PHYSICAL REVIEW B VOLUME 52, NUMBER 23 15 DECEMBER 1995-I

Kinetics of interlayer transport prior to nucleation

S. Harris*Department ofPhysics, Imperia! College, London SW72BZ, United Kingdom

(Received 9 June 1995)

We determine the probability that an adatom remains on the surface of a nucleated island in the inter-val prior to the deposition of another adatom. Our result is valid during that stage of island growth inwhich the interisland free-adatom density is in a quasisteady state. Specific (numerical) results arepresented for homoepitaxy of Ag(111) at 300 K. Experimental confirmation of our results, which pro-vide a basis for understanding the mediating effects of surfactants on growth, appears to be possible us-

ing field ion microscopy.

A clear understanding of interlayer transport duringepitaxial growth is essential before the mediating effectsof added surfactants can be fully assessed. ' In this paperwe provide an exact solution for the probability P (t) for afree adatom on the surface of a circular island to remainon the island prior to the arrival of another adatom. Thisinformation could also be directly measured using fieldion microscopy. The result we obtain depends explicitlyon the added barrier to hopping off the island, a quantitythat we expect to be much higher for metals than semi-conductors (and more accurately estimated); therefore wepresent some explicit results only for the latter case.However, the results we obtain are general, subject to re-striction imposed by the underlying assumptions of themodel we use which is discussed below.

An earlier discussion of this problem made use of di-mensional analysis to approximate P (t). If we temporari-ly assume that the island, assumed to be circular, doesnot grow appreciably in the time interval of interest, thenthis probability can be determined exactly by solving thetime-dependent diffusion equation with appropriateboundary/initial conditions. Note that in the time inter-val of interest no deposition occurs on the island top.Therefore, if p(r, t) is the adatom density (per site) and Rthe island radius at t =0, we have

p=hV p, 0(r (R, 0(t td,p(r, O)=po, p'= —sp at r =R (2)

The initial condition describes a just-arrived adatom att =0; the deposition is random with all second layer sitesequally probable (Fig. l). In the above equations alllengths are in lattice units; h is the diffusion coeKcienth =vexp —(EDlkT), s=vexp(E, ED)lkT= exp(—bs—/kT), where b,s is the difference between the step edge andsurface diffusion barriers, i.e., the additional barrier forhopping down at the step, and td is the time at whichdeposition of another particle occurs. The solution to thesystem Eqs. (I) and (2) is

sJo(aR)=ctJ, (aR ) . (4)

The probability that the adatom remains on the island attime t (td is then

P(t)=(2/R )I dr rp(r, t)0

= (4s /R ) g exp( a„h t—)[a„(s +a„)]

-E /kT

FXXZXXXXXXXXX/i

-(tK +~/kT

where the final form of Eq. (5) follows from the standardrelationships between J& and Jo.

The dependence of P(t) on b,s is complicated, since a„are given by Eq. (4). However, it is clear that P(t) de-pends on both s and h and not just the former. This sup-ports the conclusion based on dimensional analysis, ' pro-vided that the surface barrier to diffusion on top of the is-land is equal to that on the substrate. If this is not thecase, the substrate barrier can also enter the solutionthrough R, and we now investigate this possibility.

The growth of the island, and R, will occur in two dis-tinct stages. In the erst stage the free-adatom density onthe substrate, which is initially uniform and close to thecritical value for nucleation, will rapidly reorganizethrough fast (compared to deposition) diffusion into aquasisteady state. Island growth then is fed by the ada-toms close to the island perimeter attaching, lowering theadatom density in this region. After modest growth to a

p(r, t) = (2spo/R ) +exp( a„ht )Jo(ra„)—R

X [s +a„] '[Jo(Ra„)] ', (3)

where Jo are Bessel functions, and the a„are the roots ofthe equation

FIG. 1. Plane view showing (top) a freshly deposited adatomon the nucleated first-layer island of radius R; the diffusion bar-rier is ED. (Bottom) the same adatom prior to hopping down;the added barrier to hopping down is hs =ED —E, .

0163-1829/95/52(23)/16793(3)/$06. 00 52 16 793 1995 The American Physical Society

16 794 S. HARRIS

radius we (conservatively) estimate to be R =-L/10,where 2L is the average island separation, the quasi-steady-state distribution has formed on the substrate, andgrowth continues. Nucleation on the island will general-ly occur some time during this second stage. Theanalysis that follows is limited to second-stage growth forwhich we show that in the interval t & t& the island radiusis effectively constant, validating our previous results. Atthis time we are unable to extend this result to includethe first growth stage as well.

