SUPERCONDUCTIVITY
OVERVIEW OF EXPERIMENTAL FACTS
EARLY MODELS
GINZBURGLANDAU THEORY
BCS THEORY
Jean Delayen
Thomas Jefferson National Accelerator Facility
Old Dominion University
DaresburyMay June 2007
Historical Overview
Perfect Conductivity
Unexpected result
Expectation was the opposite: everything should become an isolator at
0T
Æ
KamerlinghOnnesand van derWaals in
Leiden with the helium 'liquefactor' (1908)
Perfect Conductivity
Persistent current experiments on rings have measured
15
10
s
n
σ
>
σ
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 > 10
5
years
Perfect Diamagnetism (Meissner&Ochsenfeld1933
)
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
cc
c
T
HTH
T
È˘
Êˆ
Í˙

Á˜
Ë¯
Í˙
Î˚
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 H
c1, and partial up to Hc2
Between H
c1
and H
c2
the field penetrates in the form if quantized vortices or
fluxoids
0
e
π
φ
=
=
Thermodynamic Properties
EntropySpecific Heat
EnergyFree Energy
Thermodynamic Properties
When phase transition at isof 1
st
order
latent heat
At , phase transition is of 2
nd
order
→no latent heat
→jump in heat capacity
<
c
TT
()
=
c
HHT
→
=
c
TT
()~3()
escenc
CTCT
3
(electronicspecificheat)
(reasonablefit to experimental data)
=
γ
≈α
en
es
CT
CT
Thermodynamic Properties
33
22
0
0
3
3
At:()()The entropy is continuous
Recall: (0)0 and
3
3
()()
For ()()
c
c
cscnc
T
T
es
cc
sn
cc
csn
TSTST
SC
S
TT
TTT
dtdtC
TTTT
TT
STST
TT
TTSTST
αγγ
αγ
γγ
=
∂
==
∂
ﬁ=Æ==
==
<<
Ú
Ú
⇒ superconducting state is more ordered than normal state
A better fit for the electron specific heat in superconducting state is
with 9,1.5 for
c
bT
T
escc
CaTeabTT
−
=γ≈≈
Energy Difference Between Normal and
Superconducting State
(
)
4422
2
()()Εnergy is continuous
3
()()()()
42
c
ncsc
T
nsesencc
T
c
UTUT
UTUTCCdtTTTT
T
γγ
=
==
Ú
()()
2
2
1
at000
48
c
nsc
H
TUUT
γ
π
===
2
isthecondensationenergy
8
c
H
π
2
at 0, is the free energy difference
8
c
H
T
π
π
()()
2
2
2
2
()1
1
84
c
nsncc
c
HTT
FUUTSST
T
γ
π
∆
È˘
Êˆ
Í˙
===
Á˜
Ë¯
Í˙
Î˚
()
2
1
2
()21
cc
c
T
HTT
T
πγ
È
˘
Êˆ
Í
˙
=
Á˜
Ë¯
Í
˙
Î
˚
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
cc
THM
α
α

