Rate and diffusion analyses of surface processes

J. Phys. Chem. Solids Vol. 55, No. IO, pp. 95S964, 1994 Copyright 0 1994 Elwvier Science Ltd Printed-in &eat Britain. All rights reserved tM22-3697/94 $7.00 + 0.00

Pergamon

RATE AND DIFFUSION ANALYSES SURFACE PROCESSES J. A. VENABLES,?

R. PERSAUD,

F. L. METCALFEJ

OF

R. H. MILNE

and M. AZIM School of Mathematical and Physical Sciences, University of Sussex, Brighton BNl 9QH, U.K. Abstract-The atomic processes involved in crystal growth and surface diffusion can be modelled using coupled rate and diffusion equations. Comparison of these models with selected microscopic experiments, performed as a function of deposition at temperature r,, and of annealing times at temperature T,, can be used to measure effective diffusion coefficients, from which activation energies for adsorption, diffusion and binding processes can be extracted. These energies can, in favourable cases, be compared with theory, leading to new insights into interatomic forces at surfaces. Some recent examples relating to metal-metal and metal-semiconductor systems are discussed. Keywords: A. surfaces, A. thin films, B. epitaxial growth, C. electron microscopy, D. diffusion.

1. INTRODUCTION AND BACKGROUND INFORMATION

Atomic and molecular processes which occur at surfaces are important in a wide range of materials technologies. These include the growth of thin film devices, chemical and catalytic reactions. When the materials systems studied and the experimental conditions used are “simple” enough in a scientific sense, detailed microscopic studies offer the prospect of substantial advances in our understanding of atomic processes at surfaces. Our group has concentrated on this aspect, aiming to describe the surface processes involved in thin film growth, particularly in atomistic terms, and to extract quantitative atomic level information from well defined experimental studies. Recent reviews have emphasised this approach [l, 21. Major advances in our ability to visualise surface atomic structures have been made over the last thirty years [3]. Most obviously this has been due to the revolutionary development of Scanning Tunnelling Microscopy @TM) [4], but it is also due to the evolutionary development of several other types of Ultra-High Vacuum (UHV) microscopies [5], t Also at: Dept of Physics and Astronomy, Arizona State University, Tempe, AZ 85287, U.S.A. $Present Address: Unicam Analytical Ltd, York St, Cambridge CBl 2PX, U.K. This article was presented at MRS Boston, symposium AA, November 1993 and is dedicated to Bob BallutIi on the occasion of his 70th Birthday. John Venables recognises his scientific contributions, and thanks Bob for his support and friendship during and after John’s post-doctoral period in Illinois some 30 years ago.

including Low Energy Electron Microscopy (LEEM) [6] and Field Ion Microscopy (FIM) [7]. Our own work has concentrated on the development of UHV Scanning (Transmission) Electron MicroscopySEM and STEM [8,9]. In particular, we have shown in a variety of circumstances [9-121, that the secondary electron signal is sensitive at the sub-monolayer (ML) level, especially when the sample is negatively biassed by a few hundred volts relative to its surroundings. This simple technique, biassed secondary electron imaging (b-SEI), combined with deposition through a mask in a UHV-SEM and other micro-analytic surface techniques, has been used to study the early stages of crystal growth and surface diffusion in a few deposition systems; these include and Ge(ll1) AgW(ll0) PO, 131, Ag/Si(lll) [ll, 14, 151, Ge/Si(lOO) [12] and Cs/Si(lOO) [16]. In order to abstract atomic parameters from such experiments, it is necessary that the experimental geometry be well defined, and that the experiment be performed with well controlled independent variables. In our case, using vacuum evaporation similar to Molecular Beam Epitaxy (MBE) conditions, these are the deposition rate (R, in ML/min), the substrate temperatures during deposition (7’,) and/or annealing (T,), and the deposition and annealing times (t). The dose of deposit material, 0 = Rt, is measured by a quartz oscillator, the thickness of layers estimated by Auger Electron Spectroscopy (AES), and the crystallography by Reflection High Energy Electron Diffraction (RHEED). Nucleation densities are measured by counting islands observed in the SEM.

