Kinetic Monte Carlo

The KMC algorithm

The kinetic Monte Carlo algorithm allows us to simulate the behaviour based on the microscopic kinetics.

The KMC algorithm is as follows:

For a given configuration a list of all possible events, n=1,...,N, is generated and the rate, r_n, for each event is calculated.
Exactly one event is selected, such that any event, n, is selected with probability
The seleted event is executed and time is advanced by

where u is a random number uniformly distributed in the interval [0,1[.

In our mechanism the possible events are the desorption of a A₂*-molecule and the diffusion of A₂* or A* to a neighbor site.

Usually the events are enumerated by looping over the sites in the lattice and one must be careful to list each event exactly once.

In the following we will need an algorithm to select a number, n, out of N with weight W_n or, equivalently, probability

Without loss of generality assume that the weights are non-zero. The selection can be implemented efficiently by storing
(s₁, s₂, ..., s_N)
where

We generate a random number, u, uniformly distributed in [0,1[ and the corresponding value of n from
s_n-1 < s_Nu

s_n
As the sequence ? is ordered, n can be determined by search by bisection.

During the simulation the temperature increases. Frequently this implies that the surface diffusion will go from slow to extremely fast. If we ignore this complication, the simulation will start with reasonably large timesteps, say =1 s. As the temperature increases, the diffusion rate increases, decreases and the algorithm gradually becomes intolerably slow, say =1 ns.

Fast events

This exposes a fundamental flaw in the KMC algorithm. Consider for a moment the situation of desorption at constant temperature in the absence of adsorbate-adsorbate interactions and further suppose that surface diffusion is slow. This problem can be solved both analytically and by KMC, this is our reference case. Suppose now that we speed up the surface diffusion. In the analytical approach, equation ? becomes

which can be plugged into equation ? i.e the analytical solution becomes simpler. However, the KMC simulation will use progressively shorter timesteps and the computational effort increases dramatically.

When the KMC algorithm is formulated as in ? increasing to infinity the rate of one of the steps in the mechanism leads to a dramatical increase in the computational effort rather than a fast approach to equilibrium.

The large but smooth increase in diffusion rate with temperature implies that simple approximations, such as treating the diffusion as either slow or fast independent of the temperature, will not work.

Fortunately, a smooth transtion to fast diffusion can easily be implemented by a slight modification of the KMC algorithm.

We decide on a limit between fast and slow diffusion events and then perform two scans over the system.

In the first scan we determine for each atom the set of sites that this atom can be reach through fast reactions. For each of these sites we calculate

where E_n is the energy of the system when the atom is at the n^th site and E₀ is the energy of the system prior to the move. The n^th site is selected with probability, p_n

In some cases an atom cannot move by any fast reaction, e.g because all its neighbor sites are occupied. It is not required to implement this as a special case. If the atom cannot move by any fast reaction, the set consists of only the current site of the atom. This site then has p₀=1 and is selected with probability 1.
The first scan consider only fast reactions and contributes significantly to the approach to equilibrium, but does not propagate the system in time.
In the second scan, we consider only the desorption events and the slow diffusion events and build the list of possible events as outlined above.
The second scan propagates the system in time in accordance with equation ?.

Start
Back
Next

Author Per Stoltze stoltze@fysik.dtu.dk