Adatom lifetime in film growth at solid surfaces in the framework of the Johnson–Mehl–Avrami–Kolmogorov model

Surface Science 440 (1999) L849–L856 www.elsevier.nl/locate/susc

Surface Science Letters

Adatom lifetime in film growth at solid surfaces in the framework of the Johnson–Mehl–Avrami–Kolmogorov model M. Tomellini a, *, M. Fanfoni b a Dipartimento di Scienze e Tecnologie Chimiche, Universita` di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Rome, Italy b Dipartimento di Fisica, Universita` di Roma Tor Vergata and Istituto Nazionale per la Fisica della Materia, Via della Ricerca Scientifica, 00133 Rome, Italy Received 5 April 1999; accepted for publication 15 July 1999

Abstract On the basis of the quasi-static approximation and for simultaneous nucleation the adatom lifetime, t, during film growth at solid surfaces has been computed by Monte Carlo (MC ) simulation. The quantity DN t, N and D being 0 0 respectively the cluster density and the adatom diffusion coefficient, is found to depend upon the portion of surface covered by clusters and, very weakly, on N . Moreover, a stochastic approach based on the Johnson–Mehl–Avrami– 0 Kolmogorov (JMAK ) theory has been developed to obtain the analytical expression of the MC curve. The collision factor of the mean island has been calculated and compared with those previously obtained from the uniform depletion approximation and the lattice approximation. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Adatoms; Computer simulations; Growth; Models of surface kinetics; Nucleation

1. Introduction In film formation which proceeds by nucleation and growth the latter is fundamental in determining the final morphology of the overlayer and, consequently, the technological properties of the coating [1,2]. Recently, it has been shown that, depending on the relative magnitude between deposition flux and diffusion coefficient of monomers at the cluster periphery (i.e. temperature), cluster shapes change * Corresponding author. Fax: +39-6-72594328. E-mail address: [email protected] (M. Tomellini)

from a compact disk to a fractal [3]. In these experiments cluster growth is mainly linked to the adatom surface diffusion and their incorporation into the previously formed clusters. As a matter of fact, the contribution of gas monomer impingement on the ‘lateral’ growth of the clusters may be neglected either because of the low values of the portion of surface covered by islands, S, or owing to the Ehrlich–Shwoebel barrier [4]. On the theoretical side, analytical models based on rate equations have demonstrated their potential to describe the nucleation and early stage of film growth [5–8]. In this approach a system of differential equations is solved for the surface densities of subcritical and stable clusters, namely

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N , N , …, N 1 and N=S 1 N , i1 and N being 1 2 i i>i i 1 the critical size and the surface density of adatoms, respectively. In the rate equation scheme, adatom depletion due to cluster growth, (dN /dt) , is 1 gw described through a term of the form:

A B dN

1

=−N

2 ∑

kN (1) i i 1 gw i=i +1 where gw stands for ‘growth’ and {k } are size i dependent rate coefficients. Eq. (1) can be rewritten in a quite phenomenological fashion: dt

1

A B dN

1

dt

N =− 1 t gw

(2)

which states that an atom, landing on the surface at time t=0, is captured by the growing phase, on average, at time t=t. Eq. (2) can be considered valid up to the film closure. When film growth is the only process responsible for the consumption of monomers at the surface, Eq. (2) defines the lifetime of the adatoms. Moreover, since the rate coefficients are proportional to the diffusion coefficient of monomers, D, and to the collision factors, s , the lifetime of adatoms reads [9]: i 1 −1 2 t= ∑ kN (3) = i i s: DN i=i1+1 where s: =S 1 (N /N)s is the average value of i i=i +1 i the collision factor over the cluster population. Strictly speaking this equation is only robust at the early stage of the film growth, when collisions among clusters are negligible. Nevertheless, when this is not the case Eq. (3) can still be maintained by identifying N with the total number of islands. The term island is here ascribed to any isolated collection of connected clusters, no matter how the clusters are arranged, where each cluster was produced by a nucleation event. Apparently, the density of islands is lower than the density of clusters. When two or more clusters collide they can either retain their individuality (impingement) or redistribute the matter according to some relaxation process (coalescence). The former mechanism is a pure geometrical effect. The relationship of Eq. (3) has been widely used in connection with rate equations, since it