955

J. A. VENABLES

956

This paper concentrates on the rate and diffusion equation modelling that has been used as an intermediate step in the process of extracting atomic energies. These equations are useful because, unlike direct atomistic techniques such as Monte Carlo or Molecular Dynamics, they can cope with the huge range of length and time scales involved; they are also much less demanding in programming and computer time. The disadvantage is that specific, and almost certainly considerably oversimplified, models must be used. In particular, to date we have assumed that pre-exponential frequency factors are “reasonable” in the sense that values close to the Debye frequency, consistent with the experimental vapour pressure of the deposit, have been used [17]. This then allows us to use the models, in comparison with experiments of the type described above, to extract values of the adsorption (E,), diffusion (Ed) and pair binding (E,,) energies, with an accuracy around 0.05-0.1 eV. This level of accuracy is sufficiently good to be interesting as a check on theories of atomic binding at surfaces.

2. ELEMENTS OF RATE AND DIFFUSION EQUATION MODELLING

2.1. Nucleation

and competitive

capture

Atomistic theories of nucleation, and the concept of “competitive capture” of adatoms, have been emphasised in recent papers and reviews [2, 17, 181. The value of concentrating on the fate of the adatom is that we can write a simple rate equation for the rate of change of the adatom concentration, n,, during deposition at a rate, R: dn,/dt = R - n, /z,

(1)

where the inverse adatom lifetime, r-i, can be regarded as a sum over different inverse lifetimes for all competing processes. The analogy is with resistances adding in parallel, or with people leaving a football

et al.

match by different routes; all the adatoms have to go The individual atomic processes can be linear or non-linear in the adatom concentration, n, . An example which has been discussed extensively is the case of island nucleation; this is illustrated schematically in Fig. 1. Here, we consider adatom adand de-sorption (evaporation) with a stay time r,, nucleation with a lifetime r,, and capture of adatoms by “stable” clusters 7c. Then we can write somewhere.

7-I =7-1a

+7,‘+7;‘,

(2)

and can propose specific dependencies of the various lifetimes on atomic processes. At the highest temperatures, adatoms only stay on the surface for a short time 7., during which time they migrate over the surface with diffusion coefficient D. This time is determined by the adsorption energy, E,, and is conventionally written as 7, -’

=

v,

exp( - E,/RT),

(3)

where v is an atomic vibration frequency, of order 1-10THz. A simple expression for the diffusion constant appropriate to two dimensional surface diffusion, in terms of the diffusion energy Ed and frequency vd (typically somewhat less than v,), is D = (vda2/4)exp( - E,/RT),

(4)

where a is the jump distance, of the order of the surface repeat distance, e.g. 0.2-0.5 nm. The number of substrate sites visited by an adatom in time 7, is Dt,/N,, where N, is the area1 density of such sites, of the same order as a -2. The rms displacement of the adatom from the arrival site before evaporation is /, = (D7,)li2 N u(vd/v,)1i2exp{(E,

- E,)/2RT}.

,,Evaporation(?.$

Arrival03) /

Fig. 1. Schematic illustration of the interaction between the nucleation and growth stages. The adatom density H, determines the critical cluster density n,; however, n, is itself determined by the arrival rate R in conjunction with the various loss processes described in the text. [From Ref. 17.1

(5)

Surface processes Since typically E, is several times Ed, (/,/a) can be large at suitably low temperatures. Then, in their migration over the surface, the adatoms will encounter other atoms. Depending on the size of the binding energy between these atoms, and on their area1 density n, , they will form small clusters, which may then grow to form large clusters of atoms on the surface, in the form of 2D or 3D islands. This binding energy between a pair of atoms, &, and the energy of the critical cluster, E,, are centrally important to the understanding of nucleation and growth processes on surfaces. The rate limiting nucleation step is the formation of “critical nuclei” of size i, which is defined as the size which is more likely to grow than decay. As explained in more detail elsewhere [l, 171, we can apply statistical mechanics to calculate the density of critical nuclei, ni. The atomistic expression, in terms of the single adatom density n, , and the (free) energy of the critical cluster, E,, is nilNo = C,(n, IN,,)’ exp(Ei/kT),

(6)

where C, is a statistical weight of order l-10. We can follow the density of clusters containing j atoms, nj, by writing coupled differential equations describing the rate at which nj changes due to the various processes taking place on the surface. For j > 1, these are of the form dn,/dt = U,_, - U,,

(7)

where Uj is the net rate of capture of adatoms by j-sized clusters. If only single adatoms are mobile, we may write U, as the difference between a “diffusion capture” and a “decay” term, as q = a,Dn, n, _ , - n,(v, exp{ - (AE, + Ed)/kT}). (8)

Here, uj is a “capture number” and AEj = Ej - ,?_ , . If we substitute for the diffusion coefficient D from eqn (4), and put Uj = 0 for all j < i, then by repeated application of eqn (8) we obtain eqn (6) for the density of i-sized clusters. This corresponds to the condition of “local equilibrium” between n, and n,, illustrated in Fig. 1. Equation (8) is a reminder that we need to be careful about including “back reactions” in our model, since failure to do so can result in inconsistencies with the thermodynamic limits. It is inconvenient to consider each individual atomic size for large clusters. We can therefore group clusters together, within a given size range, or even

951

more drastically, lump all clusters j > i together as “stable” clusters of density n,, resulting in dn,/dt = Vi = aiDn,ni -(loss

terms of order n, or higher).

