1. Stability and Symmetry Breaking in Metal Nanowires I: Toward a Theory of Metallic Nanocohesion Capri Spring School on Transport in Nanostructures, March 29, 2007 Charles Stafford

2. Acknowledgements Students: Chang-hua Zhang (Ph.D. 2004) Dennis Conner (M.S. 2006) Nate Riordan Postdoc: Jérôme Bürki Coauthors: Dionys Baeriswyl, Ray Goldstein, Hermann Grabert, Frank Kassubek, Dan Stein, Daniel Urban Funding: NSF Grant Nos. DMR0072703 and DMR0312028; Research Corp.

3. 1. How thin can a metal wire be?

4. Surface-tension driven instability T. R. Powers and R. E. Goldstein, PRL 78, 2555 (1997) Cannot be overcome in classical MD simulations!

5. Fabrication of a gold nanowire using an electron microscope Courtesy of K. Takayanagi, Tokyo Institute of Technology

6. Extrusion of a gold nanowire using an STM

7. What is holding the wires together? A mechanical analogue of conductance quantization?

8. 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) Corrected Sharvin conductance: T=90K Conductance histograms of sodium nanocontacts

9. 2. Nanoscale Free-Electron Model (NFEM) Model nanowire as a free-electron gas confined by hard walls. Ionic background = incompressible fluid. Most appropriate for s-electrons in monovalent metals. Regime: Metal nanowire = 3D open quantum billiard.

10. Scattering theory of conduction and cohesion Electrical conductance (Landauer formula) Grand canonical potential (independent electrons) Electronic density of states (Wigner delay)

11. Quantum suppression of Shot noise NFEM w/disorder Gold nanocontacts

12. Multivalent atoms

13. Adiabatic + WKB approximations Schrödinger equation decouples: WKB scattering matrix for each 1D channel:,

14. Comparison: NFEM vs. experiment Exp: Theory:

15. Weyl expansion + Strutinsky theorem Mean-field theory: Weyl expansion:

16. Electron-shell potential → 2D shell structure favors certain “magic radii” Classical periodic orbits in a slice of the wire

17. NFEM vs. self-consistent Jellium calculation

18. Different constraints possible in NFEM # of atoms Physical properties (e.g., tensile force) depend only on energy differences:

19. Example of the Strutinsky theorem: self-consistent Hartree approximation

20. Special case: the constant-interaction model Last term is important!

21. Semiclassical power counting Planck’s constant: → Surface energy dominates shell correction?!

22. 3. Conclusions to Lecture 1 Nanoscale Free Electron Model is able to describe quantum transport and metallic nanocohesion on an equal footing, explaining observed correlations in force and conductance of metal nanocontacts. Total energy calculations apparently not sufficient to address nanowire stability. What more is needed? See Lecture 2!