5.3 Metal nucleation and growth on insulating substrates	157

i.e. on spatial correlations which develop during nucleation and growth. Many authors (e.g. Myers-Beaghton & Vvedensky 1991, Bartelt & Evans 1992, 1994, Amar & Family 1995, Mulheran & Blackman 1995, 1996, and Zangwill & Kaxiras 1995) have evaluated size distributions during simulations, and shown that they are characteristic both of the critical cluster size, i, and of the spatial correlations. Brune et al. (1999) have made a detailed comparison of the various approximations, including coalescence terms, for the case of 2D sub-monolayer growth.

5.3Metal nucleation and growth on insulating substrates

Some results of atomistic nucleation and growth models are described in the context of speci®c experimental examples for the remainder of this chapter. We concentrate on metal deposition, on insulators in this section and on metals in section 5.4. These examples illustrate the kinds of experimental tests to which atomistic models have been subjected over almost 30 years.

5.3.1Microscopy of island growth: metals on alkali halides

An example of the use of non-UHV TEM to study nucleation and growth is shown in ®gure 5.8 from the Robins group (Donohoe & Robins 1972, Venables & Price 1975). The deposition of Au/NaCl(001) was done in UHV, but the micrographs were obtained by (1) coating the deposit with carbon in UHV; (2) taking the sample out of the vacuum; (3) dissolving the substrate in water; and (4) examining the Au islands on the carbon by TEM. By this means island densities, growth, coalescence and nucleation on defect sites could all be observed. By performing many experiments at diVerent deposition rates R, and temperatures T, as a function of coverage, their group and others have produced quantitative data of island growth and rate equation models have been tested. It is clear that this type of technique is destructive of the sample: it is just as well that NaCl is not too expensive, and that gold/silver, etc. are relatively unreactive, or the technique would not be feasible.

The work on noble metals Ag and Au deposited onto alkali halides constitutes quite a long story which can be summarized roughly as follows. A full review of early work has been given by Venables et al. (1984) and by Robins (1988), including extensive tabulation of energy values deduced from experiment using the models described in section 5.2. Typically, these values were deduced by ®rst showing that the initial nucleation rate J at high temperatures varied as R2, and so corresponded to i51 in (5.11). In this regime where n15Rta we can see that

J5dn	/dt5s	Dn	2 which is proportional to R2exp{(2E	2 E	)/kT},	(5.15)
x	i	1	a	d

so the T-dependence of the nucleation rate yields (2Ea 2 Ed). This information can be combined with the low temperature (complete condensation) nucleation density, which for i51 yields Ed, as can be seen from table 5.1. An alternative piece of data is the island growth rate at high temperature, determined by the width of the BCF diVusion zone

158 5 Surface processes in epitaxial growth

(a)

(e)

(b)

(f)

(c)

(g)

(d)

(h)

Figure 5.8. TEM micrographs of Au/NaCl (001) formed at T5250°C, R5131013 atoms cm22 s21 and deposition times of (a) 0.5, (b) 1.5, (c) 4, (d) 8, (e) 10, (f) 15, (g) 30 and (h) 85 min (from Donohoe & Robins 1972, reproduced with permission).

around each island, which leads to (5.12) for the capture numbers. The growth rate thus has a T-dependence given by (Ea 2 Ed) via an equation which depends on the 2D or 3D shape of the islands.

The energies deduced depended a little on the exact mode of analysis, but were in the region Ea50.65±0.70 and Ed50.25±0.30 eV for Au/KCl. The corresponding quantities deduced for Au/NaCl were similar; for Ag/KCl they were around 0.5 and 0.2 eV,

5.3 Metal nucleation and growth on insulating substrates	159

and for Ag/NaCl 0.65 and 0.2 eV respectively (Stowell 1972, 1974, Venables 1973, Donohoe & Robins 1976, Venables et al. 1984, Table 2). These Ea and Ed values are much lower than the binding energy of pairs of Ag or Au atoms in free space, which are accurately known, having values 1.65 6 0.06 and 2.29 6 0.02 eV respectively (Gringerich et al. 1985). We can therefore see why we are dealing with island growth, and why the critical nucleus size is nearly always one atom. The Ag or Au adatoms reevaporate readily above room temperature, but if they meet another adatom they form a stable nucleus which grows by adatom capture. This type of behavior was observed for all metal/alkali halide combinations.

