Recently, two new methods have been introduced for accelerating the dynamics of rare events such as these diffusion processes. The first, hyperdynamics [1,2], modifies the potential energy surface by adding a bias potential in potential energy basins to accelerate the escape rate from these basins, while maintaining the correct relative escape rates for multiple channels, and allowing the elapsed time to be accumulated as a statistical property. The second, parallel replica dynamics [3], involves running N independent trajectories on N processors until an event is detected on any processor, at which time all processors are notified and N new trajectories are started in the new potential basin. These independent trajectories may be computed using hyperdynamics, multiplying the computational boost factors offered by both methods. For instance, a 300 µs simulation of the smoothing of two randomly deposited layers of Cu on Cu(100), carried out over a weekend on part of a 128-node Cray/SGI Origin 2000, would have required nearly 32 years of CPU time on a single node using ordinary MD [4].

This opens up the possibility of a new technique for simulating surface growth by vapor deposition processes, bridging the gap between the currently used MD and kinetic Monte Carlo (KMC) methods. Since MD simulations are limited by atomic vibrational timescales (on the order of fs), a typical million-timestep calculation only reaches times on the order of ns. Even if only a single atom is deposited, this is still several orders of magnitude faster than the actual experimental deposition rates. Thus a more commonly used method for studying the effects of deposition conditions (deposition rate, angular distribution, impact energy, substrate temperature, ...) on the resulting surface morphology is KMC. Although offering the possibility of simulating surface growth over physically relevant timescales (seconds to hours), this approach is limited to the events which are included in the model. The growing number of complex surface diffusion mechanisms which have been discovered in recent years raises some doubts about the validity of KMC simulations which omit such pathways.

Thus we have begun to apply the combined hyperdynamics and parallel replica dynamics methods to the direct simulation of physical vapor deposition of Cu onto the Cu(100) surface. Shown below is a snapshot from a 250 K simulation after 2 monolayers have been deposited at a rate of one atom (10 eV, normal incidence) every 100 ns. One monolayer has been deposited smoothly; the second monolayer is distributed among three layers, as indicated by the three shades of grey. The simulation time is 14.4 µs, several orders of magnitude longer than would be accessible by ordinary MD. However, attempts to simulate even slower deposition rates have been limited by the ns timescale of trivial low-barrier events such as adatom diffusion along island edges. Work is currently underway on methods to overcome such events once they have been identified, and accelerate the dynamics to the next "new" event (or deposition, if no available pathway has a sufficiently low barrier).

(2 x 2 periodic simulation cells are shown here)

References:

[1] "A Method for Accelerating the Molecular Dynamics Simulation of Infrequent Events," A.F. Voter, J. Chem. Phys. 106, 4665 (1997).

[2] "Hyperdynamics: Accelerated Molecular Dynamics of Infrequent Events," A.F. Voter, Phys. Rev. Lett. 78, 3908 (1997).

[3] "Parallel Replica Method for Dynamics of Infrequent Events," A.F. Voter, Phys. Rev. B 57, 13985 (1998).

[4] "Accelerating the Dynamics of Infrequent Events: Combining Hyperdynamics and Parallel Replica Dynamics to Treat Epitaxial Layer Growth," A.F. Voter and T.C. Germann, Mat. Res. Soc. Symp. Proc. 528, 221 (1998).