monte carlo simulation - postech cmse lab. · 2013-02-19 · monte carlo – grand canonical...

Monte Carlo SimulationMonte Carlo Simulation

ByeongByeong--Joo LeeJoo Lee

Byeong-Joo Leewww.postech.ac.kr/~calphad

ByeongByeong--Joo LeeJoo Lee

Dept. of MSEDept. of MSE

Pohang University Pohang University

of Science and Technologyof Science and Technology

(POSTECH)(POSTECH)

[email protected]@postech.ac.kr

BackgroundBackground -- PhasePhase SeparationSeparation inin GaInNGaInN


Background Background –– Atomic distribution in Atomic distribution in SiGeSiGe nanowiresnanowires

0.6

0.8

1

X(G

e) 5nm

Surface Segregations


0

0.2

0.4

0.6

0 0.5 1

normalized rX

(Ge

) 5nm

2nm

bulk

Why Monte Carlo ? Why Monte Carlo ?


What is Monte Carlo ?What is Monte Carlo ?y

1

implicit integer (i-n)

implicit double precision (a-h,o-z)

call random_seed

i_circle = 0

do i = 1, 100000000

call random_number(x)

call random_number(y)

r2 = x*x + y*y

if(r2 .lt. 1.d0) i_circle = i_circle + 1


xif(r2 .lt. 1.d0) i_circle = i_circle + 1

if(mod(i,1000000) .eq. 0) then

pi = 4.0d0 * dble(i_circle) / dble(i)

write(*,*) i, pi

endif

enddo

c

stop

end

Monte Carlo Monte Carlo –– Fundamentals (from Lecture Note of Prof. V. Fundamentals (from Lecture Note of Prof. V. VitekVitek, 2002), 2002)


Monte Carlo Monte Carlo –– Distribution of states in real systemsDistribution of states in real systems

f(x)


a b x

Monte Carlo Monte Carlo –– General Monte Carlo AlgorithmGeneral Monte Carlo Algorithm


Monte Carlo Monte Carlo –– Canonical Ensemble (N,V,T)Canonical Ensemble (N,V,T)


Monte Carlo Monte Carlo –– Metropolis MethodMetropolis Method


Monte Carlo Monte Carlo –– Examples of Metropolis MethodExamples of Metropolis Method


Monte Carlo Monte Carlo –– Magnetization vs. Temperature by Ising model of spinsMagnetization vs. Temperature by Ising model of spins


Open System with a heat reservoir Open System with a heat reservoir –– Grand Canonical ensembleGrand Canonical ensemble

ii nnNNW ln~

ln~

ln ∑−=

0lnln max =−= ∑ ii ndnWd

0=∑dn 0=∑ nαδ

iri

ri

i EpEU ,

,

∑>=≡< iri

ri

EnUN ,

,

~∑=

ri

ri

nN ,

,

~∑=Number of µVT systems with Ei & Nr:

Average Energy of a system:

Average # of ptl. of a system:rri

ri

NpN ,

,

∑>=<rri

ri

NnNN ,

,

~∑>=<


0, =∑ ridn 0, =∑ rinαδ

0ln , =−++ riri NEn γβα ri NE

ri eenγβα +−−=,

0, =∑ riidnE

ri

ri

NE

ri

NEri

rie

e

N

np

γβ

γβ

+−

+−

∑==

,

,

, ~

0, =∑ riidnEβ

ri NE

ri

e

Ne

γβ

α

+−

−

∑=

,

~

ri

ri

NE

ri

NE

i

ri

e

eE

Uγβ

γβ

+−

+−

∑

∑=

,

,

0, =∑ rirdnN 0, =∑ rirdnNγ

ri

ri

NE

ri

NE

r

ri

e

eN

Nγβ

γβ

+−

+−

∑

∑>=<

,

,

Monte Carlo Monte Carlo –– Grand Canonical Ensemble (Grand Canonical Ensemble (µµ,V,T),V,T)

The number of particles is variable

- fluctuations of the concentration are allowed

Generally, select randomly one of the procedures


1) Change of the configuration

exp(-∆Ep/kT)

2) Creation of a particle

exp[-(∆Ep-µ)/kT]

3) Destruction of particles

exp[-(∆Ep+µ)/kT]

Monte Carlo Monte Carlo –– Modified Grand Canonical Ensemble (Modified Grand Canonical Ensemble (µµ,V,T),V,T)


Monte Carlo Monte Carlo –– ApplicationsApplications

Segregation to an interface

Order-disorder transition in binary alloy


Order-disorder transition in binary alloy

Morphology Evolution of an Alloy Particle

MD+MC+MS Hybrid Simulation

Phase Separation and Surface Segregation in CuPhase Separation and Surface Segregation in Cu--Ni Thin Films Ni Thin Films

298 K298 K 373 K373 K


800 K800 K473 K473 K

Order/Disorder in CoOrder/Disorder in Co--Pt System Pt System -- S.I. Park et al., Scripta Mater., 2001.S.I. Park et al., Scripta Mater., 2001.


Property Pt3Co PtCo PtCo3Cohesive Energy 5.500 5.215 4.873(eV/atom) 5.555±0.017 5.228±0.005

Lattice Constant a=3.833 3.754, c/a=.98 3.625(Å) a=3.831 3.745, c/a=.98 3.668

Transition 1070-1080 970-980 760-770Temperature (K) 1000 1100 840

Pure W W + 0.4wt% Ni Vacuum Annealing

Effect of Interface Energy Anisotropy on Morphological EvolutionEffect of Interface Energy Anisotropy on Morphological Evolution


Pure W WPure W W--14at%Ni14at%Ni

Monte Carlo Monte Carlo –– Determination of Phase DiagramDetermination of Phase Diagram


A Modified Embedded Atom Method Interatomic Potential for the Cu-Ni SystemByeong-Joo Lee and Jae-Hyeok Shim, CALPHAD 28, 125-132 (2004).

Monte Carlo Monte Carlo – Deposition or Etching


monte carlo simulation - postech cmse lab. · 2013-02-19 · monte carlo – grand canonical...

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