Nucleation and step-flow growth in surfactant mediated homoepitaxy with exchange/de-exchange kinetics

Surface Science 429 (1999) 102–116 www.elsevier.nl/locate/susc

Nucleation and step-flow growth in surfactant mediatedhomoepitaxy with exchange/de-exchange kinetics

Ivan Markov *Departamento de Fisica de la Materia Condensada, Universidad Autonoma de Madrid, Cantoblanco, 28049 Madrid, Spain

Received 27 November 1998; accepted for publication 10 February 1999

Abstract

The nucleation rate and the transition to step-flow growth are considered in detail in the two limiting cases ofsurfactant mediated homoepitaxy: (i) reversible exchange of crystal and surfactant atoms (an exchange–de-exchangeequilibrium); (ii) irreversible exchange (fast exchange of the atoms upon striking the surface and absence ofde-exchange). It is shown that in both cases the critical terrace width above which nucleation takes place displays anArrhenius behavior as has been experimentally established by Iwanari and Takayanagi [J. Cryst. Growth 119 (1992)229]. The square of the critical terrace width for step-flow growth scales with the atom arrival rate, the scalingexponents being the same, which determines the scaling behavior of the island densities in the submonolayer regimeof growth. It is shown that in the case of Sn mediated growth of Si(111) the surfactant drives the nucleation processcloser to the phase equilibrium, the critical nucleus is one order of magnitude larger than in the clean case, and thesurfactant stimulates the step-flow growth. © 1999 Elsevier Science B.V. All rights reserved.

Keywords: Exchange–de-exchange kinetics; Nucleation; Step-flow growth; Surfactant epitaxy

1. Introduction homoepitaxial growth the surfactants lead to well-pronounced layer-by-layer growth instead of themultilayer growth that is characteristic of manyDesign of novel nanoscale structures and fabri-growing surfaces [4,5]. Camarero et al. found thatcation of modern electronic devices requires thethe formation of twins during heteroepitaxialdeposition of smooth and defect-free epitaxialgrowth of Co/Cu(111) superlattices is stronglyfilms. However, owing to thermodynamic reasonssuppressed by precovering the clean Cu(111) sur-the formation of three-dimensional (3D) islandsface with a monolayer of Pb [6 ]. Suppression offrequently takes place which gives rise to imperfecttwin formation has also been observed during thelayers that are useless for microelectronics. It hasindium mediated homoepitaxial growth ofbeen found that small amounts of surface activeCu(111) [7]. Twin formation is due primarily tospecies (surfactants) can strongly influence thethe coalescence of doubly oriented two-dimen-mode of growth by suppressing 3D islanding, thussional islands and should be suppressed in casegiving rise to smoother films [1–3]. In the case ofthe film grows by attachment of atoms to stepswhich exist on crystal surfaces tilted by some small

* Corresponding author. Permanent address: Institute ofangle with respect to a low index crystal planePhysical Chemistry, Bulgarian Academy of Sciences, 1113 Sofia,(step-flow growth). de Miguel et al. establishedBulgaria. Fax: +359-2-9712688.

E-mail address: [email protected] (Ivan Markov) that Pb effectively induces step-flow growth during

0039-6028/99/$ – see front matter © 1999 Elsevier Science B.V. All rights reserved.PII: S0039-6028 ( 99 ) 00342-8

103I. Markov / Surface Science 429 (1999) 102–116

deposition of Co on Cu(111), whereas in the exchange and de-exchange processes. The case ofreversible exchange [scenario (i)] is considered inabsence of Pb step-flow growth could not be

observed in this system [8]. Detailed investigations detail in Section 3. The nucleation rate and thetransition from 2D nucleation to step-flow growthof the system Si/Si(111) in the absence and pres-

ence of Sn as a surfactant showed that the critical are treated in Sections 3.1 and 3.2, respectively,both cases of kinetic and diffusion regime ofstep distance for nucleation on terraces changes

with substrate temperature following an Arrhenius growth being considered separately. The case ofirreversible exchange [scenario (ii)] is consideredtype behavior [A exp(−E/kT )]. The critical terrace

width for step-flow growth is much larger in the in Section 4. The theory is compared with availableexperimental data in Section 5. The results arepresence than in the absence of Sn, the latter thus

stimulating the step-flow [9]. discussed in Section 6.Considering the role of the surfactants in thin

film growth it was assumed that they either influ-ence the adatom diffusivity or affect the attach- 2. Exchange and de-exchange processesment–detachment kinetics at steps [4,10–13]. In arecent paper, Kandel and Kaxiras considered a Scanning tunneling microscopy (STM) study ofmodel in which exchange and de-exchange of Si(111) growth in the presence of monolayers ofcrystal and surfactant atoms on terraces has been Ga, In, As, and Sb as surfactants showed thatexplicitly taken into account [14]. Kinetic Monte depleted zones always exist around the pre-existingCarlo simulation showed that for surfactants steps, which is an indication that the surfactantswhich passivate the pre-existing steps a high den- do not passivate them completely [15,16 ]. It hassity of islands is observed without the necessity of recently been shown that in the presence of aa reduced diffusivity. In the present paper we surfactant the work of formation of kinks at theexplore in more detail the same model considering step edges is equal to the energetic parameterseparately the two scenarios that are generallybelieved to take place. In scenario (i) the exchange v=1

2(Ecc+Ess)−Ecs (1)

and de-exchange barriers on terraces are small (orwhich determines the enthalpy of mixing, Ecc, Ess,the temperature is sufficiently high) so that theand Ecs being the works needed to separate twoatoms embedded in the surfactant layer have timecrystal (C) atoms, two surfactant (S) atoms, andto de-exchange before the deposition of a completecrystal and surfactant atoms, respectively [17].monolayer. As a result, an exchange–de-exchangeThis parameter is smaller than the work of kinkequilibrium is established in the very beginning offormation (or the energy of a dangling bond) indeposition. The concentration of adatoms on topthe clean case [18]of the surfactant layer remains significant, they

diffuse fast on the surfactant passivated surface to v0=1

2Ecc (2)

the pre-existing steps, thus promoting step-flowgrowth. The formation, growth and decay of nuclei and the concentration of kinks in the presence of

a surfactant is much larger at the same temper-occur through exchange and de-exchange pro-cesses. In scenario (ii) the impinging atoms incor- ature [19].

