ELSEVIER Materials Chemistry and Physics 49 (1997) 93-104

MATERIALS CHEM;S#Y’~D

Review

Influence of surface active species on kinetics of epitaxial nucleation and growth

Ivan Markov Institute @Physical Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria

Received 6 September 1996; a&$ed 28 October 1996

Abstract

The presence of surface active species (surfactants) on the surface of a growing crystal changes dramatically the kinetics of nucleation, the morphology of growth, the distribution of stress, the concentration of defects, etc. In the present review paper the influence of the surfactant on some of the most important growth characteristics as the rate of 2D nucleation, the saturation nucleus density, the roughness of the steps, and the mechanism of growth of hetero- and homoepitaxial films is considered in detail. It is shown that the effect of the surfactant on the nucleation and growth processes is quantitatively determined by a parameter called surfactant efficiency. The latter represents in fact the energetic parameter of the enthalpy of mixing of the surfactant and crystal materials relative to the energy of a dangling bond. A kinetic model for the surfactant mediated epitaxial growth is suggested. It is based on the well known fact that the surfactant atoms join predominantly kinks at steps which are the energetically most favored sites on the crystal surface. Thus they alter the energetics for incorporation of atoms to descending and ascending steps. As a result diffusion gradient of a kinetic origin appears that drives the atoms to join descending steps. The thermodynamic driving force for 2D-3D transformation to occur is suppressed and layer-by-layer growth takes place.

Keywords: Surfactant epitaxy; Surfactant efficiency; Nucleation; Growth; Kink formation

1. Introduction

Fabrication of devices based on heterojunctions between single crystal substrates and epitaxial films requires the epi- taxial interface to be as smooth as possible. Epitaxial films, however, often grow with formation and growth of isolated 3D islands either from the very beginning of the deposition (island or Volmer-Weber growth), or after the deposition of several (one, two or three) monolayers in a layer-by-layer (LBL) mode ( Stranski-Krastanov mechanism). The reason for islanding in the first case is the incomplete wetting of the film material by the substrate. It was expressed by Bauer [ 11 in terms of interrelation crs < (T + ui of the specific surface energies of the substrate, crs, overgrowth, o-, and the interface, cri, respectively. In the second case islanding is caused by the non-zero lattice mismatch of both materials in addition to the complete wetting, (+, > u + pi [ l-61 . It was established long ago that the mode of growth of epitaxial thin films is also strongly affected by the substrate temperature and the depo- sition rate (for a review see Ref. [ 61). Islanding is observed in the case of Volmer-Weber or Stranski-Krastanov growth at high temperatures (near-to-equilibrium conditions) whereas below a certain critical temperature the growth pro- ceeds by consecutive formation of monolayers (LBL

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growth) or by simultaneous growth of several monolayers (multilayer growth). In other words, a 2D or planar growth is observed in both cases. However, the as-grown films are unstable and aggregate into 3D islands upon annealing at higher temperatures. Films grown at low temperatures are often of poor quality. They contain many defects and could be even amorphous. That is why using the temperature as a tool to avoid islanding is not a solution to the problem in a series of cases.

According to recent observations the mode of epitaxial growth is strongly influenced by the presence of a third, surface active element called a surfactant [ 7-91. Cope1 et al. [ 71 established that predeposition of one monolayer (ML) of As suppressed islanding in Stranski-Krastanov growth of Ge on Si(OO1). Van der Vegt et al. [lo] found later that 1 ML of Sb deposited on Ag( 111) caused strong oscillations of the X-ray reflection to appear. This showed that Sb changed the growth mode from multilayer growth that is characteristic of homoepitaxial growth to LBL growth. Rosenfeld et al. [ 1 I] carried out a model study of the Ag/Ag( 111) system. They deposited anomalously high density of nuclei in the absence of a surfactant and showed that the greater the island density the stronger was the tendency for LBL growth. Voig- tlBlnder et al. [ 12,131 carried out detailed STM studies of 2D

94 I. Markov/Materials Chemistry and Physics 19 (1997) 93-l&f

nucleation of Si on clean Si ( 111) surface, and in presence of Ga, In, As and Sb as surfactants. They found, first, that the saturation island density steeply increased when group V elements As and especially Sb were used as surfactants. In contrast, In decreased the number density of the islands. Sec- ond, the nucleation exclusion (depleted) zones around the growing islands and the steps decreased in width in presence of As and Sb, and strongly increased in the case of In. Third, the islands’ shape and the form of the steps were irregular compared with the triangular shape and straight steps under clean conditions.

It was also found that the surfactant changes the energetics of the processes of incorporation of atoms into kink sites. When atoms join descending steps they should overcome an additional energetic barrier due to reduced coordination. The latter was first discovered by Ehrlich and Hudda who found that descending steps surrounding ( 1 lo), (2 11) and (321) crystal planes repulse atoms diffusing to them [ 141. The first theoretical study of the impact of this discovery performed by Schwoebel followed immediately [ 15 1. Detailed field ion microscopy (FIM) measurements of this barrier, Es, now known in the literature as a Schwoebel btier, in the case of Re, Ir and W on W( 110) gave values varying from 0.15 to 0.2 eV [ 161. Wang and Ehrlich [ 171 reported recently that incorporation of Ir atoms to a descending step on Ir( 111) surface can take place by an exchange mechanism rather than by rolling over the edge. Ab initio total energy calculations of diffusion across steps on Al( 111) surface carried out by Stumpf and Scheffler [ 181 showed that the above is not an exception. In order to explain the suppression of 3D growth Zhang and Lagally [ 191 assumed that the surfactant atoms incorporated into kink sites at steps decrease the Shwoebel barrier when the deposit atoms displace them by exchange mechanism in order to join the crystal lattice. Such an indium- induced lowering of E, has been experimentally established in the homoepitaxial growth of Cu( 001) [ 201.

