ga). Old Designation Abma, New Designation Cmca, D^ Reference Laves37 9 (1933) Bradley3 8 0 (1935) Swanson & Fuyat381 (1953) Sharma & Donohue3 8 2 (1962) Barrett Spooner38 3 (1965) Donohue3 8 4 (1972) (T ) v room-' (T=18C) (T=25C) (T ) v xroom7 (T=24C) (T ) v Aroom7 a c 4.515 4.5258 4.524 4.5258 4.5258 b a 4.515 4.5198 4.523 4.5186 4.5192 c b 7.657 7.6602 7.661 7.657 7.6586 p (z) y .159 or .1525 . 1549 . 1539 . 153 m (x) z .08 .0785 .081 .0798 (m, 0, p) Each Ga atom at (mOp) or at (Oyz) has seven nearest neighbors (m + 1/2, 1/2, p) (m + 1/2, 1/2, p) (Ref. 380) (Ref. 381) (Ref. 384) (m, 0, p) 1 at 2.437 A, (dj) 1 at 2.484 A 1 at 2.465 A (m, 1/2, p + 1/2) 2 at 2.706 A, (2) 2 at 2.691 A 2 at 2.7 A (m + 1/2, 0, p + 1/2) 2 at 2.736 A, (3) 2 at 2.73 A 2 at 2.735 A (m + 1/2, 0, p + 1/2) (iff, 1/2, p + 1/2) 2 at 2.795 A, (dA) 2 at 2.788 A 2 at 2.792 A The next closest neighbors are at 3.727 The next closest neighbors are at at 3.753 A 2 71 ICE POINT & RECORDER Thermocouple Bath I W- Wires Liquid Ga - T= CONSTANT ICE POINT & RECORDER |-C T * ^rnp ' L... I WATER RETURNS I I r~^I I I <: t. TTmp- AT IH Thermocouple II S L Interfaces n-^ ' Bath II Heaters CONSTANT ^1 \ /3/\\ Solid Ga J y CONSTANT TEMPERATURE TEMPERATURE CIRCULATOR 1 CIRCULATOR II Figure 15 Experimental set-up. 118 273 Figure A-2 The Gallium structure projected on the (100) plane; double lines indicate the short(covalent) bond distance dj* Dashed lines outline the unit cell. stop worrying and enjoy the mid-day recess; his help, particularly in my last year, is very much acknowledged. Tong Cheg Wang helped with the heat transfer numerical calculations and did most of the program writ ing. From Dr. Richard Olesinski I learned surface thermodynamics and to argue about international politics. Lynda Johnson saved me time during the last semester by executing several programs for the heat transfer calculations and corrected parts of the manuscript. I would also like to thank Joe Patchett, with whom I shared many afternoons of soccer, and Sally Elder, who has been a constant source of kindness, and all the other members of the metals processing group for their help. I have had the pleasure of sharing apartments with George Blumberg, Robert Schmees, Susan Rosenfeld, Diana Buntin, and Bob Spalina, and I am grateful to them for putting up with my late night working habits, my frequent bad temper, and my persistence on watching "Wild World of Animals" and "David Letterman." I am very thankful to my friends, Dr. Yannis Vassatis, Dr. Horace Whiteworth, and others for their continuous support and encouragement throughout my graduate work. I would also like to thank several people for their scientific advice when asked to discuss questions with me; Professors F. Rhines (I was very fortunate to meet him and to have taken a course from him), A. Ubbelohde, G. Lesoult, A. Bonnissent, and Drs. N. Eustathopoulos (for his valuable discussions on interfacial energy), G. Gilmer, M. Aziz, and B. Boettinger. Sheri Taylor typed most of my papers, letters, did me many favors, and kept things running smoothly within the group. I also thank the typist of this manuscript, Mary Raimondi. vi 331 262. S. R. Coriell and R. F. Sekerka, J. Cryst. Growth, 46 (1979) 479. 263. S. R. Coriell, R. F. Boisvert, R. G. Rehm, and R. F. Sekerka, J. Cryst. Growth, 54 (1981) 167. 264. S. C. Hardy and S. R. Coriell, J. Cryst. Growth, 3/4 (1968) 569; 5 (1969) 329; 7 (1970) 147; J. Appl. Phys., 39 (1968) 3505. 265. D. E. Holmes and H. C. Gatos, J. Appl. Phys., 52 (1981) 2071. 266. K. M. Kim, J. Cryst. Growth, 44 (1978) 403. 267. T. Sato and G. Ohira, J. Cryst. Growth, 40 (1977) 78. 268. K. Shibata, T. Sato, and G. Ohira, J. Cryst. Growth, 44 (1978) 419. 269. T. Sato, K. Ito, and G. Ohira, Trans. Jap. Inst, of Metals, 21 (1980) 441. 270. S. O'Hara and A. F. Yue, J. Phys. Chem. Solids, 28 (1967) 2105. 271. D. T. J. Hurle, J. Cryst. Growth, 5 (1969) 162. 272. J. J. Favier, J. Berthier, Ph. Arragon, Y. Malmejac, V. T. Khryapov, and I. V. Barmin, Acta Astronutica, 9 (1982) 255. 273. J. Narayan, J. Cryst. Growth, 59 (1982) 583. 274. W. J. Boettinger, D. Schechtman, R. J. Schaeffer, and F. S. Biancaniello, Met. Trans., A15 (1984) 55. 275. K. G. Davis and P. Fryzuk, J. Cryst. Growth, 8 (1971) 57. 276. J. P. Dismukes and W. M. Yim, J. Cryst. Growth, 22 (1974) 287. 277. J. C. Baker and J. W. Cahn, in: Solidification (ASM, Metals Park, OH, 1970), p. 23. 278.K. A. Jackson, G. H. Gilmer, and H. J. Leamy, in: Laser and Electron Beam Processing of Materials, C. W. White and P. S. Peercy, eds. (Academic, New York, 1980), p. 104. 279. M. J. Aziz, J. Appl. Phys., 53 (1982) 1158. 280. A. A. Chernov, Sov. Phys.-Uspekhi, 13 (1970) 101. 281. M. J. Aziz, in : Rapid Solidification Processing Principles and Technologies III, R. Mehrabian, ed. (NBS, Gaithersburg, MD, 1982), p. 113. 282. G. J. Gilmer, Mat. Sci. Engr., 65 (1984) 15. 300 Mrp = 3f 9 (AT) and p = 3f 3T. Substituting UA and into eq. (A33), we obtain a =- [-KLGL(aL b + V UT2KTmr2'1 L s L^T Ma) + 2 K a (A34) Remembering that the interface is stable when a < 0, eq. (A34) leads to results that are qualitatively similar to the previously discussed gen eral case, as long as pT PA > 0 (provided mt > 0). For the opposite case, i.e. < 0, further analysis is required since the sign of the denominator and that of the numerator depend on whether or not the liquid is supercooled (i.e. < 0). In determining the sign of p^ p^, one has to consider a particular kinetic law and examine its properties with increasing AT. For example, in the case of continuous growth kin etics, p^ p^ > 0 for small supercoolings and p^ p^ < 0 at high super coolings because of the increased melt viscosity at low temperatures (see also discussion in earlier chapters). For the case of 2DNG kinetics (PNG), the growth rate can be expressed as V = pQ ATn exp( ^) i where pQ, B are constants and n is about one. Then PT 3(AT) Mo AT exP(- t.AT n + T.AT 1 1 ma = It" = Mo ATn exp(_ (n + > 0 and M T p AT o ,n-1 exp( - B s B T.AT; T.AT i i 310 Program #3 1 REM THIS PROGRAM RECALLS THE SEEBECK EMF AND POTENT IAL STORED READINGS FROM THE INTERNAL MEMORY OF THE 3456A HP-VLTMETER. 5 DIM VS ( 350 ) DIM VI (350) ,XC350) ,Y(350) 8 CR = 0.1 ? DT = 0.1:DRDL =0.1 10 ZS = "" 30 PR# 3 40 IN# 3 50 "SCI *0 " RA" 70 "LL" 80 "LF1 90 AS = "T4RS1" 100 BS = STR" 101 PR# 1 103 PRINT "NUM. VALUE" 104 PR# 3 105 FOR NUM = 1 TO 350 106 CS = STR* (NUM) 107 DS = A* + CS + BS 108 PRINT "WT*;ZS;DS 10? PRINT "LIT6" ; ZS ; RER" 115 .PRINT "RDV" ;ZS;: INPUT VS (NUM) li VI(NUM) = VAL (VS(NUM)) 117 REM *********************-*** 118 REM NEXT THE STORED VALUES ARE PRINTED 11? PR# 1 120 PRINT N UM V1 (N UM) 121 PR# 3 122 NEXT NUM 130 PRINT "LA" 140 PRINT "UT" 170 PR# 1 175 PRINT "END OF DATA" APPENDIX III HEAT TRANSFER AT THE SOLID/LIQUID INTERFACE Heat transfer problems during solidification processes are charac terized by the existence of a moving S/L interface. The advancing of the interface is accompanied by the release of the heat of fusion, which in turn raises the temperature at the interface so that the latter is warmer than any other point in the system for growth into a supercooled melt. Except for a few idealized situations, exact solutions are not available.407408 The difficulty in solving this kind of problem either numerically or analytically lies mainly in simulating the thermal effect of releasing the heat of fusion.409 The treatment of the moving inter face as a heat source by the application of Greens' function410 and the replacement of temperature by enthalpy as the dependent variable,411 the so called enthalpy model, represent two approaches that have been devel oped during the past decade. However, both methods have some problems, among them being the determination of the time step-size.41 For the case of unidirectional solidification, the analysis re quires values of at least two of the three parameters: interface shape, interface composition, and interface velocity. In the present experi ments, for which the growth is unconstrained, the modified problem is employed which utilizes the interface shape and velocity and calculates the interface temperature (or, in reality, the difference, 6T, between the temperature of the coolant medium, T^> and the actual temperature of the interface, T.). 280 to X E n_ > en o Figure 35 The logarithm of dislocation-free (001) growth rates versus the reciprocal of the interface supercooling for 10 samples; lines A and B represent the PNG rate equations, as given in Table 7. 170 TEMPERATURE 279 Figure A-4 Ga-In phase diagram; c and c' indicate the two alloy compositions investigated. 128 junctions AB and BA are replaced by two S/L interfaces, one of which being at equilibrium (Tm) is the "hot junction," while the other super cooled by an amount AT is the "cold junction." According to eqs. (64) and (65) and taking into account the law of Magnus316 and the law of intermediate metals,317 the emf generated across the S/L interfaces is given by E s + S Â£s S .(T si m T.) (66) = S .AT si where SgÂ£ is the Seebeck coefficient of the S/L interface. When the two interfaces are at equal temperatures, then Es = 0. It should also be noted that the Seebeck coefficient of most materials is a function of temperature, but for small temperature intervals it can be approximated by a linear function, with a temperature coefficient in the order of 10-2 to 10^ pV/(C)^. Hence, according to eq. (66), the interface supercooling can be determined from the recorded emf, provided that the Seebeck coefficient Ss^ is known. This can be measured directly312 or indirectly,308311*318 if the absolute Seebeck coefficients of the solid and liquid are known, with the aid of the relation Sg?<(T) = Ss(T) SL(T) (67) where Sg and S^ are the absolute solid and liquid Seebeck coefficients. For the case of Ga, SgÂ£ was obtained in both ways, as discussed else where;1 the direct method of determining SgÂ£ will be further discussed later. In general, the coefficient SgÂ£ of a homogeneous solid is a second order tensor.319 This means that for anisotropic crystals (non- cubic symmetry), the coefficient also varies with crystal orientation. 323 98. D. Elwell, AACG Newsletter, 15 (1985) 9. 99. P. R. Pennington, S. R. Ravitz, and G. J. Abbaschian, Acta Met., 18 (1970) 943. 100. J. C. Brice and P. A. C. Whiffin, Solid State Electron., 7 (1964) 183. 101. J. A. M. Dikhoff, Solid State Electron, 1 (1960) 202. 102. T. F. Ciszek, J. Cryst. Growth, 10 (1971) 263. 103. J. H. Walton and R. C. Judd, J. Phys. Chem., 18 (1914) 722. 104. G. A. Alfintsev and D. E. Ovsienko, in: Crystal Growth, H. S. Peiser, ed. (Pergamon, Oxford, 1967), p. 757; Dokl. Akad. Nauk. SSSR, 156 (1964) 792. 105. M. Toda, R. Kubo, and N. Saito, Statistical Physics I (Springer- Verlag, Berlin, 1983), p. 118. 106. L. Onsager, Phys. Rev., 65 (1944) 117. 107. H. Muller-Krumbhaar, in: 1976 Crystal Growth and Materials, E. Kaldis and H. J. Scheel, eds. (North-Holland, Amsterdam, 1977), p. 116. 108. J. P. van der Eerden, P. Bennema, and T. A. Cherepanova, Prog. Crystal Growth Charact., 1 (1978) 219, and references therein. 109. H. J. F. Knops, Phys. Rev. Lett., 49 (1977) 776. 110. H. Van Beijeren, Phys. Rev. Lett., 38 (1977) 93. 111. J. D. Weeks, in: Ordering in Strongly Fluctuating Condensed Matter Systems, T. Riste, ed. (Plenum, New York, 1980), p. 293. 112. H. J. Leamy, G. H. Gilmer, and K. A. Jackson, in: Surface Physics of Materials, J. B. Blakely, ed. (Academic, New York, 1975), p. 121. 113. J. M. Kosterlitz and D. J. Thouless, J. Phys. C6 (1973) 1181. 