The surfactant induced roughening of the stepsporate rapidly under or between the surfactantatoms and have no time for de-exchange. All the is of crucial importance for the processses on the

surface and leads to the following consequences.processes of diffusion, nucleation and incorpora-tion to islands and steps take place under the First, one should expect a strong one- dimensional

mixing of film and surfactant atoms even whensurfactant layer. The diffusion barrier is higherthan on a clean surface and the adatoms give rise both materials are immiscible in bulk or at sur-

faces. As both kinds of atoms have different size,to a high density of 2D nuclei, thus inhibiting thestep-flow growth. The paper is organized as fol- we could expect also elastic strains along the steps.

The latter, together with the greater concentrationlows. In Section 2 we consider the possible

104 I. Markov / Surface Science 429 (1999) 102–116

of kinks, should make the exchange processesmuch easier. As a result the step will become morereactive with respect to the incoming atoms. Onthe other hand, rougher steps are better sinks foradatoms and should propagate with higher veloc-ity. All the above is valid not only in surfactantmediated homoepitaxy but in any case of heteroe-pitaxial growth. The substrate ‘does not know’whether the incoming foreign atoms are surfactantatoms or not. As a result, in every case of heteroe-pitaxial growth a ‘one-dimensional alloying’ alongthe steps and roughening of the latter should beobserved. The latter is in excellent agreement withrecent STM observations of steps on Au(100)

Fig. 1. Schematic view of the exchange (a�b�c) andbefore and after deposition of 0.05 ML of Fe atde-exchange (c�b�a) processes. The crystal and the surfactant

room temperature [20], as well as with the observa- atoms are denoted by filled and empty circles, respectively.tions of Voigtlander and Zinner in the Sb mediatedgrowth of Si/Si(111) [15]. Therefore, in the analy-sis below we accept that a single barrier, Estex, exists process (Fig. 1c). The same bond counting argu-

ments show that the barrier for de-exchange,for the exchange of a crystal with a surfactantatom at steps, irrespective of whether the atom E (0)dex , in the absence of neighboring atoms includes

in addition the stronger C–C bonds, and henceapproaches the step from the upper or the lowerterrace. In other words, we do not include a E (0)dex>E (0)ex . The above arguments are, however,

oversimplified. The difference in size between thedetailed consideration of the attachment–detach-ment kinetics owing to the existence of Ehrlich– surfactant and crystal atoms gives rise to a layer

which represents a more or less disordered mixtureSchwoebel (ES) barrier to interlayer transport[21,22]. We thus assume that the surfactant masks of surfactant and crystal atoms, and to the appear-

ance of elastic stresses. Both factors could reducethe asymmetry of the attachment–detachmentkinetics at step edges and account only for the substantially the values of E (0)dex and E (0)ex , but hardly

could reverse the above inequality.exchange and de-exchange processes.We consider next in more detail the exchange Depending on the interrelation of the exchange

and de-exchange barriers, on the one hand, andand the de-exchange processes on terraces assum-ing that a complete surfactant monolayer covers the barrier for surface diffusion, on the other, a

fraction of the incoming atoms will stay for somethe crystal surface. In analogy with the site-exchange mechanism of surface diffusion [23–25], time on top of the surfactant layer before an

exchange event. We can thus assume that nucleiwe suppose that a similar mechanism holds for theexchange and de-exchange processes with the only are formed by direct attachment of atoms through

exchange from the adatom population on top ofdifference being that two kinds of atoms areinvolved (see Fig. 1). In fact such a mechanism the surfactant layer. The exchange process on a

terrace next to a cluster consisting of i atomswas first suggested by Zhang and Lagally [11].Assuming Ecc>Esc>Ess, simple bond counting embedded in the surfactant layer requires overcom-

ing a lower barrier, E (i)ex , compared with E (0)ex owingarguments show that the activation barrier forexchange on a terrace in the absence of neighboring to the strains and disorder induced by the presence

of the cluster. If the cluster is large enough weatoms, E (0)ex , includes only the weaker S–C and S–S bonds (Fig. 1a). The final state of the exchange could accept that E (i)ex=Estex because of the similar

coordination. Hence, one could expect thatprocess (crystal atom between the surfactant atomsplus a surfactant atom on top of the surfactant E (0)ex >E (1)ex >E (2)ex …>E stex . The corresponding barri-

ers for de-exchange, E (i)dex , include the works neededlayer) is in fact the initial state for the de-exchange


to break lateral C–C bonds and thus are higherthan E (0)dex .

We introduce further two time constants whichcharacterize the exchange and the de-exchangeprocesses. First, this is the residence time of theatoms on top of the surfactant layer beforeexchange

t(0)ex=F

nexexp(bE (0)ex ) (3)

where nex is the attempt frequency for the exchangeprocess, and b=1/kT, k and T being Boltzmann’s Fig. 2. Schematic representation of both models of (a) reversible

and (b) irreversible exchange. The crystal and the surfactantconstant and the absolute temperature, respec-atoms are denoted by filled and empty circles, respectively. Notetively. Second, this is the mean residence time,that in the case of reversible exchange a considerable fractiont(0)dex , before de-exchange of an atom embedded inof the incoming atoms stay on top of the surfactant layer,

the surfactant layer whereas in the second case only displaced surfactant atoms arepresent on top of the layer consisting of mixed surfactant andcrystal atoms.

