jay amrit kapitza resistance at niobiumsuperfluid he interfaces

Kapitza Thermal Boundary Resistance

Kapitza Thermal Boundary Resistance at Niobium/Superfluid He interfaces in SRF cavities

Jay AMRIT

LIMSI-CNRS , Paris-Sud University, [email protected]

In collaboration with

Claire ANTOINE(CEA, Saclay)

Thin Films and New Ideas for RF Superconductivity, Padova, 2014

Part I : Introduction on KTBR

Why is it important?/Model predictions

Part II : Experiments with Nb: bulk purity & surface statePoly-crystals

Single crystals

Comparison and impact on SRF cavities

Part III : New analysis & ongoing work

Importance of nanoscale surface roughness

Kapitza resistance at grain boundaries in Nb

Pressure dependency of KTBR : What would happen to

cavity performance if we increased the pressure to 25 bars?

Summary & possible future studies…

Outline


Introduction: Discovery of thermal boundary resistance

Pyotr L. KAPITZA (1894-1984)

-prix Nobel 1978-

Discovered in 1941 by Kapitza

Cooling of Solids with Superfluid Helium

Superfluidity He

• Discovered 1938

•Temperatures < 2 K (-271°C)

•Quasi-infinite thermal conductivityCopper

Superfluid4He

Q

Thermal boundary resistance = Kapitza resistance

Impossible to reach zero absolute temperature by direct cooling

Fountain effect


Introduction: Fundamental interest in Kapitza resistance

TBR is an important phenomenon at low temperatures

310S

K

T

T

810L

K

T

T

Typical temperature gradient with temperature jump over atomic distances

1 mm1 mm

TL

TS

Superfluid HeSolid (Cu)

TK

x

Kcu ~1 W/(mK) KHe ~ 800 x KCu

LK TT 810

SK TT 310

For Nb :NbK TT )1000100(

Kapitza length KK RKL

kmL HeIIK 8,

cmL CuK 10,

Te

mp

era

ture


Introduction: Kapitza resistance in SRF cavities

Electrical surface resistance :

2/T

0RF

acc

RF

dt)t,z(ET

2)m/MV(EAccelerating field :

Quality factor QO :

dVE2

1U v

2o (stored energy)

Power dissipatedEnergy stored per sec

q

UωQo

(3-4 nano ohms)

Heat dissipation in inner walls : //B penetrates ~50 nm into walls

Joule effect

residualc2

s R)T/T76.1exp()T/(AR


e

He IINb

Tin

THe

KapitzaT

q

B

TinTbath

ToutRK)T(K

e

q

Ks

oacc

RK

eR

TE

1

8

2/12

Cavités ellipsoïdales : ]/[][ 4 mMVaccmT EB

2//

2

1BRq s

o

Power dissipation :

qRK

eTTT KHein

Temperature jump :

Kapitza resistance

Introduction: simple thermal model

Accelerating field dependence on K and RK

A strong RK limits the Eacc !


Heat transport in superfluid Helium

•Interaction potential between 2 atoms determines

heat transport in He

•Energy is rapidly distributed between atoms

•Only longitudinal (acoustic) phonons transport heat

•Other excitations : rotons, maxons, vortices

240Lc m.s-1

= cLk

Dispersion relation of Helium


Heat transport in superfluid Helium

240Lc m.s-1

= cLk

Dispersion relation of Helium

•Unique characteristics of heat and mass transfer

•Mixture of two fluids (& not two phases) :

•Momentum density :

Two fluid model of He II (Tisza, 1938)

Normal component : n n ns

Superfluid component : s s 0ss

sn

q

Two different sounds

First sound (240 m/s) pressure wave & both fluids move in phase

Second sound (~20m/s): temperature wave & fluids move in opposite directions

nnssJ

ssnn 0

Normal component = source & Superfluid component = sink

ns


Heat transport in Niobium in a nutshell

Lattice bcc, a =3.29 A

Atoms oscillate around their equilibrium positions, producing vibrational waves

Acoustic modes ( = vk)

•3 branches : longitudinal & 2 transverse

•Each branch has N modes

•Mode = (, k)=quantum of acoustic vibration (phonon)

111

100

L

L

T

T

J. Phys. C 2, 421 (1969)

Longitudinal Transverse


Thermal boundary resistance: how does it come about at theNb/He II interface?

