# Molecular Dynamics

Document Sample

```					Molecular Dynamics Simulations

Joo Chul Yoon

with Prof. Scott T. Dunham

Electrical Engineering
University of Washington
Contents

Introduction to MD

Simulation Setup
Integration Method
Force Calculation and MD Potential

MD Simulations of Silicon Recrystallization

Simulation Preparation
SW Potential
Tersoff Potential
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
Basic MD Algorithm

Set initial conditions ri (t0 ) and v i (t0 )

Get new forces Fi (ri )

Solve the equations of motion
numerically over a short step t
ri (t )  ri (t  t )
v i (t )  v i (t  t )

t  t  t

Is t  tmax ?

Calculate results and finish
Simulation Setup

Simulation Cell

Boundary Condition
Constructing neighboring cells

Initial atom velocities

MD Time step

Temperature Control
Simulation Cell

usually using orthogonal cells

Open boundary
for a molecule or nanocluster in vacuum
not for a continuous medium

Fixed boundary
fixed boundary atoms
completely unphysical

Periodic boundary conditions

obtaining bulk properties
Periodic boundary conditions

An atom moving out of boundary

comes back on the other side

considered in force calculation
L
rcut 
2

rcut
Constructing neighboring cells

pair potential calculation  O ( N )
2


atoms move  0.2 A per time step
not necessary to search all atoms
Verlet neighbor list

containing all neighbor atoms within rL
rc ut
updating every N L time steps                       i
rL
N L vt
where rL  rcut 
2                           skin
Constructing neighbor cells

divide MD cell into smaller subcells :   nnn
The length of subcell l    is chosen so that
L
l         rL            L: the length of MD cell
n
going through 27 NN C atom pairs
rL
instead N ( N  1)!
where N C  N / n
3

reducing it to  O( N)
26 skin cells
Simulation Setup

Simulation Cell

Boundary Condition
Constructing neighboring cells

Initial atom velocities

MD Time step

Temperature Control
Initial Velocities

Maxwell-Boltzmann distribution

The probability of finding a particle with speed
1/ 2
 m               1             
 2k T 
P( v x )               exp  mv 2 / k BT 
x
    B             2            

Generate random initial atom velocities
scaling T with equipartition theorem
3       1
k BT  mv 2
2       2
MD Time Step

Too long   t :      energy is not conserved
r/t  1/20 of the nearest atom distance
In practice   t  4 fs.
MD is limited to <~100 ns
Temperature Control

Velocity Scaling
Scale velocities to the target T
Efficient, but limited by energy transfer
Larger system takes longer to equilibrate

Nose-Hoover thermostat

Fictitious degree of freedom is added
Produces canonical ensemble (NVT)
Unwanted kinetic effects from T oscillation
Integration Method

Finite difference method
Numerical approximation of the integral over time

Verlet Method

Better long-tem energy conservation
Not for forces depending on the velocities

Predictor-Corrector

Long-term energy drift (error is linear in time)
Good local energy conservation (minimal fluctuation)
Verlet Method

From the initial ri (t ) , v i (t )
1
a(r)  F(r(t ))
m
Obtain the positions and velocities at t  t
1
r(t  t )  r (t )  v(t )t  a(r )t 2
2
1
a(t  t )  a(r (t  t ))
m
1
v(t  t / 2)  v(t )t  a(r )t
2
1
v(t  t )  v(t  t / 2)  a(t  t )t
2
Predictor-Corrector Method

Predictor Step

from the initial ri (t ) , v i (t )
1
a(r)  F(r(t ))
m
predict ri (t  t ) , v i (t  t ) using a Taylor series
a(t ) 2
r (t  t )  r (t )  v(t )t 
P
t
2
v P (t  t )  v(t )  a(t )t

a P (t  t )  a(t )  r iii (t )t

r iii : 3rd order derivatives
Predictor-Corrector Method

Corrector Step

get corrected acceleration
F (r P (t  t ))
a C (r ) 
m
using error in acceleration
a(t  t )  a C (t  t )  a P (t  t )
correct positions and velocities
t 2
r (t  t )  r P (t  t )  C0      a(t  t )
2
v(t  t )  v P (t  t )  C1ta(t  t )

C n : constants depending accuracy
Force Calculation

The force on an atom is determined by
N      u (rij )
Fi  U (r )                     ˆ
rij
j i     rij

U (r ) : potential function

N    : number of atoms in the system
rij : vector distance between
atoms i and j
MD Potential

Classical Potential

U  U1 (ri )  U 2 (ri , r j )  U 3 (ri , r j , rk )  ...