t(0)dex=F

ndexexp(bE (0)dex) (4)

is the surfactant atoms that will remain buriedwith ndex being the attempt frequency. Both time under the arriving crystal atoms.constants are normalized to the time t1=1/F to We consider now in more detail the case ofdeposit 1 ML. Obviously, if a particular time con- reversible exchange. For the concentration, n1, ofstant is greater than unity the corresponding pro- atoms embedded in the surfactant layer we cancess will not occur. write the rate equation

As t(0)dex&t(0)ex three possibilities exist. The firstone is t(0)dex%1. This case describes the reversible dn

1(h)

dh=Rex−

n1

t(0)dex−

n1

t(1)ex+

n2

t(1)dex(5)

exchange as the atoms embedded in the surfactantlayer have sufficient time to go back on top of the

where h=Ft is the deposition time in units ofsurfactant layer through a de-exchange process.number of monolayers (ML), F is the depositionAs will be shown below, an exchange–de-exchangefrequency, and n2 is the concentration of dimers.equilibrium is established. As a result a consider-The concentrations are given in units of the den-able fraction of the incoming atoms remain on topsity, N0, of the adsorption sites, and all timeof the surfactant layer and diffuse fast to the pre-constants are normalized to the time t1=1/F toexisting steps (see Fig. 2a). The formation, growthdeposit 1 ML.and decay of nuclei takes place through exchange

In Eq. (5)and de-exchange processes. This is in fact thescenario (i) mentioned above. The second possi-

Rex=nst(0)ex

(6)bility t(0)dex>1 and t(0)ex%1 gives the case of irrevers-ible exchange. The incoming atoms rapidlyexchange places with surfactant atoms and remain is the flux of atoms which exchange places with

surfactant atoms to embed themselves into theburied under the surfactant layer (see Fig. 2b). Allprocesses of diffusion, nucleation and incorpora- surfactant layer, ns being the concentration of

atoms on top of the surfactant layer. The secondtion into islands and steps occur under (orbetween) the surfactant layer. This is the scenario term in the right-hand side of Eq. (5) is the flux

of de-exchange. The third term in the right-hand(ii) mentioned above. The third possibility,t(0)dex>1 and t(0)ex>1, is excluded as it means that it side of Eq. (5) gives the rate of disappearance of


monomers due to formation of dimers through an compare the time constants t(0)dex , t(1)ex and t(1)dex . Thedissociation of dimers requires the breaking of anexchange process. The mean time of dimer forma-

tion is given by additional strong C–C bond and clearly t(1)dex ismuch larger than all other time constants. Thuswe accept that the dimer dissociation affects weaklyt(1)ex=

F

a1Dns

exp(bE (1)ex )=F

a1nns

[b(E (1)ex+Esd)]the monomer concentration. We compare nextt(0)dex and t(1)ex . The ratio t(0)dex/t(1)ex could be in prin-

(7) ciple greater or smaller than unity. However, bear-ing in mind the solution [Eq. (20)] for ns, givenwhere D and Esd are the hopping frequency andbelow we find t(0)dex/t(1)ex%1 with typical values forthe energy barrier for surface diffusion of theF, l, T, n, and with Edex=1.6 eV, Eex=0.8 eV,atoms on top of the surfactant layer, n is theand Estex=0.4 eV [14] whereas the opposite is ful-attempt frequency, and a1 is the number of possiblefilled at temperatures close to room temperature,exchange events. The product a1Dns is in fact thewhere t(0)dex&1. Physically this means that theattempt frequency of dimer formation.attempt frequency a1Dns for the exchange processFinally, the fourth term in the right-hand sideis much smaller than ndex owing to the reducedof Eq. (5) is the rate of recovering monomers byadatom concentration on top of the surfactantdissociation of dimers. This process consists oflayer and overcompensates exp[b(E (0)dex−E (1)ex )].two consecutive stages. First, one of the atoms

We conclude that the process of de-exchangegoes on top of the surfactant layer through amainly controls the concentration of adatomsde-exchange process without breaking the bondembedded into the surfactant layer. Eq. (5) thenwith the other atom. During the second stage thereduces totop atom breaks away, overcoming in addition the

barrier Esd for surface diffusion. The mean lifetimeof a dimer is then given by dn

1(h)

dh=

nst(0)ex

−n1

t(0)dex. (9)

t(1)dex=F

b2n(1)dex

exp[b(E2+E (1)dex+Esd)] (8) The solution of Eq. (9) with initial condition

n1(0)=0where E2 is the work needed to break the dimer’sbond, and b2 is the number of possible de-exchange

n1(h)=ns

t(0)dext(0)ex

(1−e−h/t(0)dex)events. Note that only one atom remains underthe surfactant layer as the other goes on top of itby a de-exchange process. In further considerations shows that a dynamic exchange–de-exchange equi-

librium is reached after deposition of approxi-we accept that all frequencies are equal and omitthe indices. mately 4–5t(0)dex ML. With the value of 1.6 eV

calculated by Kandel and Kaxiras we find thatThe above rate equation differs from that writ-ten by Venables et al.[26 ]. Owing to the mechanism t(0)dex#0.002 ML. The latter means that the

exchange–de-exchange equilibrium is reached inof direct attachment of atoms to and detachmentof atoms from the clusters embedded in the surfac- the very beginning of the deposition process – at

approximately 0.01 ML. Then for n1 one obtainstant layer accepted in the model, only the concen-trations of monomers and dimers enter Eq. (5).The concentrations of clusters consisting of more

n1=ns

t(0)dext(0)ex

=ns eb(E (0)dex−E (0)ex ) . (10)than two atoms do not enter Eq. (5) as they donot affect the monomer concentration.

