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SUPERCONDUCTIVITY
OVERVIEW OF EXPERIMENTAL FACTSEARLY MODELS
GINZBURG-LANDAU THEORYBCS THEORY
Jean Delayen
Thomas Jefferson National Accelerator FacilityOld Dominion University
USPAS June 2008 U. Maryland
Historical Overview
Perfect Conductivity
Unexpected result
Expectation was the opposite: everything should become an isolator at 0T Æ
Kamerlingh Onnes and van der Waals in Leiden with the helium 'liquefactor' (1908)
Perfect Conductivity Persistent current experiments on rings have measured
1510s
n
ss
>
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)
0Bt
∂ =∂
0B =
Perfect conductor Superconductor
Penetration Depth in Thin Films
Very thin films
Very thick films
Critical Field (Type I)
2
( ) (0) 1c cc
TH T HT
È ˘Ê ˆÍ ˙- 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 Hc1, and partial up to Hc2
Between Hc1 and Hc2 the field penetrates in the form if quantized vortices or fluxoids
0 epf =
Thermodynamic Properties
Entropy Specific Heat
Energy Free Energy
Thermodynamic Properties
( ) 1
2
When phase transition at is of order latent heat
At transition is of order no latent heat jump in specific heat
stc c
ndc
es
T T H H T
T T
C
< = fi
= fi
3
( ) 3 ( )
( )
( )
electronic specific heat reasonable fit to experimental data
c en c
en
es
T C T
C T T
C T T
ga
=
ª
∼
Thermodynamic Properties
3 3
2 200
3
3
: ( ) ( )
(0) 0
3 3
( ) ( )
( ) ( )
At The entropy is continuous
Recall: and
For
c c
c s c n c
T T
esc c
s nc c
c s n
T S T S T
S CST T
T T Tdt dt CT T T T
T TS T S TT T
T T S T S T
a g ga g
g g
=
∂= =∂
fi = Æ = =
= =
< <
ÚÚ
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 cbT
Tes c cC a T e a b T Tg
-= ª ª
Energy Difference Between Normal and Superconducting State
( )4 4 2 22
( ) ( )3( ) ( ) ( ) ( )4 2
Εnergy is continuousc
n c s c
T
n s es en c cTc
U T U T
U T U T C C dt T T T TTg g
=
- = - = - - -Ú
( ) ( )2
210 0 04 8
at cn s c
HT U U Tgp
= - = =2
8is thecondensationenergycH
p2
0,8
at is the free energy differencecHTp
π
( ) ( )222
2( ) 1 18 4c
n s n c cc
H T TF U U T S S TT
gp
È ˘Ê ˆÍ ˙= D = - - - = - Á ˜Ë ¯Í ˙Î ˚
( )21
2( ) 2 1c cc
TH T TT
pgÈ ˘Ê ˆÍ ˙= - 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 0.5c cT 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 cbT Tsc 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 descriptiveEverything 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- :
( ) = 112
14
gives =
fractionof "normal"electronsfractionof "condensed"electrons (zero entropy)
Assume: free energy
independent of temperature
Minimizationof
c
n s
n
s c
C
T T xx
F T x f T x f T
f T T
f T T
TF T xT
g
b g
< =
+ -
= -
= - =-
ÊË
( ) ( ) ( )
C
4
41/2
3
2
( ) 1 1
T3T
n sC
es
TF T x f T x f TT
C
b
g
ˆÁ ˜
È ˘Ê ˆÍ ˙fi = + - = - + Á ˜Ë ¯Í ˙Î ˚
fi =
Two Fluid Model – Gorter and Casimir
( ) ( ) ( )
( )
41/2
22
2
2
( ) 1 1
( ) 22
8
1
Superconducting state:
Normal state:
Recall difference in free energy between normal and
superconducting state
n sC
nC
c
C
TF T x f T x f TT
TF T f T TT
H
TT
b
g b
p
b
È ˘Ê ˆÍ ˙= + - = - + Á ˜Ë ¯Í ˙Î ˚
Ê ˆ= = - = - Á ˜Ë ¯
=
Ê ˆ= - Á ˜Ë ¯
( )2 2
1(0)
c
c C
H T TH T
È ˘ Ê ˆÍ ˙ fi = - Á ˜Ë ¯Í ˙Î ˚
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 vsnn : density of the normal component of velocity vn
