Citation
Growth kinetics of faceted solid-liquid interfaces

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:
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 )

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Full Text














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




Full Text
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), n£ 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 <¡>
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, Sg£, as a function of tempera
ture and orientation of the solid. Ss£ 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^> tion of interest ([hkl]) with respect to the principal crystal axes.
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 o = 2 (A35)
+ 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 £ 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 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 s£ 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 3£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(Y£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 > £/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 £, 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
oS£ 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 oS£ 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) = Ss¡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 £ (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, aS£, 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 = L£/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 og£ 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 oS£ 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
oS£ 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, og£ is obtained from measured supercooling
limits, together with a crystal-melt homogeneous nucleation theory in
which oS£ 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
s£


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 I
200 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 (£ 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 ES£ =
E££, 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 oS£ 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,
ES£, 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^ = ES£ 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 oS£ 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 (¡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)
£ {
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, Ss£. The values of
the S/L interface thermoelectric powers, Ss£, were determined directly
for each sample and were verified by the results of the previous
study.311 Direct determination of Ss£ 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 SS£ 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
PRINT
"SCI
*0
PRINT
" RA"
70
PRINT
"LL"
80
PRINT
"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
£s
S .(T
si m
T.)
(66)
= S .AT
si
where Sg£ 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, Sg£ was obtained in both ways, as discussed else
where;1 the direct method of determining Sg£ will be further discussed
later. In general, the coefficient Sg£ 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.
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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.
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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.
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348. R. W. Powell, M. J. Woodman, and R. P. Tye, Brit. J. Appl. Phys.,
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65.
350. C. J. Smithells, Metals Reference Book, 6th ed., E. A. Brandes,
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351. K. Wade and A. J. Banister, The Chemistry of Al, Ga, In, and T1
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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 C£ (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 (£) 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
) ailzi> (A15)
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 os£. 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$; "R1X"
140 PRINT "LIT*" ;Z* ; F1R2"
160 IF PEEK < 1*286) > 127 THEN GOTO 180
170 GOTO 1*0
180 NUM = NUM + 1
184 REM *^****-k*****************
185 REM THE NEXT STATEMENT SETS THE FREQUENCY OF THE
MEASUREMENTS.
190 FOR P = 1 TO N: NEXT
200 PRINT "WTX";Z*
210 PRINT "RDE";ZS;: INPUT V1*(NUM)
230 PRINT "RDV";Z*;: INPUT V2$ 240 IF PEEK < 16287) > 127 THEN GOTO 260
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 360 T(I) = 4.*7174E 3 + 25.39249 VI
+ .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
D£=
< 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)


£
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>'::>mm*
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
shielded copper cables to the leads of the nanovoltmeter (model 1506 low
thermal cable), of the current source (model 6011 Triaxial Lead), and of
the multimeter (Triaxial Shielded Cable). Extreme precautions were
taken in making all the electrical connections, as well as in connecting
the devices together, in order to minimize various noises and high con
tact resistances along the circuit which would be particularly trouble
some for the Seebeck measurements. To achieve this, the copper leads
were fusion welded to the W-electrodes and soldered to the instrument
leads via copper splice tubes by low-thermal cadmium-tin solder. All
the junctions were kept close together inside a Dewar's flask at a con
stant temperature. All instruments were connected to a common ground
and the length of the leads was kept minimum.
Interfacial Supercooling Measurements
Thermoelectric (Seebeck) Technique
A novel technique founded on thermoelectric principles was used to
directly measure the S/L interface temperature during growth from the
melt. This technique, described in detail elsewher,311_313 utilizes the
* The solubility of W in liquid Ga is negligible; for example, at 815C
is only -.001 wt%.310


V (resistance!), mm/s
Figmre 50 Comparison between optical and "resistance" growth rates; the latter were
determined simultaneously by two independent ways (see programs // 2, 3 in
Appendix IV).
202


70
where
o
1
2tty b
e
KTx
and f is a constant.
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 2iTrc from each other, arranged in a line of
length L, eq. (AO) holds with a new yQ = yQ/S when L < Airrc and yQ
2L/S when L > ATrrc. 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 as152189
V =
DL AT
(41)
4-rrT RTa V
m sx. m
Canh et al.25 have modified eq. (41) for diffuse interfaces with a
multiplicity factor |3/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 shown188 that the
effect of the stress field extends to a distance rs from the core of the
dislocation given as


114
technique is limited by the sharpness of the break in the temperature
time curve as the interface passes the thermocouple.
The thermal wave technique301 was developed to evaluate the growth
rate and interface supercooling from measurements of the attenuation of
a periodic thermal wave, induced in the liquid, as it travelled through
the S/L interface. The periodic variation at the interface allows only
for determination of the supercooling, i.e. absolute temperature mea
surements cannot be made. Aside from the experimental difficulties,
this technique has been subjected to criticism3 0 2 3 01* as it induces con
vection flows in the melt and it does not account for thermal losses
along the container walls. Moreover, it is restricted to small growth
rates (< 50 pm/s)301 and the reported kinetics using this techni
que224'225 have been conflicting. Later, a similar method was pro
posed305 that determines the growth parameters from an analysis of the
response of the interface to a periodic heat input, introduced by using
Peltier heating or cooling. Experimental results based on this tech
nique have not been reported in spite of the fact that the Peltier
effect has been widely used, particularly with semiconducting materials,
during crystal growth related experiments.306"
The single and double thermoelectric probe technique was pro
posed307 for measuring the interfacial temperature and velocity during
growth in a pure material. This method, which also disturbs the actual
growth process, is applicable under constrained growth conditions (i.e.
&L must be known). The accuracy of the technique depends on the Seebeck
* These experiments are not concerned with growth kinetics and, there
fore, are not reviewed here.


85
As mentioned previously, however, the effect of the solute on the
2DNG kinetics is two-fold; either on og or on ug. If the impurity bonds
strongly to the crystal surface (relative to the solvent atoms), the
nucleation process could be facilitated, for example, by nucleation of
clusters around the impurity. In return, this process would permit
measurable growth to take place in a region of supercoolings where the
faceted (dislocation-free) interface is essentially immobile without the
impurity atoms.206" Similarly, the adsorption of the solute atoms at
the periphery of the two-dimensional cluster can reduce the value of oe,
thus increasing the nucleation rate and, therefore, the growth rate.
Although there is not any experimental proof regarding a particular
trend concerning the effect of solute on og, it is believed70207 that,
if its concentration exceeds a certain limit, the edge free energy will
be reduced. Based on this, the loss of the faceted character of the
interface at high concentrations of solute is then understood. On the
other hand, the presence of the solute at the smooth interface (as an
adsorbed atom) can affect the spreading rate, ug, of the two-dimensional
nucleus. The overall effect on the macroscopic growth rate then depends
on the magnitude of ue.208,209 If ug is small (for example, at low
supercoolings), the solute atom will have enough time to be exchanged by
a host atom, and the growth rate will not then be affected. However, if
ug is large, the adsorbed impurities on the terrace will create an
energy barrier for the step motion, thus lowering the rate.
If the impurities enhance the nucleation process on all types of
interfaces, the growth rate would then be less sensitive to orienta
tion (smoothing agents).206


254
with the "stagnant film model," regardless of the amount of convectional
flows; at the S/L interface diffusion-advection fluxes will always domi
nate over the convection fluxes.
The same discussion applies for the .12 wt% In-doped Ga sample. In
addition, because of the ten-fold increase in CQ in comparison with the
.012 wt% alloy, (dp/dx)Q is ten times larger than those given in Table
12. Accordingly, this sample should have grown under more stable and
unstable conditions for the convection in the upwards and downwards
growth directions, respectively. However, regardless of the dopant (In)
level, the convection cannot be confirmed directly in these samples.
Nevertheless, if convection was present near the S/L interface during
growth in the downwards direction, the observed differences in the
growth kinetics for the two growth directions can be explained as fol
lows. During growth in the direction parallel to the gravity vector,
the interfacial liquid seems to be mixed by convective flows,the pattern
of which is likely to be, for example to the right of the capillary
axis, such that liquid from near the wall is brought to the interface in
a clockwise motion. Accordingly, liquid from the lower temperature and
solute concentration region transfers to the interface. Therefore, at a
given bulk supercooling and initial solute (In) concentration, the
growth rate in this direction should be higher than that for upwards
growth because the advancing interface "sees" a liquid which has less In
content than that for the interface grown under quiescent but otherwise
similar conditions. Since the higher the In content the slower the Ga
growth kinetics become, as discussed earlier, it is understood then why
the growth rates parallel to g are higher than these antiparallel to g.


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 simulations5456 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 8
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
Qe100,101 gi,63 Si,102 and H2O,103 which have L/KTm values between 2 and
4 and P4104 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 as ~ .45
L. Hence, these materials belong to a special group and it would be


252
Table 12. Solutal and Thermal Density Gradients (x1 = 7 um).
V, x 10~4 cm/s
-/+" (dp/dx' )q, g/cm3/cm
+ /-" (dp/dx1 I'p, g/cm3/cm
.5
2.2 x 10'5
1.07 x 10~6
1
-T)
1 C
O
X
CO
O'
CO
2.12 x 10"6
5
2.2 x 10~3
2.7 x 10"4
10
8.8 x 10'3
6.1 x 10-4
15
.0198
8.9 x 10"4
20
.0351
1.16 x 10'3
25
.0544
.00143
30
.0779
.00177
35
. 1054
.00211
40
. 137
.00237
Top and bottom signs apply for growth in the upward and downward direc
tion, respectively.


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 T^. 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
= h/ig)1/2
(19)
w
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
Ue hRTT
(20)
m
* For detailed derivation, see further discussion in the continuous
growth section.


58
diffusivity," D^. Alternatively, suggesting that = 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,25153
its validity has been questioned25153155 for the case of diffuse
interface, clustered, and molecularly complex liquids. These views have
been supported by recent experimental works9295156 and previously dis
cussed MD simulations of the S/L interface,505354_56 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
that has been found to be up to six orders of magnitude smaller9295
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 u0. Finally, it should be noted that AG^ also
enters the calculations of the two-dimensional nucleation rate via the
arrival rate of atoms (R) 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,


161
regardless of the supercooling range. In eq. (72), which represents the
classical spiral growth mechanism, Kq and ATC are the curve-fitting
parameters related to growth and interfacial conditions; a detailed
discussion about these will be given in the Discussion chapter. The
rate equation, as determined from the regression analysis of the data
points, as shown in Fig. 30, is given as
log V = 1.73 AT + 2.845 or
V = 700 AT1-73 (73)
where V is the growth rate in pm/s. The coefficients of determination
and correlation are, respectively, .99 and .995.
Growth at High Supercoolings, TRG Region
The results indicate that as AT increases, the relationship V (AT)
deviates from both dislocation-assisted and dislocation-free kinetic
laws presented earlier, as shown in Figs. 23 and 26. The data points
are for 15 samples; four of the samples were tested from growth rates
103-10-3 to 1-2 x 10^ pm/s, i.e. covering the entire experimental
range. The dislocation-free and dislocation-assisted rate equations for
lower growth rates are also shown in these figures. As can be seen, the
deviation from the low supercoolings laws is toward higher growth rates
at a given AT; furthermore, the deviation takes place at lower super
coolings for the dislocated interface. For the latter, the deviation
starts in the range of about 2-2.5C supercoolings, whereas for the
dislocation-free interface, the transition can be approximately located
at about 3.5C supercooling. At supercoolings higher than the above
mentioned ones, the two growth modes (SDG and PNG) approach each other


120
O
o
H-
o
U¡rj =
o 5 10 15 20 25
Vr¡ x 10 3. cm2/sec
Figure A-7 Temperature correction, T, for the (001) interface as a function of Vr. for
different values of ILr.; Analytical, Numerical calculations. 1
291


138
resulted because of the resistance change across the sample during
growth. The growth rate was then determined from an equation with a
form
AU
At
= I
(68)
where AU is the recorded potential differential drop after being cor
rected for the Seebeck emf. At is the time interval between two consec
utive measurements; I is the current and AR/A£, is the change of resist
ance along the sample with respect to unit solidified length. Typical
values of U were in the range of about 10 mV for the common initial re
sistance, in the order of 2 ohms, across the sample and the rest of the
circuit. At ranged from 1.8 to .02 seconds for the fastest growth
rates. The value of AR/A2. for each sample was calculated theoretically
using the reported resistivities of single crystals and liquid Ga322
corrected for temperature and orientation, and also determined exper
imentally from the optical growth rate measurement in the range of 500-
(1.5 2) x 103 pm/s. The agreement between the measured and calculated
values was considered more than satisfactory, with a maximum difference
of about 3%. A comparison between the experimental and calculated
values of AR/AH for the (111) interface is shown in Fig. 21.
During each milliamp current pulse, a minimum of four rate measure
ments were made; the average was then taken as the growth rate at the
mean of the supercoolings measured during the picoamp pulse before and
after the milliamp plateau, provided that the difference between the two
supercoolings was less than .025C. The maximum standard deviation
never exceeded 5% for the highest average resisistivity growth rates.


