Journal of Electron Spectroscopy and Related Phenomena 76 (1995) 283-288

Avrami's kinetic approach for describing Volmer Weber growth mode at solid surfaces studied via PES and AES

M. Fanfoni and M. Tomellini*

Dipartimento di Fisica Universita' di Roma Tor Vergata Via della Ricerca Scientifica 00133 Roma (Italy) * Dipartimento di Scienze e Tecnologie Chimiche Universita' di Roma Tor Vergata Via della Ricerca Scientifica 00133 Roma (Italy)

A model is presented for describing the time dependence of the photoelectron and/or Auger signals during the overlayer formation in the case of Volmer Weber growth mode, i.e. 3D island formation. The impingement among clusters has been taken into account in the framework of Avrami's statistical approach. A first system has been considered in which nucleation occurs at a given number of preexisting sites randomly distributed throughout the whole surface. The results obtained by numerical computations indicate that particular conditions can be indeed realized for which the PES signal is chiefly related to the kinetics of the surface fraction that is covered by islands. A more involved system has been also modeled where nucleation does not occur at preexisting sites but through the formation of stable dimers. Under this circumstance, Avrami's t reatment of island impingement can be still retained although now a system of integral differential equations has to be solved to get the kinetics. Such a modelling should be suitable for describing the metallic film growth studied by PES.

1.INTRODUCTION

Although Avrami introduced his model [11 for describing the kinetics of first order phase t r a n s i t i o n for a t h r ee d imens iona l environment, subsequently his approach has been succe s fu l l y app l i ed even for transformation occurring at solid surfaces [2]. As a mat ter of fact the theory, originally written for 3D, can be reformulated, in the same way, whatever the dimension of the space is, since the problem is essentially geometric.

The model provides the kinetics with which a new phase fills up the nD volume of the old phase, when a certain number of special points (germs), from which the nD spheres of the new phase start growing, are assigned. In 2D and in the case of random germ distribution throughout the surface, the solution reads:

S ( t ) = 1 - e -s~ ' 'x ~ t~ (1)

where S(t) and Slex(t) are the actual fraction and the so called "extended area" of the

transformed 2D phase, respectively. The concept of extended area plays a central role in Avrami's model and allows one to deal with the island impingement. It is worth underline that when two islands impinge the cluster growth is assume to cease at the common interface, whereas the growth law of the non-impinged grain portion is considered to be unaffected by the impingement event [1]. In the following this assumption will be retained in both sections 2.1 and 2.2.

Nowadays the electron spectroscopies are widely used for studying the cluster formation at surfaces [3]. Unfortunately, the measured signal, because of the finite value of the electron escape depth, does not give direct information on the kinetics of the film formation. In order to "extract" the kinetics from the PES or AES signals, it is necessary to take into account the attenuation effect on the emitted electrons.

The attenuation effect is well described by Lambert's law [4] and the kinetics of growth is usually treated by using phenomenological expressions in terms of the surface coverage [5] which, strictly speaking, is not a real kinetics.

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2.THEORY

In this paper we model the photoelectron and Auger signals for the Volmer Weber growth mode where the kinetics was faced on the basis of a microscopic growth mechanism whereas the a t tenuat ion effect was t reated in the usual way.

where dNa=(1-S)dNp.

2.1 N u c l e a t i o n a t s p e c i a l s u r f a c e s i t e s In this section we deal with a nucleation

process which fulfills Avrami's assumptions: i) stable nuclei are formed by pre-existing germs which are randomly dis t r ibuted throughout the whole surface (for example steps, kinks etc); ii) the activity of the reactive species at the island surface is considered to be constant in time. The la t te r assumpt ion is normally fulfilled in the oxide growth on paren t metal where the oxygen surface coverage in the metall ic portion of the surface is indeed one [6]. In the f ramework of Avrami's theory the PES or AES normal ized signals, for the oxidized portion of the surface, is given by the following convolution:

t dN I( t) = - J--~a s(t,x)[1-e-hCt'x)/Z ]dx

0 (2)

