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Jean Delayen
Center for Accelerator Science
Old Dominion University
and
Thomas Jefferson National Accelerator Facility
SRF FUNDAMENTALS
USPAS @ Rutgers June 2015
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Historical Overview
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Perfect Conductivity
Kamerlingh Onnes and van der Waals
in Leiden with the helium 'liquefactor'
(1908)
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Perfect Conductivity
Persistent current experiments on rings have measured
1510s
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
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Perfect Diamagnetism (Meissner & Ochsenfeld 1933)
0B
t
¶=
¶0B =
Perfect conductor Superconductor
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Penetration Depth in Thin Films
Very thin films
Very thick films
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Critical Field (Type I)
2
( ) (0) 1c c
c
TH T H
T
é ùæ öê ú- ç ÷è øê úë û
Superconductivity is destroyed by the application of a magnetic field
Type I or “soft” superconductors
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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
0e
pf =
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Thermodynamic Properties
Entropy Specific Heat
Energy Free Energy
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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
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
=
»
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Thermodynamic Properties
3 3
2 200
3
3
: ( ) ( )
(0) 0
33
( ) ( )
( ) ( )
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 CS
T T
T T Tdt dt C
T T T T
T TS T S T
T T
T T S T S T
a g ga g
g g
=
¶= =
¶
Þ = ® = =
= =
< <
òò
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
-
= » »
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( )4 4 2 2
2
( ) ( )
3( ) ( ) ( ) ( )
4 2
Εnergy is continuous
c
n c s c
T
n s es en c cT
c
U T U T
U T U T C C dt T T T TT
g g
=
- = - = - - -ò
( ) ( )2
210 0 0
4 8at c
n s c
HT U U Tg
p= - = =
2
8is thecondensationenergycH
p
2
0,8
at is the free energy differencecHT
p¹
( ) ( )
22
22( ) 1
18 4
cn s n c c
c
H T TF U U T S S T
Tg
p
é ùæ öê ú= D = - - - = - ç ÷è øê úë û
( )21
2( ) 2 1c c
c
TH T T
Tpg
é ùæ öê ú= - ç ÷è øê úë û
The quadratic dependence of critical field on T is
related to the cubic dependence of specific heat
Energy Difference Between Normal and
Superconducting State
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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-
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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 T
sc e b-
µ
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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
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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
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Material Parameters for Some Superconductors
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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
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Two Fluid Model – Gorter and Casimir
( )
( )
( )
1/2
2
2
(1 ) :
( ) = ( ) (1 ) ( )
1
2
1
4
gives
c
n s
n
s c
T T x
x
F T x f T x f T
f T T
f T T
F T
g
b g
< =
-
+ -
= -
= - =-
fractionof "normal"electrons
fractionof "condensed"electrons (zero entropy)
Assume: free energy
independent of temperature
Minimizationof
C
4
4
1/2
3
2
=
( ) ( ) (1 ) ( ) 1
T3
T
C
n s
C
es
Tx
T
TF T x f T x f T
T
C
b
g
æ öç ÷è ø
é ùæ öê úÞ = + - = - + ç ÷è øê úë û
Þ =
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Two Fluid Model – Gorter and Casimir
4
1/2
2
2
2
( ) ( ) (1 ) ( ) 1
( ) ( ) 22
8
1
n s
C
n
C
c
TF T x f T x f T
T
TF T f T T
T
H
T
T
b
gb
p
b
é ùæ öê ú= + - = - + ç ÷è øê úë û
æ ö= = - = - ç ÷è ø
=
= -
Superconducting state:
Normal state:
Recall difference in free energy between normal and
superconducting state
22 2
( )1
(0)
c
C c C
H T T
H T
é ùæ ö æ öê ú Þ = -ç ÷ ç ÷è ø è øê úë 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
