Kinetic Monte Carlo:
from transition probabilities to transition rates
With MD we can only reproduce the dynamics of the system for ≤ 100 ns. Slow thermallyactivated processes, such as diffusion, cannot be modeled. Metropolis Monte Carlo samples
configurational space and generates configurations according to the desired statistical-mechanics
distribution. However, there is no time in Metropolis MC and the method cannot be used to
study evolution of the system or kinetics. An alternative computational technique that can be
used to study kinetics of slow processes is the kinetic Monte Carlo (kMC) method.
Cartoon by Larry Gonick
As compared to the Metropolis Monte
Carlo method, kinetic Monte Carlo
method has a different scheme for the
generation of the next state. The main idea
behind kMC is to use transition rates
that depend on the energy barrier
between the states, with time increments
formulated so that they relate to the
microscopic kinetics of the system.
In Metropolis MC methods we decide whether to accept a move by considering the energy
difference between the states. In kMC methods we use rates that depend on the energy barrier
between the states.
University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
Kinetic Monte Carlo: kinetics of atomic rearrangements
Before starting kMC simulation we have to make a list of all possible events that can be
realized during the simulation and calculate rates for each event.
Input to KMC:
Fast processes – MD simulations
Slow processes – transition state theory, experiments
For example, diffusion on the surface is determined by the energy barriers for the breaking the
adatom-substrate bonds, Ea = Esaddle - Emin, and the rate constant for diffusion can be calculated
using a simplified transition state theory, e.g. [A. Voter, Phys. Rev. B 34, 6819 (1986)]:
⎛ E ⎞
k TST = n p ν exp ⎜ − a ⎟
⎝ kT ⎠
Energy
np is the number of possible jump directions ν is the
harmonic frequency. The assumptions are that the diffusion
of an adatom is a result of random uncorrelated hops
between neighboring binding sites and that the time between
hops is much longer as compared to the time for the hop.
Ea
Distance
For atoms adjacent to an island/step, additional (Ehrlich-Schwoebel) barriers/rates should be specified
for breaking bonds with the atoms of the island. As the system becomes more complex, the number of
possible events becomes larger…
University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
Kinetic Monte Carlo: kinetics of atomic rearrangements
When the rate constants of all processes are known, we can perform kMC simulation in the time
domain. In the case of a single process, the reciprocal of the rate of the process determines the
time required for the reaction to occur. This quantity can be set equal to the kMC time. In the
case of a many-particle multi-process system, however, introduction of time is less
straightforward and several modifications of kMC exist.
For example, we can use the following scheme [A. F. Voter, Phys. Rev. B 34, 6819-6829
(1986)]:
1.
The rate of all the allowed processes can be combined to obtain the total rate and the time
step is calculated as inverse of the total rate.
2.
At each time step one of all the possible processes is randomly selected with probability that
is the product of the time step and the rate of the individual process.
Other algorithms include:
• Assigning to each particle in the system independent time clock (and time step) and calculating
the real time step as an average over all independent time steps [e.g. Pei-Lin Cao, Phys. Rev.
Lett. 73, 2595-2598 (1994)]
• Choosing a single constant time step that is less than the duration of the fastest process. The
processes are chosen randomly and allowed to occur with probabilities based on individual
rates [Dawnkaski, Srivastava, Garrison, J. Chem. Phys. 102, 9401-9411 (1995)]
University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
Example: kMC simulation of diamond {001} (2x1):H surface under CVD growth conditions
[Dawnkaski, Srivastava, Garrison,
Chem. Phys. Lett. 232, 524, 1994]
First, the rates of all relevant surface
reaction have to be evaluated (from
molecular static or dynamics simulations
υi = υ exp(
0
i
υi0 [s -1 ]
− Eia
k BT
Eia [eV]
University of Virginia, MSE
4270/6270:few
Introduction
to Atomistic Simulations, Leonid Zhigilei
…………a
more reactions……………
)
Example: kMC simulation of diamond {001} (2x1):H surface under CVD growth conditions
Given the rate of each individual process in a system, the
probability of a process occurring within any specified time period
or timestep is simply the product of the timestep and the rate of the
process.
A single constant timestep can be chosen so that it is less than the
duration of the fastest process considered. The probabilities of all
the considered processes are thus between 0 and 1.
The timestep is chosen such that the acceptance probability of the
fastest process is about 0.5. This gives values of Δt = 10-5 s for
1200 K, 10-7 s for 1500 K, and 10-8 s for 1800 K.
Topographical snapshots of a growing diamond surface.