∼∼
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
/
with 1.5
c
bTT
s
ceb

µ
Two Fundamental Lengths
•London penetration depth λ
–Distance over which magnetic fields decay in
superconductors
•Pippardcoherence length ξ
–Distance over which the superconducting state decays
Two Types of Superconductors
•London superconductors (Type II)
–λ>> ξ
–Impure metals
–Alloys
–Local electrodynamics
•Pippardsuperconductors (Type I)
–ξ>> λ
–Pure metals
–Nonlocalelectrodynamics
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 –Gorterand Casimir
()
()()()
()
()
()
1/2
2
2
fractionof"normal"electrons
1:fractionof"condensed"electrons (zero entropy)
Assume:()=1free energy
1
2
1
independent of temperature
4
Minimizationof gives =
c
ns
n
sc
C
TTx
x
FTxfTxfT
fTT
fTT
T
FTx
T
γ
βγ
<=
+
=
==
Ê
Ë
()()()
C
4
4
1/2
3
2
()11
T
3
T
ns
C
es
T
FTxfTxfT
T
C
β
γ
ˆ
Á˜
¯
È
˘
Êˆ
Í
˙
ﬁ=+=+
Á˜
Ë¯
Í
˙
Î
˚
ﬁ=
Two Fluid Model –Gorterand Casimir
()()()
()
4
1/2
2
2
2
2
Superconducting state:()11
Normal state:()2
2
Recall difference in free energy between normal and superconducting state
8
1
ns
C
n
C
c
C
T
FTxfTxfT
T
T
FTfTT
T
H
T
T
β
γ
β
π
β
È
˘
Êˆ
Í
˙
=+=+
Á˜
Ë¯
Í
˙
Î
˚
Êˆ
===
Á˜
Ë¯
=
Êˆ
=
Á˜
Ë¯
()
2
2
1
(0)
c
cC
HT
T
HT
È˘
Í˙
Í˙
Î˚
Êˆ
ﬁ=
Á˜
Ë¯
The GorterCasimirmodel 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 2fluid 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
ss
ss
nn
meEE
t
Jen
Jne
E
tm
JE
υ
υ
σ
∂
=
∂
=
∂
=
∂
=
K
K
JJK
K
G
K
G
K
Model of F & H London (1935)
2
22
2
Maxwell:
0=Constant
F&H London postulated:=0
ss
ss
ss
s
s
Jne
E
tm
B
E
t
mm
JBJB
tnene
m
JB
ne
∂
=
∂
∂
—¥=
∂
Êˆ
∂
ﬁ—¥+=ﬁ—¥+
Á˜
∂
Ë¯
—¥+
G
K
G
K
K
KK
KKKK
K
KK
Model of F & H London (1935)
combine with
os
B=J
∇×µ
KG
K
()
[]
2
2
0
1
2
2
0
0
exp/
s
oL
L
s
ne
BB
m
BxBx
m
ne
µ
λ
λ
µ
KK
—=
=
È˘
=
Í˙
Î˚
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
From Gorter and Casimir twofluid model
1
1
0
1
L
s
s
C
LL
C
m
ne
T
n
T
T
T
T
λ
µ
λλ
È˘
=
Í˙
Î˚
È˘
Êˆ
Í˙
µ
Á˜
Ë¯
Í˙
Î˚
=
È˘
Êˆ
Í˙

Á˜
Ë¯
Í˙
Î˚
Model of F & H London (1935)
Model of F & H London (1935)
2
0
2
London Equation:
choose 0,0on sample surface (London gauge)
1
Note: Local relationship between and
s
n
s
s
B
JH
AH
AA
JA
JA
λ∇×=−=−
µ
∇×=
∇==
=−
λ
G
G
G
G
G
G
i
G
G
G
G
Penetration Depth in Thin Films
Very thin films
Very thick films
Quantum Mechanical Basis for London Equation
()
(
)
(
)
()
()
()
()
()
2
***
1
0
2
2
In zero field 00,
Assume is "rigid", ie the field has no effect on wave function
nnnnn
n
ee
J
rArrrdrdr
mimc
AJr
re
JrAr
me
rn
ψψψψψψδ
ψψ
ψ
ρ
ρ
G
K
=
K
G
G
G
G
Ï¸
È˘
=——
Ì˝
Î˚
Ó˛
===
=
=
Â
Ú
Pippard’sExtension of London’s Model
Observations:
Penetration depth increased with reduced mean free path
H
c
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
Nonlocal modification of London equation
()
()
4
0
0
1
Local:
3
Non local:
4
111
R
JA
c
RRAre
J
rd
cR
ξ
λ
σ
υ
πξλ
ξξ

=
È˘
¢
Î˚
=
=+
Ú
K
K
K
KK
i
K
A
London and PippardKernels
Apply Fourier transform to relationship between
()()()
()()
and :
4
c
J
rArJkKkAk=−
π
()
()
2
2
2
Specular:Diffuse:
ln1
effeff
o
o
dk
Kkk
Kk
dk
k
π
λλ
π
•
•
==
+
È
˘
+
Í
˙
Î
˚
Ú
Ú
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
22
0
1
0
s
J
E
HH
t
λµλ
∂
=—=
∂
G
G
G
G
0s
H
HJE
t
µ
∂
—¥=—¥=
∂
G
G
GG
GinzburgLandau 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
GinzburgLandau Theory
•GinzburgLandau 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
GinzburgLandau Theory
•Theory of second order phase transition is based on an
order parameter which is zero above the transition
temperature and nonzero below
•For superconductors, GL use a complex order
parameter Ψ(r) such that Ψ(r)
2 represents the density
of superelectrons
•The GinzburgLandau theory is valid close to Tc
GinzburgLandau 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
24
0
1
228
n
eh
ff
mic
β
αψψψ
π
*
*
Êˆ
=+++—+
Á˜
Ë¯
A
=
FieldFree Uniform Case
24
0
2
n
ff
β
αψψ=+
0
ff
n

