“… at each new level of complexity, entirely new properties

appear, and the understanding of this behavior requires research

which I think is as fundamental in its nature as any other”

Philip W. Anderson 1972

Si-crystalsemiconductor

MgB2 superconductor

2 atoms

　NaxCoO2

superconductor　

3 atoms

La2-xSrxCuO4

superconductor

4 atoms

DNA giant molecule

Many atoms

1 atom

From last lecture ….

Where could we find superfluidity?

np

p

He - 3

np

pnHe - 4

1 millionth of a centimetre

Helium

• Helium - 4 atoms are bosons

particles with integer spin.

• Helium - 3 atoms are fermions

particles with half integer spin.

Superfluids flow without resistance

Normal fluid Superfluid

1938 Kapitza and Allen discover superfluidity in He-4

For T < 2.4Κ – gravity ...

If the bottle containing

helium rotates for a while and

then stops, helium will

continue to rotate for ever –

there is no internal friction

(for as long as He is at T = -

269 C or lower

1938 Pyotr L. Kapitsa discovered the superfluidity of liquid Helium 4Nobel Prize in 1978

1941-47 Lev Landau formulated the theory of quantum Bose liquid - 4He superfluid liquid.  1956-58 he further formulated the theory of quantum Fermi liquid. Nobel Prize in 1962

Early 1970s David M. Lee, Douglas D. Osheroff, and Robert C. Richardson discovered the superfluidity of liquid Helium 3. Nobel Prize in 1996

Anthony Leggett first formulated the theory of superfluidity in liquid 3He in 1965.Nobel Prize in 2003

Διάστημα:3000 χιλιοστά από το απόλυτο μηδέν (-273.15

C) 5 χιλιοστά από το απόλυτο μηδέν

Χαμηλές θερμοκρασίες

LOW-TLOW-TC C Superconductors Superconductors

Lead (Pb) Mercury (Hg) Aluminum (Al) Gallium (Ga) Molybdenum (Mo) Zinc (Zn) Zirconium (Zr) Americium (Am) Cadmium (Cd) Ruthenium (Ru) Titanium (Ti) Uranium (U) Hafnium (Hf) Iridium (Ir) Beryllium (Be) Tungsten (W) Platinum (Pt)* Rhodium (Rh)

7.196 K 4.15 K 1.175 K 1.083 K 0.915 K 0.85 K 0.61 K 0.60 K 0.517 K 0.49 K 0.40 K 0.20 K 0.128 K 0.1125 K 0.023 K 0.0154 K 0.0019 K 0.000325 K

metals wood

Conductors vs. InsulatorsConductors vs. Insulators

plastics

No free electrons to carry the current

FREE ELECTRONS

The foam balls (containing small magnets) organise themselves based on

the laws of minimum energy. This arrangement mimics the crystal lattice of a

solid material.

What is Resistance?What is Resistance?

ELECTRIC FIELD VOLTAGE DIFFERENCE

IONS (+)

ELECTRONS (-)

Electrical ResistanceElectrical Resistance

• Thermal vibrations (phonons) of the ionic lattice • Lattice defects• Impurities

RESISTANCE is caused by electrons colliding with:

CationsElectrons

VI

Vs

copper

Liquid helium 4.2K (-269 ºC)

R

T

77K

273K = 0ºC

Ro

Impurities

VI

Vs

Hg

Liquid helium 4.2K (-269 ºC)

Onnes (1911)

Low -TLow -Tcc Superconductivity Superconductivity

Heike Kamerlingh Onnes

(1911)

LOW-TLOW-TC C Superconductors Superconductors

Lead (Pb) Mercury (Hg) Aluminum (Al) Gallium (Ga) Molybdenum (Mo) Zinc (Zn) Zirconium (Zr) Americium (Am) Cadmium (Cd) Ruthenium (Ru) Titanium (Ti) Uranium (U) Hafnium (Hf) Iridium (Ir) Beryllium (Be) Tungsten (W) Platinum (Pt)* Rhodium (Rh)

