A New Method for Fast Simulation of Adsorbate Dynamics
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moves to a lattice of sites, an assumption that may not be tenable for simulating systems that are amorphous or systems in which real-space phenomena (such as misfit dislocations, which arise in heteroepitaxial film growth) are important. In this paper we present a dynamical version of the Smart Monte Carlo(SMC) method [10,11]. This method has the advantage of not predetermining the rates of selected sets of dynamic events. Also this method is not lattice based and incorporates relevant real space phenomena. In this method the trial moves are generated by using a Langevin equation, and are accepted with a transition probability which maintains the detailed balance criterion. This procedure allows us to use a large time step to generate the trial
displacements. METHODOLOGY The SMC algorithm is based on the Langevin equation, in which a particle experiencing a force F is given a trial displacement Ar in a time interval At by Ar=J3AF+R
where
==I/kBT,A=At 2/2mf
(1)
,
and R is a random displacement. The random displacement
is chosen from a Gaussian distribution which depends on the temperature of the system
77 Mat. Res. Soc. Symp. Proc. Vol. 399 0 1996 Materials Research Society
[10]. For an arbitrarily calculated displacement using Equation (1), the acceptance is dictated by detailed balancing considerations. The transition probability of acceptance for a move Ar from state i to statej is given by [10, 11], P = min(1,exp(-/3p))
(2)
,
where p = Auij + - (Fi + F.) *Ar.
,
(3)
and (4)
6W =-LA((AF. )2 + 2Fi.AVi)
In the above expressions Auij is the difference in the potential energy between state i and state j, AFij is the difference in force at state i and state j, and Fi and F1 are the forces experienced by the particle in state i and state j, respectively. To implement the method, a particle is given an initial position, which is propagated via Equations (1) - (4). Time is advanced by At for each trial move. We tested this method by using 25, 50, 75 and 100 fs time steps (At) to study the diffusion of Rh and Cu on the Rh(I 11) and Cu(100) surfaces, respectively. Though this method is capable of using an order of magnitude higher time step compared to MD simulations, it still needs to be modified to reach experimental time scales. We further modified this method by implementing the umbrella sampling technique [4,121 to propagate the system. As discussed below, this sampling technique allows us to approach experimental time scales. RESULTS AND DISCUSSION The Rh/Rh(l 11) system is comprised of a surface of 3 layers with 64 atoms in each layer and a single adatom. The interatomic interactions are modeled by a LennardJones potential with parameters, o=2.47A and e/kB =7830K, with a cutoff radius of 2.5o [4]. In the case of the Cu/Cu(100) system, we have used a surface of 8 layers with 72 atoms in each layer and a single adatom. The MD/MC-CEM potential is employed for this system [13]. In all our simulations the lattice atoms are held rigid at their equilibrium positions. The trajectories of the adatoms are si
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