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

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

QuickTime™ and a YUV420 codec decompressor are needed to see this picture.

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, 216805 (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 foran 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 orbitsin 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, 165414 (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 barrierto 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, 254501 (2003)

J. Bürki, R. E. Goldstein & CAS, PRL 91, 254501 (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, 254501 (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, 254501 (2003)

Necking of a nanowire under strain

Hysteresis: elongation vs. compression

J. Bürki, R. E. Goldstein & CAS, PRL 91, 254501 (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.