Daresbury_2010_-_SC_-_Physics.pptx

Jean Delayen
Center for Accelerator Science
Old Dominion University
and
Thomas Jefferson National Accelerator Facility
SUPERCONDUCTIVITY
OVERVIEW OF EXPERIMENTAL FACTS
EARLY MODELS
GINZBURG-LANDAU THEORY
BCS THEORY
COCKCROFT INSTITUTE, DARESBURY JULY 2010

Perfect Conductivity
Unexpected result
Expectation was the opposite: everything should become an isolator at 0
T ®
Kamerlingh Onnes and van der Waals
in Leiden with the helium 'liquefactor'
(1908)

Perfect Conductivity
Persistent current experiments on rings have measured
15
10
s
n
s
s
>
Perfect conductivity is not superconductivity
Superconductivity is a phase transition
A perfect conductor has an infinite relaxation time L/R
Resistivity < 10-23 Ω.cm
Decay time > 105 years

Perfect Diamagnetism (Meissner & Ochsenfeld 1933)
0
B
t
¶
=
¶
0
B =
Perfect conductor Superconductor

Penetration Depth in Thin Films
Very thin films
Very thick films

Critical Field (Type I)
2
( ) (0) 1
c c
c
T
H T H
T
é ù
æ ö
ê ú
- ç ÷
è ø
ê ú
ë û
Superconductivity is destroyed by the application of a magnetic field
Type I or “soft” superconductors

Critical Field (Type II or “hard” superconductors)
Expulsion of the magnetic field is complete up to Hc1, and partial up to Hc2
Between Hc1 and Hc2 the field penetrates in the form if quantized vortices
or fluxoids
0
e
p
f =

Thermodynamic Properties
Entropy Specific Heat
Energy Free Energy

( ) 1
2
When phase transition at is of order latent heat
At transition is of order no latent heat
jump in specific heat
st
c c
nd
c
es
T T H H T
T T
C
< = Þ
= Þ
3
( ) 3 ( )
( )
( )
electronic specific heat
reasonable fit to experimental data
c en c
en
es
T C T
C T T
C T T
g
a
=
»

3 3
2 2
0
0
3
3
: ( ) ( )
(0) 0
3
3
( ) ( )
( ) ( )
At The entropy is continuous
Recall: and
For
c c
c s c n c
T T
es
c c
s n
c c
c s n
T S T S T
S C
S
T T
T T T
dt dt C
T T T T
T T
S T S T
T T
T T S T S T
a g g
a g
g g
=
¶
= =
¶
Þ = ® = =
= =
< <
ò
ò
superconducting state is more ordered than normal state
A better fit for the electron specific heat in superconducting state is
9, 1.5
with for
c
bT
T
es c c
C a T e a b T T
g
-
= » »

Energy Difference Between Normal and
Superconducting State
( )
4 4 2 2
2
( ) ( )
3
( ) ( ) ( ) ( )
4 2
Εnergy is continuous
c
n c s c
T
n s es en c c
T
c
U T U T
U T U T C C dt T T T T
T
g g
=
- = - = - - -
ò
( ) ( )
2
2
1
0 0 0
4 8
at c
n s c
H
T U U T
g
p
= - = =
2
8
isthecondensationenergy
c
H
p
2
0,
8
at is the free energy difference
c
H
T
p
¹
( ) ( )
2
2
2
2
( ) 1
1
8 4
c
n s n c c
c
H T T
F U U T S S T
T
g
p
é ù
æ ö
ê ú
= D = - - - = - ç ÷
è ø
ê ú
ë û
( )
2
1
2
( ) 2 1
c c
c
T
H T T
T
pg
é ù
æ ö
ê ú
= - ç ÷
è ø
ê ú
ë û
The quadratic dependence of critical field on T is
related to the cubic dependence of specific heat

Isotope Effect (Maxwell 1950)
The critical temperature and the critical field at 0K are dependent
on the mass of the isotope
(0) with 0.5
c c
T H M a
a
-

Energy Gap (1950s)
At very low temperature the specific heat exhibits an exponential behavior
Electromagnetic absorption shows a threshold
Tunneling between 2 superconductors separated by a thin oxide film
shows the presence of a gap
/
1.5
with
c
bT T
s
c e b
-
µ