We consider that each island has a catchment area ofradius L across which there is no Aux of adatoms. Thenthe free-adatom density in the catchment area R ~ r ~ L,0 ~ 0 ~ 2m is given by the solution of

1h'V p+ —=0,7

10-4 10p'=p at r =R, p'=0 at r =L, (7)

t/taep

so that

R (t) =L —[L —R (0)]exp —(t/r) . (10)

When tz /r « 1 the assumption of negligible islandgrowth in the interval is thus valid, and the analysis lead-ing to Eq. (5) and that result are also valid. Sincetz=rhrR it follows from Eq. (10) that R(t)=R (0) dur-ing second-stage growth for t & t&.

To illustrate the quantitative implications of Eq. (5), inFig. 2, we show the decay of P(t) for three differentvalues of R using data for Ag(1 1 1) (Ref. 7) at 300 K withdeposition at v=60 s (one particle per site per minute) to-gether with our calculated value of s for these condi-tions. As the island grows and more sites are availablefor deposition (-R ), tz decreases and the average dis-tance to the step increases (-R) so that P(t) decaysslower in the interval 0&t &tz For R =14. 0 (latticeunits) the result found using the full sum and only thefirst term are essentially equal, and only the latter wasused for R =35 and 70. We note that while it is temptingto try to infer the onset of nucleation on the island based

where h ' is the surface diffusion barrier on the substrate,I /r is the deposition rate, and the boundary condition atthe island perimeter, R, implies that all the adatomsreaching the island attach. Since the island density de-pends on h', L will also depend on this quantity, but weshow now that R is independent of h' in this growthstage.

The solution of the system Eqs. (6) and (7) is

p=(L /2h'Rr)+(R /2h'r)( —,'R —1)

+(L /2h'r)lnv/R v /4h'r, —

where R =R (t), i.e., the quasi-steady-state distribution isgiven by the steady-state general solution with time-dependent (through R) coefficients determined by theboundary conditions. For t & t&, we then have

, 8p(irR )=(2vrR)h' =(2nR)[(L /R ) —R]/2r,dt r R

FIG. 2. I' (t) as given by Eq. (5) for the case of Ag(111) at 300K, ~=60 s. We estimate I. =600 lattice units, and calculatedhs =180 meV (Ref. 5). For R =140 the full sum was used; forR =35 and 70 only the first term was used.

on P(tz) remaining above some threshold, we believethat such a prediction also requires a knowledge of the"accumulated" probability over several deposition inter-vals and possibly the consideration of fluctuations in thedeposition rate, neither of which are addressed here.

The results found above, and the conclusions based onthem, depend on the validity of the quasi-steady-state ap-proximation, i.e., the neglect of p from the right-handside of Eq. (6). The requirement that this remains valid isthat the order of magnitude of p as determined from Eqs.(8) and (10) is much smaller than one of the terms on theleft-hand side of Eq. (6), 1/r. If we take R (0)=aL, wefind that when the inequality

(2a ) '(L +a ) «h'ris satisfied this requirement is satisfied. The case illustrat-ed in Fig. 2, with L ))1 permitting e«1 provides astringent test. With a=10, an order of magnitudesmaller than our previous conservative estimate, the left-hand side of the inequality is 0(10 ), while h' r0(10" ),and the inequality is satisfied.

In concluding, we want to emphasize that the resultobtained for P(t), Eq. (5), supports the conclusion statedin Ref. 1 in showing an explicit dependence on both s andh. Enhancement of interlayer transport, and the attain-ment of a smoother growth front, will result from anyprocess that decreases P(t), e.g. , decreasing ED or in-creasing temperature.

I thank Professor D. D. Vvedensky and Dr. P.Smilauer for their gracious hospitality during my stay atImperial College, and Dr. Smilauer for valuable discus-sions concerning aspects of this work. I also thank theauthors of Ref. 7 for sharing their results prior to publi-cation. Support for this work was provided by NATOCollaborative Research Grant No. CRG- 931508.

KINETICS OF INTERLAYER TRANSPORT PRIOR TO NUCLEATION 16 795

*Permanent address: College of Engineering and Applied Sci-ences, SUNY, Stony Brook, NY 11794.

Z. Zhang and M. Lagally, Phys. Rev. Lett. 72, 693 (1994).2S. Wang and G. Ehrlich, Phys. Rev. Lett. 70, 41 (1993).3H. Carslaw and J. Jaeger, Conduction of Heat in Solids (Qxford

University Press, Oxford, UK, 1959).4R. Ghez, A Primer of Bi/fusion Problems (Wiley, New York,

1988).5P. Smilauer and S. Harris, Phys. Rev. B 51, 14798 (1995).sB. Lewis and J. Anderson, Nucleation and Growth of Thin

Films (Academic, New York, 1975), Sec. 3.IX.D.J. Meyer, J. Vrijmoeth, H. van der Vegt, E. Vlieg, and R.

Behun, Phys. Rev. B 51, 14 790 (1995).A. Newman, Trans. Am. Inst. Chem. Eng. 27, 203 (1931).