Misfit relaxation and formation of defects are also obsta- cles for growth of high quality heteroepitaxial films. That is why strain relaxation in surfactant mediated growth was stud- ied in detail [ 2 1,221. One curious effect is that the formation of twins during growth of Co on Cu( 111) is strongly sup- pressed in the presence of Pb as a surfactant [23]. As expected it has been found that In, Sb and H acting as surfac- tants change the equilibrium shape of 3D Ge islands deposited on the surface of Si( 001) [ 241. The last paper brings up the matter of thermodynamics in s&a&ant mediated growth. Although the importance of thermodynamic considerations is obvious as in all cases of crystal growth this question is still unresolved. As noted by Kern and Miiller [25] some authors [7] feel that the surfactants play more a thermody- namic role by shifting the Bauer criterion for 3D growth. The wetting of the substrate by the film is altered and the islanding is suppressed. Most authors, however, think that the surfac- tants affect more the kinetics of growth [ 10,11,18-201. In this paper we will consider only the effect of the surfactant on the kinetics of nucleation and the impact of the surfactant

on the attachment of atoms to steps and in turn on the kinetics of growth, Readers interested in the thermodynamics of sur- factant mediated growth are referred to the original papers [ 26,271. We note only that the chemical potential of an infi- nitely large crystal does not depend on the presence of surface active species on its surface, i.e. in the case of homoepitaxial growth [ 281. This is not the case, however, for heteroepitax- ial growth where the chemical potential of the first deposited monolayers depends in a complicated manner on the energies between the surfactant atoms and atoms of the substrate and deposit crystals [ 26,271. The paper is organised as follows. Expressions for the work for nucleus formation, the nuclea- tion rate and the saturation nucleus density in presence of a surfactant are derived in Section 2. The calculations are car- ried out for close packed crystals with central interatomic forces. The formation of kinks along the steps in the presence of a surfactant is considered in Section 3. The effect of the surfactant on the mode of growth is considered in Section 4. The results are discussed in Section 5.

2. Kinetics of nucleation

2.1. Workfor ndeusformation

It is well known from the classical theory of nucleation that the adsorption of surface active species decreases the surface energy of the 3D nuclei (or the edge energy of the 2D nuclei) owing to saturation (at least partly) of the unsat- urated dangling bonds on the crystal surface. This leads to smaller nuclei, less work for nucleus formation and higher nucleation rate [ 281. However, far from equilibrium the clas- sical theory of nucleation is not applicable as the nuclei con- sist of a very small number of atoms. Then such macroscopic thermodynamic quantities as the specific surface (or edge) energies cannot be applied to describe the process of nucle- ation. That is why we reformulate the atomistic theory of nucleation which deals with small clusters of atoms [ 29,301 to account for the presence of a surfactant.

We calculate first the Gibbs free energy for nucleus for- mation by using the following imaginable process (see Fig. 1) . The initial state is a surface covered by 1 ML of surfactant (S) atoms (Fig. 1 (a) ). First we evaporate all sur- factant atoms reversibly and isothermally. Then on the clean surface we produce a cluster consisting of i atoms (Fig. 1 (b) ) . The Gibbs free energy needed to form a cluster of an arbitrary shape on the clean surface reads [ 3 1 ] (for a review see Ref. [ 301)

AGO(i) = -i Ap+ Q, (1)

where A p is the supersaturation (the difference of the chem- ical potentials of the infinitely large ambient and new phases), and rf, is the excess surface (or edge) energy. In the case of small supersaturations, when the cluster is large enough, @ can be expressed in terms of surface, edge and apex energies. For a cluster of any size and particularly for small clusters

I. Markov/Materials Chemistry and Physics 49 (1997) 93-104 95

(a)

0)

Cc) Fig. 1. The calculation of Gibbs free energy for nucleus formation: (a) the initial surface covered with surfactant atoms denoted by filled circles; (b) the surfactant atoms are evaporated and a cluster consisting of i atoms is created; (c) the surfactant atoms are condensed back thus forming a cluster on top of the crystal cluster.

the surface energy is expressed in terms of the energy of unsaturated dangling bonds in the form [ 30,3 I]

@=iE,- U(i) (2)

where Ek is the energy needed to detach an atom from a kink (or a half-crystal) position [ 301 (the bulk energy per atom of the crystal) and U(i) is the energy of dissociation of the cluster into single atoms, including also the bonds with the substrate. In fact @ gives the number of the dangling bonds multiplied by the energy, $J2, of one dangling bond (& is the work needed to disjoin two crystal (C) atoms). As it is written above, @is equally valid for any 3D or 2D clusters with arbitrary shape and size. In case of 2D clusters @gives the energy of the dangling bonds at the periphery of the cluster, or in other words, the edge energy.

Finally we condense back all the surfactant atoms (Fig. 1 (c) ) and for the work of formation of a cluster con- sisting of i atoms we obtain [ 321

AG,(i)=-iAp+[iE,-U(i)](l-s)+@~ (3)

where

cp,= [iE,- U(i)]cr, (4)

is the edge energy of the cluster consisting of S atoms formed on top of the crystalline cluster (Fig. 1 (c)) assuming that the surfactant cluster has the same size and shape as the crystalline cluster,

QY, = As/ *cc (5)

Gss being the work needed to separate two S atoms, and

s=l-O/W0 (6)

accounts for the presence of the surfactant where

CfJ = i ( 44s + *cc) - *se ~~ (7)

and

wo = *cc/2 (8)

is the energy of a dangling bond. In the above equations & is the work needed to disjoin a S atom from a C atom.

In fact Eq. (7) represents the relation of DuprC [ 2,33,34], cri = CT + g’s - y, where the specific surface energies, c~, crs, (TV, and the specific adhesion energy, y, are taken per atom. This can be easily verified if we multiply Eq. (7) by the area occupied by one atom. It turns out that the parameter w rep- resents the interfacial energy per atom. In other words, the energy, w,= &../2, of the dangling bonds at the cluster periphery under clean conditions is replaced by the smaller interfacial energy per bond o. Note also that in fact w is the energetic parameter which determines the enthalpy of mixing the two species C and S. It must be positive in order to allow the growth of the crystal and the segregation of the mediate. As it will be shown below w> 0 is the necessary condition for layer growth of the crystal. At w 5 0 the crystal surface will be thermodynamically rough and will grow by direct incorporation of atoms to kink sites from the vapour [ 351. Moreover, the mediate will tend to mix with the crystal. Therefore, we consider the nucleation process as taking place in a two-dimensional solution on the crystal surface in which the surfactant plays the role of a solvent. In the absence of a surfactant &= Gbs,=O, o= &/2 and s=O. At the other extreme of complete efficiency of the surfactant (ICI,,+ &) I2 = I,/I~~ and s = 1. In general, ( &, + $..) 12 could be smaller than & which leads to s > 1. However, this means a negative value of w or a negative enthalpy of mixing. Thus the parameters varies from 0 at complete inefficiency to 1 at complete efficiency and is always positive. We call the parameters a s&actant eficiency. It could be calculated on the basis of thermodynamic data but, as will be shown below, it can be also experimentally measured as it determines the equilibrium structure of the steps in the presence of a surfactant.

