1 introduction to vortices in superconductors pre-ivw 10 tutorial sessions, jan. 2005, tifr, mumbai,...
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Introduction to Vortices in SuperconductorsIntroduction to Vortices in Superconductors Pre-IVW 10 Tutorial Sessions, Jan. 2005, TIFR, Mumbai, IndiaPre-IVW 10 Tutorial Sessions, Jan. 2005, TIFR, Mumbai, India
Thomas Nattermann Thomas Nattermann
University of CologneUniversity of Cologne GermanyGermany
Outline:Outline:1.1. Mean field theoryMean field theory2.2. Thermal fluctuationsThermal fluctuations3.3. DisorderDisorder4.4. MiscellaneousMiscellaneous
Reviews: Blatter et al., Rev. Mod. Phys. 1994; Brandt, Rep. Progr. Phys. 1995; Nattermann and Scheidl,, Adv. Phys. 2000.
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17th century vortex physics17th century vortex physics…whatever was the manner whereby matter was first set in motion, the vorticesvortices 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
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Superconductivity as a true thermodynamic phaseSuperconductivity as a true thermodynamic phase
Ideal conductor (Kammerling Onnes 1911)
Ideal diamagnet (Meissner-Ochsenfeld 1933)
Hg
< 10-5
Superconductivity: true thermodynamic phaseSuperconductivity: true thermodynamic phase
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9.5 K0.66 K 0.61 K 0.40 K 0.1125 K 0.0154 K 0.000325 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 7.196 K 4.15 K 3.72 K 3.41 K 1.175 K 1.083 K 0.85 K
17.5 K18.05 K23.2 K
Nb3AlNb3SnNb3Ge
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Time-line of SuperconductorsTime-line of Superconductors
JG Bednorz, KA Müller
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Fritz and Heinz London 1935Fritz and Heinz London 1935
Superconductivity = Long Range Order of MomentumSuperconductivity = Long Range Order of Momentum
perfect conductor + perfect diamagnetperfect conductor + perfect diamagnet = superconductor
F. London 1950
Fluxoid conservation and quantization
Problem : interface energy negativeExtension: anisotropy, non-locality
London penetration depth
Surface current screens bulk r£r£B= - r2 B = -2B
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Ginzburg and Landau 1950Ginzburg and Landau 1950Superconducting order parameter
T)»(T-Tc0) correlation length:
Superconductivity = broken U(1) symmetry Superconductivity = broken U(1) symmetry (ODLRO, Penrose, Onsager ´51, ´56)(ODLRO, Penrose, Onsager ´51, ´56)
Extensions:Extensions: several order parameters (e.g. s+d-wave) ~ | several order parameters (e.g. s+d-wave) ~ |D||¢¢ ||D |, |,
anisotropy |Danisotropy |D22||22,..,..
D==r - i (e - i (e**/hc) /hc) A , A ,
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Bardeen Cooper Schrieffer 1957Bardeen Cooper Schrieffer 1957attractive
Cooper pair formation (bound state of 2 electrons)electron phonon interaction:very short ranged strong in s-wave (l=0) channel
Symmetry of pairs of identical electrons:
orbital spinwave function totally antisymmetricunder particle exchange
even parity: l= 0,2,4,…, S=0 singleteven odd
odd parity: l= 1,3,5,…, S=1 tripletodd even
) e*=2e
Sigrist, Zuoz 2004Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005
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Conventional superconductivityConventional superconductivity
Order parameterstructureless complexcondensate wave function
Microscopic origin: Coherent state of Cooper pairs
Bardeen-Cooper-Schrieffer (1957)
violation of U(1)-gauge symmetry
Conventional k = independent of kpairs of electrons diametral on Fermi surface;vanishing total momentum
Sigrist, Zuoz 2004Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005
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Rescaling:
=-1 » effective chargeeffective charge
Parameters of Ginzburg-Landau-TheoryParameters of Ginzburg-Landau-Theory
~ ~ e - -
HHGLGL//TT
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Mean-field TheoryMean-field Theory
no screening
symmetric gauge A = H(-y/2, x/2,0)
For decreasing field 1st solution EFor decreasing field 1st solution En=0n=0=1 at H = H=1 at H = Hc,2 c,2 (T) = 2(T) = 21/21/2 H Hcc(T)(T)
n,m n,m
Quantum particle in magnetic field Quantum particle in magnetic field !! Landau levels E Landau levels Enn
Nattermann, pre-IV10 Tutorial Sessions, TIFR Mumnai 2005
12Abrikosov 1957Abrikosov 1957:
Lowest Landau Level ApproximationLowest Landau Level Approximation: : nn=0 only=0 only
magnetic flux penetrates SC
if
Convenient:Convenient:
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quantifized flux penetrates superconductor for quantifized flux penetrates superconductor for
Abrikosov 1957Abrikosov 1957
B(r)
(r)
r
Energy per unit length:Energy per unit length:
Vortex interaction Vortex interaction
Low field HLow field H¼¼ H Hc1c1: :
exist single vortex solution of GL-equationsexist single vortex solution of GL-equations
~ quantized flux tube~ quantized flux tube
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London ApproximationLondon Approximation
Apply r£ on 2nd GL-equation )
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B0
-4πM
Hc B0
-4πM
Hc1 Hc2
Vortex state Normal state
Superconducting state
Normal state
Type I Type II
H < Hc
M
H < Hc1
M
Hc1 < H < Hc2
Vortex
Type-I and Type-II SuperconductivityType-I and Type-II Superconductivity
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form triangular lattice
´´broken translational invariance´´
Many vortices:
Loss of perfect diamagnetism.
H
TMeissnerC66
02 ~300 T
~100 G
HC2
HC1
HC1
Bitter decoration
Abrikosov LatticeAbrikosov Lattice
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Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005
Vortices in rotating Bose-Einstein CondensatesVortices in rotating Bose-Einstein Condensates
20Crab nebula (Hubble space telescope)
Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005
Vortices in NeutronstarsVortices in Neutronstars
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Center of Crab nebula: rotating neutron star with Center of Crab nebula: rotating neutron star with vortices in its superfluid corevortices in its superfluid core
Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005
Vortices in NeutronstarsVortices in Neutronstars
23Glitches = sudden increase of rotation frequency due to depinning of vortices from outer crust
Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005
Vortices in NeutronstarsVortices in Neutronstars
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Elasticity Theory: Brandt 1977Elasticity Theory: Brandt 1977
Vortex lines: positions
Distortion from ideal positions
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Pardo et al., PRL (1997)
Hexagonal Abrikosov lattice,fragile, susceptible to plastic deformation for H close to Hc1 and Hc2
small distortions from perfect order:
Elasticity theory,
´´soft matter´´
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Dislocations in the vortex latticeDislocations 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>0
KierfeldKierfeldNattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005
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Single DislocationSingle Dislocation
•dislocation=directed stiff line•characteristic energy/length
•core energy
•stiffness
core energy long-range elastic strains ~1/r
bending energy
KierfeldKierfeldNattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005