basic principles of rf superconductivity sergio calatroni 21 july, 2011srf2011 tutorials1

SRF2011 Tutorials 1

BASIC PRINCIPLES OF RF SUPERCONDUCTIVITY

Sergio Calatroni

21 July, 2011

SRF2011 Tutorials 2

Introduction

• Surface impedance– Definition– Normal metals– Superconductors

• SC cavities for accelerators• Quality factor• Penetration depth

• Introduction to residual resistance• Thermal feedback• Trapped flux

21 July, 2011

SRF2011 Tutorials 321 July, 2011

Suggested readings• Review paper:

– J.P. Turneaure, J. Halbritter and H.A. Schwettman, J. Supercond. 4 (1991) 341

• Details:– J. Halbritter, Z. Physik 238 (1970) 466– J. Halbritter, Z. Physik 243 (1971) 201

• Books:– RF Superconductivity for Particle Accelerators by H. Padamsee, J. Knobloch

and T. Hays (Wiley, New York, 1998)– HTS Thin Films at Microwave Frequencies by M. Hein (Springer, Berlin,

1999)

• CERN Accelerator Schools: Superconductivity in Particle Accelerators “Yellow reports” n. 89-04 and 96-03

• http://w4.lns.cornell.edu/public/CESR/SRF/SRFHome.html and links therein

• Wikipedia http://en.wikipedia.org/wiki/Superconducting_Radio_Frequency !!!

http://w4.lns.cornell.edu/public/CESR/SRF/SRFHome.html

http://en.wikipedia.org/wiki/Superconducting_Radio_Frequency

SRF2011 Tutorials 4

Surface Impedance: definition

Where we use: E, H always in complex notation

E, H parallel to the surface

Our reference frame has x, y parallel to the surface of the metal, and the z axis directed towards its interior: z=0 means at the surface.

21 July, 2011

)0(

)0(

||

||

H

EiXRZ SSS

For a plane E-M wave incident on a semi-infinite flat metallic surface:

E

H

z=0

(super)conductorvacuum

SRF2011 Tutorials 5

Plane waves in vacuumWhy is it defined like this?

Plane wave solution of Maxwell’s equations in vacuum:

21 July, 2011

)(

0

00

)(0 ee tkzitkzi

μ

εEHEE

The ratio is often called impedance of the free

space and the above equations are valid in a continuous medium

7.3760

0

H

EZ

)(0

)(0 ee tkzitkzi HHEE

Where (in vacuum):0

000

2

c

k

So that:

)(

00 e tkzik

EH

and:

SRF2011 Tutorials 6

Plane waves in normal metalsMore generally, in metals:

21 July, 2011

With is the damping coefficient of the wave inside a

metal, and is also called the field penetration depth.

2

1

This results from taking the full Maxwell’s equations, plus a supplementary equation which relates locally current density and field:

ik 22

In simple cases ( << 1) is assumed frequency-independent.

In metals and the wave equations become:

z)(0

)(0 eee tzitkzi EEE

),(),( txtx

EJ eFe m

ne

vm

ne 22

0

i

1

)( 0

kZ

H

E

ik 22

SRF2011 Tutorials 7

Surface impedance in normal metals

• The Surface Impedance Zs is defined at the interface between two media. It can be calculated in a similar way as for continuous media.

• You take Maxwell’s equation, set the appropriate boundary conditions for the continuity of the waves (incident, reflected, transmitted), and you get:

• Introducing appropriate numbers:

21 July, 2011

For copper (=1/=1.75x10-8 µ.cm) at 350 MHz

Rs = 5 m

= 3.5 µm (from previous page)

SSS iXRii

H

EZ

22)0(

)0( 000

||

||


Surface Impedance – Why bother? I

The surface impedance Zs=Rs+iXs is a very useful concept, since it contains all properties of the dissipative medium. Making use of Maxwell’s equations for plane waves:

20

2000 H

2

1Re

2

1Re

2

1ss RHZP HE

• Power dissipation per unit surface (equivalent to the energy flux through the unit surface) making use of Poynting vector :

