Problems and projects for chapter 5	181

reconstruction which orders the Fe, Co or Ni islands (Voigtländer et al. 1991, Chambliss et al. 1991, Stroscio et al. 1992, Meyer et al. 1995, Tölkes et al. 1997), and Fe or Co/Cu(100), where subsurface and surface ML islands can co-exist (Kief & EgelhoV 1993, Chambliss & Johnson 1994, Healy et al. 1994). The size distributions of clusters nucleated on point defects have been studied by KMC; a broad distribution is found, with a high proportion of small islands, as shown earlier in ®gure 5.17(b) (Amar

&Family 1995, Zangwill & Kaxiras 1995, Bales 1996, Brune 1998).

All these cases take us back to ®gure 5.3, and the diYculty of making high quality

multilayers from A/B/A systems: if one interface is `good', typically an example of SK growth as described in this section, then the other interface is `bad'. These systems may formally be an example of island growth, but active participation of the substrate makes this classi®cation too naive, in some cases even at room temperature. Nuclei form by exchanging deposit and substrate atoms; clusters of deposited atoms start to form, and then tend to get covered by a substrate `skin'. Once one realizes what is happening on a microscopic scale, the evidence is already there in the classical surface science results, e.g. from AES as shown in ®gure 5.21. These data show that segregation of Ag to the surface already happens at the ML level at room temperature; at 250 °C there is widespread interdiVusion.

Further reading for chapter 5

King, D.A. & D.P. WoodruV (Eds.) (1997) Growth and Properties of Ultrathin Epitaxial Layers (The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis, Elsevier), 8.

Liu, W.K. & M.B. Santos (Eds.) (1999) Thin Films: Heteroepitaxial Systems (World Scienti®c).

Matthews, J.W. (Ed.) (1975) Epitaxial Growth, part B (Academic).

Tringides, M.C. (Ed.) (1997) Surface DiVusion: Atomistic and Collective Processes

(Plenum NATO ASI) B360.

Problems and projects for chapter 5

Problem 5.1. Growth laws and the condensation coef®cient

The rate equation (5.10) is a good approximation when the coverage of the substrate by islands, Z,,1. When Z is not so small, one might like to correct (5.13a) for direct impingement, by writing R(12 Z) in place of R. The condensation coeYcient is the ratio of the amount of material in the ®lm to the amount in the depositing ¯ux, and comes in two forms a(t) and b(t).

(a)Identify the terms in the modi®ed (5.10) which lead to the increase in size of clusters, and write down an expression for the cluster growth rate, in atoms per unit area per second.

1825 Surface processes in epitaxial growth

(b)Assuming 2D islands, express the instantaneous condensation coeYcient b(t) in terms of the rate of atom arrival and departure (per unit area per second) at time t.

(c)Use the two above expressions to derive the form of the integrated condensation coeYcient a(t), assuming deposition was started at t50.

(d)Using (5.11) in addition, we can now compute nx(t), a(t) and Z(t). However, as explained in the text, it is preferable to use Z as the independent variable. Compute

nx(Z), a(Z) and t(Z) for parameter values which illustrate the initially incomplete condensation regime for a critical nucleus size i51, and 1024,Z,0.5, and identify on the surface processes which dominate at diVerent values of Z.

Problem 5.2. Capture numbers and Bessel functions

The rate equation treatment of capture numbers needs an ancilliary diVusion equation with cylindrical symmetry. Consider the formulation of such an equation in order to understand how the solutions (5.12) arise for incomplete condensation (t5ta), and how they can be generalized to the more general case when growth also occurs (t215

ta211tc21).

(a)Express the adatom diVusion equation -n1/-t5D= 2n1 in cylindrical polar coordinates, and determine the simpli®ed equation which results when the solution does not depend on the angular variable u.

(b)Now consider a particular cluster with radius rk, centered at r50, and add the source and sink terms from the rate equation (5.13) for the adatom concentration.

Explore the relation between the resulting steady state equation for n1(r) outside the cluster and Bessel's equation.

(c)By considering the boundary condition at the edge of the cluster, show that the

concentration n1(r)5Rt(12 K0(Xk)/K0(X)), where the arguments X and Xk of the Bessel function K0 are as de®ned in the text following (5.12).

(d)Given that the derivative of K0(X)52 K1(X), derive (5.12) for the capture numbers sk and sx by considering the diVusion ¯ux J(r)52 D=n1(r) at the cluster boundary. Show this result is exact for small clusters when t5ta, and that it is a good approximation when t215ta211tc21. For complete condensation, show that the result for sx only depends on the island coverage Z.

Project 5.3. Step capture and diffusion barriers between layers

Adatoms being captured by steps can be formulated as in problem 1.3 by considering an individual terrace and the boundary conditions at the steps at either end. Consider some further 1D step capture problems along the following lines.

(a)The presence of an Ehrlich±Schwoebel barrier at a down step means that an adatom has a temperature-dependent probability to be re¯ected there. Formulate this problem and apply your equations to the data shown in ®gure 5.19. What surface processes determine the nucleation density N(x,T) shown, on the assump-

Problems and projects for chapter 5	183

tion of no intermixing between Au and Ag, and what value for the corresponding energies might you deduce from the comparison with experiment.

(b)Problems on a mesoscopic scale require a suitable mixture of atomistic and continuum modeling. One such problem is how to model the eVects of step capture on a scale large compared to the distance between the steps. Show that in this limit,

step capture contributes and extra characteristic inverse time ts to (5.13). Evaluate ts in terms of the step spacing d and the adatom diVusion coeYcient D to prove the limit given by (5.23), and see if it is valid in general.

(c)Use the results obtained from parts (a) and (b) to discuss the early stages of nucleation and growth of islands on a vicinal surface. Using such a formulation, discuss the occurrence of denuded zones parallel to the steps, and the reduction in nucleation density on vicinal surfaces. What are the eVects of step movement and island incorporation into steps at later stages?

Project 5.4. Clustering and intermixing during diffusion

The formulation of adatom diVusion in terms of hopping via (1.16) provides the simplest description of intrinsic diVusion, valid at low coverages. Consider some of the forms of the diVusion coeYcent which are appropriate to long range diVusion, valid at higher coverage and/or temperatures.

(a)The mass transport or chemical diVusion coeYcient D* is expressed here as n1D in (5.24). Consider the surface processes involved in Ostwald ripening of clusters, and

show that this expression is reasonable at low concentrations when only adatoms are mobile.

(b)At higher adatom concentrations some of the adatoms will spend part of the time in small clusters, size j, which have typically smaller (maybe zero) intrinsic diVusion

coeYcients, Dj. Show that in this case, the chemical diVusion coeYcient is concentration dependent, and is given by D*5oj j2nj Dj /{ oj j2nj}. Hence show, using the Walton relation (5.9) in its simplest form, that at non-zero concentrations D*

depends exponentially on Eb/kT as well as on Ed/kT, and may also depend on other energies due to diVusion of small clusters.

(c)At higher temperatures, surface vacancies are created in addition to adatoms, and at even higher temperatures the surface may become rough over several layers. In alloys we can have exchange diVusion with unlike species involved. Consider how these possibilities aVect the interpretation of D* in terms of individual surface (and near surface) processes.