5.3.2Metals on insulators: checks and complications

The combination of experiment and model calculations presented in the previous section is satisfying, but is it correct? What do I mean by that? Well, the experiment may not be correct, in that there may be defects on the surface which act as preferred nucleation sites. It is very diYcult to tell, simply from looking at TEM pictures such as ®gure 5.8, whether the nuclei form at random on the terraces, or whether they are nucleated at defect sites; only nucleation along steps is obvious to the eye. In the previous section

we described the classic way to distinguish true random nucleation, with i51. But there are several other ways to get J,R2, including the creation of surface defects during deposition. In this case we might have nucleation on defects (i50), but with the defects produced in proportion to R by electron bombardment. Alkali halides are very sensitive to such eVects, which were subsequently shown to have played a role in early experiments (Usher & Robins 1987, Robins 1988, Venables 1997, 1999).

As substrate preparation and other experimental techniques improved, lower nucleation densities which saturated earlier in time were observed (Velfe et al. 1982). This has been associated with the reduction in impurities/point defects, and the mobility of small clusters. From detailed observations as a function of R, T and t, some energies for the motion of these clusters have been extracted. Qualitatively, it is easy to see that if all the stable adatom pairs move quickly to join pre-existing larger clusters, then there will be a major suppression of the nucleation rate (Venables 1973, Stowell 1974). This was studied intensively for Au/NaCl(001) by Gates & Robins (1987a), who found that a model involving both defect and cluster mobility parameters were needed to explain

the results of Usher & Robins (1987). The revised values of Ea and Ed for this system are given in table 5.2; in particular, (Ea 2 Ed) has been determined in several independent experiments to be 0.33 6 0.02 eV (Robins 1988, Venables 1994).

There are several further interesting experiments, including the study of alloy deposits, which has now been performed for three binary alloy pairs, formed from Ag, Au and Pd on NaCl(100) (Schmidt et al. 1990, Anton et al. 1990). In such experiments the atoms

with the higher value of Ea, namely Au in Ag±Au, or Pd in Pd±Ag and Pd±Au, form nuclei preferentially, and the composition of the growing ®lm is initially enriched in the

element which is most strongly bound to the substrate. The composition of the ®lms was measured by X-ray ¯uorescence and energy dispersive X-ray analysis, and only approached that of the sources at long times, or under complete condensation conditions.

160 5 Surface processes in epitaxial growth

Table 5.2. Values (in eV) of Ea and Ed of Ag, Au and Pd adatoms on NaCl(001)

Alloy dE	Element	(Ea 2 Ed)	Ea	Ed
Ag±Au: 0.11 6 0.03	Ag	0.22	0.41	0.19
Au±Pd: 0.12 6 0.03	Au	0.33 6 0.02	0.49 6 0.03	0.16 6 0.02
Pd±Ag: 0.25 6 0.05	Pd	0.45	0.78	0.33

Note: Values without errors are derived from data combinations (Robins 1988, Anton et al. 1990).

Table 5.3. Calculated values of Ea and Ed of Ag and Au on alkali halide(001) surfaces

Parameter (eV)	Ag/NaCl	Au/NaCl	Ag/NaF	Au/NaF
Ea	0.27 [0.27]	0.15 [0.69]	0.26	0.18 [0.59]
	(0.41)	(0.49)		(0.63)
Ed	0.15 [0.09]	0.07 [0.22]	0.24	0.14 [0.08]
	(0.19)	(0.16)		(0.08)
(Ea 2 Ed)	0.12 [0.18]	0.08 [0.47]	0.02	0.04 [0.51]
	(0.22)	(0.33 6 0.02)

Source: From Harding et al. 1998, with experimental values (round brackets) and previous calculations [square brackets].

These experiments can be analyzed to yield energy diVerences dE, where

dE5dEx 2 dEy, and dEx, dEy5(Ea 2 Ed)x,y for the two components. Values of dE have been obtained for the pairs, namely Au±Ag, 0.11 6 0.03; Pd±Au, 0.12 6 0.03; Pd±Ag,

0.256 0.05 eV. These experiments measure, very accurately, diVerences in integrated condensation coeYcients, ax,y(t), which are determined by the BCF diVusion distances of the corresponding adatoms, as explored in problem 5.1. Coupled with nucleation density measurements, the data give accurate values for Ea and Ed for these three elements on NaCl(100), as given in table 5.2.

More recently, eVorts have been made to understand some of these energy values in terms of metal±ionic crystal bonding. Earlier estimates maintained that a considerable part of the binding was of van der Waals type, but recent work has shown that this contribution was almost certainly overestimated. In particular, Harding et al. (1998) calculated the pairwise ion±ion interactions within the relativistic Dirac±Fock approximation, the metals Ag and Au being most attracted to the halide ion; the van der Waals energy was reduced, due to overlap and better values of dispersion constants, from 0.5 to around 0.15 eV/atom. Atomic polarization within the shell model was included, and the whole assembly relaxed to equilibrium, with a claimed accuracy of 6 0.1 eV. Some results are given in table 5.3. The results suggest that the calculated values of Ed are within the quoted accuracy, but that Ea seems to be systematically underestimated, possibly because of the neglect of charge transfer in the calculation. A Hartree±Fock cluster calculation (Mejías 1996) also obtains very low values, ,0.1 eV for Ea; so this

5.3 Metal nucleation and growth on insulating substrates	161

method presumably also gives very small values of Ed, which must by de®nition be less

than Ea.

