PHYSICAL REVIEW E 91 , 042408 (2015)

Analytic scaling function for island-size distributions V. G. D ubrovskii1'2,3’* and N. V. S ibirev1’4 'St. Petersburg Academic University. Khlopina 8/3, 194021 St. Petersburg, Russia 2Ioffe Physical Technical Institute o f the Russian Academy o f Sciences, Politekhnicheskaya 26, 194021 St. Petersburg. Russia }ITMO University, Kronverkskiy prospekt 49, 197101 St. Petersburg, Russia JS?. Petersburg State Poly technical University, Politechnicheskaya 29, 195251 St. Petersburg, Russia (Received 24 December 2014; revised manuscript received 26 March 2015; published 29 April 2015) We obtain an explicit solution for the island-size distribution described by the rate equations for irreversible growth with the simplified capture rates of the form ctv(Q) a &P(a + s - 1) for all s > 1, where s is the size and 0 is the time-dependent coverage. The intrinsic property of this solution is its scaling form in the continuum limit. The analytic scaling function depends on the two parameters a and p and is capable of describing very dissimilar distribution shapes, both monomodal and monotonically decreasing. The obtained results suggest that the scaling features of the size distributions are closely related to the size linearity of the capture rates. A simple analytic scaling is obtained rigorously here and helps to gain a better theoretical understanding of possible origins of the scaling behavior of the island-size distributions. DOI. 10.1103/PhysRevE.91.042408

PACS number(s): 64.60.Q—

I. INTRODUCTION M odeling o f island-size distributions (ISDs) is param ount in term s both o f pure understanding o f physical properties o f irreversible system s far from equilibrium and o f progress toward tailoring the island m orphology in the precoalescence stage o f epitaxial processes [1-4], O ne o f the m ost interesting features o f the ISDs is the so-called scaling property w hich is usually postulated as follow s [1-5]: in the lim it o f high ratios o f the adatom diffusion constant D over the deposition rate F , A = D / F - * oo, the expected form o f the ISD over the num ber o f atom s in the island s (“size”) at tim e t is w ritten as n ( x .( s ) ,0 ) = ~ f ( x ) , x = ~ ,




for all but very short tim es. Here, (.?) and © are the tim edependent m ean size and coverage, respectively, and f ( x ) is a universal scaling function o f the scaled size s/(s) which should not depend on either (s) or 0 . Since the surface density o f islands N in the scaling lim it is related to (s) and 0 as AC(s) = 0 , the scaling function m ust obey the two norm alization conditions poc

/ J0




d x x f ( x ) = 1.



T his scaling ol the ISDs has been studied extensively by kinetic M onte C arlo (K M C) sim ulations and based on the rate equations (REs) for hom ogeneous grow th o f different is­ lands [2-23]. In m odeling, the island form ation process is usu­ ally assum ed as being irreversible or w ith a tim e-independent critical size. An alm ost irreversible character o f grow th leads to a pronounced asym m etry o f ISD shapes com pared to nucleation theory w ith decay o f islands, w here the typical ISDs are close to G aussian [24-27], Bartelt and Evans [6] noticed that, under the assum ption o f size-independent capture rates < j s = const in the REs for irreversible growth, the scaled ISD acquires a singularity for large sizes. Based on the continuum

’dubrovskii 1539-3755/2015/91 (4)/042408(5)