= S ( t ) - F z (t)

where dNa/dx is the actual nucleat ion rate [1], s(t,x) is the subs t ra te surface covered, at the time t, by a cluster which started growing at the t ime x; h(t,x) is its average height and

is the a t tenuat ion length. S(t) and Fk(t) are defined by simply expanding the integrand term. The expression for s(t,x) is [2]:

1 t dSle x ( z - x ) s( t ,x)= l _ S ( x ) f [ 1 - S ( z ) ] dz dz

X

where Slex(Z-X) is the extended surface of the cluster tha t s ta r ted growing at t ime x. By subst i tut ing it into eqn.(1) after some algebra one gets:

F;~ (t) =

t d N = [ 1 - S ( t ) ] ] dX-~xP e

h ( t , x )

Z S l e x ( t - x) (3)

t d S z dN + f dZ-~z f dX-~xP e-h(t'x)/ ~S lex(Z-X)

0 o

I(t)

s(t) I I I I I r I I W I I W I

1.0

0.8

0.6

0.4

0.2

0.0 0

1.0

0.8

0.6

0.4

0.2

0.0

• A ' ' 1 . . . . I . . . . I •

50 100 150 200 Time/s

0 50 100 150 200

284

0.2 - F( t )

0.2

0.1

0.05

Time/s

Fig. 1 In panel A the overlayer signal vs t ime is reported for n=2, Ab=10-4A'2s -1, v=10,~s -1, k=10A and [~=0.01s -1. In panel B the surface covered by is land, S(t), and the F),( t) functions are shown.

Regarding the S(t) contribution, it is given by eqn.1 in which the following convolution is required:

t dN S l e x ( t ) = ~ - ~ x P S l e x ( t - x ) d x (4)

0 In the simple case of a l inear growth for

the average cluster height, h(t,x)=v(t-x), eqn.3 assumes a form which can be simply solved numerically. In addition, according to Avrami, the extended microscopic area of the grain, Slex(t,x), and the nucleation rate are given respectively by:

slex(t,x)=b(t_x)n (5)

dN p

dt - - - Ae-fit (6)

where A, b and n are constants. A typical I(t) signal is shown in fig.lA. In panel B the surface fraction, S(t), and the F~.(t) curves are also reported. This simulation indicates tha t the PES or the AES signals can be mainly related to the fraction of surface covered by islands. In part icular, the Fx(t) te rm affects the I(t) funct ion in the early stage of the o v e r l a y e r g r o w t h (S<<1), t h r o u g h the occurrence of a small '%ump". However, such a fea ture is only obtained for a par t icular choice of the input parameters. We would like to stress tha t similar feature were observed in the init ial s tage of Ni(100) oxidation, s tudied by XPS [7]. Work is in progress to identify the conditions tha t enable the first flex in the I(t) curve to be enhanced.

2.2. N u c l e a t i o n b y s t a b l e d i m e r s In this section we consider the case in

which nucleation occurs through the formation of s table dimers , produced by the surface reactions be tween two ad-monomers as well as b e t w e e n one ad-monomer and a gas- monomer. Nucleat ion is a s sumed to occur randomly in the untransformed portion of the surface where no special sites for nucleation are present . Fur the rmore the impingement among grains j u s t affect the 2D growth of those pa r t s of the gra ins which r e su l t

2 8 5

ove r l apped . The i m p i n g e m e n t b e t w e e n is lands can be still tackled as in ref.1, by making use of the extended surface. In this case, however, part icular care must be taken to f ind the exp res s ion for the ac tua l nucleation rate. In fact, now, the concepts of

I.(X)

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0 .40

0 .20

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. . . . i . . . . i . . . . i . . . . i . . . . I . . . .

