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
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)
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:
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:
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.
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
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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