Vapour Growth and Epitaxy. Proceedings of the Third International Conference on Vapour Growth and Epitaxy, Amsterdam, The Netherlands, 18–21 August 1975

395 Pages·1975·30.92 MB·English

by  G.W. Cullen

© North-Holland Publishing Company, 1975
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Reprinted from:
JOURNAL OF CRYSTAL GROWTH 31 (1975)
Published Monthly
Printed in The Netherlands
VAPOUR GROWTH
AND EPITAXY
PROCEEDINGS OF THE THIRD INTERNATIONAL
CONFERENCE ON VAPOUR GROWTH AND EPITAXY
AMSTERDAM, THE NETHERLANDS, 18-21 AUGUST 1975
EDITED BY
G.W. CULLEN
RCA Laboratories, Princeton, N.J., U.S.A.
E. KALDIS
Eidgenössische Technische Hochschule, Zürich, Switzerland
R.L. PARKER
National Bureau of Standards, Washington, D.C., U.S.A.
C.J.M. ROOYMANS
Philips Research Laboratories, Eindhoven, The Netherlands
1975
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM
Organizing Committee
CJ.M. Rooymans Philips Research Laboratories, Eindhoven, G.J. Arkenbout Physical-Chemical Institute TNO, Zeist,
The Netherlands The Netherlands
R.C. Peters Philips Research Laboratories, Eindhoven, G.A. Bootsma Van 't Hoff Laboratory, Utrecht,
The Netherlands The Netherlands
B. Knook Kamerlingh Onnes Laboratory, Leiden, E.C. Eversteyn Philips Research Laboratories, Eindhoven,
The Netherlands The Netherlands
International Program Committee
I. Akasaki Tokyo, Kawasaki, Japan E. Kaldis Zürich, Switzerland
L.N. Aleksandrov Novosibirsk, U.S.S.R. R. Kern Marseille, France
T. Arizumi Nagoya,Japan R.A. Laudise Murray Hill, N.J., U.S.A.
H. Bethge Halle, G.D.R. J.W. Matthews Yorktown Heights, N.Y., U.S.A.
G.A. Bootsma Utrecht, The Netherlands J.B. Mullin Great Malvern, England
A.A. Chernov Moscow, U.S.S.R. R. Nitsche Freiburg, F.R.G.
G.W. Cullen Princeton, N.J., U.S.A. CJ.M. Rooymans Eindhoven, The Netherlands
F.C. Eversteyn Eindhoven, The Netherlands M. Schieber Jerusalem, Israel
P. Gibart Meudon-Bellevue, France E. Sirtl München, F.R.G.
Session Chairmen
L.N. Aleksandrov, W. Bardsley, H. Bethge, G.A. Bootsma, G.W. Cullen, F.C. Eversteyn, P.J.L. Gibart, E.A. Giess, J.G. Grabmaier, L.
Hollan, E. Kaldis, R. Kern, W.F. Knippenberg, P. Lilley, J.B. Mullin, J. Oudar, A. Rabenau, CJ.M. Rooymans, H. Schäfer, HJ. Scheel,
T.O. Sedgwick, W.T. Stacy, R. Ueda, R.J.H. Voorhoeve
Publication Committee
G.W. Cullen RCA, Princeton, N.J., U.S.A. R.L. Parker NBS, Washington, D.C, U.S.A.
E. Kaldis ETH, Zürich, Switzerland CJ.M. Rooymans Philips, Eindhoven, The Netherlands
This Conference was sponsored and financed by
Royal Dutch Chemical Society (KNCV) Municipality of Amsterdam
Dutch Crystal Growth Association (KKN) Dutch Ministry of Education and Sciences (Ministerie O & W)
International Organization for Crystal Growth (IOCG) N.V. Philips' Gloeilampenfabrieken
International Union of Crystallography (IUCr) I.B.M. Nederland
International Union of Pure and Applied Physics (IUPAP) Siemens AG, Erlangen
PREFACE
Following the international conferences in Zürich tivities, on "Materials Science and the Electronics In­
(1970) and Jerusalem (1972), the Third International dustry". He gave some inspiring thoughts on some of
Conference on Vapour Growth and Epitaxy was held the main problems our world is dealing with and the
in Amsterdam, the Netherlands, August 18-21, 1975. consequences for material and energy exploitation
The proceedings of the Zürich and Jerusalem meetings with special reference to the electronics industry. As
were also published as special volumes of the Journal an example of the future developments in our area of
of Crystal Growth. disciplines, Dr. Pannenborg predicted a shift to three-
Around 230 active participants, with some 60 com­ dimensional storage and handling of information from
panions, were in attendance at the Amsterdam Con­ the existing two-dimensional approach as now present
ference, a result clearly affected by the economic re­ in the epitactic layer structures. This might imply a
cession with its squeeze of the travel budgets at indus­ gradual change from inorganic to (bio-)organic chem­
trial and governmental laboratories alike. Over 20 dif­ istry and physics in the following decades.