What survives from previous work (Chan et al. 1977, Gates & Robins 1988) is that the metal±metal binding energies Eb are high, close to free space values, and that the diVusion energies of dimers and small clusters are very low, often as low as Ed itself. This arises because one can ®t the ®rst atom of a pair on the surface optimally, but the second one is constrained by being a member of a pair; the resulting energy surface, while quite complex, is less corrugated.

Extension of these calculations to point defects has been attempted (Harding et al. 1998), with the result that the noble metal adatoms are generally attracted to surface cation, but not anion, vacancies. This is especially the case if the adatom size is suYciently small to ®t inside the surface vacancy, the attractive energies increasing as the height of the adatom above the surface plane decreases. The role of surface charges, and their eVect on the charge state of vacancies, could be very important, as has been found for the case of Ag and other metals, particularly Pd, on MgO(001) (Ferrari & Pacchioni 1996).

5.3.3Defect-induced nucleation on oxides and ¯uorides

There are many examples in the literature where defect nucleation seems to be needed (HarsdorV 1982, 1984, Venables 1997, 1999). The transition from i50 to 1 was observed for Au/mica (Elliott 1974). In this case defect sites were used up initially, and nucleation then proceeded on the perfect terraces in the initially incomplete condensation regime. A more recent example is furnished by high resolution UHV-SEM obser-

vations of the growth of nanometer-sized Fe and Co particles on various CaF2 surfaces, typically thin ®lms on Si(111), as indicated in ®gure 5.9. In this work (Heim et al. 1996) the nucleation density, for Z0 > 0.2 close to the maximum density, was independent of temperature over the range 20±300°C. At 400°C and above the substrate

itself is unstable. This behavior is not understandable if nucleation occurs on defectfree terraces, but may be understood if defect trapping is strong enough.

Defects of various types can be incorporated into either analytical treatments or

simulations, at the cost of at least two additional parameters, the trap density nt, and the trap energy Et, or the binding energy of adatoms to steps Es, as indicated schematically in ®gure 5.10. The nucleation density of islands on defective substrates can be

derived by considering the origin of the various terms in (5.14). The right hand side of

this equation is proportional to the nucleation rate (via the term in exp(Ei/kT)), which is enhanced by a ratio Bt511At with defects present (Heim et al. 1996, Venables 1997).

We start by considering the point defect traps shown in ®gure 5.10(a), constructing a suitable diVerential equation for the number of adatoms attached to traps, n1t,

dn1t /dt5s1tDn1nte 2 n1tndexp(2(Et1Ed)/kT),

(5.16)

where nte is the number of empty traps5nt 2 n1t 2 nxt. In steady state, this equation is zero, and inserting (1.15) for D, we deduce

n1t/(nt 2 nxt )5A/(11A), with A5n1Ctexp(Et/kT),

(5.17)

162	5	Surface processes in epitaxial growth
		(a)	(c)
		20nm	20nm
		(b)	(d)
		20nm	20nm

Figure 5.9. Nucleation and growth of small Fe crystals on CaF2(10 nm)/Si(111) at (a) room temperature, (b) 140°C, (c) 300°C and (d) 400°C, observed by in situ high resolution SEM. The average thickness is between 3.1 and 3.6 ML, and the coverage of the substrate, Z,20% (from Heim et al. 1996, reproduced with permission).

where Ct is an entropic constant, which has been put equal to 1 in the illustrative calculations performed to date. Equation (5.17) shows that the traps are full (n1t5nt 2 nxt) in the strong trapping limit, whereas they depend exponentially on Et/kT in the weak trapping limit, as expected. This equation is a Langmuir-type isotherm (section 4.2.2) for the occupation of traps; the trapping time constant (tt, in analogy to 5.13) to reach this steady state is very short, unless Et is very large; but if Et is large, then all the traps are full anyway.

The total nucleation rate is the sum of the nucleation rate on the terraces and at the defects. The nucleation rate equation becomes, without coalescence, analogously to (5.11),

dnx /dt5siDn1ni1sitDn1nit,

(5.18)

where the second term is the nucleation rate on defects, and nit is the density of critical clusters attached to defects, si t being the corresponding capture number. In the simplest case where the traps only act on the ®rst atom which joins them, and entropic eVects are ignored, we have

At5 n1t /n15 (nt 2 nxt)A/[n1(11 A)].