• Averaging over one period, at the surface, recalling the definition of Zs

HES Re

SRF2011 Tutorials 9

Surface Impedance – Why bother? II

21 July, 2011

• Effective magnetic penetration depth (most general definition):

)0(d)(Re0

HzzH

Using Maxwell’s eq.:

integrating and using the fact that:

Substituting:

Remarkably similar to the expression for =1/ found previously in the case of normal metals

Hiz

E

t 0

H

E 0

0)( E

)0(d)(d)()0()(000

0 Ei

zzHzzHiEE

sXH

Ei

00

1

)0(

)0(Re


1

20

sR

Surface resistance: intuitive meaning

Since we will deal a lot with the surface resistance Rs in the following, here is a simple DC model that gives a rough idea of what it means:

Consider a square sheet of metal and calculate its resistance to a transverse current flow:

The surface resistance Rs is the resistance that a square piece of conductor opposes to the flow of the currents induced by the RF wave, within a layer

current a d

a R

d

a

a

d


More on electrical conductivity

We mentioned before the electrical conductivity and the law J=E.

• depends on the frequency:

• This is valid in only when: 0 . Called “Classical” or “local” limit.

• For low temperatures increases and the limit breaks (as function of frequency), in particular it happens

• In this case, an electron senses a variation of field during its travel between two collisions.

• The local relationship J=E between current and field does not hold anymore and a new “non-local” law has to be introduced.

)1()( 0

22

0

imv

ne

m

ne

F


Non-local current-fields relationship

• The relation between currents and fields can be expressed in terms of a non-local formula (Chambers’ formula):

Understood by Pippard, Proc. Roy. Soc. A191 (1947) 370Exact calculations Reuter, Sondheimer, Proc. Roy. Soc. A195 (1948) 336

rrRv

rERRrJ

'R

evRt't

V

RF 'd

),(

4

3),(

4

Normal skin effect

Anomalous skin effect

Asymptotic value

Skin depth

Mean free path


“Ineffectiveness”

• “Anomalous skin effect”• When cooling down a metal increases because increases. • At the same time, the penetration depth decreases.• At some moment > and the rules of the game change.• Some electrons exit the region where the current flows without acquiring

a transport velocity. Obviously they don’t contribute to the current and to the shielding of the field

.2

const

Feeff

Fe vm

ne

vm

ne 22

0

0nneff

SRF2011 Tutorials 14

Superconductivity

21 July, 2011


Meissner effect

21 July, 2011

http://upload.wikimedia.org/wikipedia/commons/b/b5/EfektMeisnera.svg


Theories of Superconductivity

• Gorter & Casimir two fluid model– London Equations– Pippard’s Coherence length

• Ginzburg-Landau– Second order phase transition of complex order

parameter

• BCS (Bardeen-Cooper-Schrieffer)– Microscopic theory– Two Fluid Model revised

• (Strong coupling – Elihasberg)

21 July, 2011


London equations

21 July, 2011

Postulated on plausibility arguments

Applying B = A

Applying Ampere’s lawto London’s 2nd eq gives:

AAJ 12

m

ens

Exponential decay of B inside SC

EJ

m

en

ts

2

BJm

ens2

20

22 1

en

m

sL

L

BB


Superconductors – Complex conductivity

• Finally: superconductors!

• You will probably be only half surprised to hear that a superconductor at T>0 has a Rs0.

• This can be easily understood in the framework of the two fluids model, where a population of normal electrons of density nn and a population of “superconducting electrons” of density ns=n0(1-T4/Tc

4) coexist such as nn+ns=1 and both give a response to the varying e.m. fields.

• Let’s “invent” the conductivity of superconducting electrons:

ee m

nei

im

ne 22

)(lim)1(

)(


Superconductors – Complex conductivity

• The conductivity of superconductors then becomes:

• Where ns=n0(1-T4/Tc4) and nn=n0(T4/Tc

4) with n0 total electron density

• Note:

214

42

04

42

0 )1()(

iT

Tm

eni

TT

m

encc

s

conductivity of “normal electrons” conductivity of

“superconducting electrons”

112

0

02

L

with0

2

L


Do you think the previous formula looks like a joke (conductivity of a perfect conductor is an imaginary number)?