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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 Et
J en
J n eE
t m
J E
u
u
s
¶= -
¶
= -
¶=
¶
=
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Model of F & H London (1935)
2
2 2
2
0 = Constant
= 0
Maxwell:
F&H London postulated:
s s
s s
s s
s
s
J n eE
t m
BE
t
m mJ B J B
t n e n e
mJ B
n e
¶=
¶
¶Ñ´ = -
¶
æ ö¶Þ Ñ´ + = Þ Ñ´ +ç ÷¶ è ø
Ñ´ +
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Model of F & H London (1935)
combine with 0 sB = JmÑ´
( ) [ ]
22 0
1
2
2
0
- 0
exp /
s
o L
L
s
n eB B
m
B x B x
m
n e
m
l
lm
Ñ =
= -
é ù= ê úë û
The magnetic field, and the current, decay
exponentially over a distance λ (a few 10s of nm)
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1
2
2
0
4
14 2
1
1( ) (0)
1
L
s
s
C
L L
C
m
n e
Tn
T
T
T
T
lm
l l
é ù= ê úë û
é ùæ öê úµ - ç ÷è øê úë û
=
é ùæ ö-ê úç ÷è øê úë û
From Gorter and Casimir two-fluid model
Model of F & H London (1935)
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Page 25
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
Ñ´ = - = -
Ñ´ =
Ñ = =
= -
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Page 26
Penetration Depth in Thin Films
Very thin films
Very thick films
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Page 27
Quantum Mechanical Basis for London Equation
2* * *
1
0
2
( ) ( ) ( )2
0 ( ) 0 ,
( )( ) ( )
( )
n n n n n
n
e eJ r A r r r dr dr
mi mc
A J r
r eJ r A r
m e
r n
y y y y y y d
y y
y
r
r
ì üé ù= Ñ - Ñ - - -í ýë û
î þ
= = =
= -
=
åò
In zero field
Assume is "rigid", ie the field has no effect on wave function
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Page 28
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
R
J Ac
R R A r eJ r d
c R
x
l
su
px l
x x
-
=
é ù¢ë û=-
= +
ò
Local:
Non local:
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Page 29
London and Pippard Kernels
Apply Fourier transform to relationship between
( ) ( ) : ( ) ( ) ( )4
cJ r A r J k K k A k
p= - and
2
2
2
( )( )ln 1
eff effo
o
dk
K kK k kdk
k
pl l
p
¥
¥= =
+ é ù+ê úë û
òò
Specular: Diffuse:
Effective penetration depth
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Page 30
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
10sJ E
H Ht l m l
¶= - Ñ - =
¶
0s
HH J E
tm
¶Ñ´ = Ñ´ = -
¶
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Page 31
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
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Page 32
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
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Page 33
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
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Page 34
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
0
1
2 2 8n
e hf f
m i c
ba y y y
p
*
*
æ ö= + + + Ñ- +ç ÷è ø
A
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Page 35
Field-Free Uniform Case
Near Tc we must have
At the minimum
2 4
02
nf fb
a y y- = +0
f fn
-0
f fn
-
yyy ¥
0a > 0a <
0 ( ) ( 1)t tb a a> = -¢
2 22
0 (1 )8 2
and cn c
Hf f H t
ay
p b- = - = - Þ µ -
2 ay
b¥ = -
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Page 36
Field-Free Uniform Case
At the minimum
2 4
02
nf fb
a y y- = +
20 ( ) ( 1) (1 ) t t tb a a y ¥> = - Þ µ -¢
22
02 8
(1 )
(definition of )cn c
c
Hf f H
H t
a
b p- = - = -
Þ µ -
2 ay
b¥ = -
It is consistent with correlating |Ψ(r)|2 with the density of superelectrons
2 (1 ) c near Tsn tl -µ µ -
which is consistent with 2
0(1 )c cH H t= -
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Page 37
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 Tn
T T n
H TT
H T H TT Tn T n
l a
l l b
a
b p
l la b
p l p l
Y= Y Þ = = -
Y
=
= - =
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Page 38
Field-Free Nonuniform Case
Equation of motion in the absence of electromagnetic
field
221( ) 0
2T
my a y b y y
*- Ñ + + =
Look at solutions close to the constant one 2 ( )
whereTa
y y d yb
¥ ¥= + = -
To first order: 21
04 ( )m T
d da*
Ñ - =
Which leads to 2 / ( )r Te
xd
-»
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Page 39
Field-Free Nonuniform Case
2 / ( )
2
(0)1 2( )
( ) ( )2 ( ) where
r T L
c L
ne T
m H T Tm T
x lpd x
la
-
**» = =
is the Ginzburg-Landau coherence length.
