1. 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 Solution to Poisson’s equation using NFEM electronic density Phenomonological, hard-core repulsion Screened Coulomb force 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. Conductance G=1G o – zigzag structure Arbitrary orientation Torsional stiffness Conductance G=3G o – helical hollow shell with four atomic strands Equilibrium structures in the canonical ensemble

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=3G o – 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 3G o wire with constant chemical potential. A) Initial number of atoms: 60. b) Ion addition radial range changed to [0,R+2R S ]. Initial number of atoms: 60.

12. Simulations with constant chemical potential also generated underfilled and overfilled structures for wires of conductance G = 3G o Underfilled wire: 4 atomic strands N o = 60; N final = 48 Overfilled wire: helical structure of 5 atomic strands, line of atoms through axis N o = 60; N final = 93

13. Conclusions Equilibrium wire structures with G = 1 and 3G o 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.