(5.19)

5.3 Metal nucleation and growth on insulating substrates	163

n1
		nxt
n1t			n1(x)
nt ±n1t	±nxt	(b)	n	x	(x)
(a)

Figure 5.10. (a) Model for nucleation at attractive random point defects, which can be occupied by adatoms, density n1t, clusters, density nxt, or can be empty; (b) schematic diagram of line defects (steps), with adatom density n1(x) and nucleation density nx(x) for position x from up-steps (with attractive forces) and down-steps (maybe repulsive forces).

A high value of A gives strong trapping, in which almost all the sites unoccupied by clusters will be occupied by adatoms; in the simplest model we assume that clusters cannot leave the traps.

However, even the simplest behavior ensures that the defect processes are not linear. The clusters which form on the defect sites get established early on and thereby deplete the adatom density on the terraces. As a result, the overall nucleation density, which appears in the left hand side of (5.14), grows only as a fractional power [typically 1/(i12.5) for complete condensation, see table 5.1] of the trap density. In this weak trapping limit, the main eVect is the reduced diVusion constant D due to the time adatoms spend at traps (Frankl & Venables 1970). Nucleation on terrace sites is strongly suppressed, due to adatom capture by already nucleated clusters. But when nx.nt, there is little eVect on the overall nucleation density. These eVects result in the s-shaped curves shown in ®gure 5.11, illustrated for nt50.01 ML, Et50.5 and 1.0 eV, and Fe/CaF2(111) parameters. If the trapping is very strong, and the diVusion energy is low, there is a large regime where nx5nt; conversely, weak trapping will lead only to a point of in¯ection at, and a change of slope above and below, the trap density.

Comparison with experiments puts bounds on the energies Ea, Eb and Et, all > 1 eV, and suggests a low value, 0.1±0.3 eV, for Ed. Note that the reason why a low value of Ed is needed is so that the adatoms can migrate far enough at low temperatures to reach the defect sites. The high values of the other energies are needed, so that something else doesn't intervene at high temperatures. For example, if Et is as low as 0.5 eV, the density does not reach nt over a large enough temperature range; if Ea is too small, condensation becomes incomplete too early. Note also that the transition from i51 to 2 is observed for Ed50.1±0.3 eV at the highest temperature; this

164 5 Surface processes in epitaxial growth

	300 200		100		0		T (oC)		300 200				100		0		T (oC)
	14.5								14.5
	(a) Et		= 0.5 eV				0.6			(b) Et			= 1.0 eV				0.6
	14.0
									14.0
	13.5						0.5										0.5
	13.0						0.4		13.5
)
2
/cm																	0.4
x	12.5								13.0
(N
					i = 1
10		0.3
Log	12.0									0.3
									12.5
	11.5		0.1
										i	= 2
									12.0
	11.0
			Parameter: Ed (eV)							0.1
	i = 2													Parameter: Ed (eV)
	10.5	2.5		3.0	3.5	4.0	4.5	5.0	11.5	2.0		2.5		3.0	3.5	4.0	4.5	5.0
	2.0
			1000/T (K±1)										1000/T (K±1)

Figure 5.11. Nucleation density on point defects predicted with trap density nt50.01 ML, trap energy (a) Et50.5 eV, and (b) 1.0 eV, parameter Ed , with Ea51.16 eV and Eb51.04 eV (recalculated after Heim et al. 1996, and Venables 1999).

means that the limiting process can become breakup of the cluster (on a trap), rather than removal of the adatom from the trap; Et is not then itself important, provided it is high enough.

This type of model thus contains several sub-cases, depending on the values of the parameters. A remarkable example is Pd/MgO(001), studied with AFM by Haas et al. (2000); this data also requires a high trapping energy Et, in agreement with the calculations of Ferrari & Pacchioni (1996) and Venables & Harding (2000) for trapping of Pd in surface vacancies, and a low value of Ed. Models involving point defects are typically indicative, rather than truly quantitative, because of the possibility of other eVects, such as cluster mobility and cluster detaching from defects, and the possibility of a range of defect binding energies. One can see qualitatively that if Et is moderate and dimer motion is easy, then the traps may become reusable at high enough temperature; this further complication, needing even more parameters, has been thought necessary on occasion (Gates & Robins 1987a, Usher & Robins 1987). This is an ongoing tension between science and technology; both need conditions to be reproducible; technology can be successful even if complicated, but science needs the models to be relatively simple: we cannot sensibly deal with too many parameters at once.