Reviews in Undergraduate Research - Issue 2
SUMMARY Molecular dynamics (MD) is a widely used tool in condensed matter physics, as well as other disciplines ranging from chemistry to high-energy physics. In MD, one integrates the equations of motion - Newton's second law for classical particles - directly, invoking no approximations. To do so requires an interaction potential energy or forces between atoms. I will discuss both integration schemes and potential types. MD potentially bridges two length scales - macroscopic and atomistic - and also links experimental results with theories. This will be emphasized through a discussion of modern research in solid-state physics, with each research application highlighting a different type of interaction potential. I will discuss fluid flow briefly and highlight some other applications of Lennard-Jones potentials, surface growth using a Stillinger-Weber potential, and defects in silicon using tight-binding potentials. I give some references regarding density-functional theory calculations of these defects. A different avenue of modern MD research
is in the method itself rather than its application. Much research
is towards developing so-called acceleration methods. By taking advantage
of the physics of condensed matter systems, acceleration methods have
been proposed which extend the time scales accessible by MD by orders-of-magnitude
in many cases. In this article, I will focus on A. F. Voter's methods,
giving their motivation, algorithm, and some derivations. INTRODUCTION Molecular dynamics (MD) is a widely used tool in condensed matter physics, as well as other disciplines ranging from chemistry (Kityk et al., 1999) to high-energy physics (Bleicher et al., 1999). This article focuses on applications to solid-state physics; however, the basic concepts presented should be valuable to any newcomer to MD, regardless of their field. One of MD's appeals is that the system in consideration is simulated directly - without assuming the nature of transitions or the types of the structures. Moreover, due to increasing computer power, MD provides atomistic detail for system sizes that are exponentially increasing to mesoscopic scales. Already, with classical potentials, simulations are performed with hundreds of millions of atoms for hundreds of picoseconds (Kadau et al, 2002) or with a more accurate tight-binding potential simulating about 64 atoms for 0.250 microseconds (Richie et al., 2002) - nearly a timescale measurable on a stopwatch! Note that MD bridges two scales - macroscopic and atomistic - and thus links experimental results, say transport measurements, with theories governing the interactions of the simulated matter's constituent atoms. Indeed, conventional "pencil-and-paper" theory generally cannot evaluate the properties that are of direct consequence in experiments. In this light, MD forms another bridge, one between experiments and conventional theory. By iterating the process of performing simulations and experiments, theories of forces between atoms are refined. With MD now appropriately evangelized, the exposition of the fundamentals begins. Traditional (non-accelerated) MD is conceptually simple. Using some potential interaction between particles, Newton's second law updates the particles' velocities and positions; repeating this, the equations of motion are integrated and the system's trajectories are obtained. In order to be relevant at finite temperature, pressure, etc., the system must be modified to ensure it stays at a constant energy, temperature or in whichever thermodynamic ensemble is desired. Methods of simulating the correct ensemble are not given here but can be found in (Rapaport, 1997). Before giving specific examples of research into new MD methods (acceleration methods), I present the more technical aspects of MD. Expositions of integration algorithms follow a background in potentials. Accelerating MD motivates the parallel-replica, hyperdynamics, and temperature-accelerated dynamics acceleration methods, as well as on-the-fly kinetic Monte Carlo. Finally, illustrative research at the forefront of MD simulation in solid-state physics is discussed; each piece of research highlights a different class of potentials. POTENTIALS To accurately propagate the system, we must have an accurate interaction potential. This potential gives the potential energy as a function of the atomic positions and velocities. Details on existing potentials are given, as well as some general ways of deducing potentials.
