Migration and capture processes in heterogeneous nucleation and growth

SURFACE SCIENCE 21 (1970) 289-306 © North-Holland Publishing Co.

MIGRATION AND CAPTURE PROCESSES IN H E T E R O G E N E O U S

NUCLEATION AND GROWTH

II. COMPARISON WITH EXPERIMENT B. LEWIS Allen Clark Research Centre, The Plessey Company Limited, Caswell, Towcester, Northants., England

Received 20 October 1969 The methods of Part I are applied to the analysis of experimental results. The nucleation of Pt on NaCI is found to be predominantly diffusion controlled, and the activation energy for surface diffusion is evaluated. Saturation at a nucleation density which increases with temperature and is independent of incidence rate, observed with Au on NaCI is now found not to be due to desorption controlled depletion, as previously claimed, but to nucleation on preferred adsorption sites. Measurements of the nucleation of Ag and Bi on C, with critical nucleus sizes, i*, between 1 and 6 atoms, are analysed, yielding values of adsorption, surface diffusion and cluster binding energies. Loss, as well as capture, by surface diffusion is found to be significant in growth behaviour when i* = 6. Maximum nucleation densities occur at the lower temperatures when the residual nucleation rate, reduced from its initial value by the depletion of single atoms and pairs by capture, is balanced by the coalescence rate due to growth. At higher temperatures saturation occurs when the nucleation rate is balanced by loss of mobile nuclei by mutual capture. 1. Introduction

In Part I x) we have f o u n d that in nucleation behaviour two cases can be distinguished. W h e n the probability of a d a t o m desorption is smaller t h a n the probability of m u t u a l capture, Pa < P . , the a d a t o m p o p u l a t i o n a n d the nucleation rate are determined by surface diffusion processes. Nucleation is too rapid for m e a s u r e m e n t s of initial nucleation rate to be possible, a n d the main interest is in nucleation densities. A good example of this case is the deposition of Pt on NaCI crystal substrates, which has been studied by Sumnere). W h e n Pa > P , , the a d a t o m p o p u l a t i o n a n d the nucleation rate are initially determined by desorption. Very complete sets of experimental results u n d e r these c o n d i t i o n s have been given by P o p p a a) for Bi a n d Ag deposits on a m o r p h o u s C a n d SiO substrates. P o p p a a n d Lewis 4) have shown that the initial nucleation rates can be analysed using the atomistic model and (less successfully) using the classical model of nucleation theory. The m a x i m u m 289

290

B.LEWIS

density o f nuclei decreases with temperature conforming qualitatively with Halpern's ~) predicted saturation behaviour. However, Lewis concluded that at the higher temperatures saturation occurs too soon, at too low a density, and at too small a nucleus size to be due to single atom depletion. Depletion of larger mobile nuclei was postulated to explain the observed behaviour. Lewis and Campbell 6) have given data for Au on NaCI which show an increase o f saturation density with temperature, following their localised depletion model, but disagreeing with predictions in Part I. These experimental results will now be used to test the adequacy of the theoretical treatment in Part I and to determine the values of the controlling material parameters. 2. Platinum on rocksalt

Sumner `)) measured the nucleation of evaporated Pt deposits on aircleaved NaCI. Pretreatment o f the substrates at 400°C and 5 x 1 0 - 7 torr r (°C) I0

400 ,

30xlO

300 ,

ns

200 ,

-.--

n mGX 20

Pt on N0C I.

g~~

"

'

~

~

( c m "2 )

Z /

5 /

/

/ ~ ~,

"~ = l O 13 sec-I i Za = "~o =

95eV IOI3 sec-I

/ 1.2

I 4

1'6

r8

-2"0 x I0 -3

PIT (oK-f ) Fig. 1. Experimental values of the density of stable nuclei ns at t ~ 15 min for Pt on NaCI, and calculated values of ns at t = 15 rain and of nmax, for the parameters shown. removed surface layers and cleavage steps by evaporation, leaving an apparently clean uniform surface. Pt deposited at 200'~C was epitaxially oriented and gave almost constant nucleation densities, n~, for deposition times between 1 and 15 min at R = 8 x 1012 cm -2 sec -1. Experimental values o f ns for temperatures between 200°C and 400°C and t = 15 min are shown in fig. 1. Following Sumner in his interpretation that n, is diffusion controlled, the depletion models o f Lewis and Campbell6), and of LoganT), which

M I G R A T I O N A N D C A P T U R E P R O C E S S E S . 11

291

assume that saturation occurs when Pc =P,, give

n~.~,= (noR~v,) ~ exp ( Ed/Zk T),

(1)

for i * = 1 (ignoring a small numerical factor in Logan's treatment). Substituting R = 8 x 1012 cm -2 see -a , v1= 1013 sec -1 and no = 1015 cm -z gives ns,, = 3 x 107 exp(Ed/2kT),