(9)

The initial evolution of the “nucleation density”, can therefore be obtained from the coupled eqns (1) and (9). In particular, the time r, represents the loss of adatoms to stable clusters. This can be formulated as a diffusion problem which can be solved in various approximations, resulting in n,(t),

r, -’ = n,Dn,,

(10)

where the capture number rrXis typically of order 5-10 [l]. To complete these coupled equations, we need to subtract the coalescence rate, U,, from eqn (9), and to express it in terms of the proportion of substrate, Z, covered by islands. Using U, = 2n,dZ/dt, with dZ/dt given by the cluster growth term [n, /tc, eqn (lo)], and the shape of the clusters, then we can derive general expressions for the maximum cluster density. These equations for the maximum cluster density are of the form n, - (R/v)~ exp(ElkT),

(11)

where p and E are tabulated elsewhere for 2D and 3D clusters in various condensation regimes [l, 171. For the cases considered here, 2D clusters and “complete” condensation, where re-evaporation is negligible, are most important. In that case, p = i/(i + 2) and E = (E, + iEd)/(i + 2). The detailed computations reported are for a “pair binding model” where the energy E, is expressed in terms of the pair binding energy Eb and the numbers of lateral “bonds” in 2D, quasi-hexagonal nuclei. The calculations have been carried through explicitly using the Einstein model of the lattice vibrations with fixed frequency factors [13, 14, 171. The calculations thus form a three-parameter fit to experimental nucleation density data obtained as a function of R and T, involving Eb and Ed, and at the highest temperatures, Ea. These nucleation calculations have been particularly informative for the case of Ag growth in the Stranski-Krastanov mode on W(llO), where Eb= 0.25 f 0.05, Ed = 0.15 9 0.10, E, = 2.2 f 0.10 eV have been measured, effectively for Ag/Ag(l 11). These values agree closely with “Effective Medium” theory, and clearly demonstrate the non-linear variation of metallic binding with co-ordination number [ 131.

958

J. A. VENABLES

More detailed calculations of the cluster size distribution could be obtained from a more extensive set of coupled equations of the form (7); some specific cases have been attempted by other authors [19]. It remains to be seen whether more atomic energies can be deduced in particular materials systems as a result.

2.2. Deposition on jinite patches Experiments involving deposition through a mask allow us to study lateral diffusion of adatoms, and the competition with evaporation and nucleation. The combined rate-diffusion equation for the adatom density can be easily generated from eqn (1) as 84

6.

t)lat

=

g

-

4

6,

tY7

+

V . [De@,

t)Vn,

(r,

t)l, (12)

where the source term g = R within the mask during deposition. The experimental observations typically consist of biassed secondary electron line-scans across the short dimension of a long, thin “patch” of typical dimension 100 pm x 20 pm. Some examples for Ag/Si(ll 1) are shown in Fig. 2 [lo]. Here one sees the increase in secondary electron yield due to the transformation of the Si(ll1)7 x 7 structure into the with layer the intermediate Ag-induced J3 x ,/3R30” (hereafter J3) structure, at deposition temperatures above around 200°C. Note also that the room temperature deposit also gives increased secondary electron emission, due to closely spaced islands on the unreconstructed 7 x 7 structure [4, 181. The diffusion term in eqn (12) corresponds to diffusion over quasi-macroscopic distances, and so includes the need for adatoms to surmount steps,

et al.

spend time in (immobile) small clusters, etc. Consequently, the diffusion coefficient is not necessarily the same as that in previous equations, where smaller distances are involved. This difference, between the “intrinsic” and “chemical” diffusion coefficients, can be explored using a model developed to describe diffusing point defects [20]. There it has been shown that, under the same assumptions of local equilibrium which were used to derive eqn (6), (13) This D, is now position and time dependent, via the cluster density nj (r, t). The simplest solution to eqn (12) is the ID steady state solution at high temperatures, when 7 = 7a and D, = D, due to the low adatom concentration and the absence of nucleation; this solution can be obtained analytically [l 11.In this case, the patch width, e(t), can be calculated from