The solution of Eq. (5) gives the connection In the case of irreversible exchange(t(0)dex>1 and t(0)ex%1) we have to use the rate equa-between the concentrations of adatoms on top of

the surfactant layer and those embedded in it. In tions written by Venables, but for the initial stagesof growth, when the concentration of clusters isorder to find a simple expression for the latter we


still negligible, we can write

n1=ns

h

t(0)ex. (11)

3. Reversible exchange kinetics

3.1. Nucleation rate

As mentioned above, the process of nucleationtakes place in fact in a 2D system which consistsof a more or less disordered monolayer of mixedC and S atoms with migrating C atoms anddisplaced S atoms on top of it (Fig. 2a). In otherwords, the nucleation could be considered astaking place in a two-dimensional solution of Catoms in a solvent consisting of S atoms. Treatingthe nucleation in alloys in this particular case is avery complicated problem. In order to circumvent

Fig. 3. The calculation of the Gibbs free energy for nucleusit without a loss of essential physics, we firstformation. (a) The initial surface covered with surfactant atomscalculate the Gibbs free energy for nucleus forma- denoted by empty circles; (b) the crystal surface after reversible

tion by using the following imaginable process and isothermal evaporation of the surfactant layer; (c) a cluster[17]. consisting of i atoms is formed; (d) the surfactant atoms are

condensed back. The crystal and the surfactant atoms areThe initial state is a crystal surface covered bydenoted by filled and empty circles, respectively.a complete monolayer of the surfactant (see

Fig. 3). We evaporate first all surfactant atomsof the 2D island [17]. The edge energy of thisreversibly and isothermally. Then on the cleancluster, Ws, increases the work for nucleus forma-surface we condense a cluster consisting of i atoms.tion by increasing the nucleus size, and in turnFinally, we condense back all surfactant atoms, oflowers the nucleation rate. It could be argued thatwhich i atoms form a cluster on the top surface ofthe surfactant atoms which go on top of thethe crystal cluster. Following further the well-surfactant layer as a result of the de-exchangeknown procedure [27–29] for the nucleation rateprocesses disperse and do not form clusters on topone obtains [17]of the nuclei. Although such a possibility exists,

Js=cN0ni1

exp{b[(1−s)Ei−Ws+siDW )]}. (12) the formation of surfactant clusters on top of the

nuclei is energetically more favorable as the S–CIn the above expression c is the flux of atomsbonds are stronger than the S–S bonds, and injoining the critical nucleus by an exchange process,addition an energy of formation of S–S bondsn1 is the density of single adatoms embedded inis gained.the surfactant layer, E

iis the work required to

On the other hand, the edge energy of thedissociate the nucleus into i single adatoms,nucleus itself is decreased as a result of the satura-DW=Ek−Ed is the work needed to evaporate a

crystal atom from a kink site onto the terrace, Ek tion of the dangling bonds along its periphery withand Ed being the work needed to detach an atom surfactant atoms. This decrease per bond is givenfrom a kink (half-crystal ) position [30,31], and by a parameter, s, called a surfactant efficiencythe desorption energy from the clean surface, given by [17]respectively.

As the surfactant ‘floats’ on top of the crystal s=1−v

v0

. (13)surface a cluster of surfactant atoms forms on top


This parameter varies from zero at complete Comparing the last two terms in the right-handside of the above expression with x=0 and assum-inefficiency of the surfactant to unity at completeing a typical value for l#300a give a value ofefficiency.E stex#6–7kT, above which the adatom concen-The flux of atoms c is given bytration has practically a constant value

c=ainns exp[−b(Esd+E (i)ex )] (14)

where ai

is the number of the possible exchange ns=nst#Fl

2DebE stex (21)

events by which an atom can join the criticalnucleus to produce a stable cluster. all over the terrace and the step propagation occurs

Combining Eqs. (12), (14), and (10) gives for in a kinetic regime of growth. In the opposite case,the nucleation rate when E stex≤2kT then

Js=ainN

0ni+1s exp{b[(1−s)E

i−Ws ns(x)#

F

2D CAl

2B2−x2D (22)+siDW+i(E (0)dex−E (0)ex )−Esd−E (i)ex ]}. (15)

and the steps advance in a diffusion regime.The above equation recovers its familiar formAs seen in Eq. (20) nst=ne when the exchange

Js=ainN

0ni+1s exp[b(E

i−Esd)] barrier at the step E stex=0. The latter is a very

small quantity at typical growth temperatures.in the absence of a surfactant (s=0).This does not necessarily mean that in the absenceIn order to find ns under condition of exchange–of a surfactant the crystal will always grow in ade-exchange equilibrium we solve the appropriatediffusion regime. Other barriers (e.g. the Ehrlich–diffusion problem for the distribution of adatomsSchwoebel barrier) exist at steps in the clean caseon top of the surfactant layer on a terrace withand determine the attachment–detachment kineticswidth l. In the case of reversible exchange it readsso that the crystals usually grow in a kinetic regimeat low temperatures [28].D

d2ns(x)

dx2+F=0 (16)

Substituting ns from either Eq. (21) or (22) inEq. (15) we obtain the corresponding expressions

subject to the boundary conditions [32] for the rate of nucleation in kinetic and diffusionregimes of growth.ns(x=±l/2)=nst (17)

3.2. Critical terrace width for step-flow growthCdns(x)

dx Dx=±l/2

=bstep [nst−ne] (18)

We calculate further the critical terrace width,where lc, above which 2D nucleation starts at a given

substrate temperature and atom arrival rate. Thebstep=[exp(bE stex)−1]−1 (19)critical number of nuclei which can cause the

is the kinetic coefficient of the step, nst is the appearance of intensity oscillations is given byadatom concentration in the near vicinity of thesteps, and ne is the equilibrium adatom concen- Ncr=P

0

1/F Js� dt (23)

tration. Note that the exchange and de-exchangefluxes do not enter Eq. (16) because at equilibrium

wherethey cancel each other.