ferent countries were represented, creating the inter­ The emphasis on the crystal growth of materials of
national climate characteristic of these conferences. interest for electronic applications was clearly re­
The site of the Conference on the premises of the Fac­ flected in the high number of both invited and con­
ulty of Sciences of the University of Amsterdam, close tributed papers on silicon, III—V and II—VI com­
to the historic city center provided excellent accom­ pounds and magnetic garnets, which formed a sub­
modation for the presentation of papers in three par­ stantial part of the total number of papers at the Con­
allel sessions, the discussions in and around the con­ ference and in this issue.
ference rooms and for the luncheons of the partici­ It became clear that the method of liquid phase
pants and their relatives. epitactic growth is by far the leading one for the mag­
The programme included twenty invited and nearly netic garnet films, while in the preparation of III—V
eighty contributed scientific papers. Manuscripts of epitaotic films both the vapour phase epitaxy and liq­
more than half of these presentations were completed, uid phase epitaxy are currently used with great suc­
refereed and processed prior to the end of the Confer­ cess. The characterization of the grown crystals and
ence, making inclusion possible in this Proceedings of the epitactic films with the underlying substrates is
issue of the Journal of Crystal Growth. Some remain­ rapidly gaining attention, X-ray topography and elec­
ing papers may be published in later, regular issues of tron microscopy being the main tools. The mechanism
the Journal. By adhering strictly to the time schedule and kinetics of the crystal growth are investigated by
it appears possible to have this Proceedings volume sophisticated spectroscopic techniques. Other, more
printed within four months after closure of the Con­ usual items, but represented by outstanding papers
ference. were the growth of single crystals from the vapour
The scope of the Conference had given special em­ phase and the nucleation and kinetics of crystal
phasis to the crystal growth aspects of the preparation, growth processes.
characterization and perfection of thin films of elec­ Subjects of special interest were the results of the
tronic interest. Both more fundamental and applied Skylab space experiments in crystal growth from the
crystal growth papers in this area were seen as objec­ vapour phase, and the use of epitactic techniques for
tives of the meeting. Fully in this line was the subject the fabrication of solar cells. The papers concerned
of the opening address: Dr. A.E. Pannenborg, Member are published in this volume. The progress made in the
of the Board of Management of N.V. Philips' Gloei- study of nucleation and growth of whiskers also
lampenfabrieken, and in charge of Philips' research ac­ formed one of the highlights of the Conference, re-
X Preface
suiting in an unusually high number of participants, contribution to both the organization of the meeting
still present and active, at the last day of the Confer­ and the editing was of essential value. We all were
ence. sorry that he could not be personally present in
The merits of the scientific programme can partial­ Amsterdam. Miss C.B. Huisken and Miss M.A. van
ly be judged from the content of this volume. Missing Marken of the Municipal Congress Bureau of the City
from the volume are not only some of the outstanding of Amsterdam and their assistants did a wonderful job
presentations, but also the vivid and lengthy discus­ in making preparations for the Conference and as­
sions inside and outside the Conference rooms. These sisting the participants during these days.
discussions were also provoked by a well organized The editors of this volume are especially grateful
social programme, which included a reception offered for the great cooperation of many reviewers, whose
by the Government and the Municipality of help made it possible to process so many papers be­
Amsterdam in the new Van Gogh Museum, where in fore and at the Conference, and to the staff members
addition to works of art an exhibition on the history assisting us in our own laboratories, who in many cases
of public transport in the Netherlands could be visited, spent considerable effort and time both reviewing and
while still being served with drinks and snacks. A sub­ processing the manuscripts. Mrs. F.Y. Verploegh
sequent evening boat trip through the illuminated Chasse not only served as an expert in the technical
canals of Amsterdam was a clear illustration of Dutch editing of the papers, she also was in Amsterdam an
individualism and initiative. Each boat chose its own ever charming hostess for the Publication Committee
routing, which resulted for one group in a very exten­ before and during the meeting. Her perseverance has
sive excursion through the harbour of Amsterdam contributed greatly to this volume.