There is indeed real physics behind it. Recall London equations, from which we can set-up the following equivalence, in the “dirty” limit:

The value of is the expression of the kinetic inductivity of the “superconducting electrons”: basically, the electron pairs have a mass, and they get accelerated by the RF wave. This is the same as what happens with the magnetic field produced inside an inductor.

London&London Proc. Roy. Soc A149 (1935) 71

Pippard Proc. Roy. Soc. A216 (1953) 547

Superconductors – London formula

Normal metal Superconductor (London theory)

EEJF

n

mv

enσ

2

0 AAJm

ens2

1


Superconductors – Equivalent circuit

Indeed if you take the time derivative ofyou get the 1st London eq:

is interpreted as a specific inductance. This justifies representing the complex conductivity of a superconductor with an equivalent circuit of parallel conductors:

tt

J

EA

)1()( 4

4

4

4

21c

n

cns T

TiT

Ti

AJ 1


Superconductors – Surface impedance I

• It is now possible, within the basic approximations we have made, to calculate the surface impedance of a superconductor.

• Take the formula for normal metals:

• Perform the substitution: n s =1-i2

• Calculate:

• In the approximation of small (small 1<2) • and 0<T<0.5Tc (nn<ns 1<2)it gives:

• Which is a good description of the experimental data

n

iZ

2)1( 0

...... 2

122

21

21

22

122

21

0

sn

s XR

Ln

NsLn

nNs XX

RR

02

021

2

31

22023

2

1 2

2

1

2


Superconductors – Surface impedance II

• How to generalise to larger ? Take: (Pippard Proc. Roy. Soc. A216 (1953) 547)

• Introduce a parameter in some way proportional to :

• Then substitute corresponding quantities into Chambers’ formula:

• Pippard used this formula to fit his experimental data, and found:

• This was the first definition ever of the coherence length 0

AAJm

ne21

AJ 1

0

'd

),(

4

3),(

40

vrARR

rJ

V

RF

R

evRt't

c

F

kT

v

15.0111

00


Superconductors – Pippard’s data

We are now ready to go into the real (“heavy”) stuff: the BCS formulation.

• In BCS theory the values for 1/n and 2/n can be calculated.• These are in turn used to calculate the surface impedance


Superconductors – BCS approach IMattis & Bardeen, Phys. Rev. 111 (1958) 412

1st step:

2nd step:

3rd step:


Superconductors – BCS approach II

• These can be approximated for low frequencies h«2 as:

• The surface impedance Z can be calculated from 1/n and 2/n• However the formulas seen until now are approximate valid only in a

very specific limit.• Calculations can be done having a full validity range (but only for

HRF<<Hc)

– Abrikosov Gorkov Khalatnikov, JETP 35 (1959) 182– Miller, Phys. Rev. 113 (1959) 1209– Nam, Phys. Rev. A 156 (1967) 470– Halbritter, Z. Phys. 238 (1970) 466 and KFK Ext. Bericht 3/70-6

kT

fkT

fnn 2

tanh )(12

ln1

1)(2 21


Superconductors – Halbritter’s theoretical predictions

In the green region (small ):

kTTRs

exp2

In the red region (large ):

kTTRs

exp23

21

21

nnn

s iZ

Z

31

21

nnn

s iZ

Z


Superconductors – CERN results on films

1 10

350

400

450

500

550

600

650

700

750

800

850

900

2 5 20

R0 B

CS [

n ]

1+0 /2l

Compilation of results from several Nb/Cu and Nb bulk 1.5 GHz RF cavities

Bulk Nb

Theoretical curve

n!It was m for copper!

Nb/Cu films


Temperature dependence of RBCS

Exponential dependence limited by a temperature independent “residual” term


Other materials than niobium

BCS surface resistance at 4.2 K and 500 MHz for films (Nb ~ 45 n)

kTTRs

exp2


HOW TO ACCELERATE A PARTICLE?