114. C. Jayaprakash and W. F. Saam, Phys. Rev., B30 (1984) 3916. 115. R. H. Swendsen, Phys. Rev., B17 (1978) 3710. 116. H. J. Leamy and G. H. Gilmer, J. Cryst. Growth, 24/25 (1974) 499. 117. G. H. Gilmer and K. A. Jackson, in: 1976 Crystal Growth and Materials, E. Kaldis and H. J. Scheel, eds. (North-Holland, Amsterdam, 1977), p. 79. 256 68. For the present experiment, the aspect ratio is larger than 200, and Rt is smaller than 3. Whether or not convection is steady during growth is a question of interest. In order to answer this question, the stagnant boundary layer 6 has to be evaluated as a function of the growth conditions. In doing so, one could adopt the following scheme: a) find out the interfacial y composition necessary to match the parallel to g growth data points V(AT, C^) with those V(AT, C^') for the antiparallel direction and b) by knowing c^ and V, one can back-calculate with the aid of eqs. (62) and (63). However, such a procedure would lack quantitative sense because 1) of the approximations in the calculations involved; 2) the differ ences in the resultant interfacial temperature by using keff instead of k in the determination of are very small. For example, asssuming 6 = 100 pm, this difference is in the order of .008C for a growth rate of 30 pm/s and cQ = .012 wt%; for = 300 pm and V = 40 pm/s, the differ ence becomes .03C, which is the order of the temperature measurements accuracy. For smaller and V this difference gets even smaller. Finally, 3) as discussed earlier, it is assumed that growth takes place under the initial transient conditions as far as segregation is con cerned. However, under diffusive-convective conditions in the liquid, the initial transient distance is much shorter than the corresponding diffusion-only case. It is given as31*1 _ 50 1 X V 2 .25 + (lybr where b = 6V/D and m^ is a constant in the order of unity which depends on k and b. For example, assuming that 6 = 100 pm/s, V = 10 pm/s, and k = .019, x is calculated as 218 pm, which is 30 times smaller than the 335 335. P. Bennema, R. Kern, and B. Simon, Phys. Stat. Sol., 19 (1967) 211. 336. J. S. Langer, R. F. Sekerka, and T. Fujioka, J. Cryst. Growth, 44 (1978) 414. 337. V. T. Borisov, I. N. Golikov, and Y. E. Matveev, Sov. Phys.-Crsty. 13 (1969) 756. 338. a) I. Gutzow, in: 1976 Crystal Growth and Materials, E. Kaldis and H. J. Scheel, eds. (North-Holland, Amsterdam, 1977), p. 379. b) I. Gutzow and E. Pancheva, Kristall. U. Technik, 11 (1976) 793. 339. W. J. Boettinger, F. S. Biancaniello, and S. R. Coriell, Met. Trans., A12 (1981) 321. 340. J. D. Verhoeven, Trans. TMS-AIME, 242 (1968) 1940. 341. W. R. Wilcox, J. Appl. Phys., 35 (1963) 636. 342. P. de la Breteque, Gallium, Bulletin d'Information et de Biblio graphic, No. 12 (Alusuisse France, Marseille, 1974), p. 11. 343. R. Kofman, P. Cheyssac, and J. Richard, Phys. Rev., B16 (1977) 5216. 344. H. E. Sostman, Rev. Scient. Instrum., 48 (1977) 127. 345. G. H. Wagner and W. H. Gitzen, J. Chem. Educ., 29 (1952) 162. 346. A. Defrain, Metaux, 417 (1960) 248. 347. G. B. Adams, Jr., H. L. Johnston, and E. C. Kerr, J. Am. Chem. Soc., 74 (1952) 4784. 348. R. W. Powell, M. J. Woodman, and R. P. Tye, Brit. J. Appl. Phys., 14 (1963) 432. 349. Y. Takahashi, H. Kadokura, H. Yokokawa, J. Chem. Therm., 15 (1983) 65. 350. C. J. Smithells, Metals Reference Book, 6th ed., E. A. Brandes, ed. (Butterworths, London, 1983). 351. K. Wade and A. J. Banister, The Chemistry of Al, Ga, In, and T1 (Pergamon Press, New York, 1974). 352. L. Bosio and C. G. Windsor, Phys. Rev. Lett., 35 (1975) 1652. 353. J. H. Perepezko and D. H. Rasmussen, in: Proc. 17th A1AA Ann. Meeting, New Orleans, LA, 1979. 78 anisotropy. In contrast, it is predicted117 that there is no growth rate difference between dislocation-free and dislocated rough inter faces. This is because a spiral step created by dislocation(s) will hardly alter the already existing numerous kink sites on the rough interface. A summary of the interfacial growth kinetics together with the theoretical growth rate equations is given in Fig. 9. Next, the growth mode for kinetically rough interfaces is discussed. Growth Kinetics of Kinetically Roughened Interfaces As discussed earlier, an interface that advances by any of the lat eral growth mechanisms is expected to become rough at increased super coolings. Evidently, the growth kinetics should also change from the faceted to non-faceted type at supercoolings larger than that marking the interfacial transition. In accord with the author's view regarding the kinetic roughening transition, the following qualitative features for the associated kinet ics could be pointed out: a) Since the interface is rough at driving forces larger than a critical one, its growth kinetics are expected to resemble those of the intrinsically rough interfaces. Thus, the growth rate is expected to be unimpeded, nearly isotropic, and proportional to the driving force. Moreover, the presence of dislocations at the inter face should not affect the kinetics, b) It is clear that the faceted interface gradually roughens with increasing AT over a relatively wide range of supercoolings. The transition in the kinetics should also be a gradual one. c) In the transitional region the growth rates should be faster than those predicted from the lateral, but slower than the 293 and growth rates of .01, .6, and 1 cm/s, the difference between the num erical and analytical results is .03, 1.917, and 10.36C, respectively. Figure A-8 shows temperature distribution in the solid and liquid sides with respect to the axial distance from the S/L interface in terms of the inside radius r^. The calculations for both (001) and (ill) interfaces were performed for the heat transfer conditions, as indicated in Fig. A-8, and for a growth rate of 1.5 cm/s. The estimate of the thermal gradients at these high growth rates was important for the interfacial stability calculations, as discussed in the next Appendix. It should be noted that the temperatures on both sides of the interface fall steeply with distance away from it. For the (111) interface, for example, the liquid at a distance 10r^ from the growth front has about the same temperature with the bulk liquid. Figure A-9 shows the ratio of the temperature correction at any point along the interface to that of the edge (inner capillary wall) for different values of U.r./K of the i i s (ill) interface. According to these calculations, the edge of the inter face is cooler than its center by 2-4.5%, depending on the lhr^/K value. Figures A-10 and A-11 compare the experimental results with the analytical and numerical for the (111) interface at low (V < .2 cm/s) and high growth rates, respectively. As can be seen, the experimental re sults are in very good agreement with the numerical and analytical calcu lations. The observed slight deviation of the experimental results from the calculations towards higher 6T corrections at large Vr^ (see Fig. A-11) could be explained as follows: the calculations are done using as a reference temperature for the properties of the coolant (30% water- ethylene glycol solution) that of 25C. At high bulk supercoolings (Tb < 123 Table 2. Mass Spectrographic Analysis of Ga (99.99999%)." Element Concentration (ppm) < A1 .03 Ba .03 Be .03 Bi .03 B .03 Cd .03 Ca .03 Cr .03 Co .03 Cu .03 Ge .03 Au .03 In .005 Fe .03 Pb .03 Mg .03 Mn .03 Hg .02 Mo .03 Ni .03 Nb .03 K .03 Si .03 S .03 Cl .03 c .03 Ag .03 Ta .03 Th .03 Sn .04 Ti .03 W .03 V .03 Zn .03 Zr .03 * Analysis as provided by the New York, NY. United Mineral and Chemical Corporation 288 written.414 Parameters that are used in these calculations are given in Table A-4 and in the print-out of the program (#5) involved in Appendix V. In the calculations of the infinite series solutions, eqs. (A38) and (A39) were truncated at n=6 term. The ratios A,/A, and B,/B. of the co t 1 oi efficients in these calculations were found to be less than 0.2%. This, in addition to the fact that A^, B^ < 10, indicates that the truncation error is negligible. The results of the present calculations, designated as numerical, as compared to those of the earlier analytical analysis,2 are summarized in Figs. A-6 through A-8. Figures A-6 and A-7 show linear plots of the temperature correction (6T) for the (111) and (001) interfaces as a function of Vr^ at different values of U.r.. The difference between the numerical and analytical re- li suits, for the same heat transfer conditions (LLr^), is denoted by the hatched areas. As noted from these figures, the two results are approxi mately the same at low growth rates but become appreciably different at high growth rates (i.e. Vr^ > 5 x 10-^ cm^/s). For the (001) interface, this difference is larger (by a factor 2.7) than that for the (ill) interface. This is expected since |K K | is much larger for the (001) interface. Accordingly, the analytical solution based on the assumption that the liquid and solid have the same thermal properties underestimates and overestimates the temperature correction during growth along the (111) and (001) interfaces, respectively. For example, for the (111) interface the difference between the numerically and analytically cal culated <5T, if it were to be used at growth rates of .075, .4, and 1 cm/s is .07, .324, and 1.4C, respectively, for conditions such that U^r^ = . 02K. On the other hand, for the (001) interface under similar conditions Growth Rate, pm/s a) b) 184 Figure 40 Effect of distance solidified on the growth rate of Ga-.01wt%In grown in the direction parallel to the gravity vector (a,b), and comparison with that grown in the antiparallel direction(a). 337 375. A. A. Karashaev, S. N. Zadumkin, and A. I. Kukhno, Russ. J. Phys. Chem., 3 (1967) 654. 376. E. F. Broome and H. A. Walls, Trans. TMS-AIME, 245 (1969) 739. 377. J. Petit and n. H. Nachtrieb, J. Chem. Phys., 24 (1956) 1027. 378. F. M. Jaeger, P. Terpstra, and H. G. K.. Westenbrink, Z. Krist., 66 (1927) 195. 379. F. Laves, Z. Krist., 84 (1933) 256. 380. A. J. Bradley, Z. Krist., A91 (1935) 302. 381. H. E. Swanson and R. K. Fuyat, Nat. Bur. Stand. Circular, No. 539, 3 (1953) 9. 382. B. D. Sharma and J. Donohue, Z. Krist., 117 (1962) 293. 383. C. S. Barrett and F. J. Spooner, Nature, 207 (1965) 1382. 384. J. Donohue, The Structure of the Elements (J. Wiley, New York, 1972). p. 236. 385. R. W. Powell, Nature, 164 (1949) 153. 386. R. W. Powell, Nature, 166 (1950) 1110. 387. L. Pauling, J. Amer. Chem. Soc., 69 (1947) 542. 388. J. C. Slater, G. F. Koster, and J. H. Wood, Phys. Rev., 126 (1962) 1307. 389. V. Heine, J. Phys., Cl (1968) 222. 390. C. G. Wilson, J. Less-Common Met., 5 (1963) 245. 391. F. J. Spooner and C. G. Wilson, J. Less-Common Met., 10 (1966) 169. 392. N. Durbec, B. Pichaud, and F. Minari, J. Less-Common Met., 82 (1981) 373. 393. N. Burble-Durbec, B. Pichaud, and F. Minari, J. Less-Common Met., 98 (1984) 79. 394. C. G. Wilson, Trans. AIME, 224 (1962) 1293. 395. P. Ascarelli, Phys. Rev., 143 (1966) 36. 396. R. Brdzel, D. Handtmann, and H. Richter, Z. Physik, 169 (1978) 374. 