A

B

represents a simple way to solve the system in closed form. Moreover, Eq. (3) is employed in the framework of the uniform depletion approximation ( UDA) where the t function is computed by solving a diffusion equation for the local density of adatoms, c(r,t). In detail, Fick’s second law is solved, self-consistently, in the presence of a circular isolated cluster [9,10]. This approach is expected to be reliable for low values of surface coverage when clusters are well separated and Eq. (3) applies. To describe film growth up to the film closure, it is necessary to deal with the process of impingement among clusters. If these are distributed at random their collisions can be treated on the basis of the Johnson–Mehl–Avrami–Kolmogorov (JMAK ) statistical theory [11–13] and, eventually, used in connection with rate equations to describe the overlayer growth up to the film closure [14]. However, in order to solve the kinetic scheme, the growth law of the film, N /t, is to be known as a 1 function of cluster dimension and adatom density. In Ref. [15] the system has been integrated by considering the cluster growth rate to be proportional to the free perimeter of the clusters; a valid assumption when the adatom mean free path is lower than the island sizes. The results of Ref. [13] show that (i) the nucleation process is quasisimultaneous and (ii) a steady state condition for the adatom surface density is established during film growth. A further improvement of the modelling proposed in Ref. [13] requires a more rigorous computation of the adatom lifetime, t, in the whole range of surface coverages (0
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where n =N /(1−S ) is the local density of the 1 1 adatoms and C(S) the film perimeter. By equating Eqs. (2) and (7) and specifying the C(S ) function in case of simultaneous nucleation of square clusters [16,17], the adatom lifetime is given by:

2. Results and discussion 2.1. Adatom lifetime in the framework of the uniform depletion approximation Until now the UDA has represented one of the most employed approaches for obtaining capture factors in closed form. The possibility of representing s through an analytical function of the surface i coverage is, however, restricted to narrow ranges of S. As a matter of fact, analytical expressions can be achieved for both low and high values of S. In the former case the well-known solution is obtained [10]: 2p s= i ln(l/r)

(4)

which holds for r%l, r being the radius of the cluster and l=앀Dt. In the model case of simultaneous nucleation all clusters start growing at t=0 and s =s: . Moreover, the cluster radius can easily i be related to the surface coverage through the JMAK model according to: S=1−e−pN0r2,

(5)

N being the number density of clusters. Using 0 Eqs. (3) and (5) in Eq. (4) we get: 1 =pt∞ e−4pt∞ ln 1−S

(6)

where t∞=DN t=1/s: . This computation is valid 0 for r%l or, alternatively, for t∞&(1/p)ln [1/(1−S)]. If the inequality is deemed to be fulfilled when the ratio between the two members is, at least, a factor of 10, the constraint gives S≤0.06. As far as the high coverage regime is concerned, the UDA provides an analytical expression for t. It was established that in the limiting case r&l the growth rate of a single cluster is proportional to its perimeter [10]. It is reasonable to exploit this result for the film case by considering the growth rate proportional to the film perimeter. Accordingly:

A B dN

1 dt

=− gw

C(S)Dn l

1 =−

C(S)lN 1 (1−S)t

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(7)

A

B

1 −1 t∞=DN t= 16 ln . 0 1−S

(8)

This is a new result which describes satisfactorily the MC curve in the range 0.4
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obtained by averaging over z according to:

Fig. 1. Monte Carlo simulation of the adatom lifetime, t∞=DN t, as a function of surface coverage for h =0.002 (full 0 0 symbols). The kinetics are compared with the results obtained by the uniform depletion approximation in the limit S<0.06 and S>0.6 (continuous lines). The adatom lifetime computed on the basis of the stochastic approach [Eq. (14)] for p=0.9 is also reported (open symbols).

to the ‘quasi-static approximation’, S is kept constant. (vi) The monomer is captured when its coordinates coincide with those of a point belonging to the periphery of the covering phase S. During each run the value of t is calculated averaging over 600 trials. Several runs were averaged to obtain the plot of Fig. 1. In the same figure the comparison between the adatom lifetime computed using the UDA and the MC output is also reported. As it appears, in the above-studied limiting cases, the diffusion model well reproduces the computer simulation for S>0.6, whereas for S<0.06, the accord is modest although the order of magnitude is correct. Monte Carlo simulations for different values of M were also performed and indicate that t∞ 0 depends, weakly, on M . In particular, simulations 0 at M /M=200−1 and 500−1 give rise to a relative 0 variation for t∞ of the order of a few per cent. 2.4. Theory In this section we will model the t∞(S ) function through a stochastic approach. Let P(z) be the probability that an adatom, landed on the surface at time zero, will not suffer a capture event by an island until time z, then −dP is the probability that the adatom will be captured in the time interval z to z+dz. The adatom lifetime, t, is

1

P

2

P(z)dz. (10) 0 In order to calculate t, the P(z) function is needed. This probability can be computed on the basis of Poisson’s process according to the following arguments. In the first place let us consider the atom at z=0 ( landing time). In the quasi-static approximation the probability that the atom is on the non-covered surface is apparently given by: t=

P(0)

P(0,t)=e−N0A(0,t)=e−N0l(t)2=1−S(t)

(11)

where A(0,t)=l(t)2 is the ‘capture area’ of the adatom which, for z=0, is just equal to the cluster area at the growing time t. By the same token, at time z after landing, the probability P(z,t) requires a knowledge of the area, A(z,t), swept by the ‘dressed’ random walker. By ‘dressed’ we mean that, during its wandering on the surface, the walker carries with it an area equal to that of a cluster at time t (Fig. 2). Therefore we get: P(z,t)=e−N0A(z,t).

(12)

The simplest modelling for A(z,t) can be achieved by exploiting the random walk results. At any time, z, the distances travelled by the walker are distributed according to a 2D Gaussian with standard deviation (sd) s (z)=s (z)=앀2Dz. Our idea is to x y associate with the walker a ‘compact’ square whose role is to replace the area swept by the adatom in its random walk. To this purpose, we need an ‘equivalent’ side that can be measured in terms of sd as 2p앀2Dz, p being a dimensionless parameter. Eqs. (10) and (12) lead to: 1

P

2

앀 e−N0(l+2p 2Dz)2 dz.

(13) e−N0l(t)2 0 Using the JMAK kinetics Eq. (13) can be rewritten as follows:

t(S)=

C

1 1 ln t∞(S)=DN t(S)= 0 4p2(1−S) 1−S ×

P

2 0

(1−S)(1+f)2f df=

D

W(S) 4p2(1−S)

.

(14)

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Fig. 2. Pictorial view of the adatom (walker) wandering on a surface partially covered by clusters (top). The lifetime of the adatom can be computed by solving the stochastic process illustrated in the bottom panel, which is equivalent to the original one: a dressed walker is considered to be captured by the nucleation centres.

Even though this equation, for p=0.9, yields a rather acceptable estimate of t∞(S ) (see Fig. 1), it can easily be improved by considering p a function of S. This is justified by the fact that the dressed walker leaves, during its roaming, patches of uncovered surface resulting in a smaller swept effective area. This effect is stronger the smaller the S value. In principle it is trivial to calculate, from the MC output, the p(S) function which

leads to a perfect description of the MC simulation itself. However, this is not the point, the important thing is that its determination allows one to confirm the weak dependence of t∞ upon N . For this 0 purpose let us exploit a property of the area A(z,t) which has to be considered, in fact, as a function of the cluster side, l=l(t), of the number of jumps done by the walker during the time interval [0,z], n=n(z), and of the jump length, a. Moreover,

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A(l,a,n) must be a homogeneous function of second order in the dimensional variables l and a and, therefore, the following formula holds: A(l,a,n)=a2A∞

A B l

a

,n ,

(15)

A∞ being the normalized area swept by the walker. Eq. (15) is important in that it suggests a particular functional form of p(S). As a matter of fact comparing the A(l,a,n) function employed in Eq. (13) and Eq. (15), we obtain: A∞

A BA l

a

,n ¬

l a

+p앀2n

B

2

(16)

which implies p¬p(l/a). Consequently, the dependence of p upon S is entirely determined by the dependence of l/a on S according to the expression [see Eq. (11)]: l a

=

1 앀h

0

A

B

1 1/2 ln 1−S

(17)

where h =N a2 is nearly equal to the number of 0 0 clusters per lattice site. This computation indicates that t∞ is, in principle, a function of the cluster density and of the covered surface. On the basis of this argument, the dependence of t∞ on N can easily be obtained by analysing a 0 single MC curve, t∞ , for a given h value MC 0 (h =h1 ). In particular, using Eq. (14) the p(S,h1 ) 0 0 0 function has been computed through a knowledge of W(S) and t∞ . Furthermore, by using Eq. (17) MC at h =h1 the argument of the function can be 0 0 evaluated and, with it, the p(l/a) universal function. The result is reported in Fig. 3 and indicates that p(l/a) is well described by two power laws with exponents equal to m #0.16 and m #0.1. This 1 2 behaviour of the p function confirms the very slight dependence of t∞ on h obtained by the MC 0 simulation. In particular, for h =0.002 and 0 h =0.005 the MC computations give, on average, 0 the relative variation Dt∞/t∞=13%, in agreement with the value predicted by the power law reported above by using the average value m#0.13. The consequences that such a dependence have on the scaling behaviour of cluster density on deposition

Fig. 3. Plot of the p(l/a) function obtained on the grounds of the MC simulation. The function is well described by two power laws with exponents m =0.16 and m =0.1. 1 2

flux and diffusion coefficient will be addressed in a subsequent paper. An issue to be discussed is the collision factor defined by Eq. (3). We remember that in this equation N is the number density of islands which, due to the collisions among clusters, is lower than N . In a previous paper we had pointed out that, 0 for simultaneous nucleation, N(S)=N F(S) where 0 F(S ) is a universal function that can be expressed through a series expansion, namely the collision series [19]: 2 P (18) F(S)= ∑ k k=1 k where P is the probability that a cluster belongs k to an island of k connected clusters, no matter how they are arranged and S2 P =1. As far as k=1 k the F(S ) function is concerned, we made use of the results of Ref. [19] where the collision series had been evaluated up to the fourth order term. Moreover, because of the rapid convergence of the collision series, the first four terms give account of the main features of the kinetics. The fact, often evaded, that the coalescence process cannot be neglected even at low coverages, can be evidenced by evaluating the probability that a cluster be subjected to at least an event of collision. From the definition of P it is easy to k see that: P (S)=1−P (S)=1−(1−S)4 c 1

(19)

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Fig. 4. Plot of the collision series, F(S ), and of the P (S ) kinetc ics. The latter being the probability that a cluster is subjected, at least, to a collision.

where we made use of the P kinetics computed 1 in Ref. [19]. From Eq. (19) one gets that P =0.2 c at S=0.05, P =0.4 at S=0.15, where the coalesc cence onset is attained [19], while about nine clusters out of 10 have suffered a collision event for S=0.4 (Fig. 4). The F(S ) curve of Fig. 4 is very well fitted by the function: F(S)=exp[−2S(1+aSb)]

(20)

with a=3.8 and b=1.86. Accordingly Eqs. (3) and (20) give: s: (S)=

1

exp[2S(1+aSb)]

(21) t(S)DN F(S) t∞(S) 0 where the slight dependence of t on N has been 0 disregarded. In Fig. 5 the s: (S) function has been reported for the MC simulation together with the capture factors computed by the UDA and the lattice approximation [9]. We observe that the lattice approximation and the UDA provide, respectively, an overestimation and an underestimation of the MC simulation; a finding which is consistent with the results of Ref. [9]. However, a close agreement between the MC result and the UDA computation is attained for 0.03
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Fig. 5. The capture coefficients evaluated through Eq. (21) by using the t∞ kinetics are reported as filled symbols. Open symMC bols (triangles) represent the s: values given by the UDA ( lower curve) and the lattice approximation (upper curve). The capture factors obtained by using Eq. (8) in Eq. (21) are also shown (circles).

Before concluding we would like to comment on the two assumptions on which our model rests, namely the hypothesis of simultaneous and random nucleation. As regards the hypothesis of a random distribution of nuclei, a more thorough discussion is needed. In a non-simultaneous nucleation the appearance of new nuclei in the ‘capture zone’ of an already formed cluster is precluded, thus leading to a non-random distribution of clusters on the whole surface. In order to estimate the extent of the capture zone, let us approximate the surface coverage of monomers around a growing cluster, h , by the expression (in units where the 1 lattice constant is one):

