COMPACT SUPERCONDUCTING CAVITIES FOR DEFLECTING AND ... · SRF FUNDAMENTALS USPAS @ Rutgers June 2015 . Page 2 Historical Overview . Page 3 Perfect Conductivity Kamerlingh Onnes and

Page 1

Jean Delayen

Center for Accelerator Science

Old Dominion University

and

Thomas Jefferson National Accelerator Facility

SRF FUNDAMENTALS

USPAS @ Rutgers June 2015

Page 2

Historical Overview

Page 3

Perfect Conductivity

Kamerlingh Onnes and van der Waals

in Leiden with the helium 'liquefactor'

(1908)

Page 4

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

Page 5

Perfect Diamagnetism (Meissner & Ochsenfeld 1933)

0B

t

¶=

¶0B =

Perfect conductor Superconductor

Page 6

Penetration Depth in Thin Films

Very thin films

Very thick films

Page 7

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

Page 8

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 =

Page 9

Thermodynamic Properties

Entropy Specific Heat

Energy Free Energy

Page 10


( ) 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

=

»

Page 11


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

-

= » »

Page 12

( )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

Page 13

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-

Page 14

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

Page 15

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

Page 16

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

Page 17

Material Parameters for Some Superconductors

Page 18

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

Page 19

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 =

Page 20

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

Page 21

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

¶= -

¶

= -

¶=

¶

=

Page 22


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´ +

Page 23


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)

Page 24

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


Page 25


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 = =

= -

Page 26

Penetration Depth in Thin Films

Very thin films

Very thick films

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

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:

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

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´ = -

¶

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

Page 32


• 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

Page 33


• 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

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

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¥ = -

Page 36


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= -

Page 37


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

=

= - =

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

-»

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

®

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=

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

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

Page 43

Magnetization Curves

Page 44

Intermediate State

Vortex lines in

Pb.98In.02 At the center of each vortex is a

normal region of flux h/2e

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.

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

Page 47

Material Parameters for Some Superconductors

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

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

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.

Page 51

BCS

The size of the Cooper pairs is much larger than their spacing

They form a coherent state

Page 52

BCS and BEC

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- £ £

=

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

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

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

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 =

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:

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

Page 60

BCS Excited States

0

2 22k

Energy of excited states:

ke x= + D

Page 61

BCS Specific Heat

10

Specific heat

exp for < ces

TC T

kT

Dæ ö-ç ÷è ø

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

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-

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

Page 65

Surface Resistance

Page 66

Surface Resistance

Page 67

Surface Resistance

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

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

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

= ´

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 =

¶ -

= ´

=

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

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

Page 74


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

Page 75


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

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


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´ = -

¶

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

-

- -

-

= +

=

= = -

=

= + = =

Page 79


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

Page 80


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

Page 81


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 »

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

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

Page 84


( )( )

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

Page 85


• 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

Page 86


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

Page 87


Page 88


Page 89


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

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