where large impressive sailing boats formed one of the Finally we would like to thank those institutions
highlights of the Amsterdam celebration summer. An and firms, which have, even in these difficult recession
informal banquet in Volendam, a small historic fishing times, contributed to a sound financial basis for the
village some 10 miles north of Amsterdam was an­ Conference. They are acknowledged elsewhere in this
other occasion at which many new relations of friend­ volume.
ship were made, and old contacts were strengthened. We are sure that many participants look forward to
It is obvious that Conference and Proceedings can­ the Fourth International Conference on Vapour
not be realized without the help of many people. Growth and Epitaxy.
Without even trying to give full coverage, several in­
dividuals deserve to be explicitly mentioned. First of
all, we mention Professor M. Schieber, whose inspiring The Editors
Journal of Crystal Growth 31 (1975) 3-19
© North-Holland Publishing Company
CRYSTAL GROWTH FROM THE VAPOUR PHASE: CONFRONTATION OF THEORY WITH
EXPERIMENT
P. BENNEMA and C. VAN LEEUWEN
Laboratory of Physical Chemistry, Delft University of Technology, Delft, The Netherlands
Starting with the BCF theory recent crystal growth models based on computer simulations are discussed. These models
are confronted with experiments for growth from the vapour concerning rates of growth and the direct observations of
step patterns, growth spirals and etch pits.
1. Introduction
In 1951 the now famous paper of Burton, Cabrera Then in part I the integration of growth units into
and Frank named "The Growth of Crystals and the a step is calculated leading to an expression for the
Equilibrium Structure of their Surfaces" was published advance velocity of steps.
[1]. Looking back over the past period of 24 years it In part II the shape of the growth spiral is cal­
can now be stated that this paper was an enormous culated and an expression for the dependence of the
breakthrough for all fields of crystal growth. This is rate of growth on the supersaturation is derived.
because for the first time a more or less complete the­ In this paper we will follow the scheme of the
ory was developed, which could serve as a basis to plan BCF paper and focus our attention on recent develop­
and to interpret experiments for growth from the va­ ments of crystal growth theories. Statistical step mod­
pour, solution, melt, etc. els and recent computer simulations of step structures
In the paper of BCF Ising models of surfaces and will be discussed. Subsequently statistical surface mod­
steps, the concept of the critical nucleus and the con­ els will be discussed together with the results of com­
cept of the screw dislocation are logically integrated puter simulations both for equilibrium and growth.
to a new theory. The BCF paper has the following Computer simulation experiments for growth will be
structure: compared with pair approximation and nucleation
Part III and IV form the statistical mechanical back­ models.
ground for the crystal growth theory which is develop­ Again following BCF recent developments in step
ed in parts I and II. propagation models and recent calculations on the
In part III a calculation of the structure of a step is shapes of a spiral will be discussed. Subsequently
carried out. Upon introducing a supersaturation the continuous step models will be mentioned leading
step becomes curved and this leads to the concept of to the well-known tanh function for the rate of
the critical nucleus. The activation energy for the cri­ growth in dependence of the supersaturation. It will
tical nucleus is calculated and it is shown that the be shown that notwithstanding the simplifications
crystal growth rate on perfect low-index faces at low of the original BCF model this tanh function still
supersaturations and low temperatures is too low to gives an excellent basis for the interpretation of ex­
be measured. periments.
Subsequently in part IV the equilibrium structure Results and implications of the BCF theory and
of a surface is calculated for a two level model, a three recent developments will be confronted with experi­
level and a many level model, respectively. The calcu­ mental data on the dependence of rates of growth on
lation of the two level model is based on the Onsager supersaturation, molecular beam experiments and
paper and is exact. the direct observation of monatomic steps and spirals.
4 P. Bennema, C. van Leeuwen / Crystal growth from the vapour phase
The experimental data will be limited to data suited same as the critical temperature in the two-dimensional
to check the theory i.e. experiments on simple clean lattice gas [1]. It has been interpreted as an estimate
monocomponent systems will be selected for which for the transition temperature for surface roughening
it is possible to define in principle the driving force. in a three-dimensional system.