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
OO
F = a S {f(uh) + Ka_2(un .u^)2} (4)
OO
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 fCu^) 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 uR. 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 () (5)
na
where z is a distance normal to the interface and the quantity


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 (interfacial) 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 Tm, 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 fee
(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) fee 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


APPENDICES
IGALLIUM 263
IIGa-In SYSTEM 278
IIIHEAT TRANSFER AT THE S/L INTERFACE 280
IVINTERFACIAL STABILITY ANALYSIS 299
VPRINTOUTS OF COMPUTER PROGRAMS 305
VISUPERSATURATION AND SUPERCOOLING 316
REFERENCES 318
BIOGRAPHICAL SKETCH 340
XI


122
Table 1.
Mass Spectrographic Analysis of Ga (99.9999%).
Element
Concentration (ppm)
<
Pb
.05
Sn
.1
A1
.05
Cu
.05
Ag
.03
Cr
.03
Fe
.05
Hg
.5
Mn
.01
Mg
.01
Si
.2
Na
.1
V
.1
Ti
. 1
Ni
. 1
Cd
.1
Zn
.1
Zr
. 1
In
. 1
* Analysis as provided by the Aluminum Company of America, Pittsburg, PA.


2
1. 5
w
\
E
n_
a
x
>
0
0
coon
S DG

O-

0.


aP ,
/
.
/

0 j AT=8
/
/


/

/
/



1. 5
AT, *C
2. 5
Figure 56 Experimental (001) dislocation-assisted growth rates as compared to the SDG Model
calculated rates (dashed lines) as a function of the interface supercooling.
229


91
example, the bulk kinetics of Ni201 are shown in Fig. 12 for bulk super
coolings up to ~240C. The ATb dependence of V in the range of 0-175C
supercoolings can be approximately described by a Kb(ATb)2 law with Kb =
.14 cm/s(C)2. In comparison, a growth law of .28 (ATb) has been
determined from another investigation223 for ATb's in the range 20-
200C. However, calculations based on heat transport limited growth
rates200 and an upper limiting kinetic coefficient based on the speed of
sound in the melt indicate that the interfacial supercooling is only a
small fraction of ATb, e.g. it is 26C when the bulk supercooling is
175C. Hence, the bulk kinetics relation in no way provides accurate
information, even in a qualitative sense, as far as crystal growth
kinetics is concerned, and may also be misleading2>2526153 when a
comparison is attempted with the existing theoretical treatments for
crystal growth. This can be understood, considering that for growth
into undercooled melts a parabolic relationship in the form of
V = Kb ATb2 (51)
is commonly obtained.63 The above mentioned relationship could repre
sent different growth mechanisms. For example, in the case of dendritic
growth, a simple solution of the governing heat flow equations (neg
lecting interfacial kinetics) predicts17 a growth rate that has an
approximate parabolic dependence on the bulk supercooling. Another
example is the growth rate by the dislocation mechanism, as discussed
previously, which can also be a parabolic function of the supercooling.
Another difficulty in using bulk kinetics is that the value of the
coefficient Kb depends on heat transfer conditions, sample and inter
face geometry, and the specifics of the experimental set-up. This


Log (V/A) (/xm-s)
1/AT,C1
Figure 36 The logarithm of the (001) low growth rates (MNG) normalized for the S/L
interfacial area plotted versus the reciprocal of the interface supercooling.
171


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-^ to 2 x 10^ pm/s at interface supercoolings
from 0.2 to A.6C, corresponding to bulk supercoolings of about 0.2 to
53C. The dislocation-free growth rates were found to be a function
xx


103
causing the solid composition to differ from the equilibrium one. The
actual distribution coefficient k is related to the equilibrium one, the
velocity and the interface supercooling of the advancing interface as
k = kQf(V, AT, n) (59)
where the function f depends on the model under consideration and has to
be determined for each model. It is obvious that for local equilibrium
conditions (i.e. V -* 0, AT -* 0) f(0, 0, n) = 1. On the other extreme,
at high growth rates or large deviations from equilibrium, no segregation
should occur according to the solute trapping theories,254277-279'' and,
therefore, f -> 1 /kQ (i.e. k -> 1) as V - <*>. For faceted materials, non
equilibrium segregation can be obtained even at low growth rates, since
large undercoolings are required for finite growth rates when the growth
mechanism is of the stepwise type. This manifestation of segregation
anisotropy during growth from the melt has been experimentally ob
served2193 in several doped semiconducting materials. This form of an
isotropy is also referred to as the facet effect that expresses the veri
fied common trend for higher solute concentration on facets than in off-
facet areas of a macroscopic interface. Several mechanisms have been
suggested to account for the interfacial segregation on faceted inter
faces.34280-283 Most of these theories involve an adsorbed layer and
are based upon the difference of the lateral and continuous growth kin
etics in order to explain the facet effect. The analytical result of
such a model,281283 is given as
V
k = k + (1 k ) exp(- r~) (60)
o o V
* The review of the solute trapping theories and related experiments is
beyond the scope of the present review.


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), ug
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
J = Z u). n.
i i
* 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.


o n
313
Program #5 continued
21 FORMAT(5X,VELOCITY \E13.6)
DO 20 1=1,6
11=6+1
B (I)=0
A(1,II)=-1
A(1,I)=l
20 CONTINUE
c xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
X FORMING COEFFICIENT MATRIX A(I,J) AND B(I) ACCORDING TO X
X EQS.(30) AND (31) X
C xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
DO 30 1=2,6
DO 40 J=1,6
JJ=J+6
X=.2*(1-1)*GAMAS(J)
Y=. 2* (I-1)*GAMAL(J)
C XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX X
C X FUNCTION BESSEL USE INFINITE SERIES EXPRESSION OF X
C X BESSEL FUNCTION TO DETERMINE VALUE OF BESSEL FUNCTIONX
C XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX X
A(I,JJ)=-BESEL(Y)
A(I,J)=BESEL(X)
40 CONTINUE
30 CONTINUE
W=V*ASL*RSL
DO 50 1=1,6
11=1+6
B(II)=V*HCT
GML=ASRI*GAMAL(I)
GMS=ASRI-VGAMAS (I)
Y2(I)=.5*W+(.25*W**2+GML**2)**.5
A(7,II)=KLS*Y2(I)
Y1(1)=-.5*V+(.25*V**2+GMS**2)**.5
50 A(7,I)=Y1(I)
DO 60 1=8,12
DO 60 J=1,6
JJ=J+6
XI=.2*(1-7)*GAMAS(J)
X2=.2*(I-7)*GAMAL(J)
A(I,J)=Y1(J)*BESEL(XI)
60 A(I,JJ)=Y2(J)*BESEL(X2)*KLS
WRITE(6,19)((A(I,J),J=1,12),1=1,12)
WRITE(6,19)B
19 FORMAT(5X,'MATRIX',/(5X,12E10.3))
C XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
C X LEQT2F IS SUBROUTINE FROM LIBRARY X
C X LIBRARY NAME IS IMSL.SP X
C X PLEASE CHECK WITH CIRCA FOR MORE INFORMATION X
C XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX


327
182. V. Bostanov, W. Obretenov, G. Staikov, D. K. Roe, and E. Budevski,
J. Cryst. Growth, 52 (1981) 761.
183. F. C. Frank, Disc. Faraday Soc., 5 (1949) 48.
184. G. W. Sears, J. Chem. Phys., 23 (1955) 1630.
185. F. C. Frank, J. Cryst. Growth, 51 (1981) 367.
186. P. Bennema, J. Cryst. Growth, 69 (1984) 182.
187. W. J. Barnes and F. P. Price, Polymer, 5 (1964) 283.
188. N. Cabrera and M. M. Levine, Phil. Mag., 1 (1956) 450.
189. K. A. Jackson, in: Crystal Growth: A Tutorial Approach, W.
Bardsley, D. T. J. Hurle, and J. B. Mullin, eds. (North-Holland,
Amsterdam, 1979), p. 139.
190. E. Budevski, G. Staikov, and V. Bostanov, J. Cryst. Growth, 29
(1975) 316.
191. A. A. Chernov, Sov. Phys. Usp., 4 (1961) 116.
192. I. Sunagawa and P. Bennema, J. Cryst. Growth, 53 (1981) 490.
193. See for example: a) B. Lewis, J. Cryst. Growth, 21 (1974) 40; b)
P. Bennema, in: Crystal Growth, H. S. Peiser, ed. (Pergamon,
Oxford, 1967), p. 43.
194.
H.
A.
Wilson, Philos
. Mag., 50
(1900) 238.
195.
J.
Frenkel, Phys. Z.
Sowjetunion, 1 (1932) 438.
196.
K.
A.
Jackson and B.
Chalmers,
Can. J. Phys., 34 (1956) 473.
197.
K.
A.
Jackson, Mat.
Sci. Engr.,
65 (1984) 7.
198.
W.
Ruhl and P. Hilsch, Z. Phys.
, B26 (1977) 161.
199.
F.
Spaepen and D. Turnbull, in:
Laser-Solid Interactions and
Laser Processing, S. D. Ferris, H. J. Leamy, and J. M. Poate,
eds., AIP Conf. Proc., 50 (1979) 73.
200. S. R. Coriell and D. Turnbull, Acta Met., 30 (1982) 2135.
201. J. L. Walker, results cited in: Principles of Solidification, B.
Chalmers (Wiley, New York, 1964), p. 114.
202. J. Q. Broughton, G. H. Gilmer, and K. A. Jackson, Phys. Rev.
Lett., 49 (1982) 1496.