The solution of the diffusion problem reads Js�=

1

l P−l/2l/2Js(x) dx (24)

ns(x)=ne+Fl

2D(ebEstex−1)+

F

2D CAl

2B2−x2D. is the nucleation rate averaged over the terrace[33]. The critical number of nuclei depends essen-tially on the instrumental resolution and is in(20)


practice an unknown parameter. However, we can It is immediately seen that the exponentialaccept Ncr=1/l2c [34], as a good approximation multiplying l2

0in Eq. (29) can be smaller or greater

for typical tilt angles of the order 0.1–1°. than unity depending on the interplay of theenergies involved in Eq. (31). If E>0 the surfac-

3.2.1. Kinetic regime of growth tant promotes the step-flow growth and vice versa.Substituting Eq. (21) into Eqs. (15), (24) and Both negative terms in the right-hand side of Eq.

(23) results in an Arrhenius type of equation as (31) are the largest ones. However, if DEsd over-found experimentally by Iwanari and Takayanagi compensated the difference E (0)dex−E (0)ex we could[9] expect that the second negative term siDW will be

in turn overcompensated. Then E>0, l2c>l20

andstep-flow growth will take place.l2c=An

FBx expA− x

ibEB (25)

where3.2.2. Diffusion regime of growth

E=(1−s)Ei−Ws+siDW+i(E (0)dex−E (0)ex ) In the other extreme of diffusion regime of

growth by using Eq. (22) one obtains+iEsd−E (i)ex+(i+1)E stex (26)

andl2c=Gx/iAn

FBx expA− x

ibEB (32)

x=2i

i+3(27)

where nowis a scaling exponent which usually appears incases of kinetic regime of growth [35,36 ]. In deriv- E=(1−s)E

i−Ws+siDW+i(E (0)dex−E (0)ex )

ing Eq. (25) all constants of the order of unity+iEsd−E (i)exhave been omitted.

In order to clarify the effect of the surfactantG=

2i(2i+3)!

a(i+1)!(i+1)!we can write the diffusion barrier as

Esd=E0sd−DEsd (28)and

where DEsd is the decrement of the diffusion barrier(relative to the barrier E0sd on the clean crystalsurface) owing to the passivation of crystal surface x=

i

i+2(33)

by the surfactant. Then Eq. (25) can be written inthe form

is the scaling exponent which appears in cases ofdiffusion limited growth [26,37].l2c=l2

0expAx

ibEB (29)

By using Eq. (28) we can write Eq. (32) in thesame form [Eq. (29)] where

where

l20=Gx/iAn

FBx expC− x

ib(E

i+iE0sd)D (34)l2

0=An

FBx expA− x

ib[E

i+iE0sd+(i+1)Est ]B

(30) is the critical terrace width for 2D nucleation inthe absence of a surfactant [36 ], and E is givenis the critical terrace width for 2D nucleation inagain by Eq. (31). Thus, in both cases of kineticthe absence of a surfactant [36 ], andand diffusion regime of growth we could expect a

E=sEi+Ws−siDW+E (i)ex−i [(E (0)dex−E (0)ex )−DEsd ]. stimulation of the step-flow provided

E (0)dex−E (0)ex <DEsd .(31)


4. Irreversible exchange kinetics between the surfactant atoms. The solution reads

As discussed above, according to scenario (ii)all incoming atoms rapidly exchange places with ns(x)=t(0)ex−(t(0)ex−nst)

cosh Ax

lsB

cosh A l

2lsB

(37)surfactant atoms and remain in between the latter(Fig. 2b). This can be supposed intuitively butnevertheless we will show it by solving the corre-

wheresponding diffusion problem.In the case of irreversible exchange (t(0)dex>1)

the de-exchange flux enters the diffusion equationand the latter reads nst=ne+(t(0)ex−ne)

1

ls(ebEst−1) tanh A l

2lsB

1+1

ls(ebEst−1) tanh A l

2lsB

.

Dd2ns(x)

dx2+F−F

nst(0)ex

=0. (35)

(38)

As seen when Est>12

(E (0)ex−Esd), ns(x) has aThis is in fact the equation of Burton et al.constant value t(0)ex%1 all over the terrace (kinetic[19], in which the desorption flux is replaced byregime of growth). (Note that both the concen-the exchange flux. The substitution Y=ns/t(0)ex−1trations and the time constants are measured inturns Eq. (35) into its familiar formone and the same units of a fraction of a mono-layer.) In the reverse case (nst�nc)

l2sd2Y

dx2=Y

ns(x)#t(0)ex C1−

cosh Ax

lsB

cosh A l

2lsBD (39)where

ls=AD

Ft(0)ex B1/2=exp[1

2b(E(0)ex−Esd)] (36)

and the steps propagate in a diffusion regimeof growth.

Assuming ls%l, ns=t(0)ex over the greater partis the mean free path of an adatom on top of theof the terrace with the exception of narrow stripessurfactant layer before exchange. It is difficult toalong the steps with a width of the order of lsevaluate this quantity because of the lack of reli-[19]. From Eq. (11) it follows that in the case ofable data of the energies involved.irreversible exchange n1=0, i.e. all incoming atomsHowever, taking for example E (0)ex =0.8 eV [14],exchange places with S atoms upon striking theand Esd=0.5–0.6 eV [38,39] we obtain ls#5–15asurface and remain buried under the surfactant(or 15–50 A) in the temperature interval 600–layer. The concentration of adatoms, ns, on top900 K. These values are much smaller than theof the surfactant layer is negligible. Thus in thetypical terrace width of 500–1000 A. We conclude case of fast exchange and strongly inhibited

that the impinging atoms are buried under the de-exchange [irreversible exchange, scenario (ii)]surfactant layer after making several jumps. all the processes of diffusion, incorporation into