Two Fundamental Lengths
• London penetration depth λ
– Distance over which magnetic fields decay in
superconductors
• Pippard coherence length ξ
– Distance over which the superconducting state decays

Two Types of Superconductors
• London superconductors (Type II)
– λ>> ξ
– Impure metals
– Alloys
– Local electrodynamics
• Pippard superconductors (Type I)
– ξ >> λ
– Pure metals
– Nonlocal electrodynamics

Material Parameters for Some Superconductors

Phenomenological Models (1930s to 1950s)
Phenomenological model:
Purely descriptive
Everything behaves as though…..
A finite fraction of the electrons form some kind of condensate
that behaves as a macroscopic system (similar to superfluidity)
At 0K, condensation is complete
At Tc the condensate disappears

Two Fluid Model – Gorter and Casimir
( )
( ) ( ) ( )
( )
( )
( )
1/2
2
2
1- :
( ) = 1
1
2
1
4
gives =
fractionof "normal"electrons
fractionof "condensed"electrons (zero entropy)
Assume: free energy
independent of temperature
Minimizationof
c
n s
n
s c
C
T T x
x
F T x f T x f T
f T T
f T T
T
F T x
T
g
b g
< =
+ -
= -
= - =-
æ
è
( ) ( ) ( )
C
4
4
1/2
3
2
( ) 1 1
T
3
T
n s
C
es
T
F T x f T x f T
T
C
b
g
ö
ç ÷
ø
é ù
æ ö
ê ú
Þ = + - = - +ç ÷
è ø
ê ú
ë û
Þ =

Two Fluid Model – Gorter and Casimir
( ) ( ) ( )
( )
4
1/2
2
2
2
2
( ) 1 1
( ) 2
2
8
1
Superconducting state:
Normal state:
Recall difference in free energy between normal and
superconducting state
n s
C
n
C
c
C
T
F T x f T x f T
T
T
F T f T T
T
H
T
T
b
g
b
p
b
é ù
æ ö
ê ú
= + - = - +ç ÷
è ø
ê ú
ë û
æ ö
= = - = - ç ÷
è ø
=
æ ö
= - ç ÷
è ø
( )
2 2
1
(0)
c
c C
H T T
H T
é ù æ ö
ê ú Þ = - ç ÷
è ø
ê ú
ë û
The Gorter-Casimir model is an “ad hoc” model (there is no physical basis
for the assumed expression for the free energy) but provides a fairly
accurate representation of experimental results

Model of F & H London (1935)
Proposed a 2-fluid model with a normal fluid and superfluid components
ns : density of the superfluid component of velocity vs
nn : density of the normal component of velocity vn
2
superelectrons are accelerated by
superelectrons
normal electrons
s s
s s
n n
m eE E
t
J en
J n e
E
t m
J E
u
u
s
¶
= -
¶
= -
¶
=
¶
=

2
2 2
2
0 = Constant
= 0
Maxwell:
F&H London postulated:
s s
s s
s s
s
s
J n e
E
t m
B
E
t
m m
J B J B
t n e n e
m
J B
n e
¶
=
¶
¶
Ñ ´ = -
¶
æ ö
¶
Þ Ñ ´ + = Þ Ñ ´ +
ç ÷
¶ è ø
Ñ ´ +

combine with 0 s
B = J
m
Ñ´
( ) [ ]
2
2 0
1
2
2
0
- 0
exp /
s
o L
L
s
n e
B B
m
B x B x
m
n e
m
l
l
m
Ñ =
= -
é ù
= ê ú
ë û
The magnetic field, and the current, decay
exponentially over a distance λ (a few 10s of nm)

( ) ( )
1
2
2
0
4
1
4 2
1
1
0
1
From Gorter and Casimir two-fluid model
L
s
s
C
L L
C
m
n e
T
n
T
T
T
T
l
m
l l
é ù
= ê ú
ë û
é ù
æ ö
ê ú
µ - ç ÷
è ø
ê ú
ë û
=
é ù
æ ö
ê ú
- ç ÷
è ø
ê ú
ë û

