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