Simulating atomic structures of metallic nanowires in the grand canonical ensemble

1. 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

2. Overview • Motivation • Modeling the nanowire • Monte-Carlo simulated annealing • Simulated annealing in the grand canonical ensemble • Results: Equilibrium structures • Conclusions

3. 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]

4. 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]

5. 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

6. 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:

7. 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

8. 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

9. 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

10. 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:

11. 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.

12. 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

13. 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

14. 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.