Substituting Eq. ( 1) in Eq. (3) gives

AG,(i) =AG&i) -s[iEk-U(i)] +QS (9)

It becomes immediately obvious that in the case of surfac- tant mediated growth the Gibbs free energy for nucleus for- mation contains two more terms which have opposite signs and thus compete with each other. The s-containing term accounts for the decrease of the edge energy of the cluster due to saturation of the dangling bonds by the surfactant atoms. The energy, @St,, of the dangling bonds of the periphery of the cluster, consisting of surfactant atoms, which is una- voidably formed on top of the nucleus due to segregation of the mediate, increases the work of cluster formation. It is important to note that the parameters s and cy, are independent of each other, although both equal zero by definition in the absence of a surfactant. This means that at nearly complete inefficiency of the surfactant, s = 0, cr, has a finite value and the work for nucleus formation and the rate of nucleation change discontinuously.

Fig. 2 shows the dependence of AG,( i) relative to the crystal bond strength $., on cluster size, i, for the (111) surface of f.c.c. metals with cr, = 0.2, different values of the

1. Markov /Materials Chemistq and Physics 49 (1997) 93-104

Fig. 2. The change of Gibbs free energy for cluster formation relative to the work needed to disjoin two crystal atoms, A G,( i) / &, vs. the number of atoms on ( 111) surface of a fee crystal. The values of the surfactant effi- ciency, S, are denoted by figures at each curve. The structure of the nucleus is given by the filled circles. The empty circles denote the atoms which turn the critical nuclei into stable clusters. The supersaturation APL= 1.1 QCC for all curves. As seen the increase of efficiency of the surfactant leads to decrease of the critical nucleus size and the work needed for its formation. The only exception is at small values of s at which the edge energy of the surfactant cluster, QS, overcompensates the decrease of the edge energy of the crystalline cluster. In this case i* and AG,(i*) are larger than in the clean case.

surfactant efficiency, s, and at a constant supersaturation A p = 1.1 &. s is denoted by figures at each curve. As it could be seen AG,(i) represents a broken line (as should be expected at small number of atoms) displaying a maximum at i = i*. The latter represents the size of the critical nucleus. Under clean conditions (s= 0) the critical nucleus consists of 2 atoms. When s approaches zero ( = 0.05)) the number of atoms in the critical nucleus equals 6 owing to the contri- bution of the edge energy of the surfactant cluster, QS. The work of formation of the critical nucleus also increases. Increasing the surfactant efficiency to 0.3 leads to a decrease of the nucleation work and i” becomes again equal to 2. At some greater value of s ( = 0.7) i* equals 1 and the work of nucleus formation decreases drastically.

In the case of 2D nucleation and close packed lattices

U(i*) = E” + iiEd,. (10)

where Edes is the activation energy for desorption from the clean surface and E” is the work required to dissociate the cluster into single adatoms.

The work of formation of the critical nucleus then reads

(11)

We have to bear in mind that the supersaturation does not depend on the presence of a surfactant on the crystal surface. This follows from the definition of the supersaturation as a difference of the chemical potentials of the infinitely large ambient and new phases, and as has been mentioned above, the chemical potential of an infinitely large crystal does not

0.0 0.2 S”RFACk4T E&C&Y

1.0

Fig. 3. Variations of the Gibbs free energy with the surfactant efficiency of formation of a critical nucleus relative to the same quantity under clean conditions at two different values of cy,. The jump at s = 0 is due to the finite value of Q9. The singular points divide regions with different number of atoms in the critical nucleus denoted by the figures at each part. A p= 1.1 I& for both curves.

depend on the presence of surface active species on its sur- face. Fig. 3 shows the work needed to form the critical nucleus AG,( i*) relative to the work for nucleus formation, A G,( i*), under clean conditions as a function of surfactant efficiency, s, at two different values of a,. A decreasing bro- ken line is seen which is due to the change of i* at some particular values of S. At s = 0 AG,( i*) increases discontin- uously and becomes larger than A GO( i*) , owing to the edge energy, djS, of the surfactant cluster. The larger the value of Q5 is the stronger is this effect. At small values of tr, (small QS) the plot of AG,( i*) on s tends to cross the zero at some value of s close to unity. This means that at low edge energy of the surfactant cluster the supersaturation overcompensates the former at the very beginning of nucleus formation. It follows that at very high efficiencies of the surfactants the crystallization will take place without any need to overcome a barrier for nucleus formation. On the contrary, at high values of 9 the supersaturation cannot overcompensate the edge energy of the surfactant cluster and the work for nucleus formation has a finite value even at s = 1.

2.2. Nucleation rate

In order to calculate the nucleation rate we follow the usual procedure [ 301. The nucleation rate is given by

J,=@*NJexp[ -AG,(i*)lkTf (121

where ,B* is the flux of atoms joining the critical nucleus, N,, is the density of adsorption sites on the crystal surface, and r ( z 0.1) is the non-equilibrium Zeldovich factor. The flux of atoms @ * is given by

/3*=cw(n,lNo) exp[ -(EsdiAESd+AiJ)/kT] (13)

In the above equation a is the number of ways by which an adatom can join the critical nucleus [ 301, v is the vibra- tional frequency of the adatoms, nS is the adatom concentra-

I. Markov / Materials Chemistry and Physics 49 (1997) 93-104

tion, A U is the kinetic barrier an adatom should overcome to displace a surfactant atom in order to join the critical nucleus, Esd is the activation energy for surface diffusion on a clean surface in the absence of a mediate, and AEsd is the additional energy an atom has to overcome when diffusing on a surface covered by a monolayer of surfactant atoms. The plus sign refers to the case of inhibited diffusion (diffusion in between the surfactant atoms) whereas the minus sign refers to the case of surfactant-facilitated diffusion (diffusion on top of energetically smoother surfactant layer).

It seems reasonable to argue that the energies AEsd and A lJ depend on the interrelation of the works needed to break C-C, S-S and S-C bonds, or in other words, on the value of s. The greater the value of s, the more difficult are the dis- placement processes and the larger the values of AEsd and A U should be. The simplest way to express the s-dependen- ciesofAE,,andAUistowriteAE,,=sAE$andAU=sA@ where A/?$ and A U’ are some maximum values of AEsd and A U at s = 1. One can assume that A,!& and A @ could have values even greater than that of Esd.

