1

In a vortex system a force applied on a vortex by all the others can be written within London appr. as a sum:

IV. FLUX LINE LATTICE

1

2 iji j

F V

i ijj

f f

The total energy of the vortex system is:

A. General overview of phase diagram and magnetic

properties

2

minimal magntic field H at which vortices appear

To minimize this energy distribution is likely to be as homogeneous as possible : triangular or (sometimes) square lattice (FLL).

maximal magnetic field H at which vortices disappear

0

1 2 ( )4

cH T LogT

0

2 22

cH TT

Critical fields

H

T

Mixed

Meissner

Hc2

Hc1

Tc

Normal

3

Steep second order phase transition at Hc1.

B

H1cH 2cH

H

M4

Magnetization curves

1cH 2cH

4

Magnetization of BSCCOMagnetization of BSCCOB

- H

[G]

H [G]

Bm(75K)=92G

Bm(70K)=140G

Bm(60K)=250G

T = 60, 70, 75, 80 KT = 60, 70, 75, 80 K

5

214

2c

c

HM B H H Log

H

For the “strongly bound” lattice (London)

2 2c cfraction of H H H

For the “sparse” lattice (London)

1 1c cH H several H

For “weakly superconducting” lattice (LLL)

1 2c cH H H

2 11

1

cc

c

HB H Log

H H

22

2

4 12

c

c c

H H TM

H T

6

B. Relatively sparse lattice. London appr.

a

1. Lower critical field.

7

1cH

2 0

0

cBa

B

Due to flux quantization, the fluxon density is

When first vortices appear at , the distances are large and interactions exponentially negligible for

triangular

012 ca B H

square1

2

3

c

c

The transition field therefore will be determined by the energy magnetic field has to overcome to push vortices through the material.

8

Meissner no first Abrikosovfluxon fluxon

G G G G

BHdFG 3

4

1

A - Surface area

-line energyLH

The condition of the phase transition is:

9

2

0 01 2

0 0

4 4( ) ( )

4 4

A M

c

G G

H Log Log

31

1 0

1

41

4

A M c

M c

G G A L H B d x

G H A L

One therefore obtains an expression for the lower critical field

10

Since for a very sparse lattice, (near Hc1) there is just exponentially small repulsion,

only nearest neighbors contribute significantly

2. FLL energy and magnetization just above Hc1

a

Number of vortices

20

int 02 20 2 8

B z aF A K

4,6 口zz

z- coordination number ( # of nearest neighbors)

11

1/ 4 5/ 4 0int

1~ expF z

cc B

B

Since this is exponential in c and just linear in z, the hexagonal (triangular) FLL is preferable energetically, at least for sufficiently small B. We will see that at larger fields B this feature still holds for rotation invariant interactions (Kepler’s theorem).

Using the asymptotic expression for large separations and the expression for the lattice spacing a one obtains:

12

int0

[ ] 1[ ]

4

G B A BF B A B H

L

For a given external field H, the induction B should minimize G[B]

Since for , B=0 is the minimum.

int

0

0 04

dFdG H

dB dB

int 0F 10

4cH H

However for H>Hc1, the minimum is nontrivial and is defined by:

Magnetization

13

01( )1 int

1~4

leadingcin a B

c Bc

H H dFH e

dB

2 11

1

cc

c

HB H Log

H H

14

3. Intermediate vortex densities. London eqs. And energy.

a The “logarithmic repulsion” dominates FLL for

Magnetic field becomes quite homogeneous

15

In high Tc SC this region is very wide. Example: In high Tc SC this region is very wide. Example: BSCCOBSCCO at at T=70KT=70K

B [G

]

H [G]

to Bc2(70K)

Normal

materia

l: Bcen

ter = B ext

Hc1(70K)

MeissnerB=0

16

Q

rQiz eQBrB

)()(

)(2

ynxma

Q

口

口a

2 x

y

Unit cell

For the square array

In this case one still can use London approximation neglecting cores, but not only nearest neighbors now interact. Summations over whole lattice can be performed using Fourier transform.

London equations

17

For the hexagonal FLL array two reciprocal lattice vectors perpendicular to hexagonal translation symmetry vectors are:

1 2

1 2

2( ),

ˆ 2,

3 3

Q mQ nQa

y yQ x Q

x

y 2Q

1Q

18

London’s equations are:

2 20

2

ˆ( ) ( ),

( ) ,

ilattice

iQ ri

lattice Q

B B z r r

r r e

2 2( ) ( ) iQ r iQ r

Q Q

B Q Q B Q e B e

2 2 2 2( ) ( )

1 1

iQ r

zQ

B eB Q B r B

Q Q

Solution is a sum over reciprocal lattice

(*)

Where is the density of vortices

19

2 2 2 2int

2

02 2

1[ ( ) ]

8

1( )

8 8 1ilattice Q

F B B d x

BB r A

Q

Using (*) for one obtains::0r

2

int2 2

1

8 1Q

F B

A Q

London interaction energy

20

It logarithmically diverges at due to

neglect of the core cut off

max

1Q

Q

Minimization of the Gibbs energy with respect to B gives

Q QB

Bd

dFH

221

14

4. A problem with London approximation and its solution

21

2 20

11

1Q

H BQ

Since the field is rather uniform, especially for

the term dominates and better to be separated.

