computer simulations of dislocationsibilion/...md simulation of dislocations: initial configuration,...

CCOORRNNEELLLL U N I V E R S I T Y

MAE 715 – Atomistic Modeling of Materials

N. Zabaras (04/04/2012) 1

References

Computer Simulations of Dislocation, V. V. Bulatov and W Cai

Theory of Dislocations, Hirth and Lothe

Introduction to Dislocations, D Hull and D J Bacon

Introduction to Solid State Physics, Kittel

Computer Simulations of Dislocations



N. Zabaras (04/04/2012) 2

Review

Dislocation: line defect in crystals.

It defines great many properties.

Line sense, Burgers vector.

Edge dislocation, screw dislocation, and mixed dislocation.

Peach-Koehler formula

Plastic strain rate

Conservative motion: glide; nonconservative motion: climb.

Peierls stress: The minimum stress required to make a straight dislocation move

at zero temperature.

Atomistic simulations: MD, MC.

MD simulation of dislocations: initial configuration, relaxation, thermalization,

dislocation motion under stress at constant temperature.

Extract dislocation velocity, drag coefficient.



N. Zabaras (04/04/2012) 3

5. More About Periodic Boundary Conditions

W. Cai, V. V. Bulatov, J. Chang, J. Li, and S. Yip. Periodic image effects in dislocation modelling.

Philosophical Magazine, 83(5): 539–567, 2003.



N. Zabaras (04/04/2012) 4

Application of PBC in all three directions holds an important advantage when the

behavior in the bulk is to be examined.

Important technical issues:

A fully periodic simulation cell can accommodate only such dislocation

arrangements whose net Burgers vector is zero. The minimal number of

dislocations is two (dipole).

Dipole and their periodic images interaction pollution.

The artifacts of PBC can be quantified through linear elasticity theory.

How to take full advantage of PBC when the displacement field induced by a

dislocation is to be calculated.

How to calculate the dislocation’s core energy.

How to calculate Peierls stress.

The common theme is an attempt to construct a solution of the elasticity

equations in a periodic domain by superimposing a periodic array of solutions of

an infinite domain.

5. More about periodic boundary conditions



N. Zabaras (04/04/2012) 5

Similar to the example in 3.1, a

dislocation is to be introduced by

displacing the atoms according to an

appropriate solution for the

displacement field of a dislocation

dipole in the periodic domain, .

5.1 setting up an initial configuration

The analytical solution for the displacement field of a dislocation dipole in an

periodic domain, , is not available. It should be constructed from the known

solution , in the infinite domain.

Isotropic elasticity

superposition

( , )2

zu x y b

Discontinuity across the cut

area.



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( , ) , , 0,2

0

z x yu x y b u u

except x y

does not satisfy the periodic boundary condition in y direction.




N. Zabaras (04/04/2012) 7

Because the equations of

elasticity are linear, it should

be possible to construct a

periodic displacement field

from the non-periodic solution

using superposition.

Each individual term

Corresponds to the

contribution from an image

dipole shifted from the

primary dipole by

Summation in practice, is

truncated over a finite number.

Better, but still

not periodic in

y direction




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Why the field obtained by superposition of fields due to a periodic array of sources

turns out to be non-periodic?

The sum depends on how the summation is truncated – conditional convergent.

For a sum to be absolutely convergent, it is necessary and sufficient that the sum

of the absolute value of its terms is convergent. However, for this summation

a

for large r

for large R

diverge




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The good news is that the “right” sum does exist and can be obtained from the

“wrong” sum

The summation of the derivatives is absolutely convergent. The limit of this

absolutely convergent sum produces the derivatives of a periodic displacement

field

Integrate twice both sides

deviation

Non-periodic

Remedy: (1) evaluate sum using arbitrary truncation. (2) measure the linear

spurious part of the resulting field by comparing its value at four points

in the periodic supercell . (3) subtract the linear term

from to obtain a corrected solution .




N. Zabaras (04/04/2012) 10

In some simulation, the repeat vectors of periodic supercells are not mutually

orthogonal to compensate for the internal stress produced by dislocations.

In a supercell with fixed periodicity vectors, an increment in plastic strain will be

compensated by an oppositely signed increment of elastic strain of the same

magnitude . There will be an internal back-stress eliminating the source of

the strain, i.e. the dislocation dipole.