7.196 K 4.15 K 1.175 K 1.083 K 0.915 K 0.85 K 0.61 K 0.60 K 0.517 K 0.49 K 0.40 K 0.20 K 0.128 K 0.1125 K 0.023 K 0.0154 K 0.0019 K 0.000325 K

BCS TheoryBCS Theory

John BardeenLeon CooperJohn Schrieffer(1957)

No collisions Zero resistance

Meissner EffectMeissner Effect

• 1933 – Walther Meissner and Robert Ochsenfeld

• T<Tc: external magnetic field is perfectly expelled from the interior of a superconductor

The energy gap and Bardeen-The energy gap and Bardeen-Cooper-Schrieffer theoryCooper-Schrieffer theory

The key point is the existence of energy gap between ground state and quasi-particle excitations of the system.

cg kTE 528.3)0(2)0(

1. Existence of condensate.

2. Weak attractive electron-

phonon interaction leads to

the formation of bound pairs

of electrons, occupying states

with equal and opposite

momentum and spin.

3. Pairs have spatial

extension of order .

The electron-electron attraction of the Cooper pairs causedthe electrons near the Fermi level to be redistributed aboveor below the Fermi level. Because the number of electrons remains constant, the energy densities increase around the Fermi level resulting in the formation of an energy gap.

E s s e n t i a l d e t a i l s :F . a n d H . L o n d o n ( 1 9 3 5 ) p r o p o s e d a s i m p l e t h e o r y t o d e s c r i b e t h e e l e c t r o d y n a m i c s

o f s u p e r c o n d u c t o r s . T h e y a s s u m e d t h a t s u p e r c o n d u c t i v i t y i s g e n e r a t e d b y

s u p e r e l e c t r o n s , w h i c h a r e n o t s c a t t e r e d b y e i t h e r i m p u r i t i e s o r l a t t i c e v i b r a t i o n s ,

t h u s a r e n o t c o n t r i b u t i n g t o t h e r e s i s t i v i t y . T h e y s t a r t e d f r o m t h e e q u a t i o n o f

m o t i o n o f a f r e e e l e c t r o n i n a n a p p l i e d e l e c t r i c a l f i e l d E

sm e v E

( 1 . 1 )

w h e r e sv i s t h e v e l o c i t y o f t h e s u p e r e l e c t r o n s a n d m a n d - e a r e t h e i r m a s s a n d

c h a r g e , r e s p e c t i v e l y . H e n c e t h e s u p e r c u r r e n t d e n s i t y i s g i v e n b y

s sn e J v ( 1 . 2 )

h e r e sn i s t h e d e n s i t y o f s u p e r e l e c t r o n s . S u b s t i t u t i n g ( 1 . 2 ) i n t o ( 1 . 1 ) , t h e y d e r i v e d ,

t h e s o - c a l l e d f i r s t L o n d o n e q u a t i o n

2

s

m

n eE J

. ( 1 . 3 )

T a k i n g t h e c u r l o f b o t h s i d e s o f ( 1 . 3 ) , a n d u s i n g M a x w e l l ' s t h i r d e q u a t i o n

( F a r a d a y ' s l a w ) , t h e y o b t a i n e d

2

s

m

n e B J

Ñ . ( 1 . 4 )

E q u a t i o n ( 1 . 4 ) c a n b e i n t e g r a t e d w i th r e s p e c t t o t im e a n d o b t a in

0 2s

m

n e 0B B J JÑ ( 1 . 5 )

w h e r e 0B a n d 0J , r e la t e d b y 0 0 0 B JÑ , a r e t h e m a g n e t i c f i e l d a n d c u r r e n t d e n s i t y

a t 0t , r e s p e c t i v e ly . H o w e v e r , a c c o r d in g t o t h e M e i s s n e r e f f e c t ( M e i s s n e r a n d