It is different from, but related to, the Pippard coherence length. ( )
0
1/22
( )1
Tt
xx
-
GL parameter: ( )
( )( )
L TT
T
lk
x=
( ) ( )
( )
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
®
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Page 40
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
TmT
e T T
bl
a
*æ ö= ç ÷ -¢è ø
1/22
( )4
c
c
TT
m T Tx
a*æ ö
= ç ÷ -¢è ø
The critical field is directly related to those 2 parameters
0( )2 2 ( ) ( )
c
L
H TT T
f
x l=
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Page 41
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 lp
l
p
x
p
é ù-ë û
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Page 42
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
cH H
s
x l s
xl x s
l
>>
>
1:
2
1:
2
area to volume ratio
quantized fluxoids
More exact calculation (from Ginzburg-Landau):
= Type I
= Type II
lk
x
lk
x
®
<
>
2 21
8cH Hs x l
pé ù-ë û
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Page 43
Magnetization Curves
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Page 44
Intermediate State
Vortex lines in
Pb.98In.02 At the center of each vortex is a
normal region of flux h/2e
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Page 45
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
csh
c
c c
cc
c
c
H
HH
H
H H
HH
H
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.
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Page 46
Superheating Field
0.9
1
Ginsburg-Landau:
for <<1
1.2 for
0.75 for 1
csh
c
c
HH
H
H
kk
k
k >>
The exact nature of the rf critical
field of superconductors is still
an open question
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Page 47
Material Parameters for Some Superconductors
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Page 48
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
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Page 49
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
Dw
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Page 50
Cooper Pairs
One electron moving through the lattice attracts the positive ions.
Because of their inertia the maximum displacement will take place
behind.
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Page 51
BCS
The size of the Cooper pairs is much larger than their spacing
They form a coherent state
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Page 52
BCS and BEC
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Page 53
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
0
q k if and
otherwise
D DV x w x w- £ £
=
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Page 54
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 åH
( ) 0q q q qq
a b c c f* *
-Y = +P
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Page 55
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
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Page 56
BCS
2
2 2
0
1
(0)
0
0, 1 0
1, 0 0
2 1
21
sinh(0)
q q q
q q q
q
q
q
VDD
a b
a b
b
e
V
r
x
x
x
x
ww
r
-
= = <
= = >
= -+ D
D =é ùê úë û
Is there a state of lower energy than
the normal state?
for
for
yes:
where
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Page 57
BCS
( )
11.14 exp
( )
0 1.76
c D
F
c
kTVN E
kT
wé ù
= -ê úë û
D =
Critical temperature
Coherence length (the size of the Cooper pairs)
0 .18 F
ckT
ux =
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Page 58
BCS Condensation Energy
2
0
2
0 00
0
4
(0)
2
8
/ 10
/ 10
s n
F
VE E
HN
k K
k K
r
e p
e
D- = -
æ öD- D =ç ÷è ø
D
F
Condensation energy:
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Page 59
BCS Energy Gap
At finite temperature:
Implicit equation for the temperature dependence of the gap:
2 2 1 2
2 2 1 20
tanh ( ) / 21
(0) ( )
D kTd
V
w ee
r e
é ù+ Dë û=
+ Dò
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Page 60
BCS Excited States
0
2 22k
Energy of excited states:
ke x= + D
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Page 61
BCS Specific Heat
10
Specific heat
exp for < ces
TC T
kT
Dæ ö-ç ÷è ø
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Page 62
Electrodynamics and Surface Impedance
in BCS Model
0
4
( , )
[ ] ( , , )
ex
ex i i
ex
rf c
R
l
H H it
eH A r t p
mc
H
H H
R R A I R T eJ dr
R
ff f
w-
¶+ =
¶
=
<<
×µ
å
ò
There is, at present, no model for superconducting