If we allow the simulation box to adjust its shape during energy minimization, it

could reach a state of zero average internal stress. However, the dislocations can

move significantly during the relaxation. To avoid such unwanted behaviors, the

shape of the supercell is intentionally adjusted before relaxation to set the average

internal stress to zero.


Plastic strain produced by insertion.

from



N. Zabaras (04/04/2012) 11

In the case of the example, the internal stress induced by the dipole can be

removed by

Adjusting c2 amounts to adding an extra term

to the solution

linear


is introduced intentionally to remove the internal stress induced b the

insertion of a dislocation dipole.

c2 is adjusted. The periodicity is not impaired.



N. Zabaras (04/04/2012) 12

5.2 Dislocation core energy

An example:

Construct a perfect diamond cubic crystal in a

periodic supercell (lattice constant 5.4301A)

Then introduce a screw dislocation dipole with Burgers

vector , and the separation between two

dislocations .

To eliminate average internal stress, the supercell is tilted

Using the Stillinger-Weber potential1, the relaxation is proceeded by CGR.

[1] F. H. Stillinger and T. A. Weber. Computer simulation

of local order in condensed phases of silicon. Physical

Review B, 31(8): 5262–5271, 1985.

0.08eV

higher

A/A0=1/2



N. Zabaras (04/04/2012) 13

Total energy

Excess energy (per unit length of the dislocation dipole).

Cohesive energy (energy per atom in the perfect crystal). Here is -4.63eV

Residual stress, after tilt – corresponds to energy associated with internal

stress.

is dependent on supercell geometry. An more invariant property, core energy,

which is independent of the simulation cell geometry will be extracted.

dipole In an infinite isotropic elastic medium

The difference between periodic elastic energy and infinite elastic energy is

Conditional convergent

The yet unknown contribution from the periodic boundary conditions. The interaction

energy between the primary dislocation dipole with an infinite array of image dipoles.




N. Zabaras (04/04/2012) 14

In this example

The interaction energy between

two identical dislocation dipoles

separated by R.

The interaction energy between two dislocation dipoles can be written as the

stress field produced by one dipole integrated over the cut plane of the other

dipole.

Introduce four test or “ghost” dislocations with Burgers vectors

such that is separated from by , is separated from by and

Denoting as the interaction energy between the dislocation

dipole at R and the four “ghost” dislocations,

MadSum


When R is large, ~ 1/R2. Sum is conditional convergent.

Corrected image energy

http://micro.stanford.edu/~caiwei/Forum/2004-08-08-MadSum/



N. Zabaras (04/04/2012) 15

Eatm

Eel

2Ecore

C11=161.6GPa, C12=81.6 Gpa, C44=60.3GPa.

rc=b=3.84A, Ecore=0.565 eV/A, independent of c1



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5.3 Peierls stress

Peierls stress: the stress necessary to move a dislocation at zero temperature.

Position the dislocation dipole perpendicular to the glide

plane. Apply a constant stress using the Parrinello-

Rahman method (applies stress to the periodic supercell by

dynamically adjusting its shape so that the virtual stress

fluctuates around a specified value). This stress exerts equal

but opposite forces on the two dislocations in x and –x

directions.

Gradually increase the applied stress. Each increment is

followed by a relaxation.

Isotropic elasticity

Anisotropic elasticity

MD simulation Eatm

http://micro.stanford.edu/~caiwei/CSD-Book/source/chap05/sect5.1/scripts/mo-screw-dipole.gif



N. Zabaras (04/04/2012) 17

The slope of E is the image force that the dislocations see in addition to the PK

force exerted by the applied stress. It introduces an error in the Peierls stress

computed in PBC. This error is minimized when x=0 or x=c1/2. But a second order

error still exists.

c2 varies, c1 invariant

When , computed at x=0 overestimates , because the next lattice

position has a slightly higher energy.

c1

c2

5.3 Peierls stress



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6. Free-energy calculation



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6.1 Introduction to free energy

Free energy is of central importance for understanding the properties of physical

systems at finite temperatures. When an open system exchanges energy with the

outside world and maintains a constant temperature, its evolution proceeds

towards minimizing its free energy.

