The Peierls Instability in Metal Nanowires

Daniel Urban (Albert-Ludwigs Universität Freiburg, Germany)

In collaboration with C.A.Stafford and H.Grabert

Peierls Instability

• is a distortion energetically favorable?• max energy gain for

EF

This model requires:• good charge screening• almost spherical Fermi-surface

NFEM is suitable for s-orbital-metals (alkali metals, gold)

Nanoscale Free-Electron Model (NFEM)

• free electrons + confining potential

• ions = incompressible homogeneous background• nanowire = quantum waveguide

• open system connected to reservoirs

scattering problem

• eigenenergies

NFEM: Nanowire = Waveguide

• transverse wave function

(modes, channels)

• wave function

EF

quantized motion inx-y-plane

free motion in z-direction

kF,1kF,n

Difference from standard Peierls theory:

• no periodic boundary conditions

Peierls Instability at Length L

Cylindrical wire + perturbation

Pseudo gap, energy gap only for

• nanowire with finite length L

• system = nanowire + leads

Surface Phonons

• Ions = incompressible fluid

• Born-Oppenheimer approximation

• Phonon frequency

mode stiffness

mode inertia

Grand canonical potential:

Scattering Matrix Formalism

density of states grand canonical potential

Grand canonical potential

mode stiffness

Mode Stiffness

Cylindrical nanowire + perturbation

LC: critical length

Dispersion Relation

CDW Correlations

Crossover:L<LC: small fluctuations about cylindrical shapeL>LC: CDW with quantum fluctuations, no long-range order

Finite-size Scaling

Scaling of the mode stiffness:

Length scale

Energy scale

critical length

Critical point and

Correlation length ξn

ξ is material dependent & tunable by applying strain

singular part of the mode stiffness

Summary

• Peierls instability in metal nanowires at L=LC~ξ

Further reading:

• DFU, Stafford, Grabert, cond-mat/0610787

• DFU, Grabert, PRL 91, 256803

• Hyperscaling of the singular part of the free energy

• CDW in metal nanowires should beexperimentally observable under strain