0
ff
n

ψ
ψ
ψ
•
0α>0α<
Near Tc
we must have
At the minimum
0()(1)
tt
βαα>=¢
2
2
2
0
and (1)
82
c
nc
H
ffHt
α
ψ
πβ
==ﬁµ
2
α
ψ
β
•
=
FieldFree Uniform Case
24
0
2
n
ff
β
αψψ=+
At the minimum
2
0()(1) (1)ttt
βααψ
•
>=ﬁµ
¢
2
2
0
(definition of )
28
(1)
c
nc
c
H
ffH
Ht
α
βπ
==
ﬁµ
2
α
ψ
β
•
=
It is consistent with correlating Ψ(r)2 with the density of
superelectrons
2
c
(1) near T
s
nt
λ

µµ
which is consistent with
()
2
0
1
cc
H
Ht
=
FieldFree Uniform Case
Identify the order parameter with the density of superelectrons
2
2
2
2
22
2
2
22
24
2
24
()
1(0)1()
()()
(0)
()
1()
since
28
()()
()()
()and
4(0)4(0)
L
s
LL
c
cc
LL
LL
T
T
n
TTn
HT
T
HTHT
TT
nTn
Ψ
Ψ
Ψ
=ﬁ==
=
==
∼
λα
λλβ
α
βπ
λλ
αβ
πλπλ
FieldFree NonuniformCase
Equation of motion in the absence of electromagnetic field
2
2
1
()0
2
T
m
ψαψβψψ
*
—++=
Look at solutions close to the constant one
2
()
where
T
α
ψψδψ
β
••
=+=
To first order:
2
1
0
4()mT
δδ
α
*
—=
Which leads to
2/()rT
e
ξ
δ

ª
FieldFree NonuniformCase
2/()
2
(0)
12
where ()
()()
2()
rT
L
cL
n
eT
mHTT
mT
ξ
λ
π
δξ
λ
α

*
*
ª==
is the GinzburgLandau coherence length.
It is different from, but related to, the Pippardcoherence length.
()
0
1/2
2
()
1
T
t
ξ
ξ

GL parameter:
()
()
()
L
T
T
T
λ
κ
ξ
=
Both () and () diverge as but their ratio remains finite
() is almost constant over the whole temperature range
Lc
TTTT
T
λξ
κ
Æ
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
eTT
β
λ
α
*
Êˆ
=
Á˜
Ë¯
¢
1/2
2
()
4
c
cT
T
mTT
ξ
α
*
Êˆ
=
Á˜
Ë¯
¢
=
The critical field is directly related to those 2 parameters
0
()
22()()
c
L
HT
TT
φ
ξλ
=
Surface Energy
22
2
2
1
8
:Energy that can be gained by letting the fields penetrate
8
:Energy lost by "damaging" superconductor
8
c
c
HH
H
H
σξλ
π
λ
π
ξ
π
È˘

Î˚
Surface Energy
c
22
Interface is stable if >0
If >0
Superconducting up to H where superconductivity is destroyed globally
If >><0for
Advantageous to create small areas of normal state with large
c
HH
σ
ξ>>λσ
ξ
λξσ>
λ
area to volume ratio
quantized fluxoids
More exact calculation (from GinzburgLandau):
1
=:Type I
2
1
=:Type II
2
→
λ
κ<
ξ
λ
κ>
ξ
22
1
8
c
HHσξλ
π
È
˘