1cB H0Q

“Rounding” the Brillouin zone gives a good approx.:

0Q

minQmaxQ

Approximation of the “round Brillouin zone”

22

Performing the integral one obtains

max max

min min

0

0

2( ) ( ) ( )

/ 2

Q Q

Q Q Q

f Q f Q QdQ f Q QdQarea site B

2

min0

4B

Q

2max

min

0

1/ 20 0

2 2 2 24

0 0 1 22 2

( )

2 1 4 1

4 log4 2

Q

BQ

c c

Q dQ d QH B B

Q Q

H HB Log B Log

B B

Where the expression for the upper critical field was used (see next section) 0

2 22cH

23

C. Dense lattice. The LLL appr. and Hc2.

a Order parameter is greatly reduced and only small

“islands” between core centers remain superconducting. Still superconductivity dominates electromagnetic properties of the material

24

Some historySome history

1967BitterDecoration

U. EssmannH. Trauble

1989STM

H.F. HessBell labs

A.A. Abrikosov 1957 2003

25

2cB H H

Supercurrent in turn is very small since it is proportional to 2 2

0

Since magnetization is small we replace the field inside superconductor B by external field H which is essentially homogeneous.

magnetization is relatively small (smaller than field)

We will use the Landau gauge

0x

y

A

A H x

extA A

26

Since order parameter is small everywhere Naively we can drop nonlinear higher powers of

1. Linearization of the GL eqs. Near Hc2.

2

22 2 2

2

*12

2

1

2

cm T Tx xi

y

t

22

0

20

2 * ci A T Tm

This is the usual linear Schrodinger equation of quantum mechanics.

20

1 2 H

27

In the Landau gauge it still has the manifest translation symmetries in both z and y directions, while the x translation invariance is “masked” by the choice of gauge. Therefore one can disentangle variables:

( , , ) ( )y zik y ik zx y z const e e f x

getting the shifted harmonic oscillator equation

22

4 2

1 1 1''( ) ( )

2 2 22

zktf x x X f f f

eigenvalue2yX k

28

2

, 2

( )exp

2N X

x X x Xf const

solved by the Landau harmonics

Eigenvalues of harmonic oscillator are (for any shift):

2

2 2

1 1 1

2 2 2z

N

ktN

20 0

2 2

1( , )

2 (2 1)2z z

tH N k k

N

The corresponding magnetic fields are

0, 0zk N The highest such value of H is for

29

For yet higher fields the only solution of GL equations is normal:

2 2

02 2

1

2

2

c c

c

typeII

c c

H T H t

H

H H

0

This is Hc2 – the highest field for which a nontrivial solution exists obtained for

Hc2 and the LLL appr.Hc2 and the LLL appr.

0

/ ct T T

1

2 2/ /c cb B H H H

1

normal

lattice0

30

1t b

The two pair breaking phenomena, temperature and magnetic field appear symmetrically, the second order transition line being a straight line

For N=0 and one has and the Larmor orbital center X still can be anywhere inside the sample.

0zk 2 22 / 1 t T

2

22

( )exp

2 / 1X

x Xf

t

Eigenvalues of the linearized GL equation however are still highly degenerate (the Landau degeneracy):

How should we chose the correct one and its normalization?

31

Below the nonlinear term will lift the degeneracy. It is reasonable to try an Ansatz: most general state on the lowest Landau level (LLL):

42cH

However the number of the variational parameters is unmanageble. To narrow possible choices of coefficients one has to utilize all the symmetries of the lattice solution.

2. The Abrikosov lattice solution. Heuristic approach

X XX

C

32

2

2 /

222

22;

2exp / 2

y y yik y a ik y ik ay n

i n y an n n

n n

Tne e e k X n

a a

C C e

Tx n T

a

Periodicity in the y direction with lattice constant a means that X is quantized:

Periodicity in the x direction is possible when absolute values of coefficients C are the same and in addition phases are periodic in n.

33

For the square FLL:

For the hexagonal (also called sometimes triangular) FLL:

1n nC C C

1 0

2n n

C iC

C C

Geometry + flux quantization gives us, as before in units of lattice spacing a instead of customary :

2 2exp ( )i yn

n

C e x n

22 0

2

2

1

2

c

a TH

T

a

Square lattice

22n

TX n a n

a

34

)1,( yx

),( yx ( 1, )x y The x direction

translation takes a form

2 2

2 ( 1) 2 2

( 1, ) exp ( 1 )

exp ( ) ( , )

iyn

n

iy n iy

n

x y C e x n

C e x n e x y

This is “regauging” which generally accompany a symmetry transformation.

35

The symmetry is not “manifest”, for example in our Landau gauge the symmetry along the x direction is “hidden”. The gauge invariant quantities like modulus is invariant and, in particular, zeros are repeated.