I l(t)-fe,=

L

20

lJnl

I

electron line-scans for Ag/Si(l 11) under various deposition conditions. The contrast level is around 7% for 1 ML deposit. The irregular spikes on the bottom two curves correspond to scanning across Ag islands. [From Refs 10.1

Fig. 2. Biassed secondary

D(Vn,) *dt’, s 10

(14)

where the gradient is evaluated at (+)8(t), and to is the time required for the formation of the intermediate layer, whose coverage is 9 ML. This solution is only a function of the total dose (i.e. Rt in ML units) and a single diffusion length ed = (D7,)‘12. This diffusion length, which is the same as in the classic Burton-Cabrera-Frank (BCF) problem [21], is governed by the activation energy (E. - Ed)/2. Within the high temperature limit of the Einstein model we find

=

t---=-i

-;

(ukT/2nmv,v,)exp[(E,-

E,)/kT],

(15)

where the constant a = (a2No)/4. This version of eqn (5) was used for explicit calculations. The experimental results for Ag/Si(lll) shown in Fig. 3 correspond essentially to this situation for 770 < Td < 850 K. The patch width depends strongly on temperature, but not on R, nor on crystal orientation or imperfections. The use of a steady state formula is not limited to the BCF case, but can be used more generally for a constant, or effective, 7 consistent with eqn (1). The data requires that the effective diffusion distance, measured in the high temperature region, vary from around 4Opm at T = 750 K to 5 pm at 850 K. Using eqn (15) with the given pre-exponential factor, the high temperature diffusion data resulted in (E, - Ed) = 2.00 eV with an error around kO.05 eV [14].

Surface processes

-.* Eb=O.

A x +

1 OeV Eb=O.O%V R-0.43 ML/mio Rsm0.50 ML/m% R-1.41 ML/min

-\.

X %.%.\, -2._,

X

--..._, -.. X

201 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65

1000/T

(1/K)

Fig. 3. Width of the J3 “patch“ on AgjSi(I11) after a 5 ML dose through a 20 /tm wide mask at various deposition temperatures, Td, showing a peak at around 770 K, with the patch width around 100 &rn. Superimposed calculations for R = 1.41 ML/min, Es = 2.45, Ed = 0.45 eV and Eb as indicated [14].

The maximum in patch width in Fig. 3 at 750 K corresponds to the temperature at which nucleation of small clusters becomes important. However, no large islands were actually observed on these finite patches for T s- 705 K, although on an unmasked substrate, they were seen at the same deposition rate up to 790 K. This depression of the onset temperature is due to the long range diffusion lowering the adatom concentration. Since the nucleation rate given by eqn (9) is extremely sensitive to n, , the onset of nucleation depends critically on both Ed and the lateral binding energy Eb. These effects are illustrated in the computations of Fig. 4.

959

Figure 4 shows one set of computations for T=XlOK the deposition and rate R = 1.41 MLmin-’ corresponding to the best data set of Fig. 3. The curves correspond to dose increments of l/3 ML on top of the J3 layer, with an assumed 2j3 ML coverage, and the final curve is for 5 ML dose (including the 213 ML). The profiles of ni(x) are shown on the left hand, and n,(x) on the right hand panels. Computations where the energies used differed by no more than 0.05 eV gave significantly different results, due to the close balance of competing effects. As a result, values of Eb, Ed and .Ea were determined as 0.05 f 0.03, 0.40 r 0.05 and 2.45 t_ 0.05 eV, respectively [14]. These values remain to be explained theoretically. With these parameter values, the maximum value of n, at x = 0 is of order 0.1 ML. The above calculations have been performed using a diffusion coefficient, tt, in eqn (13), which includes the effects of clustering into pairs. If Eb and n, are substantial, there can be a sizeable difference between D and De,,, in a direction which slows up the diffusion, reduces the predicted patch width and promotes nucleation. The effect is most pronounced when the patch width is large, since this arises when the adatom density is not limited either by re-evaporation or nucleation. 2.3. D@iiion

during annealing

In the quantitative parts of the above discussion, we have assumed that clusters with size j > i are “stable”, and that the initial nucleation events occurred at random positions. These assumptions are good for relatively short times on defect free terraces, but as the deposition proceeds, and even more as the

Ea=2.45

eV

EdG0.45

eV

Eb-0.05

eV

Fig. 4. Calculated profiles during deposition for dose increments A@= l/3 ML, starting at 9 = 2/3 ML, for R = 1.41 ML min- ‘, T, = 700 K, and the energy values shown: (a) n, (x, 8); (b) n, (x, 0). The nucleation density values shown are just suiiicient to observe nucleation at the centre of the patch. The horizontal scale is the width of the patch in micrometres. See text for discussion [I4].