surface resistanc
is treated as a smal
e
l perturbation
at high rf fie
simil
ld
ar to
( ) ( ) ( )4
(0) 0 :
cJ k K k A k
K
p= -
¹
Pippard's model
Meissner effect
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Page 63
Penetration Depth
2
4
2( )
( )
dkdk
K k k
TT
T
lp
l
=+
æ öç ÷è ø
ò
c
c
specular
1Represented accurately by near
1-
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Page 64
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
Frequency dependence
is a good approximation
c
c
s
tT t
t
TT
AR
T kT
l
w
w
D
--
- <
Dæ ö-ç ÷è ø
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Page 65
Surface Resistance
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Page 66
Surface Resistance
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Page 67
Surface Resistance
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Page 68
Surface Impedance - Definitions
• The electromagnetic response of a metal,
whether normal or superconducting, is described
by a complex surface impedance, Z=R+iX
R : Surface resistance
X : Surface reactance
Both R and X are real
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Page 69
Definitions
For a semi- infinite slab:
0
0
0
(0)
( )
(0) (0)
(0) ( ) /
Definition
From Maxwell
x
x
x x
y x z
EZ
J z dz
E Ei
H E z zw m
+
¥
=
=
= =¶ ¶
ò
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Page 70
Definitions
The surface resistance is also related to the power flow
into the conductor
and to the power dissipated inside the conductor
212 (0 )P R H -=
( )0
0
0
1/2
0
(0 ) / (0 )
377 Impedance of vacuum
Poynting vector
Z Z S S
Z
S E H
m
e
+ -=
= W
= ´
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Page 71
Normal Conductors (local limit)
Maxwell equations are not sufficient to model the
behavior of electromagnetic fields in materials.
Need an additional equation to describe material
properties
( ) 0
14
1
3 10 secFor Cu at 300 K,
so for wavelengths longer than infrared
J JE
t i
J E
sss w
t t wt
t
s
-
¶+ = Þ =
¶ -
= ´
=
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Page 72
Normal Conductors (local limit)
In the local limit
The fields decay with a characteristic
length (skin depth)
( ) ( )J z E zs=
/ /
0
1/2
00
( ) (0)
(1 )( ) ( )
(0) (1 ) (1 )(1 )
(0) 2 2
z iz
x x
y x
x
y
E z E e e
iH z E z
E i iZ i
H
d d
m w d
m wm wd
s d s
- -=
-=
æ ö+ += = = = + ç ÷è ø
1/2
0
2d
m ws
æ ö= ç ÷è ø
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Page 73
Normal Conductors (anomalous limit)
• At low temperature, experiments show that the surface
resistance becomes independent of the conductivity
• As the temperature decreases, the conductivity s increases
– The skin depth decreases
– The skin depth (the distance over which fields vary) can
become less then the mean free path of the electrons (the
distance they travel before being scattered)
– The electrons do not experience a constant electric field
over a mean free path
– The local relationship between field and current is not
valid ( ) ( )J z E zs¹
1/2
0
2d
m ws
æ ö= ç ÷è ø
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Page 74
Normal Conductors (anomalous limit)
Introduce a new relationship where the current is related to
the electric field over a volume of the size of the mean
free path (l)
Specular reflection: Boundaries act as perfect mirrors
Diffuse reflection: Electrons forget everything
/
4
( , / )3( , )
4with
F R l
V
R R E r t R vJ r t dr e R r r
l R
s
p
-é ù× -¢ë û
= = -¢ ¢ò
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Page 75
Normal Conductors (anomalous limit)
In the extreme anomalous limit
( )1/3
2 2
091 08
31 3
16
1 :
: fraction of electrons specularly scattered at surface
fraction of electrons diffusively scattered
p p
lZ Z i
p
p
m w
p s= =
æ ö= = +ç ÷
è ø
-
2
2
31
2 cl
l
d
æ öç ÷è ø
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Page 76
1/3
5 2/3( ) 3.79 10l
R l ws
- æ ö®¥ = ´ ç ÷è ø
For Cu: 16/ 6.8 10 2ml s -= ´ W×
1/3
5 2/3
0
3.79 10(4.2 ,500 )
0.12(273 ,500 )
2
K MHz
K MHz
l
R
R
ws
m w
s
- æ ö´ ç ÷è ø
= »
Does not compensate for the Carnot efficiency
Normal Conductors (anomalous limit)
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Page 77
Surface Resistance of Superconductors
Superconductors are free of power dissipation in static fields.
