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)

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, 165414 (2003)

A>0

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.

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, 245414 (2006)

3. Stable elliptical nanowires

D. F. Urban, J. Bürki, C.-H. Zhang, CAS & H. Grabert, PRL 93, 186403 (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, 186403 (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, 186403 (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, 186403 (2004)Discussion: D. F. Urban et al., Solid State Comm. 131, 609 (2004)

D. Urban, J. Bürki, CAS & H. Grabert PRB 74, 245414 (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, 245414 (2006)

Higher-m deformations less stable due to increasedsurface energy.

5. Material dependence of stability

Na

AuRelative stability of deformed structuresdepends on surface tension in naturalunits:

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/0703589

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/0703589

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.

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

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/0703589

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)