PHY 598 (Venables) Sect E2
## Notes for PHY 598 Sect E2 (Venables)

### © *Arizona Board of Regents for Arizona State University
and John A. Venables*

**Click to download this document in Microsoft Word 6.0 Format**
Lecture notes by John A. Venables. Lectures given 28 March 96
and updated 12 Dec 96. I then gave a series of lectures at EPFL
in the Fall of '97, and EPFL Lecture #4
is on closely related topics.

## E2. Atomistic Models and Rate Equations

Rate Equations, Controlling Energies, and Simulations
We have considered simple rate equations for adatom
concentrations in section 1.3, and in problem 2, adding a
diffusion gradient in problem 3. Now we need to add non-linear
terms to describe clustering and nucleation of 2D or 3D
islands. These equations are governed primarily by energies,
which appear in exponentials, and also by frequency and
entropic preexponential factors.

This is the main advantage of such ‘mean field’ models.
They are known not to describe fluctuations very well, so
various quantities, such as size distributions of clusters,
are not described accurately. In current research, using fast
computational techniques such as ‘Kinetic Monte Carlo’ (KMC),
the early stages can be simulated on moderate size lattices.
These KMC ‘experiments’ using the same assumptions can then
be used to check whether mean field treatments work for a
particular quantity. The CSSS seminar on April 15th is by
one of the leaders in this area, Dr Steve Bales (Sandia,
Livermore). This will be a good chance to compare these
different approaches.

The emergence of computer simulation as a third way between
experiment and theory is clearly a growth area of our time.
To make progress in this area, one has to start with the
simplest models, and stick with them until they are really
understood. You need to beware generating more heat than
light, and in particular of generating special cases which
may or may not be of real interest. Simulations can however
be very illuminating, and may suggest inputs for simple
models that one hadn’t thought of. Animations are immediately
appealing, and if Spielberg can do it, why shouldn’t we? The
problem lies only in the subsequent claims for correspondence
with reality; then a measure of self-discipline is needed,
both from the lecturer/writer and the listener/reader.

Elements of Rate Equation Models
But with the above provisos, here goes with the atomistic
models. We consider rate equations for the various sized
clusters and then try to simplify them. If only isolated
adatoms are mobile on the surface, we have

nucleus size. In its simplest form, this means that a) we
can consider all clusters of size > i to be ‘stable’, in
that another adatom usually arrives before the clusters
(on average) decay; the reverse is true for clusters of
below critical size; b) these subcritical clusters are in
local equilibrium with the adatom population.

The capture numbers are related to the size, stability and
spatial distribution of islands, and solving this problem
has caused a lot of words to be spilled; it isn’t over yet.
The simplest mean field model, which I and others worked on
long ago, and which several people are working on now,
considers a typical cluster of size k immersed in the
average density of islands of all sizes. Then one can set
up a diffusion equation for the adatom concentration in
the vicinity of the k-cluster (size specific), or x-cluster
(the average size cluster), which has a Bessel Function
solution.

Regimes of Condensation
Bales and coworkers have been studying this model using a
combination of rate equations and KMC techniques. So far the
comparison for i = 1 has been published (G.S. Bales and D.C.
Chrzan, Phys. Rev B50 (1994) 6057); other treatments are in
the pipeline. My 1987 PRB paper concentrated on including
vibrations in a self-consisent way, within the mean field
framework outlined above. In the last few years there have
been many related treatments by several groups, mostly in
response to the new UHV STM-based experimental results. We
describe some of these results in section E3.

Continue to section E3
Return to Lecture list