1

L.D. Landau ( 1937 )

A second order phase transition is generally well described phenomenologically if one identifies:

a). The order parameter field

b) Symmetry group G and its spontaneous breaking pattern.

A. General phenomenological approach to second order phase

transitions

)(xi

I. OVERVIEW of the VORTEX PHYSICS

1. The order parameter field and spontaneous symmetry breaking

2

Rest of degrees of freedom are “irrelevant” sufficiently close to the critical temperature Tc. Later, the “relevant” part, namely the symmetry breaking pattern and dimensionality was termed “the universality class”.

An example: XY- (anti) ferromagnet

Consider in-plane (planar) classical spins of fixed length

1S

defined on the D-dimensional lattice (the type of lattice and other micriscopic details are also irrelevant).

3

0 SM

1. T=0

well ordered

2. 0<T<Tc

ordered

3. T>Tc

disordered

large

small

0M

0 SM

4

),( yx MMM

yx iMM or, using complex numbers,

a. Order parameter : magnetization.

b . Symmetry : 2D rotations

cos sin '

sin cos '

x x y x

y x y y

M M M M

M M M M

5

)'()( MFMF

Symmetry means that the energy of the rotated state is the same as that of the original (not rotated) one.

ie Using complex numbers the symmetry transformation becomes U(1):

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2)*(

2

)(*)(* Tb

TaxdF D

Higher order terms

2. Effective free energy near the phase transition

Most general functional symmetric under

and space rotations, with lowest possible powers of

ie

,....)*(,* 32 are expected to be smaller close enough to Tc.

and lowest number of gradients is

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( ) ( ) ...,

( ) ...ca T T T

b T

F

cT T

F

cT T

Now we apply this general considerations to the superconductor – normal metal phase transition.

The remaining coefficients can be expanded around Tc:

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B. Ginzburg – Landau description of the SC-normal transition

1. Symmetry and order parameter in terms of the microscopic degrees of freedom

( ) ( ), ( )x x k c k c k

The complex order parameter is “amplitude of the Cooper pair center of mass”:

which is “the gap function” of BCS or any other (no matter how “unconventional”) microscopic theory .

a. Order parameter

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( )( ) *( ) i xsx n x e

( )x

( )*( )

2s

s

n xn x

density of the Cooper pairs, the Bose condensate

the superconductor (the Josephson, the global U(1) ) phase

We disregard the quantum fluctuations of the “bound” state (Cooper pair) and therefore consider the “Bose condensate” amplitude as a classical field.

The symmetry content of this complex field can be better specified via modulus and phase:

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b. Symmetry

The broken symmetry is charge U(1) mathematically the same symmetry as that of the XY magnet.

Without external magnetic field the free energy near transition therefore is:

23 2[ ] * ( ) * ( * )

2 * 2

* 2

c

e

F d x T Tm

m m

Ginzburg and Landau (1950) postulated a reasonable way to generalize this to the case of arbitrary magnetic field :)(xB

One is using the “principle” of local gauge invariance of electrodynamics.

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( )( ) ( )* 2

( ) ( ) ( )*

i xx e xe ec

A x A x xe

Electrodynamics is invariant under local gauge transformations:

x

2. Influence of magnetic field.

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*( ) ( )

eD x i A x

c

This invariance although not a symmetry (only the “global” part of it is) dictates the charge fields coupling to magnetic field.

To ensure local gauge invariance one makes the “minimal substitution”, namely replaces any derivative by a covariant derivative:

The local gauge invariance of the gradient term follows from linearity of the transformation of the covariant derivative:

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( )

( )

( )

* ( ) ( ) ( )

*

*

i x

i x

i x

ie cD x A x x e

c e

ieA e

c

D e

i i

Magnetic field

kjijki ABAB

is also gauge invariant.

2 2| ( ) | | ( ) |D x D x

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Minimizing the free energy with covariant derivatives one arrives at the set of GL equations: the nonlinear Schrödinger equation (variation with respect to )

22* ** *

( )4 2 * *s

c ie eB J A

m m c

222 *

( ) 02 * c

ei A T T

m c

and the supercurrent equation (variation of A):

Ginzburg – Landau equations

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)(x)(xB

3. Two characteristic scales

characterizes variations of , while the penetration depth characterizes variations of

( ) ,2 * ( )c

Tm T T

*

( )* 4 c

c mT

e T T

Both diverge at T=Tc.

GL equations possess two scales. Coherence length

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The only dimensionless parameter is the ratio of the two lengths which is temperature independent:

( ) *

( ) * 2

T m c

T e

2

1

Abrikosov (1957)

Ginzburg – Landau parameter

(the type II superconductivity ) there exist “topologically nontrivial” solutions – the Abrikosov vortices.

Properties of solutions crucially depend on the GL parameter. When

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1. Why look for an asymmetric “singular” solution? Normally one doesn’t look for inhomogeneous solutions in a homogeneous physical situation. Also one prefers to consider smooth regular one rather than singular solutions of field equations. Examples: Maxwell, Schrödinger eqs. etc. However the type II superconductor case is very special. Homogeneous external magnetic field does penetrate a sample as an array.

One has to look for these solutions due to combination of four facts. Two crucial and two more technical.

C. Abrikosov Vortices

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Type I :

Minimal area of domain walls.

H H

Type II:

Maximal area of domain walls.

a. Interface energy is negative for type II superconductors, while positive for the type I.

Mixed state under applied magnetic field

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b. Flux quantization.

( ) ( )0 0

*0i x i xe

D x De i A ec

0

( )cT T

To minimize the potential term far from isolated vortex (where B=0 ), one has to optimize the modulus of the order parameter:

The phase however is free to vary. In order to minimize the (positive) gradient term, one demands:

0gradF

Division into domains stops due to this.

*eAc

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0 *C C

cds A ds n

e

0

integer

* 2

n

hc hc

e e

C

c. n=1 is energetically favored over n>2

d. The normal core region shrinks to a point.

0

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2. Shape of the vortex solution

Vortex – a linear topological defect.

0 B

r

The “singularity” line 0

core

J

22

KT pair

Tonomura et al PRB43,7631 (1991)

Tonomura et al PRL66,2519 (1993)

Abrikosov vortices in type II superconductors as seen by electron beam tomography.

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D. Overview of properties of vortices and systems of vortices

(vortex matter)

To create a vortex, one has to provide energy per unit length ( line tension )

2

0 log4

Line energy

1. Inter-vortex repulsion and the Abrikosov flux line lattice

Therefore vortices enter an infinite sample only when field exceeds certain value

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They interact with each other via a complicated vector-vector force. Parallel straight vortices repel each other forming highly ordered structures like flux line lattice (as seen by STM and neutron scattering).

Pan et al (2002)

S.R.Park et al (2000)

Interactions between vortices

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01 24cH

Two critical fields

02 22cH

first vortex penetrates.

cores overlap

H

T

Mixed

Meissner

Hc2

Hc1

Tc

Normal

As a result the phase diagram of type II SC is richer than that of the two-phase type I

222

21

22 1c

c

H

H

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1cH

Two theoretical approaches to the mixed state

London appr. for infinitely thin lines

Just above vortices are well separated and have very thin cores

2cHJust below vortex cores almost overlap. Instead of lines one just sees array of superconducting “islands”

Lowest Landau level appr. for constant B

Tc

Mixed

Meissner

Hc2

Hc1

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In high Tc SC due to higher Tc, smaller and high anisotropy thermal fluctuations are not negligible. Thermally induced vibrations of the flux lattice can melt it into a “vortex liquid”.

Normal

Vortex liquid

2cH

1cH Meissner

FLL

H

TcT

The phase diagram becomes more complicated.

2. Thermal fluctuations and the vortex liquid

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First order melting of the Abrikosov lattice

Schilling et al Nature (1996,2001)

Zeldov et al Nature (1995)

Magnetization Specific heat

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Metastable states: zero field cooled and field cooled protocols result in different states.

Neutron scatering in NbLing et al (2000)

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Vortices can entangle around each other like polymers, however due to vectorial nature of their interaction they can also “disentangle” or “cut each other”.

There are therefore profound differences compared to the physics of polymers

Vortex “cutting” and entanglement

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Vortices are typically pinned by disorder. For vortex systems pinning create a glassy state or viscous entangled liquid. In the glass phase material becomes superconducting (zero resistance) below certain critical current Jc.

Columnarpoint

3. Disorder and the vortex glass

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Disappearance of Bragg peaks as the disorder increases

STM of both the pinning

centers (top) and vortices

(bottom)

Pan e alPRL (2000)

Gammel et al PRL (1998)

Imperfections act as pinning centers of vortices

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Vortices move under influence of external current (due to the Lorentz force).

The motion is generally friction dominated. Energy is dissipated in the vortex core which is just a normal metal. The resistivity of the flux flow is no longer zero.

Field driven flux motion probed by STM on NbSe2 A.M.Troianovski (2004)

4. Vortex dynamics

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Vortex loops, KT pairs and avalanchesCurrent produces expanding vortex loops even in the Meissner phase leading to non-ohmic “broadening” of I-V curves

/const T IVe

I

extJ

In 2D thermal fluctuations generate a curious Kosterlitz – Thouless vortex plasma exhibiting many unique features well understood theoretically

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Unstable normal domain under homogeneous quench splits into vortex-antivortex (KT) plasma

Polturak, Maniv (2004) Scanning SQUID magnetometer

Kirtley,Tsuei and Tafuri (2003)

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Spontaneous flux in rings

Kirtley,Tsuei and Tafuri (2003)

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Vortex front propagation is normally shock wave like, but occasionally creates avalanches

Magneto-optics in YBCO films, 10K, B=30mT, size 2.3x1.5 mm

Boltz et al (2003)

after 50 ns

after 10 s

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Disorder profoundly affects dynamics leading to the truly superconducting vortex glass state in which exhibits irreversible and memory dependent phenomena (like aging).

It became perhaps the most convenient playground to study the glass dynamics

Magneto-optics in NbJohansson et al (2004)

5. Vortex dynamics in the presence of disorder

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Transport in Nb

FC

FCW

B

T4.2K

0.01 0.1 10

10

20

30

80%

FC

FCW

4.90 K

Vol

tage

(V

)

Time(ms)0.01 0.1 10

10

20

30

FC

FCW

6.30 K

Time(ms)

Reversible region

Irreversible region

Dependence on magnetic history: the field cooled and the field cooled with return protocols result in different states.

Andrei et al (2004)

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Summary

1. In extreme type II superconductors the “topological” vortex degrees of freedom dominate most of the macroscopic magnetic and transport properties.

2. One can try to use the GL theory to describe these degrees of freedom.

3. Experiments suggest that in new high Tc SC thermal fluctuations are important as well as disorder.

4. The vortex matter physics is quite unique, well controlled experimentally and may serve as a “laboratory” to test a great variety of theoretical ideas.