POSITION INTEGRATION ALGORITHMS Some details of potentials and their calculation are now familiar. With the potential and initial conditions Newton's second law can be integrated for all the particles. The verlet and predictor-corrector (PC) methods are common for performing the integration. The integration algorithm usually does not need to be extremely accurate. Because an extremely slight position displacement at any time (or an equivalent round-off error) can cause huge differences in the atom's trajectory at all later times after a certain time, only quantities which are insensitive to exact trajectories "matter." This is not particular to MD, but is characteristic of the natural process itself. The verlet method gives positions at a short time t after the time corresponding to the supplied positions. Each is derived in a straightforward manner from the Taylor expansion, in time, of the atomic coordinates. The formula for the verlet propagator is (Rapaport, 1997) . The verlet propagator is commonly used for its simplicity and tendency to conserve energy. The PC methods are more accurate than verlet, but they are not used as frequently as the simpler verlet-class propagators. The primary advantage of the PC method over the verlet-like algorithms is in the ability to change t on the fly, which may be useful in systems where one set of particles inherently move faster than others. PC is also useful when constraints (on, say, bond length or angles) are placed on the system (Rapaport, 1997). The PC method predicts positions based upon Adams-Bashforth extrapolation, which is exact if they follow monic polynomials. After prediction, correction is made via a different set of formulae. For the (relatively complex) equations see (Rapaport, 1997). BETTER (FASTER!) MOLECULAR DYNAMICS Although MD is an increasingly mature field, there are still continuous advances in methods. Voter et al. have developed several methods for increasing MD's performance by many orders of magnitude. Each method requires some assumptions - usually forms of transition state theory (TST) (Voter, 2002 or Lombardo, 1991) - on the nature of transitions. However, the assumptions are minimal and their validity can be checked. There are three common families of acceleration techniques, namely parallel-replica (PR or par-rep), temperature-accelerated dynamics (TAD), and hyperdynamics. The families can be utilized simultaneously for multiplicative performance boosts. Voter gives an excellent, accessible review concentrating on these acceleration methods (Voter, 2002). The limitation that keeps one from simulating long time scales is the fact that MD is a multi-scale problem (for most solid-state systems). Specifically, one must use a small enough integration time step to reproduce the dynamics of the fast vibrational modes. Since these vibrations occur on the order of 1013 - 1014 times a second, the time step for accurate integration must be on the order of femtoseconds. A typical time step falls in the range of 1-5 fs. On the other hand, transitions occur infrequently; time scales between interesting transitions range from picoseconds (quickly diffusing surface atoms) to seconds (dislocation motion under shearing (Haasen, 1996)). One can conceive of watching interesting behavior for minutes or hours (for example, in crystal growth), however the longest MD simulations can now run for only microseconds. It must be kept in mind that the acceleration methods discussed below only apply to infrequent event systems, systems in which the time for transitions between 'sites' or 'structures' is much longer than the vibrational period.
RECENT SIMULATIONS
CONCLUSIONS This review does not cover all classes of research using MD - references are representative rather than exhaustive. However, it should have given the reader an idea of some typical applications - fluid dynamics, surface growth, and defect dynamics. Through examples, connections between experiment, theory, and MD are emphasized. One now hopefully has a feeling for the variety and advantages of some potential types. Also one should now be
aware of some of the growing number of molecular dynamics acceleration
methods - parallel-replica, temperature-accelerated dynamics, hyperdynamics,
and on-the-fly Monte Carlo. Perhaps the most important information
presented is the necessary background to understand what goes into
an MD simulation. With this and the references, a motivated individual
could probably code a simplistic MD simulator in a matter of a week
(though given the increasing number of sophisticated, fast MD programs,
writing one from scratch is probably not advisable except as a teaching
tool). ABOUT THE AUTHOR Kaden Hazzard is a third-year
undergraduate at The Ohio State University planning on graduating
in the spring of 2004. He plans on pursuing a doctoral degree in condensed
matter physics. He intends to pursue a research career at a university
or at a national lab. He has been involved in computational condensed
matter physics with Professor John Wilkins's research group since
the summer of 2000. He has been researching a variety of topics, mainly
defect evolution in silicon; surface growth in silicon; and structure
recognition, characterization, and data mining of defects in solids.