O c h s e n f e ld 1 9 3 3 ) t h e m a g n e t i c f lu x i n s id e a s u p e r c o n d u c t o r i s c o m p le t e ly e x p e l l e d ,

i r r e s p e c t i v e o f w h e th e r t h e m a g n e t i c f i e ld w a s a p p l i e d b e f o r e o r a f t e r c o o l i n g b e lo w

cT , i . e . 0 0B . T h e r e f o r e ( 1 . 5 ) l e a d s t o t h e p o s tu la t e d s e c o n d L o n d o n e q u a t i o n

2s

m

n e B JÑ . ( 1 . 6 )

The field distribution within a superconductor is calculated from (1.6) in combination

with Maxwell's fourth equation 0B J to obtain

22L

B

B (1.7)

where

2

0L

s

m

en

(1.8)

is called the London penetration depth. Equation (1.7) implies that the magnetic field is

exponentially screened from the interior of a sample within a distance L(typically

0.1m ). Therefore, if the sample size is much larger than L, the whole specimen will

be effectively screened.

T h e G i n z b u r g - L a n d a u t h e o r y

G i n z b u r g a n d L a n d a u ( 1 9 5 0 ) i n t r o d u c e d a c o m p l e x p s e u d o - w a v e f u n c t i o n

( ) ( ) e x p ( i ) r r a s a s u p e r c o n d u c t i n g o r d e r p a r a m e t e r . T h e t h e o r y a s s u m e s t h a t t h e

l o c a l d e n s i t y o f s u p e r c o n d u c t i n g c a r r i e r s i s g i v e n b y

2* ( )sn r . ( 1 . 9 )

T h e r e f o r e , t h e o r d e r p a r a m e t e r ( ) r i s z e r o a b o v e cT a n d i n c r e a s e s c o n t i n u o u s l y a s t h e

t e m p e r a t u r e d e c r e a s e s . F o r s m a l l a m p l i t u d e s a n d s l o w v a r i a t i o n i n s p a c e o f ( ) r , t h e

f r e e e n e r g y d e n s i t y f c a n b e e x p a n d e d i n s e r i e s o f t h e f o r m

2

2 4 2

0

1 1i 2

2 4 2nf f em

B

r r A= + + + - Ñ - ( 1 . 1 0 )

w h e r e nf i s t h e f r e e e n e r g y d e n s i t y i n t h e n o r m a l s t a t e , A i s t h e v e c t o r p o t e n t i a l w h i c h i s

r e l a t e d t o t h e l o c a l m a g n e t i c i n d u c t i o n B b y t h e f o r m u l a A BÑ . I n e q u a t i o n ( 1 . 1 0 ) i ti s a s s u m e d t h a t t h e s u p e r c o n d u c t i n g c a r r i e r s a r e e l e c t r o n p a i r s ( C o o p e r p a i r s ) w i t h m a s sa n d c h a r g e e q u a l t o 2 m a n d 2 e ( 0 )e , r e s p e c t i v e l y ( B a r d e e n , C o o p e r a n d S c h r i e f f e r1 9 5 7 ) .

For a small range of temperatures near cT the parameters and are approximately

given by

0 1c

T

T

(1.11)

constant (1.12)

where 0 0 is temperature independent.

If the free energy density is integrated over all space and minimised with respect to local

changes in Aand , two coupled differential equations are obtained. These govern the

equilibrium variation of A and with position, given particular boundary conditions,

and are known, respectively as the first and second Ginzburg-Landau equations

2 21i 2 0

4e

m AÑ (1.13)

2 2

2* * 2i2

2

ee ee

m m m

AJ A

Ñ Ñ Ñ (1.14)

where is the phase of the order parameter.

T h e u p p e r c r i t i c a l f i e l d a n d c o h e r e n c e l e n g t h

F o r s u f f i c i e n t l y h i g h f i e l d s , s u p e r c o n d u c t i v i t y i s d e s t r o y e d a n d t h e f i e l d i s u n i f o r m i n t h e

s a m p l e . I f t h e f i e l d i s c o n t i n u o u s l y r e d u c e d , a t a c e r t a i n f i e l d 2cB = B , c a l l e d t h e u p p e r

c r i t i c a l f i e l d , s u p e r c o n d u c t i n g r e g i o n s b e g i n t o n u c l e a t e s p o n t a n e o u s l y . I n t h e r e g i o n s

w h e r e t h e n u c l e a t i o n o c c u r s , s u p e r c o n d u c t i v i t y i s j u s t b e g i n n i n g t o a p p e a r a n d

t h e r e f o r e i s s m a l l , a n d e q u a t i o n ( 1 . 1 3 ) b e c o m e s

21i 2

4e

m A Ñ . ( 1 . 1 5 )

E q u a t i o n ( 1 . 1 5 ) i s i d e n t i c a l t o t h e S c h r ö d i n g e r e q u a t i o n f o r a p a r t i c l e o f c h a r g e 2 e a n d

m a s s 2 m i n a u n i f o r m m a g n e t i c f i e l d . F o r a n a p p l i e d f i e l d B a l o n g t h e z - a x i s , t h e h i g h e s t

s o l u t i o n c o r r e s p o n d i n g t o t h e u p p e r c r i t i c a l f i e l d i s

02 22cB

( 1 . 1 6 )

a n d t h e c o r r e s p o n d i n g o r d e r p a r a m e t e r

y

2i k 0

2e x p

2zy k z x x

e

( 1 . 1 7 )

with

(0)

4 1cm TT

(1.18)

where 0 2he is the flux quantum, 0 02yxk B, and 0(0) 4m is the

value of at 0T. Equation (1.17) shows that is the characteristic length overwhichcan vary appreciably. The parameter is called the Ginzburg-Landaucoherence length.

T h e p e n e t r a t i o n d e p t h

I n t h e c a s e w h e r e t h e d i m e n s i o n o f t h e s a m p l e a r e m u c h g r e a t e r t h a n , t h e n

0B i n s i d e t h e s a m p l e . T h e n i s c o n s t a n t , i f v a r i e d t h e g r a d i e n t t e r m i n ( 1 . 1 0 )

w o u l d m e a n t h a t t h e f r e e e n e r g y i n c r e a s e d . T h e c o n s t a n t v a l u e o f i s g i v e n f r o m

e q u a t i o n ( 1 . 1 3 ) :

2 2 0

0 1c

T

T

. ( 1 . 1 9 )

S i n c e t h e o r d e r p a r a m e t e r i s c o n s t a n t , i . e . 2 0 Ñ , e q u a t i o n ( 1 . 1 4 ) b e c o m e s

2

2

0

2 e

m J A . ( 1 . 2 0 )

T a k i n g t h e c u r l o f b o t h s i d e s o f e q u a t i o n ( 1 . 2 0 ) , a n d s u b s t i t u t i n g f o r t h e v e c t o r p o t e n t i a l B AÑ y i e l d s t o

2

22

m

e B J0 Ñ . (1.21)

Equation (1.21) is identical to the second London equation (1.6) with a penetration depth

given by

22

0 0

(0)

12 c

m

T Te

(1.22)

where 20 0(0) 2m e is the penetration depth at zero temperature. The above

equation, in contrast to the expression (1.8) of the London penetration depth, contains

the temperature dependent parameter, 2

0 , which is defined in terms of ( )T .

T h e t h e r m o d y n a m i c c r i t i c a l f i e l d

T h e e x i s t e n c e o f t h e M e i s s n e r e f f e c t , w h e r e t h e m a g n e t i c f l u x i s c o m p l e t e l y e x p e l l e d

f r o m a t y p e - I s u p e r c o n d u c t o r , i m p l i e s t h a t t h e s u p e r c o n d u c t i n g s t a t e h a s a l o w e r f r e e

e n e r g y t h a n t h e n o r m a l s t a t e . T h e r e f o r e , t h e t h e r m o d y n a m i c c r i t i c a l f i e l d )( TB c r e q u i r e d

t o d e s t r o y t h e s u p e r c o n d u c t i n g s t a t e , i s d e f i n e d f r o m t h e c o n d i t i o n w h e n t h e w o r k d o n e i n

m a g n e t i c e x p u l s i o n e q u a l s t h e z e r o f i e l d f r e e e n e r g y d i f f e r e n c e b e t w e e n t h e n o r m a l a n d

s u p e r c o n d u c t i n g s t a t e s , o r i n t e r m o f f r e e e n e r g i e s d e n s i t i e s a s

2

02c

n s

Bf f f

. ( 1 . 2 3 )

T h e q u a n t i t y f , c a l l e d t h e c o n d e n s a t i o n e n e r g y d e n s i t y , i s t h e e n e r g y p e r u n i t v o l u m e

r e l e a s e d b y t r a n s f o r m a t i o n f r o m t h e n o r m a l i n t o t h e s u p e r c o n d u c t i n g s t a t e . I n t h e c a s e o f

z e r o a p p l i e d m a g n e t i c a n d s m a l l v a r i a t i o n o f t h e o r d e r p a r a m e t e r , t h e s o l u t i o n ( 1 . 1 9 )

c a n b e s u b s t i t u t e d i n t o ( 1 . 1 0 ) , a n d t h e m i n i m u m f r e e e n e r g y d e n s i t y c o r r e s p o n d i n g t o t h e

s u p e r c o n d u c t i n g s t a t e a t z e r o f i e l d w i l l b e g i v e n b y

21

2s nf f

= . ( 1 . 2 4 )

C o m p a r i n g ( 1 . 2 4 ) t o ( 1 . 2 3 ) , a n d u s i n g t h e e x p r e s s i o n s o f t h e p e n e t r a t i o n d e p t h ( 1 . 2 2 ) , a n d

t h e c o h e r e n c e l e n g t h ( 1 . 1 8 ) , t h e t h e r m o d y n a m i c c r i t i c a l f i e l d )( TB c c a n b e w r i t t e n i n t h e

f o r m

0( )2 2

cB T

. ( 1 . 2 5 )

F o r a t h i n f i l m o f t h i c k n e s s d i n a n e x t e r n a l m a g n e t i c f i e l d B a p p l i e d p a r a l l e l t o t h e

p l a n e o f t h e f i l m a n d h a v i n g t h e s a m e v a l u e a t b o t h f a c e s , t h e G i n z b u r g - L a n d a u e q u a t i o n s

h a v e t h e s o l u t i o n ( T i n k h a m 1 9 9 6 )

2 2

2 2

0 2 21

2 4 c

d B

B

( 1 . 2 6 )

w h e r e a n d cB a r e t h e p e n e t r a t i o n d e p t h a n d t h e r m o d y n a m i c c r i t i c a l f i e l d o f t h e b u l k

m a t e r i a l , r e s p e c t i v e l y . T h u s , t h e f i l m b e c o m e s n o r m a l , i . e . 2

0 , w h e n 2 / /cB B ,

g i v e n b y

02 / /

1 22 6

2c

c

BB

d d

. ( 1 . 2 7 )

H e r e 2 / /cB i s k n o w n a s t h e p a r a l l e l u p p e r c r i t i c a l f i e l d f o r a t h i n f i l m . T a k i n g i n t o a c c o u n t

t h e t e m p e r a t u r e d e p e n d e n c e o f t h e c o h e r e n c e l e n g t h , ( 1 . 2 7 ) c a n b e w r i t t e n i n t h e f o r m

02 / /

1 21

2 ( 0 )c cB T Td

. ( 1 . 2 8 )

I t i s c l e a r f r o m ( 1 . 2 8 ) t h a t t h e t e m p e r a t u r e d e p e n d e n c e o f t h e p a r a l l e l u p p e r c r i t i c a l f i e l d

i s f o l l o w i n g a p o w e r l a w . T h i s i s i n c o n t r a s t t o t h e l i n e a r d e p e n d e n c e o f e q u a t i o n ( 1 . 1 6 )

o f t h e u p p e r c r i t i c a l f o r a b u l k s a m p l e .