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/ /

J / /

l, I . . . . I , , I , I . . . . I . . . . I , ,,l

0,5 I.{} 1.5 2.0 2.5 3.1)

e I M L max

Fig.2 I(0ma x) and S(0ma x) functions obtained by numer ica l computa t ion of the sys tem r e p o r t e d in eqn .7 . T he s i m u l a t i o n p a r a m e t e r s are: X=2A, a=2A, rm=l.& , T=300K and Q=0.69eV.

germ capture by the growing phase and of nuclei feeding by ad-monomers, coincide. This is because an ad -monomer can be considered a germ for the format ion of a dimer, namely a nuc leus . A more involved expression is found for the actual nucleation rate [8] when compared to the one obtained by Avrami [1] for a nucleation occurring at a pre-ex is t ing d i s t r ibu t ion of germs. When

286

nucleation proceeds through the formation of stable surface dimers, a system of integral differential equations has to be solved to obtain the kinetics. We denote by N and 01 the surface coverages of stable clusters and ad-monomers; by 0o and S the fraction of surface sites available for adsorption and the surface fraction occupated by clusters, respectively. Considering clusters having a circular base (Sex=nrex 2) the system reads[8]:

" d N ~: I = J°01 + (1 -S----~ 01

d 01 _ Jo (0o - 01) - I~_101 - 2 I~1 dt (1 - S )

+ 2 z D 01 t[dN(x) r,,~,t.~, ]dx

a '~ ~ ~ [ l : S ( x )

0 o = 1 - S - O I

d S = [ 1 - S ( t ) ] 2 z ×

t × ~ dN(x) r,.~ (t, ~) dr,.~ ~t,

drex (t. x) _ Jose dt a2ps

D 01 ( t ) + - - a 3 p [ 1 - S ( t ) ] r , , ~ . , x ,

(7)

where Jo , ~1 and r,_ 1 are the rate constants for gas monomer adsorption at a bare site, for dimer formation by ad-monomer diffusion and for ad-monomer desorption, respectively. In addition: D=a2K1 is the surface diffusion coefficient, a is the lattice space, rex is the extended radius of the cluster, s (Sc) is the

sticking coefficient at the substrate (cluster) surface and p is the density of monomers in the cluster.

In the rate equation for 91 the integral term is related to the stable cluster growth by the ad-monomer capture at the cluster edges. The last equation of the system is the microscopic growth law of the extended radius of the cluster, rex(t,x), where t is the running time and x is the time at which the stable dimer started growing. In particular, the island size is allowed to increase because of the capture of both the ad-monomers at the cluster edge and the gas-monomer at the island surface. However, the detai led derivation of each rate equation is reported in a more extended and comprehensive work

8 ~ - which is still in progress. The kinetic problem is therefore solved once the numerical solution of the system reported in eqn.7, in the N(t), 01(t), 0o(t), S(t) and rex(t,x) variables, is found.

The next step is the evaluation of the PES and AES signals. For this purpose the convolution product between the nucleation kinetics and Lambert 's a t tenuat ion term has to be evaluated. To simplify the complexity of the mathematical computation an approximation can be done in which the average cluster height is used. In the case of a cubic packing of the monomers into the cluster the average height, h(t), is given by the following mass balance:

8r 3 Jo t h(t) = "' (8)

a 2 [ S ( t ) + O ( t ) ]

where r m is the monomer radius. The PES and AES normalized signals of the electrons emitted from the substrate, is:

I ( t )={1-[S( t )+O1( t )] (1-e-h( t ) /~ )t (9)

where ~ is the at tenuation length of the electrons. It is worth noticing that the kinetics of the thin film formation enters in the eqns.8, 9 through the S and the 01 terms.

The system (7) was numerically solved, using Euler's method, in the limit of total

287

condensat ion of the ad-monomers (~_1=0). The S(t) and the 01( t ) o u t p u t s we re successively used in eqn.9 to eva lua te the PES signal. The rate constant J0 is given by J0=s0max/td, where td is the deposition time for the maximum coverage 0max. The typical

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2.0 3.0 4.0 5.0 6.0 7.0 8.0

Time/s

Fig.3 The kinet ics of nucleat ion, N(t), is shown in panel A (right scale, full symbol)) jointly to the t ime evolution of the surface cove rage of a d - m o n o m e r s . S i m u l a t i o n parameters are the same as in fig.2. In panel B the PES/AES signal is r epor ted as a function of t ime for k=2A.

t rends of the I(0max) and the S(Omax) functions are shown in fig.2A and 2B for cons tan t va lues of J O = 0 . 2 5 M L s "1 and s=l . The d i f fus ion coe f f i c i en t is in the fo rm D=va2exp[-Q/kT], where Q is the activation energy for su r face hopp ing and v t h e vibrational frequency (usually v=5 1012s'l).