He has implemented several routines in the group's multi-scale materials
simulator (OHMMS), and a new Monte Carlo method for calculating free
energies. He also spent the summer of 2002 at Los Alamos National
Labs doing low-temperature experimental work. ACKNOWLEDGEMENTS Everyone in Professor John Wilkin's research group who I have had the fortune to work with deserves thanks for their continued guidance, discussions, and ideas. In particular, I would like to thank Professor Wilkins for his continuous suggestions for improving this manuscript and my writing in general. I would also like to thank Angela K. Hartsock for her gracious help with the figures. FURTHER READING A
standard text regarding MD in general, along with many techniques
especially for simulations of liquids is For those interested in
finding out more about density-functional theory, a preprint for a
good review article accessible to undergraduates who have taken some
quantum mechanics, I suggest: REFERENCES Arai, N., Takeda , S., and Kohyama, M. (1997) Self-Interstitial Clustering in Crystalline Silicon Physical Review Letters 78 4265 Baskes, M.I. (1997) Calculation of the behaviour of Si ad-dimers on Si(001) Modelling Simul. Mater. Sci. Eng. 5 p. 149-158. Berthier, L. and Barrat, J.L. (2002) Shearing a Glassy Material: Numerical Tests of Nonequilibrium Mode-Coupling Approaches and Experimental Proposals Physical Review Letters 89 95702 Birner, S., Kim, J., Richie, D.A., Wilkins, J.W., Voter, A.F., and Lenosky, T. (2001) Accelerated dynamics simulations of interstitial-cluster growth. Solid State Communications 120 (7-8) Bleicher, M., Zabrodin, E., Spieles, C., Bass, S.A., Ernst, C., Soff, S., Bravina, L., Belkacem, M., Weber, H., Stöcker, H., and Greiner, W. (1999) Relativistic Hadron-Hadron Collisions in the Ultra-Relativistic Quantum Molecular Dynamics Model (UrQMD) J.Phys. G25 1859-1896 Brooks, B. R., Bruccoleri,
R. E., Olafson, B. D., States, D. J., Swaminathan, S., and Karplus,
M. (1983) CHARMM: A Program for Macromolecular Energy, Minimization,
and Dynamics Calculations. J. Comp. Chem. 4, 187-217 Chen, D. and Boland, J. J. (2002) Chemisorption-induced disruption of surface electronic structure: Hydrogen adsorption on the Si(100)-2x1 surface Physical Review B (Condensed Matter and Materials Physics). vol.65, no.16 15. p. 165336/1-5. Collings, A.F., Watts, R.O., and Woolf, L.A. (1971) Thermodynamic properties and self-diffusion coefficients of simple liquids. vol.20, no.6 p. 1121-33. Colombo, L. (2002) Tight-binding theory of native point defects in silicon. Annu. Rev. Mater. Res., Vol. 32: 271-295 Daw, M.S. and Baskes, M.I. (1984) Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals. Phys. Rev. B 29, 6443 Doye, J.P.K., Miller, M.A., and Wales, D.J. (1999) Evolution of the Potential Energy Surface with Size for Lennard-Jones Clusters. , J. Chem. Phys. 111, 8417-8428 Estreicher, S. K., Hastings, J. L., and Fedders, P. A. (1999) Hydrogen-defect interactions in Si. Materials Science & Engineering B (Solid-State Materials for Advanced Technology) E-MRS 1998 Spring Meeting, Symposium A: Defects in Silicon: Hydrogen v 58 n 1-2 p.31-5 Fabricius, G. and Stariolo, D.A. (2002) Distance between inherent structures and the influence of saddles on approaching the mode coupling transition in a simple glass former. Phys. Rev. E 66, 031501 Galli, G., Kim, J., Canning, A., and Haerle, R. (1998) Large scale quantum simulations using Tight-Binding Hamiltonians and linear scaling methods Tight-Binding Approach to Computational Materials Science, Eds. P. Turchi, A. Gonis and L. Colombo, 425 (1998). Gawlinski, E.T. and Gunton, J.D. (1987) Molecular-dynamics simulation of molecular-beam epitaxial growth of the silicon (100) surface. Phys. Rev. B 36, 4774-4781 Haasen, Peter. (1996) Physical Metallurgy. (Cambridge University Press) pp. 282-283 Henkelman, G., Jóhannesson, G., and Jónsson, H. (2000) Methods for Finding Saddle Points and Minimum Energy Paths, Progress on Theoretical Chemistry and Physics, Kluwer Academic Publishers, Ed. S. D. Schwartz and references therein. Henkelman, G. and Jónsson, H. (2003), Multiple time scale simulations of metal crystal growth reveal importance of multi-atom surface processes, Phys. Rev. Lett. Jansen, R. W., Wolde-Kidane, D. S., and Sankey, O. F. (1988) Energetics and deep levels of interstitial defects in the compound semiconductors GaAs, AlAs, ZnSe, and ZnTe. Journal of Applied Physics v 64 n 5 p.2415 Jiang, M., Choy, T.-S., Mehta, S., Coatney, M., Barr, S., Hazzard, K., Richie, D., Parthasarathy, S., Machiraju, R., Thompson, D., Wilkins, J., and Gaytlin, B. (2003) Feature Mining Algorithms for Scientific Data Proceedings of SIAM Data Mining Conference (to appear) Kadau, K., Germann, T. C., Lomdahl, P. S., and Holian, B. L. (2002) Microscopic view of structural phase transitions induced by shock waves Science. vol.296, no.5573 31 p. 1681-4. Katircioglu, S. and Erkoc, S. (1994) Adsorption sites of Ge adatoms on stepped Si(110) surface Surface Science. vol.311, no.3 20 p. L703-6. Kim, J., Kirchhoff, F., Wilkins, J.W., and Khan, F.S. (2000) Stability of Si-interstitial defects: from point to extended defects Physical Review Letters 84, 503 Kityk, I. V., Kasperczyk, J., and Plucinski, K. (1999) Two-photon absorption and photoinduced second-harmonic generation in Sb/sub 2/Te/sub 3/-CaCl/sub 2/-PbCl/sub 2/ glasses Journal of the Optical Society of America B (Optical Physics). vol.16, no.10 p. 1719-24. Klimeck, G., Oyafuso, F., Boykin, T. B., Bowen, R. C., and von Allmen, P. (2002) Development of a Nanoelectronic 3-D (NEMO 3-D) simulator for multimillion atom simulations and its application to alloyed quantum dots Computer Modeling in Engineering & Sciences. vol.3, no.5 p. 601-42. Kohyama, M. and Takeda (1999) S. First-principles calculations of the self-interstitial cluster I/sub 4/ in Si, Physical Review B v. 60 issue 11, p. 8075. Laradji, M., Toxvaerd, S., and Mouritsen, O.G. (1996) Molecular Dynamics Simulation of Spinodal Decomposition in Three-Dimensional Binary Fluids Phys. Rev. Lett. Vol. 77, 2253-2256 Lee, B.J., Baskes, M. I., Kim, H., Cho, Y.K. (2001) Second nearest-neighbor modified embedded atom method potentials for bcc transition metals Physical Review B. vol.64, no.18 1 p. 184102/1-11. Lee, G.