REs, V vedensky [22] and D ubrovskii and Sibirev [23] showed that this singularity pertains w h e n as o c s^ fo rO < p < l,w ith the second singularity em erging at x = 0 when fi > 0. Such nonanalytic behaviors are not com m only observed in sim ula­ tion results such as shown, e.g., in Refs. [1-4,10.11,14,15,18] and thus can hardly satisfy the scaling hypothesis. N onanalytic ISD shapes have been obtained within the m ean-field approxim ation for the capture rates w here islands are considered isolated [6,15,16,22,23,28], This approach fails in the scaling lim it o f high adatom diffusivities because neigh­ boring islands always com pete for the diffusion flux [1-4], A nalyses o f tw o-dim ensional (2D) island growth based on the Voronoi tessellation for the mean capture zones [7,8] and direct KM C sim ulations [14,19,20] reveal the com m on feature o f the capture ra te s-a linear increase o f as w ith s at large enough s. The coefficient in this linear correlation is alm ost independent o f 0 for com pact islands but changes w ith increasing 0 for point and fractal islands [14], This property is distinctly different from the m ean-field situation and shows that larger islands have larger capture zones surrounding such islands and that the strength o f a given island to capture adatom s is roughly proportional to its surface area [7,14], A ccording to the current view [14], the REs can quantitatively reproduce the scaled ISDs but only w ith the correct self-consistent determ ination o f the capture rates crs( 0 ) . However, analytic scaling functions proposed previously, e.g., by A m ar and Fam ily for the ISDs [9] or Pim pinelli and Einstein for the distribution o f capture zones [21], are based on sem iem pirical considerations rather than direct solutions to the REs. Linear and © -independent correlation as = a + s in the REs has been considered once [29] in the case o f heterogeneous nucleation. Such a size linearity o f the capture rates yields sim ple analytic ISDs in the form o f a Polya (or G am m a) distribution w hich autom atically acquires the scaling form in the continuum lim it [29], On the other hand, purely heterogeneous nucleation, for exam ple, form ation o f linear chains o f metal adatom s on (2 x 1(-reconstructed Si(100) surfaces with a high concentration o f C-type defects [30,31], applies only to som e growth scenarios and is not a general situation studied for the scaling behavior o f the ISDs.


©2015 American Physical Society

PHYSICAL REVIEW E 91,042408 (2015)


Consequently, in this work we closely examine homoge­ neous irreversible growth based on the REs assuming that the capture rates are given by —1). In view of Eq. (4) and N (s) = 0 , we have N {cr) = c®p+],


ers.( 0 ) = a (0 )(a + 5 — 1), 5 > 1,

with an arbitrary function a (0 ), which agrees with the results of Refs. [7,8,14] for large 5 as discussed above. Adding the direct impingement terms to Eq. (3) (which are proportional to F) can modify the solutions but does not much change the scaling properties in the limit of D / F —»• oo [7]. According to Refs. [28,29], analytical study of the REs given by Eqs. (3) is simplified by introducing the timedependent variable z by the definition ^ = Dn\a, z(t = 0) = 0, (5) dt which linearizes Eqs. (3). A unique exact solution in the particular case of the size-linear capture rates given by Eqs. (4) can be obtained by using the generating function

n(z - y)e~(a+V)y n(x,z) ■ a / d y---J o ' [ 1 - x ( l - e - y ) ] fl+i

0 = 0 oe+



The choice of the initial condition ©o will be discussed shortly. Using Eq. (14) in Eq. (13) at small 0 , we obtain the exponential dependence n i (z) of the form e ~ {P + ] ) z

= veHp+l)z.



cA 0 q+1

This result is central for what follows. For self-consistent determination of the ©o value, we use the integral representation of the generating function given by Eq. (7). From Eq. (6) it is clear that n (l,z) = N at n\ -» 0. Using Eq. (15) in Eq. (7) at x = 1 and integrating, we obtain N =



(P + 1 )’

i.e., the island density in our model saturates to a constant. From (cr) = Q / N and using Eqs. (14) and (16), the mean size should be

(5 )= ^ + 1— av

= -(p

+ \ )®%+2A ez


On the other hand, from the definition of the mean size and Eq. (6) it follows that -VvS-l

N(s) =


s >

c A ® p+' ’

Here, the last expression applies when © 0


A N {a)'

We consider irreversible homogeneous growth of 2D immobile islands As from mobile adatoms (“walkers”) A\ via the scheme As + A i -* As+], s ^ 1, described by the REs for the dimensionless island concentrations ns of the form


where (cr) denotes the mean value of as. Using Eq. (11) in the known reduced RE in the scaling limit A -> oo which has the form [14]


(Iyi — - = Dn\(os-]ns-\ - asns),s > 2. dt The capture rates are taken as


dn(x,z) dx

(18) X= ]

at «i —y 0. Using again Eqs. (15) and (7), Eq. (18) gives

Here, B(.y,a)

r(5)T(fl) r(5 + a ) '