Combining Eqs. (9)-( 11) gives [ 321

Js=cuVOv~(n,/NO)‘*+’ exp{[i*(E,-E,,,)s+(l-s)E*

-Q-(E,,+sAE$)-sA@]lkT} (14)

In deriving Eq. (14) we used the fact that the supersatu- ration is given by

A~=kTln(n,ln,,)

where the equilibrium adatom concentration, rise, reads [ 361

n ,,=f% exp[ - (&-Edes)~kTl

It becomes immediately obvious that in absence of a sur- factant s = 0, Gs = 0, Eq. ( 12) takes the familiar form [ 301

JO=crNOv~(n,lNO)‘*+‘exp[(E*-E,,)lkTj (15)

Comparing Eqs. (14) and (15) shows that the presence of the surfactant leads above all to a sharp decrease by orders of magnitude of the nucleation rate owing to the edge energy, Qs, of the surfactant cluster (Fig. 4). After that the nucleation rate increases with s which stems from the decrease of the edge energy as in the classical theory of nucleation. The dependence is represented again by a broken line which reflects the discontinuous decrease of the number of atoms in the critical nucleus.

Note that in Eqs. (14) and (15) the nucleation rate is expressed as a function of the adatom concentration, n,. In the case of incomplete condensation (adsorption-desorption equilibrium) the latter is given by n,=Rrs where rs is the mean residence time before desorption of the adatoms on the crystal surface and R (cmB2 s-‘) is the atom arrival rate. At the other extreme of complete condensation (absence of re- evaporation) the adatom concentration increases linearly with time, n, = Rt or is given by n, = Rr where r is the mean free time of the adatoms before incorporation into islands or steps. As given above the equations for the nucleation rate are suitable for any particular case mentioned above.

97

Fig. 4. Dependence of the nucleation rate, J,, relative to the rate, JO, under clean conditions on the surfactant efficiency at two different values of (Y,. The broken line reflects the variation of the Gibbs free energy on the size i* of the critical nucleus denoted by the figures at each section. A /L = 1.1 I)- for both curves.

2.3. Saturation nucleus density

Once the supersaturation is ‘switched on’ the adatom con- centration increases linearly with time. At sufficiently high temperature a dynamic adsorption-desorption equilibrium is gradually reached and n, = Rr, so that the nucleation process takes place at a constant supersaturation. At low temperatures (which is usually the case) the adatom concentration increases with time during the nucleation process. If the sur- face diffusion is still fast enough depleted zones appear around the growing islands and in the near vicinity of the steps. The system in them is undersaturated and the nucleation process is arrested. After some time the depleted zones over- lap and cover the surface completely. This results in satura- tion of the number density of the nuclei [ 371. If the surface diffusion is slow, a large number of islands is formed at the start of the process and they begin to coalesce at an early stage of deposition. The saturation nucleus density is then determined by coalescence of the islands [ 38,391, The ada- tom concentration is again constant, n,=Rr, and is deter- mined by the balance of atoms arriving from the vapor and joining the islands and/or the steps on the crystal surface (for a review see Ref. [ 391) . We consider here the saturation of the island density which is due to the overlapping of depleted (nucleation exclusion) zones. There are two reasons for this. First, it has been shown in numerous papers that the surface diffusion has a considerable rate at extremely low tempera- tures (see Ref. [40] ). Second, the overlapping of the depleted zones always precedes in time the coalescence process.

We assume that there is no re-evaporation, i.e. we consider the case of complete condensation and follow the procedure developed by Stowell 11371. A depleted zone is formed around each growing nucleus with a radius following the time law r F (D,t) 1’2, where D, is the surface diffusion coefficient [ 371. Outside the depleted zones the adatom concentration increases linearly with time, II, = Rt. Nuclei are progressively

98 I. Markov/Materials Chemistry and Physics 49 11997) 93-104

formed outside the depleted zones with a rate J(t) at’*+‘. The saturation nucleus density is given by [41-43]

Substituting Eq. ( 14) in Eq. ( 16) and making the integra- tion gives [ 321

(3

<i*+ l)/(i*+3)

N, = qN; (i*-l)/(i*+3) R

D

X exp [

i*(Ek-Edes)s+(l-s)E”-@S-sA~ (i*+3)kT 1 ( 17)

where LJ is nearly a constant and has a value of about 0.1. Under clean conditions (s = 0) Eq. ( 17) becomes

R (i*+l)/(i*+3)

NfEqN;(‘*-‘)/(‘*+3) D (3 (18)

Note that in the absence of a surfactant the surface diffusion coefficient, 0: differs from that in a surfactant mediated growth, D,, by a factor of exp( is Aed/kT).

Eqs. ( 17) and ( 18) are comparedin Fig. 5. As can be seen the increase of s up to 0.8 leads to an increase of N, by about three orders of magnitude in cases when surface diffusion is inhibited by the presence of the surfactant. In the opposite case the nucleus density remains of the same order of mag- nitude. The change of the number of atoms in the critical nucleus leads to some characteristic discontinuities. Fig. 6 represents Arrhenius plots of the nucleus density under clean conditions (the line denoted by @) and mediated by surfac- tams which either facilitate or inhibit the surface diffusion. The intervals of the temperature are chosen so that i” = 2 for all curves.

lEt4 3 I I 1 I I

lEt2

lE+O

Fig. 5. Variations of the saturation island density, N,, relative to the same quantity, Nz, under clean conditions at a different impact of the surfactant on the rate of surface diffusion. The number of atoms in the critical nucleus is denoted by figures at each section of the curve. The number of atoms in the critical nucleus in the absence of a surfactant is i* = 2. I& = AE$ = AU”=O.deV, &,,,=0,44eV. R=2X10’3cm-‘s-‘, No=1X10’5cm’, r~=lXlO’“s-’ Cr.=05 75 .

lE+K?.

3

/ A&i = * sAEzd

lEtI

lE+I 1, 1.0 1.5 2.0 2.5 3.0 3.5

IOOOrr Fig. 6. Arrhenius plots of the saturation nucleus density, N,, at a different influence of the surfactant on the rate of surface diffusion. x denotes the curve obtained under clean conditions. The values of the quantities are the same as in Fig. 5.