Theories will be applicable to more complicated sys­ Stepped surfaces can also be simulated with the
tems as well. But then more theoretical work must Monte -Carlo technique. Recently, Leamy and Gil­
be carried out to generalize the models. This is, how­ mer [4] studied the influence of steps on the excess
ever, beyond the scope of this paper. surface energy. They found that the transition tem­
perature for surface roughening is somewhat higher than
predicted by the two-dimensional models.
2. Statistical step and surface models Using the Monte-Carlo technique Van Leeuwen and
Mischgofsky [5] were able to bring into consideration
2.1. Step models the role of overhangs on the properties of a step. In
their approach a ledge was defined by a unique path
A frequently used model for the crystal-vapour that could be followed as a line separating a higher and
interface system is the lattice-gas model. The sys­ a lower part of the surface of the three-dimensional
tem is divided into an imaginary lattice of cells, each system. They compared the properties of the BCF
cell being either filled or empty. When attractive SOS step model which can be calculated exactly and
forces are present between filled cells, phase separa­ the computer generated step model — which auto­
tion occurs below the critical temperature. The bound­ matically contains overhangs — by studying jump
ary between the dense phase and the gaseous phase density distributions, jump correlations and ledge
is the surface of the crystal. Whenever filled cells are overhang densities.
only allowed on top of other filled cells (the solid-on- In fig. 1 a high temperature surface with a rough
solid (SOS) constraint) the gaseous phase is completely containing overhangs is presented. The average profile
empty and the dense phase, the crystal, does not con­ of a step can be calculated for an SOS step model.
tain empty cells. The model is then equivalent to the Some actual step profiles obtained from computer-
Kossel crystal. In equilibrium the surface of a Kossel simulation experiments are shown in fig. 2.
crystal is rather flat at low temperatures. An edge on So far only recent work on equilibrium step struc­
such a surface is then similar to the interface of a two- tures has been mentioned. The statistical SOS model
dimensional lattice—gas system. Here the SOS con­ has been generalized by BCF by imposing a super-
ditions can also be introduced. In such a constrained saturation (Δμ) on the step. Then a step becomes
two-dimensional lattice—gas system: (i) one part of curved and BCF were able to calculate the properties
the system is completely filled, the other part is com­ of a critical nucleus. Recently, an alternative deriva­
pletely empty ;(ii) edge overhangs are excluded. tion of some of the BCF formulas was given using
BCF [1] described the interface of a constrained the formalism of ref. [3] and the model was extended
two-dimensional lattice—gas system, applying the to an anisotropic system [6]. Implications for the
principle of detailed balancing. Temperley [2] and shape, size and the edge free energies were calculat-
Leamy, Gilmer and Jackson [3] showed that the
same formulas can be obtained for equilibrium edges
by evaluating .the grand partition function. They fixed
the position of one end of the interface. This corre­
sponds physically to a step emanating from a screw
dislocation, ending on the surface.
The model has a "critical" temperature: the tem­
perature at which the step free energy becomes zero.
At higher temperatures the model becomes inconsistent,
because the system can reduce its free energy by gen­
Fig. 1. A rough step on a surface at a relatively high tempera­
erating new steps. This temperature appears to be the
ture; a = 4.
P. Bennema, C. van Leeuwen / Crystal growth from the vapour phase 5
Fig. 2. Illustration of the effect of the step length on the width of the step in the SOS step model. The step lengths shown are
L = 40, 80, 160 and 480 for a = 5.6 (top line) and a = 4 (L is the length of the step in block units). The long step at the bottom
of the figure has a length L = 960 for a = 4.0.
ed and presented in an easily accessible form. ed in the one-layer model of Burton, Cabrera and
The dependence of the shape of the critical nu­ Frank [1]. A new "breed" of models, which belongs
cleus on the temperature is presented in fig. 3 for to Type 1, are the already mentioned computer simul­
an isotropic system. At a given temperature, the di­ ation models [11,12]. These models allow an "in­
mensions of the critical nucleus change with the super- finite" number of layers and automatically bring clus­
saturation, but its shape does not. tering into account.
