GROWTH KINETICS OF FACETED SOLIDLIQUID 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.
freezingmeltingevaporationcondensation).
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 ReedHill 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 solidliquid interface. His collaboration with me in
the laboratory is often missed.
The financial support of this work, provided by the National
Science Foundation (Grant DMR8202724), 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 midday 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 tothem 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
Twodimensional nucleation assisted growth (2DNG) .............. 58
Twodimensional nucleation ........................ ........... 59
Mononuclear growth (MNG) ..................................... 62
viii
Polynuclear growth (PNG) ..................................... 64
Screw dislocationassisted 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 SetUp ..................... ........................... 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
DislocationFree Growth Kinetics ................................. 150
MNG region .............................. ....................... 155
PNG region ....................................... .............. 156
DislocationAssisted Growth Kinetics ............................. 159
Growth at High Supercoolings, TRG Region ......................... 161
(001) Interface .................................................... 164
DislocationFree Growth Kinetics ................................. 166
MNG region ..................................................... 166
PNG region ....................................... .......... 172
DislocationAssisted Growth Kinetics ............................. 173
Growth at High Supercoolings, TRG Region ......................... 174
InDoped (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
InDoped 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 GaIn 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 InDoped Ga Samples ......................... 141
TABLE 6 Seebeck Coefficients of S/L InDoped (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 Ai Physical Properties of Gallium .......................... 265
TABLE A2 Metastable and High Pressure Forms of Ga ................ 267
TABLE A3 Crystallographic Data of Gallium (aGa) ................. 271
TABLE A4 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 crosssectional 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 setup .................................... 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 Indoped 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
Dislocationfree and Dislocationassisted 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
Dislocationfree (111) low growth rates versus the inter
facial supercooling for 4 samples, two of each with the
same capillary tube crosssection 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 ..................
Dislocationassisted (111) growth rates versus the inter
face supercooling; line represents the SDG rate equation,
as given in Table 7 .....................................
Dislocationfree and Dislocationassisted 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 dislocationfree (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) dislocationassisted growth
rates and the SDG Model calculations shown as dashed
lines ................................................... 227
Experimental (001) dislocationassisted 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 Ai
Figure A2
Figure A3
Figure A4
Figure A5
Figure A6
Figure A7
Figure A8
Figure A9
Figure A10
Figure A11
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
GaIn 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 heattransfer 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 A12
Figure A13
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 A14 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 SOLIDLIQUID 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 insitu the interfacial conditions during unconstrained crystal
growth of Ga crystals from the melt and to measure the solidliquid
(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 dislocationfree and dislocationassisted growth kinetics of
(111) and (001) interfaces of high purity Ga, and Indoped Ga, as a
function of the interface supercooling (AT) were studied. The growth
rates cover the range of 103 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 dislocationfree growth rates were found to be a function
xx
of exp(1/AT) and proportional to the interfacial area at small super
coolings. The dislocationassisted 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
twodimensional 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 facetednonfaceted 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 Indoped 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 solidliquid (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 insitu 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 setup 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, GaIn 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 solidvapor (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.610 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 solidlike to liquid
like properties. On the other hand, a sharp interfaces10 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 criterion812 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 "brokenbonds" models. Based on this criterion, the interface is
either smooth (singular, 13 faceted) or rough (nonsingular, 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 Kfaces, respectively.13
8
function of orientation plot (Wulff's plot"4 or yplot15). 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 crosssectional 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 kinksites.
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 (ac)
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 twodimensional 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 viceversa. 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) + Ka2(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 socalled 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
'44
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.2527 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 PT diagram has not been discovered yet, the quantities f(un) and
the gradient energy are hard to qualify for the solidliquid 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 closepacked 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 manylevel 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 BraggWilliams35 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. twolevel or twodimensional 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 solidlike and liquidlike 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 BraggWilliams 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.
101
A
102 A
103
104
105 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" apriori. 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 cooperative phenomena in a twodimensional 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 twolevel
model interface: as such it classifies the molecules into "solidlike"
and "liquidlike" ones, b) it considers only the nearest neighbors, and
c) it is based on BraggWilliams statistics.
The main concluding point of the model is that the roughness of the
solidliquid 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 closepacked 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) LennardJones, 126,
potential.42 The LJ 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) hardsphere,
b) computer simulations (CS); molecular dynamics (MD), or Monte
Carlo (MC), and
c) perturbation theories.