2
superelectrons are accelerated by
superelectrons
normal electrons
s s
s s
n n
m eE Et
J en
J n e Et m
J E
u
u
s
∂ = -∂= -
∂=
∂
=
Model of F & H London (1935)
2
2 2
2
0 = Constant
= 0
Maxwell:
F&H London postulated:
s s
s ss s
ss
J n e Et m
BEt
m mJ B J Bt n e n e
m J Bn e
∂=
∂
∂—¥ = -∂
Ê ˆ∂fi —¥ + = fi —¥ +Á ˜∂ Ë ¯
— ¥ +
Model of F & H London (1935)combine with 0 sB = Jm—¥
( ) [ ]
22 0
12
20
- 0
exp /
s
o L
Ls
n eB B
m
B x B x
mn e
m
l
lm
— =
= -
È ˘= Í ˙Î ˚
The magnetic field, and the current, decay exponentially over a distance λ (a few 10s of nm)
( ) ( )
12
20
4
14 2
1
10
1
From Gorter and Casimir two-fluid model
Ls
sC
L L
C
mn e
TnT
T
TT
lm
l l
È ˘= Í ˙Î ˚
È ˘Ê ˆÍ ˙μ - Á ˜Ë ¯Í ˙Î ˚
=È ˘Ê ˆÍ ˙- Á ˜Ë ¯Í ˙Î ˚
Model of F & H London (1935)
Model of F & H London (1935)
2
0
2
0, 0
1
London Equation:
choose on sample surface (London gauge)
Note: Local relationship between and
s
n
s
s
BJ H
A H
A A
J A
J A
lm
l
—¥ = - = -
—¥ =
— = =
= -
i
Penetration Depth in Thin Films
Very thin films
Very thick films
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 nn
e eJ r A r r r dr drmi mc
A J r
r eJ r A r
mer n
y y y y y y d
y y
y
r
r
Ï ¸È ˘= — - — - - -Ì ˝Î ˚Ó ˛
= = =
= -
=
ÂÚ
Pippard’s Extension of London’s ModelObservations:
-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-
34
1 1 1
Local:
Non local:
R
J Ac
R R A r eJ r d
c R
x
l
s upx l
x x
-
=
È ˘¢Î ˚=-
= +
Úi
London and Pippard KernelsApply Fourier transform to relationship between
( ) ( ) ( ) ( ) ( ):4
and cJ r A r J k K k A kp
= -
( ) ( )2
2
2
ln 1Specular: Diffuse:eff eff
o
o
dkK k k K k
dkk
pl lp
•
•= =
+ È ˘+Í ˙
Î ˚
ÚÚ
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
22 2
0
1 0sJ E H Ht l m l
∂= - — - =
∂
0sHH J Et
m ∂—¥ = —¥ = -∂
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
• 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
Ginzburg-Landau Theory
• 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 22 4
01
2 2 8ne hf f
m i cba y y y
p
*
*
Ê ˆ= + + + — - +Á ˜Ë ¯
A
Field-Free Uniform Case
Near Tc we must have
At the minimum
2 40 2nf f ba y y- = +
0f fn- 0f fn-
•
0 > 0<
0 ( ) ( 1)t tb a a> = -¢
2 22
0 (1 )8 2
and cn c
Hf f H ta yp b
- = - = - fi μ -
2 ayb• = -
Field-Free Uniform Case
At the minimum
2 40 2nf f ba y y- = +
20 ( ) ( 1) (1 ) t t tb a a y•> = - fi μ -¢
22
0 2 8(1 )
(definition of )cn c
c
Hf f H
H t
ab p
- = - = -
fi μ -
2 ayb• = -
It is consistent with correlating |Ψ(r)|2 with the density of superelectrons
2 (1 ) cnear Tsn tl-μ μ -
which is consistent with ( )20 1c cH H t= -
Field-Free Uniform Case
Identify the order parameter with the density of superelectrons
222
22 2
22
2 22 42
2 4
( )(0)1 1 ( )( ) ( ) (0)
( )1 ( )2 8
( ) ( )( ) ( )( )4 (0) 4 (0)
since
and
Ls
L L
c
c cL L
L L
T TnT T n
H TT
H T H TT Tn T n
l al l b
ab p
l la bp l p l
Y= Y fi = = -
Y
=
= - =
∼
Field-Free Nonuniform Case
Equation of motion in the absence of electromagnetic field
221 ( ) 02
Tm
y a y b y y*- — + + =
Look at solutions close to the constant one2 ( )where Tay y d y
b• •= + = -
To first order:21 0
4 ( )m Td d
a* — - =
Which leads to 2 / ( )r Te xd -ª
Field-Free Nonuniform Case
2 / ( )2
(0)1 2( )( ) ( )2 ( )
where r T L
c L
ne Tm H T Tm T
x lpd xla
-**
ª = =
is the Ginzburg-Landau coherence length.