2
0
2
0, 0
1
London Equation:
choose on sample surface (London gauge)
Note: Local relationship between and
s
n
s
s
B
J H
A H
A A
J A
J A
l
m
l
Ñ ´ = - = -
Ñ ´ =
Ñ = =
= -

Quantum Mechanical Basis for London Equation
( ) ( ) ( )
( )
( )
( )
( )
( )
2
* * *
1
0
2
2
0 0 ,
In zero field
Assume is "rigid", ie the field has no effect on wave function
n n n n n
n
e e
J r A r r r dr dr
mi mc
A J r
r e
J r A r
me
r n
y y y y y y d
y y
y
r
r
ì ü
é ù
= Ñ - Ñ - - -
í ý
ë û
î þ
= = =
= -
=
å ò

Pippard’s Extension of London’s Model
Observations:
-Penetration depth increased with reduced mean free path
- Hc and Tc did not change
- Need for a positive surface energy over 10-4 cm to explain
existence of normal and superconducting phase in
intermediate state
Non-local modification of London equation
( )
( )
4
0
0
1
-
3
4
1 1 1
Local:
Non local:
R
J A
c
R R A r e
J r d
c R
x
l
s
u
px l
x x
-
=
é ù
¢
ë û
=-
= +
ò

London and Pippard Kernels
Apply Fourier transform to relationship between
( ) ( ) ( ) ( ) ( )
:
4
and
c
J r A r J k K k A k
p
= -
( ) ( )
2
2
2
ln 1
Specular: Diffuse:
eff eff
o
o
dk
K k k K k
dk
k
p
l l
p
¥
¥
= =
+ é ù
+
ê ú
ë û
ò
ò
Effective penetration depth

London Electrodynamics
Linear London equations
together with Maxwell equations
describe the electrodynamics of superconductors at all T if:
– The superfluid density ns is spatially uniform
– The current density Js is small
2
2 2
0
1
0
s
J E
H H
t l m l
¶
= - Ñ - =
¶
0
s
H
H J E
t
m
¶
Ñ ´ = Ñ ´ = -
¶

Ginzburg-Landau Theory
• Many important phenomena in superconductivity occur
because ns is not uniform
– Interfaces between normal and superconductors
– Trapped flux
– Intermediate state
• London model does not provide an explanation for the
surface energy (which can be positive or negative)
• GL is a generalization of the London model but it still
retain the local approximation of the electrodynamics

• Ginzburg-Landau theory is a particular case of
Landau’s theory of second order phase transition
• Formulated in 1950, before BCS
• Masterpiece of physical intuition
• Grounded in thermodynamics
• Even after BCS it still is very fruitful in analyzing the
behavior of superconductors and is still one of the
most widely used theory of superconductivity

• Theory of second order phase transition is based on
an order parameter which is zero above the transition
temperature and non-zero below
• For superconductors, GL use a complex order
parameter Ψ(r) such that |Ψ(r)|2 represents the
density of superelectrons
• The Ginzburg-Landau theory is valid close to Tc

Ginzburg-Landau Equation for Free Energy
• Assume that Ψ(r) is small and varies slowly in
space
• Expand the free energy in powers of Ψ(r) and its
derivative
2
2
2 4
0
1
2 2 8
n
e h
f f
m i c
b
a y y y
p
*
*
æ ö
= + + + Ñ - +
ç ÷
è ø
A

Field-Free Uniform Case
Near Tc we must have
At the minimum
2 4
0
2
n
f f
b
a y y
- = +
0
f fn
- 0
f fn
-


¥
0
 > 0
 <
0 ( ) ( 1)
t t
b a a
> = -
¢
2 2
2
0 (1 )
8 2
and
c
n c
H
f f H t
a
y
p b
- = - = - Þ µ -
2 a
y
b
¥ = -

At the minimum
2 4
0
2
n
f f
b
a y y
- = +
2
0 ( ) ( 1) (1 )
t t t
b a a y ¥
> = - Þ µ -
¢
2
2
0
2 8
(1 )
(definition of )
c
n c
c
H
f f H
H t
a
b p
- = - = -
Þ µ -
2 a
y
b
¥ = -
It is consistent with correlating |Ψ(r)|2 with the density of superelectrons
2
(1 ) c
near T
s
n t
l -
µ µ -
which is consistent with ( )
2
0 1
c c
H H t
= -