163
and eventually fall under the same kinetics equation. It should also be
noted that, although the dislocation-free rates seem to deviate more
drastically than those of the dislocated interface in the (V, AT) range
before they meet, the latter still moves much faster. The scatter of
the (V, AT) points in this range is generally larger than those of the
lower supercooling regions. No variations in experimental circumstances
could be found which would explain this effect. The estimated errors
involved in determining the data points were no more than 5% and 2.5%
for the growth velocities and the supercoolings, respectively. These
error limits are smaller than the observed scatter in the data. As men
tioned previously, even though the growth rates were measured by the
resistivity change technique, the interface was still directly observed
at the high growth rates region. Nucleation ahead of the interface was
never detected, even for the highest bulk supercooling. However, be
cause of the small magnifications involved (7-1 OX) and the rapid rates,
it cannot be claimed that the faceted character of the interface is
retained for growth rates larger than about 1 cm/s.
The scatter in the data, as indicated earlier, may be attributed to
the combined SDG + 2DNG growth mode or to the common sluggish behavior
of transitional kinetics, as in most transformations. Linear regression
analysis of data points for growth rates higher than about 6500 um/s in
dicates that the data points fall between two almost parallel upper and
lower boundaries. The analysis points out that they may be correlated


65
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)l/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.1/4 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) t3^2 (i.e. the cluster area increases linearly


287
and radial temperature profiles can be determined. Furthermore, the
temperature correction 6T can be calculated from any of equations (A28)
or (A29) after setting zs or z^ equal to zero. Accordingly, then
0. =
T. T,
i b
6T
N
i T Tu
m b
= E A J (y r')
1-1, n o sn
(A32)
m b n=l
Before proceeding, the differences of the present analysis from the
former two2181 are briefly explained. First, both of the previous anal
yses assume that the thermal properties and densities of the solid and
the liquid are equal; for the conductivities it is assumed that
K + Kt
K = KT =
s L
= K.
Hence, equations (A26) and (A27) become identical, which implies that
Ysn = = Y Consequently, from equation (A30) it is concluded that
An = Bn; the latter can then be obtained from equation (A31) and, if sub
stituted into equation (A32), yields 6T. Michaels et al.181 had shown in
their work the parameter Vr./2K is much smaller than y (this is true for
i n
small growth rates and high values of Ur^/K) so that the right hand side
of equation (A31) is drastically simplified. Abbaschian and Ravitz,2 who
solved the problem without incorporating the later assumption, found that
the two analyses yield temperature corrections that are essentially the
) ^ cm^y
the calculations of Michaels et al.181 were shown to overestimate T.177
same for Vr^ less than 4 x 10 ^ cm^/s, but not at high growth rates where
Results and Discussion
In order to compare the numerical solution with the experimental re
sults obtained by the Seebeck technique, as described in the Experimental
chapter, a computer program based on the above mentioned analysis was


332
283. R. N. Hall, Phys. Rev., 88 (1952) 139.
284. M. J. Aziz, in: MRS Symp. Proc., Vol. 23, J. C. C. Fan and N. M.
Johnson, eds. (Elsevier, New York, 1984), p. 369.
285. W. A. Tiller, K. A. Jackson, J. W. Rutter, and B. Chalmers, Acta
Met., 1 (1953) 428.
286. V. G. Smith, W. A. Tiller, and J. W. Rutter, Can. J. Phys., 33
(1955) 723.
287. J. A. Burton, R. C. Prim, and W. P. Slichter, J. Chem. Phys., 21
(1953) 1987.
288. J. R. Carruthers, in: Preparation and Properties of Solid State
Materials, Vol. 3, W. R. Wilcox and R. A. Lefever, eds. (Marcel-
Dekker, New York, 1977), p. 1.
289. W. R. Wilcox, in: Preparation and Properties of Solid State
Materials, Vol. 1, W. R. Wilcox and R. A. Lefever, eds. (Marcel-
Dekker, New York, 1977), p. 41.
290. D. T. J. Hurle, in: Crystal Growth 1971, R. A. Laudise, J. B.
Mullin, and B. Mutaftschiev, eds., (North-Holland, Amsterdam,
1972), p. 39.
291. J. S. Turner, Buoyancy Effects in Fluids (Cambridge Univ. Press,
Cambridge, England, 1973).
292. F. K. Moore, Theory of Laminal Flows (Princeton Univ. Press,
Princeton, NJ, 1964).
293. I. G. Currie, Fundamental Mechanics of Fluids (McGraw-Hill, New
York, 1974).
294. W. R. Lindberg and R. D. Haberstroh, AIChE Journal, 18 (1972) 243.
295. S. R. Coriell, M. R. Cordes, W. J. Boettinger, and R. F. Sekerka,
J. Cryst. Growth, 49 (1980) 13.
296. J. D. Verhoeven, K. K. Kingery, and R. Hofer, Met. Trans., B6
(1975) 647.
297. K. Shibata, T. Sato, and G. Ohira, J. Cryst. Growth, 44 (1978)
435.
298. R. M. Sharp and A. Hellawell, J. Cryst. Growth, 8 (1970) 29; J.
Cryst. Growth, 12 (1972) 261.
299. R. G. Pirich, in: Materials Processing in the Reduced Gravity
Environment of Space, G. E. Rindone, ed. (Elsevier, New York,
1982), p. 593.


12
the "work done to create unit area of interface." Analytically og^ can
be given by
asl = us£ TSs£ + pvi ~ us2, TSsZ (2)
where US£ is the surface energy per unit area, Ss£ is the surface en
tropy per unit area, and the surface volume work, PV^, 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, aS£.
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


80
60
40
20
0
5
10 15 20
Vr, x 10'? cm2/*ec
25
30
-11 Comparison between the (111) Experimental results (O.D) and the Model
( Analytical, Numerical) calculations as a function of Vr. for
given growth conditions. 1
296


370
330
3*0
400
405
40
41 0
420
430
440
450
460
470
430
490
307
Program //I continued
PRINT I,T(I),V1(I)
NEXT I
PR# 0
GOTO 30
REM a"*"*--******-********-'*-*'*'*-*-*'*
REM THE NEXT STATEMENTS STORE THE DATA ON A
UNDER THE FILENAME "FITDATA".
CDS = CHRS ( 4 > :F $ = "FITDATA"
PRINT CDS;"OPEN" ;FS: PRINT CDS ; LIR I TE" ; FS
PRINT NUM 2: REM NUMBER OF DATA VALUES
FOR I = 1 TO NUM
PRINT TCI)
PRINT V1CI)
NEXT I
PRINT CDS;"CLOSE";FS
NUM = 0: GOTO 30
)I SK


Growth Rate. |jm/s
188
Figure 43
Growth behavior of Ga-.12 wt% In (111) interface; a) Growth
rates as a function of distance solidified, b) Growth front
X40; solid shows as darker regions


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,34 which
can be studied directly on the microscopic scale by several experimental
techniques.5 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


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 £ 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
15 0
Based on kinetic measurements, it was initially suggested that
undergoes a transition from faceted to non-faceted growth at supercool
ings between 1-9C.151 However, this was not confirmed by a later study
by the same authors, who reported that grew with faceted dendritic
form at high supercoolings.41
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


116
actual interface temperature has been used in this study as described
the next chapter.
in


241
coefficient of the (001) normal growth VC/AT = .63 cm/s-C (see Results
chapter) a value of 3 = .205 is found. Note that this value, being
consistent with that of the upper limit calculated for the (ill) inter
face and the interfacial diffusivity values, could indicate, according
to the diffuse interface model, that the (001) linear growth at the
highest supercoolings is due to a continuous growth mechanism. This
was, anyhow, expected since the (001) interface at this range of super
coolings (AT > 2.2C) is rough, as indicated from the fact that oe is
zero at this range.
In conclusion, despite the fact that the experimental data are in
qualitative agreement with most of the diffuse theory's predictions, the
lack of quantitative agreement makes one to believe that the theory does
not explain the magnitude of the observed transition from the lateral
growth; an agreement could be possible if the model would allow for a
supercooling dependent interfacial diffuseness, g. On the other hand,
the estimated 3 values agree well with the experimental results and sup
port the conclusions of this study that the interfacial diffusivity is
smaller than the Ga liquid self-diffusion coefficient.
The possibility of morphological breakdown of the planar interface
(changing to a cellular or dendritic form) was also investigated using
the morphological stability criteria to determine the conditions under
which the interface may become unstable, as discussed in detail in
Appendix IV. The analysis indicates that the (ill) S/L interface, for
example, should be stable up to about .8 cm/s if the perturbation
wavelength is equal to the interface diameter; for smaller perturbations
the interface should be stable at higher rates. Therefore, since the


283
supercooled liquid Ga at a constant velocity V. The capillary with an
inside radius r. and wall thickness x is assumed to be of infinite ex-
i 8
tent in axial direction z on both sides of the interface. Under these
assumptions, the temperature distribution in the system is described by
the following equations and boundary conditions:
Heat conduction in the solid region,
(92T /9r2) + (1/r) (9T /9r) = (V/k ) (9T /9z ) (92T /9z 2) (Al)
s s s s s s s
Heat conduction in the liquid region,
(92TL/9r2) + (1/r) (9TL/9r) = (Vps/klpl) (9Tl/9zl) (A2)
Boundary conditions:
(1) Far away from S/L interface condition,
Tt = T = T, as zT or z -* <*>, 0 < r < r.
L s b L s i
(2) Symmetry condition,
(9Tr/9r) = (9T /9r) = 0 at r = 0, for 0 < zT ,z <
L s L s
(3) Condit ion at the inner wall of capillary tube
-Kt (9T /9r) = U. [ (T ) T ] 0 < z <
L L r=r. i L r=r. b L
i i
-K (9T /9r) = U.[(T ) T J 0 < z <
s s r=r i s r=r d s
i i
(A3)
(A4)
(A5)
(A6)
where is the overall heat-transfer coefficient of the glass wall and
the boundary layer; it is assumed to be the same for the solid and the
liquid region and is given by
1/U. = (1/h) (r./r ) + (x /K ) (r./r ) (A7)
i i o g g i e.m
In the above expression, K is the thermal conductivity of the capil-
s
lary, r is the mean logarithmic radius, and h is the heat transfer


255
Low concentrations for the onset of solutal convection during
growth have been predicted theoretically.295 For example, during growth
of Pb-Sn under solutally unstable conditions, at a growth rate of 1 pm/s
and a temperature gradient of 50C/cm, solutal convection will occur at
all compositions above 1 x 10-^ wt% Sn; at smaller gradients the crit
ical concentration is predicted to be even less.
However, a surprising feature of the solutal convection is the fact
that the small (.028 cm) tube diameter seems to be unable to suppress
convection. Since the length (D/V) is the characteristic scale of com
positional inhomogeneities in the liquid, the solute Rayleigh number is
based on this length for the prediction of solutal convection. However,
D/V is smaller than the radius of the tubes used in this study only for
growth rates larger than 9 pm/s.
The absence of convection during upwards growth appears obvious.
The solutal convection is eliminated and hence the solute diffusion pro
cess is not interrupted. On the other hand, the very small thermal
gradient (G^ < 2.5C/cm) near the interface does not promote thermal
convection. For typical parameters of the present experiments" a ther
mal Rayleigh number Rt = a g r^/vic for this geometry, where a is the
thermal expansion coefficient, g the acceleration of gravity, r the tube
radius, v the kinematic viscosity, and k the thermal diffusivity. The
Rayleigh number is less than 68, which is the predicted value for crit
ical Rayleigh number for tubes of finite length and of aspect ratios
greater than 22.339,340 For smaller aspect ratios, R^r is larger than
* a = 1.58 x 10 3oC 3, g = 980 cm/s3, v = 3.46 x 10-3 cm3/s, D = 1.35 x
10-3 cm3/ s; see more also in Appendix II.


40
or, according
as113
to a different model, £ diverges in the vicinity of TR
£ oc exp (C/(-^ )1/2) (T < Tr)
R
5 (T > Tr)
(17)
where C is a constant (about 1.5Al3 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 E, 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 £. 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
oe <* exp
(-C/0
T T
R
>1/2)
(18)
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.