(2a)

n, = (3 _+ 0.5) x l09 exp[(0.18 + O.02)/kT],

(2b)

whereas Sumner found

where the units of k are eV °K -1. Thus eq. (1) does not adequately describe the experimental results. The usual value of adsorption site density, n0=1015 cm -z, is a mean value appropriate for most substrates. Possible adsorption sites for metal atoms on NaC1 have been discussed by Gillet and Gillet s) who show that the observed epitaxy indicates adsorption above the Na + ions and between 4 C I - ions, giving a = 3 . 9 / ~ and n o = 6 . 7 x 1014 cm -2. These values will be used, but their only influence is on the value found necessary for vI with which they are associated in the equations. Sumner's micrographs of 200°C and 300°C deposits show nuclei of fairly uniform size, as expected with rapid nucleation. Equating the volume RtV of incident atoms with M'r3ns for values of the mean radius r and density ns measured on the micrographs indicates M'-~ 12 for both temperatures. For spherical or other equiaxed nucleus shapes M'~-4, so for these shapes ~] of the incident material appears to have been lost. A similar loss, which was independent of incidence rate and temperature, was observed by Lewis and Campbell 6) with Au on NaCI. Incomplete thermal accommodation, i.e. failure of a proportion of incident material to be adsorbed, is a possible explanation, but no allowance for this will be made in the analysis. The shape factor M'=4 gives M=M'a3/V= 12 for cluster radii q measured in units of hop distance a when a = 3.9 A and V= 15.3 A 3. The value of M makes very little difference to the calculated nucleation behaviour and adatom depletion, since the capture rate wq is only weakly size dependent. The computer programme used in Part I was modified to use T, Ea, v0 E d and vl as input data, instead of Pa and v. i*--1 was written into the programme since the supersaturation ratio is so high, 1036 at 200°C, that pairs must be stable. A value of E a was assumed, sufficiently high for desorption to be negligible. Values of vI and Ed were chosen to give agreement with the experimental densities at 200°C and 300°C. The full lines in fig. I give//max and ns at t = 1 5 min for Ea=0.45 eV and vI =1013 sec-1; the times tma~ at which nm~ is reached are indicated. Sumner measured the

292

B.LEWIS

variation ofn~ with time at 200°C and found a maximum of 3 × 10 l° cm -z sec-~ at t = 15 min. The calculated variation of n~ with time is fairly close to these figures. Thus substantial agreement with experiment at 200°C and 300°C is obtained with vI =1013 sec-t and Ed=0.45 eV. These values are interdependent and the fit with v~ = 1014 sec -~, Ed=0.5 eV and with v~ = 1012 sec -1, Ed =0.4 eV is almost as good. The experimental density at 400°C is about half the calculated density with no desorption. Agreement can readily be obtained, as shown by the broken line in fig. 1, by assigning a value for E a which allows some desorption. With E,=0.95 eV and Vo= 1013 sec -~, at 200"C, p, is still small cf. p,, so that the nucleation behaviour is unchanged; at 300°C, pa>p, and the initial nucleation rate is reduced, but pc=p,, after about 10 sec and the value ofn~ at t = 15 min is not changed; at 400°C, p,>>p.,, the initial nucleation rate is 104 times lower and remains almost constant for the first five minutes and then decreases slowly. The spatial distribution of nuclei at 400"C would be expected to be fairly uniform with a range of sizes arising from the prolonged nucleation. In Sumner's micrographs for 400°C about a third of the island shapes indicate agglomeration of two or more nuclei; whereas tbr 200°C most clusters have simple centrosymmetric shapes and there is little evidence of agglomeration, despite the closer mean spacing. This suggests that at 400°C clusters are mobile. The irregular nucleus shapes which persist after coalescence show that the surface mobility of Pt atoms must be low, as would be expected from the high melting point. Vapour pressure data for NaCI indicates an evaporation rate of about I A s e c - ' at 400°C. The observed deterioration of the orientation of the Pt deposit with increasing temperature can be ascribed to instability of the substrate surface. If substrate evaporation allows a mean cluster mobility of less than 1 A sec-1 this would account for the reduction of numbers of separate nuclei compared with the full line curve of fig. 1. In the absence of experimental data of the variation of numbers and sizes of nuclei with time at 400~C it is not possible to estimate the relative importance of desorption of adatoms and of mobility of stable clusters in reducing the density. 3. Gold on rocksalt

Experimental results with Au on vacuum cleaved NaCI were presented by Lewis and Campbell 6) in support of their localised depletion saturation model. At 150~C a saturation density which was independent of incidence rate was rapidly established. Robins and Rhodin 9) interpreted similar behaviour with Au on MgO as nucleation on preferred adsorption sites, tentatively identified as impurity atoms. With Au on NaCI, the densities

MIGRATION AND C A P T U R E PROCESSES. 11

293

were too high to be due to impurity and depletion-controlled saturation at the density giving pc =pa was proposed. This gives n , , = no exp [ - (E~ - Ed)/kT ]

(3)