21 July, 2011


Metamorphosis of a RLC resonator

“Pillbox cavity” 0 = 2.405 c /(2**radius)LC

2

10


Cavities are resonators• An accelerating RF cavity is indeed a resonator for an electromagnetic

wave. We already mentioned the idea of comparing an RF cavity to an RLC circuit.

• Two important quantities characterise a resonator (remember your first physics course..): the resonance frequency f0 and the quality factor Q0

cP

U

f

fQ 00

0

• Where U is the energy stored in the cavity volume and Pc/0 is the energy lost per RF period by the induced surface currents

)exp(0 LtUU


• Let’s expand on the latter definition:

• Thus:

• depends only on the shape of the cavity, and is generally calculated numerically with computer codes (Superfish, HFSS, -wave studio...)

• For a “pillbox” cavity is 257 . For an accelerating cavity for relativistic particles (=1) like those used at LHC is in 270 .

• The Q0 factor of a SC cavity is in the 109÷1010 range!

• Leff is the effective inductance of the equivalent RLC circuit

Quality factor and surface resistance

cP

UQ 0

0

sRPvUS

sc

V

d2

1 d

2

1 22

0 HH

s

eff

s

S

V

s R

L

Rv

v

RQ 0

2

2

000

d

d

H

H


Performance of cavities I

Standard + BCP

Large grain + EPDESY TTC-Report 2011-01

sRQ

0


Thin films

• Copper cavities coated with a thin niobium film had been chosen at CERN because of several potential advantages:

– Thermal stability: the thermal conductivity of pure OFE copper at low temperature is at least one order of magnitude better than that of the purest niobium (a superconductor is a poor thermal conductor) (more on this later)

– Cost: ultrapure Nb costs more than 500 EUR per kg– Performance: niobium films at 4.2 K have a two times smaller RBCS

than pure niobium (this no longer applies at 1.7 K, more on this later)– Insensitive to external magnetic fields: savings on shielding

• 288 cavities were used in LEP providing 2.7 GV accelerating voltage, 16 in the LHC

• De-coupling the mechanical and thermal functions of the substrate from the superconducting properties of the film leaves the option of coating other innovative materials such as NbTiN, V3Si, MgB2...


Thin films – Sputtering (1.5 GHz cavities)

“Standard” niobium coating:• Discharge current stabilized at 3 A.• Sputter gas pressure of 1.5x10-3 mbar,

corresponding to ~ 360 V.• Coating temperature is 150 °C.

“Standard” film characteristics:• Thickness: 1.5 µm• Grain size: 200 ± 80 nm• RRR: 11.5 ± 0.5• Argon content: 400 ppm

• The technique can be easily extended (by introducing also nitrogen in the sputter gas) to make “reactive sputtering” to produce coatings of materials such as NbTiN


Performance of cavities II

109

1010

1011

0 5 10 15 20 25

Q0 (1

.7 K

)

Eacc

[MV/m]

Nb/Cu cavities 1.5 GHz at 1.7 K (Q0=295/Rs)

Q slope


Again on skin depth

21 July, 2011

• London penetration depth

– dependence on mean free path

– dependence on T (T>Tc/2)

– define

• London limit >>1 (mfp “small”)

• Pippard limit <<1 (mfp “large”)– (non-local effects, non-exponential decay of fields, field

reversal for 1)

21)0,(),( 0 TT LL

4

11),0(),(

cLL T

TT

111

2)0,0(),(/),(),(

00

FFL TTT

),(),( TT LSC

3/1),(),(),( TfTT LsLSC


Surface impedance and skin depth

• We are ready to meet again our friend Xs :

• Where is the resonant frequency if the cavity walls were perfectly conducting, in which case the penetration depth would vanish.