326
162. J. L. Katz, in: Interfacial Aspects of Phase Transformations, B.
Mutaftschiev, ed. (D. Reidel Publ. Co., Dordrect, Netherlands,
1982), p. 261.
163. B. Lewis, J. Cryst. Growth, 21 (1974) 29.
164. W. B. Hillig, Acta Met., 14 (1966) 1808.
165. J. P. Hirth, Acta Met., 7 (1959) 755.
166. A. M. Ovrutskii, Sov. Phys.-Cryst., 26 (1981) 242.
167. V. V. Voronkov, in: Modern Theory of Crystal Growth, A. S.
Chernov and H. Muller-Krumbhaar, eds. (Springer-Verlag, Berlin,
1983), p. 75.
168. A. E. Nielsen, Kinetics of Precipitation (Pergamon, New York,
1964), p. 46.
169. A. N. Kolmogorov, Izv. Akad. Nauk, SSSR, Ser. Math., 3 (1937) 355.
170. M. Hoyashi, J. Phys. Soc. Japan, 35 (1973) 614.
171. V. Bertocci, Surf. Sci., 159 (1969) 286.
172. M. S. Viola, K. A. Van Wormer, and G. D. Botsaris, J. Cryst.
Growth, 47 (1979) 127.
173. G. H. Gilmer, J. Cryst. Growth, 49 (1980) 465.
174. G. H. Gilmer, J. Cryst. Growth, 42 (1977) 3.
175. D. Kashchiev, J. Cryst. Growth, 40 (1977) 29.
176. C. Van Leeuwen and J. P. Van der Eerden, Surf. Sci., 64 (1977)
237.
177. G. J. Abbaschian, Ph.D. Thesis, Univ. of California-Berkeley
(1971).
178.W. B. Hillig, in: Growth and Perfection of Crystals, R. H.
Doremus, B. W. Roberts, and D. Turnbull, eds. (John Wiley, New
York, 1959), p. 350.
179.
A. M.
374.
Ovrutskii
and T
. M. Mal'chenko,
Sov. Phys. Cryst., 23 (1978)
180.
V.
V.
Voronkov,
Sov.
Phys. Cryst., 17
(1972) 807.
181.
A.
S.
Michaels,
P. L.
T. Brian, and P,
, R. Sperry, J. Appl. Phys.,
37 (1966) 4649.


108
segregation process. This form of convection is called solutal convec
tion. When convection is caused simultaneously by thermal and concen
tration gradients, it is usually termed as thermosolutal. Other convec
tive phenomena that occur during crystal growth are due to a) surface
tension gradients along free surfaces (Marangoni convection), b) thermal
diffusion of species in a solution in the presence of temperature gradi
ents (Soret effect),'' and c) externally applied body-forces other than
gravity.
A fluid in a vertical configuration is statically stable if the
density decreases with height, and is unstable for the reverse case. A
statically stable density gradient does not cause convection. On the
other hand, an unstable profile will cause convection when the density
gradient is larger than a critical value necessary to initiate flow
(i.e. to overcome the viscous forces). Following the thermal, solutal,
and thermosolutal convection during unidirectional growth of a dilute
alloy, that grows parallel or antiparallel to the gravity vector and
whose solute density is higher than that of the solvent will be exam
ined. This case is related to the present experiments on In doped-Ga,
as will become apparent later. For different cases under the same
principle, reference is made elsewhere.294295
For unidirectional solidification of an undercooled alloy melt that
grows upwards, as shown in Fig. 14a, the liquid is heated from below
(i.e. hot at the interface and cold away from it) and the temperature
* Note that the compositional differences caused by the Soret effect
will not be maintained when convection begins.


217
which the polynuclear growth takes over. The expression satisfying
these conditions can have the following form
A 1+1/2mj 2m/2m+l
-1/m 1/m
m u
(89)
e
According to the above discussion, at low supercoolings, when the dimen
sionless term in the denominator is much less than unity, eq. (89) re
duces to the MNG rate equation of V = hJA. For the reverse case, it is
realized that the general eq. (91) reduces to the form of eqs. (33)-(35)
for the PNG mechanism. Assuming that m = 3/4, eq. (89) can be rewritten
as
V =
h J A
o + a5/3 j ,,,)3/5
1.46 u
4/3'
e
As may be surmised already, the exponent m = 3/4 was chosen because, as
indicated earlier, this particular PNG submodel satisfactorily predicts
the experimental slope ratios between the MNG and PNG kinetics for both
interfaces. At first glance, after recalling the expression for the 2DN
rate, J (eq. (27)), equation (89) is identical to the "fitted" one, eq.
(69). In the latter, is the product of the pre-exponential term of
the 2D nucleation rate and the step height (h); K2, in turn, expresses
the ratio of K^/1.46 h ue^^. Generally, the edge velocity ue is
assumed to be proportional to AT. Based on this assumption, K9 should
then vary as about AT However, thorough evaluation of the factors K-^
and K2 suggests that K2 is indeed independent of the supercooling, which
implies that ue AT^^.


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
v


BIOGRAPHICAL SKETCH
The author was born on March 22, 1957 in Lamia, Greece. He re
ceived his diploma in metallurgical and mining engineering from the
National Technical University, Athens, Greece, in 1980. Afterwards, he
attended the George Washington University in Washington, D.C. and re
ceived a Master of Science degree in materials engineering and solid
mechanics in 1982. He entered the University of Florida in August, 1982
and has been a candidate for the Doctor of Philosophy degree in
materials science and engineering since October, 1983. He is a memher
of Tau Beta Pi, Alpha Sigma Mu, American Association of Crystal Growth,
Materials Research Society, American Institute of Mining, Metallurgical
and Petroleum Engineers, American Society of Metals, and the Technical
Chamber of Greece.
340


267
Table A-2.
Metastable and High Pressure Forms of Ga
Property
Metastable Forms at Normal Pressure
0 <5 Y
melting point, C
density at Tm, g/cm^
crystal structure
-16.3 -19.4 -35.6
6.22 6.21 6.2
monoclinic trigonal orthorhombic
pressure atmosphere
crystal structure
High Pressure Forms
II III
28 26
cubic tetragonal


269
contrast with two other studies where it was found that the liquid does
not perfectly wet the solid and the contact angle between the solid and
the liquid was estimated to be about 7(332a>b while it is supported by
another study3320 which concluded that the S/L contact angle is 0.
The self diffusion coefficient in liquid Ga has been studied
72
experimentally by using Ga isotope and three independent techni
ques.329376377 The Arrhenius parameters of the diffusion coefficient
/ O
equation (D = D exp (-Q/RT)) are D = 3.45 10 cm /s and Q = 1.85
o o
Kcal/mol in the range of 280-680 K. Self diffusion in Ga crystals has
67 72
also been measured using Ga3373 and Ga.337*5 In both investigations
it was found that the process is characterized by very low self diffu
sion coefficients, about 10 ^3 3 8 and 10. ^3 3 7 cm^/s at 20C, in crystal
orientations [100], [010], and [001]. A certain degree of anisotropy of
the activation energy for self diffusion is also indicated.
Crystal Structure of Gallium
Like the rest of its properties, the crystal structure of Ga is
quite unusual for a metal; Ga crystallizes in the base centered ortho
rhombic system in a complex manner and is very anisotropic. The latter
property, as well as most of its other unique characteristics, is
attributed to the coexistence of metallic and directed non-metallic (co
valent) binding within the crystal.
The crystal structure of Ga was first reported incorrectly as
tetragonal.378 Later, it was shown that the structure is orthorhombic,
with eight atoms in the unit cell, although, as in the earlier work, it
was considered that two of the crystal axes were approximately equal.379


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.1
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


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 = Tm, 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 C2Clg 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 hep ^He crystals has been experi
mentally found for at least three crystal orientations ((0001), (1100),
(1101)129130). 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 oe(T).
At TR it was shown that ag vanished as
exp (-C/(-£_ )1/Z)
R
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.131132 The transition, which was studied
isothermally, took place in the composition range where important


315
Program it5 continued
SUBROUTINE TEMP(B,I,T,Z,TB,ASRI)
COMMON/GAMMA/GAMAS(6),GAMAL(6),Y1(6),Y2(6)
DIMENSION T(10,100)
DIMENSION B(12)
DO 10 L=1,100
ZZ=(L-1)*Z/99
DO 10 M=1,10
T(M,L)=0.
RR=(M-1)/9 .
DO 5 N=1,6
IF(I.EQ.1)GO TO 20
NN=N+6
Y=-Y2(N)*ZZ/ASRI
GAMA=GAMAL(N)*RR
GO TO 30
20 NN=N
Y=-Y1(N)*ZZ/ASRI
GAMA=GAMAS(N)*RR
30 CONTINUE
IF(Y.EQ.0.)Y=0.000001
5 T(M,L)=T(M,L)+B(NN)*BESEL(GAMA)*EXP(Y)
10 T(M,L)=T(M,L)*(29.7-TB)+TB
RETURN
END
FUNCTION BESEL(X)
A0= 1 .
X2=X**2
BESEL=1.
IF(X.EQ.0. )GO TO 30
DO 10 1=1,30
I2=I**2
AN=-AO*X2/4./l2
AO=AN
ANN=ABS(AN)
IF(ANN .LE.O.00001)GO TO 30
10 BESEL=BESEL+AN
30 CONTINUE
RETURN
END


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,5153 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 fee 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 (ill) interfaces, the interfacial potential energies and
S/L surface energies were found to be nearly equal (within 5%).51


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


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).

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
v

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

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 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 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

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

APPENDICES
IGALLIUM 263
IIGa-In SYSTEM 278
IIIHEAT TRANSFER AT THE S/L INTERFACE 280
IVINTERFACIAL STABILITY ANALYSIS 299
VPRINTOUTS OF COMPUTER PROGRAMS 305
VISUPERSATURATION AND SUPERCOOLING 316
REFERENCES 318
BIOGRAPHICAL SKETCH 340
XI

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

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

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 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

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

Figure 47 Initial (111) growth rates of Ga-.Ol wt% In ( O )
and Ga-.12 wt% In ( X <^ ) grown in the direction
parallel ( X D ) and antiparallel (0,0 ) to
the gravity vector; continuous line represents the
growth rate of pure (111) Ga interface 193
Figure 48 The logarithm of the (111) rates versus the reciprocal
of the interfacial (open symbols) and bulk supercooling
(closed symbols) for two samples sizes 196
Figure 49 Absolute thermoelectric power of solid along the three
principle Ga crystal axes and, liquid Ga as a function
of temperature 199
Figure 50 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
Figure 51 Comparison between the (111) experimental growth rates
and calculated, via the General 2DNG rate equation, as
a function of the supercooling 214
Figure 52 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
Figure 53 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 (ill) interface 222
Figure 54 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 and supercooling dependent oe 226
Figure 55 Comparison between the (ill) dislocation-assisted growth
rates and the SDG Model calculations shown as dashed
lines 227
Figure 56 Experimental (001) dislocation-assisted growth rates as
compared to the SDG Model calculated rates (dashed lines)
as a function of the interface supercooling 229
Figure 57 The (ill) growth rates versus the interface supercooling
compared to those determined from CS on the solid/vapor
interface (Ref. (117)) 232
Figure 58 The (ill) growth rates versus the interface supercooling
compared to the combined mode of 2DNG and SDG growth
rates (dashed line) at high supercoolings 234
XVII

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

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

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-^ to 2 x 10^ pm/s at interface supercoolings
from 0.2 to A.6C, corresponding to bulk supercoolings of about 0.2 to
53C. The dislocation-free growth rates were found to be a function
xx

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

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

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.1
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

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.

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,34 which
can be studied directly on the microscopic scale by several experimental
techniques.5 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 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

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

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)

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 = 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

12
the "work done to create unit area of interface." Analytically og^ can
be given by
asl = us£ TSs£ + pvi ~ us2, TSsZ (2)
where US£ is the surface energy per unit area, Ss£ is the surface en
tropy per unit area, and the surface volume work, PV^, 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, aS£.
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
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
s£

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

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
OO
F = a S {f(uh) + Ka_2(un .u^)2} (4)
OO
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 fCu^) 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 uR. 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 () (5)
na
where z is a distance normal to the interface and the quantity

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.

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

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 oS£ 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 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.

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.

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

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 ES£ =
E££, 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 oS£ 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)

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 £ 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 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

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 £, which could be expected to make sense only

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

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 os£. 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 (interfacial) 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 Tm, 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 fee
(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) fee 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,5153 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 fee 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 (ill) interfaces, the interfacial potential energies and
S/L surface energies were found to be nearly equal (within 5%).51

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

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

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 og£ 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 oS£ 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
oS£ 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, og£ is obtained from measured supercooling
limits, together with a crystal-melt homogeneous nucleation theory in
which oS£ 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.