(9) 042408-2

, v ~ (fl + 1)(P + 1) z W (P + 2)



PHYSICAL REVIEW E 91, 042408 (2015)

Com parison o f Eqs. (17) and (19) yields the self-consistent ©o in the form

a(a + 1 )1

@ /’ + 2


( 20 )

c(p + 2) A '

It is seen that 0 O —1, and thus our solution indeed applies for all but very short times. On the other hand, from Eqs. (15) and (20) the ratio o f the diffusion term over the direct im pingem ent term scales as A n , ~ a (p+2)/(p+1) and tends to infinity at A —> oo. This confirm s negligible influence o f the direct im pingem ent on the scaling behavior in our model. T he integral in the right-hand side o f Eq. (8) for the ISD is reduced to the incom plete Beta function w henever the m( z) dependence is exponential, as given by our Eq. (15). Using Eq. (19) in Eq. (8), after som e sim ple calculations we find the discrete ISD in the form

ns+l((s),@) (P + l ) b p+l B ( s , a - p )



a( s) p

h-b/{s)(s,a - p),

B (s,a)


FIG. 1. (Color online) Comparison of the numerical solutions to the REs with size-linear capture rates at a = 2, p = 0.5, c = 10, A = I05 and different times (relating to different 0 and ($)) (symbols) and the analytic scaling function (line) in the scaling variables.



{a + 1)(p + 1)

( 22 )

(P + 2) Here,

Iy(s,q) =


B (s,q) Jo

d t t s~ \ \ -


is the regularized incom plete Beta function in standard notations [33]. The distribution obtained is tw o-param etric and is controlled by the dim erization rate a and the pow er exponent p. T he latter defines the coverage dependence o f the capture rates and equals zero w hen as values are independent of 0 . III. CONTINUUM APPROXIMATION AND ANALYTIC SCALING FUNCTION The m ost im portant observation regarding the ISD given by Eq. (21) is its natural reduction to the scaling form at la rg e.? and (s). This feature follow s from the know n asym ptotic behavior o f the incom plete Beta function at s ^ oo, (s) -> oo, and finite b and a — p [33]:

I\-b/(*)(s,a - p ) = y

p\ .

Obviously, the obtained analytic function f ( x ) is universal, i.e., does not depend on either (s) or 0 . It is easy to prove that f i x ) satisfies both norm alization conditions given by Eqs. (2) for any a and p. The collapse o f the discrete ISDs calculated num erically from the discrete REs given by Eqs. (3) w ith size-linear capture rates defined by Eqs. (4) and (10) at a = 2, p = 0.5, c = 10, A = 105 and replotted in the scaling variables (s)2ns/@ and s / (s ) to the continuum analytic scaling function is shown in Fig. 1. It is seen that Eq. (26) provides an excellent fit to the results o f num erical integration starting already from © = 0.03 and (s) = 9. Figure 2 shows the com parison o f the KM C sim ulation results for fractal islands from Ref. [14] and our model. The fits o f the nonscaled ISDs are obtained from Eqs. (8) and (15) at a = 12, c = 3.52, p = 0.8, and A = 107 and show reasonable correlation w ith the KM C results for large enough 0 ^ 0.1. However, these fits require a high a value o f 12 w here the scaling shape given by Eq. (26) is less


Here, 1

f° °

y M = Twl


is the regularized incom plete G am m a function. U sing also B(.s,a — p) /B( s, a) = s p T(a — p ) / at large s a, we get


n(x,{s),@) = ~ f { x ) , (s)~ f i x ) = ( p + 1)bp+] ~ ~ ^ x py ( b x , a - p).


FIG. 2. (Color online) Fits of the KMC results for ISD ns for fractal islands of Ref. [14] obtained from Eqs. (8) and (15) at a = 12, c = 3.52, and p = 0.8, for the same values of A = 107 and 0 .


