kMC

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Kinetic Monte Carlo: from transition probabilities to transition rates

With MD we can only reproduce the dynamics of the system for ≤ 100 ns. Slow thermally-activated 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.

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.

Cartoon by Larry Gonick

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.


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 simulationsSlow processes – transition state theory, experiments

⎟⎠⎞

⎜⎝⎛ −ν=

kTEexpnk a

pTST

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…

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)]:

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

Energy


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)]:

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)]

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.


Example: kMC simulation of diamond {001} (2x1):H surface under CVD growth conditions

[Dawnkaski, Srivastava, Garrison, Chem. Phys. Lett. 232, 524, 1994]

…………a few more reactions……………

First, the rates of all relevant surface reaction have to be evaluated (from

molecular static or dynamics simulations

)exp(0

TkE

B

ai

ii−υ=υ

[eV] aiE][s -10

iυ


Example: kMC simulation of diamond {001} (2x1):H surface under CVD growth conditions

[Dawnkaski, Srivastava, Garrison, Chem. Phys. Lett. 232, 524, 1994]

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.


-6.0

-5.5

-5.0

-4.5

Ln (V

j)

24x10-3

222018

1/kT (1/meV)

Example: Growth of fractal structures in fullerene layersMD and kinetic Monte Carlo simulations by Hui Liu (term project for MSE 6270)

STM images of C60 film growing on graphite

MD simulations – finding the energy barriers, attempt

frequencies, and probabilities of diffusion jump events

νi = νoexp(-Ei/kBT)

All possible thermally-activated events have to be considered


Example: Growth of fractal structures in fullerene layersMD and kinetic Monte Carlo simulations by Hui Liu (term project for MSE 627)

STM images of C60 film growing on graphite


Example: Diffusion of Ge adatoms on a reconstructed Si(001) substrateMD 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)



1/kBT

Ln(D

[cm

2 /sec

])8 10 12 14 16

-15

-14

-13

-12

-11

D - MDMean square displacements for Ge adatoms on a Au-covered Si(001) substrate, and logarithmic plot of the diffusion coefficient from MD simulations

Ge adatom trajectories on a Au covered Si(001) substrate

MD simulations of Ge adatom diffusion on a Au covered Si(001) substrate

)/exp(0 TkEDD Bd−=Dttr 4~)( 2rΔ



STM, Mo et al., Phys. Rev. Lett. 66, 1998 (1991)

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

kMC

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.


Kinetic Monte Carlo: limitationsOne (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 activation energies are changing during the simulation?

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 = ES +nEN + ED

⎟⎠⎞

⎜⎝⎛ −=

kTtrEkTEk ),(exp),( 0

r


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)

http://www.lce.hut.fi/publications/annual2000/node22.html

islands on a homogeneous substrate

islands on a substrate with nanoscale patterning (a checkerboard structure)

Blue denotes the substrate and green the deposited atoms.

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.


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 conditionsare 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 dynamic events, and calculating the rates associated with these processes. Dynamic events include the production/annihilation of double kink pairs, as well as the lateral motion of the existing kinks.


Summary: MD, Metropolis MC and kinetic MCWith MD we can only reproduce the dynamics of the system for ≤ 100 ns. Slow thermally-activated 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.

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.

Cartoon by Larry Gonick


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.

Summary: MD, Metropolis MC and kinetic MC

kMC

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