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

5. Constructing neighboring cells Simulation Cell Boundary Condition Initial atom velocities MD Time step Simulation Setup Temperature Control

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 considered in force calculation

8. pair potential calculation atoms move per time step Constructing neighboring cells not necessary to search all atoms Verlet neighbor list containing all neighbor atoms within updating every time steps where skin

9. Linked cell method Constructing neighbor cells divide MD cell into smaller subcells : The length of subcell is chosen so that : the length of MD cell going through 27 atom pairs instead where 26 skin cells reducing it to

10. Constructing neighboring cells Simulation Cell Boundary Condition Initial atom velocities MD Time step Simulation Setup Temperature Control

11. Initial Velocities The probability of finding a particle with speed Maxwell-Boltzmann distribution Generate random initial atom velocities scaling T with equipartition theorem

12. MD Time Step 1/20 of the nearest atom distance In practice fs. MD is limited to <~100 ns Too long : energy is not conserved

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. Verlet Method From the initial, Obtain the positions and velocities at

16. : 3rd order derivatives Predictor-Corrector Method from the initial, predict, using a Taylor series Predictor Step

17. : constants depending accuracy get corrected acceleration using error in acceleration correct positions and velocities Corrector Step Predictor-Corrector Method

18. 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

19. MD Potential Classical Potential : Single particle potential Ex) external electric field, zero if no external force : Pair potential only depending on : Three-body potential with an angular dependence

20. Using Classical Potential Born-Oppenheimer Approximation Consider electron motion for fixed nuclei ( ) Assume total wavefunction as : Nuclei wavefunction : Electron wavefunction parametrically depending on The equation of motion for nuclei is given by (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 : energy and length units Pair potential function

23. 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 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 : environment-dependent parameter weakening the pair interaction when coordination number increases cluster-functional potential

26. repulsive part attractive part potential cutoff function where 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

29. MD Simulation Setup 5 TC layer 1 static layer 4 x 4 x 13 cells Initial Setup

30. MD Simulation Setup Ion Implantation(1 keV) Cooling to 0K System Preparation

31. Recrystallization 1200 K for 0.5 ns

32. Recrystallization Crystal Ratea/c interface displacement SW Potential 1200K

33. 6 TC layer MD Simulation Setup Initial Setup 5 x 5 x 13 cells 1 static layer

34. System Preparation Ion Implantation(1 keV) Cooled to 0K MD Simulation Setup

35. Recrystallization 1900 K for 0.85 ns

36. Recrystallization Tersoff Potential 1900K Crystal Ratea/c interface displacement

37. Recrystallization Tersoff Potential 1900KSW Potential 1200K Crystal Rate

38. Recrystallization Tersoff Potential 1900KSW Potential 1200K a/c interface displacement

39. 6 TC layer MD Simulation Setup Initial Setup 2 x 2 x 13 cells 1 static layer

40. Recrystallization 1800 K for 20 ns

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

43. Tersoff Potential As in recrystallized Si : 0.82 in amorphized Si 0.20 in crystalline Si

44. Summary Review Molecular Dynamics MD simulation for recrystallization of Si with SW, Tersoff with As