T h e G i n z b u r g - L a n d a u p a r a m e t e r

T h e s u r f a c e e n e r g y , , o f a s u p e r c o n d u c t i n g - n o r m a l b o u n d a r y i s d e f i n e d a s t h e

d i f f e r e n c e b e t w e e n t h e G i b b s f r e e e n e r g y p e r u n i t a r e a b e t w e e n a h o m o g e n e o u s p h a s e

( e i t h e r a l l n o r m a l o r a l l s u p e r c o n d u c t i n g ) a n d a m i x e d p h a s e . A s s u m i n g t h a t t h e

s u p e r c o n d u c t i n g p h a s e i s l o c a t e d i n t h e h a l f - s p a c e 0x , a n d t h e n o r m a l p h a s e i n t h e

o t h e r s i d e ( f i g u r e 1 . 1 ) , a n d u s i n g t h e G i n z b u r g - L a n d a u f r e e e n e r g y d e n s i t y e x p r e s s i o n

( 1 . 1 0 ) , t h e s u r f a c e e n e r g y i s g i v e n b y ( e . g . T i n k h a m 1 9 9 6 )

2 42

0 0

12

c

c

B Bd x

B

. ( 1 . 2 9 )

H e r e , t h e t e r m t o t h e l e f t - h a n d s i d e o f t h e s q u a r e b r a c k e t i n ( 1 . 2 9 ) r e p r e s e n t s t h e p o s i t i v e

c o n t r i b u t i o n t o t h e s u r f a c e e n e r g y a s s o c i a t e d w i t h t h e d i a m a g n e t i c s c r e e n i n g e n e r g y . T h e

t e r m o n t h e r i g h t r e p r e s e n t s t h e n e g a t i v e c o n t r i b u t i o n t o t h e s u r f a c e e n e r g y a s s o c i a t e d

w i t h t h e c o n d e n s a t i o n e n e r g y . H e n c e , i t c a n b e s e e n f r o m ( 1 . 2 9 ) t h a t t h e s i g n o f i s

d e t e r m i n e d f r o m t h e b a l a n c e o f t h e p o s i t i v e m a g n e t i c e x p u l s i o n a n d t h e n e g a t i v e

c o n d e n s a t i o n e n e r g i e s .

D e t a i l e d n u m e r i c a l c a l c u l a t i o n s o f ( 1 . 2 9 ) s h o w t h a t t h e s i g n o f t h e s u r f a c e e n e r g y , ,

d e p e n d s o n t h e v a l u e o f , c a l l e d t h e G i n z b u r g - L a n d a u p a r a m e t e r . T h e s u r f a c e

e n e r g y i s p o s i t i v e f o r m a t e r i a l s w i t h 1 2 , c a l l e d t y p e I s u p e r c o n d u c t o r s , a n d

n e g a t i v e f o r m a t e r i a l s w i t h 1 2 , c a l l e d t y p e I I s u p e r c o n d u c t o r s . T h e m a g n e t i c

b e h a v i o u r o f t h e s e m a t e r i a l s i s s h o w n i n f i g u r e 1 . 2 .

Type I superconductors completely exclude magnetic flux from their interior, i.e. are in

the Meissner state, for all applied magnetic field below the thermodynamic critical field

cB. The superconducting elements, with the exception of niobium, are all type I.

Type II superconductors allow the penetration of the magnetic flux when the applied field

exceeds a value referred to as the lower critical field, 1cB. For increasing applied fields

above 1cB, the magnetic field penetrates partially forming what is called a mixed state.

Eventually, when the applied field reached the value of the upper critical field 2cB, the

material becomes normal. The superconducting alloys and compounds are type II.

Figure 1.1: Diagram of variation of B and in a domain wall. The case refers to a type I

superconductor (positive surface energy); the case refers to a type II superconductor

(negative surface energy).

Bc

0

B

0 x

superconductingnormal

B

B

0

c

B

0 x

superconductingnormal

Type I Type II

(a ) ( b)

F ig u re 1 .2 : M agnetic p hase d iag ram fo r (a) T yp e-I and (b ) typ e-II sup erco nd ucto r.

