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

1. How thin can a metal wire be?

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

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

Extrusion of a gold nanowire using an STM

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, (2001) Conductance histograms for sodium nanocontacts Corrected Sharvin conductance: T=90K

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.

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

Comparison: NFEM vs. experiment Exp: Theory:

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

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.

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

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

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

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

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

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.

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)

“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

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.

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:

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

Propagation of a surface instability: Phase separation ↔

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

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

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

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

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

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

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.

Quantum suppression of Shot noise NFEM w/disorder Gold nanocontacts

Multivalent atoms