(when the capture factor b = l ) , in which n~at is independent of rate and increases with temperature, as experimentally observed. At 300°C, the saturation density was 10times higher than at 150°C. However, the nucleation rate J = 8 x 10 ~° cm -2 sec -~, together with R = 2 x l0 ~3 c m - 2 s e c - 1 , i * = I, n o = 7 X 10~4cm -2 and v=1013 sec -~, in eq. (16b) of Part I, gives 2E a - E d = 1.4 eV. Then, for E d = .~E~, E ~ - E d = 0.56 eV and eq. (3) gives n , a t = 7 x 109cm -2, whereas the actual value was 4 x l0 '1 cm -2. Conversely, the value of E ~ - E d which satisfies eq. (3) at 300°C would give J = l06 cm -2 sec- ~ which is far too low. Alternativevalues of i*, n o, v or Ed do not remove these discrepancies. It was shown in P a t t i that the present model predicts that nucleation density always increases with incidence rate and decreases with increasing substrate temperature. The experimental results for Au on NaCI do not show this behaviour. The observed variation of nucleation rate and saturation density is compatible with nucleation on preferred adsorption sites. The increase of n~t with temperature is then due to an increase in the density of preferred sites, possibly as a result of thermal etching. Sumner's pretreatment of NaCl at 400°C apparently gave a uniformly high density of adsorption sites of depth 0.45 eV. For Au on NaCl the variation of site density between specimens precludes estimation of the depth of either shallow or deep adsorption sites. 4. Silver on carbon

In Poppa's a) experiments with Ag on amorphous C, he observed the variation of numbers of nuclei with time for several different temperatures and incidence rates. Supersaturation ratios ranged from 104 to 15, with pa>>p., SO that the initial nucleation rate J is given by eq. (16) of Part I. Then, i* can be found from the slope of log J versus log R, or from the intercept at I / T = 0 of log J versus I/T. Experimental values of J are plotted in fig. 2. The broken line is for i * = 2 throughout, as suggested by Poppa. However, this interpretation was based partly on a misplotted 1/T value at 873°K, and fig. 2 shows that a transition to i * = 3 at 811°K and R = 6 . 3 × 1013 c m - 2 s e c - 1 allows a better fit, particularly at 811 °K. In order to obtain the theoretical lines in fig. 2, the computer programme was formulated for a variable critical nucleus size and for Ea, (Ea-Ed), and (AEi+Ea) for i = 1 to 6, as input data. The values of AEi+Ea determine

294

B. LEWIS

R ic~ ~ ~ I )

Temperature (o K ) 87~ 847 8H 775 742

{010

I (cm2~ec)l09

(c;~ 2 secl)q IO-

b}

22

IO IO

.

i8,,d]

Ag on C ,.,,,/ -

-

6E31"Eo-2IbeV

13

6E3eEa= 2 1 5 e V ~ / /

108

L*:3jI ~°7

I'/

E3+4eo-ed I

I0

7

mob~ F i05

I

I2 I/T

I~

(°K-I) (a)

-~ 1"4 xlO

IO S

IT: BII*KI 3 ,o R (cm 2 sec I}

3o;,o '~

(b)

Fig. 2. Experimental values of initial nucleation rate J for Ag on C and calculated values, for the parameters shown.

the transition points from one critical size to the next. The nucleation rate depends on E~+(Ea-I:~) when i * = 1 , E~+(E~-Ed)+(AE2+Ea) when i * = 2 , and so on, and also on no, Vo, vl and the capture factor b i given by eq. (12) of Part I. A cause of uncertaint2~ in the experimental nucleation rates is an induction time of up to 90 sec before nuclei were observed. As discussed by Poppa, there is first the time required to establish the equilibrium density of critical nuclei. This is of the order of the decay time, (l/v) exp [(Ei,+Ed)/kT] and is only about 10-1 sec for the highest possible values of Er +E~. The more significant component of induction time is the time required for a nucleus to grow to visible size. The computer programme was arranged to give the number of nuclei nv which exceeded 30 A diameter. Using M = 2 (hemispherical nuclei) and trial values of the input data, the variation ofnv with time was calculated and compared with the experimental results, as shown in fig. 3, to obtain data values giving the best fit. The difficulties of assessing an initial nucleation rate from plots with a rapidly changing slope, or an indeterminate induction time were thereby alleviated. The "theoretical" initial nucleation rates, for the best-fit data, are plotted as the full lines in tig. 2. Poppa found that with % = v, = 1 0 1 3 sec-i a very high value of a, i.e. a low value of no, was needed to fit the data. Lewis 4) showed that no--1015cm -2 could be used with v o = v 1=10 l ~ s e c - ' . The present analysis, using the procedure above, was satisfactory with % = 10's

295

MIGRATION AND C A P T U R E PROCESSES. II

cm -2, V o = V 1 = 1 0 1 2 s e c - t for both Ag and Bi on C, and these values have been adopted for all the calculations. The transition from i * = 2 to i * = 3 at T=811 °K and R = 6 . 3 x 1013cm -2 sec-i gives A E 3 + E a =2.15 eV. If i*---2 at 742 °K then 1.95 eV is the highest possible value of E 2 + E a. The experimental nucleation rate for t < 3 0 sec between 742°K and 811°K (i.e. when i * = 2 ) gives E2+3Ea-Ed=3.4eV. This term is the sum of E a - E d which also controls the growth rate, and E 2 + 2 E a. Increasing Ea-Ed, while keeping E2+3Ea-Ed constant, causes faster growth and earlier coalescence at a lower nucleus density. Thus by 12

II

IO

IO R = 6 3 x IO 13 crn ? sec- I

nV

(cry2)

T = 811*K

nv

LI

IO

~:~rK

Ea ='8SeV Ed- 2SeV

18 x 1 0 1 4 c m 2 s e c I

,~o

1o

~

c

m ~6~

2 secI ~O crn

1I

g

10 7

# 0

I

I

200

300

~I

I00

t

(sec) (o)

F i g . 3.