B

T Tc

0

Normal state

Bc2

(T)

Bc1

(T)Meissner phase

Mixed state

Type-II

B

T Tc

Normal state

Bc(T)

Meissner phase

Type-I

T h e a n i s o t r o p i c G i n z b u r g - L a n d a u t h e o r y

A n i s o t r o p i c s u p e r c o n d u c t o r s , s u c h a s N b S e 2 , t h e h i g h t e m p e r a t u r e s u p e r c o n d u c t o r s , a n d

a r t i f i c i a l l y p r e p a r e d s u p e r c o n d u c t i n g m u l t i l a y e r s , d i f f e r f r o m i s o t r o p i c m a t e r i a l s i n m a n y o f t h e i r

p r o p e r t i e s . A s s e e n i n t h e p r e v i o u s s e c t i o n s , t h e p r o p e r t i e s o f i s o t r o p i c s u p e r c o n d u c t o r s a r e

d e s c r i b e d i n t e r m o f t h e p e n e t r a t i o n d e p t h w h i c h i s p r o p o r t i o n a l t o m ( e q u a t i o n ( 1 . 2 2 ) ) , a n d

t h e c o h e r e n c e l e n g t h p r o p o r t i o n a l t o 1 m ( e q u a t i o n ( 1 . 1 8 ) ) , w h e r e m i s t h e m a s s o f t h e

s u p e r e l e c t r o n s . T h e s i m p l e s t w a y t o e x t e n d t h e G i n z b u r g - L a n d a u t h e o r y t o t h e c a s e o f m a t e r i a l s

w i t h a n i s o t r o p i c s u p e r c o n d u c t i n g p r o p e r t i e s i s b y i n t r o d u c i n g a p h e n o m e n o l o g i c a l a n i s o t r o p i c

m a s s t e n s o r i km i n s t e a d o f t h e i s o t r o p i c m ( C l e m 1 9 8 9 ) . T h i s m a s s t e n s o r i s d i a g o n a l , a n d t h e

d i a g o n a l e l e m e n t s ( , , )im i a b c a r e n o r m a l i s e d s u c h t h a t 1 / 3

1 2 3 1m m m , w h e r e a , b , a n d c a r e

t h e t h r e e p r i n c i p a l c r y s t a l d i r e c t i o n s . T h e c o h e r e n c e l e n g t h s a n d p e n e t r a t i o n d e p t h s a l o n g t h e

i d i r e c t i o n a r e g i v e n b y i im a n d i im , r e s p e c t i v e l y , w i t h t h e n o r m a l i s a t i o n

p r o p e r t i e s 1 / 3

a b c a n d 1 / 3

a b c , a n d t h e G i n z b u r g - L a n d a u p a r a m e t e r i s d e f i n e d

a s i i .

Hence, within the mass tensor approach, an anisotropic superconductor is characterised

by two average lengthsand , and two mass ratios, for example /a cmm and /b cmm,

the third mass being determined from the above normalisation. In this theory, the

thermodynamic critical field is similar to the isotropic case and is given by

0 0

22 22c

i i

B

. (1.30)

The upper critical field along principal axis i can be written as

02// 2

2c i i cj k

B B

(1.31)

where i im , j and k are the coherence lengths along the j and k-axis,respectively.

However, m ost of the superconducting m ultilayers and the high tem perature

superconductors are uniaxial or alm ost uniaxial m aterials. In this case the

superconducting properties are uniquely defined by the in-plane a b abm m m and axial

cm effective m asses, and equations (1.30) and (1.31) becom e

0 0

2 2 2 2c

ab ab c c

B

(1.32)

02 // 22c c

ab

B

(1.33)

02 // 2c ab

ab c

B

. (1.34)

The anisotropy ratio , which describes the degree of anisotropy of uniaxial

superconductors, is defined from the formula 1 2

c ab c ab ab cm m . This

number enters the expressions of many anisotropic quantities, such as the ones describing

the vortex matter in layered superconductors (see next chapter). The magnitude of

depends on the different classes of superconductors, for example 3.3 for NbSe2

(Morris et al 1972), 7.7 for YBa2Cu3O7- (Farrell et al 1990), and 150 for

Bi2Sr2CaCu2O8+ (Okuda et al 1991).

