1. 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, 186403 (2004)

2. Electron-shell potential

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

4. 1. Linear stability analysis of a cylinder (m=0) Mode stiffness: Classical (Rayleigh) stability criterion:

5. F. Kassubek, CAS, H. Grabert & R. E. Goldstein, Nonlinearity 14, 167 (2001) Mode stiffness α(q)

6. Stability under axisymmetric perturbations C.-H. Zhang, F. Kassubek & CAS, PRB 68, 165414 (2003) A>0

7. Stability of nanocylinders at ultrahigh current densities C.-H. Zhang, J. Bürki & CAS, PRB 71, 235404 (2005) ! Generalized free energy for ballistic nonequilibrium electron distribution. Coulomb interactions included in self-consistent Hartree approximation.

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

9. General stability analysis of a cylinder D. Urban, J. Bürki, CAS & H. Grabert PRB 74, 245414 (2006)

10. 3. Stable elliptical nanowires D. F. Urban, J. Bürki, C.-H. Zhang, CAS & H. Grabert, PRL 93, 186403 (2004)

11. 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)

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

13. 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)

14. D. Urban, J. Bürki, CAS & H. Grabert PRB 74, 245414 (2006) 4. Quadrupolar cross sections

15. 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)

16. Higher multipole deformations D. Urban, J. Bürki, CAS & H. Grabert PRB 74, 245414 (2006) Higher-m deformations less stable due to increased surface energy.

17. 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.

18. Special case: Aluminum A. I. Mares, D. F. Urban, J. Bürki, H. Grabert, CAS & J. M. van Ruitenbeek, cond-mat/0703589 Two different types of histograms (history dependent) Crossover from electronic to atomic shell effects at

19. Extracting individual conductance peaks A. I. Mares, D. F. Urban, J. Bürki, H. Grabert, CAS & J. M. van Ruitenbeek, cond-mat/0703589

20. Linear stability analysis for Aluminum A. I. Mares, D. F. Urban, J. Bürki, H. Grabert, CAS & J. M. van Ruitenbeek, cond-mat/0703589 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.

21. A. I. Mares, D. F. Urban, J. Bürki, H. Grabert, CAS & J. M. van Ruitenbeek, cond-mat/0703589 Electron-shell structure: Theory vs. Experiment

22. Superdeformed nanowires A. I. Mares, D. F. Urban, J. Bürki, H. Grabert, CAS & J. M. van Ruitenbeek, cond-mat/0703589 cf. Physics of superdeformed nuclei

23. 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…

24. 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)