Aq on C

(cm-?)l~l

400

0

13

-2

sec

-I

i

i

i

I00

200

300

400

t (sec) (b)

Experimental values of the density of visible nuclei nv for Ag on C and calculated values, for the parameters shown here and in fig. 2.

comparison of calculated and experimental values of nv against time at 742 ° K for t > 30 sec the individual best fit values Ea - Ed = 0.6 eV, E2 + 2E, = = 2 . 8 eV were found. Ifwe assume E 2 +Ea = 1.95 eV, we then obtain Ea=0.85 eV, Ed=0.25 eV and E2 = 1.1 eV. E 2 could be lower (and Ea and E d correspondingly higher) but the direct measurement of the binding energy of diatomic Ag as 1.7 eV t0) justifies selection of the highest possible value. Then AE3+Ea=2.15 eV gives AE3= 1.3 eV and E 3 = 2 . 4 eV. These are the data used in figs. 2 and 3. At 775 and 811 °K there are notable discrepancies between the theoretical and experimental densities, but since the deviations are in opposite directions for t < 50 sec this may be due to experimental error. It is more significant that except at 742°K the theoretical curves in fig. 3 show a continued rise beyond the highest values reached experimentally. We therefore conclude

B. LEWIS

296

that the experimental saturation above 7 4 2 ° K is due to some mechanism other than single atom depletion and coalescence due to growth.

5. Bismuth on carbon

In Poppa's experiments with Bi on amorphous C, supersaturation ratios ranged from 10s to il30. As shown in fig. 4(a) the gradient and intercept of l o g J versus I/T for R = l . 7 × 1 0 1 4 c m - 2 s e c -1 and T = 5 0 1 ° K , 538°K and 573°K establish the value i*=1 and give 2 E ~ - E d = 0 . 9 e V . Fitting R (crn -2 sec H)

T (~KI Sc~O

IO

i

573

538

w

501

iO

L

12

I 7

2

2-4 x

IO14

Bi on C I

iO It

,o

AE6+Eo:

I.SeV

I

( c,~2sec I)

(¢m2sec I) '9 IO

iO 9 ~-b /

2Ea- Ed='qeV 8

8

IO

I0

Gradient = 7 I0

I0

~

m--~

7

/

7

~O i4

-2

R= I ~

I0 cm

i

i

i

I7

18

I'q

b

- I sec

T = 5900K b

i

b

F i g . 4.

Eb- 7 E a - E d : 8 4eV

2xlO "S--

IO

I

I

2

2S

I/r (°~-~)

R (cn~2 sec-L)

(a)

(b)

3 I0 ~4

Experimental values of initial nucleation rate J for Bi on C and calculated values, for the parameters shown.

calculated values to the experimental nv versus t results at 501 °K then gives E~=0.65 eV, E , - E d = 0 . 2 5 eV and Ed----0.4 eV. Between 573~K and 590°K with R = 1.7 × 1014 cm -a sec-1 the nucleation rate falls by a factor 10. Poppa assumed there was a single transition to a larger critical size at 573°K and his l o g J versus I/7" plot then indicated i * = 3 at 590~K. However, provided that the values of AEg+Ea lie between 1.46 eV and 1.52 eV there may be one or more transitions anywhere between 573 and 590°K. From eq. (16) of Part I the slope of log J versus log R is (i*+ 1). Experimental values at 590°K plotted in lig. 4(b) give i* between 4 and 8. For a close-packed structure, in growth from a pair to a six-atom cluster, each additional atom (if it stays on the substrate) can only make two nearest-neighbour cluster bonds, so it is not unreasonable for all the increments AEg for i = 3 to 6 to be nearly equal. A seventh atom can make

297

MIGRATION AND CAPTURE PROCESSES. II

three cluster bonds and is thus expected to be more stable. We will assume that i = 7 is stable and that the critical size at 590°K is i* =6. If i* = 6 for R = 2 . 4 x l 0 ta c m - 2 sec -1 at 590°K then z t E 6 - E ~ 1.5 eV (for Vo= 1012 sec-l), and if Ea=0.65 eV then AE6~0.85 eV. A region with i * = 2 would occur between i * = I and i * = 6 if AE2
Wq=no-/Zq2+

-n0p b , - 2nqvt ex p

bq = ~,rtq' K1 (q')/Ko(q').

(

- - ~-1; /[1 - - f ( q +

1)], (4)

The first term represents direct impingement and is the smallest term when q is small and the largest when q is large. The second term represents surface diffusion capture and is initially almost constant and then increases steadily with q. The decay term varies rapidly with q when q is small, and then more gradually. in Poppa's experiments with Bi on C at T = 5 9 0 ° K and R = l . 7 x l 0 t4 c m - z sec- t, nuclei were first detected after about 70 sec. The growth of a single nucleus, as observed in successive micrographs is plotted in Poppa's fig. 15. These data are replotted in fig. 5 and compared with theoretical

298

B. LEWIS

ck

/

(at°ms~'

]

T = 5qO°K

Bi or C

/7

f,Jll hnes ar'G e×per~mento! DOi~ts R = ' 7 × tO 14 <;n2 sec-!