Free energy , where the average energy E of the system is

The potential energy of a typical many-body system

has many local minima. Each one corresponding to

a set of configurational microstate . The

entire configurational space can be divided into a

set of non-overlapping energy basins. When the

thermal energy is much lower than the energy

barriers, the system’s dynamics can be viewed as a

sequence of jumps from one basin to the next,

separated by long residence periods .

Each configurational state includes

all microstates that map to the same

underlying energy minimum



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During the residence periods , the system continues to interact with the

thermostat and may have enough time to achieve partial thermal equilibrium. The

relative probabilities of various microstates within the basin obey Boltzmann’s

distribution. The free energy of a given configurational state (basin) can be

where the partial statistical integral is obtained by integration over the volume

of the energy basin

Partial free energies of configurational states are related to the total free energy

through the sum of partial statistical integrals

The calculations of the total free energy F is impossible due to the large number of

distinct configurational states. The calculations of the free energy of a given

configurational state are more manageable. When the system reaches complete

thermal equilibrium, the distribution of microstates in the entire phase space

satisfies Boltzmann’s distribution. The probability of the system can be found in a

configurational state is

6.1 Introduction to free energy



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6.2 Harmonic approximation

For several particularly simple systems, their statistical integral Q can be obtained

analytically, i.e. one-dimensional harmonic oscillator:

Integrate over x and p

The free energy of a collection of N independent harmonic oscillators (not atoms)

Consider a solid containing N atoms, the Hamiltonian can be written as

Assuming is a local minimum of the potential function V, its Taylor

expansion around is

are the components of Hessian matrix.

HQ e dxdp

2

d , 0axe x aa



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Diagonalize the Hessian matrix

Denoting new transformed variables

The Hamiltonian can be rewritten as

In a crystal at temperatures much lower than its melting point, the amplitude of

atomic vibrations is small so that the anharmonic terms can be ignored.

Example: using the harmonic approximation to compute the free energy of a

vacancy in the dislocation core.

1. Build a 192-atom fragment of a perfect diamond-cubic crystal in a periodic

supercell with dimensions

2. Make a cut-plane along one of the (111) planes and insert a dipole of 30

partial dislocations with Burgers vector and line direction along

.


3N independent

harmonic oscillators



N. Zabaras (04/04/2012) 23

3. Relax the configuration to a local energy minimum using the CGR algorithm

and the SW interatomic potential function for silicon.

4. Remove the atom from site A near the core and relax the resulting

configuration to its energy minimum .

5. Compute the Hessian matrix K. This can be done numerically by adding a

small displacement to each atomic coordinate one at a time, and

computing and recording the resulting forces on all other coordinates

.

6. Diagonalize K and obtain its 573 eigenfrequencies . The

three lowest ones should be zero, corresponding to

translations of the crystal as a whole in three directions.

They should be excluded from the free-energy calculations;

this is equivalent to fixing the position of the system’s center

of mass.

The free energy of the system can be straightforwardly

computed by summing over 570 harmonic oscillators by

[111]

[110]

[112]

A




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By itself, the magnitude of the free energy of a given configuration has little

significance. It becomes useful only when compared to the free energy of another

configuration of the same system.

Formation free energy of a vacancy in the dislocation core: the change in the free

energy of the system containing the core vacancy relative to the free energy of the

same system without the core vacancy.

Cannot compute the free energy of system with atom A still in and do subtraction,

because the two system are not the same (191 atoms vs. 192 atoms).

The right remedy is transport the removed atom to a surface ledge of a large

crystal. This operation is mimicked by adding to the free energy of the vacancy

system the free energy per atom in a perfect crystal.

The energy minimum and free energy of the system without vacancy at T=600K

are




N. Zabaras (04/04/2012) 25

The per-atom free energy of a perfect crystal can be computed as follows:

1) Build a small diamond-cubic crystal in a periodic box.

2) Use the SW potential to compute the Hessian matrix and diagonalize it.

3) Compute the free-energy.

The per-atom free energy at T=600K is

Finally, the energy of vacancy formation in the dislocation core is

The corresponding free energy is

Another important parameter of the core vacancy is its excess entropy.

is independent of temperature within harmonic approximation, where the

thermal energy of any configurational state, defined as the average energy

minus the potential energy at the minimum microstate is simply


The cohesive energy of the perfect crystal



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It is of interest to compare the above results with the vacancy formation energy in

the bulk.