The k inet ics of nuc lea t ion , N vs t, is reported in fig.3A for Jo=0.25MLs 1, s= l and Omax=4ML. The same panel also shows the t ime dependence of the surface coverage of monomers , 01, which exhibits a max i mum tha t is typical of the kinetics of t r a n s i e n t species.

The proposed model has been also used to get an insight into the impor tan t issue regarding the existence of scal ing laws, for the PES/AES signals. We recall tha t scaling

0.80

0.70

.~_ 0.60 §

"~ 0.50

0,40

0'300. 0

0 =IML max

I e o ~

• , o S I I I l l l

. . . . I . . . . I I I I I I . . . . I . . . . i . . . . 1 , 1 1 i

1,0 2.0 3.0 4.0 5.0 6,0 7.0

In(m)

Fig.4 A t t enua t i on s ignals I and sur face fraction S that is covered by islands vs In(m). The d iment ionless term, m, is defined as m=( Omax/a2)D/Jo .

laws are frequently encountered in the theory of thin film formation at solid surfaces and they are succesfully used to extract kinetic quant i t ies from the exper imenta l da ta [9]. Our model indicates tha t the solution of the s y s t e m (7) sca le s in t e r m s of t he d i m e n s i o n l e s s v a r i a b l e m = t d D / a 2 = =(emax/a2)D/J0 at given values of 0rex, a and r m. In other words, the kinet ics of cluster formation, as well as the I(t) function, only depend on m whatever the values of both D and Jo are.

The results are shown in r iga where the I and the S quan t i t i e s are r epor ted as a function of ln(m). In part icular , s imulat ions were performed at T=300K, k=2A and for different values of Q in the range 0.5-0.8 eV. Differences in the values of N, I, S and 01 has

288

been obtained, at a given 0m,x , a and rm terms, by changing m. To the author 's knowledge this is the first time that such scaling plots, regarding the at tenuation signals, are reported. We stress that this kind of plots can indeed be used to extract the activation energy for the ad-monomer surface diffusion from PES and/or AES experiments.

3.CONCLUSIONS

The fo l lowing conc lus ions can be summarized: i) when nucleation occurs at special sites, part icular conditions can be realized for which the PES signals is related to the S(t) kinetics. Moreover, the model reproduces the characteristic features of the AES signal for oxide nucleation on parent metal; ii) in the case of nucleation by stable dimers, the model indicates that scaling laws, in term of the temperature dependent variable m=(Omax/a2)D/Jo are satisfied by the PES/AES signals.

ACKNOWLEDGEMENTS One of the authors (M.T.) wishes to thank the European Community for the finantial support (contract BRE2.CT93.0448).

R E F E R E N C E S 1. M. Avrami, J. Chem. Phys. 7 (1939) 1103; 8 (1940) 212. 2. M. Tomellini, J. Appl. Phys. 72 (1992) 1589. 3. M. Zinke-Allmang, L. C. Feldman, M. H. Grabaw, Surf. Sci. Reports 16 (1992) 377. 4. R. Memeo, F. Ciccacci, C. Mariani, S. Ossicini, Thin Solid Film 109 (1983) 167. 5. A. Balzarotti, M. Fanfoni, F. Patella, A. Sgarlata, R. Sperduti, Phys.Rev.B49 (1994) 9103. 6. P. H. Holloway, J. B. Hudson, Surface Sci. 43 (1974) 123; C.R. Brundle, H. Hopster, J. Vac. Sci. Technol. 18 (1981) 663. 7. P.R. Norton, R.L. Tapping and J.W. Goodale, Surface Sci. 65 (1977) 13. 8. M. Fanfoni, M. Tomellini, in preparation. 9. J. W. Evans, M.C. Bartelt, J. Vac. Sci. Technol. A 12 (1994) 1800.