2.4. 20 nucleation on n foreign substrate

The formation of 2D nuclei on the surface of an unlike crystal in the presence of a surfactant is easy to consider following the same procedure as above. Two additional terms i*(Ec,ea- Esub) and i* ( Es,sub - ES,+J enter the expression for the work for nucleus formation. Correspondingly, the expres- sions for the nucleation rate and for the saturation nucleus density should be multiplied by exp{ - [ P(Edes - Esub) + i*(ES,sub -Es,dep) 1 /kTJ and WI - [i*c(L --Ld +i*(Es,sub - Es,dep) ] /(i* + 3)kT}, respectively. The first difference Edes -Esub accounts for the wetting of the deposit material by the substrate, Esub being the energy of desorption of a deposit atom from the substrate surface, or in other words the adhe- sion energy. The second term is the difference between the energy of desorption of a surfactant atom from the substrate surface, Es,sub, and from the deposit surface, Es,dep. In general the wetting can be either complete, Edes < &,, or incomplete, Edes > .&,, so that the first term can in principle increase or decrease the nucleation work. Complete wetting gives rise to smaller nucleation work and higher nucleation rate and vice versa. Although data for desorption energies are scarce in the literature one could predict at least qualitatively the sign of the first term and its influence on the kinetics of nucleation. For example, when depositing Ge on Si one could assume that Ge atoms are more strongly attracted by the Si substrate than by its own Ge substrate and thus EdCS < Es”,,. The eval- uation of the second term is, however, much more difficult. As the energies Es,sub and Es,dep are of the order of several eV their difference could be large and can affect significantly the kinetics of nucleation. Some information concerned at least with the sign of Es,sub - Es,drp could be obtained as a result of a comparative study of nucleation on like and unlike substrates in the presence of one and the same surfactant.

3. Work for kink formation

The work for kink formation determines the roughness of the steps and in turn the state of the crystal surfaces and the

I. Markov / Materials Chemistn, and Physics 49 (1997) 93-104 99

mechanism of growth [36] (for a review see Ref. [35]). Modem techniques for surface analysis such as the scanning tunnelling microscopy (STM) allow direct observation of steps and measurement of step roughness. Burton and Cabrera [ 441 were the first to define and calculate the work for kink formation by thermal excitation in the case of growth in vapor. They considered a straight step and removed consec- utively two adjacent atoms embedded in the step and placed them at the step edge thus producing four kinks. The spent work was 2&, and the work, wO, for kink formation was equal to half the work needed to break a nearest neighbour bond, or wg = I$J 2. A generalization was later given for the case in which the crystal surface was in contact with its own melt (for a review see Ref. [ 451) . The latter can be directly used for the case of surfactant mediated growth. As it can be se& from Fig. 7 simple calculations based on the principle of conservation of bonds and the lattice model of the surfactant layer show that the work of kink formation is given precisely by the parameter w. Following Burton et al. [36], we can write an approximate expression for the density, p, of kinks in the presence of a surfactant

(19)

where a is the interatomic distance. Neglecting the kink-kink interaction the Gibbs free energy

of the step reads [ 361

G,,= -“Tin a (20)

where 7~ = exp( - wlkT) . What follows is that the surfactant efficiency must be

always positive (positive enthalpy of mixing, w > 0). Ifs 2 1 the work for kink formation and in turn the free energy of the step will equal zero. The steps will disappear and the crystal surface will become rough. Then the crystal surface will grow by direct incorporation of atoms from the vapour phase with- out any need to form 2D nuclei [ 34-361.

(a) (al

(b) Fig. 7. Calculation of the work needed for kink formation: (a) a straight step without kinks; (b) a step with four kinks. The crystal and surfactant atoms are denoted by empty squares and filled circles, respectively. The energetic difference between both configurations yields the work, ~OJ= 2&-t 2&,- 4&,, required to create four kinks.

4. Kinetics of growth

As discussed above based on the observations of Ehrlich and Hudda [ 141 Schwoebel assumed that incorporation of atoms to kinks is asymmetric, being easier at ascending rather than at descending steps [ 151. Whereas an atom approaching an ascending step joins it upon striking, an atom approaching a descending step has to overcome an additional barrier, Es, in order to join a kink at the step [ 141 as shown in Fig. 8(a). The observations of Wang and Ehrlich [ 171 and the total energy calculations of Stumpf and Scheffler [ 181 showed that incorporation of an atom at a descending step can take place by an exchange mechanism rather than by rolling over the edge. In particular Wang and Ehrlich [ 171 found that if the island is large enough ( > 20 atoms) the adatoms at the step edge become trapped for some time and prefer to join the step rather than to diffuse backwards. Therefore, they proposed the potential diagram shown in Fig. 8(b) in which the last barrier that the atoms have to overcome in order to join the step is shifted downwards by an amount E, below the level of the surface diffusion barrier. Thus the last poten- tial trough becomes asymmetric and E, is a measure of its asymmetry. The latter means that the step becomes attractive

E sd

4

(4

E sd

4

” E,,, I ,

Fig. 8. Schematic potential diagrams for atoms diffusing toward ascending and descending steps: (a) traditional view; (b) according to the results of Wang and Ehrlich (solid line) [ 171; E, is the Schwoebel barrier. The additional barrier E, at the ascending step in the presence of a mediate is shown by the dashed line. A W is the work needed to transfer an atom from a kink site at the step on the terrace. Note that the measure of asymmetry E,

is accounted for from the level of the surface diffusion barrier. Surfactant and crystal atoms are denoted by filled and empty circles, respectively.

100 I. Markov/Materials Chemistq and Physics 19 (1997) 93-103

rather than repulsive for atoms arriving from the upper terrace.

The presence of a surfactant should change the asymmetry of incorporation. The surfactant atoms should predominantly join the kink sites at the steps as the latter are the energetically most favoured sites on the crystal surface [ 3.51. Then the deposited atoms must displace them in order to join the crystal lattice. An atom approaching a descending step can join a kink displacing a mediate atom by the exchange mechanism [ 171. One could expect that the Schwoebel barrier will be reduced by an amount Z( &- &), where Z is the coordi- nation number of the atoms in a kink site, as shown by the broken line in Fig. S(b) [ 191. Moreover, if the Schwoebel barrier has the form shown in Fig. 8(a) in absence of a sur- factant its value can be reduced considerably and become negligible in presence of a surfactant [ 201. On the other hand, an atom that approaches an ascending step (Fig. 8(b)) should displace a surfactant atom in the same plane. This will give rise to an additional barrier of kinetic origin of amount Em as shown by the dashed line in Fig. 8(b) [ 351. One can speculate that the additional barrier Em is of the order of the amount of work needed to break a bond $s-,c between a surfactant and a crystal atom.