The key is to determine

the transmission of

phonons

Very small overlap

in wave vectors

Dispersion relation of Nb and He

0

1

2

3

4

5

6

0 0,5 1 1,5 2 2,5

Nb dispersion relation

freq (THz)Freq (Thz)Freq (THz)

q (A-1

)

Nb (111)

L

T

He II


Number of phonons of wave vector k incident at an angle q per unit time :

Bose-Einstein distribution

Heat energy transmitted :

Energy incident per branch and for a given k :

Thermal resistance

q

cosc

4

ddk)k(g)T,(nN b,1b,1

1Tk1e)T,(n B

3b

2

3

3

c

d4

)L/2(

kddk)k(g

N° of modes with wavevector k for a given branch

b,1N

321 T

ARK

Thermal boundary resistance: formal approach

bb

K

dcdT

Tng

T

q

R

q

qqq

0

2/

00 sincos2

),()(

4

11

Specific heat

21121

4

11

D

K

CT

q

R

b

b dcdgTnQQq

q

qqq0

2/

001221 sincos)(),(

2

1

Average transmission

(phonons /)


Kapitza resistance model : Acoustic mismatch model (AMM) (Khalatnikov, 1952)

•Parallel component of momentum

•Frequency

Conservation laws at the interface:

//S

//L pp

SL

Snell’s lawS

S

L

L

c

sin

c

sin qq

S

L

SL

SLAM

Z

Z

ZZ

ZZ 44)(

2

q

Transmission coefficient is due to discontinuity in sound velocity & density

,LLL cZ

,SSS cZ

Lq

LL kp

SS kp

jcoskisinkk SSSSS

qq

jkikk LLLLL

qq cossin

Sq

240Lc m.s-1

53cLq

Critical cone in superfluid limits transmission of phonons !!

5068, LNbc m.s-1 m.s-12092, TNbc002.04

)(20

qq d

c

AMAM

Nb

He


Kapitza resistance models : diffuse mismatch model (DMM)

“Trick” to increase phonon transmission in model

Phonons loose “memory” of former states

No physical restrictions on scattering mechanism !!

NO dependency on interface properties

)(t1

)(t2 )(r2

)(r1

1)(r)(t 11

1)(r)(t 22

2211 )()( qtqt

5.0~q

q1)(t

1

2

11

)(t)(r 21

He II

Nb

DMM phonon transmission is increased by two orders of magnitudeHow the diffuse scattering comes about is not explained in this model!

(at equilibrium)

Swartz & Pohl 1989


0

1

2

3

4

5

40

60

80

100

120

1,5 1,6 1,7 1,8 1,9 2 2,1

Khalatnikov & Dm for NbHeII

RKhalat (cm2K/W)Rdm (cm2K/W)

T (K)

AM model

DM model

Theoretical Model predictions of Kapitza resistance for Nb/HeII

Depends on properties of liquid He

Depends on properties of Nb

ExperimentsNb/HeII

The Kapitza resistance at a solid/ superfluid He II interface is anomalous !


Experimental Cell

Nb disc sampleHeater

Carbon Thermometers

Q

superfluid

To

T1 4He Filling line

Temperature profile across a sample

K : thermal conductivitye : sample thickness

K

e

Q

)TT(SR

.K2

01

To

T1

RKQTo

RK

T

Experiments : cell configuration & method


Polycrystalline Niobium: 4 different bulk & surface treatments

100 µm

chemically etched

250 µm

annealed and chemically

etched

Light Heavy

250 µm

electropolished

50 µm

annealed, chemically

etched & electropolished

Micrographics of surfaces

1

2

3

4

RRR 178

RRR 178

RRR 647

RRR 647

4.08.1σ

25.085.0σ

4.03.1σ

1.02.0σ

)mμ(σK2.4

K300

ρ

ρRRR

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

RK

(cm

2K

.W-1

)

Temperature (K)

1

3

4

2

Polycrystalline Niobium : RK results

Sample, RRR Surface Treatment s (µm)# RK (cm2K4W-1)

#1, 178 CP(~30 µm) 1.8 + 0.4 10.7T-3.55

#2, 178 EP 0.85 + 0.25 21.3T-4.11

#3, 647 Annealed +CP 1.3 + 0.4 16.1T-3.93

#4, 647▲ Annealed +EP 0.2 + 0.1 19.1T-3.61

Bulk purity :

Change in K by a factor of 5 (annealing)

RK changes by ~15% only

(1 & 3)

Surface Roughness :

Smaller surface roughness s leads to

higher RK (independent of bulk purity)


Polycrystalline Niobium : RK compared to total thermal resistance

20

30

40

50

60

70

80

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

T(K)

4

3

2

1

K

eR

R

K

K

RRR 647

annealed

RRR 178

•RK constitutes ~70% of total thermal resistance

•Higher bulk purity and lower temperatures lead to a stronger impact of RK

•RK is the key parameter for cooling cavities


Polycrystalline Niobium : Which samples are best?

