Stability and Symmetry Breaking in Metal Nanowires II: Linear Stability Analyses

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Stability and Symmetry Breaking in Metal
Nanowires II Linear Stability Analyses
Charles Stafford
D. F. Urban, J. Bürki, C.-H. Zhang, C. A.
Stafford H. Grabert, PRL 93, 186403 (2004)
Capri Spring School on Transport in
Nanostructures, March 29, 2007
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Electron-shell potential
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1. Linear stability analysis of a cylinder
Mode stiffness
Classical (Rayleigh) stability criterion
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1. Linear stability analysis of a cylinder (m0)
Mode stiffness
Classical (Rayleigh) stability criterion
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Mode stiffness a(q)
F. Kassubek, CAS, H. Grabert R. E. Goldstein,
Nonlinearity 14, 167 (2001)
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Stability under axisymmetric perturbations
Agt0
C.-H. Zhang, F. Kassubek CAS, PRB 68, 165414
(2003)
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Stability of nanocylinders at ultrahigh current
densities
Generalized free energy for ballistic
nonequilibrium electron distribution. Coulomb
interactions included in self-consistent Hartree
approximation.
!
C.-H. Zhang, J. Bürki CAS, PRB 71, 235404
(2005)
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2. General linear stability analysis
General cross section
Free energy
Stability requires

1. Stationarity
2. Convexity

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General stability analysis of a cylinder
D. Urban, J. Bürki, CAS H. Grabert PRB 74,
245414 (2006)
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3. Stable elliptical nanowires
D. F. Urban, J. Bürki, C.-H. Zhang, CAS H.
Grabert, PRL 93, 186403 (2004)
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Combining cylindrical and elliptical structures
Theory of shell and supershell effects in
nanowires
D. F. Urban, J. Bürki, C.-H. Zhang, CAS H.
Grabert, PRL 93, 186403 (2004)
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Combining cylindrical and elliptical structures
Theory of shell and supershell effects in
nanowires

• Magic cylinders 75 of most-stable wires.
• Supershell structure most-stable elliptical
  wires occur at the nodes
• of the shell effect.

D. F. Urban, J. Bürki, C.-H. Zhang, CAS H.
Grabert, PRL 93, 186403 (2004)
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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 93, 186403 (2004) Discussion D.
F. Urban et al., Solid State Comm. 131, 609
(2004)
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4. Quadrupolar cross sections
D. Urban, J. Bürki, CAS H. Grabert PRB 74,
245414 (2006)
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Elliptical vs. quadrupolar cross sections
Quadrupole favored for large deformations due to
reduced surface energy. For e lt 1.3, quadrupole
ellipse. No generically preferred shape
can be positive or negative. ? Integrable cross
sections not special (except cylinder)
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Higher multipole deformations
Higher-m deformations less stable due to
increased surface energy.
D. Urban, J. Bürki, CAS H. Grabert PRB 74,
245414 (2006)
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5. Material dependence of stability
Na
Au
Relative stability of deformed structures depends
on surface tension in natural units Absolute
stability also depends on ? Lecture 3.
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Special case Aluminum
Two different types of histograms (history
dependent)
Crossover from electronic to atomic shell
effects at
A. I. Mares, D. F. Urban, J. Bürki, H. Grabert,
CAS J. M. van Ruitenbeek, cond-mat/0703589
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Extracting individual conductance peaks
A. I. Mares, D. F. Urban, J. Bürki, H. Grabert,
CAS J. M. van Ruitenbeek, cond-mat/0703589
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Linear stability analysis for Aluminum
Trivalent metal Fermi surface free-electron like
in extended-zone scheme. Physics of Al clusters
suggests NFEM applicable for
? Same magic sequence, but relative stability of
deformed wires enhanced.
A. I. Mares, D. F. Urban, J. Bürki, H. Grabert,
CAS J. M. van Ruitenbeek, cond-mat/0703589
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Electron-shell structure Theory vs. Experiment
A. I. Mares, D. F. Urban, J. Bürki, H. Grabert,
CAS J. M. van Ruitenbeek, cond-mat/0703589
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Superdeformed nanowires
cf. Physics of superdeformed nuclei
A. I. Mares, D. F. Urban, J. Bürki, H. Grabert,
CAS J. M. van Ruitenbeek, cond-mat/0703589
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6. Conclusions

• Cylinders are special
• Only generically stable shape.
• Analogy to shell-effects in clusters and nuclei
• quantum-size effects in thin films.
• Open questions
• Structural dynamics (Urban seminar, Lecture 3)
• Putting the atoms back in

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Putting the atoms back in
Left (experiment) Y. Kondo K. Takayanagi,
Science 289, 606 (2000) Right (theory) Dennis
Conner, Nate Riordan, J. Bürki CAS (unpublished)