960

J.

A. VENABLES er al.

deposited film is left at elevated temperature for long times, a degree of “self-organisation” asserts itself. Small differences in free energy can make themselves felt via subtle effects. For example, during annealing after deposition g = 0 in eqn (12) nominally; but this equation can be generalised to include re-evaporation (in 2D, i.e. onto the adsorbed layer) of islands. This corresponds to having a further source term nle/7= when large islands are present, whose sublimation energy is L,, where nl, = K exp{-_(& - &)/kT},

(16)

with K as an entropy factor, of order l-10. The value of n,, will be small, except during annealing at high temperatures, unless (& - E,) is unexpectedly small. A similar term occurs in Ostwald ripening and other coarsening processes, where we require that the sublimation energy, L, becomes size dependent; large clusters grow, and small clusters disappear, via exchange of adatoms. This is the Gibbs-Thomson effect on the (2D) vapour pressure of the island, and depends on the island radius r, in contact with the substrate as n,,(r) = n,,(l + 2yRlkTr),

(17)

where R is the atomic volume of the deposit, y is a suitable surface energy, which depends on cluster shape, and the density n,, is as in eqn (16). When the 2D evaporation rate of the islands is diffusion, rather than interface limited, the rate of atom transfer is proportional to (n,,D), and the activation energy for evaporation of large islands is simply (L, - E, + Ed) [18]. As we are not considering size distributions of islands in this paper, the reduction in this activation energy due to the finite size of the islands is not explicitly included; eqn (16) adequately describes the annealing of large islands. The same approach can also be used to discuss the 2D evaporation of surface phases, as discussed in the next section. 3. EXAMPLES OF METAL-SEMICONDUCTOR SYSTEMS 3.1. Ag/Ge(l

11) in comparison

with Ag/Si(l

11)

Both these systems form an intermediate layer with the ,/3 structure, though the structures themselves are thought to be slightly different [22]. Lower coverage structures are also observed, both during adsorption and desorption of the J3 layer. A 3 x 1 structure is observed for Ag/Si(lll) and a 4 x 4 structure for Ag/Ge(l 11) in certain coverage and temperature ranges. There are, of course, also major

differences in the (111) surface reconstruction of the clean surfaces. Si(ll1) has the famous 7 x 7 structure over all the temperature range of interest, whereas Ge(ll1) transforms from the low temperature (c2 x 8) to a disordered 1 x 1 structure at around 300°C. The intruiging feature of a comparison between these two systems is therefore the contrast between the similarity of the bulk materials, and the major differences in surface properties. The exact metal coverage of these J3 structures has been controversial. Micro-AES measurements between the islands of the J3 Ag/Si( 111) layer have shown that 0 varies between 213 and 1 ML depending on deposition and annealing conditions [14]. This was explained in terms of competing kinetic processes taking place in the formation of the layer. A model was developed in which there is an activation barrier for the third atom (per unit cell, on average) to enter the J3 structure, E,, with the Ag atoms making a net gain in energy, E,, upon entering the ,/3 layer. Micro-AES intensities obtained as a function of Td, and of annealing temperature T, and time t, yielded a substantial E, = 1.90 f 0.05 eV and E, = 0.60 + 0.05 eV. Subsequent LEEM observations suggested that this temperature dependence was rather caused by the different phase formation routes [23]; direct from 7 x 7 to ,,/3, with a fine scale microstructure at lower temperatures, and via the 3 x 1 phase at higher temperatures. In this case, the value of E, may be related to the nucleation of the 3 x 1 phase, but the other conclusions are unchanged. The coverage of the J3 Ag/Ge(lll) has been measured as 0 = 0.85 + 0.04 ML [24] and 0.82 f 0.05 ML [25]. These values were obtained from AES break points, after deposition at temperature T, above 150°C. Such values have been confirmed by measuring the integrated intensity of the fractional order LEED spots as a function of Ag dose. The 4 x 4 faded to zero, and the ,/3 intensity saturated at 8 = 0.85 ML [26]. Our micro-AES 1.1 and measurements gave values between 1.Of 0.06 ML, decreasing slightly with increasing Td between 200 and 500°C. This suggests that kinetic barriers to forming the J3 layer are much lower on Ge than Si(ll1) [15]. No direct structural analyses of the Ag/Ge 4 x 4 or Ag/Si 3 x 1 structure have been reported, though the ideal coverages are thought to be l/4 and l/3 ML, respectively. The saturation coverage of the 4 x 4 has been calculated to be 0.27 f 0.03 ML from LEED intensity measurements [26]; although this phase was observed to extend from 0.17 to 0.35 ML, the 4 x 4 reached a maximum, and ,/3 reflections began to appear above l/4 ML [24]. In the following we assume that the coverage of the ,/3 phase in

Surface. processes Ag/Ge( 1 11), & = 1 ML, and that of the 4 x 4 phase,

961

(a>

8, = 0.25 ML. Here we comment on previously published diffusion results [15], obtained by depositing patches of Ag/Ge(l 11) at room temperature and annealing at elevated temperature, T,. The raw data is as shown in Fig. 5(a). On annealing when islands are present, the ,/3 layer broadens as t’j2. When the islands have evaporated (in 2D) and if T, is high enough, the diffusion front splits into two, with the lower coverage structure (the 4 x 4) moving ahead of the ,/3 phase. We can then calculate an effective diffusion constant, D,, from (e(t) - ei)* = 2D,(t - ti), 180

I

‘60 -

I

I

I

-

I

100

-

P _

P

-

r

0

10

20

30

40

t1/2

50

l

60

70

Fig. 6. Diffusion model for the analysis of Ag/Ge(lll) annealing: (a) when Ag islands are present; (b) when the ,/3 structure (coverage 0,) splits into the 4 x 4 structure, which has a lower coverage 04. See text for discussion. where /, and ti correspond to the initial values of C and f. Experiments with two different mask widths, and three different T,s gave the values of De plotted in Fig. 5(b). Assuming particular frequency factors, as explained below, activation energy values for the spreading of the ,/3 and 4 x 4 layers were determined

f

80

11, (0

I

35OT 20 /ml

a

03

:

(18)

140 120

: I (t)

as Q, = 0.78 f 0.04 eV, and Q4 = 0.87 + 0.04 eV, respectively.

80

(sl/2)

Fig. S(a) T (“C)

1o-’

The model for these diffusion processes can be understood in relation to Fig. 6(a). The mask width is 2[,. The islands, at high temperatures, and during the later stages of annealing, are very close to the centre of the patch. In this case, eqn (16) applies, with n,,(O, t) = n,,. Putting a sink for adatoms at x = l(t), the edge of the patch, we find the steady state solution

h

10-a

-i

-$

n, (x, t) = nle . sinh[(/(t) Inserting

nm

1

this solution

10-J 1.4

1.5

1.6

1000/T

1.7

1.8

(19)

in the flux eqn (14) gives

ocash(X)

lo-'0

- x)/e,]/sinh[~(t)/dd].

1.9

(K-l)

Fig. 5(b) Fig. 5. Experiments on Ag/Ge(lll): (a) Width of RT deposited patch, as a function of time t at annealing temperature T, = 350°C (v = inner width, ,/3 structure, V = outer width, 4 x 4 structure); (b) Effective diffusion coefficients deduced from outer and inner widths as a function of 1000/T, (lower scale); upper scale in “C. See text for discussion. [From Ref. 151.

- cosh(Xi)

= (n,,D/@;)(t

- ti),

(20)

where X = e(t)/, and Xi = e,/ed. Equation (20) is in the form of a diffusion length equation, with an effective diffusion constant (rq,D/O,). This equation shows clearly that, in the low temperature limit where the dependence on ed disappears, the patch broadening is determined by n,,D, with an activation energy Q,, for large islands, given by (& - E, + Ed). This model gives a good fit to the data with an activation energy of 0.78 + 0.04 eV; the corresponding value for Ag/Si(l 11) is 0.75 + 0.05 eV [14]. Thus with the Ag sublimation energy L, = 2.95 eV, we have an estimate

J. A. VENABLES et al.

962

of (E, - E& = 2.17 f 0.05 eV for Ge, compared with 2.20 + 0.05 eV for Si. Thus the diffusion length e,, , and its activation energy (E, - E,)/2, are very similar for the ,/3 structures in Ag/Ge and Ag/Si(l I 1). These estimates are in fact upper bounds, because the Ag islands are actually quite small, with a binding energy L somewhat less than L,. The best estimate of (E, - Ed)> for Ag/Si(l 11) is 2.05 f 0.05 eV [14]. We can use eqn (17) to estimate the radius of the islands necessary to cause a lowering of the binding energy, L, by 0.1 eV. Using a value of y of order 1 J m-*, r is found to be around 1 nm. Thus the islands do have to be really small to affect the above arguments significantly. But such small islands will undoubtedly coarsen during annealing at T, around 700 K. The breakup of the ,,/3 layer to form the 4 x 4 layer follows an equation similar to (20) where it can be shown [ 151, using the conservation of Ag adatoms in complete condensation, that

where ti corresponds to the time where the (extrapolated) straight-line segments on Fig. 5(a) cross. The effective diffusion coefficient is now D, = n,,D/t?,, where n,4 is the equilibrium adatom concentration on the 4 x 4, 0, is an effective coverage = 0,(9, - &)/Q), and the activation energy Q4 = L, - (E, - Ed)&. This energy is determined to be 0.87 f 0.04 eV. Further deductions are presently hampered by lack of precise knowledge of the binding energy of the ,/3 phase, L3. A previous model, based on isothermal spectroscopy [24], gave L, = desorption 3.30 + 0.13 eV, but with a large, and possibly unrealistic prefactor. This means that we have determined (E, - Ed)4 to be less than 2.43 + 0.14 eV, and that this is a true upper bound. We are currently modelling the high temperature deposition data, and hope that this will give closer bounds on these values. It is certain, as has been noticed previously [25], that diffusion over the 4 x 4 structure is very rapid. The above

analysis shows that this could be due either to a low adatom diffusion energy (&) or to a high adatom concentration, ni4, which is caused by a low activation energy (L - Ea)4 in eqn (18). 3.2. Caesium on silicon (100) The b-SE1 technique has been used to study the diffusion of Cs on Si(lO0) with the 2 x 1 reconstruction. In this case, the large change in work function (Acp = 3.7 eV between 8 = 0 and 0.5 ML) makes bSE1 extremely sensitive, allowing coverage changes down to 0.5% ML to be detected readily. By correlating the SE signal with the known work function change, concentration profiles across the surface can be obtained. Patches of Cs, 100pm square, corresponding to coverages of 0.09, 0.13, 0.23,0.32 and 0.44 ML, have been deposited at room temperature [16]. The patches were stable at room temperature, but diffusion occurred when the samples were annealed between 50 and 100°C. Above 100°C desorption effects were noticed, which restricted the usable temperature range. For initial concentrations < l/6 ML, the diffusion profiles were fairly linear. Above this concentration the profiles showed wings that were fairly linear below l/l2 ML. Some diffusion profiles at 6O”C, for initial concentrations of 0.13 and 0.23 ML, are shown in Fig. 7. If the width of the patch is plotted against tliz, straight lines are obtained, as expected for classical diffusion. However, for initial coverages between l/6 and l/3 ML, the slopes of the lines are almost the same at the same annealing temperature; i.e. the diffusion is almost independent of the patch concentration. How can we model such results? For coverages less than 0.5 ML, it is generally accepted that the Cs atoms occupy the pedestal sites, at the centre of two Si dimers, and that the atoms avoid the sites between the dimer rows, which have a much reduced adsorption energy [27,28]. LEED results show that there is a stable structure at If6 ML, where every third pedestal site is occupied [29]. Below

(4

t=15

min

1 t=15

min

Fig. 7. Biassed secondary electron linescans for Cs/Si(lOO),before and after annealing at 4O*Cfor the times indicated. Coverage 8 =(a) 0. I3 ML; (b) 0.23 ML. The contrast level for the higher coverage is around 90%. Scale ticks = 50pm. [From Ref. 16.)