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
AT =
2o T
si m
Lr
(15)
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 relation7780"
o .45 L (units of Ca^).
si 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 Tm,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 oS£
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,81^3 b) whether a crystal nucleus (of
atomic dimensions, a few hundred atoms)/melt interface can be adequately
described with og^ of an infinite interface, which is a macroscopic
* A slope of .45 has also been proposed80 for the empirical relation of
the ratio a /a (a is the grain boundary surface tension).
si 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 osis
isotropic,83'' and e) the fact that the values obtained represent some
average interfacial energy over all orientations. In spite of these
limitations, the og^ 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 os^ 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 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.

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 simulations5456 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 8
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
Qe100,101 gi,63 Si,102 and H2O,103 which have L/KTm values between 2 and
4 and P4104 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 as ~ .45
L. Hence, these materials belong to a special group and it would be

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

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 cp/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 solved106
to obtain the partition function and the transition temperature Tc
(Curie temperature) for the ferromagnetic phase transition (paramagnetic
-* 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.

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.

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)
£ {
IT TCH
|TC T|
-v
(T < T)
(16)

40
or, according
as113
to a different model, £ diverges in the vicinity of TR
£ oc exp (C/(-^ )1/2) (T < Tr)
R
5 (T > Tr)
(17)
where C is a constant (about 1.5Al3 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 E, 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 £. 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
oe <* exp
(-C/0
T T
R
>1/2)
(18)
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.

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.

wi

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,
ES£, 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^ = ES£ 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.

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

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 = Tm, 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 C2Clg 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 hep ^He crystals has been experi
mentally found for at least three crystal orientations ((0001), (1100),
(1101)129130). 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 oe(T).
At TR it was shown that ag vanished as
exp (-C/(-£_ )1/Z)
R
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.131132 The transition, which was studied
isothermally, took place in the composition range where important

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

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.

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 ^He (see an earlier discussion), for which sev
eral transitions have already been discovered in the hep phase. Whether
the superfluid ^He 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 revealed117145146 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 < T^, 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 fn m, the system cannot be solved.
Indeed, all the existing kinetic SOSmodels 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

50
a)
Figure 6 Kinetic Roughening. After Ref. (117). a) MC interface
drawings after deposition of .4 of a monolayer on a (001)
face with KT/c|) = .25 in both cases, but different driving
forces (Ay). b) Normalized growth rates of three different
FCC faces as a function of Ay, 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.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.

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 £ 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
15 0
Based on kinetic measurements, it was initially suggested that
undergoes a transition from faceted to non-faceted growth at supercool
ings between 1-9C.151 However, this was not confirmed by a later study
by the same authors, who reported that grew with faceted dendritic
form at high supercoolings.41
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 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 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

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 T^. 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
= h/ig)1/2
(19)
w
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
Ue hRTT
(20)
m
* For detailed derivation, see further discussion in the continuous
growth section.

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

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

58
diffusivity," D^. Alternatively, suggesting that = 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,25153
its validity has been questioned25153155 for the case of diffuse
interface, clustered, and molecularly complex liquids. These views have
been supported by recent experimental works9295156 and previously dis
cussed MD simulations of the S/L interface,505354_56 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
that has been found to be up to six orders of magnitude smaller9295
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 u0. Finally, it should be noted that AG^ also
enters the calculations of the two-dimensional nucleation rate via the
arrival rate of atoms (R) 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), ug
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
J = Z u). n.
i i
* 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, R^, and the surface area
of the cluster, S), n£ 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

61
J
N_ fLAT_vl/2 3£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

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 > £/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.

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.)

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.

65
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)l/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.1/4 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) t3^2 (i.e. the cluster area increases linearly

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

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

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 (£) 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,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)

70
where
o
1
2tty b
e
KTx
and f is a constant.
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 2iTrc from each other, arranged in a line of
length L, eq. (AO) holds with a new yQ = yQ/S when L < Airrc and yQ
2L/S when L > ATrrc. 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 as152189
V =
DL AT
(41)
4-rrT RTa V
m sx. m
Canh et al.25 have modified eq. (41) for diffuse interfaces with a
multiplicity factor |3/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 shown188 that the
effect of the stress field extends to a distance rs from the core of the
dislocation given as

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

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.

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.

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.

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

76
V/[l exp (- £pp)]
m
as a function of l/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 experimentally2526*63198 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;199200 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
= 1 f3K]\l/2
A ^ m 1
[1 exp (-
LAT
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. 2 0 2 > 2 0 3
Other approaches for continuous growth are mostly based on the kin
etic SOS model for a Kossel crystal in contact with the vapor.117145
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 (£ 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

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

INTERFACIAL KINETICS
THEORETICAL GROWTH RATE EQUATIONS
Smooth Interface
Rough Interface
2-D Nucleatlon, (2DNG)
r*.
Ve;keDLAT
M (L DL AT^expi-so;2!,/ )
V 2a T-K hT AT'
A<
Mononuclear, (MNG)
V.- h J A or
V; k, A AT/z expi-Maj2/TAT)
Dislocation Assisted (SDG)
V: k. AT. tanhlAT, )
3 'AT. yAT
or
AT
A >(ve/j)^
/
Va -^AT" n <2
Polynuclear, (PNG)
V: ch (J ve2 )3
Vr k2 AT5/6exp(-Mct2/3TAT)
Continuous
V : f.DLa/x(l -exp(-LAT/KTT )] ,
or
v=dlbatl/ktt a at
Figure 9 Interfacial growth kinetics and theoretical growth rate equations.

80
continuous growth. Although referred to continuous growth, it would be
possible, to differentiate between two linear growth mechanisms, A) the
continuous growth which takes place at all supercoolings for a rough
interface and B) the normal growth which is the upper limit of the lat
eral growth, due to the kinetic roughening of the initially faceted
interface. The former, at low AT's,'' is represented by a line (V =
KCAT) in a V(AT) plot that passes through the origin.
For growth from the melt, the only indication of a kinetic transi
tion of both 2DNG and SDG kinetics has become available from a previous
detailed kinetic study for the Ga (001) interface.2 It was shown that,
with increasing AT, the 2DN and SD growth rates approach each other and
finally fall under the same kinetics. Analogous qualitative results
have been published for salol,63 where it was found that, at high AT's,
the growth velocities scatter between a maximum and a minimum; the first
with parabolic dependence on AT and the second with exponential depend
ence. However, the large scatter did not allow for any quantitative
conclusion.
The transition from lateral to continuous growth kinetics with
increasing AT is also predicted from the diffuse interface theory, as
shown in Fig. 10. As discussed previously, this theory assumes that
classical LG should prevail as long as
which with the aid of eqs.
LG kinetics as
. h e
W < rc r < MG"
Vg V
(48)
(38) and (l) determines the range of AT's for
* Note that for metallic melts any effects because of decreased atomic
mobility are not to be expected up to relatively high supercoolings.

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

82
O < AT =
a g V T
si m m
Lh
(49)
On the other hand, the critical driving force for continuous growth is
derived from eq. (9) as
dg c sl
G +o f = 0 -*> -AG =
v o dx v h
or that continuous growth should operate for supercoolings such that
AT > it AT* (50)
At intermediate supercoolings, i.e. AT'' < AT < it AT", there is a
transitional regime where growth deviates off those extrapolated from
the low AT's regime towards the direction of faster growth rates.
To account for such a deviation, it was suggested that within the
transitional region the step edge free energy should decrease with AT
and finally become zero when the growth becomes continuous. For the SDG
kinetics, starting from inequality (48), it was also concluded that the
departure from the classical SDG law ( AT^) should occur at the same
supercooling AT" as for 2DNG kinetics. Note that here the width of the
step is compared to rc and not to yQ, the distance between the steps.
The latter would result in an increased AT" by a factor 2tt .
In spite of the fact that the theory attempts to quantify the
transition in the kinetics, it leaves unexplained the following key
points: a) in accordance with their proposed analytical equation for oe,
i.e. oe = h oS£ Vg, it is hard to rationalize how it is possible for oe
to decrease with AT as long as g," os and h are constant. However, if
* It is obvious from their analysis and discussion of experimental growth
kinetics25 that g is a constant number.

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

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.

85
As mentioned previously, however, the effect of the solute on the
2DNG kinetics is two-fold; either on og or on ug. If the impurity bonds
strongly to the crystal surface (relative to the solvent atoms), the
nucleation process could be facilitated, for example, by nucleation of
clusters around the impurity. In return, this process would permit
measurable growth to take place in a region of supercoolings where the
faceted (dislocation-free) interface is essentially immobile without the
impurity atoms.206" Similarly, the adsorption of the solute atoms at
the periphery of the two-dimensional cluster can reduce the value of oe,
thus increasing the nucleation rate and, therefore, the growth rate.
Although there is not any experimental proof regarding a particular
trend concerning the effect of solute on og, it is believed70207 that,
if its concentration exceeds a certain limit, the edge free energy will
be reduced. Based on this, the loss of the faceted character of the
interface at high concentrations of solute is then understood. On the
other hand, the presence of the solute at the smooth interface (as an
adsorbed atom) can affect the spreading rate, ug, of the two-dimensional
nucleus. The overall effect on the macroscopic growth rate then depends
on the magnitude of ue.208,209 If ug is small (for example, at low
supercoolings), the solute atom will have enough time to be exchanged by
a host atom, and the growth rate will not then be affected. However, if
ug is large, the adsorbed impurities on the terrace will create an
energy barrier for the step motion, thus lowering the rate.
If the impurities enhance the nucleation process on all types of
interfaces, the growth rate would then be less sensitive to orienta
tion (smoothing agents).206

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,

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

88
heat away from the interface through the crystal and/or the undercooled
liquid.
It is well recognized that macroscopic mass and heat transport play
a very important role in crystal growth processes. Up to now, the
microscopic processes were examined, i.e. how molecules cross the S/L
interface and attach themselves to the crystal and how this process is
governed by the driving force and the nature of the interface. However,
for the molecules to reach the growth sites, they are transported in the
liquid by diffusive and/or convective fluxes over macroscopic distances.
The interface velocity at the same time depends on how fast the heat of
fusion is transported away from the growth front, on the concentration
of species present, and the nature of the interface. Hence, the rate
with which the interface advances is a coupled interfacial kinetics and
macroscopic transport process. This section is concerned only with the
macroscopic transport processes. The subjects that will be discussed
are interfacial heat transfer, segregation, and morphological stability
in relation to the growth of both the pure and doped materials.
Heat Transfer at the S/L Interface
The rate at which the heat of fusion can be removed away from the
growth front depends on the specifics of the experimental set-up and the
thermal properties of the solid and liquid. Whether or not this rate is
greater than the interfacial kinetics determines whether the growth is
limited by the interfacial kinetics (kinetics controlled) or by heat
transfer (heat flow controlled). However, in many occasions during
growth from a supercooled melt, regardless of the thermal arrangements,
the growth is a heat flow-interface kinetics coupled process. Including

89
the growth kinetics in the transport (Stefan219) problem, the coupled
boundary conditions at the interface can be described as220
AT = ATb hcV, ATb > AT > 0
V = f(n, T), f(, Tm) = 0
where hc is a heat transfer parameter to be described later and the
function f depends upon the specific growth mechanism by which the
interface advances. For faceted materials, the interface supercooling
is appreciable so that the assumption of local equilibrium (T^ = Tm)
does not apply. The interfacial supercooling AT may be a small fraction
of other temperature differences within the system and, most import
antly, of the bulk supercooling, ATb, as shown in Fig. 11a. The inter
facial conditions are even more complicated when growth involves addi
tional matter transport, e.g. during growth from a supersaturated solu
tion, schematically shown in Fig. lib. Since the interfacial tempera
ture (T^) is higher than the bulk temperature (Tb), the growing inter
face has a higher equilibrium concentration (c) than that of the bulk
(ce). At the same time, the solute diffuses from the supersaturated
solution (cM) to the growing interface, resulting in an interfacial com
position of c^. Although the magnitude of c^ depends, among other fac
tors, on the growth rate and convection, it is the difference Ac = c-
c^ which governs the interfacial processes and, therefore, has to be
evaluated in order to determine the growth kinetics.221
Since a direct measurement of the interfacial supercooling (S/L) is
a difficult problem, most experimental investigations disregard the
essential role of interfacial supercooling and describe growth kinetics
in terms of the "Bulk Kinetics," i.e. the relationship V(ATb).222 As an

90
(B)
Figure 11 Heat and mass transport effects at the S/L interface.
a) Temperature profile with distance from the S/L inter
face during growth from the melt and from solution. b)
Concentration profile with distance from the interface
during solution growth.

