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Shells and Supershells in Metal Nanowires

Charles Stafford. Shells and Supershells in Metal Nanowires. Research supported by NSF Grant No. 0312028. NSCL Workshop on Nuclei and Mesoscopic Physics , October 23, 2004. 1. How thin can a metal wire be?. Surface-tension driven instability. Cannot be overcome in classical MD simulations!.

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Shells and Supershells in Metal Nanowires

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  1. Charles Stafford Shells and Supershells in Metal Nanowires Research supported by NSF Grant No. 0312028 NSCL Workshop on Nuclei and Mesoscopic Physics, October 23, 2004

  2. 1. How thin can a metal wire be?

  3. Surface-tension driven instability Cannot be overcome in classical MD simulations! T. R. Powers and R. E. Goldstein, PRL 78, 2555 (1997)

  4. Fabrication of a gold nanowire using an electron microscope Courtesy of K. Takayanagi, Tokyo Institute of Technology

  5. Extrusion of a gold nanowire using an STM Courtesy of K. Takayanagi, Tokyo Institute of Technology

  6. What is holding the wires together? Is electron-shell structure the key to understanding stable contact geometries? Conductance histograms for sodium nanocontacts Corrected Sharvin conductance: T=90K A. I. Yanson, I. K. Yanson & J. M. van Ruitenbeek, Nature 400, 144 (1999); PRL 84, 5832 (2000); PRL 87, 216805 (2001)

  7. Model nanowire as a free-electron gas confined by hard walls. Ionic background = incompressible fluid. Appropriate for monovalent metals: alkalis & noble metals. Regime: Metal nanowire = 3D open quantum billiard. 2. Nanoscale Free-Electron Model (NFEM)

  8. Scattering theory of conduction and cohesion Electrical conductance (Landauer formula) Grand canonical potential (independent electrons) Electronic density of states (Wigner delay)

  9. Comparison: NFEM vs. experiment Exp: Theory:

  10. Weyl expansion + Strutinsky theorem Mean-field theory: Weyl expansion:

  11. Semiclassical perturbation theory for an axisymmetric wire • Use semiclassical perturbation theory in λ to express δΩ in terms of classical periodic orbits. • Describes the transition from integrability to chaos of electron motion with a modulation factor accounting for broken structural symmetry: • Neglects new classes of orbits ~ adiabatic approximation.

  12. Electron-shell potential Classical periodic orbits in a slice of the wire → 2D shell structure favors certain “magic radii”

  13. 3. Linear stability analysis of a cylinder Mode stiffness: Classical (Rayleigh) stability criterion:

  14. 3. Linear stability analysis of a cylinder (m=0) Mode stiffness: Classical (Rayleigh) stability criterion:

  15. Mode stiffness α(q) F. Kassubek, CAS, H. Grabert & R. E. Goldstein, Nonlinearity 14, 167 (2001)

  16. Stability under axisymmetric perturbations A>0 C.-H. Zhang, F. Kassubek & CAS, PRB 68, 165414 (2003)

  17. Stability analysis including elliptic deformations: Theory of shell and supershell effects in nanowires • Magic cylinders ~75% of most-stable wires. • Supershell structure: most-stable elliptical wires occur at the nodes • of the shell effect. • Stable superdeformed structures (ε > 1.5) also predicted. D. F. Urban, J. Bürki, C.-H. Zhang, CAS & H. Grabert, PRL (in press)

  18. Comparison of experimental shell structure for Na with predicted most stable Na nanowires Exp: A. I. Yanson, I. K. Yanson & J. M. van Ruitenbeek, Nature 400, 144 (1999) Theory: D. F. Urban, J. Bürki, C.-H. Zhang, CAS & H. Grabert, PRL (in press)

  19. “Lifetime” of a nanocylinder Instanton calculation using semiclassical energy functional. Cylinder w/Neumann b.c.’s at ends + thermal fluctuations. Universal activation barrier to nucleate a surface kink

  20. Stability at ultrahigh current densities Generalized free energy for ballistic nonequilibrium electron distribution. Coulomb interactions included in self-consistent Hartree approximation. ! C.-H. Zhang, J. Bürki & CAS (unpublished)

  21. 4. Nonlinear surface dynamics • Consider axisymmetric shapes R(z,t). • Structural dynamics → surface self-diffusion of atoms: • Model ionic medium as an incompressible fluid: • Born-Oppenheimer approx.→chemical potential of a surface atom: .

  22. Chemical potential of a surface atom J. Bürki, R. E. Goldstein & CAS, PRL 91, 254501 (2003)

  23. Propagation of a surface instability: Phase separation ↔ J. Bürki, R. E. Goldstein & CAS, PRL 91, 254501 (2003)

  24. Evolution of a random nanowire to a universal equilibrium shape → Explains nanofabrication technique invented by Takayanagi et al. J. Bürki, R. E. Goldstein & CAS, PRL 91, 254501 (2003)

  25. What happens if we turn off the electron-shell potential? Rayleigh instability!

  26. Thinning of a nanowire via nucleation & propagation of surface kinks Sink of atoms on the left end of the wire. Simulation by Jérôme Bürki

  27. Thinning of a nanowire II: interaction of surface kinks Sink of atoms on the left end of the wire. Simulation by Jérôme Bürki

  28. Necking of a nanowire under strain J. Bürki, R. E. Goldstein & CAS, PRL 91, 254501 (2003)

  29. Hysteresis: elongation vs. compression J. Bürki, R. E. Goldstein & CAS, PRL 91, 254501 (2003)

  30. 5. Conclusions • Analogy to shell-effects in clusters and nuclei, • quantum-size effects in thin films. • New class of nonlinear dynamics at the nanoscale. • NFEM remarkably rich, despite its simplicity! • Open questions: • Higher-multipole deformations? • Putting the atoms back in! • Fabricating more complex nanocircuits.

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