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 ri (t0 ) and v i (t0 ) Get new forces Fi (ri ) Solve the equations of motion numerically over a short step t ri (t ) ri (t t ) v i (t ) v i (t t ) t t t Is t t max ? 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 L rcut
2 rcut Constructing neighboring cells 2 pair potential calculation O( N ) atoms move 0.2 A per time step not necessary to search all atoms Verlet neighbor list containing all neighbor atoms within rL
rcut updating every N L time steps where rL rcut N L vt 2 i rL skin
Constructing neighbor cells Linked cell method divide MD cell into smaller subcells : The length of subcell l L l rL n n n n is chosen so that L : the length of MD cell
going through 27 NN C atom pairs instead N ( N 1)! 3 where N C N / n rL reducing it to O( N ) 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 m P( v x )
2k BT 1/ 2 1 exp mv 2x / k BT 2 Generate random initial atom velocities scaling T with equipartition theorem 3
1 k BT mv 2 2 2 MD Time Step Too long t : energy is not conserved r/t 1/20 of the nearest atom distance
In practice t 4 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 for 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 ri (t ) , v i (t ) 1
a(r ) F (r (t )) m Obtain the positions and velocities at t t 1 r (t t ) r (t ) v(t )t a(r )t 2 2 1 a(t t ) a(r (t t )) m 1 v(t t / 2) v(t )t a(r )t 2 1
v(t t ) v(t t / 2) a(t t )t 2 Predictor-Corrector Method Predictor Step from the initial ri (t ) , v i (t ) 1 a(r ) F (r (t )) m predict ri (t t ) , v i (t t ) using a Taylor series a(t ) 2 r (t t ) r (t ) v(t )t t
2 v P (t t ) v(t ) a(t )t P a P (t t ) a(t ) r iii (t )t r iii : 3rd order derivatives Predictor-Corrector Method Corrector Step get corrected acceleration P F (
r (t t )) C a (r ) m using error in acceleration a(t t ) a C (t t ) a P (t t ) correct positions and velocities 2 t r (t t ) r P (t t ) C0 a(t t )
2 v(t t ) v P (t t ) C1ta(t t ) Cn : constants depending accuracy Force Calculation The force on an atom is determined by N u (rij ) j i rij
Fi U ( r ) rij U (r ) : potential function N : number of atoms in the system rij : vector distance between atoms i and j
MD Potential Classical Potential U U1 (ri ) U 2 (ri , r j ) U 3 (ri , r j , rk ) ... i i, j i , j ,k U1 : Single particle potential Ex) external electric field, zero if no external force
U 2 : Pair potential only depending on Fij r U (rij ) i U 3 : Three-body potential with an angular dependence Fi i (Vij V ji ) V jki j k j Using Classical Potential
Born-Oppenheimer Approximation Consider electron motion for fixed nuclei ( me M 0) Assume total wavefunction as (R i , r ) (R i ) (r , R i ) (R i ) : Nuclei wavefunction
(r , R i ) : Electron wavefunction parametrically depending on R i The equation of motion for nuclei is given by Pi 2 H N U (R i ) i 2M i (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
U U 2 (ri , r j ) U 3 (ri , r j , rk ) i, j i , j ,k U 2 (rij ) f 2(rij / ) U 3 (ri , r j , rk ) f3 (ri / , r j / , rk / ) , : energy and length units Pair potential function
f 2(r ) A( Br p r q ) exp[(r a ) 1 ] , r a 0 , r a Stillinger-Weber Potential Three body potential function f3 (ri , r j , rk ) h(rij , rik , jik ) h(rji , rjk , ijk )
h(rki , rkj , ikj ) h(rij , rik , jik ) exp[ (rij a) 1 (rik a ) 1 ] 1 (cos jik ) 2 3 Stillinger-Weber Potential 1 2 (cos )
Limited by the cosine term jik 3 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 U U repulsive ( rij ) bijkU attractive (rij ) bijk : environment-dependent parameter weakening the pair interaction when coordination number increases Tersoff Potential
U ij f C (rij )[aij f R (rij ) bij f A (rij )] where repulsive part f R ( r ) Ae 1r attractive part f A ( r ) Be 2 r potential cutoff function f C (r )
1, r R D 1 1 (r R) sin , R D r R D 2 2 D
2 0, r R D Tersoff Potential bij (1 n ijn ) 1/ 2 n ij f C (rik ) g ( jik ) exp[33 (rij rik )3 ] k i , j
c2 c2 g ( ) 1 2 2 d d (h cos ) 2 aij (1 nijn ) 1/ 2 n ij f C (rik ) exp[33 (rij rik )3 ] k i , j 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 4 x 13 cells MD Simulation Setup System Preparation Ion Implantation(1 keV) Cooling to 0K
Recrystallization 1200 K for 0.5 ns Recrystallization SW Potential 1200K Crystal Rate a/c interface displacement MD Simulation Setup
Initial Setup 6 TC layer 1 static layer 5 x 5 x 13 cells MD Simulation Setup System Preparation
Ion Implantation(1 keV) Cooled to 0K Recrystallization 1900 K for 0.85 ns Recrystallization Tersoff Potential 1900K Crystal Rate
a/c interface displacement Recrystallization Crystal Rate SW Potential 1200K Tersoff Potential 1900K Recrystallization a/c interface displacement SW Potential 1200K
Tersoff Potential 1900K MD Simulation Setup Initial Setup 6 TC layer 1 static layer 2 x 2 x 13 cells
Recrystallization 1800 K for 20 ns 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 Tersoff Potential As in recrystallized Si :
o2 0.82 A2 / 100 ps in amorphized Si o A / 100 ps in crystalline Si 0.20 Tersoff Potential As in recrystallized Si : o2 A / 100 ps 0.82
in amorphized Si 2 o 0.20 A / 100 ps in crystalline Si Summary Review Molecular Dynamics MD simulation for recrystallization of Si with SW, Tersoff with As