• This formula (Slater’s theorem) in its differential form is particularly useful to measure the penetration depth

• A non-zero penetration depth results in a larger resonating volume, thus a lower resonating frequency

• Changes in (T) with the temperature result in changes of 0,

• Recall the two-fluids (London limit) formula for (T) :

20sX

4

0

1

1

21)0,0(),(

c

LL

T

TT


Surface impedance and skin depth

• Expressing the frequency shift as a function of (T) we can write:

• Non-linear-fitting it to experimental data (measuring frequency as a function of temperature, approaching Tc) one can calculate the mean free path assuming reasonable values for the material (Nb) parameters:

)()(

)(0

20 T

Tt

L 36 nm

0 40 nm

* 0.28 x 10-15 m2

phononic(300K) 14.5 x 10-8 m


1.495 GHz cavity E8.7• Our most studied “thin film” cavity

• Thanks to Alessandro d’Elia for the recent re-fitting

21 July, 2011

(0 K) 7.51 nm

19nm

imp 1.47 x 10-8 Ohm m

RRR (14.5x10-8+ 1.47x10-8 )/1.47x10-8 = 10.8

Sergio Calatroni - 4.5.2011

Film analysesStandard films Oxide-free films

0.5 µm 0.5 µm

FIB cross sections courtesy: P. Jacob - EMPA

EuCARD SRF Annual Review

43

Standard Oxide-freeRRR 11.5 ± 0.1 28.9 ± 0.9

TC 9.51 ± 0.01 K 9.36 ± 0.04 K

Ar cont. 435 ± 70 ppm 286 ± 43 ppm

Texture -110 (110), (211), (200)

Hc1 85 ± 3 mT 31 ± 5 mT

Hc2 1.150 ± 0.1 T 0.73 ± 0.05 T

a0 3.3240(10)Å 3.3184(6) Å

Da^/a^ 0.636 ± 0.096 % 0.466 ± 0.093 %

Stress -706 ± 56 MPa -565 ± 78 MPa

Grain size 110 ± 20 nm > 1 µm


Deviations from BCS• The first notable fact is that the surface resistance Rs is not constant

with field. (The Mattis-Bardeen equations are a zero-field perturbation theory, and this should not come as a surprise)

• Second, even when lowering temperature the surface resistance does not reduce as foreseen. This is the residual surface resistance Rres

• Third, other phenomena take place at high fields: electron field emission, thermal breakdown (quench)

• Fourth, in the case of films, there is a stronger decrease of Q0 factor (increase of Rs) with field compared to bulk cavities

• In fact the surface resistance is generally:

Rs(T, Erf , HDC)=RBCS(T, Erf , 0)+Rres(0, Erf , 0)+Rfl(T, Erf , HDC)


BCS surface resistance – Field dependence

• The field dependence of the BCS surface resistance can be due to several factors:– Intrinsic: with a higher-order perturbation expansion of the

BCS formulas one could describe some field dependence– Magnetic: variation of the gap parameter with the Hrf field

– Thermal: higher fields mean higher dissipated power thus a larger T gradient across the cavity walls (between the liquid helium bath and the cavity surface).

• Thermal feedback model is the most studied and better understood (Ciovati + Halbritter Physica C 441 (2006) 57–61)

• Typical form of this dependence is:


Thermal feedback

21 July, 2011

2 K 1.37 K

Linear term and quadratic term are both present depending on conditions


Thermal feedback films vs bulk

100

1000

0 10 20 30 40 50

RB

CS (

4.2

K)

[n

]

HRF [mT]

Surface resistance of 1.5 GHz Nb bulk and Nb/Cu cavities at 4.2 KRBCS(T, Erf )=RBCS(T, 0)(1 + E2) =0.0084 for both

See also S.C. SRF2001 (Tsukuba) proceedings


Causes of the residual resistance

• Typical values of residual resistance are of the order of 10÷20 n, but in several experiments values as low as 1÷2 n have been reached, and no intrinsic limitation should exist.