Î
˚
Magnetization Curves
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
Type IThermodynamic critical field
Superheating critical field
Field at which surface energy is 0
Type IIThermodynamic critical field
2
1
(ln.008)(for 1)
2
c
c
sh
c
cc
c
c
c
c
H
H
H
H
HH
H
H
H
H
κ
=κ
κ+κ
κ
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 H
sh>Hc
→metastablesuperheating region in which the
material may remain superconducting for short times.
Superheating Field
GinsburgLandau:
0.9
for <<1
1.2 for 1
0.75for 1
κ
κ
κ
κ>>
∼
∼∼
∼
c
sh
c
c
H
H
H
H
The exact nature of the rf critical
field of superconductors is still
an open question
Material Parameters for Some Superconductors
BCS
•What needed to be explained and what were the clues?
–Energy gap (exponential dependence of specific heat
–Isotope effect (the lattice is involved)
–Meissnereffect
Cooper Pairs
Assumption: Phononmediated attraction between
electron of equal and opposite momentalocated
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
ω
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 and BEC
BCS Theory
()
0,1 :states where pairs (,) are unoccupied, occupied
,:probabilites that pair (,) is unoccupied, occupied
BCS ground state
01
Assume interaction between pairs and
qq
qq
qq
qq
q
qk
qq
abqq
ab
qk
V
Ψ=+
=
Π
G
G
G
GG
G
G
qk
if and
0otherwise
DD
V−ξ≤ωξ≤ω
=
==
BCS
•Hamiltonian
•Ground state wave function
destroys an electron of momentum
creates an electron of momentum
number of electrons of momentum kkqkqqkk
kqk
k
q
kkk
nVcccc
ck
ck
ncck
ε
**

*
*
=+
=
ÂÂ
H
(
)
0qqqq
q
abcc
∗∗
−
Ψ=+φ
Π
G
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
22
0
1
0
0
Is there a state of lower energy than
the normal state?
0,1for0
1,0for0
yes:21
where
2
1
sinh
0
qqq
qqq
q
q
q
V
D
D
ab
ab
b
e
V
−
ρ
==ξ<
==ξ>
ξ
=−
ξ+∆
ω
∆=ω
ρ
=
=
BCS
(
)
()
1
1.14exp
01.76
cD
F
c
kT
VNE
kT
ω=
È
˘
=
Í
˙
Î
˚
D=
Critical temperature
Coherence length (the size of the Cooper pairs)
0
.18
F
c
kT
υ
ξ
=
=
BCS Condensation Energy
()
2
0
2
00
0
0
4
F
0
Condensation energy:
2
8
/10
/10
sn
F
V
EE
H
N
kK
kK
ρ∆
∆
∆
επ
ε
=
Êˆ
=
Á˜
Ë¯
D
BCS Energy Gap
At finite temperature:
Implicit equation for the temperature dependence of the gap:
()
(
)
(
)
12
22
12
22
0
tanh/2
1
0
D
kT
d
V
ω
ε
ε
ρ
ε
È
˘
+D
Í
˙
Î
˚
=
+D
Ú
=
BCS Excited States
0
22
k
Energy of excited states:
2
k
εξ
=+D
BCS Heat Capacity
es
Heat capacity
Cexpfor<
10
c
T
T
kT
D
Êˆ

Á˜
Ë¯
Electrodynamics and Surface Impedance
in BCS Model
(
)
[]
()
0
4
,
is treated as a small perturbation
,,
similar to Pippar
ex
exii
ex
rfc
R
l
HHi
t
e
HArtp
mc
H
HH
RRAIRTe
Jdr
R
φ
φφ
ω
There is, at present, no model for superconducting surface resistance at high rffield
=

∂
+=
∂
=
<<
◊
µ
Â
Ú
()
()()
()
d's model
4
00:Meissner effect
c
JkKkAk
K
π
=
π
Penetration Depth
()
2
c
4
c
2
(specular)
1
Represented accurately by near
1
dk
dk
Kkk
T
T
T
λ=
π+
λ
⎛⎞
⎜⎟
⎝⎠
∫
∼
Surface Resistance
()
()
4
3
2
2

kT
2
2
Temperature dependence
close to :dominated by change in
1
for : dominated by density of excited statese
2
exp
kT
Frequency dependence
is a good approximation
c
c
s
t
Tt
t
T
T
A
R
T
∆
−λ
−
−<
∆
ω−
ω
∼
∼
Surface Resistance
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