1. Classical Molecular Dynamics CEC, Inha University Chi-Ok Hwang

2. Perspectives Empirical methods - classical molecular dynamics - tight-binding methods First-principles methods - tight-binding methods - density-functional theory - exact methods; quantum MC

3. Molecular Dynamics: General Solving classical equations of motion for a system of N molecules interacting via a potential V V ≈ Σ V 1 (r i ) + Σ Σ V eff 2 (r ij ) Lennard-Jones 12-6 potential V IJ (r)= 4ε ((σ/r) 12 -(σ/r) 6 )

4. 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)

5. 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

6. 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; }

7. Molecular Dynamics: General Potential truncation Cell method: linked list and non-overlapping nearby cell sweeping Thermodynamic quantities - kinetic temperature

8. Molecular Dynamics: General - pressure

9. Molecular Dynamics: General Mean square displacement: Einstein relation radial distribution function g(r) Green-Kubo relation

10. Ion Implantation Simulation size: cascade size (10 -25 cm 3 (M.-J. Caturla etc, 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, 16683, 1996) Si density: 5 x 10 22 /cm 3 (5.43Å unit cell, 8/unit cell) ion dose: 10 17 ions/cm 2

11. Ion Implantation ion implantation Potential: BCA - nuclear stopping power; elastic collision V ij (r) = Z i Z j e 2 /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

12. Ion Implantation Simulations of ion implantation - Full MD - Recoil Interaction Approximation (RIA) (1- 100 keV) - BCA: valid for low-mass ions at incident energies from 1-15 keV (M.-J. Caturla, etc, PRB 54, 16683, 1996)

13. 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 Microscopy ((EF)TEM)

14. 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 its 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 distance difference in the mean range about 300 eV

15. MDRANGE nuclear stopping power - ZBL-type as default - Mazzone (Morse+harmonic well) for tetrahedral semiconductors 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 file - Puska-Echenique-Nieminen-Richie (PENR) model (MDRANGE3.0) - Brandt-Kitakawa (BK) model (MDRANGE3.0)

16. MDRANGE Dose range: Damage build-up modeling (2002) - Amorphization level is proportional to the nuclear deposited energy in that depth region - Electronic stopping ① point-like ion and a spherically symmetric electron distribution ② maximum distance of the charge distribution of Si, 1.47 Å - Using 20 predamaged boxes

17. MDRANGE Time step Δt min = min( k t /v, E t /Fv, 1.1 Δt old ) - k t, E t are proportional constants - inversely proportional to the recoil velocity - inversely proportional to the product of the total force F the recoil atom experiences and its velocity v - not to increase more than 10% from its previous value

18. MDRANGE Simulation cell - less than r 0 (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

19. MDRANGE Recoil event calculation - placing the recoil atom of desired energy and velocity a few Å outside the simulation cell with z=0 - recoil atom threshold energy, 1 eV - electronic stopping (S e ): Δv=Δt S e /m where m is the ion mass - calculating nuclear and electronic deposited energies - energy losses of the recoil atom are evaluated for each time step and stored in arrays as a function of the depth - nuclear energy loss = total energy loss - electronic energy loss

20. etc Jeong-Won Kang’s local damage accumulation model E ion : deposited energy in a unit cell R D : dose rate (neglected) M 1 : target material atom weight M 2 : projectile atom weight R X : relaxation and recombination effects n coil : coil events rate

21. 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