963

Surface processes 0.10,

(b)

I

0.1

0.0

I

0.2

0.3

0.4

5

0.5

X Fig. 8. Diffusion profiles for Cs/Si(lOO) simulated by a two-phase model with (a) fast diffusion in the dilute phase (coverage BD< Ii6 ML); (b) a concentrated phase (coverage 0,. which breaks up to supply atoms for the faster diffusing dilute phase. See text for discussion. this

coverage,

no

ordered

structures

have

been

observed at RT; at higher coverages the next ordered structure probably consists of pairs of Cs adatoms separated by one unoccupied site. The interaction between adatoms for these two stable phases will be considerably different. For the l/6 ML structure there will be strong dipole-dipole repulsion. But when two adatoms are adjacent, the repulsive forces will be considerably

diminished

by depolaris-

ation. These general considerations allow the experimental data to be interpreted in terms of two “phases”. In the dilute phase, there are at least two empty spaces between adatoms, and the repulsive forces give rise to a large diffusion coefficient. The second to the formation of chains, “phase” corresponds consisting of at least two adatoms in neighbouring sites. The dilute phase will dominate the diffusion, with the chains tending to break up, so acting as a source for the dilute phase. As outlined below, this model explains the large diffusion coefficient, and its lack of dependence on coverage. For coverages above 5 113 ML, the breakup of the chains becomes the rate-limiting

step

and

the amount

of diffusion

is

reduced. However, if the concentration is below l/6ML, there is no source of atoms from the denser phase, so the linear diffusion profile needs to be explained in terms of the interaction between the adatoms. It can be shown, using a Boltzmann-Matano analysis, that a linear profile requires a concentration-dependent diffusion coefficient of the form D(6) a O(l - 6). This form also arises from thermodynamic arguments, where we additionally assume that the adsorption energy Ea decreases linearly with concentration. If we follow the reasoning given by Reed and Ehrlich [30], with E,(0) = E,(O) - A6, then D(0) = D, + Dr(A/kT)[Ql

- 0)],

(22)

where D, is the intrinsic adatom diffusion coefficient, valid at low coverage, and the enhanced diffusion is a result of two factors: (1) the thermodynamic driving force, where diffusion is biassed because lower coverages have higher adsorption energy (A /kT); (2) the lowering (if any) of the activation energy for motion which would give D, > D,. A simple two phase model is being explored which has the above characteristics. As shown in Fig. 8, this explains the main features of the data obtained so far. The details of this calculation and the full range of experimental data will be published elsewhere. Once we are convinced that this model is sufficiently general to describe the form of all the experimental results, it will be useful to produce a detained microscopic explanation from which the various activation energies can be determined.

4. DISCUS!5lON AND CONCLUSIONS This paper has briefly described the ingredients of rate and diffusion equation modelling as applied to atomic processes involved in nucleation, growth and surface

diffusion.

Examples

from previous

work on

Ag/W( 110) and Ag/Si( 111) were used to illustrate the use of these equations, when applied to UHV-SEM experiments, to deduce accurate values for adsot-ption, diffusion and binding energies, E,, E,,and E,, with a precision around 0.054 I eV. At this level, the results form a useful check on our understanding of atomic interactions at surfaces. Work in progress on two systems, Ag/Ge( 111) and Cs/Si(lOO), was described. The Ag/Ge(l II) study shows that a detailed comparison can be made with Ag/Si(l 1I), which has very similar material properties, but very different surface structures. The great difference between deposition at elevated temperature, T,,and RT deposition followed by annealing at elevated temperature T,, has been demonstrated.

964

J. A. VENABLES et al.

During deposition, the adatom concentration can be high, especially if evaporation and nucleation are difficult. During annealing, the adatom concentration is typically much lower, and is determined by the binding energy of islands of more strongly bound surface phases. The study of Cs/Si(lOO) shows that the biassed secondary electron technique used is extremely sensitive, and that coverages down to 0.5% ML can be detected easily. Diffusion of layers below 0.5 ML has been studied, and evidence of repulsive forces between the Cs adatoms obtained from straight-line profiles. Analytic arguments are given, which aliow us to abstract effective diffusion coefficients, and express their activation energies in simple form. However, further development of coupled rate and diffusion equations is needed to study situations, exemplified in these two systems and others, where more than one phase, or diffusion channel, is present simultaneously on the surface. are grateful to the organisers of the Robert W. Balluffii Fest for the invitation to present this paper, and to the SERC for supporting the work. M. Azim thanks the Government of the Islamic Republic of Pakistan for financial support. Acknowledgements-We

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Futamoto M., Hanbucken- &%.,&land C. J.,‘ Jones G. W. and Venabla J. A., Surface Sci. 150,430 (1985); Jones G. W. and Venabl& J.” A., Ultram&rosc~py l& 439 (1985). 11. Doust T. N., Metcalfe F. L. and Venables J. A., Ultramicroscopy 31, 116 (1989); Venables J. A., Doust T. N. and Kariotis R., Proc. M.R.S. Symp. 94,3 (1987);

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