Eq. (35) has to be solved subject to the same the pre-existing steps and formation and growthboundary conditions [Eqs. (17) and (18)]. Note of nuclei take place under the surfactant layer.that the exchange barrier, E stex , at the steps should The crystal atoms have again to exchange placesbe replaced by another barrier Est which arises with surfactant atoms in order to join pre-existingfrom the necessity of breaking stronger S–C bonds steps or 2D islands, but laterally in the same plane.when the crystal atom displaces the surfactant All processes of diffusion, nucleation and incor-

poration to islands and steps occur in a mixedatom at the step edge diffusing to the latter in


layer of crystal and surfactant atoms with the behavior l2c=A exp(−E/kT ). The prefactorA=a2(n/F )x scales with the reciprocal arrival rate,displaced surfactant atoms on top. Therefore wethe scaling exponents being the same which deter-have to solve the same diffusion problem [Eq.mine the island densities in the submonolayer(16)] for the atoms which are buried under (orregime of growth. Thus, measuring the dependenceintermixed with) the surfactant atoms. In this caseof the critical terrace width on the atom arrivalD will be the diffusion coefficient of the atoms inrate one can deduce the mechanism of growth andthe mixed layer (presumably much smaller thanthe size of the critical nucleus. The measurementson a clean surface), and ns should be replaced byof the temperature dependence of the critical ter-the concentration n1 of adatoms in it. As therace width will give erroneous results as both thenucleation process occurs in this layer the sameenergy term and the prefactor depend on theconcentration enters both Eqs. (12) and (14). Innucleus critical size and in turn on the temperature.the case of irreversible exchange the latter readsSuch measurements could be carried out only in a

c=ainn1

exp(−bEsd). (40) narrow temperature interval in which the criticalsize remains the same.As a result we obtain for the nucleation rate

As has been noted in Ref. [35] the scalingthe same expression as Eq. (15)exponent x=i/(i+2) varies with i from 1/3 to 1,

Js=ainN

0ni+11

exp{b[(1−s)Ei−Ws+siDW−Esd ]} whereas x=2i/(i+3) which is characteristic for

kinetically controlled growth has values larger than(41)unity already at i>2. The value of the ratio n/F is

with the only difference that all exchange and of the order 1015–1016 in typical MBE experiments,de-exchange barriers vanish, and ns is replaced by whereas the area a2 occupied by an atom is of then1. order of 10−15 cm2. Then the prefactor A varies

We write now the diffusion barrier as approximately from 10−10 to 1 cm2 in diffusionEsd=E0sd+DEsd where DEsd is the increment of controlled growth, [x=i/(i+2)], whereas in thethe barrier E0sd due to the inhibited diffusion in other extreme [x=2i/(i+3)] it varies from 10−7 tobetween the surfactant atoms. 1015 cm2 at i�2.

In the kinetic regime of growth the equation Iwanari and Takayanagi measured the temper-for l2c has the same form as Eq. (25) with the only ature dependence of the critical terrace widthdifference that the term i(E (0)dex−E (0)ex ) is omitted. in the case of growth of Si(111) precovered withWe again can write the resulting equation for l2c Sn, and under clean conditions [9]. From therelative to l2

0with x=2i/(i+3) where now corresponding Arrhenius plots they found that

the prefactor A on clean Si(111) surface,E=sEi+Ws−siDW−iDEsd . (42)

7.35×10−4 cm2, is 17 orders of magnitude smallerIn the case of diffusion regime of growth the than that, 9.82×1013 cm2, on Sn precovered sur-

expression for l2c again coincides with Eq. (29) in face. It becomes immediately obvious that the Snwhich the energy is given by Eq. (42). mediated growth of Si takes place in the kinetic

As seen in Eq. (42) the two largest energy terms limit with x=1.8. It should be pointed out thatare negative and improbably overcompensated by the same value 1.76 has been found in a quitethe first two terms. Hence, we can always expect different system by Hwang et al., who measuredthat the surfactant will stimulate the 2D nucleation the saturation nucleus density versus depositionrather than the step-flow growth in the case of rate in a STM study of the 2D nucleation of Geirreversible exchange. on Pb precovered surface of Si(111) [40]. In the

case of growth on clean surface the value of theprefactor lies in the interval in which both limiting

5. Interpretation of experimental data cases overlap. With the value of x=0.75 we con-clude that the nucleation process takes place either

As follows from Eqs. (25) and (32) the square in a diffusion regime with i=6 or in the kineticlimit with i=2. It should be noted that the aboveof the critical terrace width displays an Arrhenius


value of the scaling exponent agrees excellently expected as E6 is the energy to break six first-neighbor bonds. Owing to the very narrow temper-with that, 0.85, found in the same system by

Voigtlander et al. from STM study of the island ature interval the energy term in the Sn mediatedgrowth agrees with the theoretically expected one.density vs. deposition rate [16 ]. What follows is

that in Sn mediated growth of Si(111) the size of With the slope 2.6 eV measured by the authors weobtain for the sum of energies in Eq. (26) a valuethe critical nucleus (i#25–30) is much larger than

that in growth of clean surfaces. of the order of 35–45 eV. In order to dissociate aSi cluster consisting of 25–30 atoms we have toMoreover, Iwanari and Takayanagi observed

that the critical step distance in Sn mediated break more than 30 Si–Si bonds, but in this caseEi

is multiplied by 1−s which is expected to begrowth is much longer than in the clean case atone and the same temperature. Thus, at 573 K the much smaller than unity (see Section 6).

We could conclude that Sn mediated growth ofvalues of lc are 3300 and 1200 A, whereas at 623 Kthey are 2800 and 2200 A for Si/Sn/Si and Si/Si, Si/Si(111) most probably takes place in a kinetic

regime under conditions of exchange–de-exchangerespectively. From Eq. (29) we find for E/(i+3)the values of 0.05 and 0.14 eV at 573 and 623 K, equilibrium. A definite conclusion concerning the

growth in the absence of a surfactant cannot berespectively. We can assign the increase of thevalue of E/(i+3) with temperature to the decrease drawn. However, it seems more likely that the

growth of clean Si(111) surface occurs in a kineticin nucleus size. From the above we conclude thatthe sum of energies in Eq. (31) is positive and regime with i=2 bearing in mind the comparatively

low growth temperatures (<700 K) and that Si isthe nucleation takes place under conditions ofreversible exchange (exchange–de-exchange a very strongly bonded material. A two-atom