Identify the order parameter with the density of superelectrons
2
2
2
2
2 2
2
2
2 2
2 4
2
2 4
( )
(0)
1 1 ( )
( ) ( ) (0)
( )
1 ( )
2 8
( ) ( )
( ) ( )
( )
4 (0) 4 (0)
since
and
L
s
L L
c
c c
L L
L L
T T
n
T T n
H T
T
H T H T
T T
n T n
l a
l l b
a
b p
l l
a b
p l p l
Y
= Y Þ = = -
Y
=
= - =

Field-Free Nonuniform Case
Equation of motion in the absence of electromagnetic
field
2
2
1
( ) 0
2
T
m
y a y b y y
*
- Ñ + + =
Look at solutions close to the constant one
2 ( )
where
T
a
y y d y
b
¥ ¥
= + = -
To first order:
2
1
0
4 ( )
m T
d d
a
*
Ñ - =
Which leads to 2 / ( )
r T
e x
d -
»

Field-Free Nonuniform Case
2 / ( )
2
(0)
1 2
( )
( ) ( )
2 ( )
where
r T L
c L
n
e T
m H T T
m T
x l
p
d x
l
a
-
*
*
» = =
is the Ginzburg-Landau coherence length.
It is different from, but related to, the Pippard coherence length.
( )
0
1/2
2
( )
1
T
t
x
x
-
GL parameter:
( )
( )
( )
L T
T
T
l
k
x
=
( ) ( )
( )
Both and diverge as but their ratio remains finite
is almost constant over the whole temperature range
L c
T T T T
T
l x
k
®

2 Fundamental Lengths
London penetration depth: length over which magnetic field decay
Coherence length: scale of spatial variation of the order parameter
(superconducting electron density)
1/2
2
( )
2
c
L
c
T
m
T
e T T
b
l
a
*
æ ö
= ç ÷ -
¢
è ø
1/2
2
( )
4
c
c
T
T
m T T
x
a
*
æ ö
= ç ÷ -
¢
è ø
The critical field is directly related to those 2 parameters
0
( )
2 2 ( ) ( )
c
L
H T
T T
f
x l
=

Surface Energy
2 2
2
2
1
8
:
8
:
8
Energy that can be gained by letting the fields penetrate
Energy lost by "damaging" superconductor
c
c
H H
H
H
s x l
p
l
p
x
p
é ù
-
ë û

Surface Energy
2 2
c
Interface is stable if >0
If >0
Superconducting up to H where superconductivity is destroyed globally
If >> <0 for
Advantageous to create small areas of normal state with large
c
H H
s
x l s
x
l x s
l
>>
>
1
:
2
1
:
2
area to volume ratio
quantized fluxoids
More exact calculation (from Ginzburg-Landau):
= Type I
= Type II
l
k
x
l
k
x
®
<
>
2 2
1
8
c
H H
s x l
p
é ù
-
ë û

Intermediate State
Vortex lines in
Pb.98In.02
At the center of each vortex is a
normal region of flux h/2e

Critical Fields
2
2
1
2
2
1
(ln .008)
2
Type I Thermodynamic critical field
Superheating critical field
Field at which surface energy is 0
Type II Thermodynamic critical field
(for 1)
c
c
sh
c
c c
c
c
c
c
H
H
H
H
H H
H
H
H
H
k
k
k k
k
=
+
Even though it is more energetically favorable for a type I superconductor
to revert to the normal state at Hc, the surface energy is still positive up to
a superheating field Hsh>Hc → metastable superheating region in which
the material may remain superconducting for short times.

Superheating Field
0.9
1
Ginsburg-Landau:
for <<1
1.2 for
0.75 for 1
c
sh
c
c
H
H
H
H
k
k
k
k >>
The exact nature of the rf critical
field of superconductors is still
an open question

BCS
• What needed to be explained and what were
the clues?
– Energy gap (exponential dependence of
specific heat)
– Isotope effect (the lattice is involved)
– Meissner effect

Cooper Pairs
Assumption: Phonon-mediated attraction
between electron of equal and opposite
momenta located within of Fermi surface
Moving electron distorts lattice and leaves
behind a trail of positive charge that attracts
another electron moving in opposite direction
Fermi ground state is unstable
Electron pairs can form bound
states of lower energy
Bose condensation of overlapping
Cooper pairs into a coherent
Superconducting state
D
w

Cooper Pairs
One electron moving through the lattice attracts the positive ions.
Because of their inertia the maximum displacement will take place
behind.