~, , . 5 ~ / / ' f ~ .\0/.,~ "/ ,,

Fo - E ~ : ~s~v

-.~->"

40 k

~f"

2Q

/

I0

//

0

}

E~

l ',00

/ . IbC

~ ~.~ S>

/ /

~

0

/"

~

bd = I 2eV

r ?00

; .'D,"3

0 qO©

Fig. 5. Experimental values of nucleus radius q for Bi on C and calculated values, for the parameters shown: line 1, direct impingement only; line 2, direct impingement and surface diffusion capture; line 3, direct impingement, surface diffusion capture and detachment; lines 4 and 5, as line 3, for the incidence rates shown.

growth relations obtained from t q3 ~ q03 "IL IIIV~; wq dt.

(5)

0

For surface diffusion capture, the smallest stable cluster size ( i = 7 ) is q o = 2 , since atoms ariving at this radius will be captured. For decay, qo = 1 gives 2~ edge atoms, which is close to the actual number of 6. However, taking q0--2 allows a correction factor 2 for the lower mean binding energy of edge atoms of very small clusters, compared with the large cluster value Ee. For direct impingement only, eq. (5) gives the linear relation shown in fig. 5 as line 1, and is remarkably close to experimental points. Addition of the surface diffusion term for E ~ - E d = 0 . 2 5 eV gives line 2 which is much higher than the experimental growth rate. The observed low growth rate cannot, in this case, be ascribed to thermal misaccommodation because nucleation depends entirely on surface diffusion and if a given set of parameters satisfy the observed nucleation rates at different temperatures they should also satisfy the observed surface diffusion growth rates. Although an accommodation coefficient less than unity would impose a first order correction on the value of E~, the derived quantities J, n, and w~ would

299

M I G R A T I O N A N D C A P T U R E PROCESSES. n

only require second order corrections. Our main objective is to obtain quantitative agreement between theory and experiment with one set of parameters and for this purpose, we will continue to assume complete thermal accommodation. Line 3 in fig. 5 includes the detachment term of eq. (4) and shows the calculated growth for the value of E e which gives the best fit with the experimental results. The surface diffusion capture and decay rates are almost balanced for all values of q. The value of Ee is 0.95 eV which is 0.1 eV higher than LIE6, and indicates a factor 5 difference in stability between edge atoms of large and of critical nuclei at 590°K. Growth behaviour at R = 2 x 1 0 1 4 and 2 . 4 x 1 0 1 4 c m - 2 s e c - l , calculated from eqs.(4) and (5) with E a - E a = 0 . 2 5 eV and E~=0.95 eV is shown as the broken lines 4 and 5, in fig. 5. There is no experimental data in Poppa's paper against which to check these results. The modified growth relation was used for the calculation of the variation ofnv with t. The results are shown in fig. 6, together with Poppa's experimental data. Omission of the decay term in eq. (4) would have reduced the maximum density values at 573°K (to 1.4x 10 ~ cm -z) and at 590°K (to 4, 6 and 10x 10 ~° cm -2 at 1.7, 2 and 2.4x 10 ~4 cm -z sec -1, respectively) but they would still have been substantially higher than the experimental values. The agreement of the maximum density at 501 °K, the induction times at 590°K, and all the initial nucleation rates derives directly from the choice of energy parameters. At 501 °K, after 50 sec, depletion of adatoms by capture

5x I 0 II nv

I

R=

I0 x I0I0 I . T x l O 14 cm 2 sec"I

[

(cm-2)

0 ~

O

0

IO0 r

/s~:)

(o)

2OO

300

/ 2 -24xsI0

II°"T~A

0

f

J

100

200 t

'

300

400

(sec)

(b)

Fig. 6. Experimental values of the density of visible nuclei nv for Bi on C and calculated values, for the parameters shown here and in figs. 4 and 5.

300

a. LEWIS

halves nl and coalescence assists in reducing the net nucleation rate to 0.1 of its initial value. After 100 sec, the predicted surface coverage is high and continued growth causes a sharp reduction of nucleation density, due to coalescence. At higher temperatures adatom depletion by capture is less because of the higher probability of desorption and because there are fewer capturing nuclei. The calculated nucleation rates remain almost constant until high surface coverage causes coalescence and the density of nuclei then falls. This is also true for Ag on C but is not so evident in fig. 3 because of the log-linear scaling. Experimentally, except tbr R = 2 . 4 x 1 0 t 4 c m - 2 s e c - 1 , T=590OK which shows a maximum of nv, there appears to be a progressive decrease in nucleation rate as the density of nuclei approaches a saturation value. For R = l . 7 x 1 0 1 4 c m - 2 s e c -1, T = 5 9 0 ~ K there is direct experimental evidence that the observed saturation is not due to adatom depletion or to coalescence. The growth rate is so close to direct impingement growth that depletion of adatoms by capture must be negligible. Poppa's fig. 4 (third micrograph) shows this case at 290 sec. The mean spacing of the nuclei is several times larger than the mean nucleus size and the coalescence rate is obviously negligible. Poppa (private communication) has stated that a study of the complete sequence of micrographs reveals only very few coalescence events. The experimental results for Bi on C thus necessitate some other depletion mechanism to account for the low observed values of maximum density of nuclei.