nucleus can be thought to consist of an atom inequilibrium).As has been mentioned above the Arrhenius the lower half layer and an atom belonging to the

upper half layer, both connected by a first-neighborplot of l2c vs. 1/T could be highly misleading. Thereason is that the critical island size and in turn bond. The third atom which stabilizes the nucleus

belongs again to the lower half layer and is con-the scaling exponents vary with temperature anda single straight line should not be observed in a nected by two first-neighbor bonds with the upper

atom of the nucleus and with the underlyingwide temperature interval. A better approach is tostudy the dependence of lc on the deposition rate. bilayer. This configuration is very stable as a

barrier of about 3.5–4.5 eV should be overcomeIn the experiments of Iwanari and Takayanagi thetemperature interval in which the dividing line in order to break simultaneously two Si–Si bonds.between 2D nucleation and step-flow lies is about100 K in the clean case, and about 25 K on Snprecovered surface. One could expect that the 6. Discussioncritical nucleus size remains the same in suchnarrow intervals. However, this might be the Irrespective of the discrepancy of the experimen-reason for the discrepancy of the measured energy tally estimated energy term in the deposition on aterms in the data of Iwanari and Takayanagi with clean surface with the theoretically expected valuethe theoretically expected ones. Assuming a kinetic one could conclude that the critical size of thecontrol of the nucleation with x=2i/(i+3) and nucleus in Sn mediated growth of Si(111) is ani=2 on clean Si(111) surface results in order of magnitude greater than that in depositionE2+2E0sd+3Est=1.9 eV (the energy term mea- on clean surfaces. The small nucleus size in the

sured by the authors from the slope of the clean case is confirmed by independent measure-Arrhenius plot is 0.76 eV ). A much higher value ments [16 ]. What follows is that the surfactantis expected as only the work, E2, needed to break drives the nucleation process closer to the phasea Si–Si bond is of the order of 1.7–2.3 eV [41,42]. equilibrium.Assuming further a diffusion limited nucleation The critical nucleus size is a thermodynamicwith x=i/(i+2) and i=6 we obtain E

6+ quantity [43]. It is determined as the size at which

the Gibbs free energy change connected with a6E0sd=6.08 eV which is again much lower than


formation of a cluster of the new phase has a as [19]maximum, or in other words, at which the negative

kc=koc−TS (43)volume term, −iDm (Dm being the difference inchemical potentials of the infinitely large new and where koc=Ecc/2a (by definition) [28,29] is theambient phases) overcompensates the positive specific energy of the perfectly straight step, a iswork needed to create a new surface (or edge) interatomic spacing, anddividing both phases. One of the reasons for largercritical nuclei in surfactant mediated growth is the S=

k

aln A1+g

1−gB (44)formation of clusters on top of the nuclei consistingof displaced S atoms [17]. Another reason could

is the configurational entropy of the step, wherebe the reduced supersaturation as only a fractiong=exp(−v/kT ). In the absence of a surfactant vof atoms which are embedded into the surfactanthas to be replaced by the energy of the danglinglayer (or, which is the same, under the latter)bonds vo and if the bond strength of the materialunder conditions of reversible exchange gives riseis sufficiently large (or the temperature is low) theto a steady state size distribution of clusters. Westeps will be straight and the entropy term couldcan explore this problem by making use of thebe omitted. Then kc=koc . In the surfactant medi-classical (capillary) theory of nucleation. It hasated nucleation the entropy term should be clearlybeen shown that although not quantitatively appli-accounted for.cable to small nuclei, the classical theory behaves

Assuming for simplicity a circular shape of thequalitatively correctly when the nucleus size isnucleus the Gibbs free energy change of formationdecreased. The chemical potential of the smallof a 2D cluster consisting of i C atoms covered bycluster becomes a discrete rather than a continuousa 2D cluster consisting of i displaced S atoms infunction of the nucleus size i, but the tendencythe capillary limit reads

remains the same [29,44,45]. We analyze belowthis problem in more detail in order to illustrate DGs(i )=−iDms+2a(pi )1/2[(1−s)kc+ks ] (45)the essential physics. In no case should the figures

where Dms=kT ln(n1/ne) is the supersaturation inwhich are obtained be taken as accurate andthe surfactant layer, and kc and ks are the edgereliable.energies of the crystalline and surfactant clusters,As mentioned at the beginning of Section 2, therespectively, the former being given by Eq. (43).edges of the steps are more rough in the presenceThe multiplier 1−s<1 accounts for the decreaseof a surfactant at the same temperature.in edge energy of the nucleus owing to the satura-Concerning the nucleation the latter leads to twotion of the dangling bonds along its periphery withvery important consequences. First, the shape ofsurfactant atoms. The condition dDGs(i)/di=0the nuclei will be more or less circular in theleads to the Thomson–Gibbs equation which givespresence of a surfactant, whereas it will be wellthe critical size of the 2D nucleus as a function ofpolygonized under the clean conditions. Thethe supersaturation. In the case of a surfactantdifference in shape leads to a correction of themediated formation of a critical nucleus it reads

order of unity when writing the periphery of thenucleus and can be neglected. What is more impor-

is=pa2 [(1−s)kc+ks ]2

Dm2s. (46)tant is that owing to the increased density of kinks

in the presence of a surfactant along the nucleusedges the specific energy of the edge per unit length The familiar form of the Thomson–Gibbs equa-will be smaller compared with the clean case due tion for the 2D nucleus in the absence of ato the increased configurational entropy. Following surfactant [19]the ideal gas approximation (non-interactingkinks) of Burton et al. we can write the specific ic=

pa2ko2cDm2c

(47)edge energy of a step (the overhangs are excluded)


is recovered with s=0, ks=0, and g%1. Here difference of the exchange and de-exchange barri-ers and thus stimulates the step-flow growth. ItDmc=kT ln(ns/nc) is the supersaturation in the

absence of a surfactant. Note that ns should be follows that the supersaturation in the surfactantlayer is reduced owing to the de-exchange process.given by Eq. (20) in which the value of the

diffusion coefficient is taken for the clean surface It is sufficient for the latter to be five times smallerin order to increase 10 times the number of atomsand Estex is replaced by another step energy barrier