-D., Wang, C. Z., Lu, Z. Y., and Ho, K. M. (1998) Ad-Dimer Diffusion between Trough and Dimer Row on Si(100). Physical Review Letters 81 5872 Lombardo, S.J. and Bell, A.T. (1991) A review of theoretical models of adsorption, diffusion, desorption, and reaction of gases on metal surfaces. Surface Science Reports Volume 13, Issues 1-2, p. 3-72 Machiraju, R., Parthasarathy, S., Wilkins, J., Thompson, D.S., Gatlin, B., Richie, D., Choy, T., Jiang, M., Mehta, S., Coatney, M., Barr, S., Hazzard, K. (2003) Mining of Complex Evolutionary Phenomena. In et al H. Kargupta, editor, Data mining for Scientific and Engineering Applications. MIT Press. MacKerell Jr., A.D., Brooks, B., Brooks III, C.L, Nilsson, L., Roux, B., Won, Y., and Karplus, M. (1998) CHARMM: The Energy Function and Its Parameterization with an Overview of the Program, in The Encyclopedia of Computational Chemistry, 1, 271-277, P. v. R. Schleyer et al., editors (John Wiley & Sons: Chichester) Montalenti, F., Sorensen M.R., Voter A.F. (2001) Closing the Gap between Experiment and Theory: Crystal Growth by Temperature Accelerated Dynamics Phys. Rev. Lett. 87 126101 1-4 Montalenti, F., Voter, A. F., Ferrando, R. (2002) Spontaneous atomic shuffle in flat terraces: Ag(100) Physical Review B (Condensed Matter and Materials Physics). vol.66, no.20 15 p. 205404-1-7. Rapaport, D.C. (1997). The Art of Molecular Dynamics Simulation (Cambridge University Press) Richie, D.A., Kim, J., Hennig, R., Hazzard, K., Barr, S., Wilkins, J.W. (2002) Large-scale molecular dynamics simulations of interstitial defect diffusion in silicon Materials Research Symposium Proceedings, vol.731, p.W9.10-5 Roland, C. and Gilmer, G.H. (1992) Epitaxy on surfaces vicinal to Si(001). I. Diffusion of silicon adatoms over the terraces. Physical Review B 46 13428-13436 Scopigno, T., Ruocco, G., Sette, F., and Viliani, G. (2002) Evidence of short-time dynamical correlations in simple liquids Phys. Rev. E 66, 031205. Shirts, M.R. and Pande, V.S. (2001) Mathematical Analysis of Coupled Parallel Simulations Physical Review Letters 86, 22 Shirts, M.R. and Pande V,S. (2000) Computing - Screen Savers of the World Unite. Science 290, 1903-4 Sprague, J.A., Montalenti, F., Uberuaga, B.P., Kress, J.D., and Voter, A.F. (2002) Simulation of growth of Cu on Ag(001) at experimental deposition rates Physical Review B (Condensed Matter and Materials Physics) v 66 n 20 p.205415-1-10 Stillinger, F.H. and Weber, T.A. (1985) Computer simulation of local order in condensed phases of silicon Phys. Rev. B 31, 5262-5271 Tang, M., Colombo, L., Zhu, J., and Diaz de la Rubia, T. (1997) Intrinsic point defects in crystalline silicon: Tight-binding molecular dynamics studiesof self-diffusion, interstitial-vacancy recombination, and formation volumes. Physical Review B - 1 Volume 55, Issue 21 pp. 14279-14289 Uberuaga, B.P., Henkelman, G., Jónsson, H., Dunham, S., Windl, W., and Stumpf, R. (2002) Theoretical Studies of Self-Diffusion and Dopant Clustering in Semiconductors, Physica Status Solidi B, 233, 24 Voter, A.F. (1998) Parallel replica method for dynamics of infrequent events. Physical Review B 57 pp. 13985-88. Voter, A.F., Montalenti, F., and Germann, T. C. (2002) Extending the Time Scale in Atomistic Simulation of Materials Annual Reviews in Materials Research, vol. 32, p. 321-346 Windl, W., Stumpf, R., Masquelier, M., Bunea, M., and Dunham, S.T. (1999) Ab-initio pseudopotential calculations of boron diffusion in silicon. International Conference on Modeling and Simulation of Microsystems, Semiconductors, Sensors and Actuators 1999 p.369-72 Zagrovic, B., Sorin, E.J.,
and Pande, V.J. (2001) Beta-hairpin folding simulations in atomistic
detail using an implicit solvent model. J. Mol. Biol. 313:151-69 |

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