Since , vacancies have a tendency to segregate at the dislocation core.




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6.3 Beyond Harmonic Approximation

Many cases HA is inaccurate. More accurate methods is needed.

Although free energy cannot be expressed as a statistical average, the difference

between two systems can be written in the form of a statistical average. If the free

energy of one system is known, then the free energy of the other system can be

obtained by computing the difference between the two.

Consider systems 1 and 2 with the same number of atoms but described by two

different Hamiltonians

By the definition of free energy

Not so accurate if fluctuation of is

large, when H1 and H2 are far from each other.



N. Zabaras (04/04/2012) 28

Improvement:

Consider a family of Hamiltonians . For a simple case

The free energy of the system is

Taking derivative with respect to

The derivative of the absolute free energy with respect to is equal to the

average of function over the canonical ensemble generated by the

“mixed” Hamiltonian . is much better for numerical computation.

The free-energy difference can then be obtained by integration




N. Zabaras (04/04/2012) 29

When

The above method is based on thermodynamic integration (TI). It requires a lot of

equilibrium simulations (at least one for each value of ) to obtain accurate results.

The method of adiabatic switching (AS) derives from TI is often more efficient. It is

a nonequilibrium measurement of quasistatic work.

Define the generalized force conjugate to variable as

Then is the thermodynamic work required to drive the system from

to , against the generalized force . It becomes possible to compute the free-

energy difference using a single non-equilibrium (MD or MC) simulation, in which

the Hamiltonian gradually changes from H1 to H2. For every increment , the

instantaneous work of switching is . The total work accumulated in one such

simulation gives an estimate of the free-energy difference F2-F1.


M. de Koning, W. Cai, A. Antonelli, and S. Yip. Efficient free-energy calculations by the simulation of nonequilibrium

processes. Computing in Science and Engineering, 2(3): 88–96, 2000.



N. Zabaras (04/04/2012) 30

In order to use MD to sample statistical ensembles of the mixed Hamiltonians, it

is necessary to define a switching function such that, at the beginning of the

simulation and the at end .

Strictly speaking, this equation is valid only in the limit of

One widely used reference system is the Einstein crystal equivalent to 3N

harmonic oscillators,

The parameters should be determined to make the reference system as close as

possible to the system of interest.

Another complication arises in MD (but not in MC) if H1 or H2 is exactly or nearly

harmonic is that MD sampling becomes non-ergodic. A more sophisticated version

of the Nose-Hoover thermostat, namely a Hose-Hoover chain, can be used.


G. J. Martyna, M. L. Klein, and M. Tuckerman. Nosé–Hoover chains: the canonical ensemble via continuous dynamics.

Journal of Chemical Physics, 97(4): 2635–2643, 1992.



N. Zabaras (04/04/2012) 31

In the standard Metropolis MC algorithm, the acceptance probability at step n

should be computed using the mixed potential energy

Since the kinetic energies of the two Hamiltonians H1 and H2 are the same, only

the difference between two potential energies V2-V1 is computed and added to

the accumulated work of switching

approaches only in the limit of the total number of MC steps

approaching infinity. In practice, overestimates

since some part of it is dissipated as heat during the

switching, . Switching along the same

trajectory but in the opposite direction, from V2 to V1,

should produce a different amount of work,

. Therefore, a more accurate value of the free-

energy difference is




N. Zabaras (04/04/2012) 32

Revisit the free energy of a core vacancy using MC adiabatic switching simulation

without relying on harmonic approximation of H1 and H2.

1) Run switching simulations from system 0,1,2 to Einstein crystal with the same

number of atoms. – Not good due to statistical and systematic errors.

2) Obtain the free-energy difference by direct switching from system 1 to system 2.

The issue that 1 and 2 do not have the same number of atoms is resolved by

augmenting system 1 with an extra atom in a harmonic potential, whose free-

energy contribution is known, that does not interact with other atoms.

Extra atom

Potential of augmented system 1

To reduce dissipation while switching between systems 1’ and 2, we slightly

modify1’ to make it structurally similar to 2. Assume and are the

minimum-energy configurational states of and , respectively.

