Stability analysis of metallic nanowires: Interplay of Rayleigh and Peierls Instabilities Daniel Urban and Hermann Grabertcond-mat/ 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 leads cylindrical leads L 2R 0 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 R 0 is stable if and for arbitrary deformations a is called stability coefficient stability criterion: > 0 for all q 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) 1m1m Au nanowires: Kondo et al. Science 289, p.606 (2000) Images of gold nanowires show approx. cylindrical geometries with certain “magic” radii Conductance quantization : Histogram of conductance values, measured for Potassium at T=4.2K with 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 (1997) Conductance measurements for Au in a STM-setup, showing four different contacts at T=4.2K. Untied et al. Phys. Rev. B 56, p (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 Lk F =1000 at three different temperatures. Red areas are unstable ( <0) at T=0.05T F, the solid and the dotted lines show the contours of the unstable regions for T=0.01T F and T=0.005T F respectively. The dashed blue line shows the criterion for the Rayleigh instability (qR 0 = 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 Lk F =1000 at zero temperature. Red areas are unstable ( <0). The dashed blue line shows the criterion for the Rayleigh instability (qR 0 =1), the dotted lines show the criterion for the Peierls instability (q = 2k F ( ) ). (b)Extent of the regions of instability as a function of wire length Stability diagrams: