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.

Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

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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

Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

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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 ,

Nattermann, pre-IVW10 Tutorial Sessions, TIFR Mumbai 2005

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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

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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)

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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

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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