The surface area is a square of side 25 A and the plots is
shaded by layer number with the highest layer being the
lightest in shade.
[Dawnkaski, Srivastava, Garrison, Chem. Phys. Lett. 232, 524, 1994]
University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
Example: Growth of fractal structures in fullerene layers
MD and kinetic Monte Carlo simulations by Hui Liu (term project for MSE 6270)
All possible thermally-activated
events have to be considered
STM images of C60 film
growing on graphite
-4.5
νi = νoexp(-Ei/kBT)
Ln (Vj)
-5.0
-5.5
MD simulations – finding the
energy barriers, attempt
frequencies, and probabilities of
18
20
22
24x10
diffusion
events
University of Virginia, MSE 4270/6270: Introduction
Leonidjump
Zhigilei
1/kT (1/meV) to Atomistic Simulations,
-6.0
-3
Example: Growth of fractal structures in fullerene layers
MD and kinetic Monte Carlo simulations by Hui Liu (term project for MSE 627)
STM images of C60 film
growing on graphite
University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
Example: Diffusion of Ge adatoms on a reconstructed Si(001) substrate
MD and kinetic Monte Carlo simulations by Avinash Dongare
Calculation of the potential energy
surface for a Ge adatom using procedure
described by Roland and Gilmer in
Phys. Rev. B 46, 13437 (1992)
University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
Example: Diffusion of Ge adatoms on a reconstructed Si(001) substrate
MD and kinetic Monte Carlo simulations by Avinash Dongare
MD simulations of Ge adatom
diffusion on a Au covered
Si(001) substrate
Ge adatom trajectories on a Au covered Si(001) substrate
-11
D - MD
-12
2
Ln (D [cm /sec])
Mean square displacements
for Ge adatoms on a Aucovered Si(001) substrate,
and logarithmic plot of the
diffusion coefficient from
MD simulations
-13
-14
r
Δr (t ) 2 ~ 4 Dt
D = D0 exp(− Ed / k BT )
-15
8
10
12
1/kBT
University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
14
16
Example: Diffusion of Ge adatoms on a reconstructed Si(001) substrate
MD and kinetic Monte Carlo simulations by Avinash Dongare
kMC
Surface structures after
deposition of 0.07 ML of Si
on a Si(001) substrate at
400 K with a deposition
rate of 0.1 ML/min
STM, Mo et al., Phys. Rev.
Lett. 66, 1998 (1991)
Surface structures predicted in kMC
simulations of the deposition of 0.10
ML of Ge on a Au patterned Si(001)
substrate (76 nm x 76 nm) at 600 K with
deposition rate of 9 ML/min.
University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
Kinetic Monte Carlo: limitations
One (main?) problem in kMC is that we have to specify all the barriers/rates in advance,
before the simulation.
But what if we have a continuous variation of the activation energies in the system? What if the
r
activation energies are changing during the simulation?
⎛ E (r , t ) ⎞
k ( E , T ) = k 0 exp ⎜ −
⎟
kT
⎝
⎠
Example: strain on the surface can affect the diffusion of adatoms and nucleation of islands.
There could be many possible origins of strain, e. g. buried islands, mesas, dislocation patterns in
heteroepitaxial systems. One can try to introduce the effect of strain on the activation energies
for the diffusion of adatoms. For example, Nurminen et al., Phys. Rev. B 63, 035407, 2000, tried
several approaches.
In one approach, they introduced a spatial dependence of the
adatom-substrate interaction on a patterned surface, E = ES(x,y) +
nEN, where E is the diffusion activation energy, ES is the
contribution due to the interaction with substrate, and EN is the
energy of interaction with other adatoms.
In another approach, an additional hop-direction dependent
diffusion barrier ED is introduced to describe the long-range
interaction between adatom and domain boundaries:
E =toEAtomistic
ED
University of Virginia, MSE 4270/6270: Introduction
Leonid Zhigilei
S +nEN + Simulations,
Kinetic Monte Carlo: example
Kinetic Monte Carlo simulation of island growth on a homogeneous substrate and a substrate
with nanoscale patterning (by Nurminen, Kuronen, Kaski, Helsinki University of Technology)
islands on a homogeneous substrate
Blue denotes the substrate
and green the deposited atoms.
islands on a substrate with nanoscale patterning
(a checkerboard structure)
http://www.lce.hut.fi/publications/annual2000/node22.html
The growing heteroepitaxial islands by themselves can locally modify diffusion barriers.
Relaxation of nanostructures introduces local strains that constantly change the energy landscape
and corresponding probabilities of Monte Carlo events. Modified approach based on the locally
activated Monte Carlo techniques (Kaukonen et al. Phys. Rev. B, 61, 980, 2000) has been
proposed to account for local strains.
University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
Example: kMC in simulations of dislocation dynamics
[Karin Lin and D. C. Chrzan, Phys. Rev. B 60, 3799 (1999)]
Simulation of evolution of the collective behavior of a large number of dislocations requires
velocity vs. stress law. Typically, in Dislocation Dynamics simulations an empirical law
involving a damping term is assumed. Kinetic Monte Carlo can provide the needed velocity vs.
stress relationship. In particular, the simulations may reflect the stochastic aspects associated with
overcoming the Peierls barrier, interactions with vacancies, etc. in a natural way.
Dislocation dynamics method can be used (potentially) to connect atomic scale calculations with
macroscopic continuum description of plasticity. Kinetic Monte Carlo simulations can be
parameterized based on atomic scale MD studies of the properties of dislocation cores, kinks, etc.
The model discussed in this paper includes dislocation segment interactions in the isotropic
elasticity theory limit. The Peierls potential is also included and free surface boundary conditions
are used (all image forces and surface tractions are reflected in the energetics governing the
dynamics). Dislocations are assumed to be composed entirely of screw and edge segments.
The algorithm involves cataloging all of the possible
kinetic events, and calculating the rates associated with
these
processes.
Kinetic
events
include
the
production/annihilation of double kink pairs, as well as the
University of Virginia, MSE 4270/6270: Introduction
to of
Atomistic
Simulations,
lateral motion
the existing
kinks. Leonid Zhigilei
Summary: MD, Metropolis MC and kinetic MC
With MD we can only reproduce the dynamics of the system for ≤ 100 ns. Slow thermallyactivated processes, such as diffusion, cannot be modeled. An alternative computational
techniques for slow processes are Monte Carlo methods.
Monte Carlo method is a common name for a wide variety of stochastic techniques. These
techniques are based on the use of random numbers and probability statistics to investigate
problems in areas as diverse as economics, nuclear physics, and flow of traffic. There are many
variations of Monte Carlo methods. In this lecture we will briefly discuss two methods that are
often used in materials science - classical Metropolis Monte Carlo and kinetic Monte Carlo.
Metropolis Monte Carlo – generates
configurations according to the desired
statistical-mechanics
distribution.
There is no time, the method cannot be
used to study evolution of the system.
Equilibrium properties can be studied.
Cartoon by Larry Gonick
Kinetic Monte Carlo – can address kinetics. The main idea behind KMC is to use transition
rates that depend on the energy barrier between the states, with time increments formulated so
that they relate to the microscopic kinetics of the system.
University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
Summary: MD, Metropolis MC and kinetic MC
Molecular Dynamics – based on the solution of the equations of motion for all particles in the
system. Complete information on atomic trajectories can be obtained, but time of the simulations
is limited (up to nanoseconds) – appropriate for fast processes (e.g. FIB local surface
modification, sputtering, implantation) or for quasi-static simulations (e.g. stress distribution in
nanostructures).
Metropolis Monte Carlo – generates random configurations with probability of each
configuration defined by the desired distribution P(rN). This is accomplished by setting up a
random walk through the configurational space with specially designed choice of probabilities of
going from one state to another. Equilibrium properties can be found/studied (e.g. surface
reconstruction and segregation, composition variations in the surface region due to the surface or
substrate induced strains, stability of nanostructures).
Kinetic Monte Carlo – when the rate constants of all processes are known, we can perform kMC
simulation in the time domain. Time increments are defined by the rates of all processes and are
formulated so that they relate to the microscopic kinetics of the system. This method should be
used when kinetics rather than equilibrium thermodynamics dominates the structural and/or
compositional changes in the system.
Kinetic Monte Carlo vs Metropolis Monte Carlo: in MMC we decide whether to accept a
move by considering the energy difference between the states, whereas in kMC methods we use
rates that depend on the energy barrier between the states. The main advantages of kinetic
Monte Carlo is that time is defined and only a small number of elementary reactions are
considered, so the calculations are fast.
University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei