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MolecularDynamics.ppt
1. Joo Chul Yoon
with Prof. Scott T. Dunham
Electrical Engineering
University of Washington
Molecular Dynamics Simulations
2. Simulation Setup
Force Calculation and MD Potential
Integration Method
Introduction to MD
MD Simulations of Silicon Recrystallization
SW Potential
Tersoff Potential
Contents
Simulation Preparation
3. Introduction to Molecular Dynamics
Calculate how a system of particles evolves in time
Consider a set of atoms with positions /velocities
and the potential energy function of the system
Predict the next positions of particles
over some short time interval
by solving Newtonian mechanics
4. Basic MD Algorithm
Set initial conditions and
Get new forces
Solve the equations of motion
numerically over a short step
Is ?
Calculate results and finish
)
( 0
t
i
r )
( 0
t
i
v
)
( i
i r
F
)
(
)
( t
t
t i
i
r
r
)
(
)
( t
t
t i
i
v
v
t
t
t
t
max
t
t
6. Simulation Cell
Open boundary
for a molecule or nanocluster in vacuum
not for a continuous medium
usually using orthogonal cells
Fixed boundary
fixed boundary atoms
completely unphysical
Periodic boundary conditions
obtaining bulk properties
7. Periodic boundary conditions
An atom moving out of boundary
comes back on the other side
cut
r
considered in force calculation
2
L
rcut
8. pair potential calculation
atoms move per time step
)
N
( 2
O
Constructing neighboring cells
A
2
.
0
not necessary to search all atoms
Verlet neighbor list
containing all neighbor atoms within
updating every time steps
2
t
N
r
r L
cut
L
v
L
r
L
N
where
i
L
r
cut
r
skin
9. Linked cell method
Constructing neighbor cells
divide MD cell into smaller subcells :
The length of subcell is chosen so that
n
n
n
L
r
n
L
l
l
L: the length of MD cell
3
/ n
N
NC
going through 27 atom pairs
C
NN
instead )!
1
(
N
N
where
26 skin cells
)
N
(
O
reducing it to
L
r
11. Initial Velocities
The probability of finding a particle with speed
Maxwell-Boltzmann distribution
T
k
m
T
k
m
P B
x
B
x /
2
1
exp
2
)
( 2
2
/
1
v
v
Generate random initial atom velocities
2
2
1
2
3
v
m
T
kB
scaling T with equipartition theorem
12. MD Time Step
1/20 of the nearest atom distance
t
r/
In practice fs.
4
t
MD is limited to <~100 ns
Too long : energy is not conserved
t
13. Temperature Control
Velocity Scaling
Nose-Hoover thermostat
Scale velocities to the target T
Efficient, but limited by energy transfer
Larger system takes longer to equilibrate
Fictitious degree of freedom is added
Produces canonical ensemble (NVT)
Unwanted kinetic effects from T oscillation
14. Verlet Method
Integration Method
Predictor-Corrector
Finite difference method
Numerical approximation of the integral over time
Better long-tem energy conservation
Not for forces depending on the velocities
Long-term energy drift (error is linear in time)
Good local energy conservation (minimal fluctuation)
15. ))
(
(
1
)
( t
m
r
F
r
a
2
)
(
2
1
)
(
)
(
)
( t
t
t
t
t
t
r
a
v
r
r
t
t
t
t
t
)
(
2
1
)
(
)
2
/
( r
a
v
v
))
(
(
1
)
( t
t
m
t
t
r
a
a
t
t
t
t
t
t
t
)
(
2
1
)
2
/
(
)
( a
v
v
Verlet Method
From the initial ,
)
(t
i
r )
(t
i
v
Obtain the positions and velocities at t
t
16. ))
(
(
1
)
( t
m
r
F
r
a
2
2
)
(
)
(
)
(
)
( t
t
t
t
t
t
t
P
a
v
r
r
t
t
t
t
t
P
)
(
)
(
)
( a
v
v
t
t
t
t
t iii
P
)
(
)
(
)
( r
a
a
: 3rd order derivatives
Predictor-Corrector Method
from the initial ,
)
(t
i
r
)
( t
t
i
v
predict , using a Taylor series
)
( t
t
i
r
)
(t
i
v
iii
r
Predictor Step
17. )
(
2
)
(
)
(
2
0 t
t
t
C
t
t
t
t P
a
r
r
)
(
)
(
)
( 1 t
t
t
C
t
t
t
t P
a
v
v
: constants depending accuracy
)
(
)
(
)
( t
t
t
t
t
t P
C
a
a
a
m
t
t
P
C ))
(
(
)
(
r
F
r
a
get corrected acceleration
using error in acceleration
correct positions and velocities
n
C
Corrector Step
Predictor-Corrector Method
18. ij
N
i
j ij
ij
i
r
r
u
U r̂
)
(
)
r
(
F
The force on an atom is determined by
: potential function
: number of atoms in the system
: vector distance between
atoms i and j
Force Calculation
)
r
(
U
N
ij
r
19. MD Potential
Classical Potential
...
)
,
,
(
)
,
(
)
(
,
,
3
,
2
1
k
j
i
k
j
i
j
i
j
i
i
i
U
U
U
U r
r
r
r
r
r
: Single particle potential
Ex) external electric field, zero if no external force
: Pair potential only depending on
: Three-body potential with an angular dependence
)
( ij
ij r
U
i
r
F
j k
jki
j
ji
ij
i
i V
V
V )
(
F
1
U
2
U
3
U
20. Using Classical Potential
Born-Oppenheimer Approximation
Consider electron motion for fixed nuclei ( )
Assume total wavefunction as )
,
(
)
(
)
,
( i
i
i R
r
R
r
R
: Nuclei wavefunction
: Electron wavefunction
parametrically depending on
)
( i
R
)
,
( i
R
r
i
R
0
M
me
The equation of motion for nuclei is given by
i
i
i
i
N U
M
P
H )
(
2
2
R (approximated to classical motion)
21. Empirical Potential
Semi-empirical Potential
Ab-initio MD
MD Potential Models
functional form for the potential
fitting the parameters to experimental data
Ex) Lennard-Jones, Morse, Born-Mayer
calculate the electronic wavefunction
for fixed atomic positions from QM
Ex) EAM, Glue Model, Tersoff
direct QM calculation of electronic structure
Ex) Car-Parrinello using plane-wave psuedopotential
22. Stillinger-Weber Potential
works fine with crystalline and liquid silicon
)
,
,
(
)
,
(
,
,
3
,
2 k
j
i
k
j
i
j
i
j
i
U
U
U r
r
r
r
r
)
/
(
)
( 2
2
ij
ij r
f
r
U
)
/
,
/
,
/
(
)
,
,
( 3
3
k
j
i
k
j
i f
U r
r
r
r
r
r
: energy and length units
,
Pair potential function
)
(
2 r
f
a
r
a
r
r
Br
A q
p
,
]
)
exp[(
)
( 1
a
r
,
0
23. )
,
,
(
)
,
,
(
)
,
,
(
3 ijk
jk
ji
jik
ik
ij
k
j
i r
r
h
r
r
h
f
r
r
r
)
,
,
( ikj
kj
ki r
r
h
]
)
(
)
(
exp[
)
,
,
( 1
1
a
r
a
r
r
r
h ik
ij
jik
ik
ij
2
)
3
1
(cos
jik
Three body potential function
Stillinger-Weber Potential
24. too low coordination in liquid silicon
incorrect surface structures
incorrect energy and structure for small clusters
Bond-order potential for Si, Ge, C
Limited by the cosine term
not for various equilibrium angles
2
)
3
1
(cos
jik
forces the ideal tetrahedral angle
Stillinger-Weber Potential
bond strength dependence on local environment
Tersoff, Brenner
25. Tersoff Potential
environment dependence without
absolute minimum at the tetrahedral angle
The more neighbors, the weaker bondings
)
(
)
( ij
attractive
ijk
ij
repulsive r
U
b
r
U
U
: environment-dependent parameter
weakening the pair interaction
when coordination number increases
ijk
b
cluster-functional potential
26. )]
(
)
(
)[
( ij
A
ij
ij
R
ij
ij
C
ij r
f
b
r
f
a
r
f
U
r
R Ae
r
f 1
)
(
r
A Be
r
f 2
)
(
repulsive part
attractive part
)
(r
fC
D
R
r
,
1
D
R
r
D
R
D
R
r
,
)
(
2
sin
2
1
2
1
D
R
r
,
0
potential cutoff function
where
Tersoff Potential
27. n
n
ij
n
ij
b 2
/
1
)
1
(
]
)
(
exp[
)
(
)
( 3
3
3
,
ik
ij
jik
ik
j
i
k
C
ij r
r
g
r
f
2
2
2
2
2
)
cos
(
1
)
(
h
d
c
d
c
g
n
n
ij
n
ij
a 2
/
1
)
1
(
]
)
(
exp[
)
( 3
3
3
,
ik
ij
ik
j
i
k
C
ij r
r
r
f
Tersoff Potential
28. Simulation Setup
Force Calculation and MD Potential
Integration Method
Introduction to MD
MD Simulations of Silicon Recrystallization
SW Potential
Tersoff Potential
Contents
Simulation Preparation
41. Tersoff Potential
Melting temperature of Tersoff: about 2547K
Potential energy per particle versus temperature:
the system with a/c interface is heated by
adding energy at a rate of 1000K/ns
42. Tersoff Potential
As in recrystallized Si :
0.82 in amorphized Si
0.20 in crystalline Si
ps
100
/
A
2
o
ps
100
/
A
2
o
43. Tersoff Potential
As in recrystallized Si :
0.82
in amorphized Si
0.20
in crystalline Si
ps
100
/
A
2
o
ps
100
/
A
2
o