6. Depletion of mobile clusters Although it is usual to consider only single atoms as mobile there is no basis for this assumption. For the movement of a pair, simultaneous hops by both atoms would require an activation energy 2Ed. However, movement of one atom at a time would require an energy, Ed2 say, smaller than 2E d. Since there are two atoms, the j u m p rate is 2v I exp(-Ed:/kT ) but the restriction on hop direction imposed by attachment of the two atoms as a pair, reduces the effective hop rate to about half this value. Also, the pair as a whole moves only half a site for each atomic hop, and for a sequence of random hops the mean number of hops for a migration distance of one site is four. Thus the pair mobility is )v I exp ( - E d , / k T ). Direct evidence that pairs can have quite high mobility is found in the nucleation of molecular species, e.g. AgC111). For i = 3 to 6 there are i atoms which can jump, but peripheral migration now requires breakage of a cluster bond. If the energy required is Edp then the j u m p rate per atom is vt exp(-Edp/kT) and the effective mobility in

M I G R A T I O N A N D C A P T U R E P R O C E S S E S . 1I

301

sites sec-i is I/2i 2 times the total jump rate, giving v, = (Vl/2i) exp ( - Edp/k T ) .

(6)

Depletion of mobile unstable nuclei by capture by stable nuclei density n~, decreases the nucleation rate by a factor A i, say, determined by the probabilities of capture and of decay. For i ~ i * it can be shown that Ai

(

=

vI ~

E i -I- E d - - Edp

n~bq .... exp 2inov o

I +

k-T

>

(7)

'

where the capture rate b~ is given by eq. (12) of Part I, using appropriate values of q for the capturing nucleus and p, the probability of loss during migration one site, for the mobile nucleus. Since the detachment energy AE~+Ed is necessarily greater than the peripheral migration energy Edp (and A E z + E d > Ed,_ for pairs), the depletion factor A~ decreases with temperature and it is also independent of incidence rate. Thus it cannot account for the observed low saturation densities at high temperatures and low incidence rates. Depletion of mobile stable nuclei must be considered.

1014 873 847 811 •

'

775

"

T(oK)

742

590 573

538

ill

501 1014 ~'n i v i

n sat

i012

Aq

C c..-2 sec-I 0..,,--_.__.__

on

R:6.~xlO

(cm"2 ) I0 I0

1012 3

0

0

ic~°

0

o

, ~

z

0 i0 8

13

106

]

v

(era-2 see-l)

J

I

I

I

/

[dp = lev

- ~ ' ~ "

~

--

X

BJ on C R : 1 7 x lOl4cm -2 sec -I

, o8

vp . sp

S

~sites sec "l )

I000

•

1o'

"

dp

1 I

( ~ segll I00

102

~

.....

v

s

I000 ,

,

,

2

13

,

;oo,o,2qo:5o~ 15

lit

-......~

1 I

,

I

'

I000 ~ 1.6

7

"

"

~ 8

49

IO

2 x IO - 4

(o£1J

Fig. 7. Values of J, nsat, ~ n i v t = noJ/nsat and of stable cluster mobility for A g and Bi on C. 6~ - ~ n~vl/nsat is the mean value of mobility which would account for the observed nsat by mutual capture of mobile n u c l e i . . ~ = 6ttta is the corresponding point-to-point migration speed. A size-dependent mobility v~ - vpi ~/a was used to obtain figs. 8 and 9, with the values o f vp shown here. The corresponding values of VlO00 and the migration speeds sp and sl0oo are also shown.

302

B. LEWIS

The loss rate Jm of stable nuclei by mutual capture can be written

Jm = (n~lno) Y~ n,~,,

(8)

in which the s u m m a t i o n is from i* + 1 to i,,~ and a mean value bq = 1 has been assumed for the capture factors, giving n~ rather than ~ n~b~ for the capturing nuclei. Saturation due to mobile nucleus depletion alone will occur when dm = J i.e. when d = (ns~,/no) ~. nlvi. (9) Experimental values of J and n~at are plotted against I / T in fig. 7, and the expected correspondence between them is clear. Values of ~. nlv~ obtained from eq. (9) are plotted as solid squares in fig. 7. The mean stable nucleus mobility ~,. at saturation, defined by ItsatOi ~- ~. tlil~ i can now be found and is plotted as the solid circles in fig. 7. The open symbols give ge obtained in the same way for other incidence rates, s3g is, of course, a r a n d o m walk mobility and corresponds to a mean point-to-point migration speed •~i = ~ a . Fig. 7 has scales for both v and s. Since saturation is partly due to single a t o m and unstable nucleus depletion and coalescence, the values of stable nucleus mobility plotted in fig. 7 are m a x i m u m values. They have been obtained as directly as possible from the experimental data, with no assumption as to the size dependence of the mobility. In order to put mobile nuclei into the nv, t calculation, it is necessary to use a size dependent mobility. For large hemispherical nuclei which have 5~ peripheral atoms, eq. (6) becomes