235 associated with the kinetic roughening recently described334 for the growth of naphthalene from solution. This is because, as shown before, the growth data in this region can still be expressed via a 2DNG or SDG mechanism. Moreover, dislocations in this region, yet appear to effect although in a minor way, the growth rates as clearly shown for the (ill) interface in Fig. 23. This is expected since for the (ill) interface the highest growth rates are for supercoolings in the range of 4.5- 4.6C, which is below the estimated roughening supercooling of 4.75C. The (001) growth kinetics beyond the roughening supercooling (~2.2C) are different than those of the TRG regions for both inter faces, as can be seen in Fig. 31. Indeed, the linear growth curve for V > 1.4 cm/s, if extrapolated to zero growth rates, essentially passes through the origin. On the other hand, the determined kinetic coeffi cient of .63 cm/s*C seems to be in excellent agreement with that of the continuous growth theory, as discussed later. Disagreement Between Existing Models for High Supercoolings Growth Kinetics and the Present Results As it was discussed earlier, the growth rates at high supercoolings deviate from the rate equations expressing both the spiral and bi- dimensional growth mechanisms at lower supercoolings. Nevertheless, as shown in the previous section, the kinetics are well described by a gen eral lateral growth model based on the classical ideas, but corrected for and oe(AT). Several features of the experimental growth data curve, as shown in Fig. 59, are in qualitative agreement with the diffuse interface theory25 such as: i) lateral growth at low supercoolings regardless of the values of "a" factor for each interface, ii) the growth curve at 96 a = V{- KTGT(aT- ) K G (a + ) 2 KT TiA + 2KmG a(a (a pj) l} LLLkt sssk m c D Hr L s L V + 2KmG a(a p^) ^ v c D with (eq. (54)) x rfXU 2 a, 1/2 2D + 2D + W + D r V rr v 'i2 -u 2 x i1 /2 aL (2^> + 1(2^> + + - r v 'i x r r v '2 4. 2 x o 11/2 a = -(^) + [(r) + id + J S K K K S S S (K a + KTaT ) s s L L (2K) K = K + Kt s L p = 1 k v where Gc is the solid thermal gradient, KT and kt are the liquid and solid thermal conductivities and diffusivities, respectively, Lv is the latent heat of fusion per unit volume, and Tm is the melting point in the absence of a solute. noonnnnnonnnnnnnnnnnnn 312 Program //5 C c c c c c c xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx X THIS PROGRAM SOLVES A(SUB N) AND B(SUB N) OF EQ.(30) OF THE X XANALYTICAL MODEL BY TRUNCATING THE INFINITE SERIES AT 6TH TERMX XAND SOLVING EQS.(30) AND (31) SIMULTANEOUSLY AT 6 VALUES OF R X XR+N/5*RADIUS,N=0,1,...,5. X XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX X INPUT PARAMETER DESCRIPTION X NOTE; USE CCS UNIT : CM,GM,SEC X KLS=K(SUB L)/K(SUB S) ; RATIO OF LIQUID AND SOLID HEAT COND X TIVITIES X ASL=KAPPA(SUB S)/KAPPA(SUB L); RATIO OF THERMAL DIFFUSIVITIESX RSL=DENSITY OF SOLID/DENSITY OF LIQUID X VW=INTERFACE VELOCITY X HCT=HEAT OF FUSION/BATH SUPERCOOLING/HEAT CAPACITANCE SOLID X Z= AXIAL LENGTH OF CA WHERE TEMP. DISTRIBUTION ARE TO BE X CALCULATED/I.R. OF CAPILLARY TUBE x TB= BATH TEMPERATURE IN DEC. CENTIGRADE X ASRI=KAPPA(SUB S)/I.RADIUS OF CAPILLARY TUBE X X GAMAS AND CAMAL ARE VALUES OF GAMA (SUB S) AND GAMA(SUB L) X X EVALUATED FROM EQ.(26) AND EQ.(27) WITH TABLE FROM 'CONDUC X X TION HEAT TRANSFER' BOOK X XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX X OUTPUT OF THIS PROGRAM CONTAINS A LIST OF INPUT PARAMETERS X X AND THE MATRIX FOR SOLVING A(SUB N) AND B(SUB N), THE COEFFICX X IENTS A(SUB N) AND B(SUB N) AND TEMPERATURE DISTRIBUTION IN X X BOTH SIDES OF THE INTERFACE ACROSS THE CAPILLARY TUBE X XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX DIMENSION A(12,12),B(12),WKAREA(160),TS(10,100),TL(10,100) COMMON /GAMMA/CAMAS(6),CAMAL(6),Y1(6),Y2(6) REAL KLS M= 1 IA=1 2 IDGT=0 N = 12 IDX=0 999 CONTINUE IDX=IDX+1 READ(5,10)KLS,ASL,RSL,VW,HCT,Z READ(5,10)TB,ASRI WRITE(6,16)KLS,ASL,RSL,VW,HCT,Z,TB,ASRI READ(5,10)(GAMAS(I),1=1,6) READ(5,10)(CAMAL(I),1=1,6) WRITE(6,17)CAMAS WRITE(6,18)CAMAL 15 FORMAT(5X,'CASE NO. ,15) 16 FORMAT(5X,'INPUT PARAMETERS; KLS,ASL,RSL,V,HCT,Z,TB,ASRI', 1/9X.8E13.6) 17 FORMAT(5X,'GAMAS ',6E13.6) 18 FORMAT(5X,'GAMAL .6E13.6) V=VW APPENDIX I GALLIUM Physical Properties of Gallium Gallium, which was discovered in 1875 and was named from Gallia in honor of its discoverer's homeland,341 is a unique element in many ways. Although the solid has the characteristic silvery (slightly bluish) appearance of a metal, the liquid is more white than silver, with a shiny surface that resembles Hg to a great extent; it has some very par ticular properties uncharacteristic of metals. For example, it has an extremely low melting point, 29.78C, and a very high boiling point, about 2370C; it has the second longest range of all the elements. Its vapor pressure is very low even at elevated temperatures, and it expands upon solidification (3.2%) a property shared by only three other ele ments: Ge, Bi, and Sb. Its crystal structure, as discussed later in this appendix, is unusual for a metal; black P, Br, and I have the same structure. Furthermore, it displays marked anisotropy on its electri cal, thermal, and mechanical properties. For instance, the ratio be tween its largest and smallest electrical conductivity is about 7, the highest value among all metals.342 Most of its unusual properties and strong anisotropy are usually attributed to the existence of Ga^ mole cules and the combined metallic and covalent bonding in the crystal. Current applications for Ga are primarily in compound form, mostly III/V compounds (GaAs, GaP), used in optoelectronic devices, coherent 263 264 electroluminescence, photovoltaic conversion, Schottky barrier switch ing, magnetic bubbles, and superionic conduction. Since its vapor pres sure is so low at high temperatures, it is particularly suited as a sealant in high temperature manometers. A new use of Ga is as a thermo electric standard and in the form of a chloride solution for neutrino radiation measurements. Ga is also useful as an alloying agent. Table A-l summarizes important physical properties of Ga, together with the relevant references. Ga, a member of the B-Al family, is very active chemically; at a given temperature, liquid Ga is believed to be the most corrosive sub stance to almost any metal.351 Only W, Nb, and Ta show good resistance to Ga up to temperatures of about 500C. Liquid Ga penetrates very quickly into the crystal structure of certain metals, thus having a haz ardous embrittling property, particularly for aluminum. It scarcely re acts with water and glass at low temperatures,31*5 but it is easily oxi dized by such oxidizing agents as aqua regia and H^SO^ when it is hot. It also readily reacts with halogens upon heating. Ga wets almost all surfaces, especially in the presence of oxygen, which promotes the form ation of a fine Ga suboxide film (by which it is protected from air oxi dation at ambient temperatures); the oxide film causes the loss of its mirror-like surface appearance and it can be removed by treating the oxidized metal with dilute HC1 or simply by draining the metal through a capillary tube. When it is free of oxides, it no longer wets glass and other surfaces, as experienced during this study. Figure 59 Comparison between the (001) growth curves and those predicted by the diffuse interface model.6 236 Figure 60 Normalized (111) growth rates as a function of the nor malized supercooling for interface supercoolings larger than 3.5C; continuous line represents the universal dendritic law growth rate equation.336 243 Figure 61 Density gradients as a function of growth rate 253 Figure A-l The gallium structure (four unit cells) projected on the (010) plane; triple lines indicate the covalent (Ga2) bond 272 Figure A-2 The gallium structure projected on the (100) plane; double lines indicate the short (covalent) bond distance d^. Dashed lines outline the unit cell 273 Figure A-3 The gallium structure projected on the (001) plane; double lines indicate the covalent bond and dashed lines outline the unit cell 274 Figure A-4 Ga-In phase diagram 279 Figure A-5 Geometry of the interfacial region of the heat transfer analysis; Lf is the heat of fusion 282 Figure A-6 Temperature correction <5T for the (111) interface as a function of Vr for different heat-transfer conditions, U^r; Analytical calculations (K^ = Ks = K), Numerical calculations 290 Figure A-7 Temperature correction <5T for the (001) interface as a function of Vr^ for different values of U^r^; Anal ytical, Numerical calculations 291 Figure A-8 Temperature distribution across the S/L (ill) and (001) interfaces as a function of the interfacial radius; Analytical model calculations, Numerical calcula tions 292 Figure A-9 Ratio of the Temperature correction at any point of the interface to that at the edge as a function of r' for different values of Ur^/Ks 294 Figure A-10 Comparison between the (111) Experimental results ( O ) and the Model ( Analytical, Numerical) calcula tions, at low growth rates (V < .2 cm/s) 295 Figure A-ll Comparison between the (ill) Experimental results (0,D) and the Model ( Analytical, Numerical) calcula tions as a function of Vr^ for given growth conditions .. 296 xvii i 4 Figure 25 The logarithm of the (111) growth rates plotted as a function of the logarithm of the interfacial and bulk supercoolings; the line represents the SDG rate equation given in Table 7. 152 124 (001) Figure 16 Gallium monocrystal X 20 94 Until now, two fundamentally different theoretical approaches have been used to describe the interface stability. The first is the consti tutional supercooling (CS) theory237238 which is based on an equilib rium thermodynamics argument describing the solute-rich (or depleted) liquid adjacent to the S/L interface. The stability criterion of this static analysis, which assumes a constant growth velocity (V) and no convection and solute diffusion in the solid is expressed as Gl CÂ£ (1 k) (-m) T > D (stable) (52) Here is the thermal gradient in the liquid, D is the solute diffusion in the melt, and k is the equilibrium distribution coefficient, assumed to be independent of growth rate and kinetics. Here, the case of k < 1 only is considered and, thus, the liquidus slope m is negative in sign. C' is the liquid composition at the interface, which, for the equilib rium steady state conditions assumed by the CS theory, is given as c" C C" = _s = l k k Here, C" is the solid composition and CQ is the initial composition at the interface of the melt. Hence, eq. (52) can be rewritten as Gt C (1 k) (-m) > -2 V Dk or as GL > m Gc (53) where Gc is the composition gradient at the interface and is given as _ V (1 k) c Dk o In the case of the solidification of pure material, Gc = 0, so eq. (53) can be written as APPENDIX VI SUPERSATURATION AND SUPERCOOLING The supersaturation o during vapor growth is defined as o = a 1, a = P/PQ (A38) where P is the actual vapor pressure, PQ is the equilibrium pressure, and a is called the saturation ratio. In the case of solution growth, the saturation ratio is given as a = C/Ce, where C is the actual concentra tion of the solution and Ce is the concentration in equilibrium at the temperature T. Note that a can also be written as a = 1 + AC/Ce or a = 1 + AP/P0 where AC = C C0 and AP = P PQ. In growth from solution, L (T T) lna = S T (A39) e where Ls is the enthalpy of solution. For small saturations (a < 1.1), also note that lna = a 1 = a. Correspondingly, in growth from the melt L T> .T KT T Iff (A40) m m where L is the heat of fusion per atom. In this case, for small super coolings the supersaturation o is proportional to AT/Tm. An example of correspondence between supersaturation and supercooling is given next. As mentioned earlier, the critical 2D radius for vapor growth is given as _ hy r KTlna (A41) 316 54 the site adjacent to the first atom rather than an isolated site. From