154 5 Surface processes in epitaxial growth

Table 5.1. Parameter dependencies of the maximum cluster density

Regime	3D islands	2D islands
Extreme incomplete	p52i/3	i
	E5(2/3)[Ei1(i11)Ea 2 Ed]	[Ei1(i11)Ea 2 Ed]
Initially incomplete	p52i/5	i/2
	E5(2/5)[Ei1iEa]	[Ei1iEa]
Complete	p5i/(i12.5)	i/(i12)
	E5(Ei1iEd)/(i12.5)	(Ei1iEd)/(i12)

about how surfactants might work, especially in relation to semiconductor growth; at this stage a surfactant can be thought of as any foreign species which remains at the surface as growth proceeds.

5.2.4General equations for the maximum cluster density

The ®nal question following this type of reasoning is to ask whether there is a general equation for the maximum cluster density, which yields these three regimes as limiting cases. The answer is yes, but the resulting equation is not especially simple. We can see that (5.11) will lead to a maximum in the stable cluster density at the point where the (positive) nucleation term is balanced by the (negative) coalescence term. At this point,

dnx/dt50 and the coverage of the substrate by islands Z5Z0. If we make substitutions for all the various terms, then we will obtain an explicit expression for nx(Z0). As a practical point, we can calculate the Z-dependence of nx, within each of the condensation regimes, and obtain pre-exponentials h(Z,i) for each of these regimes, as illustrated for

3D and 2D islands in complete condensation in ®gure 5.6. The following arguments are aided by the fact that these pre-exponentials, which multiply the parameter dependencies of table 5.1, are only weakly dependent on both Z and the critical size i.

Although the coverage Z0 depends on the formula chosen for the coalescence term Ucl, which in turn depends on the spatial correlations that develop during growth, none of this in¯uences the exponential terms in the equations. Here we use the coalescence

expression due to Vincent (1971), where Ucl52nxdZ/dt. The rest is algebra, starting from (5.11). Inserting the Walton equation (5.9) for ni, the steady state equation (5.13) for n1 (neglecting the nucleation term Rtn), and specializing to 2D islands, nx is then given, after considerable rearrangement, by

n	x	(g1r)i (Z	1r)5f (R/D)i {exp (E	/kT)}(s	Dt )i 11.	(5.14)
		0	i	x	a

The slowly varying numerical functions f and g involve the capture numbers si and sx. For 3D islands, nx is changed to nx3/2 on the left hand side of (5.14), and the constants change a little (Venables 1987).

A point that may have been misunderstood is the following: the arguments of this section have been carried through on the assumption that the critical nucleus size is i. The actual critical nucleus size is that which produces the lowest nucleation rate and density; it is only then that the critical nucleus is consistent with the highest free energy

5.2 Atomistic models and rate equations	155

Figure 5.6. Pre-exponential factors h(Z,i) in the complete condensation regime: (a) 3D islands for i51, 2 and 3 with sx evaluated in the lattice (full line) and uniform depletion (dashed line) approximations; (b) 2D islands for i51±20, with sx approximated by 4p/(2 lnZ) which is very close to the uniform depletion approximation (Venables et al. 1984, reproduced with permission).

DG( j), for j5i, as discussed in section 5.1.3. The critical nucleus size is thus determined self consistently as an output, not an input, of an iterative calculation for all feasible assumed critical sizes. In complete condensation the ratio r; ta/tc is much greater than both g and Z0, corresponding to adatom capture being much more probable than reevaporation. In the extreme incomplete regime both r and g,,Z0. In between we have Z0,r,g, where most cluster growth occurs by diVusive capture, at least initially. It is a straightforward exercise to check that these conditions on (5.14) lead to the parameter dependencies given in table 5.1; keeping track of all the pre-exponential terms requires patience, and cross checks with the original literature.

5.2.5Comments on individual treatments

The argument of the last section indicates that nucleation equations are only soluble if

we know the Ei to enter into (5.11) or (5.14); however, we have also noted that the value of i is itself determined implicitly. This means that the predictions are only explicit if

156 5 Surface processes in epitaxial growth

(a)	1000
θ)(s	100
	10	(iii)	(ii)	(i)
	1
	10 −5	10 −3		10 −1
		θ

(b)			θ =0.05
	1.0
)				θ =0.1
5
)(x10	0.5					θ=0.15
(θ
s
n
	0
	0	50	100	150	200	250	300	350
				s

Figure 5.7. Comparison of rate equation (solid lines) and KMC calculations (dashed lines) of

(a) the average number of atoms (sÅ) in a monolayer island, for the ratio (D/R)5(i) 105, (ii) 107 and (iii) 109; (b) the size distribution of islands Kns(u)L as a function of coverage u, for u50.05, 0.1 and 0.15 (after Bales & Chrzan 1994, reproduced with permission).

we can calculate, within the model, the binding energy Ej for all sizes j. This is the principal reason why a pair binding model is invoked: life is too complicated otherwise. For 2D clusters, this simpli®cation allows us to estimate Ej5bjEb, where bj is the number of lateral bonds in the cluster, each of strength Eb. However, by retaining Ea as the verti- cal bonding to the substrate, we have enough freedom to model large diVerences between vertical and lateral bonding, within a three-parameter ®t to nucleation and growth data. This feature of the pair binding model is important in giving us enough latitude to mirror, in the simplest way, the diVerent types of bonding which actually occur in the growth of one material on another.

In developing the above model (Venables 1987), vibrations were included in a selfconsisent way within the mean ®eld framework outlined above. It is very easy to construct a model which is inconsistent with the equilibrium vapor pressure of the deposit unless the vibrations are treated reasonably carefully. This paper builds on the Einstein model calculations which we have done as problems in chapter 1, and is the basis for subsequent model calculations described here.

In the past few years there have been many related treatments by several groups, mostly in response to the new UHV STM-based experimental results, which have studied nucleation and growth down to the sub-monolayer level. Recent papers include a detailed comparison of rate equations and KMC simulations for i51 in the complete condensation limit (Bales & Chrzan 1994). The KMC work is important for checking that the rate equation treatment works well for average quantities, such as the nucleation density, nx, or the average number of atoms in an island, wx, as shown in ®gure 5.7(a); but it also shows that the treatment does not do a good job on quantities such as size distributions, shown in ®gure 5.7(b), which are dependent on the local environment,