In microwave fields, the time-dependent magnetic field in the
penetration depth will generate an electric field.
The electric field will induce oscillations in the normal
electrons, which will lead to power dissipation
BE
t
¶Ñ´ = -
¶
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Page 78
Surface Impedance in the Two-Fluid Model
In a superconductor, a time-dependent current will be carried
by the Copper pairs (superfluid component) and by the
unpaired electrons (normal component)
0
2
0 0
0
2
2
0
(
2( )
2 1
Ohm's law for normal electrons)
with
n s
i t
n n
i t i tcs e c
e
i t
cn s s
e L
J J J
J E e
n eJ i E e m v eE e
m
J E e
n ei
m
w
w w
w
s
w
s
s s s sw m l w
-
- -
-
= +
=
= = -
=
= + = =
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Page 79
Surface Impedance in the Two-Fluid Model
For normal conductors 1sR
sd=
For superconductors
( ) 2 2 2
1 1 1n ns
L n s L n s L s
Ri
s s
l s s l s s l s
é ù= Â =ê ú
+ +ë û
The superconducting state surface resistance is proportional to the
normal state conductivity
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Page 80
Surface Impedance in the Two-Fluid Model
2
2
2
0
3 2
1
( ) 1exp
( )exp
ns
L s
nn s
e F L
s L
R
n e l Tl
m v kT
TR l
kT
s
l s
s sm l w
l w
Dé ù= µ - =ê ú
ë û
Dé ùµ -ê ú
ë û
This assumes that the mean free path is much larger than the
coherence length
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Page 81
Surface Impedance in the Two-Fluid Model
For niobium we need to replace the London penetration depth with
1 /L ll xL = +
As a result, the surface resistance shows a minimum when
lx »
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Page 82
Surface Resistance of Niobium
Surface Resistance - Nb - 1500 MHz
1
10
100
1000
10000
10 100 1000 10000
Mean Free Path (Angstrom)
nohm
2.0 K
4.0 K
4.2 K
3.5 K
3.0 K
2.5 K
1.8 K
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Page 83
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 it
eH A r t p
mc
H H H
R R A I R T eJ dr
R
ff f
w-
¶+ =
¶
=
<<
×µ
å
ò( ) ( ) ( )
( )4
0 0 :
d's model
Meissner effect
cJ k K k A k
K
p= -
¹
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Page 84
Surface Resistance of Superconductors
( )( )
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
kT
Frequency dependence
is a good approximation
c
c
s
T
tt
t
TT
AR
T
l
w
w
D
-
-
- <
Dæ ö-ç ÷è ø
( )2
59 10 exp 1.83
A reasonable formula for the BCS surface resistance of niobium is
GHzc
BCS
f TR
T T
- æ ö= ´ -ç ÷è ø
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Page 85
Surface Resistance of Superconductors
• The surface resistance of superconductors depends on
the frequency, the temperature, and a few material
parameters
– Transition temperature
– Energy gap
– Coherence length
– Penetration depth
– Mean free path
• A good approximation for T<Tc/2 and ω<<Δ/h is
2 exps res
AR R
T kTw
Dæ ö- +ç ÷è ø
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Page 86
Surface Resistance of Superconductors
2 exps res
AR R
T kTw
Dæ ö- +ç ÷è ø
In the dirty limit
In the clean limit
Rres:
Residual surface resistance
No clear temperature dependence
No clear frequency dependence
Depends on trapped flux, impurities, grain boundaries, …
1/2
0 BCSl R lx -µ
0 BCSl R lx µ
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Page 87
Surface Resistance of Superconductors
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Page 88
Surface Resistance of Niobium
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Page 89
Surface Resistance of Niobium
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Page 90
Super and Normal Conductors
• Normal Conductors
– Skin depth proportional to ω-1/2
– Surface resistance proportional to ω1/2 → 2/3
– Surface resistance independent of temperature (at low T)
– For Cu at 300K and 1 GHz, Rs=8.3 mΩ
• Superconductors
– Penetration depth independent of ω
– Surface resistance proportional to ω2
– Surface resistance strongly dependent of temperature
– For Nb at 2 K and 1 GHz, Rs≈7 nΩ
However: do not forget Carnot