Stability and Symmetry Breaking in Metal Nanowires II: Linear Stability Analyses Capri Spring School on Transport in Nanostructures, March 29, 2007 Charles.

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Stability and Symmetry Breaking in Metal Nanowires II: Linear Stability Analyses Capri Spring School on Transport in Nanostructures, March 29, 2007 Charles Stafford D. F. Urban, J. Bürki, C.-H. Zhang, C. A. Stafford & H. Grabert, PRL 93, (2004)

Electron-shell potential

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

1. 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 of nanocylinders at ultrahigh current densities C.-H. Zhang, J. Bürki & CAS, PRB 71, (2005) ! Generalized free energy for ballistic nonequilibrium electron distribution. Coulomb interactions included in self-consistent Hartree approximation.

2. General linear stability analysis Stability requires: General cross section: i)Stationarity ii)Convexity Free energy:

General stability analysis of a cylinder D. Urban, J. Bürki, CAS & H. Grabert PRB 74, (2006)

3. Stable elliptical nanowires D. F. Urban, J. Bürki, C.-H. Zhang, CAS & H. Grabert, PRL 93, (2004)

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, (2004)

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, (2004) Magic cylinders ~75% of most-stable wires. Supershell structure: most-stable elliptical wires occur at the nodes of the shell effect.

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, (2004) Discussion: D. F. Urban et al., Solid State Comm. 131, 609 (2004)

D. Urban, J. Bürki, CAS & H. Grabert PRB 74, (2006) 4. Quadrupolar cross sections

Elliptical vs. quadrupolar cross sections Quadrupole favored for large deformations due to reduced surface energy. For ε < 1.3, quadrupole ≈ ellipse. No generically preferred shape; can be positive or negative. → Integrable cross sections not special (except cylinder)

Higher multipole deformations D. Urban, J. Bürki, CAS & H. Grabert PRB 74, (2006) Higher-m deformations less stable due to increased surface energy.

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.

Special case: Aluminum A. I. Mares, D. F. Urban, J. Bürki, H. Grabert, CAS & J. M. van Ruitenbeek, cond-mat/ Two different types of histograms (history dependent) Crossover from electronic to atomic shell effects at

Extracting individual conductance peaks A. I. Mares, D. F. Urban, J. Bürki, H. Grabert, CAS & J. M. van Ruitenbeek, cond-mat/

Linear stability analysis for Aluminum A. I. Mares, D. F. Urban, J. Bürki, H. Grabert, CAS & J. M. van Ruitenbeek, cond-mat/ 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/ Electron-shell structure: Theory vs. Experiment

Superdeformed nanowires A. I. Mares, D. F. Urban, J. Bürki, H. Grabert, CAS & J. M. van Ruitenbeek, cond-mat/ cf. Physics of superdeformed nuclei

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…

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)