91
example, the bulk kinetics of Ni201 are shown in Fig. 12 for bulk super
coolings up to ~240C. The ATb dependence of V in the range of 0-175C
supercoolings can be approximately described by a Kb(ATb)2 law with Kb =
.14 cm/s(C)2. In comparison, a growth law of .28 (ATb) has been
determined from another investigation223 for ATb's in the range 20-
200C. However, calculations based on heat transport limited growth
rates200 and an upper limiting kinetic coefficient based on the speed of
sound in the melt indicate that the interfacial supercooling is only a
small fraction of ATb, e.g. it is 26C when the bulk supercooling is
175C. Hence, the bulk kinetics relation in no way provides accurate
information, even in a qualitative sense, as far as crystal growth
kinetics is concerned, and may also be misleading2>2526153 when a
comparison is attempted with the existing theoretical treatments for
crystal growth. This can be understood, considering that for growth
into undercooled melts a parabolic relationship in the form of
V = Kb ATb2 (51)
is commonly obtained.63 The above mentioned relationship could repre
sent different growth mechanisms. For example, in the case of dendritic
growth, a simple solution of the governing heat flow equations (neg
lecting interfacial kinetics) predicts17 a growth rate that has an
approximate parabolic dependence on the bulk supercooling. Another
example is the growth rate by the dislocation mechanism, as discussed
previously, which can also be a parabolic function of the supercooling.
Another difficulty in using bulk kinetics is that the value of the
coefficient Kb depends on heat transfer conditions, sample and inter
face geometry, and the specifics of the experimental set-up. This

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

93
complicates interpretation of results and may explain the contradictory
conclusions reached by various investigators on the growth mode and
kinetics. The disagreement between kinetic coefficients found from
various investigations for the same material is very common indeed.
(See, for example, the reported kinetics laws for melt growth of
Sn.63,2223,224-231)
Morphological Stability of the Interface
The purpose of this section is to present an overview of theoret
ical and some experimental work on the morphological stability of the
S/L interface during melt growth. The emphasis will be on the assump
tions, conclusions, and predictions (in analytical forms) rather than
the mathematical details of the topic, which are available in several
reviews.232-236 A detailed discussion of the stability analysis will be
given only for the case of supercooled pure melt. An understanding of
the morphological stability of the S/L interface during solidification
in terms of all the pertinent, i.e. thermodynamic, kinetic, thermal, and
hydrodynamic parameters, is essential for predicting the interface mor
phology on a microscopic scale.
Instability of a planar interface occurs whenever the shape of a
smooth growing interface develops protrusions, depressions, or
undulations, primarily because of the "point effect" of heat and/or sol
ute diffusion. From the practical point of view, stability theories are
to decide whether a growing flat interface will remain flat or will be
come irregular, cellular, or dendritic and therefore structurally and
compositionally inhomogeneous under given growth conditions.

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 C£ (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

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

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.

97
In eq. (5A) both terras in the denominator are always positive (k <
1). Therefore, the planar246-252'' interface is unstable or stable de
pending on whether or not the numerator ever becomes positive. The num
erator consists of four terms proportional to Gs, G^, T, and Gc. For the
usual case of positive temperature gradients (solidification of super
heated liquids), the thermal terms (first two terms) are negative, thus
promoting stability. The surface tension term (third term) is always
negative, therefore also favoring stability. The compositional gradient
term (fourth term) is positive. Hence, instability happens when the de
stabilizing effect of the solute term is large enough to overcome the
stabilizing influences of thermal gradients and surface tension.
In most practical cases, a sufficient and necessary condition for
stability (after neglecting the surface tension term that becomes im
portant at high growth rates as discussed later) is240
KTGT + K G
L L s s p (55)
K + K > m Gc
s L
which is the modified constitutional supercooling (MCS) criterion as
compared with the original CS criterion given in eq. (53). The MS
stability criterion is usually also expressed as
2Kt
L Gt
D
K + K. (1 k) v2Kt V (-m) C
s L L o
> S(A,k)
(56)
where S is the stability function244 that ranges from 0 to 1 and A is a
dimensionless parameter given as
, 2
T
rv m
1 k D (-m) C
* There is a large number of theoretical work regarding the stability of
a non-planar interface; the two geometries that have been studied the
most are the spherical246-249 and the cylindrical.250-252

98
Both CS and MS theories are very similar for the case of A < 0
(neglecting capillarity effect or low growth rates) and S = 1 since both
stability expressions (56) and (53) become identical by setting = Ks
in eq. (56). The theories differ appreciably however, for S < 1 and in
particular for large A (high growth rates); when S -> 0, the interface is
stable for all G^/V below a critical concentration, as predicted by the
absolute stability criterion that will be discussed next.
Absolute stability theory during rapid solidification
The situation where the CS and MS theories differ is the case of
rapid solidification, where the MS theory21*0 and its extended
forms253254 predict an increased form of stability known as absolute
stability. At high growth rates because of the limited time for solute
diffusion away from the interface, instability can only occur for per
turbations at the S/L interface with very short wavelengths.240 This,
however, requires such a large increase in the area of the interface that
the perturbations are stabilized by surface tension forces. The critical
velocity is given by the expression
mD(1 k)C
V = = 2 (57)
k T r
m
Whenever V exceeds the value given by eq. (57), a planar S/L interface is
stable. There are, however, several restrictions255 on using eq. (57).
First, eq. (57) is only valid if the interface is at local equilib
rium and k is the equilibrium partition coefficient. When k depends on
the growth rate and/or the growth kinetics, which is the case during
rapid solidification and solidification of facet forming materials (see
the discussion in the next section), eq. (57) has to be modified

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

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

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

102
solidification273274 are also other studies supporting the MS theory.
It is generally agreed that, at low G^/V values, experimental findings
agree with the CS theory (or MCS criterion), while at high G /V values,
i-i
the stability of the interface is in accord with the MS theory;275276
however, the experimental verification of the effect of non-equilibrium
interfacial conditions upon the interfacial stability is still pending
from the experimental point of view, mainly due to the lack of knowledge
regarding the relationships V(AT) and k(V,AT).
Segregation
Partition coefficients
During solidification in the near-local equilibrium limit, the com
position of the solid and liquid at the interface may be represented by
the equilibrium phase diagram. The segregation coefficient kQ is then
defined as
C
where kQ is generally a function of temperature, but can be treated as
constant when the solidus and liquidus lines of the phase diagram are
nearly straight. When the interface is not planar but has a curvature,
R, the equilibrium coefficient kQ is expected to scale as
k0(R) = kQ(l + TR)
where T is the capillarity constant.
In the case of finite growth rates, however, the interfacial compo
sition on either side of the S/L interface can no longer be represented
by the equilibrium phase diagram since the solid forms at a temperature
lower than that of the equilibrium because of the interfacial kinetics
discussed earlier. Furthermore, solute trapping also may take place,

103
causing the solid composition to differ from the equilibrium one. The
actual distribution coefficient k is related to the equilibrium one, the
velocity and the interface supercooling of the advancing interface as
k = kQf(V, AT, n) (59)
where the function f depends on the model under consideration and has to
be determined for each model. It is obvious that for local equilibrium
conditions (i.e. V -* 0, AT -* 0) f(0, 0, n) = 1. On the other extreme,
at high growth rates or large deviations from equilibrium, no segregation
should occur according to the solute trapping theories,254277-279'' and,
therefore, f -> 1 /kQ (i.e. k -> 1) as V - <*>. For faceted materials, non
equilibrium segregation can be obtained even at low growth rates, since
large undercoolings are required for finite growth rates when the growth
mechanism is of the stepwise type. This manifestation of segregation
anisotropy during growth from the melt has been experimentally ob
served2193 in several doped semiconducting materials. This form of an
isotropy is also referred to as the facet effect that expresses the veri
fied common trend for higher solute concentration on facets than in off-
facet areas of a macroscopic interface. Several mechanisms have been
suggested to account for the interfacial segregation on faceted inter
faces.34280-283 Most of these theories involve an adsorbed layer and
are based upon the difference of the lateral and continuous growth kin
etics in order to explain the facet effect. The analytical result of
such a model,281283 is given as
V
k = k + (1 k ) exp(- r~) (60)
o o V
* The review of the solute trapping theories and related experiments is
beyond the scope of the present review.

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

Composition
Figure 13 Solute redistribution as a function of distance solidified during
unidirectional solidification with no convection.
105

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,

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.

108
segregation process. This form of convection is called solutal convec
tion. When convection is caused simultaneously by thermal and concen
tration gradients, it is usually termed as thermosolutal. Other convec
tive phenomena that occur during crystal growth are due to a) surface
tension gradients along free surfaces (Marangoni convection), b) thermal
diffusion of species in a solution in the presence of temperature gradi
ents (Soret effect),'' and c) externally applied body-forces other than
gravity.
A fluid in a vertical configuration is statically stable if the
density decreases with height, and is unstable for the reverse case. A
statically stable density gradient does not cause convection. On the
other hand, an unstable profile will cause convection when the density
gradient is larger than a critical value necessary to initiate flow
(i.e. to overcome the viscous forces). Following the thermal, solutal,
and thermosolutal convection during unidirectional growth of a dilute
alloy, that grows parallel or antiparallel to the gravity vector and
whose solute density is higher than that of the solvent will be exam
ined. This case is related to the present experiments on In doped-Ga,
as will become apparent later. For different cases under the same
principle, reference is made elsewhere.294295
For unidirectional solidification of an undercooled alloy melt that
grows upwards, as shown in Fig. 14a, the liquid is heated from below
(i.e. hot at the interface and cold away from it) and the temperature
* Note that the compositional differences caused by the Soret effect
will not be maintained when convection begins.