Models of Type II [13] arising from the world of
2.2. Surface models metallurgy lead to the important result that if the
Within the solid vapour interface models two types temperature becomes higher than a critical tempera­
can be distinguished: Type I models, in which over­ ture T, the surface disappears. This phenomenon is
c
hangs and vacancies of vapour in the solid and solid in not of interest for growth from the vapour; this is
the vapour phase respectively are excluded and Type one of the reasons that Type II models will not be
II models, in which these configurations are allowed. discussed in this paper. In Type I, Bragg-Williams
Both types of models can be further divided into models two factors a and ß play an essential role.
groups on the basis of the following criteria: (1) the For the (001) face of a Kossel crystal a is defined as
number of layers taken into consideration, (2) the
approximations introduced. In most cases the zeroth a=-20ss//:r, (1)
order or Bragg—Williams approximation is used with­
where 0 is the solid—solid first neighbour pair po­
in each layer (random distribution of solid and fluid SS
tential. ß is defined as:
blocks). This rules out preferential clustering. Type I
Bragg—Williams models are the two layer model of
ß = An/kT=ln(p/p ) = ln(l + σ) « σ , (2)
Q
Jackson [7] and Mutaftschiev [8] and the infinite
layer model of Temkin [9]. Clustering is exactly treat- where Δμ is the difference in chemical potential be­
tween the vapour and the solid phase, p the actual
andp the equilibrium pressure of a mono-compo­
0
nent vapour and σ the relative supersaturation.
Now, it follows from Type I Bragg-Williams mod­
els that if a is higher and β is lower than certain val­
ues, a real minimum in the excess free energy with
respect to the compositions of the solid and fluid in
each layer of the surface occurs; below and above
these values for a and β respectively, no minimum
occurs. It is said that in the first case growth only
can occur by step growth or two-dimensional nuclea-
tion. In the second case no thermodynamical bar­
riers exist and "continuous" growth occurs.
The physical relevance of these results is obscured
Fig. 3. This figure shows the shape (not on scale) of the criti­ due to the Bragg-Williams approximation, among
cal nucleus for several values of a: a = 80,40, 8, 5.28,4 and
others. Therefore these models are now used in con­
3.52 going from the outer nucleus to the inner nucleus. It can
junction with computer simulation of Type I mod­
be seen that increasing temperature results in rounding off the
corners. els.
6 P. Bennema, C. van Leeuwen / Crystal growth from the vapour phase
2.3. Computer simulation of crystal growth; two-di
mensional nucleation
/ a=Z5//
Recently growth for a Type I model was simulated.
Rate (R) versus supersaturation curves (the supersa-
turation was given by ß [see eq. (2)] were measured
[12,14]. These measurements were repeated with a
high degree of accuracy using a special purpose com­
puter. For limited ranges of a and ß measured (R, ß)
curves could be fitted by (here simplified formulae
are used (see ref. [31]):
R=Aß5/e exp(-B/ß), (3)
where
A=(ivA'C Y'3(2xJa)C (4)
0
Fig. 4. Dimensionless growth rate versus β (= Αμ/kT) for
B=\ny/kT, (5)
0<β< 0.5: (■) results of Gilmer and Bennema [11,12], (o)
special purpose computer results [14], dashed line empirical
C = dk+ ~ [P /(27rmkT)112] . (6) relation [14]; solid lines nucleation formula similar to eq. (3);
Q 0
uppermost line, maximal Wilson-Frenkel law occurring if the
surface is in an equilibrium state, giving an equilibrium back
A B and C are constants defined by eqs. (4), (5), (6)
t flux.
respectively, C is the fraction of the flat surface oc­
Q
cupied by separate units, A' is an empirical factor In order to check the growth process in more de­
smaller than unity, x is the mean diffusion distance tail and to check the domain of validity of various nu­
s
of a growth unit on the surface, a is the distance be­ cleation models, the growth of a new layer on the (001)
tween adjacent blocks in a crystal, γ the edge free surface of a Kossel crystal can be followed by using the
energy per growth unit in the edge of a critical nuc­ Monte-Carlo simulation technique. An example is given
leus, d the interplanar distance and k$ the kinetical in fig. 5 where the time dependence of the degree of cov­
coefficient for the creation of an adunit on the sur­ erage Θ is shown.
face in equilibrium (this quantity is used in the com­ In this example we started with a surface with an
puter). For growth from the vapour C is given by eq. equilibrium structure. Almost immediately after im­
(6). Here p is the equilibrium vapour pressure. posing a supersaturation, a metastable state is reached.