We consider now the kinetics of deposition of the epitaxial films. Atoms strike the substrate surface, accommodate ther- mally, migrate over the surface, and collide with each other to give rise to 2D nuclei. The latter grow further as 2D islands. Atoms arrive on their surface and an adatom population is formed whose concentration has a maximum value on the middle of the islands [ 3,6]. The maximum value of the ada- tom concentration on top of the 2D islands increases with the square of the island size and at some moment 2D islands of the second monolayer will most probably form around the middle of the lower islands, thus giving rise to flat pyramids of growth as shown in Fig. 9 [ 3,5]. The upper islands grow at the expense of the atoms arriving on their own surfaces and on the terraces between upper and lower islands, whereas the lower islands grow at the expense of atoms on the terraces and these on the surface between the pyramids. Islandgrowth will be observed when the surface transport on the terrace is directed towards the edges of the upper islands, or in other words, when the intralayer diffusion dominates. As a result the upper islands will grow faster and will catch up with the steps surrounding the lower islands, thus transforming the pyramids into bilayer (3D) islands. The thermodynamic driving force for intralayer transport is the difference of the chemical potentials of both monolayers [4-6,45]. The latter is due to the difference in bonding of the atoms to kink sites at the upper and lower steps, on one hand, and to the distri-

I I j P-21

/ PI

0 Pz P, Fig. 9. Pyramid of growth consisting of two monolayer islands with radii p, and pz, and chemical potentials pL1 and pL? (p2 < CL,).

bution of the misfit strain, on the other [ 61. Such amechanism of formation of 3D islands was recently observed by Corcoran et al. [47] in the case of Stranski-Krastanov growth of Ag on Au( 111) . LBL growth will be observed when the surface transport is directed downwards towards the descending steps surrounding the lower islands, or in other words, when the interlayer diffusion is faster than intralayer diffusion.

We consider a pyramid of growth consisting of two 2D islands with radii p, and p,, one on top of the other (Fig. 9). We assume that the chemical potential k* of the upper island is lower than that, p,, of the lower one. In fact, the inequality &-<p, represents the thermodynamic criterion for island growth [4--6,&l. We further assume that adatoms joining a descending step have to overcome a barrier higher than that for surface diffusion Esd by an amount Es minus an amount due to the presence of a surfactant as shown in Fig. 8(b) . It is worth noting that the same final result will be obtained if we assume that an adatom should overcome the barrier Es shown in Fig. 8(a) but taking into account that it is strongly reduced in the presence of a surfactant. The necessary and sufficient condition for faster interlayer diffusion is that the barrier E, which inhibits the intralayer diffusion be higher than Es.

The adatom concentration n,(r) on the terrace in the case of complete condensation (absence of re-evaporation) can be found by solving the diffusion equation

d*n,(r) I 1 &Jr) 1 R -o

d? r dr D,

The solution reads

n,(i.)=C1--&?+CZln(r) 5

(21)

(22)

The integration constants C, and C, can be found from the boundary conditions ji = + D, [h,(r) ldr] r= pr ( i = 1,2), with the minus sign referring to i= 1 [ 151, where j, andj, are the net fluxes of atoms towards the lower and upper steps. In order to join the upper step the atoms have to make a single jump over the barrier E, and

j2=Mayexp[ -E,,/kT](n,-n;) (23)

where M = exp( - E,,,lkT), n2 is the adatom concentration in the near vicinity of the upper step, and ng is the adatom concentration which is in equilibrium with the step. The latter is given by 1361

nB=n,, exp( -AW,IkT) (24)

where nzk is the density of detachable atoms at the step (pre- sumably atoms at kink sites), and A W, is the work to transfer an atom from a kink position along the step on the terrace.

In the case of negligibly small Schwoebel barrier at the descending step ( Es < Esd) we obtain for j, the same expres- sion as above. The problem becomes more complicated when the potential at the step has the shape shown in Fig. 8(b) . The incorporation of atoms to the descending step takes place

2. Mmkov /Materials Chemistry nrd Physics 49 (1997) 93-104 101

in two consecutive stages, n, -nuenlk, where n,, n,,, and nlk are the concentrations of the atoms in the near vicinity of the step, the atoms trapped in the last potential trough before joining the step, and the atoms detachable from the step. Assuming a steady state concentration of the atoms trapped in the last potential trough (dn,ldt = 0) , and bearing in mind the potential diagram in Fig. 8(b) , forj, one obtains [ 481

2 J1 =jq-y exp 1 1 -2 (nl-n;) (25)

where A = exp( - E,/kT) is the factor accounting for the asymmetry of the last potential trough, n;=n,, exp( - A W, lkZ’) as above, A W, being the work to transfer an atom from a kink site at the lower step on the terrace. Note that A W, <A IV, is in fact equivalent to h < ,LL~. Note also that the Schwoebel factor S = exp( - E,/kT) does not enter the flux when the potential has the shape shown in Fig. 8 (b) . If the last potential trough was symmetric, A = 1, and the flux should be exactly equal to the fluxj,, in absence of a Schwoe- be1 effect.

The adatom concentration on top of the upper island, which is surrounded by a descending step only, can be found by the same way from the solution of Eq. (21), n,(r)=C- R?-/4D,, subject to the corresponding boundary condition with a flux analogous to j,. The exact expressions for the adatom concentrations are very cumbersome, so we do not give them; instead, we proceed further and calculate the rates of growth of the 2D islands. In order to calculate the rate of growth dprldr of the lower island we assume that atoms arriving on the crystal surface between the growth pyramids in case of complete condensation are equally distributed among the islands and the flux is j, =R( 1 -rpfN,) IN,, where N, is the density of the pyramids. Through the substi- tutions C?Ji = 1~pfN~ (i = 1,2) and T= Rt/N,, where N,, is the density of the adsorption sites, a set of equations for the surfaces coverages, Oi and O,, of the lower and upper mon- olayers as a function of the number, T, of deposited monolay- ers can be written in the form [ 481

with

F( 0,,@2) =

(27)

where

p = 4~WVn? - 4) R

and

(28)

200 400 600 800 TEMPERATURE (K)

Fig. 10. Schematic representation of the temperature dependence of the parameters P and Q. Pa exp ( - A W/klkT) where A W is the work needed to transfer an atom from a kink position on the surface. A W is of order of several eV and thus P is very sensitive to the temperature. QaNj” and hence Qctexp(E/2(i*+3)kT), where E=&+E*l(i*+ 1). At i*= 1, Qaexp(E,,/Skr) and is very weak function of the temperature.