G

A B

dh 1 h (x)= 1 dx

h (x)=h: 1 1

x

0
x=0 x>x1

(22)

where x is the distance from the cluster edge, h: is the average density of adatoms and 1 x1=h: [(dh /dx)| ]−1. On the other hand, the 1 1 x=0 derivative (dh /dx)| is directly linked to the 1 x=0 cluster growth law, l(h ,h ), through the relationt z ship: dh 1 dx

K

h: (h ) 1 J dl(h ,h ) 0 t z = 1 t = x1 2 D dh x=0 t

A B

(23)

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where h =J t#S(t) and h are, respectively, the t 0 z coverages at time t and z (with z
P

x1 0

4[h: −h (x)](l+2x)dx 1 1

(24)

and 4x12 A(l,h )=2lx1+ t 3

(25)

which defines the capture area as a function of cluster size and actual coverage. To get the order of magnitude of x1 in the nucleation stage, i.e. for h #S
A = T S1

P

S1 0

V(S1) A (h)dh< T S1

P

S1

A(h)dh.

0 (27)

The results of the kinetic Monte Carlo simulations (figs. 2 and 3 in Ref. [18]) have been employed to evaluate Eq. (27) through the use of Eqs. (23) and (25). Specifically, at D/J = 0 109 [V(S1)=2×10−4, S1=0.03 (see below)], the inequality Eq. (27) gives rise to A <0.075, i.e. T the assumption of a random distribution of clusters brings an uncertainty that is lower than 10%. Similar conclusions are reached for the other two cases, D/J =105 and D/J =107. Moreover, in 0 0

respect to the hypothesis of simultaneous nucleation, the kinetics of fig. 2 in Ref. [18] are enlightening since they allow one to evaluate the surface coverage, S1, at which the nucleation rate becomes negligible. For the three studied cases (D/J =105, 107 and 109) S1 is, in fact, independent 0 of D/J and equivalent to about 0.03 ML. This 0 shows that the nucleation process is complete at the very beginning of the deposition process. Therefore, as far as high coverage regimes are concerned, the nucleation rate can be reasonably approximated by a Dirac delta function. Finally, it is worth stressing that the effect on nucleation of a ‘capture zone’ around the growing clusters could also be included in the JMAK model by making use of the approach proposed by Markov and Kashchiev [20]. Such a phenomenon should lead to a shift of the ‘coalescence onset’ towards larger coverage values.

References [1] J.M. Howe, in: Interfaces in Materials, Wiley, New York, 1997. [2] L.-C. Dufour, M. Perdereau, in: L.-C. Dufour, C. Monty, G. Petot-Ervas (Eds.), Surface and Interfaces of Ceramic Materials, Kluwer, Dordrecht, 1989, p. 419. [3] H. Roeder, E. Hahan, H. Brune, J.P. Bucher, K. Kern, Nature 366 (1993) 141. [4] H. Brune, K. Kern, in: D.A. King, D.P. Woodruff (Eds.), The Physical Chemistry of Solid Surfaces, Vol. 8, Elsevier, Amsterdam, 1997, Chapter 5. [5] G. Zinsmeister, Vacuum 16 (1966) 529. [6 ] G. Zinsmeister, Thin Solid Films 2 (1968) 497. [7] G. Zinsmeister, Thin Solid Films 4 (1969) 363. [8] G. Zinsmeister, Thin Solid Films 7 (1971) 51. [9] J.A. Venables, Philos. Mag. 27 (1973) 697. [10] V. Halpern, J. Appl. Phys. 40 (1969) 4627. [11] I. Markov, D. Kashchiev, J. Crystal Growth 16 (1972) 170. [12] D. Kashchiev, J. Crystal Growth 40 (1977) 29. [13] M. Fanfoni, M. Tomellini, Il Nuovo Cimento 20 (1998) 1171. [14] M. Fanfoni, M. Tomellini, Phys. Rev. B 54 (1996) 9828. [15] M. Volpe, M. Tomellini, M. Fanfoni, Surf. Sci. 423 (1999) 258. [16 ] S. Stoyanov, Surf. Sci. 199 (1988) 226. [17] M. Tomellini, M. Fanfoni, Surf. Sci. 349 (1996) L191. [18] G.S. Bales, D.C. Chrzan, Phys. Rev. B 50 (1994) 6057. [19] M. Fanfoni, M. Tomellini, Appl. Surf. Sci. 136 (1998) 338. [20] I. Markov, D. Kashchiev, J. Crystal Growth 13/14 (1972) 131.