v i = 5i-~vl exp ( - Ea~,/kT ),

(10)

in which we assume that m o v e m e n t of peripheral atoms is the predominant mobility process and that Ed, is independent of cluster size and temperature. Nuclei formed during successive time intervals were considered batch by batch. If a batch contains n(l.2) nuclei with a size range i I to i2, and if within the batch we assume there are equal numbers of each size, then E nlvi is integrable and writing eq. (10) as v, = vpi ~ we find ~2 il

n (1,2)

n~v: = .

12 - -

.

r, 3(t~

- i 2 ~).

(12)

il

At each stage in the growth calculation the capture rate din(l,2) of nuclei in each batch was found from eqs. (8) and (12). The for a time increment At, the n u m b e r lost by capture was din(l,2)At and this was substracted from n(l,2), n~ and n,.. Loss of nuclei up to i = 6 was allowed for in the calculation of the nucleation rate J 6 , using equations similar to eq. (7)

303

M I G R A T I O N A N D C A P T U R E PROCESSES. [I

but without the restriction to i ~ i*. The coalescence rate Jc, due to growth of nuclei was subtracted from J6 to give the net nucleation rate used during the next time increment. The results of this procedure for Ag and Bi on C are shown in figs. 8 and 9. At the lower temperatures, ~ Jm at first rises with n~, but as adatom and unstable cluster depletion progressively reduce the numbers of small

'°I 12

iO 12 R - b3 x iO 13 c m 2 s e c i nV

(c.~ 2)

742*K

i0 II

cm2

)

I

L.Bx I0 cm sec

l

O

~

m

-2 sec I

,0 9

8

I0

7 I0

IOO

0

200

300

400

O

1OO

200

t (sec)

(a)

Fig. 8.

300

400

t (sec)

(b)

Experimental values of the density of visible nuclei nv for Ag on C and calculated values allowing for large cluster mobility, for the parameters shown.

5xlO

Ii

tOxlO

I0

R " I 7 r 11~4cn~2 sec -I

T = SgO*K

nv

nv

(c ~ 2 )

Old,solo

Eo : 7eV ~) Ed : 45eV

(cm-2 )

Bi on C

A 8

z,

~" 2 4xLO14 c m 2 se(~ t

Ed2: 65eV b

Edp= 8 5 e V 538*K

0

Fig. 9.

0

iO0 200 f (sec) (a)

300

0

iOO

I 200

I 300

400

t (sec)

(b)

Experimental values of the density of visible nuclei nv for Bi on C and calculated values allowing for large cluster mobility, for the parameters shown.

304

a. LEWIS

nuclei ~ J m decreases and is relatively small when ns=nmax. At the higher temperatures ~ J m increases steadily until it causes saturation when

~ Jm +J~=Jo. For Bi on C, when pairs are stable, their depletion by capture can significantly reduce the nucleation rate. The observed maximum density at 501°K is obtained theoretically with E~=0.7eV, E d = 0 . 4 5 e V and En2 =0.65 eV. However, the initial nucleation rate is now rather too high, so that 0.65 eV must be accepted as the lowest possible value for the pair diffusion energy, although compared with Ed=0.45 eV for single atoms it is higher than might be expected. At 590°K, pairs are unstable and their depletion by capture has less effect. Saturation is mainly due to mutual capture of larger nuclei and the observed saturation value at T = 5 9 0 ° K and R = l . 7 x 1014cm-Z sec -1 is obtained with Eap=0.85eV. The corresponding values of vp, v;=vpi --~ and s~=v~a for i=1000 atoms, 2qa= =nucleus d i a m e t e r = 5 0 A are plotted as lines in fig. 7. At 590°K the low growth rate maintains a relatively high proportion of small nuclei and the migration speed required is only about 10 A sec -1 for a 50 A nucleus, or i A sec -1 for a 120 A nucleus. This is quite low, and Poppa's fig. 4 shows at least one nucleus which has moved about 100 A in 130 sec while growing from 50 to 200 A. The agreement between theory and experiment in fig. 9 is substantially better than in fig. 6 and no longer shows any systematic errors. The remaining discrepancies can fairly be attributed to experimental error. For Ag on C, Ea=0.95 eV, E d = 0 . 4 e V , Ed2=0.7 and Edp= l eV give agreement between theory and experiment at 742°K. Again this value of Ed2 is about the lowest permissible value. At 775°K and 811°K the theoretical plots in fig. 8 flatten off at about the same values of t as the experimental curves, but the saturation values are still too high. Evaluating Edp from the data for either of these temperatures would give a lower theoretical saturation value at 742°K. For the higher temperatures and the lowest rate at 811 °K, the theoretical plots are still rising at 300see, whereas the experimental values have flattened off. The theoretical densities at 300 sec could be adjusted by choice of AE2 and AE3, but the best agreement for both the initial rise and the saturation values would require a slightly higher nucleation rate and a higher stable nucleus mobility. The need for an increased mobility at these temperatures is also shown by the values of t:i and .~i in fig. 7. Bassett 12) has observed translation and rotation of Ag nuclei on amorphous C in the electron microscope, but the mobility at 7 0 0 ' K is not high. However, Stowell (private communication) has reported that Pb nuclei are very mobile when they are liquid, and Gladkich, Neidermayer and Spiegel lz) have found that Ag on C nucleates in the liquid phase above 823 °K. Thus the cluster mobilities required to account for the observed