(29)

Note that the parameter P contains the thermodynamic driving force nT -n; for the 2D-3D transformation to occur as the equilibrium adatom concentrations ny and nz, (n; > n;), are determined by the corresponding chemical potentials through np = ne exp [ ( pj - ,LL~) /kTj (i = 1,2) [ 61, where p- is the chemical potential of the infinitely large deposit crystal andne = No exp( - A WIkT) is theequilibrium adatom concentration on its surface [36]. As shown in Fig. 10 the parameter P decreases steeply with decreasing temperature but is always smaller than unity. In contrast, the parameter Q increases with decreasing temperature varying approximately from 1 X 10-l at very low (room) tempera- tures to 1 X 10W3-1 X 10m4 at high temperatures.

In the absence of a surfactant A4 = 1. At high temperatures P=0.2-0.3,Q<l,andF(@,,@,) turnsinto [3,6]

(30)

The solution of the system of Eqs. (26) with Eq. (30) (P = 0.25) is shown in Fig. 11 (curves 1’ and 2’). As seen at temperatures high enough so that P is large, the solution for the surface coverage of the first monolayer, 0,) increases initially, displays a maximum, and decreases. This means that at some stage of growth the rate of advance of the first mono- layer islands, dp,ldt, becomes negative or, in other words, the lower islands decay and the atoms feed the upper islands. The edges of the latter catch up with the edges of the former and islands with double height result. The double steps prop- agate more slowly than the single steps [ 351, and after some time the single steps of the third monolayer islands catch up with the edges of the double-height islands, thus pro- ducing islands with triple height. As a result, island growth takes place. At low temperatures P +c 1, Q = 0.1, and

102 I. Markols / Materials Chemistv and Physics 49 (1997) 93-l 04

0.0 NUMBER OF~ONOLAYERS

1.0

Fig. 11. Dependence of the surface coverages of the first (curves 1 and 1’) and second (curves 2 and 2’) monolayers on the amount of material depos- ited in number of monolayers in the case of heteroepitaxial growth. Curves 1’ and 2’: growth in the absence of a surfactant or at very high temperatures [F(@,,@,) is given by Eq. (30)]; curves 1 and 2: surfactant-mediated growth with F( O,,O,) = 0,. The insert between curves 1’ and 2’ shows the shape of the island at the coverage at which both curves nearly touch each other. The insert above curve 1 shows the shape of the island in surfactant mediated epitaxy at the completion of the first monolayer.

F( O,,O*) = ( olo*)l’*. The solution of Eqs. (26) (not shown) demonstrates the typical behaviour of multilayer growth: 0, reaches unity when 0, reaches a value of about 0.3. The thermodynamic driving force for 2D-3D transfor- mation is suppressed and the atoms on the terrace are equally shared between both islands. As a result planar growth is observed.

In the case of surfactant mediated growth, Mf 1. At very high temperatures Q K 1 and MG 1. F( 0, ,@) is given by Eq. (30) and island growth should be observed irrespective of the presence of a surfactant. As the temperature decreases Q increases and M decreases so that the terms containing Q/M become dominant. Then F( O,,@) = 0, and the solution of Eqs. (26) subject to the initial conditions T= 0, 0, = @ and 0, = @ reads [ 481

@,=@+T-@[exp(T)---11; @,=@exp(T) (31)

Eqs. (3 1) are plotted in Fig. 11 (curves 1 and 2). As seen 0, reaches unity while 0, is still negligible. The 2D islands grow in a kinetic regime [ 341, the thermodynamic driving force for 2D-3D transformation being totally suppressed. The interlayer transport dominates owing to the fact that the upper step repulses the atoms whereas the lower step attracts them. A diffusion gradient on the terrace appears, which drives the atoms towards the lower step. On one hand, this prevents the transformation of the pyramid to a 3D island. On the other hand, the upper islands grow only at the expense of the atoms adsorbed on their surfaces. They remain small and the nucle- ation on top of them is also suppressed. As aresult the growth is nearly LBL owing to kinetic reasons.

The case of homoepitaxial growth [e.g., Ag/Ag( 111) / Sb] [ 5 ] is treated in the same way with P = 0 ( pLI = Pi). The same result will be obtained if the lower step also repulses

0.0 NUMBER OF’%NOLAYERS ’

,O

Fig. 12. Dependence of the surface coverage of the first (upper curves) and second (lower curves) monolayers on the amount of material deposited in number of monolayers in the case of homotaxial growth, e.g. Ag/Ag( 111). The surface coverage of the second monolayer reaches a value of about 0.4 when the first monolayer is completed.

the atoms but weaker than the upper step. This is shown in Fig. 12 where the surface coverage of the second monolayer reaches a value of about 0.4 at the moment of completion of the first monolayer in the absence of a surfactant. In the presence of a surfactant the surface coverage of the second monolayer is negligible and nearly LBL growth is observed.

5. Discussion

As shown above, the formation of a 2D nucleus in a sur- factant medium leads unavoidably to formation of a cluster consisting of surfactant atoms owing to the segregation of the latter. As a result the work for nucleus formation contains in addition the edge energy, @$, of the surfactant cluster. The work for nucleus formation decreases as a result of the lower ‘edge energy’ but increases as aresult of the surfactantcluster formation. The latter increases the number, i”, of atoms in the critical nucleus at lower values of surfactant efficiency. Thus at one and the same temperature the size i* can be larger or smaller in the presence of surfactant than that in the clean case depending on the interrelation of s and a,. The greater the value of cts, the greater is the possibility that i” will be larger in surfactant mediated epitaxy and vice versa. It is important to note that the parameter (p, will equal zero if the concentration of the smfactant is much less than one mono- layer. If this is the case the surfactant atoms will prefer to join kink sites at the island edges where the bonding is stronger than at other sites [ 351.

The value of cr, can be evaluated qualitatively as a ratio of the energies of the dangling bonds of the C and S crystals

a, = ( o-v2’3) ,I ( o-v*‘3) c

calculated from the respective specific surface energies, c+, and molar volumes, u. Thus in the case of Sb-mediated growth of Ag(ll1) with nSb=395 erg cm-‘and ~,,=I140 erg cm-* one obtains (Y, z 0.5. In the case of Sb-mediated growth

I. Markov /Materials Chemists and Physics 49 (1997) 93-104 103

ofSi(lll) ((~~~=1240ergcm-‘) [2] cu,=00.42.Theseval- ues are large enough (see Fig. 3) and the contribution of @> should not be overlooked when comparing the experimental data with the theory even at very low temperatures. However, one should bear in mind how unreliable the measurements of the surface energies are and should consider the above values as approximate. In any case, they should be of the order of 0.5.

One very important consequence of the decrease of the work for kink formation due to the presence of a surfactant is the change of shape of the growing islands. Steps which are straight under clean conditions will become rough in surfactant mediated growth. As a result the well polygonized islands under clean conditions will be rounded in the presence of a surfactant. The above prediction is in agreement with the observations of Voigtlander and Zinner [ 121 on the growth of Si ( 11 I ) The shape of the islands and the form of the steps are irregular in Sb-mediated growth compared with the tri- angular island shape and straight steps under cleanconditions. A comparison of experimental observations on the structure of single steps in the absence and presence of a surfactant such as the STM studies carried out by Swartzentruber et al. [ 491, with Eq. ( 19) can give direct information about the value of the surfactant efficiency. This is particularly impor- tant in the case of SA steps on Si( OOl)-2 X 1 surface which are straight under clean conditions. Thus the presence of a surfactant in the growth of Si(OO1) can drastically change the mechanism of growth by step flow and prevent formation of double steps.

The saturation nucleus density is derived under the assumption of overlapping nucleation exclusion (depleted) zones around the growing nuclei. It is valid before the occur- rence of significant coalescence. An indication of applicabil- ity of the expressions derived above are the denuded zones observed around the steps as in the case of Si( 111) growth [ 12,131. Thus surface diffusion plays a dominant role in determination of the saturation nucleus density. Owing to the influence of the presence of surfactant on surface diffusion the saturation nucleus density can become much greater or smaller in comparison with growth under clean conditions. The schematic Arrhenius plot shown in Fig. 6 is in a good qualitative agreement with the data on nucleation on Si ( 111) [13], althoughEqs. (17) and (18) arenotvalidfor (111) surfaces of crystals with diamond lattice. There are two rea- sons for that. First, Si( 111) grows by complete bilayers [ 501 and Eq. (14) is not valid because only half of the atoms of the bilayer are bound to the atoms of the underlying crystal plane. Second, a clean Si( 111) surface is reconstructed in 7 X 7 unit cells below 83O”C, whereas a surface covered with Sb or As recovers the 1 X 1 bulk structure [ 12,131. The latter means that we have to deal with two completely different crystal surfaces on which the mechanisms and the activation energies for surface diffusion differ considerably. Thus the experimental results obtained for both surfaces are not com- parable. There is, however, experimental evidence which shows that both surfaces 7 X 7 and 1 X 1 do not differ consid-

erably with respect to surface diffusion and nucleation. Ellip- sometric measurements of Latyshev et al. [ 5 1 ] showed that during the phase transition from 7 X 7 to 1 X 1 reconstruction around 830°C an excess of atoms of about 0.2 monolayers become adsorbed on the 1 X 1 surface. Assuming these atoms are adsorbed on T4 sites the 1 X 1 surface will resemble very much the 7 X 7 with the exception of the dimers and the stacking faults which characterize the dimer-adatom-stacking fault structure of the 7 X 7 unit cell. This is probably the reason that measurements of the diffusivity on both 7 X 7 and 1 X 1 surfaces did not show any difference [ 521.

The increase of the saturation nucleus density in the pres- ence of a surfactant easily explains the experimentally observed transition from multilayer to LBL growth in Sb- mediated epitaxy of Ag( 111) [ lo]. The rate of nucleation on top of 2D islands depends on their size through the adatom concentration [ 3,6]. The bigger the island, the higher the adatom concentration on top of it will be and the higher the nucleation rate. When a large population of small islands is formed as a result of the presence of a surfactant, the nucle- ation rate on top of them remains negligible up to the begin- ning of coalescence and the critical surface coverage for nucleation in the upper monolayer is large [ 3,531. This results in a LBL growth. In the absence of a surfactant a small population of big islands is formed and nucleation on top of them begins long before the surface is completely covered by them. Consequently, two and more monolayers grow together and a multilayer growth takes place.

The main problem in the comparison of theory with exper- imental data is the evaluation of the surfactant efficiency, s, or in other words, of the energetic parameter w. In principle it could be evaluated if the enthalpy of mixing of the two species is known. However, w is, in a general case, dependent upon both composition and temperature [ 541. Moreover, it is not clear whether a value of w which is evaluated from the equilibrium phase diagram of a three-dimensional alloy and thus depends on the environment could be applied to treat a two-dimensional problem. It seems that the best way to meas- ure w is by studying the equilibrium structure of single steps. It is the surfactant efficiency which determines the kink den- sity. The slope of an Arrhenius plot of kink density should directly give the value of surfactant efficiency and a compar- ison with the experiment can be carried out without unnec- essary speculations. On the other hand the problem of kink density in the presence of a surfactant is very important in itself. The equilibrium structure of the steps determines the mechanism of growth and the transition from step-flow growth to growth by 2D nucleation.

We conclude further that in addition to the influence on the saturation island density the effect of the smfactant for suppressing island growth can be explained on the basis of the following assumptions: (i) the atoms of the surfactant occupy the kink sites on the crystal surface as they provide the strongest bonding, and (ii) the deposited atoms should displace the surfactant atoms in order to join the crystal lattice, which in turn gives rise to an additional energy barrier at

104 I. Markov /Materials Chemistq and Physics 49 (1997) 93-104

ascending steps ofa kinetic origin, a fact which is well known in crystal growth [34], and most probably reduces the Schwoebel barrier at descending steps. The descending steps are thus attractive rather than repulsive (or still repulsive but much weaker than the ascending steps) as believed previ- ously. It should be stressed that the traditional view of the potential diagram as shown in Fig. 8(a) with a considerable Schwoebel barrier leads in the absence of a surfactant to F( O,, 0,) = 0, and thus to 3D growth at low temperatures, which contradicts the experimental observations.

Acknowledgements

The financial support of the Volkswagenstiftung (Ger- many) and the Bulgarian National Fund for Scientific Research is gratefully acknowledged.

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