In summary …In summary … Characteristic lengths in SCCharacteristic lengths in SC

London equation:

The Pippard coherence length:

Penetration depth is the characteristic length of the fall off of a magnetic field due to surface currents.

Ginzburg-Landau parameter:

for pure SC far from Tc temperature-dependent Ginzburg-Landau coherence length is approximately equal to Pippardcoherence length

Coherence length is a measure of the shortest distance over which superconductivity may be established

The London equation shows that the magnetic field exponentially decays to zero inside a SC (Meissner effect)

Magnetic propertiesMagnetic properties

Dependences of critical fields on temperature.Phase boundaries between

superconducting, mixed and normal states of type I and II SC.

Intermediate state Intermediate state (SC of(SC of type I type I))(Type I SC show a reversible 1st order phase transition with a latent heat when the applied field (Type I SC show a reversible 1st order phase transition with a latent heat when the applied field

reached Breached Bcc. At this particular field relatively thick Normal and SC domains running parallel to the . At this particular field relatively thick Normal and SC domains running parallel to the field can coexist, in what is known as the field can coexist, in what is known as the intermediate stateintermediate state))

Intermediate state of a mono-crystalline tin foil of 29 m thickness in perpendicular magnetic field (normal regions are dark)

A distribution of superconducting and normal states in tin sphere (superconducting regions are shaded)

Mixed state (SC of type II)(In type II SC finely divided quantized flux vortices or flux lines enter the material over a range of applied fields below Bc, and remain stable over a range of applied fields, in

what became known as the mixed state. If these flux lines are pinned by lattice defects or other agencies, type II SC can carry a large super-current: see development

of useful high-field SC magnets.)

Abrikosov: [1957]

One vortex carries one quantum of the flux:

Triangular lattice of vortex lines going out to the surface of SC Pb0.98In0.02 foil in perpendicular to the surface magnetic field

Supercurrent Normal core

Normal regions are approximately 300nm

Closer packing of normal regionsoccurs at higher temperatures orhigher external magnetic fields

Vortex characteristicsVortex characteristics

• Magnetic field of a vortex

e

hc

20 A quantum of magnetic flux is

Normal core

Vortex state of type II superconductorsVortex state of type II superconductors

• Type II

Phase of GL pseudo-wave function

changes by 2 when going around spatial

lines where is zero

0

1

||

Normal core

Vortex state of type II superconductorsVortex state of type II superconductors

• Type II

In type-II SC field penetrates to the bulk of material in the form of vortices (or magnetic flux lines, or fluxons)

Phase of GL pseudo-wave function

changes for 2 when going around spatial

lines where is zero

Each vortex represents magnetic flux quantum

B/Beq

0

1

||

Critical current densityCritical current density

Critical current is the maximum current SC materials can carry, above which they stop being SCs. If too much current is pushed through a SC, the latter will become normal, even though it may be below its Tc. The colder you keep the SC the higher the current it can carry.

Three critical parameters Tc, Hc and Jc define the boundaries of the environment within whicha SC can operate.

Fig. demonstrates relationship between Tc, Hc and Jc (a criticalsurface). The highest values for Hc and Jc occur at 0K, while thehighest value for Tc occurs when H and J are zero.

Josephson effectJosephson effect (see also hand-out)(see also hand-out)

In 1962 Josephson predicted Cooper-pairs can tunnel through a weak link at zero voltage difference. Current

in junction (called Josephson junction – Jj) is then equal to:

21sin cJJ

Electrical current flows between two SC materials - even when they are separated by a non-SC or insulator. Electrons "tunnel" through this non-SC region, and SC current flows.

The Meissner-Ochsenfeld EffectThe Meissner-Ochsenfeld Effect

Walter MeissnerRobert

Ochsenfeld (1933)

T>TCT<TC

Superconductor

Magnet

DIAMAGNETISMDIAMAGNETISM

?