Stability analysis of metallic nanowires:Stability analysis of metallic nanowires:Interplay of Rayleigh and Peierls InstabilitiesInterplay of Rayleigh and Peierls Instabilities

Daniel Urban and Hermann Grabert cond-mat/0307279

Jellium model:• Free electron gas confined by a hard wall potential (given by the wire geometry)• Ions = incompressible jellium background• Open system, electron reservoirs at both ends• Volume conservation constraint

This model requires:• Strong delocalisation of the valence electrons• Good screening• Nearly spherical Fermi surface

Conditions satisfied for s-orbital metals like the alkali metals and partially gold

Results:Stability analysis:perturbated cylindrical

wire: R(z)cylindrical

leadscylindrical

leads

L

2R0

Matching the wave functions on the boundaries determines the S-matrix that describes the perturbated wire.

Which cylindrical configurations are stable?

• look at axisymmetric deformations

• calculate scattering matrix in orders of

• calculate density of states• calculate grand canonical potential

• cylindrical wire with radius R0 is

stable if

and

for arbitrary deformations

q

iqzqebRzR 1)( 0

)()( 3)2(2)1()0(

0)1( 0)2(

L

nq

2

is called stability coefficient

stability criterion: > 0 for all q

LLRqb

L

LR

qq

1),,(

),(0

20

)2(

0)1(

sm

chms

surf LRqRqLRq,

0)(

0)(

0 ),,(),(),,(

Ohnishi et al. Nature 395, p.780 (1998)

Fabrication:

3D metallic nanowires can be produced by i. Scanning tunneling microscopy (STM)ii. Mechanically controllable break junctions (MCBJ)iii. Electron-beam irradiation of thin metal films

Kondo et al. Phys. Rev. Lett 79, p.3455 (1997)(a)Yanson et al. Nature 400, p.144 (1999)(b) Scheer et al. Nature 394, p.154 (1998)

(b)

(a)

1m

Au nanowires:

Kondo et al. Science 289, p.606 (2000)

Images of gold nanowires show approx. cylindricalgeometries with certain “magic” radii

Conductance quantization:

Histogram of conductance values,measured for Potassium at T=4.2Kwith an MCBJ-device.Yanson, PhD thesis, Universiteit Leiden (2001)

Conductance histogram for STM gold nanowires at T=4.2K. Inset: typical conductance vs. time staircase. Costa-Krämer et al. Phys. Rev B 55, p.12910 (1997)

Conductance measurements forAu in a STM-setup, showing fourdifferent contacts at T=4.2K.Untied et al. Phys. Rev. B 56, p. 2154 (1996)

Peierls instability:• Known from

quantum mechanics of 1d systems

• Linear chains lower their energy by dimerization

•Known from classical

continuum mechanics

•Due to surface tension

•For a length L > 2 R

Rayleigh instability: Why are these

wires stable at all?

??????

(a)Stability diagram for a cylindrical wire of length LkF=1000 at three different temperatures. Red areas are unstable (<0) at T=0.05TF, the solid and the dotted lines show the contours of the unstable regions for T=0.01TF and T=0.005TF respectively. The dashed blue line shows the criterion for the Rayleigh instability (qR0 = 1).

(b)Extent of the regions of instability as a function of temperature

(a) (b)(a) (b)

(a)Stability diagram for a cylindrical wire of length LkF=1000 at zero temperature. Red areas are unstable (<0). The dashed blue line shows the criterion for the Rayleigh instability (qR0=1), the dotted lines show the criterion for the Peierls instability (q = 2kF

() ). (b)Extent of the regions of instability as a function of wire length

Stability diagrams: