Chemistry Reference
In-Depth Information
hop on the surface at the correct rate. Second, if an atom can hop out of a site
along two directions with different local rates,
k
high
and
k
low
, then the ratio of
hops along the fast direction to hops along the slow direction is
k
high
k
low
. The
same argument can be applied to two atoms sitting in different sites. These
ideas show that different processes occur with the correct ratio of rates and
that the absolute rate of the processes is correct—in other words this algorithm
defines physically correct dynamics. A kMC simulation is intrinsically sto-
chastic, so two simulations cannot be expected to generate the same sequence
of events, even if they begin from the same initial conditions. It is typically
necessary to average over the evolution of multiple simulations to get reliable
average information.
There are two important drawbacks of kMC simulations that should be kept
in mind. The first is related to the numerical efficiency of our algorithm. If the
catalog of rates includes a wide range of absolute rates, then the ratio
k
hop
=
/
k
max
in step 4 of the algorithm can be very small for the slowest rates. For example,
if the activation energies of the slowest and fastest rates are 1.0 and 0.2 eV and
the rates for these processes have the same prefactors, then this ratio is
10
2
13
at room temperature and
10
2
7
at 300
C. This means our algorithm is highly
inefficient; many millions of iterations of the algorithm may pass with no
change in the state of our system. This problem is relatively simple to solve
by using one of a variety of more sophisticated kMC algorithms that have
been developed over the years. The further reading section at the end of the
chapter can point you to literature about these methods.
The second drawback of kMC is more serious: kMC results can only be cor-
rect if the catalog of rates used to define the method is complete. To be specific,
let us return to the diffusing Ag atoms in the example above. Our algorithm is
well-defined if the Ag atoms on the surface are isolated, but what happens if
two Ag atoms are found in two neighboring surface sites? We need to add
information to our lattice model to define what can happen in this situation.
From a physical point of view, we do not want an Ag atom to hop into a
site that is already filled by another Ag atom. This is easy enough to achieve
by simply defining the hopping rate to be zero if the site the atom is hopping
into is already filled. But how fast will one of the Ag atoms hop away from an
adjacent pair of atoms? It is very likely that these rates are different (and lower)
than the rates for an isolated Ag atom on the surface since it is typically ener-
getically favorable for metal adatoms to group together on a surface. We could
include these rates in our kMC algorithm if we take the time to use DFT cal-
culations to compute the rates. You can see from Fig. 6.13 that two adjacent Ag
atoms can sit on a variety of sites. To be complete we would need to compute
the rates for each of these situations separately. Similarly, if we are considering
random doping of the surface by Pd atoms, then there will be locations on the
surface where two Pd atoms are adjacent in the surface layer. To account for
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