In the Bernal model (hardsphere),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 closepacked 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 LJ 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 "channeledlike" structure
28
of the first 23 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 45 planes. However, these trajectory maps, in
terms of characterizing the atoms as liquid or solidlike, 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
solidlike ordering. The interfacial thickness was estimated quite
large (1015 layers) and the observed density profile oscillations were
less sharp than those observed4750 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 45 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 threedimensionally 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 latticegas model using the cluster variation
method. In addition, it was shown that for the nonclosepacked 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. lowenergy dif
fraction, Auger spectroscopy, and probes like xrays61) 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 crystalmelt 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
GibbsThomson 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 gat).
sz gatom
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
approaches8487 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 nonzero 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 KCs
* 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).9496 The
ice experiments94'95 have shown that a "structure" builds up in the
liquid adjacent to the interface (1.46 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 criticalpoint 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 liquidvapor
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 lightscatter
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. nonfaceted 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) semimetallic properties; c) some of their
interfaces have been found to be nonwetted 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 nonequilibrium 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 19491951.10,36 The problem then was to calcu
late how rough a (S/V) interface of an initially flat crystal face
(closepacked, lowindex 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 (orderdisorder,
secondorder 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 cooperative 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 twodimensional lat
tice gas model.* The Ising model is a square twodimensional 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 lowhigh temperature expan
sions methods.
39
twodimensional 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 ITTcIP 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 TIV (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 twodimensional Ising model, indicates that E diverges by
a power law, while the latter of the KosterlitzThouless113 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 KosterlitzThouless 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 solidonsolid (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 = Klhrhr'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
solidsolid 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 nonmetallic 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 closepacked 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 closepacked
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 nonexistent experimentally. A
faceted to nonfaceted transition, however, has been observed for a
metallic solidsolution (other liquid metals or alloys) interface in the
ZnIn and ZnBiIn 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 solventsolute 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 twophase 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 firstorder phase transition
and the latter to secondorder 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 + higherorder 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 PokrovskyTalapov transition139 and
smaller than the prediction of the meanfield 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 KosterlitzThouless trans
ition in the twodimensional 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 nonlinear,
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 nonexistent.
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)
AD (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 nonfaceted (smooth spherulitic) transition
was observed for three high melting entropy (L/KTm 67) organic sub
stances, salol, thymol, and Oterphenyl. 47 The transition that took
place at bulk supercoolings ranging from 3050C 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 0terphenyl,
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 IIII in cyclohexanol with increasing supercool
ing. The morphological transition was associated with the change in
growth kinetics, as indicated by a nonlinear 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 1.93.7, 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 nonfaceted growth at supercool
ings between 19C.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
twodimensional nucleation process or by dislocations whose Burgers vec
tors intersect the interfaces; the growth mechanisms associated with
each are, respectively, the twodimensional nucleationassisted and
screw dislocationassisted, 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 twodimensional 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 selfdiffusion 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 + g1/2) DLAT (21)
e hRTT
Here B corrects for orientation and structural factors; it principally
relates the liquid selfdiffusion 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 secl), 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
liquidlike 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
36 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 twodimensional nucleation rate via the
arrival rate of atoms (Ri) at the cluster, which is discussed next.
Twodimensional nucleationassisted growth (2DNG)
As indicated earlier, steps at the smooth interface can be created
by a twodimensional 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 twodimensional nuclea
tion theory.
Twodimensional nucleation. The prevailing twodimensional nucle
ation theory is based on fundamental ideas formulated several decades
ago.158161 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 supercritical 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 nonequilibrium fac
tor which corrects for the depletion of the critical nuclei when nuclea
tion and growth proceed. Z has a typical value of about 102,163 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 disklike 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 twodimen
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 preexponential 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
selfdiffusion 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, twodimen
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 twodimensional clusters nucleate at ran
dom positions at the interface before the layer is completed, or on the
top of already growing twodimensional islands, resulting in a hill and
valleylike interface, as shown in Fig. 7b. Assuming that the clusters
are circular and that ue is independent of the twodimensional 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 polynuclearmonolayer 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 twodimensional 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 11.4. 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 34 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 Squarelike172 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 preexponential 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 (103 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 preexponential 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,63177181 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 dislocationassisted 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
nonmetallic 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 twodimensional 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 = fv 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 dislocationassisted 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 (WF). 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 selfdiffusion
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 selfdiffusion coefficient
in the liquid. A similar expression can be derived based on the melt
viscosity, n, by the use of the StokesEinstein 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 glassforming 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
glassforming 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 LennardJones
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 WF
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 dislocationfree 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 nonfaceted 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
