Shells and Supershells in Metal Nanowires
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Transcript of Shells and Supershells in Metal Nanowires
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.