Then the system described by the potential energy function,

will have the same minimum-energy structure as , and the same free-energy as




N. Zabaras (04/04/2012) 33

For a fixed total number of Monte Carlo steps, the switching trajectory should be

chosen so that the efficiency is improved by reducing both statistical errors and heat

dissipation. A reasonable choice is




N. Zabaras (04/04/2012) 34

1) MC metropolis: system should be equilibrated at the temperature where the free

energy is interested.

2) Start switching simulation. Run multiple chains to minimize the statistical

fluctuation error. The area under curve is

3) To eliminate the systematic error due to dissipation, switch in the opposite

direction, i.e. from 2 to 1’, for multiple chains. The area under curve is

4) Average

5) Subtract the free-energy difference between systems 1’ and 1. And add the free

energy per atom in a perfect lattice.

where


In Einstein model k=20eV/A. nmax=5x105 MCS.

The free energy of the auxiliary atom in

a 3D harmonic potential

The free energy is lower than that predicted from the harmonic approximation.



N. Zabaras (04/04/2012) 35

7. Finding transition pathways



N. Zabaras (04/04/2012) 36

7.1 The rare event problem

In MD simulations of solids , the integration step is usually of the order of one

femtosecond (10-15s). The time horizon of MD simulations of solids rarely exceeds

one nanosecond (10-9s). Dislocation behaviors of interest typically occur on time

scales of milliseconds or (10-3s) larger. The disparity of time scales can be traced to

certain topographical features of the potential energy function, i.e. deep energy

basins separated by high energy barriers. Direct MD and MC simulations spend

most time tracing the unimportant fluctuations within the energy basins.

Points S1 and S2 on the ridge are more special because their energies are locally

minimal among all the points on the ridge; they are called saddle points (unstable

equilibrium, Highest along the MEP).

Minimum

energy path



N. Zabaras (04/04/2012) 37

In crossing from one basin to another, the system is more likely to pass through

saddle points. Quantitatively, the free energy of the dividing surface (ridge) is

related to the rate of the corresponding transition. When multiple MEPs exist, the

most likely escape route will be the one that crosses the saddle point with the

lowest free energy. In the limit of low temperatures, the free energy of the dividing

surface reduces to the energy of the lowest saddle point.

7.2 Transition state theory

Transition state theory provides an alternative approach

to computing transition rates, which is applicable even

when the intervals between transitions exceed the total

time duration of the atomistic simulation.

The transition rate from A to B is

is the probability that a system initially in basin will cross the dividing

surface towards the neighboring basin during a short time interval

Here is the statistical integral over basin and is the statistical integral

over the sub-region of microstates that cross from A to B within time .



N. Zabaras (04/04/2012) 38

Rewrite statistical integrals using the step function

1, 0( )

0,

xx

otherwise

The meaning of is that for a microstate at points s<0 to cross

within time , its velocity in the s direction has to be greater than .

In the limit of

Define the statistical integral over the dividing surface as

The rate of crossing from A to B

7.2 Transition state theory

k true transition rate

The probability of the system is

on the left side and give enough

kinetic energy to pass the barrier.



N. Zabaras (04/04/2012) 39

7.3 Local Path Optimization

Constrained minimization: assume the minimum energy path is close to a straight

line connecting the energy minima of two metastable states.

To allow the candidate path to deviate from the straight line and to better track an

actual MEP, the path is modified as

The exact shape of can be obtained by minimizing the energy at every point

along the path.

Deepest descent

Constrained minimization



N. Zabaras (04/04/2012) 40

One way to prevent the system from moving along the constraint direction

during optimization is to remove from the gradient vector g its component parallel

to


The minimization of step 5 is performed using CGR modified to enforce the

orthogonality constraint



N. Zabaras (04/04/2012) 41

The chain of states method:

The constrained minimization may fail in certain situations when the MEP are

curved somewhere along the path may become nearly orthogonal to the straight-

path direction.

The nudged elastic band (NEB) method assumes that the MEP is a smooth curve

that can be approximated by a chain of states, each state representing a copy of

the whole system. The end states of the chain are fixed at the energy minima of

interest .