MIGRATION AND CAPTURE PROCESSES. 11

305

saturation densities of visible nuclei at the higher temperatures are not unreasonably high. 7. Conclusions The growth, capture, depletion and nucleation relations, developed in Part I, have proved capable, with some extensions, of dealing comprehensively with experimental data. When mutual capture of adatoms is more probable than desorption, nucleation and growth processes depend on only one energy parameter, the activation energy for surface diffusion, Ed. For Pt on NaCI the analysis gives E d = (0.45 + 0.05) eV. When desorption occurs, the adsorption energy Ea becomes significant; at low supersaturations dissociation involves other energy terms. Values of the energy parameters found for Ag on C and Bi on C are given in table 1. AE~ is the binding energy per atom of bulk condensate, obtained from vapour pressure dataa). Where three-figure values are quoted, in the text and figures and in table 1, this is to satisfy a specific relationship, such TABLE I

Values of energy parameters (eV) System

Ea

Ea - Ed

Ed

Ed2

Eao

.4E2 + Ea

AEz

AE.j + Ea

AE3

AE~

Ag on C Bi on C

0.95 0.7

0.55 0.25

0.4 0.45

0.7 0.65

1 0.85

1.95 1.5

1 0.8

2.15 1.5

1.2 0.8

2.8 1.9

as a critical size transition. The parameters are extensively inter-related and depend on the values of vibrational frequency vo and vl, the adsorption site density no, the shape factor M, and on the thermal accommodation coefficient ~, as well as on the experimental data. Consequently the absolute o/ values are only reliable to perhaps +20~o. However, used with Vo= vI = 1012 s e c - ' , n o = l0 is cm -2 and ~ = 1 they provide a mutually consistent set of energy parameters. The values given for Ed2 and Edp for Ag on C do not apply above about 820 °K. The values of E d are of particular interest. It has been usual to assume, for example Ed~E~ (ref. 3) or Ed=¼Ea (ref. 4) but we now find that for Ag and Bi on C Ed~-½E~. The former analyses of these experiments gave lower values of E a as a direct consequence of the assumed low value of E d. The quantitative treatment of Poppa's detailed measurements of nucleation and growth behaviour, has shown that the mobility of stable Ag and Bi nuclei is significant at high temperatures. There is no reason to suppose that Ag and Bi are unusual in this respect, so that limitation of nucleation

306

i~.

I.EWIS

density by stable nucleus mobility can be expected to operate universally at high temperatures. Hence saturation occurs at a density t/sat which is lower than n,,,xin fig. 6 of Part I; the width of the b a n d between n~, and n,,~, is roughly one order of magnitude. Since there are no analytical solutions of the equations, the only satisfactory method of evaluating material parameters is by c o m p a r i s o n of experimental and calculated results. Copies of the c o m p u t e r p r o g r a m m e s used here are available from the author. Experimental data of the variation of n u m b e r s and sizes of nuclei with time, for several incidence rates and temperatures, is desirable for the simple cases and essential for the more complex cases.

Acknowledgements The a u t h o r is again grateful to Dr. M. R. J o r d a n and Mr. R. W. Allen for their assistance. The work was supported by the Ministry of Technology and is published by permission of the Plessey C o m p a n y .

References I) 2) 3) 4) 5) 6) 7) 8) 9) 10)

B. Lewis, Surface Sci. 21 (1970) 273. G. G. Sumner, Phil. Mag. 12 (1965) 767. H. Poppa, J. Appl. Phys. 38 (1967) 3883. B. Lewis, J. Appl. Phys. 41 (1970) 30. V. Halpern, J. Appl. Phys. 40 (1969) 4627. B. Lewis and D. S. Campbell, J. Vacuum Sci. Technol. 4 (1967) 209. R. Logan, Thin Solid Films 3 (1969) 59. E. Gillet and M. Gillet, Compt. Rend. (Paris) 262 (1966) 359. J. L. Robins and T. N. Rhodin, Surface Sci. 7. (1964) 346. G. Verhaegan, F. E. Stafford, P. Goldfinger and M. Ackermann, Trans. Faraday Soc. 58 (1962) 1926. 1 I) G. I. Distler and V. P. Vlasov, Thin Solid Films 3 (1969) 333. 12) G. A. Bassett, in: Proc. Intern. Syrup. on Condensation and Evaporation o f Solids, Eds. E. Rutner, P. Goldfinger and J. P. Hirth (Gordon and Breach, New York, 1964) p. 599. 13) N. T. Gladkich, R. Niedermayer and K. Spiegel, Phys. Status Solidi 15 (1966) 181.