0
Eq. (3) results from the so-called birth and spread Disregarding statistical fluctuations Θ does not change
models [16—21]. In these models critical nuclei are with time until rc. After rc, Θ increases about linearly
continuously formed on top of previously formed with time. Inspection of the surface structure showed
nuclei. that growth took place by spreading of one large clus­
In fig. 4 (R, β) curves resulting from general purpose ter. The structure of the surface layer at rc is shown in
computer experiments and experiments with a special fig. 6. More detailed information will be presented in a
purpose computer are presented [14]. It was found following paper [22]. Phenomena of this kind are de­
that for values of a> 3.6 and values of β < 0.5, the scribed by two-dimensional nucleation theories.
nucleation formula (3) gives a satisfactory descrip­ Several nucleation theories can be distinguished: (i)
tion of the Monte-Carlo data. For values of a < 3.1 in a probabilistic two-dimensional nucleation theory
linear growth kinetics occur. This behaviour corre­ [23] the existence is assumed of a class of clusters
sponds to exceeding a surface roughening transition with a equal probability to grow or to vanish. There­
temperature as found by Leamy and Gilmer [4] (sec­ fore spreading starts as a result of the formation of
tion 2.1.). a series of clusters with this property, all but the last
P. Bennema, C. van Leeuwen / Crystal growth from the vapour phase 7
size of the nuclei is large enough, thermodynamics
f\
can be applied to obtain their properties.
For small nuclei, application of the thermodynamics
•
of small systems [25J or even an atomistic evaluation
— -
* [26] may be give the desired properties. According to
V the statistical treatment of BCF [1], the critical nu­
* cleus contains a fixed number of units and has a speci­
•i·
-- _.j fic average shape. Their treatment, however, allows
/*'"
numerous shapes of the edge on an atomic level.
•
Nucleation theories model reality on two levels.
y/ First of all a system is defined which may be a simpli­
? fkj& fication of a real system (for instance a perfect (001)
2V$ ^ *&
face of a Kossel crystal is considered). Secondly, pre­
suppositions are introduced concerning the nuclea­
Fig. 5. The time dependence of the degree of coverage Θ, re­
tion path for this system. It is an advantage of Monte-
sulting from a Monte-Carlo simulation experiment. At t = 0
an equilibrium surface was present. The surface structure at Carlo simulation experiments that the system of the
the time t = r is given in fig. 6. Here a - 4 [eq. (1)] and (simulation) experiment and the (nucleation) model
Q
ß = 0.1 [eq. (2)] ;xs = a, where xs is the mean displacement can be made equivalent. A confrontation may con­
[eq. (10), section 3.2.1].
firm or reject the presuppositions. Models in conjunc­
tion with Monte-Carlo simulations may then be used
one vanish; (ii) in a kinetic two-dimensional nuclea-
for interpreting experimental results.
tion theory, the nucleation path is assumed to lead
As mentioned above, birth and spread models only
over a barrier [4]. Generally, the existence of num­
give a satisfactory description of Monte-Carlo data for
erous nucleation paths may be assumed, all having
a limited range of a and ß values. In order to obtain
their own barrier and therefore their own activation
a model, which describes (R, ß) curves for larger values
free energy. To all barriers correspond "activated"
of ß and lower a values recently Gilmer et al. [ 15] in­
clusters, with the property that their rates of increas­
troduced a pair approximation method for a (100)
ing and decreasing in size are equal.
surface of a Kossel crystal. In this method pair cor­
The essential clusters in the probabilistic and kinet­
relations are treated explicitly and the method is anal­
ic approach are equivalent; they are called nuclei.
ogous to the Bethe approximation for equilibrium.
From a thermodynamic point of view these nuclei are
Condensation and evaporation are considered explicitly
in (labile) equilibrium with the fluid phase. When the
in a system of master equations applied to a pair of
sites. The effect of neighbouring sites is treated with
I ■ 1 a set of parameters, which are determined self-con-
I .i sistently.
Numerical integration of the equations yields the
■
time evolution of the surface. The results of high a
values indicate a region of metastable states (zero
growth rate) for small driving forces and the instan­
taneous growth rate is a periodic function of time
for larger ß values. The period of the variations is the
time required to grow a monolayer and the amplitude
is less than that obtained by Temkin using a mean
field approximation in a dynamic version [27]. The
variations appear as artifacts which are inherent to
the pair approximation. The average growth rate is
Fig. 6. The structure of the surface at t = T (see fig. 5). The
Q
solid units in the growing layer are black. Growth took place less than the Wilson-Frenkel law for all growth con­
by spreading of the largest cluster. ditions studied. This results since the surface becomes