It is different from, but related to, the Pippard coherence length. ( )0
1/22( )
1T
t
xx-
GL parameter:( )( )( )
L TTT
lkx
=
( ) ( )
( )
Both and diverge as but their ratio remains finite
is almost constant over the whole temperature range
L cT T T T
T
l x
k
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
cL
c
TmTe T Tbla
*Ê ˆ= Á ˜ -¢Ë ¯
1/22
( )4
c
c
TTm T T
xa*
Ê ˆ= Á ˜ -¢Ë ¯
The critical field is directly related to those 2 parameters
0( )2 2 ( ) ( )c
L
H TT Tf
x l=
Surface Energy
2 2
2
2
18
: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 lp
lpxp
È ˘-Î ˚
Surface Energy
2 2
c
Interface is stable if >0If >0
Superconducting up to H where superconductivity is destroyed globally
If >> <0 for
Advantageous to create small areas of normal state with large
cH H
sx l s
xl x sl
>>
>
1 :2
1 :2
area to volume ratio quantized fluxoids
More exact calculation (from Ginzburg-Landau):
= Type I
= Type II
lkxlkx
Æ
<
>
2 218 cH Hs x lpÈ ˘-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
12
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
csh
c
c c
cc
c
c
HHH
H
H H
HHH
H
k
k
k kk
=
+
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
csh
c
c
HH
HH
kk
kk >>
∼
∼ ∼∼
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)
– 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 overlappingCooper pairs into a coherentSuperconducting state
Dw
Cooper PairsOne 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 ,-
, : ( ,- )
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 qq qq
qk
q q
a b q q
a b
q k
V
Y = +
=
P
0q k if and
otherwiseD DV 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 kk qk
k
q
k k k
n V c c c c
c k
c k
n c c k
e * *- -
*
*
= +
=
 ÂH
( ) 0q q q qqa 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 20
10
0
0, 1 0
1, 0 0
2 1
21sinh0
Is there a state of lower energy than the normal state?
forfor
yes:
where
q q q
q q q
q
VDD
a b
a b
b
e
V
r
xx
x
x
w w
r
-
= = <
= = >
= -+ D
D =È ˘Í ˙Î ˚
BCS
( )( )
11.14 exp
0 1.76
c DF
c
kTVN E
kT
wÈ ˘
= -Í ˙Î ˚
D =
Critical temperature
Coherence length (the size of the Cooper pairs)
0 .18 F
ckTux =
BCS Condensation Energy
( ) 20
20 0
0
0
4
02
8/ 10
/ 10F
Condensation energy: s n
F
VE E
HN
k K
k K
r
e p
e
D- = -
Ê ˆD- D =Á ˜Ë ¯
D
BCS Energy GapAt finite temperature:
Implicit equation for the temperature dependence of the gap:
( )( )( )
1 22 2
1 22 20
tanh / 210
D kTd
V
w ee
r e
È ˘+ DÍ ˙Î ˚=+ DÚ
BCS Excited States
0
2 22k
Energy of excited states:
ke x= + D
BCS Specific Heat
10
Specific heat
exp for < ces
TC T
kTDÊ ˆ-Á ˜Ë ¯
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
Rl
H H it
eH A r t pmc
HH H
R R A I R T eJ dr
R
ff f
w-
∂+ =∂
=
<<
◊μ
Â
Ú( ) ( ) ( )( )
40 0 :
d's model
Meissner effect
cJ k K k A k
Kp
= -
π
Penetration Depth
( ) 2
4
2 ( )
c
c
specular
1Represented accurately by near
1-
dk dkK k k
TTT
lp
l
=+
Ê ˆÁ ˜Ë ¯
Ú
∼
Surface Resistance
( )( )
4
32 2
2
:1
:2
exp
- kT
2
Temperature dependence
close to dominated by change in
for dominated by density of excited states e
kTFrequency dependence
is a good approximation
c
c
s
tT tt
TT
ART
l
w
w
D
--
- <
DÊ ˆ-Á ˜Ë ¯
∼
∼
Surface Resistance
Surface Resistance
Surface Resistance
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