this simplified atomistic picture, it is obvious that atoms not only prefer to "group" upon arrival, but also choose such sites on the sur face as to lower the total free energy. These sites are the ones next to the edges of the already existing clusters of atoms. The edges of these interfacial steps (ledges) are indeed the only energetically favorable growth sites, so that steps are necessary for growth to pro ceed (stepwise growth). The interface then advances normal to itself by a step height by the lateral spreading of these steps until a complete coverage of the surface area is achieved. Although another step might simultaneously spread on top of an incomplete layer, it is understood that the mean position of the interface advances one layer at a time (layer by layer growth). Steps on an otherwise smooth interface can be created either by a two-dimensional nucleation process or by dislocations whose Burgers vec tors intersect the interfaces; the growth mechanisms associated with each are, respectively, the two-dimensional nucleation-assisted and screw dislocation-assisted, which are discussed next. Prior to this, however, we will review the atomistic processes occurring at the edge of steps and their energetics, since these processes are rather independent from the source of the steps. Interfacial steps and step lateral spreading rate (uQ) In both lateral growth mechanisms the actual growth occurs at ledges of steps, which, like the crystal surface, can be rough or smooth; a rough step, for example, can be conceived as a heavily kinked step. For S/V interfaces it has been shown107112 that the roughness of 66 with time) as in a diffusion field, the growth rate equation is derived as175176 V c'h (JUe2)1/2 (34) where c1 is a constant close to unity. Indeed, growth data (S/V) of a MC simulation study were represented by this model.176 Alternatively, if the growth of the cluster is assumed to be such that its radius in creases with time as r(t) t + t^2 (i.e. a combined case of the above mentioned submodels), it can be shown that the growth rate takes the form of V c"h (Jue2)2/5 (35) where c" is a constant. Therefore, according to these expressions, the power in the growth rate equation varies from 1/3 to 1/2.177 A faceted interface that is dislocation free grows by any of the two previously discussed 2DN growth mechanisms. At low supercoolings the kinetics are of the MNG mode, while at higher supercoolings the interface advances in accord with PNG kinetics. The predicted growth rate equations (eqs. (30) and (32)) can be rewritten with the aid of eqs. (27), (26), and (20) as (MNG) V = Kx A (|V/2 exp (- ^|-) (36) Mo 2 (PNG) V= K2 (|^)5/6 exp (- ^Jj) (37) Here, K^, K2, and M are material and physical constants whose analytical expressions will be given in detail in the Discussion chapter. The growth rates as indicated by eqs. (36) and (37) are strongly dependent upon the exponential terms, and therefore upon the step edge free energy LIST OF FIGURES Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Interfacial Features, a) Crystal surface of a sharp interface; b) Schematic cross-sectional view of a diffuse interface. After Ref. (17) Variation of the free energy G at Tm across the solid/liquid interface, showing the origin of osg_. After Ref. (22) Diffuse interface model. After Ref. (6). a) The sur face free energy of an interface as a function of its position. A and B correspond to maxima and minima figuration; b) The order parameter u as a function of the relative coordinate x of the center of the inter facial profile, i.e. the Oth lattice plane is at -x .... Graph showing the regions of continuous (B) and lateral (A) growth mechanisms as a function of the parameters 3 and y, according to Temkin's model.7 Computer drawings of crystal surfaces (S/V interface, Kossel crystal, SOS model) by the MC method at the indicated values of KT/d>. After Ref. (112) Kinetic Roughening. After Ref. (117). a) MC inter face drawings after deposition of .4 of a monolayer on a (001) face with KT/cp = .25 in both cases, but differ ent driving forces (Ap). b) Normalized growth rates of three different FCC faces as a function of Ap, showing the transition in the kinetics at large supersaturations Schematic drawings showing the interfacial processes for the lateral growth mechanisms a) Mononuclear, b) Poly nuclear. c) Spiral growth. (Note the negative curva ture of the clusters and/or islands is just a drawing artifact. ) Free energy of an atom near the S/L interface. and Qs are the activation energies for movement in the liquid and the solid, respectively. is the energy required to transfer an atom from the liquid to the solid across the S/L interface xiii Page 9 13 con- 16 21 42 50 63 74 53 an "a" factor close to the theoretical borderline of 2 are suitable for testing. Even in such cases the transition cannot be substantiated and quantified in the absence of detailed and reliable growth kinetics anal ysis . Interfacial Growth Kinetics Lateral Growth Kinetics (LG) It is generally accepted that lateral growth prevails when the interface is smooth or relatively sharp; this in turn implies the fol lowing necessary conditions for lateral growth: 1) the interfacial temperature is less than Tr and 2) the driving force for growth is less than a critical value necessary for the dynamic roughening transi tion, and/or the diffuseness of the interface. The problem of growth on an atomically flat interface was first considered by Gibbs,20 who suggested that there could be difficulty in the formation of a new layer (i.e. to advance by an interplanar or an interatomic distance) on such an interface. When a smooth interface is subjected to a finite driving force (i.e. a supercooling AT), the liquid atoms, being in a metastable condition, would prefer to attach them selves on the crystal face and become part of the solid. However, by doing so as single atoms, the free energy of the system is still not de creased because of the excess surface energy term associated with the unsatisfied lateral bonds. Thus, an individual atom, being weakly bound on the surface and having more liquid than solid neighbors, is likely to "melt" back. However, if it meant to stay solid, it would create a more favorable situation for the next arriving atom, which would rather take Figure 57 The (111) growth rates versus the interface supercooling compared to those determined from CS on the solid/vapor interface (lief. (117)). 232 47 singularities of the free energy r(T,n)136 that determines the ECS phase diagram.137 The shape of the smooth edge varies y = A(x xc) + higher-order terms where xc is the edge position; x, y are the edge's curvature coordin ates. The critical exponent 0 is predicted to be as 0 = 2136 or 0 = 3/2. 1 37 1 38 The 3/2 exponent is characteristic of a universality class139140 and it is therefore independent of temperature and facet orientation as long as T < T^. Indeed, the 3/2 value has been reported from experimental studies on small equilibrium crystals (Xe on Cu sub strate141 and Pb on graphite134). For the equilibrium crystal of Pb grown on a graphite substrate, direct measurements of the exponent 0 via SEM yielded a value of 0 1.60, in the range of temperatures from 200- 300C, in close agreement with the Pokrovsky-Talapov transition139 and smaller than the prediction of the mean-field theory.137 Sharp edges have also been seen in some experiments, as in the case of Au,142,143 but they have received less theoretical attention. At the roughening transition, the crystal curvature is predicted to jump from a finite universal value for T = Tp+ to zero for T = Tfl-,130138144 as contrasted to the prediction of continuously vanish ing curvature.136 Similarly, the facet size should decrease with T and vanish as T + Tp", like exp (-C/VCT^ T)),113 as opposed to the behav ior as (T^ t)^V2.136 -phe j^p the crystal curvature has been ex actly related59 to the superfluid jump of the Kosterlitz-Thouless trans ition in the two-dimensional Coulomb gas.113130134141 In addition, the facet size of Ag2S crystals128 was found (qualitatively) to de crease, approaching Tp, in an exponential manner. 101 growth rates (oj V); at high growth rates interfacial stability will depend on the competitive effects of the thermal and the capillarity fields. Incorporation of the effect of interfacial kinetics on the stability leads to conclusions analogous to those mentioned earlier that the stability-instability demarcation is virtually unaffected by the kinet ics. Slow kinetics are expected to enhance stability, while rapid kin etics will have little effect on it. The mathematical analysis that leads to the above mentioned conclusions will be given in Appendix III. Experiments on stability The commonly used procedure to verify the CS and MS predictions is to plot G^/V vs. CQ and determine the demarcation line between the cell ular or dendritic substructure region and that with no substructure. The slope of the experimental line can then be compared to those of the CS and MS theories according to eqs. (53) and (55). However, the theoret ical slopes are related to the diffusion coefficient D, which is often poorly known, and to the partition ratio k. Because of the above, and also the fact that the predictions of both theories are almost identical at low growth rates (or small G^/V), it is difficult to discriminate the CS and MS theories as far as agreement with the experimental results is concerned. Nevertheless, there are several experiments which are sup portive of dynamic theories. These include direct observation of the interface shape during evolution of instability264 265 and determination of the onset of the instability while varying the growth condi tions.266-269 The influence of thermal diffusion270 (Soret effect) at large thermal gradients, convection,271 thermosolutal convection under microgravity conditions,272 and recent experimental results during rapid 68 rates; 3) the required close control of the interfacial supercooling and, therefore, its accurate determination; and 4) the problems associ ated with analyzing the growth data analysis when the experimental range of AT's is small or it falls close to the intersection of the two MNG and PNG kinetic regimes for a given sample size. Nevertheless, there are a couple of experimental studies which rather accurately have veri fied the 2DN assisted growth for faceted metallic interfaces.26399>182 Screw dislocation-assisted growth (SPG) Most often crystal interfaces contain lattice defects such as screw dislocations and these can have a tremendous effect on the growth kinet ics. The importance of dislocations in crystal growth was first pro posed by Frank,183 who indicated that they could enhance the growth rate of singular faces by many orders of magnitude relative to the 2DNG rates. For the past thirty years since then, researchers have observed spirals caused by growth dislocations on a large variety of metallic and non-metallic crystals grown from the vapor and solutions,16 and on a smaller number grown from the melt.184 When a dislocation intersects the interface, it gives rise to a step initiating at the intersection, provided that the dislocation has a Burgers vector (Â£) with a component normal to the interface.185 Since the step is anchored, it will rotate around the dislocation and wind up actually in a spiral (see Fig. 7c). The edges of this spiral now pro vide a continuous source of growth sites. After a transient period, the spiral is assumed