Kinetic Monte Carlo Simulation of Dopant Diffusion in Crystalline
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Kinetic Monte Carlo Simu lation of Dopant Diffusi on in Crystalline CEC, Inha University Chi-Ok Hwang
description
Kinetic Monte Carlo Simulation of Dopant Diffusion in Crystalline. CEC, Inha University Chi-Ok Hwang. Kinetic Monte Carlo (KMC). MD vs KMC -MD time-spanning problem: automatic time increment adjustment in KMC -KMC (residence-time or n-fold way or Bortz-Kalos-Liebowitz (BKL) ) - PowerPoint PPT Presentation
Transcript of Kinetic Monte Carlo Simulation of Dopant Diffusion in Crystalline
Kinetic Monte Carlo Simulation of Dopant Diffusion in Crystalline
CEC, Inha UniversityChi-Ok Hwang
Kinetic Monte Carlo (KMC)
• MD vs KMC -MD time-spanning problem: automatic time increment adju
stment in KMC -KMC (residence-time or n-fold way or Bortz-Kalos-Liebowitz
(BKL) )• KMC conditions (J. Chem. Phys. 95(2), 1090-1096) - dynamical hierarchy - proper time increments for each successful event - independence of each possible events in system
KMC
• Markovian Master Equation: time evolution of probability density
- : transition probability per unit time- : successive states of the system• Detailed balance
i i
tPWtPWt
tPfififi
f
),()(),()(),(
)( fiW
fi and
),()(),()( tPWtPW fififi
Poisson Distribution
• Three assumptions of Poisson distribution - 1.
- 2.
- 3. Events in nonoverlapping time intervals are statistically independent
0
)(),;1(
t
ttrtttP
,...3,2,0
0),;(
kt
tttkP
KMC time increment
t
nr
t
lim
,0
평균적 발생 확률
t 시간 동안 ne 번의 사건이 발생할 확률
rt
e
n
nnn
ene
en
rt
rrn
nnP
e
ee
!
)(
)1()()( lim0,
KMC time increment
• KMC time increment
0
1
RdtRtet Rt
R
U
edtT RRt
ln
1Re'(0
'
RteRtp )(
Example
• Jump over the barrier due to thermal activation: Boltzmann distribution
- ω0: attempt frequency, vibration frequency of the atom (order of 1/100 fs) independent of T in solids
•
- D: diffusivity - λ: jump distance
TkE BbeTR /)(
)/6()( 2DTR
End
Example
Parameter settingSet the time t =0
Initialize all the rates of all possible transitions in the system
Calculate the cumulative function Ri
Select next event randomly
Carry out the event
Get a uniform random number
Update configuration & time increment
Desired time is reached ?
Start