Est. in the critical nucleus. We conclude further thatSn is a very good surfactant from a thermodynamicDividing both equations givespoint of view as it saturates effectively the danglingbonds along the periphery of the Si nucleus. OnDms

Dmc=Aic

isB1/2C(1−s)A1−

TS

kocB+ ks

kocD. (48)

the other hand, the edge energy of the clusterconsisting of displaced Sn atoms on top of the

An approximate evaluation of the ratio of thenucleus contributes significantly to the increase of

edge energies could be made by acceptingthe nucleus size.

ks/koc#(ss/sc)(vs/vc)1/3 where ss, sc, vs and vc areIn the case of irreversible exchange the supersat-

the corresponding specific surface energies anduration is expected to be the same as in the clean

molecular volumes. The surfactant efficiency cancase as practically all atoms take place in the

be evaluated by using the approximationnucleation process. Then, as follows from Eq. (48)

Ecs#(EssEcc)1/2 [46 ]. One obtains an expressionthe critical nucleus size should be practically thesame or even smaller in surfactant mediated epi-

s#2AEssEccB1/2− Ess

Ecctaxy. Thus, it is the critical nucleus size whichwould allow us to distinguish with some certaintybetween both cases of reversible and irreversiblewhich usually overestimates the value of s, and

where Ess/Ecc#(ss/sc) (vs/vc)2/3 [17]. exchange.Another effect which could be expected if theWe will illustrate the above considerations with

the same example of growth of Si on Sn precovered exchange–de-exchange kinetics govern the processof nucleation is the transient (time dependent)surface of Si(111). With s(Sn)=680 erg/cm2 and

s(Si)=1240 erg/cm2 [47], one obtains ks/kc#0.6 character of the latter. The steady state nucleationrate is reached after some induction period whichand s#0.96. The entropy TS is always smaller

than the enthalpy koc of the step under the roughen- is inversely proportional to the flux c of atoms[Eqs. (14) and (40)] joining the critical nucleusing temperature [28,29], so that the first term

(1−s)(1−TS/koc ) in the square brackets in Eq. [28,29,50–53]. As a result the dependence of thenucleus density on the deposition time, Ns(t),(48) is smaller than 0.04 and can be neglected in

comparison with the ratio ks/koc#0.6. Assuming should not begin from the onset of deposition butwill be shifted in time. Significant induction timesic/is#0.1 gives Dms/Dmc#0.2. From the relative

difference (Dms−Dmc)/Dmc=−0.8 we find {by are usually observed in crystallization in condensedphases in which the transport of atoms or mole-making use of Eqs. (10), (21) and (28), and

assuming one and the same equilibrium adatom cules is strongly inhibited. This is usually the caseof crystallization of glass-forming melts whereconcentration, ne, in the absence and presence of

a surfactant [48]} (E (0)dex−E (0)ex )−DEsd#−0.8Dmc . induction periods as long as hours are measured[54]. In the case of vapor deposition the inductionThe supersaturation Dmc can be evaluated from

Eq. (47). Then we obtain (E (0)dex−E (0)ex )− period is of the order of microseconds and is thusundetectable [28,29,52,55,56 ]. In surfactant medi-DEsd#−1.0 eV. This value is not unreasonable

bearing in mind the crude approximations made ated growth we could expect detectable inductionperiods as the nucleation process practically takesabove. The surface diffusion barrier on a clean

Si(111)7×7 surface calculated ab initio is about place in a condensed 2D phase. As follows fromEq. (14) the diffusion barrier is low but the2 eV [49], so that DEsd could be well above 1 eV.

The latter means that DEsd overcompensates the exchange barrier is high, whereas in Eq. (40) the


exchange barrier has vanished but the diffusion passivation of the crystal surface by the surfactant.The critical terrace width for step-flow growthbarrier is much higher than in the clean case.

Hence, experimentally detectable induction periods shows an Arrhenius behavior and the values ofthe prefactors could allow us to distinguishshould be expected in both cases of reversible and

irreversible exchange. Induction times of the order between kinetic and diffusion regimes of growth.This quantity scales with the atom arrival rate, theof seconds and minutes depending on the atom

arrival rate have indeed been measured recently scaling exponents being the same, which determinethe scaling behavior of the island density in theby Hwang et al. in deposition of Ge on Pb

precovered Si(111) surface [40]. They could be submonolayer regime of growth. Thus measuringthe critical terrace width as a function of the atomexplained in terms of the classical nucleation

theory if the critical nucleus is sufficiently large. arrival rate we could determine the critical nucleussize. It is also shown that in surfactant mediatedThe analysis of the data could give additional

information concerning the effect of the surfactant epitaxy we could expect a well-pronounced tran-sient character of the nucleation rate and in turnon the kinetics of nucleation.

It should be emphasized that the two limiting of the nucleus density as a function of time.cases of reversible and irreversible exchange cannotbe assigned to different systems. As t(0)dex increasessteeply with decreasing temperature, (E (0)dex>1 eV ) Acknowledgements[14], it could happen that varying the temperatureboth cases of reversible and irreversible exchange The financial support of the Direccion Generalcan take place in one and the same system. de Ensenanza Superior (Spain) is gratefullyObviously, films of strong bonded materials are acknowledged. The author is greatly indebted tomore probably grown at typical growth temper- the anonymous referees for the highly professionalatures under conditions of irreversible exchange, analyses in their reports which helped immenselyand vice versa. However, the possibility of trans- to improve the paper.ition from reversible to irreversible exchange withdecreasing temperature should not be overlooked.

In summary we have shown that the surfactantReferences

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Documents

Nucleation and step-flow growth in surfactant mediated homoepitaxy with exchange/de-exchange kinetics