Molecular Dynamics Simulations Joo Chul Yoon with Prof

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 and Get new forces Solve the equations of motion numerically over a short step Is ? 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

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

Constructing neighbor cells Linked cell method divide MD cell into smaller subcells : The length of subcell is chosen so that : the length of MD cell going through 27 instead where atom pairs reducing it to 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 Generate random initial atom velocities scaling T with equipartition theorem

MD Time Step Too long : energy is not conserved 1/20 of the nearest atom distance In practice 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 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 , Obtain the positions and velocities at

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

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

Force Calculation The force on an atom is determined by : potential function : number of atoms in the system : vector distance between atoms i and j

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

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)

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 : energy and length units Pair potential function

Stillinger-Weber Potential Three body potential function

Stillinger-Weber Potential Limited by the cosine term 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 : environment-dependent parameter weakening the pair interaction when coordination number increases

Tersoff Potential where repulsive part attractive part potential cutoff function

Tersoff Potential

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 13 cells

MD Simulation Setup System Preparation Implantation(1 ke. V) Cooling to 0 K

Recrystallization 1200 K for 0. 5 ns

Recrystallization SW Potential 1200 K Crystal Rate a/c interface displacement

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

MD Simulation Setup System Preparation Implantation(1 ke. V) Cooled to 0 K

Recrystallization 1900 K for 0. 85 ns

Recrystallization Tersoff Potential 1900 K Crystal Rate a/c interface displacement

Recrystallization Crystal Rate SW Potential 1200 K Tersoff Potential 1900 K

Recrystallization a/c interface displacement SW Potential 1200 K Tersoff Potential 1900 K

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

Recrystallization 1800 K for 20 ns

Tersoff Potential Melting temperature of Tersoff: about 2547 K Potential energy per particle versus temperature: the system with a/c interface is heated by adding energy at a rate of 1000 K/ns

Tersoff Potential As in recrystallized Si : 0. 82 in amorphized Si 0. 20 in crystalline Si

Tersoff Potential As in recrystallized Si : 0. 82 in amorphized Si 0. 20 in crystalline Si

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