140 likes | 271 Views
Simulating atomic structures of metallic nanowires in the grand canonical ensemble. Corey Flack Department of Physics, University of Arizona Thesis Advisors: Dr. Jérôme Bürki Dr. Charles Stafford. Overview. Motivation Modeling the nanowire Monte-Carlo simulated annealing
E N D
Simulating atomic structures of metallic nanowiresin the grand canonical ensemble Corey Flack Department of Physics, University of Arizona Thesis Advisors: Dr. Jérôme Bürki Dr. Charles Stafford
Overview • Motivation • Modeling the nanowire • Monte-Carlo simulated annealing • Simulated annealing in the grand canonical ensemble • Results: Equilibrium structures • Conclusions
Nanowires are of principal interest for applications in nanotechnology • What is their atomic structure? • Early simulations predicted non-crystalline structures of either icosahedral packing or a helical multishell • TEM video by Takayanagi ‘s group suggests helical structures • Classical structural models lead to Rayleigh instability • Need quantum mechanics! Atomic scale TEM image of a gold nanowire. Diagram courtesy of Ref. [2]
Cylindrical nanowires are found to be stable with a number of conductance channels equal to magic conductance values • Predicted by nanoscale free-electron model (NFEM) • Confinement potential generated by the electron gas Conductance quanta: Stability diagram for cylindrical nanowires. Diagram courtesy of Ref. [4]
The total energy of the ions Confinement Interaction Energy • Phenomonological, hard-core repulsion • Screened Coulomb force • Solution to Poisson’s equation using NFEM electronic density • Kinetic energy is neglected for an equilibrium state
Monte-Carlo simulated annealing methods use random displacements with a slow cooling method • Attempt to reach a minimum energy configuration • Beginning at high temperatures – high thermal mobility • As T is lowered, atoms are frozen into a minimum energy configuration • Metropolis algorithm: new configurations are generated from random displacements of the ions • Accepted with a probability of:
Simulated annealing in the canonical ensemble • V, N, and T are externally controlled parameters • Initial random configuration of uniform density • Random, isotropic displacement of one atom imitating Maxwellian velocity distribution • Acceptance of moves according to Boltzman factor: Decreases in energy automatically accepted Increases accepted with finite probability
Equilibrium structures in the canonical ensemble • Conductance G=1Go – zigzag structure • Arbitrary orientation • Torsional stiffness Conductance G=3Go – helical hollow shell with four atomic strands
Canonical ensemble does not represent the physical reality of a wire suspended between two contacts • Canonical ensemble: difficult to anneal out defects at the wire ends • Grand canonical ensemble allows for atom interchange with the contacts • V, T, and μ are externally controlled parameters Conductance G=3Go – trapped defect at wire end
Implementation of the grand canonical ensemble allows for two new Monte-Carlo moves: addition and removal of atoms • Probability of move acceptance is given by the Gibbs factor: • Probability of trying removal is dependent on position • Placement of additional atoms determined by:
Simulations were run for various constant chemical potentials • Rise at N=60: region of canonical ensemble • Disposition to atom removal: • Kinetic effects • Atom addition method • Non self-consistent confining potential Final number of atoms in 3Go wire with constant chemical potential. A) Initial number of atoms: 60. b) Ion addition radial range changed to [0,R+2RS]. Initial number of atoms: 60.
Simulations with constant chemical potential also generated underfilled and overfilled structures for wires of conductance G = 3Go • Underfilled wire: 4 atomic strands • No = 60; Nfinal = 48 • Overfilled wire: helical structure of 5 atomic strands, line of atoms through axis • No = 60; Nfinal = 93
Conclusions • Equilibrium wire structures with G = 1 and 3Go were obtained within canonical and grand canonical simulations • Advantages of grand canonical simulation: • Correct physical ensemble • Defects not trapped at wire ends • Difficulties of grand canonical ensemble: • μ not known a priori • Achieving detailed balance between atom addition and removal • Further directions: • Further experimentation within the grand canonical ensemble • Exploring equilibrium structures for higher conductance wires
Acknowledgements Dr. Bürki Dr. Stafford All atomic structure images generated by Jmol: an open-source Java viewer for chemical structures in 3D. http://www.jmol.org/ References: [1] O. Gülseren, F. Ercolessi, E. Tosatti, Phys. Rev. Lett. 80, 3775 (1998) [2] Y. Kondo, K. Takayanagi, Sci. 289, 606 (2000) [3] C.A. Stafford, D. Baeriswyl, J. Bürki, Phys. Rev. Lett. 79, 2863 (1997) [4] J. Bürki, C.A. Stafford, Appl. Phys. A 81, 1519 (2005) [5] N.W. Ashcroft, N.D. Mermin. Solid State Physics. (1976) [6] Numerical Recipes in C, 10.9, pp.444-455 [7] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, J. Chem. Phys. 21, 1087 (1053) [8] Numerical Recipes in C, 7.2, pp. 289-290 [9] D. Conner, Master Thesis (2006), N. Rioradan, Independent studies with C.A. Stafford (2007) [10]D. Frenkel. Introduction to Monte Carlo Methods.