• Many sources of residual surface resistance have been identified over the years and when possible cured. Several act at any field range, and a few have an influence only at high field. Among these are:

– Surface defects: inclusions, dust, roughness, etc.– Hydrogen– Oxides– Grain boundaries

• Which will be mostly addressed in other Tutorials. Here:

– Trapped magnetic flux


Type I and II• Two types of superconductors defined by G-L

• <1/2 and >1/2

21 July, 2011

)(/)( TT

MHB

(Goodman relation)


Flux quanta• Flux penetrates type II superconductors in the form of flux quanta, fluxons

• Due to demagnetization effects, defects, inhomogeneites, etc., when cooling a cavity in presence of an external magnetic field (even low), flux quanta remain trapped in the surface (no flux line lattice)

• The flux quanta dissipate RF energy, either through the higher surface resistance of the normal core, or through oscillation of the fluxons

• Most models result in a dependence of the type:

21 July, 2011

e

h

20

nc

extfl R

H

HR

2

0


Trapped magnetic flux

21 July, 2011

Weingarten, Arnolds-Mayer

• No magnetic shielding needed for Nb-film cavities


Fluxon-induced losses

1 10

10

100

Rfl0 [

n/G

]

1+0 /2l

1 10

1

10

Rfl1 [

n/G

/mT

]1+

0 /2l

Bulk Nb

Nb/Cu best films

Bulk Nb

Nb/Cu best films

• A parametrisation of the form Rfl (1.7 K) = (Rfl0 + Rfl

1 HRF) Hext fits very well the data independently of the values of HRF and Hext

• Simple picture is clearly not sufficient as can be seen from the experimental data, which follow the BCS pattern as a function of mean free path!


THANK YOU!

... Questions?

21 July, 2011


Some useful definitions

21 July, 2011


Maxwell’s equations• Gauss’ Law

• Faraday's law of induction

• Gauss Law for magnetism

• Ampère's circuital law

• With:

• Continuity equation

21 July, 2011

t

DJH

B

t

BE

D

0

0

tJ

HBED


Why use superconducting cavities?

• The reason is very simple: to obtain very high accelerating fields in continuous mode, the power dissipated by normal conducting (i.e copper) cavities becomes too large. Remember that there can be a 105 difference in Rs, thus in dissipated power, for the same surface fields.

• This is valid also taking into account the efficiency of the cryogenic plant. The Carnot efficiency of a perfect cooler working between 300 K and T is:

• For T=4.2 K, c=0.014. A modern cryoplant has a technical efficiency that can reach about 30% of the ideal one, thus real=0.0042. Even taking this into account the energy savings are huge.

• Moreover, SC cavities allow designs with larger beam pipes which, are beneficial for beam stability although reducing Ra. This would be unacceptable for NC cavities, but owing to the huge Q0 the performance of SC cavities don’t suffer much.

• LEP case: Cu hypothesis 3MW/m @ 5 MV/m accelerating fieldNb case: 30 W/m @ 5 MV/m accelerating field

T

Tc

300


SC vs NC• A brief summary for comparing SC cavities (Nb @ 4.2 K)

and NC cavities (Cu at 300 K), both at 500 MHz

Superconducting Normal conducting

Q0 2x109 2x104

Ra/Q0 [/m] 330 900

Dissipated power [kW/m](for Eacc=1MV/m)

0.0015 56

Total AC power [kW/m](for Eacc=1MV/m)

0.36 112


Grain boundaries• Grain boundaries in polycrystalline materials behave as Josephson

Junctions. Under RF currents, there might be and added dissipation when cooper pairs tunnel through the junction

• Even in “simple” materials like Nb the grain boundaries may play a role, in particular at low temperature, due to the very small contribution from the BCS surface resistance

22

1

2

1

sR

Standard BCS dirty limit

22

1

2

1

r

effsjR

BCS dirty limit plus grain boundaries contribution


Effect of grain boundaries on NbTiN films at 500 MHz

1086420109

1010

1011

Best Nb/Cu(Nb0.55Ti0.45)N/Cu

E[MV/m]

Q0

X

basic principles of rf superconductivity sergio calatroni 21 july, 2011srf2011 tutorials1

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