1

1. GL equations in a rotationally invariant situation

Straight vortex line has symmetries z-translations + xy rotations. A clever choice of gauge should utilize these symmetries. Since physical quantities depend on the distance from the center r only the cylindrical (polar) coordinates is the natural choice.

A. The isolated vortex solution

III. VORTICES and THEIR INTERACTIONS in LONDON

APPROXIMATION

2

)(1

)( rAdr

d

rrB

0 ( )

ˆ( )

if r e

A A r

0

1( ) ' ' ( ')

r

A r r dr B rr

JA

azimuthal vector field

tangential vector field

Using polar coordinates one chooses the following Ansatz (which includes a choice of the “unitary” gauge):

3

The vector potential

Details: polar coordinates

Partial derivatives

siny r cosx r 22 yxr

yarctg

x

( )sinxA A r ( ) cosyA A r

2 2 2 2

1sin cos

r y x

x x x r x y rx y

r r

4

1cos sin

y r r

Magnetic field

2 2 2 2 '

1sin ( cos ) cos ( cos )

1cos ( sin ) sin ( sin )

1 1cos cos ' sin sin

1 1' ( )

y xB A A A Ax y r r

A Ar r

A A A Ar r

dA A A r

r r dr

B is indeed a function of r only

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Supercurrent

22* *

2

0

22 2

0

222 2 2

0 0

2 20

0

* *( )

2 * 2 *

* 1sin cos . .

2 *

*sin

2 *

* 1 *sin sin

2 * 2 *

* 1 2

*

x x x x

i i

ie eJ A

m m c

iefe fe c c

m r r

eA f

m c

e e2 f A f

m r m c

e Af

m r

sin

Current therefore flows around the vortex.

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Supercurrent equation has the azimuthal component only

220

0

1 * 1 2( )

4 4 *

c dB c d d e AJ rA f

dr dr r dr m r

2

3 2

0

1 2 10

A d dff f f r

r r dr dr

Similarly the nonlinear Schroedinger equation takes a form

This should be supplemented by a set of four boundary conditions at the center and far away.

GL equations

7

Near the center one expects a maximum of magnetic field B(0) leading to linear potential:

rB

rAnear 2

)0()(

A

r

2. Boundary condition and asymptotics near the vortex core.

0rAsymptotics of the order parameter at

( ) , 0mnearf r cr m

is assumed to be a power

8

Substituting this single vortex Ansatz into the NLSE one obtains:

2

3 3 2 2 2

0

1 (0)m m m mBcr c r r cr m cr

r

Leading terms are two:

The order parameter therefore vanishes at the center of the vortex core as r for a single fluxon vortex.

Near r=0, we can use an expansion in r.

2 21 1mm r m

9

00 *

( ) 0 ( ) , .2far

hcB r A r

r e

The order parameter therefore exponentially approaches its bulk value in SC

Far away flux quantization gives

/( ) 1 rfarf r const e

Using the four boundary conditions and linearity of both A and f at origin one can effectively use the “shooting” method to find the vortex solution

3. Boundary conditions outside the core. Numerical solution

10

( ) tanhr

f r

A good simple fit for order parameter all r is available:

A simple expression for the magnetic field distribution can be obtained in phenomenologically important case of strongly type II superconductors using the London approximation

Exercise 2: transform the GL equations for a single vortex into a dimensionless form and solve it numerically using the shooting method for

11

Far enough from the vortex cores one generally makes the London appr. (even for many vortices)

4. The London electrodynamics outside vortex cores. Magnetic field of a vortex for

,0, i x yx y e

,0

*

*( , )

x x

i x yx

eD i A

x c

ei x y A e

x c

1

Covariant derivative

12

In this case the supercurrent equation takes a London form:

* *

20

22 20 0

*( )

2 ** *

( , )*

* *( , )

* *

ieJ D D

me ei A x ym c

e ei x y Am m c

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Supercurrent and Londons’ eqs

Taking 2D curl of the Maxwell equation

Jc

B 4

13

2

2 20

2002

4

4 4 *( )

* *4 *

ˆ ( )*

B Jcc e

x Be m c

ez x B

m c

This is transformed into Londons’ equations in the presence of a straight vortex:

one obtains for a single vortex phase field:

2 202 2

1ˆ ( )B B z x

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The eqs. are mathematically identical to the those for the Green’s function of the Klein-Gordon eqs and therefore can be solved by Fourier transform.