BCS
The size of the Cooper pairs is much larger than their spacing
They form a coherent state

BCS Theory
( )
0 , 1 ,-
, : ( ,- )
0 1
:states where pairs ( ) are unoccupied, occupied
probabilites that pair is unoccupied, occupied
BCS ground state
Assume interaction between pairs and
q q
q q
q q
q q
q
qk
q q
a b q q
a b
q k
V
Y = +
=
P
0
q k
if and
otherwise
D D
V x w x w
- £ £
=

BCS
• Hamiltonian
• Ground state wave function
destroys an electron of momentum
creates an electron of momentum
number of electrons of momentum
k k qk q q k k
k qk
k
q
k k k
n V c c c c
c k
c k
n c c k
e * *
- -
*
*
= +
=
å å
H
( ) 0
q q q q
q
a b c c f
* *
-
Y = +
P

BCS
• The BCS model is an extremely simplified model of reality
– The Coulomb interaction between single electrons is
ignored
– Only the term representing the scattering of pairs is
retained
– The interaction term is assumed to be constant over a
thin layer at the Fermi surface and 0 everywhere else
– The Fermi surface is assumed to be spherical
• Nevertheless, the BCS results (which include only a very
few adjustable parameters) are amazingly close to the real
world

BCS
( )
( )
2
2 2
0
1
0
0
0, 1 0
1, 0 0
2 1
2
1
sinh
0
Is there a state of lower energy than
the normal state?
for
for
yes:
where
q q q
q q q
q
q
q
V
D
D
a b
a b
b
e
V
r
x
x
x
x
w
w
r
-
= = <
= = >
= -
+ D
D =
é ù
ê ú
ë û

BCS
( )
( )
1
1.14 exp
0 1.76
c D
F
c
kT
VN E
kT
w
é ù
= -
ê ú
ë û
D =
Critical temperature
Coherence length (the size of the Cooper pairs)
0 .18 F
c
kT
u
x =

BCS Condensation Energy
( ) 2
0
2
0 0
0
0
4
0
2
8
/ 10
/ 10
F
Condensation energy: s n
F
V
E E
H
N
k K
k K
r
e p
e
D
- = -
æ ö
D
- D =
ç ÷
è ø
D

BCS Energy Gap
At finite temperature:
Implicit equation for the temperature dependence of the gap:
( )
( )
( )
1 2
2 2
1 2
2 2
0
tanh / 2
1
0
D kT
d
V
w e
e
r e
é ù
+ D
ê ú
ë û
=
+ D
ò

BCS Excited States
0
2 2
2
k
Energy of excited states:
k
e x
= + D

BCS Specific Heat
10
Specific heat
exp for < c
es
T
C T
kT
D
æ ö
-
ç ÷
è ø

Electrodynamics and Surface Impedance
in BCS Model
( )
[ ] ( )
0
4
,
, ,
There is, at present, no model for superconducting
sur
is treated as a small pe
face resistance at high rf
rturbation
similar to Pippar
field
ex
ex i i
ex
rf c
R
l
H H i
t
e
H A r t p
mc
H
H H
R R A I R T e
J dr
R
f
f f
w
-
¶
+ =
¶
=
<<
×
µ
å
ò
( ) ( ) ( )
( )
4
0 0:
d's model
Meissner effect
c
J k K k A k
K
p
= -
¹

Penetration Depth
( ) 2
4
2
( )
c
c
specular
1
Represented accurately by near
1-
dk
dk
K k k
T
T
T
l
p
l
=
+
æ ö
ç ÷
è ø
ò

Surface Resistance
( )
( )
4
3
2 2
2
:
1
:
2
exp
-
kT
2
Temperature dependence
close to dominated by change in
for dominated by density of excited states e
kT
Frequency dependence
is a good approximation
c
c
s
t
T t
t
T
T
A
R
T
l
w
w
D
-
-
- <
D
æ ö
-
ç ÷
è ø

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