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.
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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
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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
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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
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Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005
Time-line of Superconductors
JG Bednorz, KA Müller
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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
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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 |D2|2,..
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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
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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
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Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005
Parameters of Ginzburg-Landau-Theory
Rescaling:
B ~ e - HGL/T
/=-1
»
effective charge
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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)
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Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005
Lowest Landau Level Approximation:
n=0 only
Convenient:
Abrikosov 1957:
magnetic flux penetrates SC
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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
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Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005
London Approximation
Apply r£ on 2nd GL-equation )
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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
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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
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Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005
Vortices in rotating Bose-Einstein Condensates
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Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005
Vortices in Neutronstars
Crab nebula (Hubble space telescope)
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Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005
Vortices in Neutronstars
Center of Crab nebula: rotating neutron star with
vortices in its superfluid core
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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
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Elasticity Theory: Brandt 1977
Vortex lines:
positions
Distortion from ideal positions
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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´´
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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
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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
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