Hexagonal lattice

)1,( yx

),( yx( 3 / 2, 1/ 2)x y

22 0

2

1/ 4

32

2

3

2

c

a TH

T

a

22 3

2n

TX n a n

a

36

2

2

2

2

2 3exp

23

2 3exp

23

i yn

n even

i yn

n odd

C e x n

i e x n

The y direction translation is the same with different a as a unit of length:

The translation in diagonal direction is

2

2 1/ 2

2

2 1/ 2

3 1 2 3 3, exp

2 2 2 23

2 3 3exp

2 23

in y

n even

in y

n odd

x y C e x n

i e x n

37

2

2 ' 1 1/ 2

'

2

2 ' 1 1/ 2

'

3 1 2 3, exp '

2 2 23

2 3exp '

23

i n y

n odd

i n y

n even

x y C e x n

i e x n

Shifting n by 1 switches even and odds:

2

2 1/ 2 ' 2 '

'

2

' 2 '

'

2 3exp '

23

2 3exp '

23

i y in in y

n odd

in in y

n even

Ce e e x n

i e e x n

38

2

2 2 '

'

2

2 '

'

3 1 2 3, exp '

2 2 23

2 3exp '

23

iy in y

n odd

in y

n even

x y Ce e x n

i e x n

2 iyie

Again regauging.

Exercise 3: construct the Abrikosov lattices for the rhombic symmetry with an arbitrary angle

[0,1]

[0,1]

[1,0]

39

First let us write the GL for constant magnetic field using dimensionless variables with as the unit of length, as unit of magnetic field as the unit of energy and

221 10

2 2

tD

2

2

0

1,

2

h

hc c

H a

T B t bt b a

T H

,

2cH cT

3. Abrikosov lattice solution: a systematic expansion approach

2 2 20/ 2

40

Here21

2 2

bH D

, (0, , 0),D iA A bx ��������������������������������������������������������

is the (shifted) Schroedinger operator of an electron in magnetic field with nonnegative Landau spectrum:

NE b N

With the hexagonal symmetric eigenfunctions that we will normalize as

2( , ) 1Nunit cellx y

41

For the positive and small the energy of a LLL lattice

ha

becomes negative provided it is proportional to the Abrikosov solution.

ha

2 41

2hG H a

t

1

b

1

normal

lattice

2 ha

42

20 1 2 .... .h h ha a a

One therefore develops perturbation theory in small around the Hc2 line:

The leading ( ) order equation gives the lowest LLL restriction already motivated in the heuristic approach:

0 00H C

order equation is:

ha

3/ 2ha

With normalization undetermined. The next to leading

22 *1 0 0 0 0H C C C

ha

43

Multiplying it with and integrating one obtains *

2* 2 *1 0 0 0 0

unit cell

H C C C

0

1

A

C

2 4

01 0unit cell

C

4

22

1.16

1.18A

Exercise 4: Fix the normalization of andcalculate the Abrikosov ratio for the square lattice.

44

Free energy of the leading order solution is indeed negative

2 4

222 4

0 0

1

2

11

2 2 8

unit hcell

hh A

A A

g unit cell vol H a

t baa C C

A To find the most favorable lattice symmetry (the minimum of free energy) one should minimize

Free energy

45

Therefore since

the hexagonal lattice energy is slightly (by about 2%) lower than that of the square lattice. This sound small, but for comparison typical latent heat at melting (difference between lattice and homogeneous liquid) is of the same order of magnitude.

One can show that energies of other lattices are also higher than that of the hexagonal. Autler et al (67)

One also observes that while the first derivative with respect to T is smooth, while the second jumps:

0 0;2

h

c A c A

g a gS C

T T

46

2 2 2 222

( ),

2 2 4c h

A

H a b hG

To calculate magnetization we return to standard units for energy ensity and add the Maxwell term

Minimizing G with respect to b one obtains2

2

22 2 22 2

( )0

2 2

( )4 , ( ) 1

2

h h

A A

h cc c c

A c

a b h ab h

a H T H TB H M H H T H

T

Magnetization

47

22 2 2

2

4 12

h cc

A c c

a H T HM H

T H

48

Multiplying the GL equation with and integrating over unit cell, then using orthonormality relations one obtains:

* , 0N N

Higher order correction will include higher Landau level contributions

Corrections

1 10

NN

N

C

2* *1 1 0 0

3/ 220

1

1NN N

NN

period

H N bC C C

C

N b

C

49

The LLL component is found from the order etc.

All the corrections are very small numerically partially due to factor 1/n with multiples of 6 contributing due to symmetry

5 / 2ha

(6) (12)1 1

0.279 0.025;

6 12c c

b b

LLL leading LLL next to leadings 6th LL

2/ 0.5 / 0.1 0.2c c hT T B H a

50

ha Therefore the perturbation theory in

LLL is by far the leading contribution above this line.

converges well up to surprisingly low fields

and temperatures, roughly above the line

11

13b t

51

Summary

1. Vortex lattice is a strongly bound elastic medium. Not too close to Hc1 the field becomes homogeneous due to the vortex overlaps.

2. The diamagnetism in the mixed is far from ideal but still very strong compared with the normal state one.

3. One can rely on standard solid state methods far from Hc2.

4. One must use a delicate bifurcation perturbation theory in an interesting region not far from Hc2.