Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Introduction to Vortices in Superconductors Pre-IVW 10 Tutorial Sessions, Jan. 2005, TIFR, Mumbai, India Thomas Nattermann University of Cologne Germany Outline: 1. Mean field theory 2. Thermal fluctuations 3. Disorder 4. Miscellaneous Reviews: Blatter et al., Rev. Mod. Phys. 1994; Brandt, Rep. Progr. Phys. 1995; Nattermann and Scheidl,, Adv. Phys. 2000. 1 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 17th century vortex physics …whatever was the manner whereby matter was first set in motion, the vortices into which it is divided must be so disposed that each turns in the direction in which it is easiest to continue its movement for, in accordance with the laws of nature , a moving body is easily deflected by meeting another body… I hope that posterity will judge me kindly, not only as to the things which I have explained, but also to those which I have intentionally omitted so as to leave to others the pleasure of discovery. Rene Descartes 1644 2 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Superconductivity as a true thermodynamic phase Ideal diamagnet (Meissner-Ochsenfeld 1933) Ideal conductor (Kammerling Onnes 1911) Hg < 105 Superconductivity: true thermodynamic phase 3 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Carbon (C) Lead (Pb) Mercury (Hg) Tin (Sn) Indium (In) Aluminum (Al) Gallium (Ga) Zinc (Zn) 15 K 7.196 K 4.15 K 3.72 K 3.41 K 1.175 K 1.083 K 0.85 K Niobium (Nb) Osmium (Os) Zirconium (Zr) Titanium (Ti) Iridium (Ir) Tungsten (W) Rhodium (Rh) Lead (Pb) 9.5 K 0.66 K 0.61 K 0.40 K 0.1125 K 0.0154 K 0.000325 K 7.2 K Nb3Al 17.5 K Nb3Sn 18.05 K Nb3Ge 23.2 K 4 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Time-line of Superconductors JG Bednorz, KA Müller 5 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Fritz and Heinz London 1935 London penetration depth Surface current screens bulk r£r£B= - r2 B = -2B perfect conductor + perfect diamagnet = superconductor Superconductivity = Long Range Order of Momentum Fluxoid conservation and quantization F. London 1950 Problem : interface energy negative Extension: anisotropy, non-locality 6 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Ginzburg and Landau 1950 Superconducting order parameter D=r - i (e*/hc) A , (T)»(T-Tc0) correlation length: Superconductivity = broken U(1) symmetry (ODLRO, Penrose, Onsager ´51, ´56) Extensions: several order parameters (e.g. s+d-wave) ~ |D|¢ |D |, anisotropy |D2|2,.. 7 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Sigrist, Zuoz 2004 Bardeen Cooper Schrieffer 1957 attractive electron phonon interaction: Cooper pair formation (bound state of 2 electrons) very short ranged strong in s-wave (l=0) channel ) e*=2e Symmetry of pairs of identical electrons: wave function totally antisymmetric under particle exchange orbital spin even parity: l= 0,2,4,…, S=0 singlet even odd odd parity: l= 1,3,5,…, S=1 triplet odd even 8 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Sigrist, Zuoz 2004 Conventional superconductivity structureless complex condensate wave function Order parameter Microscopic origin: Coherent state of Cooper pairs Bardeen-Cooper-Schrieffer (1957) violation of U(1)-gauge symmetry pairs of electrons diametral on Fermi surface; vanishing total momentum Conventional k = independent of k 9 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Parameters of Ginzburg-Landau-Theory Rescaling: B ~ e - HGL/T /=-1 » effective charge 10 Nattermann, pre-IV10 Tutorial Sessions, TIFR Mumnai 2005 Mean-field Theory no screening symmetric gauge A = H(-y/2, x/2,0) = fn,m n,m Quantum particle in magnetic field ! Landau levels En For decreasing field 1st solution En=0=1 at H = Hc,2 (T) = 21/2 Hc(T) 11 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Lowest Landau Level Approximation: n=0 only Convenient: Abrikosov 1957: magnetic flux penetrates SC 12 if Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Abrikosov 1957 y(r) Low field H¼ Hc1: B(r) x l r exist single vortex solution of GL-equations ~ quantized flux tube Energy per unit length: Vortex interaction quantifized flux penetrates superconductor for 13 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 14 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 London Approximation Apply r£ on 2nd GL-equation ) 15 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Type-I and Type-II Superconductivity Type II Type I Superconducting state -4πM -4πM Normal state Hc M Vortex state B0 Hc1 Normal state Hc2 B0 M Vortex 16 H < Hc H < Hc1 Hc1 < H < Hc2 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Abrikosov Lattice Many vortices: form triangular lattice ´´broken translational invariance´´ Loss of perfect diamagnetism. Bitter decoration H H C2 ~f 0 /x 2 ~300 T H C1 C66 Meissner H C1~100 G T 17 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 18 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Vortices in rotating Bose-Einstein Condensates 19 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Vortices in Neutronstars Crab nebula (Hubble space telescope) 20 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Vortices in Neutronstars Center of Crab nebula: rotating neutron star with vortices in its superfluid core 22 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Vortices in Neutronstars Glitches = sudden increase of rotation frequency due to depinning of vortices from outer crust 23 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Elasticity Theory: Brandt 1977 Vortex lines: positions Distortion from ideal positions 24 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Hexagonal Abrikosov lattice, fragile, susceptible to plastic deformation for H close to Hc1 and Hc2 Pardo et al., PRL (1997) small distortions from perfect order: Elasticity theory, ´´soft matter´´ 25 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Kierfeld Dislocations in the vortex lattice screw dislocation loop •entanglement screw dislocations •loss of translational order, edge dislocations •topological line defect, charge = Burgers vector b •planarity constraint: dislocations cannot climb out of b-H plane (no "vortex ends") •mobile dislocations r>0 26 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Kierfeld Single Dislocation •dislocation=directed stiff line •characteristic energy/length •core energy •stiffness core energy long-range elastic strains ~1/r bending energy 27

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