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

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

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

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

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

7. 2. Nanoscale Free-Electron Model (NFEM) 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.

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

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

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

17. Stability analysis including elliptic deformations: Theory of shell and supershell effects in nanowires D. F. Urban, J. Bürki, C.-H. Zhang, CAS & H. Grabert, PRL (in press) 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.

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 C.-H. Zhang, J. Bürki & CAS (unpublished) !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

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

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 ↔

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

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

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.

31. Quantum suppression of Shot noise NFEM w/disorder Gold nanocontacts

32. Multivalent atoms