i            i, j             i , j ,k

U 1 : Single particle potential
Ex) external electric field, zero if no external force

U 2 : Pair potential only depending on Fij   ri U ( rij )

U 3 : Three-body potential with an angular dependence
                          
Fi  i  (Vij  V ji )  V jki 
 j                j k     
Using Classical Potential

Born-Oppenheimer Approximation

Consider electron motion for fixed nuclei (
me
M    0 )

Assume total wavefunction as  (R i , r )   (R i ) (r , R i )
(R i )      : Nuclei wavefunction
 (r , R i ) : Electron wavefunction
parametrically depending on R i
The equation of motion for nuclei is given by
Pi 2
HN          U (R i )   (approximated to classical motion)
i 2M i
MD Potential Models

Empirical Potential
functional form for the potential
fitting the parameters to experimental data
Ex) Lennard-Jones, Morse, Born-Mayer
Semi-empirical Potential
calculate the electronic wavefunction
for fixed atomic positions from QM
Ex) EAM, Glue Model, Tersoff
Ab-initio MD
direct QM calculation of electronic structure
Ex) Car-Parrinello using plane-wave psuedopotential
Stillinger-Weber Potential

works fine with crystalline and liquid silicon

U  U 2 (ri , r j )  U 3 (ri , r j , rk )
i, j                  i , j ,k


U 2 (rij )   f 2(rij /  )

U 3 (ri , r j , rk )   f3 (ri /  , r j /  , rk /  )

 , : energy and length units

Pair potential function


A( Br  p  r  q ) exp[( r  a) 1 ] , r  a
f 2( r ) 
0, r a
Stillinger-Weber Potential

Three body potential function

f3 (ri , r j , rk )  h(rij , rik , jik )  h(rji , rjk , ijk )

 h(rki , rkj , ikj )
h(rij , rik , jik )   exp[ (rij  a)1   (rik  a)1 ]
1
 (cos jik  ) 2
3
Stillinger-Weber Potential

1
Limited by the cosine term (cos jik  ) 2
3
forces the ideal tetrahedral angle
not for various equilibrium angles
too low coordination in liquid silicon
incorrect surface structures

incorrect energy and structure for small clusters

Bond-order potential for Si, Ge, C

bond strength dependence on local environment
Tersoff, Brenner
Tersoff Potential

cluster-functional potential
environment dependence without
absolute minimum at the tetrahedral angle

The more neighbors, the weaker bondings

U  U repulsive (rij )  bijk U attractive (rij )

bijk : environment-dependent parameter
weakening the pair interaction
when coordination number increases
Tersoff Potential

U ij  f C (rij )[ aij f R (rij )  bij f A (rij )]

where

repulsive part         f R ( r )  Ae  1r

attractive part        f A ( r )   Be  2 r

potential cutoff function


1,       r  RD
1 1      (r  R) 
f C (r )          sin            , R  D  r  R  D
2 2    2     D 
0,        r  RD
Tersoff Potential

bij  (1    )      n     n 1/ 2 n
ij

 ij     fC (rik ) g ( jik ) exp[3 (rij  rik )3 ]
k i , j
3

c2        c2
g ( )  1  2  2
d   d  (h  cos ) 2
aij  (1    )      n    n 1/ 2 n
ij

ij        fC (rik ) exp[3 (rij  rik )3 ]
k i , j
3
Contents

Introduction to MD

Simulation Setup
Integration Method
Force Calculation and MD Potential

MD Simulations of Silicon Recrystallization

Simulation Preparation
SW Potential
Tersoff Potential
MD Simulation Setup

Initial Setup

    5 TC layer

1 static layer

4 x 4 x 13 cells
MD Simulation Setup

System Preparation

Ion Implantation(1 keV)      Cooling to 0K
Recrystallization

1200 K for 0.5 ns
Recrystallization

SW Potential 1200K

Crystal Rate                  a/c interface displacement
MD Simulation Setup

Initial Setup

     6 TC layer

1 static layer

5 x 5 x 13 cells
MD Simulation Setup

System Preparation

Ion Implantation(1 keV)       Cooled to 0K
Recrystallization

1900 K for 0.85 ns
Recrystallization

Tersoff Potential 1900K

Crystal Rate                    a/c interface displacement
Recrystallization

Crystal Rate

SW Potential 1200K             Tersoff Potential 1900K
Recrystallization

a/c interface displacement

SW Potential 1200K             Tersoff Potential 1900K
MD Simulation Setup

Initial Setup

     6 TC layer

1 static layer

2 x 2 x 13 cells
Recrystallization

1800 K for 20 ns
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
Tersoff Potential

As in recrystallized Si :
o2
0.82 A / 100 ps in amorphized Si
o2
0.20 A / 100 ps in crystalline Si
Tersoff Potential

As in recrystallized Si :
o2
0.82 A / 100 ps
in amorphized Si
o2
0.20 A / 100 ps
in crystalline Si
Summary

Review Molecular Dynamics

MD simulation for recrystallization of Si
with SW, Tersoff with As

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 3 posted: 9/2/2011 language: English pages: 44
How are you planning on using Docstoc?