Chemistry Reference
In-Depth Information
Figure 6.13
Schematic illustration of a lattice model for diffusion of Ag atoms on Pd doped
Cu(100). The diagrams on the right show the fourfold surface sites in terms of the four surface
atoms defining the site. The section of the surface shown on the left includes two well separated
Pd atoms in the surface. The Pd atoms are located at the centers of the two grey squares in the
diagram on the left.
It is easiest to describe the kMC method by giving a specific example of an
algorithm for the system we have defined above. Let us imagine we have a
large area of the surface divided into fourfold sites of the two kinds shown
in Fig. 6.13 and that we have computed all the relevant rates. We will
denote the largest of all the rates in our catalog of rates as
k
max
. To begin,
we deposit
N
Ag atoms at random positions on the surface. Our kMC
algorithm is then the following:
1. Choose 1 Ag atom at random.
2. Choose a hopping direction (up, down, left, or right) at random. Using
our catalog of rates, look up the rate associated with this hop,
k
hop
.
3. Choose a random number,
1
, between 0 and 1.
4. If
1
,
k
hop
=k
max
, move the selected Ag atom in the selected direction.
Otherwise, do not move the atom.
5. Regardless of the outcome of the previous step, increment time by
Dt ¼
1
=
(4
Nk
max
).
6. Return to step 1.
We can verify that this algorithmwill generate a physically correct sequence
of events by thinking about two simple situations. First, if the surface is pure
Cu, then every local hopping rate has the same rate,
k
, so the hops in step 4 of
the algorithm would always be performed. The average time between hops for
any individual atom in this case will be
Dt ¼
1
4
k
. This means that the atoms
