“… at each new level of complexity, entirely new properties appear, and the understanding of...
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“… 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.
Electrical ResistanceElectrical Resistance
• Thermal vibrations (phonons) of the ionic lattice • Lattice defects• Impurities
RESISTANCE is caused by electrons colliding with:
CationsElectrons
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
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.
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