The Peierls Instability in Metal Nanowires Daniel Urban (Albert-Ludwigs Universität Freiburg,...
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Transcript of The Peierls Instability in Metal Nanowires Daniel Urban (Albert-Ludwigs Universität Freiburg,...
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