1 Introduction to Vortices in Superconductors Pre-IVW 10 Tutorial Sessions, Jan. 2005, TIFR, Mumbai, India Pre-IVW 10 Tutorial Sessions, Jan. 2005, TIFR, Mumbai, India Thomas Nattermann University of Cologne Germany GermanyOutline: 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 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

2 17th century vortex physics vortices …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 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

3 Superconductivity as a true thermodynamic phase Ideal conductor (Kammerling Onnes 1911) Ideal diamagnet (Meissner-Ochsenfeld 1933) Hg <  Superconductivity: true thermodynamic phase Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

4 9.5 K 0.66 K 0.61 K 0.40 K K K K 7.2 K Niobium (Nb) Osmium (Os) Zirconium (Zr) Titanium (Ti) Iridium (Ir) Tungsten (W) Rhodium (Rh) Lead (Pb) Carbon (C) Lead (Pb) Mercury (Hg) Tin (Sn) Indium (In) Aluminum (Al) Gallium (Ga) Zinc (Zn) 15 K K 4.15 K 3.72 K 3.41 K K K 0.85 K 17.5 K K 23.2 K Nb 3 Al Nb 3 Sn Nb 3 Ge Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

5 Time-line of Superconductors JG Bednorz, KA Müller Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

6 Fritz and Heinz London 1935 Superconductivity = Long Range Order of Momentum perfect conductor + perfect diamagnet perfect conductor + perfect diamagnet = superconductor F. London 1950 Fluxoid conservation and quantization Problem : interface energy negative Extension: anisotropy, non-locality London penetration depth Surface current screens bulk r£r£ B= - r 2 B = -2 B Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

7 Ginzburg and Landau 1950 Superconducting order parameter  T)»(T-T c0 ) correlation length: Superconductivity = broken U(1) symmetry (ODLRO, Penrose, Onsager ´51, ´56) Extensions: several order parameters (e.g. s+d-wave) ~ |   | ¢ |   |, Extensions: several order parameters (e.g. s+d-wave) ~ |D   | ¢ |D   |, anisotropy |D  2  | 2,.. anisotropy |D  2  | 2,.. = - i (e * /hc) A, D= r - i (e * /hc) A, Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

8 Bardeen Cooper Schrieffer 1957 attractive Cooper pair formation (bound state of 2 electrons) electron phonon interaction: very short rangedstrong in s-wave (l=0) channel Symmetry of pairs of identical electrons: orbitalspin wave function totally antisymmetric under particle exchange even parity: l= 0,2,4,…, S=0 singlet evenodd odd parity: l= 1,3,5,…, S=1 triplet oddeven ) e * =2e Sigrist, Zuoz 2004 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

9 Conventional superconductivity Order parameter structureless complex condensate wave function Microscopic origin: Coherent state of Cooper pairs Bardeen-Cooper-Schrieffer (1957) violation of U(1)-gauge symmetry Conventional  k =  independent of k pairs of electrons diametral on Fermi surface; vanishing total momentum Sigrist, Zuoz 2004 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

10 Rescaling:  =  -1 » effective charge Parameters of Ginzburg-Landau-Theory  ~ - H GL / T    ~ e - H GL / T Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

11 Mean-field Theory no screening symmetric gauge A = H(-y/2, x/2,0) For decreasing field 1st solution E n=0 =1 at H = H c,2 (T) = 2 1/2  H c (T)  n,m  n,m Quantum particle in magnetic field ! Landau levels E n Nattermann, pre-IV10 Tutorial Sessions, TIFR Mumnai 2005

12 Abrikosov 1957 Abrikosov 1957: Lowest Landau Level Approximation: n=0 only magnetic flux penetrates SC if Convenient: Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

13 quantifized flux penetrates superconductor for Abrikosov 1957 Energy per unit length: Vortex interaction Low field H ¼ H c1 : exist single vortex solution of GL-equations ~ quantized flux tube Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

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15 London Approximation Apply r£ on 2nd GL-equation ) Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

16 B0B0 -4πM HcHc B0B0 H c1 H c2 Vortex state Normal state Superconducting state Normal state Type I Type II H < H c M H < H c1 M H c1 < H < H c2 Vortex Type-I and Type-II Superconductivity Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

17 form triangular lattice ´´broken translational invariance´´ Many vortices: Loss of perfect diamagnetism. Bitter decoration Abrikosov Lattice Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

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19 Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Vortices in rotating Bose-Einstein Condensates

20 Crab nebula (Hubble space telescope) Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Vortices in Neutronstars

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22 Center of Crab nebula: rotating neutron star with vortices in its superfluid core vortices in its superfluid core Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Vortices in Neutronstars

23 Glitches = sudden increase of rotation frequency due to depinning of vortices from outer crust Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005 Vortices in Neutronstars

24 Elasticity Theory: Brandt 1977 Vortex lines:positions Distortion from ideal positions Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

25 Pardo et al., PRL (1997) Hexagonal Abrikosov lattice, fragile, susceptible to plastic deformation for H close to H c1 and H c2 small distortionsfrom perfect order: Elasticity theory, ´´soft matter´´ Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

26 Dislocations in the vortex lattice entanglement  screw dislocations screw dislocation loop 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>0Kierfeld Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

27 Single Dislocation dislocation=directed stiff line characteristic energy/length core energy stiffness core energy long-range elastic strains ~1/r bending energy Kierfeld Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005