to reach a steady state, becoming isotropic, or, in terms of continuous mechanics, an archimedian spiral. This further means that the spiral becomes completely rounded since anisotropy of the 119 maximum resolution of 4 pV/cm. The usually selected 20 pV full range resulted in a temperature reading accuracy of .0125C. In addition, depending on the experimental procedures, as it will be discussed later, the thermocouple outputs were also read by the nanovoltmeter or the multimeter. After the sample had been positioned inside the observation baths, it was electrically connected to a Keithley-181 model nanovoltmeter which measured the thermoelectric emf output of the sample with a pre cision of 5nV; the sample was also electrically connected to a Hewlett- Packard model 3456A Voltmeter and to a Keithley model 220 programmable current source. The latter instruments and the nanovoltmeter were interfaced to an Apple lie microcomputer using an IEEE-488 (GPIB) inter face bus card. The heaters 1, 2, and 3, shown in Fig. 15, were used to station the two S/L interfaces, respectively, in a desired position during the preliminary steps of an experimental run; they were turned off during the growth kinetics measurements. The heaters were made out of Kanthal wire ($ = .051 cm, .0708 Q/cm resistance), which was wound into a two or three turn coil, were connected to a 12V battery through a variable re sistor. The leads of the heater 3 were inserted into two-hole ceramic tube such that the coil could be moved up and down the observation bath. The cell of the observation bath consisted of a copper frame (32 x 5x1 cm) with two circulating fluid inlets and outlets on its sides; the front and the back of the cell was enclosed by transparent plexi glass plates (.6 cm thick). A stereoscopic zoom microscope Nikon model SMZ-10 was used to observe the S/L interface with a magnification range APPENDIX IV INTERFACIAL STABILITY ANALYSIS The morphological stability of the S/L Ga interface is discussed in this Appendix. The analysis follows the linear perturbation theory form ulated by Coriell and Sekerka245253 and includes non-local equilibrium conditions at the interface (i, ,e. kinetics). Calculations have been per- formed based on the Ga growth data; the thermal gradients have been cal- culated from the heat transfer analysis presented in Appendix III. The stability criterion in terms of the real part of the time constant, o (see eq. (54) of the text), for the amplitude of a perturbation for a pure material is given as v b Li o = - K G (a + )] UA -2KT ra>2a} s s s k A m Lvvu? (A33) with UA = 1 MA/l,T L>T 1 yA + 2K a mt ^vmt The other parameters involved in eq. (A33) have been described previ- ously, except for p^ and p^, which are defined below. The extra terms in eq. (A33), as compared to the criterion for the dilute binary alloy (eq. (54)), account for the interfacial kinetics. Assuming that the growth rate can be expressed as V = = f(T^, AT), the coefficients p^ and p^ are given as 299 240 The second parameter 3 of the theory, which rather meant to "mod ify" the liquid self-diffusion coefficient for interfacial transport, can be obtained from the continuous growth rate equation as h RT2V c LD AT (95) where Vc is the growth rate in the continuous regime. However, since the theory assumes that continuous growth prevails above the break point in the growth rate curve (at AT = 3AT") and that the product Vn should be linear with AT in this regime, then the present (ill) data should be still in the transitional regime. Thus, a lower limit to the quantity 3 can be obtained as b > 5!_S P D LAT where V is the actual growth rate. For the (ill) interface at AT = 4C, the measured rate is about 1.25 cm/s. Thus, 3 must be at least .08 according to the above inequality. It is expected that for symmetrical molecules 3 should be in the order of ten. If this is the case, then the kinetic coefficient of the linear growth in the order of about 40 cm/s would be an almost acceptable high value. An upper limit for 3 could be estimated from the slope of the experimental high growth data linear equation, assuming that the continuous growth predicted by the theory should pass through the origin of a V vs. AT plot. Then 3 D L < .98 or 3 < .25 h RTZ Indeed, this upper value of 3 agrees well with the previously calculated values of interfacial diffusivity. Furthermore, if 3 is calculated via eq. (95) for the (001) interface, utilizing the experimental kinetic 2 E H > CD O 0 -1 -2 (111) Ga-. 01 wt%In V//g v//-g . 3 . 4 . 5 . 6 1/AT. *C"1 Figure 42 Comparison between the growth rates of Ca-.Ol wt% In in the direction parallel ( ) and antiparallel ( O ) to the gravity vector as a function of the interface super cooling; line represents the growth rate of pure Ca. 186 213 of ten for both interfaces, or, according to the diffuse interface growth model, by a factor of 10 P/g. Generalized Lateral Growth Model The two-dimensional nucleation assisted growth kinetics over both the mononuclear (MNG) and polynuclear (PNG) regimes (supercoolings from 1.5 to 3.5C and .6 to 1.45C and growth rates from 10-8 to 1500 pm/s and 10~- to 600 pm/s for the (111) and (001) interfaces, respectively) are well expressed by the following rate equation V = A (AT)1/2 exp(- |j) (, Z 7717172 .5/3 B ,.3/5 (1 + K2 (AT) A exp(- )) (69) Here Ki, K-?, and B are assumed, for the time being, to be independ ent of the growth parameters and A is the S/L interfacial area. It should be noted that B is a weak function of supercooling within the above mentioned range of supercoolings, but becomes strongly dependent on AT at higher supercoolings, as indicated later. The values of Kp K2, and B found by fitting (ill) and (001) crystal growth data to the proposed eq. (69) are given in the following Table. Table 10. Growth Rate Parameters of General 2DNG Rate Equation (111) Kl = 1.39 x 1017 K2 = 7.5 x 1018 B = 58.76 (001) K1 = 3.8 x 1017 K2 = 4.6 x 1019 B = 25.43 A in cm V in pm/sec A comparison between the experimental data and the calculated ones, by using eq. (69) in conjuction with the parameters in Table 10, is shown in Figs. 51 and 52 for the (ill) and (001) interfaces, 19 form or rule for prediction of the diffuseness of the interface for a given material and crystal direction. However, the model predicts6 that the resistance to motion is greatest for close-packed planes and, thus, their diffuseness will comparatively be quite small. 3) The theory, which has been reformulated for a fluid near its critical point30 (and received experimental support2431), provides a good description of spinodal decomposition3233 and glass formation.34 The present author believes that this theory's concept is very rea sonable about the nature of the S/L interface. Indeed, recent studies, to be discussed next, indirectly support this theory. However, there are several difficulties in "following" the analysis with regard to the motion of the interface, which stem primarily from the fact that it a) does not explicitly consider the effect of the driving force on the dif fuseness of the interface, and b) conceives the motion of the interface as an advancing averaged profile rather than as a cooperative process on an atomic scale, which is important for smooth interfaces. In a later development7 about the nature of the S/L interface, many aspects of the original diffuse interface theory were reintroduced via the concept of the many-level model/' Here the thickness of the inter face, i.e. its diffuseness, is considered a free parameter that can ad just itself in order to minimize the free energy of the interface (F); the latter is evaluated by introducing the Bragg-Williams35 approxima- * As contrasted to other models where the transition from solid to liquid is assumed to take place within a fixed and usually small num ber of layers, e.g. two-level or two-dimensional models. 104 where Vp is the diffusive speed (i.e. Vq = D/h). The above equation pre dicts that k -> kQ when V D/h (~5 m/s for Ga (ill) interface) and k -> 1 when V D/h. Although this model has been shown to agree with experi ments of high growth rates (V > 1 m/s),284 it cannot explain the observed increase283 in k at much lower rates (~1 pm/s) than the diffusive speed, assuming that D = D. It is clear that k depends more strongly on the interfacial supercooling (or growth rate) rather than the interface ori entation. For example, if a macroscopic interface grows at an average constant rate (e.g. Czochralski technique), its faceted and non-faceted regions will have equal growth rates. Accordingly, the facets will re quire a much higher supercooling than the off-facet area if it grows by the 2DNG mechanism; the larger driving force, in turn, results in a higher k value. Alternatively, for a given growth rate, the growth di rection "determines" the magnitude of the required driving force; there fore, orientation affects k indirectly through growth kinetics. Other factors that are expected to affect k are282 i) the relative mobility of the solute and solvent atoms and ii) the bonding strength of the solute atom to the crystal. Solute redistribution during growth This section is related to the bulk mass transfer during unidirec tional growth when the melt is convection free or that the solute trans port in the liquid is purely diffusive. The composition of solid and liquid as a function of distance solidified is shown in Fig. 13. The initial region of the solid before reaching CQ composition (steady state) is termed transient with a characteristic distance in the order of D/kV. The last part of the solidified ingot is the final transient with a 1/AT. T' Figure 44 Initial (111) growth rates of Ga-.12 wt% Tn a a function of the interface super cooling; (<>) effect of distance solidified on the growth rate, and ( ) growth rate of pure Ga. 189 150 the two growth modes, which accounts for the relatively large scatter of the data points for rates higher than about 6500 pm/s. The growth rates of the (111) interface as a function of the inter face and bulk supercooling for several samples are shown on a linear and log-log scale in Figs. 24 and 25, respectively. As can be seen, the bulk supercooling is higher than the interfacial one at growth rates higher than about 1 pm/s; for example, for growth rates in the order of 3, 350, and 1.9 x 10^ pm/s, the bulk supercooling is about .015, 1.6, and 45C, respectively, larger than the corresponding interfacial super cooling. At low growth rates, less than about 1 pm/s, the two super coolings are nearly equal, as revealed in Figs. 24 and 25. The differ ence between dislocation-free and dislocation-assisted kinetics is easily revealed from Fig. 26, where the growth rates are plotted on semi-log scale versus the reciprocal of the interfacial supercooling. Note that for graphical clarity the x-axis is shown in two different scales in this figure. The data are for several samples, some with cross-sectional area of A and others with 4.5A. The kinetics data for each growth mode, dislocation-free and dislocation-assisted, are pre sented separately in more detail in the following section. Dislocation-Free (111) Growth Kinetics The dislocation-free data for the (ill) interface, as shown in Figs. 23 and 26, represent the growth behavior of a total of 15 samples'' * In reality, this is the number of samples whose kinetics data extend at least two orders of magnitude in growth rates; otherwise, the num ber of samples tested far exceeds the above mentioned one. Further more, it should be noted that all the (111) graphs represent growth data from 15 samples, except where otherwise stated. 231 the anisotropy 6 = r-p/r^ is about .1, oe is calculated for the (ill) interface to be about 6 ergs/cm*- at AT = 1.5C. Nevertheless, since oe is assumed to be independent of supercooling, the criterion AG" = KT is only satisfied at supercoolings much higher than 10C. In conclusion, based on the experimentally determined exponential terms of the 2DNG (111) and (001) rate equations, the criterion AG" = KT fails to explain the observed deviation in the growth kinetics. The case of a supercooling dependent edge free energy was examined earlier. It was shown that as AT increases oe decreases and finally be comes zero in a fashion analogous to that for the thermal roughening transition. At first glance, such behavior seems to be in the opposite direction from what one would expect; since T^ is expected to be not very far from Tm (but larger), og should increase with decreasing T, or at least it should remain constant. However, this idea abandons the dynamic morphology of the interface because of the lateral growth pro cess, as well as the state of the liquid near the spreading steps. At high supercoolings, an interface that is growing by any of the stepwise mechanisms (2DNG or SDG) is not only covered by of 2D clusters and therefore of heavily kinked edges, but also extends itself over several atomic planes regardless of its diffuseness. While the former is due to the increased 2DN rate, the latter is because of the nature of the lateral growth processes. Therefore, the top layers of the interface would "look" alike to the MC simulations computer drawings of Fig. 6 as well as the resultant growth kinetics, as shown in Fig. 57. Under such conditions, an interfacial step, which is a rather unique feature in the background of the interface at small supercoolings, cannot be 281 This Appendix deals with the heat transfer problem during steady- state unconstrained growth into a supercooled melt. Its basic concept is that the heat evolved at the interface (in proportion to the growth rate) must be transported away from and into the heat sink (coolant) via a thermal resistance; the latter, as shown in Fig. A-5, consists of the Ga, the wall of the capillary tube, and a cooling fluid boundary layer surrounding the tube. The analytical model is based on the original formulation'' of Michaels et al.181 for their experiments on the growth kinetics of the (0001) ice/water interface in capillary tubes. It was later modified by Abbaschian and Ravitz,2 who used it to determine the interface supercooling during a previous Ga growth kinetics study. Both analyses .have been augmented with the assumption that solids and liquids have the same thermal properties, equal to the average properties of the two, as discussed in more detail later. In the present calculations, this assumption was removed and the calculated results at various growth rates have been compared with the actual interface temperature measure ments obtained by the Seebeck technique. Analytical Model for Heat Flow Calculations The geometry of the system used for the heat transfer analysis is shown in Fig. A-5. It is assumed that the S/L interface is planar and normal to the axis of the capillary tube and that it advances into the * Actually, the original analysis was given by Hillig,178 who assumed that the temperature of the outer wall of the capillary was equal to T^. Michaels et al.181 removed this assumption by introducing the heat transfer coefficient between the tube surface and the bulk of the cool ing bath; the latter is available for certain geometries such as for a cylinder in cross-flow.413 338 397. S. F. French, D. J. Sanders, and G. W. Ingle, J. Phys. Chem., 42 (1938) 265. 398. W. J. Svirbely and S. M. Selis, J. Phys. Chem., 58 (1954) 33. 399. R. M. Evans and R. I. Jaffee, Trans. AIME, 194 (1952) 153. 400. J. P. Denny, J. H. Hamilton, and J. R. Lewis, Trans. AIME, 194 (1952) 39. 401. M. Hansen, Constitution of Binary Alloys, 2nd ed. (McGraw-Hill, New York, 1958), p. 745. 402. A. Gokhale, private communication. 403. P. E. Eriksson, S. J. Larsson, and A. Lodding, Z. Naturforsch, 29a (1974) 893. 404. K. Suzuki and 0. Vemura, J. Phys. Chem. Sol., 32 (1971) 1801. 405. B. Predel and A. Ernn, J. of the Less-Common Metals, 19 (1969) 385. 406. 0. J. Kleppa, J. Chem. Phys., 18 (1950) 1331. 407. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford Univ. Press, London, 1959). 408. F. C. Frank, Disc. Faraday Soc., 5 (1949) 189. 409. T. W. Clyne, Mat. Sci. Eng., 65 (1984) 111. 410. R. J. Schaeffer and M. E. Glicksman, J. Cryst. Growth, 5 (1969) 44. 411. N. Shamsundar and E. M. Sparrow, J. Heat Transfer, 97 (1975) 333. 412. C. G. Levi Rodriguez, Ph.D. Thesis, Univ. of Illinois, at Urbana- Champaign (1981). 413. E. R. G. Eckert and R. M. Drake, Jr., Heat and Mass Transfer, 2nd Ed. (McGraw-Hill, New York, 1959), p. 239. 414. T. C. Wang, unpublished work, Univ. of Florida (1985). 415. L. J. Briggs, J. Chem. Phys., 26 (1957) 784. 416. C. Y. Ho, R. W. Powell, and P. E. Liley, J. of Phys. and Chem. Ref. Data, 1 (1972) 279. 417. CRC Handbook of Chemistry and Physics, 65th ed., R. C. Weast, ed. (CRC Press, Boca Raton, FL). 172 the quantity log (V/A) as a function of the 1/AT results in a straight line for four samples with different capillary cross sections. The equation for the regression line, as determined from least square anal ysis, is given as log j = 17.4702 11.0438 AT with a coefficient of determination and correlation of 0.99 and 0.995, respectively. The growth rate equation is, therefore, determined as V = 2.948 x 109 A exp (- 25.428/AT) (75) where V is the growth rate in pm/s and A is the S/L interfacial area in pm^. The features of this region, as well as the form of the growth rate equation, as indicated by eq. (75), show that the growth behavior of this region is in good qualitative agreement with the mononuclear growth theory. PNG region In this region, the data points are still exponential functions of (1/AT), but with a smaller slope than that for the MNG region, as shown by the plot of log (V) vs. 1/AT in Fig. (35). However, in contrast with the (111) PNG region, the growth data for the (001) interface of 10 samples fall onto two approximately parallel lines A and B; line A is composed from data of four samples and line B from six samples. The growth rate equation as determined from the regression analysis are line A: V = 6.03 x 10-* exp (- 9.7/AT) (76) line B: V = 2.4 x 105 exp (- 9.78/AT) where V is the growth rate in pm/s. The coefficients of determination and correlation are, respectively, .991 and .995 for line A, and .984 325 139. V. L. Pokrovsky and A. L. Talapov, Phys. Rev. Lett., 42 (1979) 65. 140. E. E. Gruber and W. W. Mullins, J. Phys. Chem. Solids, 28 (1967) 875. 141. M. Jaubert, A. Glachant, M. Bienfait, and G. Boato, Phys. Rev. Lett., 46 (1981) 1679. 142. J. Metois and J. C. Heyraud, J. Cryst. Growth, 57 (1982) 487. 143. J. C. Heyraud and J. J. Metois, Surf. Sci., 128 (1983) 334. 144. D. S. Fisher and J. D. Weeks, Phys. Rev. Lett., 50 (1983) 1077. 145. J. D. Weeks and G. H. Gilmer, Adv. Chem. Phys., 40 (1979) 157. 146. J. P. Van der Eerden, C. van Leeuwen, P. Bennema, W. L. van der Kruk, and B. P. Th. Veltman, J. Appl. Phys., 48 (1977) 2124. 147. C. E. Miller, J. Cryst. Growth, 42 (1977) 357. 148. J. R. Green and W. T. Griffith, J. Cryst. Growth, 5 (1969) 171. 149. W. T. Griffith, J. Cryst. Growth, 47 (1979) 473. 150. T. Watanable, J. Cryst. Growth, 50 (1980) 729. 151. M. E. Glicksman and R. J. Schaefer, J. Cryst. Growth, 1 (1967) 297. 152. W. B. Hillig and D. Turnbull, J. Chem. Phys., 24 (1956) 914. 153. D. Turnbull, J. Chem. Phys., 66 (1962) 609. 154. D. Turnbull, Solid State Phys., 3 (1956) 279. 155. A. G. Walton, in: Nucleation, A. C. Zettlemoyer, ed. (Marcel- Dekker, New York, 1969), p. 245. 156. D. Froschhammer, H. M. Tensi, H. Zoller, and V. Feurer, Met. Trans., Bll (1980) 169. 157. R. Becker, Disc. Faraday Soc., 5 (1949) 45. 158. R. Becker and W. Doring, Ann. Physik (Leipzig), 24 (1935) 719. 159. R. Kaischev and I. M. Stranskii, Z. Phys. Chem., A170 (1934) 295. 160. M. Volmer and M. Marder, Z. Phys. Chem., A154 (1931) 97. 161. J. B. Zeldovich, Acta Physicocima, USSR, 18 (1943) 1. 284 coefficient across the tube-coolant boundary. The latter is given by the following empirical equation413 h = Nu.K,/2r (A8) d b o where Nu^ is the Nusselt number for the tube and is the thermal con ductivity of the coolant fluid. (4) Condition at the S/L interface T = T at z. = z =0, for 0 < r < r. (A9) s L L s i (5) Heat balance condition at the interface -Vp L = -Kt s L (3V3VZl=o K (3T /3Z ) n s s s z =0 s (A10) Introducing the following dimensionless variables, 6s>t (Ts,L V/(T V- zs,l/U. and r' = r/r. (All) equations (A1) (A10) become dimensionless. Assuming that the solu tions to equations (A1) and (A2) in the dimensionless form have the fol lowing forms 0 (r', z' ) = R (r1) Z(z') (A12) s s s s 0^(r' z) = Ra(r') Zt(zp (A13) the heat conduction equations (A1) and (A2) become (R /R ) + (R /r'R ) = (Vr./ic ) (Z /Z ) (Z /Z ) (A14) ss ss isss ss and Since the right hand sides and the left hand sides of the above equa tions are functions of different variables, the only way they can be equal is for the expressions of either side of both equations to be equal to some constants. To assure real solutions for the radial parts of the two equations these two constants must be negative. 210 the (111) and (001) interfaces, the SDG kinetics are better correlated by an equation in the form of AT2 ATc V = KD AT t3nh (AT } (88) c Here the parameters Kp and ATC are constants, given below in Table 9, as determined by curve-fitting the SDG (111) and (001) experimental growth data in eq. (88). This rather illustrates the problem with using the O parabolic law, V c AT over a limited experimental range to describe the SDG kinetics. As a matter of fact, most of the experimental studies on SDG kinetics conclude on relationships in teh form of V ATn with 1.5 < n < 2.5, which is not surprising based on teh form of eq. (88). Certainly, the dislocation-assisted growth data can be fitted, within isolated (V, AT) ranges, to an equation in the form of V = K ATn with n close to 2. Nevertheless, such an interpretation of the kinetics is of limited importance since the growth data are proportional to AT tanh (1/AT) over the entire experimental range, as discussed later. Table 9. Experimental and Theoretical Vlaues of SDG Parameters Interface (pm/sec'C) ate (c) (111) Theoretical Experimental 1.775 7.3 x 104 1422 (001) 5.7 x 104 1968 1.1 175 V = 6300 AT (78) with a coefficient of correlation equal to .97. The rate equations of the dislocation-assisted and dislocation-free growth data for both (111) and (001) interfaces up to supercoolings of about 3.5 and 1.5C, respectively, are summarized in Table 7. The ex perimental growth kinetics for supercoolings higher than the above men tioned ones are quantitatively described in the Discussion chapter. In-Doped (111) Ga Interface The In-doped Ga growth rates have been measured as a function of distance solidified and interface supercooling for two dopant levels, 0.01 and 0.12 wt% In. In addition, the effect of growth direction, with respect to the gravity vector, was also determined by allowing the growth to proceed parallel or antiparallel to the gravity vector. For each composition, the results are presented in the next section in accord with the above mentioned order of the solidification rate vari ables. It should be noted that all the growth rates mentioned here are dislocation-free rates; also, unless indicated otherwise, the growth direction is antiparallel to the gravity force, g. Ga .01 wt% In As mentioned earlier, the growth rates of the doped Ga, unlike those of the pure Ga, were a function of the distance solidified at a constant bulk supercooling. The results for three constant but differ ent bulk supercoolings are shown in Fig. 37. It can be seen that the growth rate decreased gradually as the interface moved along the capil lary until interfacial breakdown, indicated by arrows in Fig. 37, had Composition Figure 13 Solute redistribution as a function of distance solidified during unidirectional solidification with no convection. 105 26 deposited spheres on the crystalline substrate. Under this concept, the model441,5 shows how the disorder gradually progresses with distance from the interface into the liquid. The beginning of disorder, on the first deposited layer, is accounted by the existence of "channels"44 (p. 6) between atom clusters, whose width does not allow for an atom to be placed in direct contact with the substrate. As the next layer is de posited, new sites are eventually created that do not continue to follow the crystal lattice periodicity, which, when occupied, lead to disorder. However, the very existence of the formed "channels" is explained by the peculiarity of the hep or fee close-packed crystal face that has two interpenetrating sublattices of equal occupation probabilities.4 The density profiles calculated at the interface also show a minimum associ ated with the existence of poor wetting; on the other hand, perfect wet ting conditions were found when the atoms were placed in such a way that no octahedral holes were formed.46 Thermodynamic calculations from these models allow for an estimate of the interfacial surface energy (se,), which are in qualitative agreement with experimental findings. In conclusion, these models give a picture of the structure of the interface which seems reasonable and can calculate osÂ£. However, they neglect the thermal motion of atoms and assume an undisturbed crystal lattice up to the S/L interface, eliminating, therefore, any kind of interfacial roughness. Computer simulation of MC and MD techniques are linked to micro scopic properties and describe the motion of the molecules. In contrast with the MD technique, which is a deterministic process, the MC tech nique is probabilistic. Another difference is that time scale is only 306 Program //I 1 REM THIS PROGRAM RECORDS THE BULK TEMPERATURE WITH THE HP-VOLTMETER 3456A AND THE SEEBECK EMF WITH THE K EITH L EY-NAN OVO LTM ET E R 181. 5 DIM 01(500),V2<500),T<1000) DIM V1* <50 0) ,V2* < 50 0) 10 N = 2000 30 PRINT "MAKE DATA<1) OR SAVE DATA(2)"INPUT K 40 IF K = 1 THEN GOTO 65 50 IF K = 2 THEN GOTO 410 0 GOTO 30 65 ZS = CHRS (26) 66 D* = ": REM DS= 7 REM 68 REM IN THE NEXT SECTION OF THE PROGRAM THE DATA AR E REITREVED FROM THE ABOVE MENTIONED INSTRUMENTS. 70 PR# 3 SO IN# 3 90 PRINT "SCI" 100 PRINT "RA" 110 PRINT "LL" 120 PRINT LFl" 130 PRINT "WTX." ;Z$; "R1X" 140 PRINT "LIT*" ;Z* ; F1R2" 160 IF PEEK < 1*286) > 127 THEN GOTO 180 170 GOTO 1*0 180 NUM = NUM + 1 184 REM *^****-k***************** 185 REM THE NEXT STATEMENT SETS THE FREQUENCY OF THE MEASUREMENTS. 190 FOR P = 1 TO N: NEXT 200 PRINT "WTX";Z* 210 PRINT "RDE";ZS;: INPUT V1*(NUM) 230 PRINT "RDV";Z*;: INPUT V2$ 250 GOTO ISO 260 PRINT "LA" 270 PRINT "UT" 280 PR# 0 290 IN# 0 295 REM ********)************* 29* REM THE FOLLOWING STATEMENTS SET-UP THE PRINTER A NB PRINT THE DATA. 300 PR# 1 310 PRINT CHRS (9);"120N" 320 PRINT CHR$ (27);"Q" 330 FOR I = 1 TO NUM 340 VI(I) = VAL ( MID* < V1 < I ) 5,1 5 ) ) 1000000 350 V2 = VAL + .0492097 V2 3 .43769 V2 83 the diffusiveness of the interface increases with AT (i.e. g decreases), then ae should decrease. b) The growth kinetics in the transitional regime are not described quantitatively. c) The quantitative parameter, AT^, of the theory is not predicted, instead it has to be obtained ex perimentally. Nevertheless, this model is the only existing phenomeno logical approach attempting to describe the lateral to continuous growth transition at high AT's. Growth Kinetics of Doped Materials The presence of impurities in the melt is expected to affect growth kinetics in several ways. However, the role and the mechanism of the influence of the impurity on the interfacial growth processes have not yet been studied in detail. Needless to say, this question is of im portance and considerable interest because small quantities of impur ities are almost always present in the melt, intentionally in the case of doped semiconducting substances or unintentionally in other cases. As discussed later, the possibility of negligible amounts of impurities in the melt and its influence on kinetics has created questions regard ing the reliability of reported crystal growth kinetics for supposedly pure materials. Moreover, the effects of these impurities and their understanding would help to better understand the crystal growth mechan ism of pure materials. Thus, as a whole, the complicated problem of solute influence on growth kinetics requires further attention and in vestigation. This problem will be reviewed rather qualitatively since for growth from melt the existing theories are mostly empirical or just deal with the diffusion-diffusionless growth mode at high growth rates. Especial 2 1. 5 w \ E 3. 1 o I X > . 5 0 0 .5 1 1. 5 2 2. 5 3 AT, *C Figure 31 Dislocation-free and Dislocation-assisted growth rates of the (001) interface as a function of the interface supercooling; dashed curves represent the 2DNG and SDG rate equations, as given in Table 7. 165 Step Edge Free Energy, ergs/cm* SteP Ed9e FrcG Energy, ergs/cm2 13 <0D1) 12 11 10 0 0 0 0 CD o o o O 0 o o % % o o 9 .5 1.0 1.5 2. Interfacial Supercooling, *C 22 21 20 19 18 17 16 1 2 3 4 5 Interfacial Supercooling. 'C Figure 53 The step edge free energy as a function of the interfacial supercooling. a) Ge (AT) for steps on the 001 interface b) a0 (AT) for steps on the (111) interface. 233 distinguished from its many similar neighbors whose effect can also be thought as a "catalytic" one. Similarly, the addition of the peripheral area of a new cluster on the multistepped interface would hardly alter its total energy. Furthermore, from the diffuseness point of view, the liquid is ex tensively clustered in both directions normal and parallel to the inter face, which would result in steps of quite large width, and, therefore, of negligible step edge energy. As shown earlier, it was found that oe goes to zero at supercool ings of 4.75 and 2.2C for the (ill) and (001) interfaces, respectively. The present experimental results were described well in the previous section by taking into account the kinetic roughening of the interfaces (i.e. assuming that oe depends on AT). It was also shown that approach ing the roughening transition, the dislocation-assisted growth rates become comparable to the dislocation-free. Such a behavior cannot be explained otherwise; for example, assuming that the high AT's growth rates are the sum of the 2DNG and SDG rates determined from the lower supercoolings.335 This comparison is shown in Fig. 58 for the (111) interface, where it is realized that the actual growth rates are higher than the calculated ones, shown as the dashed line in this figure. At supercoolings below that of the kinetic roughening (TRG region), the mixed (2DNG/SDG) rates depend approximately linearly on the super cooling, as shown in Figs. 23 and 31 and as discussed earlier. This linear growth curve extrapolates to the right of the origin on the supercooling axis. It should be noted, however, that this linear (V, AT) curve neither implies a different growth mechanism nor can be 308 Program #2 5 REM THIS PROGRAM TAKES A SEEBECK EMF READING WITH T HE KEITHLEY-181 NANOVOLTMETER WHEN THE CURRENT SURC E IS OFF; TURNS ON THE KEITHLEY-220 CURRENT SOURCE A ND RECORDS THE POTENTIAL ACROSS THE SAMPLE WITH THE HP-3456A VOLTMETER. 20 N = 300 25 M = 300 30 DIM V1 M ) V2* < M) V3 < 500) DU < 50 0 > 35 DIM VI<50CO ,02(500) 40 2.% CHR$ i 2o) 50 D* = " : REM DÂ£= < CTRL-D > 55 REM ************************** 56 REM THE NEXT STATEMENTS SET-UP THE INSTRUMENTS 0 PR# 3 70 IN# 3 SO PRINT "SCI" 90 PRINT RA" 1 0 0 PRINT "LL" 1 10 PRINT "LFl" 1 20 PRINT "WTX" ; Z* ; "R2X" 1 30 PRINT "WT 6" ;z*; " F1 R 2" 1 40 PRINT "WT 6" ; ; " + 100 E-1 ST I" 1 42 PRINT "WT." ; Z%; "ROP1FOX" ;"I5E-3" ;"VI " 1* V < A 1 50 IF PEEK ( - 16 236) > 127 THEN GOTO 1 70 160 GOTO 150 1 70 NUM = NUM + 1 1 75 REM ************************* 176 REM STATEMENTS 130,210 SET THE READINGS INTERVAL. 130 FOR P = 1 TO N: NEXT 135 REM ************************* 186 REM NEXT THE SEEBECK EMF IS RETREVED. 190 PRINT "WTX";ZS 200 PRINT "RDE";Z*;: INPUT VIS(MUM) 203 REM ************************* 204 REM NEXT THE CURRENT SOURCE OPERATES. 205 PRINT "WT,"; Z* ; "FIX" 210 FOR P = 1 TO M: NEXT 215 REM ************************* 216 REM NEXT THE POTENTIAL DROP ACROSS THE SAMPLE IS RE CORDED 220 PRINT RDV";Z*;: INPUT V2*(NUM) 221 REM ************************ 222 REM NEXT THE CURRENT IS OFF 225 PRINT "WT,";Z$;"FOX" 230 IF PEEK < 1287) > 127 THEN GOTO 250 240 GOTO 170 245 REM ************************ 246 REM FOLLOWING THE DATA ARE PRINTED tCODED FORM) 0 N THE APPLE I Ie SCREEN 250 PRINT "LA" 2 (/) \ E zl r O i X > 1. 5 S e eP Q h. d 11) Â£ s / 4] B E0 0 P 0 0 0 W 0 S / Q eQ S/ 0 0 ' g I 0 0 0 n ffer 0 7a 0 EKBBFlFHFffi et0^ B/ /Q 00 EP Figure 54 The (111) and (001) growth rates as a function of the interfacial supercooling. The dashed lines are calculated in accord with the general 2DN0 rate equation "corrected" for and supercooling dependent 0 . 226 20 tion, and depends on two parameters of the model, namely 3 and y given as AG o v AW 3 KT and y KT here W = EsZ (Ess + EZZ )/2 is the mixing energy, Es is the bond energy between unlike molecules and Ess, E^ are the bond energies between solid-like and liquid-like molecules, respectively; K is the Boltzman's constant. Numerical calculations show that the interface under equilibrium is almost sharp for y > 3 and increases its diffuseness with decreasing y. It can also be shown that the roughness of the interface defined as1036 U U S = U (12) o where UQ is the surface energy of a flat surface and U that of the act ual interface. The latter increases with decreasing y, with a sharp rise at y ~2.5. This is expected since U is related to the average num ber of the broken bonds (excess interfacial energy).37 When the interface is undercooled, AGV < 0, the theory shows a pro nounced feature. The region of positive values of the parameters 3 and y can be divided into two subregions, as shown in Fig. 4. In region A there are two solutions, each corresponding to a minimum and a maximum of F, respectively, while in region B there are no such solutions. In * The Bragg-Williams or Molecular or Mean Field approximation35 of stat istical mechanics assumes that some average value E can be taken as the internal energy for all possible interfacial configurations and that this value is the most probable value. Then, the free energy of the interface becomes a solvable quantity. Qualitatively speaking, this approximation assumes a random distribution of atoms in each layer; therefore, clustering of atoms is not treated. Log CV), pm/s 1/AT, C Figure 46 Initial (111) growth rates of Ga-.12 wt% In growth in the direction parallel to the gravity vector as a function of the interface supercooling; () effect of distance solidified, and ( ) growth rate of pure Ga. 192 177 Figure 37 Growth rates as a function of distance solidified of Ga-.01wt%In at different bulk supercoolings; ( t ) indicates interfacial breakdown. 