2

22

( ) ( )

1( ) ( )

(2 )

ikx

ikx

B k e B x d x

B x e B k d k

2 2 02

( )k B k

Field of a single vortex

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02 2 2

2 cos0 0

02 2 2 20 0

( )(2 )

(2 ) 2

ikx

ikr

eB x

k

k e rd dk K

k

0K

which has a pole. Inverse Fourier transform therefore will fall off exponentially:

where is the Hankel function

0

2 2 2( )B k

k

16

r

)(rB

02log

2

02 ( / )

2Log r

1/ 2/0

22 2re

r

r

r

Exponential tail

The core cutoff

Most of the flux for passes through the region. The core region fraction is insignificant:

2 002 2

log

2 2

17

/

~ 0

* 1( ) * ( )

4 *s

r

r

c dB eJ r e v r r

dr m r

e r

Taking a derivative the supercurrent is calculated

One observes a rather long range decrease of the supercurrent between the coherence length and the penetration depth distances.

The supercurrent distribution

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Then in the Laplacian we will have to replace

5. Vortex carrying multiple flux quanta

0 ( )inf r e

r

n

r

21

nrf～and asymptotics at r=0 changes to:

Core is much larger. As a result these vortices have

larger energy and are difficult to find.

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Neglecting the core and the condensation energy, we have:

E. The line energy and interaction between vortices

2[ ]grad magn

out ofcore D

f g d x

2C

1C

D

1 1. Line Energy forThe vortex line energy density is defined as the Gibbs energy of vortex solution minus .

0s ng g g

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In the London limit ( ) cov. gradient is proportional to supercurrent:

1 2

2 2

2

1[ ( )]

8

( )8

D

C C

B B B d x

z B B dS

0 *

2 2 22*

0

2 2 2 2

1 2

8

1[ ( ) ]

8

s

D

D

mJ B d x

e

B B d x

This replaces the Maxwell energy. Integration by parts gives

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Using the Londons equation

22 2

0

1( ) ( ) 2

8 8D r

BB x d x B r r

r

2 202 2

1ˆ ( )B B z x

One sees that the bulk integral vanishes and the inner boundary gives

To calculate the derivative one uses magnetic field in the intermediate region

0 02 2

1

2 2|r

dB

dr r

22

20

02

2220 0

22 2

0 0

1( )2 (0)

8 2 8

4 4

4 ( ) 4 ( ) ( )8c

B B

Log Log

HLog g Log Log

Consistency check: contribution of the core to energy is indeed small for but just logarithmically

20 0g the core area g

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The London equation is linear in magnetic field. Therefore within range of its validity

1 2( ) v vB x B x x B x x

r

2. Interaction between two straight vortices

Consider two parallel straight vortices

1x

x

2x 2 1r x x

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1 2 1 2( , ) ( ) ( )F F x x F x F x

The interaction line energy (potential) between two straight vortices is defined by

Neglecting cores and using the trick of integration by part as before one obtains from the London equation with two sources

1 2

21 2 0 1 2

2

1( , ) ( )[ ( ) ( )]

2

[ ( )8

D

C C

F x x B x x x x x d x

z B B dS

25

2C

D

1C

Since we will always (while using Londons appr.) assume r>> the last term which is Powerwise small in will be dropped

To estimate the multiple internal boundary contribution, we first approximate the derivatives

1 11 2

012

| ( ) ( ) |

1 1

2

C CB B x B x

Cr

26

20

12 1 22

01 1 1 2 2 1 2 2

[ ( ) ( )]8

[ ( ) ( ) ( ) ( )]8 v v v v

F B x B x

B x x B x x B x x B x x

2

0 00 0 02 2

( ) / /4 8vF B r K r K r

=

The interaction energy is

The two solitons energy is therefore proportional to magnetic field

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Force per unit length:

Parallel vortices repel, anti-parallel attract, however the picture is more complicated than that: the force between curved vortices is of the vector-vector type

dForce F

dr

1,

r

1/ 2/ ,

2 2re

r

r

r

1( )x 2 ( )x 2 ( )x

1( )x

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Curved Abrikosov vortices in London approximation are infinitely thin elastic lines with interaction energy

( )x

Interaction is therefore mainly magnetic, hence pair - wise (superposition principle).

1 2 1 2

1 2

1 21 2 / /0

int1 2 1 2,2

x x x xd x d xd x d x

E e ex x x x

Brandt, JLTP (1991)

3. Vortices as line - like objects

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4. Lorentz force of a current on the fluxon.

cxJf L

)(

0ˆ B

J

f

Magnetic field affects current (moving charges) via the Lorentz force

Current consequently applies a force in the opposite direction on fluxon due to Newton’s 3rd law.

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Ao, Thoules, PRL 70, 2159 (93)

1x

J

JV

FL

0

FL

In particular, force of vortex at on vortex at can be written as:

cxJf

)( 2112

1x

2x

The same logic leads to repultion between a vortex and an antivortex pointing to a vector – vector type of interaction

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5. Flux flow and dissipation.

The Lorentz force on vortices which causes their motion is balanced in the stationary flow state by the friction force due to gapless excitations in the vortex cores. The vortex mass is negligible. Phenomenologically the friction force is described (in 2D) by:

dissipation

df x v

dt

32

00

Bv t L vLB t

The overdamped dynamics results in motion of vortices with a constant velocity

across the boundary of length L. It produces the flux change

0dissipation L

JBf v f J v

c c

Leading, using Maxwell eqs., to the voltage

02

1 1 vV v L B E B J B

c t c c c

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which in turn implies a finite flux flow Ohmic resistivity

02

Bc

Unless some other force like pinning obstructs the motion, the SC loses its second “defining” property: zero conductivity

34

Let us assume that the dissipation which happens mainly in the normal cores is the same as in normal metal with resistivity . The fraction of area covered by the cores is proportional to B:

n

20

2

2 /c

BB

H

The phenomenological Bardeen – Stephen model

The resistivity therefore is the same fraction of the normal state resistivity

2n

c

B

H

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When the magnetic field reaches the cores cover the whole area and one is supposed to recover the whole normal state conductivity. This fixes the coefficient.Now the friction constant can be estimated:

2cH

0 0 22 2

2

cn

c n

B B H

c H c

We will return to this later using the time dependent GL eqs.

Within the Bardeen – Stephen model the vortex velocity is

2

n

c

cv J

H

How fast vortices can move?

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For the Nb films5 2

2

10 /

5n

c

A cm

H T

One gets velocities of 20m/sec and 600m/sec for the critical and the depaitring current values of J respectively

6 2

5 2

310 /

10 /

d

c

J A cm

J A cm

37

For YBCO film 4 222 10 / ; 100n cA cm H T

One gets velocities of 20m/sec and 200km/sec.

8 2 4 210 / ; 10 /d cJ A cm J A cm

Boltz et al (2003)

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6. Simulation of vortex arrays

0 0( )a a a b pin a ax J x K x x U xc

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Given all the forces one can simulate the vortex system using Runge – Kutta … When random disorder or thermal fluctuations are important they are introduces via random potential or force respectively (the Langeven method). The problem becomes that of mechanics of points or lines.

Here the gaussian (usually white noice) Langeven random force represents thermal fluctuations

39

Pinning force is assumed to be well represented by a gaussian random pinning potential with certain

correlator:

i j ija b abT

pin pinU x U y k x y

Fangohr et al (2001)

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Some sample results in 2D

The I-V curves at different temperatures. Critical current

Dynamical phase diagram in 2D

Koshelev (1994)

41 Hellerquist et al (1996)

Structure functions and hexatic order

Fangohr et al (2001)

42

Summary1. An isolated Abrikosov vortex carries in most cases on

e unit of magnetic flux. It has a normal core of radius and the SC magnetic “envelope” of the size carrying a vortex of supercurrent.

2. It has a small inertial mass and the creation energy (chemical potential)

3. Parallel vortices repel each others, while curved ones interact via direction dependent vector force.

4. Interact with electric current in the mixed state. The current might induce the flux flow with finite resistance.

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Details: Singular functions

The polar angle function

( ) 0x y y x

y

xarctgyx ),(

For singular functions generally

To prove this let us take integral over arbitrary circle.

xd 2

(2)2 ( )ij i j x

In particular

F

CR

at the origin x=y=0 and has a “mild” singularity- a cut at y=0.

44

2

2

( )ij i j i i

area circumference

A C

d x F F x dx

or d x F F dx Stokes Theorem

F

for function

Using derivatives formula in polar coordinates one finds that the line integral is:

2 0

0RThis is true for any