Each point r(s) along the MEP should be a local energy minimum in the

hyperplane normal to the local tangent direction t(s). The MEP can be found by

iterative relaxation of the trial chain. The local tangent to the path at each state si,

i=1,…,Nc-1, is approximated by

The MEP can be found by an iterative steepest-descent

relaxation with a step size




N. Zabaras (04/04/2012) 42

Computationally demanding

1

2

3



N. Zabaras (04/04/2012) 43

Case study: Kink migration in Silicon

Because of the strong covalent bonding between the atoms, dislocation in

diamond-cubic semiconductors consist of very long straight segments connected

by rather narrowly localized kinks (30° partial dislocation).

N=6912, spatial DOF: 3N=20736

Reduce the dimensionality by focusing on a

smaller number of atoms that are close to the

defect, while treating the rest of the atoms in

a simplified manner.

n=109, reduced DOF: 3n=327

Relax positions of the surrounding (N-n) atoms to

their minimum energy with no further constraints.

Searching for a saddle in 3n subspace. The

components of the gradient corresponding to the

surrounding (N-n) atoms are all zero.

Reducing the computational cost and memory requirements, better control of the

transition process. Constrained minimization is worth trying first. Neither is global

optimal when there are multiple MEPs and saddle points connecting the same

two energy minima. Only can apply when the destination state B is known.

2) 0.48ev

S=0.45

1)0.47ev

S=0.5



N. Zabaras (04/04/2012) 44

7.4 Global path optimization

“finite-temperature”– simulated annealing.

A path represented by a chain of Nc+1 microstates: r(si), with fixed at (0) Ar r

and . Maximum length of the path Lmax=Ncr0, substantially longer than the

distance between its two end states, |rB-rA|. Here, (1) Br r

Energy barrier of a path: the highest energy Emax among all states along a given

path. The goal is to find the optimal path whose energy barrier is the lowest among

all paths with the same Nc.

Need to decide a chain first



N. Zabaras (04/04/2012) 45

Case study: the two-dimensional potential

2D potential

where

The new term introduces an additional saddle point at S0=(0,0), whose energy is

higher than that of the other two saddle points

If the initial trial path is chosen to be the straight line through A, B, and S0, the

local path optimization will be trapped at the high-energy saddle S0.

Initial chain path is along the

straight line from A to B, with

100 equally spaced states.

S0 S1 or S2

The location of the saddle point can

be further refined starting from the

maximum energy state on the final

path using local path optimization or

Sinclair-Fletcher algorithm [1].

[1] J. E. Sinclair and R. Fletcher. A new method of saddle-point location for the calculation of defect migration energies.

Journal of Physics C (Solid State Physics), 7(5): 864–870,1974.



N. Zabaras (04/04/2012) 46

Case study: kink migration in Silicon revisited

To reduce the computational burden, only 39 atoms round the kink are selected for

the MC moves. Positions of the remaining N-39 atoms are linearly interpolated

along the chain from state A to state B. The initial path is obtained from a NEB

relaxation using a chain with 100 segments.

n=6x104

n=2x105

n=0

Start at a local minimum



N. Zabaras (04/04/2012) 47

7.5 Temperature-accelerated sampling

Destination state B is unknown beforehand.

Temperature-accelerated sampling (TAS) is used to identify B.

At high temperatures the transitions between metastable states occur frequently.

Because the potential energy landscape does not depend on temperature, it should

be possible to use MD or MC simulations at a fictitiously high temperature (T2) to

explore the energy landscape and identify neighboring metastable states and

transition pathways that might be relevant at a lower temperature of interest (T1)

Successful application of this algorithm depends on the choice of the T2.

Find B first at high T,

then determine the

MEP at normal T.



N. Zabaras (04/04/2012) 48

Case study: Kink Migration in Silicon

The TAS algorithm is applied only to n=612 atoms around the kink while the

remaining N-612 atoms are held fixed at their respective positions in state A.

T2=2000K

rc=0.12A

To reduce the incidence of high-energy

transition events, set T2=1800K. 10

transition events are recorded. The energies

of the destination states are:

-0.4054eV (5 times), -0.3919eV (3 times),

0.7956eV (once), and 1.4670eV (once).

T2=2000K

rc=1.5A

Many transition events

Destination states with

energies range from -0.4eV to

1.7 eV

computer simulations of dislocationsibilion/...md simulation of dislocations: initial configuration,...

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