109
^soluce > ^solvent
+ T -
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 tnan the
density of the solvent.

no
gradient, G^, is negative. Assuming that the alloy is also of such com
position that k < 1, there is more concentration of solute at the inter
face upon growth; hence, the solute gradient, Gc, is also negative.
Considering the negative temperature gradient separately, it is realized
that it would result in a positive density gradient (i.e. increases
with distance from the interface) that makes the liquid statically un
stable. The negative solute gradient alone would cause a negative den
sity gradient (since the solute is assumed to be more dense than the
solvent), stabilizing the liquid. The net liquid density profile then
depends on the relative magnitude of the thermal and solutal gradients.
Since the solute concentration decreases as exp(-D/V) away from the
interface, the compositional gradient is important only within the decay
distance (D/V). The temperature gradient on the other hand, decays with
ic^/V, which is much larger than the previous decay distance. According
ly, convection may occur, depending on how large the temperature gradi
ent is.
The second case to examine is the reverse of the first one, where
the interface moves downward, as shown in Fig. 14b. Here, the positive
thermal gradient (hot on top of cold) results in a negative density
gradient, while the positive solute gradient leads to a positive density
gradient. Thus, solutal convection may arise in this case, depending on
the relative magnitudes of Ap/Ac and Ap/AT, only within a distance in
the order D/V (or, more precisely, 2D/V, which is the distance at which
C¡£ reaches .93 CQ) away from the S/L interface.
Next, consider a disturbance of the bottom fluid288295 that makes
a fluid element rise for both configurations. Since the thermal

til
diffusivity is much greater than the chemical diffusivity, the fluid
will gain or lose heat much faster than solute. Thus, in the first case
(upwards growth), the element will still be more dense than its new sur
rounding and will tend to return to its original position, i.e. at the
interface. This is the case of an oscillatory instability. However,
for the case of downward growth, the element that rises from the bottom
is less dense than its surroundings because of the segregation profile,
and, therefore, it will continue rising. Hence, convection may occur
even though the overall density profile is stable.
Studies of thermosolutal convection in crystal growth experiments
rely upon measuring segregation profiles and comparing them with those
predicted theoretically. For example, macrosegregation profiles ob
served during unidirectional growth of Pb-Sn (Pb rich),296 Al-Ti and
Al-Cr,297 Al-Cu and Al-Mg,298 and Mn-Bi299 have been correlated with
thermosolutal convection. In these studies, all but the last study
(Mn-Bi), growth took place upwards and for conditions of destabilizing
solute gradients and stabilizing thermal gradients. For the Pb-Sn sys
tem, the boundary layer thickness (6) was found to be independent of
growth rate (.8-5.7 pm/s) and sample size (.3-.6 cm diameter) and equal
to ~65 pm (< D/V for the rates studied). For the Al-Ti and Al-Cr exper
iments, it was concluded that convection is only important at low growth
rates. This was based on the fact that, although for the Al-Cr experi
ments the gradient dp/dx was larger than that for the Al-Ti system, con
vection affected solute segregation of the former at the growth rates
around 15 pm/s, while for the latter convection was significant at much
lower solidification rates (<1.8 pm/sec).

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

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, aS£, 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

114
technique is limited by the sharpness of the break in the temperature
time curve as the interface passes the thermocouple.
The thermal wave technique301 was developed to evaluate the growth
rate and interface supercooling from measurements of the attenuation of
a periodic thermal wave, induced in the liquid, as it travelled through
the S/L interface. The periodic variation at the interface allows only
for determination of the supercooling, i.e. absolute temperature mea
surements cannot be made. Aside from the experimental difficulties,
this technique has been subjected to criticism3 0 2 3 01* as it induces con
vection flows in the melt and it does not account for thermal losses
along the container walls. Moreover, it is restricted to small growth
rates (< 50 pm/s)301 and the reported kinetics using this techni
que224'225 have been conflicting. Later, a similar method was pro
posed305 that determines the growth parameters from an analysis of the
response of the interface to a periodic heat input, introduced by using
Peltier heating or cooling. Experimental results based on this tech
nique have not been reported in spite of the fact that the Peltier
effect has been widely used, particularly with semiconducting materials,
during crystal growth related experiments.306"
The single and double thermoelectric probe technique was pro
posed307 for measuring the interfacial temperature and velocity during
growth in a pure material. This method, which also disturbs the actual
growth process, is applicable under constrained growth conditions (i.e.
&L must be known). The accuracy of the technique depends on the Seebeck
* These experiments are not concerned with growth kinetics and, there
fore, are not reviewed here.

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

116
actual interface temperature has been used in this study as described
the next chapter.
in

CHAPTER III
EXPERIMENTAL APPARATUS AND PROCEDURES
Experimental Set-Up
The apparatus used to study the growth kinetics of Ga as well as
In-doped Ga is shown in Fig. 15. It consisted of two constant temper
ature circulators, two observation baths, temperature measuring devices,
and a computer interfaced to a nanovoltmeter, a multimeter, and a cur
rent source. The constant temperature bath circulators, (manufactured
by Lauda, model K-4/R) were used in order to circulate and control the
temperature of the liquid flowing inside the observation baths; the
circulators could cool thirteen liters of fluid down to -30C with a
temperature control accuracy of 0.02C, and they feature a flow regu
lating valve to control the rate of liquid circulation. The circulating
liquids used were water and aqueous ethylene glycol solutions (10-50 wt%
HOCH2CH2OH) for the subzero temperatures. The cooling rate of the con
stant temperature circulators depended on several factors, such as en
vironmental conditions, circulating fluid, and temperature range. Typi
cal cooling rates for temperatures around 27, 5, and -10C were .85, .7,
and .2C/ min, respectively. The temperature of each observation bath,
constant within .025C, was monitored using Chromel-Alumel thermo
couples (I and II) in conjunction with an Omega model TRC-III ice point
cell (0.1C). The output of the thermocouples was recorded (one at a
time through a switching device) by a Gould-110 dual channel model strip
chart recorder with a high gain multi-span DC Preamplifier providing a
117

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

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

120
of 6.6-40X. The graduated eyepiece of the microscope was calibrated
against a .01 mm standard micrometer slide by Bausch and Lomb. The
thickness of the graduation lines was about 3 mu. Under ordinary exper
imental conditions a change in the position of the interface of about 10
pm could be resolved. The microscope was mounted on a hydraulic jack
which allowed for continuous vertical movements while in focus.
Sample Preparation
The growth kinetics measurements were made on Ga single crystals
contained in borosilicate glass capillary tubes with inside diameter and
wall thickness ranging from .595 to .24 mm and .39 to .07 mm, respec
tively. The capillaries were freshly drawn, for each sample, from
Corning or Kimble glass tubes of .8 cm o.d. (.6 cm i.d.) which were
first thoroughly washed with aqua regia and then rinsed with distilled
water; finally, the tubes were dried under vacuum and/or a high pressure
flow of inert gas (Ar). Subsequently, the glass tube was subjected to
gas flame heating for drawing. The two ends of a capillary tube, about
25 cm long, were connected to polyethylene tubes, with i.d.'s ranging
from .7-1.1 mm, which had been cleaned by the previously mentioned pro
cedure. Finally, the whole capillary assembly (65-75 cm long) was
passed through the observation baths so that the glass portion of the
tubing extended from about 5 cm above the top bath (II) to about 1-2 cm
below the lower bath (I). The capillary, in place, was then filled with
99.9999% or 99.99999% purity Ga (spectrographic analyses are given
* It should be noted that very few preliminary runs of this study were
done on six 9's purity Ga; more than 95% of the reported experimental
results are for the seven 9's purity Ga.

121
respectively in Tables 1 and 2). The as received Ga ingots (25g),
sealed in polyethylene bags, was stored under vacuum until a portion of
it was melted with a heating lamp and sucked into a polypropylene ster
ile syringe with a prewashed polyethylene tube, instead of a needle,
attached to the needle hub. The molten metal contained in the syringe
was subsequently injected smoothly in the capillary under ambient con
ditions. The tube was filled in excess in order to discard the first
part of the liquid Ga which had come in contact with the atmospheric
air.
Immediately following the capillary filling, solidification was
initiated at one end of the sample by touching the liquid Ga with a
fresh seed crystal. The Ga seeds were grown in advance by slowly cool
ing molten Ga, protected from oxidation by a dilute HC1 (5-10%) solu
tion. The monocrystal, floating on the surface of the liquid Ga, was
separated from the bulk liquid by a tweezer whose ends were covered with
polyethylene. The characteristic shape of the monocrystals are shown in
Fig. 16, where the faces of interest (001) and {111} can be seen.
The seeding of the capillary tubes was done in such a way that the
desired crystal face, (111) or (001), was essentially normal to the tube
axis. Following the seeding, crystal growth proceeded through the top
observation bath (II) and it was stopped when the S/L interface had
reached about the middle of the lower bath (I). The first solidified
end of the sample was then melted down so another S/L interface would
form inside the top bath.
The above mentioned procedure resulted in the formation of a con
tinuous sample consisting of two S/L interfaces, one inside each of the

122
Table 1.
Mass Spectrographic Analysis of Ga (99.9999%).
Element
Concentration (ppm)
<
Pb
.05
Sn
.1
A1
.05
Cu
.05
Ag
.03
Cr
.03
Fe
.05
Hg
.5
Mn
.01
Mg
.01
Si
.2
Na
.1
V
.1
Ti
. 1
Ni
. 1
Cd
.1
Zn
.1
Zr
. 1
In
. 1
* Analysis as provided by the Aluminum Company of America, Pittsburg, PA.

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

124
(001)
Figure 16 Gallium monocrystal
X 20

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
shielded copper cables to the leads of the nanovoltmeter (model 1506 low
thermal cable), of the current source (model 6011 Triaxial Lead), and of
the multimeter (Triaxial Shielded Cable). Extreme precautions were
taken in making all the electrical connections, as well as in connecting
the devices together, in order to minimize various noises and high con
tact resistances along the circuit which would be particularly trouble
some for the Seebeck measurements. To achieve this, the copper leads
were fusion welded to the W-electrodes and soldered to the instrument
leads via copper splice tubes by low-thermal cadmium-tin solder. All
the junctions were kept close together inside a Dewar's flask at a con
stant temperature. All instruments were connected to a common ground
and the length of the leads was kept minimum.
Interfacial Supercooling Measurements
Thermoelectric (Seebeck) Technique
A novel technique founded on thermoelectric principles was used to
directly measure the S/L interface temperature during growth from the
melt. This technique, described in detail elsewher,311_313 utilizes the
* The solubility of W in liquid Ga is negligible; for example, at 815C
is only -.001 wt%.310

126
Figure 17 Thermoelectric circuits. a) Seebeck open circuit.
b) Seebeck open circuit with two S/L interfaces.

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

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
£s
S .(T
si m
T.)
(66)
= S .AT
si
where Sg£ 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, Sg£ was obtained in both ways, as discussed else
where;1 the direct method of determining Sg£ will be further discussed
later. In general, the coefficient Sg£ 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.

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

130
remained constant; values of it for several samples are listed in Table
3. c) The Seebeck coefficient of the S/L interface, Ss£. The values of
the S/L interface thermoelectric powers, Ss£, were determined directly
for each sample and were verified by the results of the previous
study.311 Direct determination of Ss£ 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 SS£ 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

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

Seebeck emf, pV
Figure 18 The seebeck emf as a function of temperature for the (111) S/L interface.
132

133
Figure 19 Seebeck emf of an (001) S/L Ga interface compared with the
bulk temperature

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

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

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

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

138
resulted because of the resistance change across the sample during
growth. The growth rate was then determined from an equation with a
form
AU
At
= I
(68)
where AU is the recorded potential differential drop after being cor
rected for the Seebeck emf. At is the time interval between two consec
utive measurements; I is the current and AR/A£, is the change of resist
ance along the sample with respect to unit solidified length. Typical
values of U were in the range of about 10 mV for the common initial re
sistance, in the order of 2 ohms, across the sample and the rest of the
circuit. At ranged from 1.8 to .02 seconds for the fastest growth
rates. The value of AR/A2. for each sample was calculated theoretically
using the reported resistivities of single crystals and liquid Ga322
corrected for temperature and orientation, and also determined exper
imentally from the optical growth rate measurement in the range of 500-
(1.5 2) x 103 pm/s. The agreement between the measured and calculated
values was considered more than satisfactory, with a maximum difference
of about 3%. A comparison between the experimental and calculated
values of AR/AH for the (111) interface is shown in Fig. 21.
During each milliamp current pulse, a minimum of four rate measure
ments were made; the average was then taken as the growth rate at the
mean of the supercoolings measured during the picoamp pulse before and
after the milliamp plateau, provided that the difference between the two
supercoolings was less than .025C. The maximum standard deviation
never exceeded 5% for the highest average resisistivity growth rates.

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.

141
Table 5. Analysis of In-Doped Ga Samples.
Sample Intended Composition
(wt% In)
Analyzed Composition"
(wt% In)
1 .01
.012
2 .1
.12
As determined by Applied Technical Services, Inc., Marietta, GA by
atomic absorption technique.

142
Table 6. Seebeck Coefficients of S/L In-Doped (ill) Ga Interfaces.
Composition (wt% In)
ss!i (mV/C)
Comments
.01
1.72
24 hrs equilibration
1.75
end of a run; after the
interface was melted back
1.81
after interfacial break
down
1.628
V = 5.6 pm/s; distance
solidified, x = 3500 pm"
1.56
V = 21.3 pm/s; x = 2200
pm''
.12
1.49
24 hrs equiliation
1.31
V = 1.9 pm/s; x = 2100
1.27
V = 2.8 pm/s; x = 3500
1.3
V = .88 pm/s; x = 2900
1.468
after breakdown
x Ss£ measured upon heating at the end of the run.

143
values are lower than those of the (ill) interface for the pure mater
ial. Moreover, the "noise" of the Seebeck emf was higher than that of
the pure and the offset emf was not as stable throughout an experimental
cycle. These effects are attributed to the "non-homogeneity" of the
sample caused by a) diffusion of the In towards or away from the inter
faces; and b) by the solute (In) rich liquid bands entrapped along the
solid portion of the sample.
In spite of these difficulties, the Seebeck emf was recorded during
growth of the doped material so as to provide a continuous and in-situ
detection of the interfacial conditions; it was revealed that the
Seebeck technique also detects morphological instability of the inter
face and entrapment of second phase deposits during growth. This can be
seen in Fig. 22, where the Seebeck emf generated across the S/L (111)
interface of a Ga-0.01 wt% In doped sample is shown. For a low level of
dopant concentration, the growth mechanism of the doped Ga is similar to
that of the pure, except the growth rates are slightly lower at a par
ticular AT, as discussed later. As growth proceeds, the solute build-up
ahead of the interface reaches a critical profile, causing morphological
instability or interfacial breakdown. During the breakdown, the inter
face moves faster than prior to the instability and an In-rich layer is
entrapped. The abrupt changes in the interfacial conditions (V, AT, and
C^) are reflected in the momentary changes in the Seebeck potential.
After the formation of the In-rich band, the interface becomes faceted
again and the supercooling changes back to its original position.
The interfacial supercooling during growth for the doped material
was calculated from the solutions of the associated heat transfer model

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.

145
(see calculations in Appendix III), the validity of which has been veri
fied for the case of pure Ga. Moreover, the interfacial supercoolings
were calculated also by the Seebeck method. The agreement then between
the two AT values was quite satisfactory. Furthermore, as discussed
later and in Appendix III, at low growth rates, i.e. V < 10 pm/s, the
bulk supercooling is approximately the same as the interfacial super
cooling (ATjj/AT > .98). Thus, at rates less than 1 pm/s, the values of
SS£ and EQ£f can be back calculated with the aid of AT^. This procedure
provided an extra check in determining the AT values.
All the growth rate measurements were made via the optical micro
scope. Initially, the interface, II or I, depending on whether the
growth was antiparallel or parallel to the gravity vector, was super
cooled by .5-.7C below the liquidus temperature; the sample was held at
this supercooling, where the interface was practically stationary, for a
period of 24-48 hours. Afterwards, the water flow was turned off and
the circulator temperature was set to a lower desired temperature. When
the temperature of the circulator had become constant, the flow was then
resumed and the rates measuring procedure started. At a given bulk
supercooling the growth rates were measured as a function of the dis
tance solidified (i.e. the distance from the initial equilibrium posi
tion of the interface); the latter ranged from .5-3 cm. After the com
pletion of these measurements, the interface was melted back to about
its original position, where it was held for 24-48 hours, and the rate
measurements were repeated at another bulk supercooling. It should be
noted that the rate measurements along the parallel and antiparallel to
the gravity vector growth direction were performed on the same sample.

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

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

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

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

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.

GROWTH RATE, mm/sec
Figure 24 Growth rates of the (111) interface as a function of the interfacial and
the bulk supercooling.
151

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

Figure 26 The logarithm of the (111) growth rates versus the reciprocal of the
interfacial supercooling; A is the S/L iterfacial area.
153

154
tested. By observing these figures, and particularly the semilogarith-
mic plot of Fig. 26, it is realized that the growth rate of the perfect
interface is a strong function of the reciprocal of the interface super
cooling. However, its growth behavior does not appear to be monotonic-
ally related to the supercooling within the investigated experimental
range. Indeed, as can be seen in Fig. 26, the growth rates at the lower
range of supercoolings increase much faster with the supercooling than
those in the intermediate range, but still slower than those at super
coolings higher than 3.5C. The latter feature is better revealed in
Fig. 23. It is easily depicted from Fig. 26 that at least up to super
coolings of about 3.5C the growth rates depend upon the interface
supercooling in an Arrhenius fashion (e.g. V exp(- 1/AT)), and at
supercoolings less than about 1.9C, the rates also depend on the sample
size. As may be surmised, already these features provide a basis for
"grouping" the growth behavior of the dislocation-free interface within
a given range of supercoolings. Such grouping is intended for a more
detailed discussion of the results, rather than identifying and/or
implying kinetically distinct regions. Indeed, the dislocation-free
kinetics, which have all the characteristic signs of two-dimensional
nucleation assisted growth, are well expressed over the whole experi
mental range by the rate equation
i / 9
Kx A (AT) exp(- B/AT)
(1 + K2 A5/3 AT1/2 exp(- B/AT))3/5
(69)

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

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

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

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

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

Growth Rate
.25 30 35 .40 .45 50 55 60 65
Figure 29 PNG (111) growth rates versus the reciprocal of the interface supercooling;
solid line represents the PNG rate equation, as given in Table 7.
160

161
regardless of the supercooling range. In eq. (72), which represents the
classical spiral growth mechanism, Kq and ATC are the curve-fitting
parameters related to growth and interfacial conditions; a detailed
discussion about these will be given in the Discussion chapter. The
rate equation, as determined from the regression analysis of the data
points, as shown in Fig. 30, is given as
log V = 1.73 AT + 2.845 or
V = 700 AT1-73 (73)
where V is the growth rate in pm/s. The coefficients of determination
and correlation are, respectively, .99 and .995.
Growth at High Supercoolings, TRG Region
The results indicate that as AT increases, the relationship V (AT)
deviates from both dislocation-assisted and dislocation-free kinetic
laws presented earlier, as shown in Figs. 23 and 26. The data points
are for 15 samples; four of the samples were tested from growth rates
103-10-3 to 1-2 x 10^ pm/s, i.e. covering the entire experimental
range. The dislocation-free and dislocation-assisted rate equations for
lower growth rates are also shown in these figures. As can be seen, the
deviation from the low supercoolings laws is toward higher growth rates
at a given AT; furthermore, the deviation takes place at lower super
coolings for the dislocated interface. For the latter, the deviation
starts in the range of about 2-2.5C supercoolings, whereas for the
dislocation-free interface, the transition can be approximately located
at about 3.5C supercooling. At supercoolings higher than the above
mentioned ones, the two growth modes (SDG and PNG) approach each other

0.1 1 10
AT, C
Figure 30 Dislocation-assisted (111) growth rates versus the interface supercooling;
line represents the SDG rate equation, as given in Table 7,
162

163
and eventually fall under the same kinetics equation. It should also be
noted that, although the dislocation-free rates seem to deviate more
drastically than those of the dislocated interface in the (V, AT) range
before they meet, the latter still moves much faster. The scatter of
the (V, AT) points in this range is generally larger than those of the
lower supercooling regions. No variations in experimental circumstances
could be found which would explain this effect. The estimated errors
involved in determining the data points were no more than 5% and 2.5%
for the growth velocities and the supercoolings, respectively. These
error limits are smaller than the observed scatter in the data. As men
tioned previously, even though the growth rates were measured by the
resistivity change technique, the interface was still directly observed
at the high growth rates region. Nucleation ahead of the interface was
never detected, even for the highest bulk supercooling. However, be
cause of the small magnifications involved (7-1 OX) and the rapid rates,
it cannot be claimed that the faceted character of the interface is
retained for growth rates larger than about 1 cm/s.
The scatter in the data, as indicated earlier, may be attributed to
the combined SDG + 2DNG growth mode or to the common sluggish behavior
of transitional kinetics, as in most transformations. Linear regression
analysis of data points for growth rates higher than about 6500 um/s in
dicates that the data points fall between two almost parallel upper and
lower boundaries. The analysis points out that they may be correlated

164
by a linear relationship, in contrast with a geometric (V ATn) or par
abolic relationship. The rate equation is given as
V 9500 (AT 2.7), AT > 3.5 (74)
with a coefficient of correlation of about .93.
(001) Interface
The growth behavior of the (001) interface in contact with the
supercooled liquid, as shown in Fig. 31, was qualitatively similar to
that of the (111) interface, except that the threshold supercooling nec
essary to obtain a measurable growth rate (~10-^ pm/s) for the undis
turbed (001) interface was about 0.6C; only the dislocated (001) inter
face was mobile below this threshold supercooling. It should be men
tioned that the rate measurements for this interface were more difficult
than those for the (111) interface. This is because when dislocation(s)
would intersect the growth front, the (001) would frequently change into
two or more {111} type interfaces or it would grow out of the interface,
leaving only a (ill) face. Such behavior was also very common at higher
supercoolings and particularly when the initial (001) was not perfectly
normal to the growth direction. This is understood since the (ill)
face, if dislocation-free, is immobile for supercoolings up to about
1.5C and it moves much slower than the (001) face, as discussed later.
In contrast with the earlier described growth behavior for the dislo
cated interfaces, restoration of the initial (001) face was rarely done
by itself. Hence, in order to bring out the initial (001) interface, it
was necessary to melt the sample back and allow it to grow again.

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

166
The dislocation-free and dislocation-assisted data for the (001)
interface are also shown on a log-log scale in Fig. 32. Comparing the
(001) with the (111) growth rates, as shown in the composite linear plot
of Fig. 33, it is realized that the (001) interface moves much faster
than the (ill) at any interface supercooling. For example, at AT =
1.8C, the dislocation-free (001) interface is growing by a factor of
about 1.5 x 103 faster than the (ill) interface, whereas with AT = .8C,
the dislocation-assisted (001) interface is growing by a factor of about
two faster than the (ill) dislocated interface.
Dislocation-Free Growth Kinetics
The dislocation-free (001) data, similar to the (111) interface,
can be well correlated up to supercoolings of about 1.5C, by the gen
eral 2DNG equation, eq. (69), presented earlier. Similarly, for a more
detailed discussion of the results, the (001) data can be divided into
three regions, as shown in Fig. 34. The first region, called MNG, ex
tends from about 0.6-0.8C interface supercoolings and growth rates up
to about 1 pm/s. The second region, PNG, covers the range of about 0.9-
1.45C supercoolings. The growth rates in these two regions are expo
nential functions of (- 1/AT). The sample size affects only the rates
in Region I, but not in Region II. It should be noted that the data in
Region II also fall on two separate lines. As will be discussed later,
the two rate equations are not due to the effect of the interfacial
area. Region III, TRG, follows region II at supercoolings larger than
about 1.5C.
MNG region
The proportionality of the low growth rates (< 1 pm/s) upon the S/L
interfacial area is shown in Figs. 35 and 36. In the latter, a plot of

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

2
1. 5
\
E
n_
r* 1
O
i
X
>
. 5
0
0 1 2 3 4 5
AT, 'C
Figure 33 Growth rates of the (001) and (111) interfaces as a function of the interfacial
supercooling.
168

Figure 34 The logarithm of the (001) growth rates versus the reciprocal of the interface
supercooling.
169

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

Log (V/A) (/xm-s)
1/AT,C1
Figure 36 The logarithm of the (001) low growth rates (MNG) normalized for the S/L
interfacial area plotted versus the reciprocal of the interface supercooling.
171

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

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