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Classical Molecular Dynamics
CEC, Inha UniversityChi-Ok Hwang
Perspectives
• Empirical methods - classical molecular dynamics - tight-binding methods• First-principles methods - tight-binding methods - density-functional theory - exact methods; quantum MC
Molecular Dynamics: General
• Solving classical equations of motion for a system of N molecules interacting via a potential V
V ≈ Σ V1(ri) + Σ Σ Veff2(rij)
• Lennard-Jones 12-6 potential V IJ(r)= 4ε ((σ/r)12-(σ/r)6)
Molecular Dynamics: General
• Algorithms 1: Verlet algorithm r(t+δt)=r(t) + δtv(t)+1/2 (δt)2 a(t) (1) r(t-δt)=r(t) - δtv(t)+1/2 (δt)2 a(t) (2) from the above two equations, we get r(t+δt)= 2r(t) - r(t-δt) + (δt)2 a(t) v(t) = (r(t+δt) - r(t-δt))/(2δt)
Molecular Dynamics: General
• Algorithms 2: Leap-Frog algorithm r(t+δt)=r(t) + δt v(t+δt/2) v(t+δt/2) = v(t-δt/2) + δt a(t); update first• Algorithms 3: Velocity Verlet algorithm r(t+δt)= r(t) + δt v(t) + (δt)2/2 a(t) v(t+δt) = v(t) + δt (a(t) + a(t+δt))/2
Molecular Dynamics: General
• Periodic boundary conditions 1) for( i=1;i <= Cell_N_x; i++){ Cell_P[i] = i+1; Cell_M[i] = i-1; } Cell_P[Cell_N_x] = 1; Cell_M[1] = Cell_N_x; 2) while( (*xnew) < 0 ){ *xnew = *xnew + Sx; }
Molecular Dynamics: General
• Potential truncation• Cell method: linked list and non-
overlapping nearby cell sweeping• Thermodynamic quantities - kinetic temperature
N
iii tvm
dNN
tK
dtkT
tKtkTd
N
1
2 )(1)(2
)(
)()(2
Molecular Dynamics: General
- pressure
jiijij
jiijij
ideal
tFtrdNkTNkT
PV
tFtrdV
tkTV
NtP
tkTV
NP
))()((1
1
)()(1
)()(
)(
Molecular Dynamics: General
• Mean square displacement: Einstein relation
• radial distribution function g(r)
• Green-Kubo relation
i ij
ijrrN
Vrg )()(
2
DttR 6)(2
0
)()0(3
1tvvdtD
Ion Implantation• Simulation size: cascade size (10-25 cm3 (M.-J. Caturla et
c, PRB 54, 16683, 1996) ) - 1000 atoms (J.B. Gibson etc, PR 120(4), 1229, 1960) - a few hundreds of thousands of atoms (J. Frantz etc, PRB
64, 125313 , 2001)• Time scales - thermal vibration periods of atoms in solids: 0.1 ps (10-13
sec) or longer - cascade lifetime: 10 ps (M.-J. Caturla etc, PRB 54, 1668
3, 1996) • Si density: 5 x 1022 /cm3 (5.43Å unit cell, 8/unit cell) • ion dose: 1017 ions/cm2
Ion Implantation
• ion implantation Potential: BCA - nuclear stopping power; elastic collision Vij(r) = Zi Zje2 /r Φ(r) Φ(r); screening of the nuclei due to the electron cloud ① Thomas-Fermi ② ZBL; universal screening potential - electronic stopping power; frictional force ③ Stillinger-Weber potential
Ion Implantation
• Simulations of ion implantation - Full MD - Recoil Interaction Approximation (RIA) (1-1
00 keV) - BCA: valid for low-mass ions at incident en
ergies from 1-15 keV (M.-J. Caturla, etc, PRB 54, 16683, 1996)
Ion Implantation• Three phases of collision cascade- collisional phase (0.1-1 ps) - thermal spike (1 ns) - relaxation phase (a few thousands of fs)• Measurements of depth profiling - Rutherford Backscattering Spectroscopy (RBS) - Secondary Ion Mass Spectroscopy (SIMS) - (Energy-Filtered) Transmission Electron Microscop
y ((EF)TEM)
MDRANGE
• Calculating range profiles of ions implanted into crystalline materials as a histogram of the maximum penetration depths of 10,000-20,000 ions
• Modification of full MD MOLDY code • Recoil interaction approximation (RIA) - considering only interactions between the recoil ion and i
ts nearest neighbors within a certain distance - better than BCA but about ten times slower than BCA • 0.1-100 keV energy range: one fourth of the interatomic d
istance difference in the mean range about 300 eV
MDRANGE• nuclear stopping power - ZBL-type as default - Mazzone (Morse+harmonic well) for tetrahedral semicond
uctors in initial state calculation - Morse-type potential for metals in initial state calculation• Electronic stopping power models - Non-local electronic stopping power read in from input fil
e - Puska-Echenique-Nieminen-Richie (PENR) model (MDRAN
GE3.0) - Brandt-Kitakawa (BK) model (MDRANGE3.0)
MDRANGE
• Dose range:• Damage build-up modeling (2002) - Amorphization level is proportional to the nuclea
r deposited energy in that depth region - Electronic stopping ① point-like ion and a spherically symmetric elec
tron distribution ② maximum distance of the charge distribution o
f Si, 1.47 Å - Using 20 predamaged boxes
MDRANGE
• Time step Δtmin = min( kt /v, Et /Fv, 1.1 Δtold )
- kt, Et are proportional constants
- inversely proportional to the recoil velocity - inversely proportional to the product of the total f
orce F the recoil atom experiences and its velocity v
- not to increase more than 10% from its previous value
MDRANGE• Simulation cell - less than r0 (2-3 Å in ZBL) - a simulation cell with a side length of 10-15 Å (a
cell containing 50-100 atoms) - translation method• Structure of the sample - atom coordinates of all atoms except the recoil
atom are read in from a file at the beginning of the simulation
- polycrystalline materials: grain size is calculated using a Gaussian distribution and orientation of each grain is selected randomly
- multilayered structures with depth regions and the same size of the simulation cell
MDRANGE• Recoil event calculation - placing the recoil atom of desired energy and velocity a f
ew Å outside the simulation cell with z=0 - recoil atom threshold energy, 1 eV - electronic stopping (Se): Δv=Δt Se /m where m is the ion
mass - calculating nuclear and electronic deposited energies - energy losses of the recoil atom are evaluated for each ti
me step and stored in arrays as a function of the depth - nuclear energy loss = total energy loss - electronic energ
y loss
etc
• Jeong-Won Kang’s local damage accumulation model
Eion: deposited energy in a unit cell
RD: dose rate (neglected)
M1: target material atom weight
M2: projectile atom weight
RX: relaxation and recombination effects
ncoil: coil events rate
),,,,,( 21 recoilXDionD nRMMREf
Future Studies
• Area and ion dose criteria where local accumulated damages affect implanted ion range
• Damage accumulation model• Different stopping powers• Full MD criteria for ultra-low energy implantation• Ion-beam amorphization modeling (of silicon)• Multi-ion recoil approximation