PHYSICAL REVIEW E 91, 042408 (2015)


Scaled size s/

FIG. 3. (Color online) Continuum scaling function at a = 1 and different p. accurate as it assumes s )§> a. We also note that the KMC results of Ref. [14] for the capture rates are better fitted by crs(0 ) — a a + a (0 )(s - 1) with a 0 independent of 0 rather than by Eq. (4) in our model, which explains the observed discrepancy in Fig. 2. Generally, the function / ( x ) starts from zero at x = 0 when p > 0, equals 1 at x = 0 when p = 0 and is infinite at x = 0 when p < 0 (Fig. 3). The /( x ) has a maximum in x at p > 0 and is monotonically decreasing at p < 0. Therefore, it can qualitatively reproduce both monomodal [10,11,14,15,18] and monotonically decreasing [32] ISD shapes. This is regulated by the sign of p, i.e., whether the capture rates increase or decrease with the coverage. It should be noted, however, that the dependence (7,(0) = c&p cannot be valid at very small 0 ->■ 0, which may explain why our model does not describe the monomodal ISDs which do not vanish atx = 0 [1-4]. Figure 4 shows how the scaling function changes with fixed p = 0.5 and varying a. It is seen that the ISDs acquire a more pronounced monomodal and symmetric quasi-Gaussian shape as a increases. However, very large a values are not allowed in our model in view of the assumption s a made earlier to justify Eq. (26). Indeed, the formal limit of infinitely large a would yield a size-independent capture rate for which the

analytic scaling is known to be broken [6,23]. Finally we note that, whenever a - p = 1, our solution simplifies to f { x ) = [a°/ r(a )]x fl“ lexp(-ax), which is the continuum form of the Polya ISD [29]. Therefore, the exact scaling solution obtained previously in heterogeneous growth [29] is just a particular case of our ISD. This is further reduced to the exponential scaling function exp(—x) at a = 1. In conclusion, we have obtained an explicit solution to the REs for irreversible growth with size-linear capture rates whose coverage dependence has a power-law form. Our solution is given by a discrete distribution expressed through the regularized incomplete Beta function. The intrinsic property of the discrete ISD is its analytic scaling form in the continuum limit which depends on the two control parameters a and p. The corresponding scaling function contains a regularized incomplete Gamma function and the additional power-law factor x p. Quite surprisingly, our solution does not seem to follow from the continuum first-order RE in scaling variables [7], This requires a separate study. ACKNOWLEDGMENTS

This work was partially supported by Grants No. 13-0212405 and No. 13-02-00662 of the Russian Foundation for Basic Research, and the FP7 Project NanoEmbrace (Grant Agreement No. 316751). APPENDIX: SOLUTIONS FOR GENERATING FUNCTION AND DISCRETE ISD

When the as are given by Eq. (4) of the main text, we linearize the REs in Eq. (3) by using the ansatz defined by Eq. (5) which yields dn

— - = (a + s —2)ns-\ — (a + s — 1)ns, s ^ 2. (Al) dz Using this in Eq. (6), after simple calculations we obtain the closed differential equation in partial derivatives for the generating function, 9n(x,z) dn(x,z) -------------(x — 1)x---------dz dx = (x — l)(a + l)n(x,z) + an\(z)-


This should be solved with the initial condition n(x,z — 0) = 0. The equivalent system of ordinary differential equations for Eq. (A2) is written as dz 1

dx (x — l)x

dn (x — l)(a + l)n + an\


From the first equation, we get the first integral of the form = z — ln[x/(l —x)]. Integrating the second equation with zero initial condition, we obtain Eq. (7) of the main text. The Taylor series expansion of this n(x,z) is performed by applying the known formula i/t]

1 (1 - T )a+1

FIG. 4. (Color online) Continuum scaling function at p = 1/2 and different a.


r ( q + s + l ) yI

r(« + i)s!


for Y = x(l —e~y) in Eq. (7) of the main text. The result yields the discrete ISD given by Eq. (8) of the main text.



PHYSICAL REVIEW E 91, 042408 (2015)

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Analytic scaling function for island-size distributions.

We obtain an explicit solution for the island-size distribution described by the rate equations for irreversible growth with the simplified capture ra...
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