36 37 38 39 40 41 42 43 44 45 46 171 177 179 181 184 185 186 188 189 191 192 The logarithm of the (001) low growth rates (MNG) nor malized for the S/L interfacial area plotted versus the reciprocal of the interface supercooling Growth rates as a function of distance solidified of Ga-.01 wt% In at different bulk supercoolings; (f ) indicates interfacial breakdown Photographs of the growth front of Ga doped with .01 wt% In showing the entrapped In rich bands (lighter region) X 40 Initial (ill) growth rates of Ga-.01 wt% In as a func tion of the interface supercooling; (O ') effect of distance solidified on the growth rate, and ( ) growth rate of pure Ga Effect of distance solidified on the growth rate of Ga-.01 wt% In grown in the direction parallel to the gravity vector (a,b), and comparison with that grown in the antiparallel direction (a) Initial (111) growth rates of Ga-.01 wt% In grown in the direction parallel to the gravity vector; () effect of distance solidified on the growth rate, and ( ) growth rate of pure Ga Comparison between the growth rates of Ga-.01 wt% In in the direction parallel ( ) and antiparallel ( O ) to the gravity vector as a function of the interface super cooling; line represents the growth rate of pure Ga Growth behavior of Ga-.12 wt% In (111) interface; a) Growth rates as a function of distance solidified, b) Growth front of Ga-.12 wt% In, X 40; solid shows as darker regions Initial (111) growth rates of Ga-.12 wt% In as a function of the interface supercooling; (O) effect of distance solidified on the growth rate, and ( ) growth rate of pure Ga Initial (ill) growth rates of Ga-.01 wt% In ( O ) and Ga-.12 wt% In ( <^> ) as a function of the interface supercooling; line represents the growth rate of pure Ga Initial (ill) growth rates of Ga-.12 wt% In growth in the direction parallel to the gravity vector as a function of the interface supercooling; () effect of distance solidified, and ( ) growth rate of pure Ga xv i GROWTH RATE, mm/sec Figure 24 Growth rates of the (111) interface as a function of the interfacial and the bulk supercooling. 151 10 (or ledges) of these terraces that are characterized by a step height h; c) the kinks, or jogs, which can be either positive or negative; and d) the surface adatoms or vacancies. From energetic considerations, as understood in terms of the number of nearest neighbors, adatoms "prefer" to attach themselves first at kink sites, second at edges, and lastly on the terraces, where it is bonded to only one side. With this line of reasoning, then, atoms coming from the bulk liquid are incorporated only at kinks, and as most crystal growth theories imply,18 growth is strongly controlled by the kink-sites. Although the above mentioned features are understood in the case of an interface between a solid and a vapor where one explicitly can draw a surface contour after deciding which phase a given atom is in, for S/L interfaces there is considerable ambiguity about the location of the interface on an atomic scale. However, the interfacial features (a-c) can still be observed in a diffuse interface, as shown schematically in Fig. lb. Thus, regardless of the nature of the interface, one can refer, for example, to kinks and edges when discussing the atomistics of the growth processes. Thermodynamics of S/L Interfaces Solidification is a first order change, and, as such, there is dis continuity in the internal energy, enthalpy, and entropy associated with the change of state.19 Furthermore, the transformation is spatially discontinuous, as it begins with nucleation and proceeds with a growth process that takes place in a small portion of the volume occupied by the system, namely, at the interface between the existing nucleus (crys tal seed or substrate) and the liquid. The equilibrium thermodynamic 25 for flat planes or simple structures, but not for some complex struc tures. 4 1 d) In spite of the limitations of this model, the success of its predictions is generally good, particularly for the extreme cases of very smooth and very rough interfaces.262734 Other models The goal of most other theoretical models of the S/L interface is the determination of the structural characteristics of the interface that can then be used for the calculation of thermodynamic properties which are of experimental interest; the majority of these models follow the same approaches that have been applied for modeling bulk liquids.4 Therefore, these are concerned with spherical (monoatomic) molecules that interact with the (most frequently used) Lennard-Jones, 12-6, potential.42 The L-J potential, which excludes higher than pair contri bution to the internal energy, is a good representation of rare gasses and its simple form makes it ideal for computer calculations. The model approach can be classified into three groups:4 a) hard-sphere, b) computer simulations (CS); molecular dynamics (MD), or Monte Carlo (MC), and c) perturbation theories. In the Bernal model (hard-sphere),43 the liquid as a dense random packing of hard spheres is set in contact with a crystal face, usually with hexagonal symmetry (i.e. FCC (111), HCP (0001)). Computer algor ithms of the Bernal model have been developed4 based on tetrahedral packing where each new sphere is placed in the "pocket" of previously Growth Rate, /m/s 2 x 10 1.5 x 10 10 5 x 10' 2DNG + SDG (111) OO o 8 / o o 0 o ft 8 / > / / o o o ft o o 3 GO o/ /o 6 o / / 0/< / S' oO Qd o Dislocation Assisted OO o o o L ^rnnrf00 <~* 6 8 CD O Dislocation Free (mini i>' m GCP OOO OO Figure 58 AT, C The (111) growth rates versus the interface supercooling compared to the combined mode of 2DNG + SDG growth rates (dashed line) at high supercoolings. 234 73 (39)), as discussed later, for large supercoolings reduces to an equa tion in the form V = A' AT B' (43) where A' and B' are constants. Note: if eq. (43) is extrapolated to V = 0, it does not go through the origin, but intersects the AT axis at a positive value. It should be mentioned that none of the above discussed transitions has ever been found experimentally for growth from a metallic melt. The parabolic to linear transition in the BCF law has been verified through several studies of solution growth.181193 Continuous Growth (CG) The model of continuous growth, being among the earliest ideas of growth kinetics, is largely due to Wilson194 and Frenkel195 (W-F). It assumes that the interface is "ideally rough" so that all interfacial sites are equivalent and probable growth sites. The net growth rate then is supposed to be the difference between the solidifying and melt ing rates of the atoms at the interface. Assuming also that the atom motion is a thermally activated process with activation energies as shown in Fig. 8, and from the reaction rate theory, the growth rate is given as15 4 >19 6 V = Vq exp (- j^) [1 exp (- ||^) ] (44) m where VQ is the equilibrium atom arrival rate and is the activation energy for the interfacial transport. As mentioned earlier, for practi cal reasons, is equated to the activation energy for self-diffusion in the liquid, Q^, and VQ av^ where a is the jump distance (interlayer spacing/interatomic distance) and is the atomic vibration frequency. 100 Furthermore, anisotropic interfacial kinetics leads to the translation of the perturbations parallel to the interface as they grow, with their peaks at an angle to the growth direction.259 This conclusion may ex plain to the existence of preferred directions for cellular and dendritic growth. Stability of undercooled pure melt During solidification of a pure liquid, morphological instability of the planar growth front can occur when the melt is supercooled. Insta bility then arises from thermal supercooling rather than the constitu tional supercooling; this is because the outflow of the latent heat into the supercooled liquid is aided by the protrusions and impeded by the in trusions at the interface ("point effect"). During solidification of an undercooled melt, the CS criterion al ways predicts instability, in contrast with experimental observations. According to the morphological stability theory, however, the interface can be stable despite the melt supercooling (G^ < 0) if Gs is suffi ciently large (see eq. (55)). Providing that the thermal steady state approximation holds (K^G^ + KSGS >0) and the kinetics effects are neg ligible, the original MS criterion can be used to predict morphological instability conditions of the interface by setting Gc equal to zero. The remaining terms then in the stability criterion are the destabilizing thermal field and the stabilizing capillarity term. Under conditions for which K^G^ + KSGS < 0, detailed analysis shows that the thermal field is stabilizing for large wavelength perturbations (a) -> 0) and is destabilizing for small wavelengths (co -> ). Since the capillarity term is always stabilizing and is rather important for large a), it is concluded that the interface will most likely be stable at low 84 attention will be given to the possible effects of the solute on the two-dimensional nucleation assisted growth, since they will be utilized in discussing the results of the current experiments on the influence of the In dopant on the 2DNG kinetics of Ga. The overall crystal growth rate will depend on the interaction of the solute with the "pure interfacial processes. The effect of impur ities on the 2DNG kinetics mainly comes from its influence5' on the step edge free energy (oe) and on the lateral spreading rate of the steps (ue). The first effect will alter the two-dimensional nucleation rate, while the latter will interfere with the coverage rate of the inter facial monolayer; hence, both MNG and PNG kinetics are expected to be affected by the impurity. Moreover, the additional transport process occurring at the interface, as compared with those encountered in the moving "pure" interface must be considered because of interfacial segre gation, as discussed elsewhere. The transport process is concerned with the diffusion of solute away from the interface on both the liquid and solid sides. On the other hand, the presence of the solute rich (or depleted, depending on the value of the partition coefficient) layer on the growth front alters the diffusional barrier for the host atoms in crossing the nucleus/L interface. The thickness of the interfacial sol ute rich layer, among other factors (i.e. steady or transient growth conditions) depends on the growth rate and the segregation coefficient k, as discussed later. * Note that the impurity influence on the equilibrium thermodynamics of the system (e.g. melting point temperature, heat of fusion, etc.) are not considered here. Since this study is concerned with very dilute solutions, these effects are quite small. 125 observation baths, linked by a single crystal of Ga of a specific crys tal orientation. The sample thus formed a circuit similar to those shown in Fig. 17. The last step of the sample preparation was its electrical con nection to the nanovoltmeter, multimeter, and the current source. This was achieved by inserting tungsten" electrodes (
liquid ends of the sample which were in turn connected via coaxial
liquid ends of the sample which were in turn connected via coaxial |