20

30

40

50

60

70

80

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

T(K)

4

3

2

1

K

eR

R

K

K

RRR 647

annealed

RRR 178

30

40

50

60

70

80

90

1,5 1,6 1,7 1,8 1,9 2 2,1

T(K)

3

4

1

2

2/1

1

K

eR

E

K

acc

Annealed and CP is best !


•RK constitutes ~70% of total thermal resistance

•Higher bulk purity and lower temperatures lead to a stronger impact of RK

•RK is the key parameter for cooling cavities

µm

-14

-13

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

µm

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

1

1,5

2

2,5

3

3,5

4

4,5

1,5 1,6 1,7 1,8 1,9 2 2,1

Temperature (K)

Damaged

Layer

Chemically

polished

Single Crystal (111) Niobium : Impact of surface state on RK

Niobium ingot ( = 5 cm) , RRR = 300

Crystallographic orientation (111) – EBSD technique

2 sample surfaces :

Damaged Layer sample• Electrical discharge machining (EDM)• large dumps (25µm)• impurities (O2, Cu, Zn,…)• velvet-like texture, s ~ 7µm

Chemically polished sample• BCP- removal of 30µm• s ~ 1µm


Single Crystal Niobium : Phonon-dislocation interactions in skin layer

1

1,5

2

2,5

3

3,5

4

4,5

1,5 1,6 1,7 1,8 1,9 2 2,1

Temperature (K)

Damaged

Layer

Chemically

polished

EDM creates dislocations within narrow layer ( µm)

Thermal Resistivity Model for Nb (W. Wasserbäch)

(Philos. Mag. A.38 401 (1978))

Random distribution of dislocations:

(cm3K3/W)2

91005.3T

NR d

dp

ddpCPKDLK RRR ,,Analysis :

1~d

Plausible dislocation density which explains results

12109~ dN cm-2

Important result indicating scattering/reflection of energy back into Nb due to dislocations &

impurities (contradicts theoretical ideas)

~ 40%


Relative importance of RK

compared to (RK

+ e/K)

0.5

0.6

0.7

0.8

0.9

1.5 1.6 1.7 1.8 1.9 2 2.1

RK/[R

K+(d/K)]

T(K)

(a)

(b)

(c)

(d)

polycrystallineRRR = 647

single crystal

Polycrystalline and Single crystal Nb

CP

EP

DL

CP

Single crystals :

Relative importance of RK increases with T

Polycrystalline Nb :

Relative importance of RK decreases with T

RK constitutes ~75% at T~1.8K for

chemically polished polycrystalline

& single crystals


0,4

0,5

0,6

0,7

0,8

0,9

1,5 1,6 1,7 1,8 1,9 2 2,1

Nb crystal with Damaged LayerNb crystal _chemically polishedCP polycrystalline Nb (sample 3)

Temperature (K)

2/1

1

K

eR

E

K

acc

Comparing polycrystalline to single crystals

Annealed & CP polycrystalline

is strictly equivalent to CP

single crystal

Surface roughness are the

same in both cases!

Thermal point of view …


- roughness length

- root mean roughness height

- inclination of roughness

New analysis using resonant scattering at Nb/HeII interfaceI. N. Adamenko & I. M. Fuks, JETP, 32, 1123 (1971)

s

He II

Nb surface

)r(ζ

Nature of phonon scattering is defined by s, l and Amplification of heat flux due to Multiple resonant phonon scattering

Solid surface is characterized at a given scale length by :

σ

σ2γ

(roughness)

ideal interface

l = phonon wavelength

Phonon wavelength =

Transmission coefficient =

2

2311l

s AM

)(

3

8.3)(

KT

nm

Tk

hcnm

B

L l

2

169

l

s

l

sf

l

s fQQ o

2

2

11


Impact of Nanoscale surface roughness on RK

(new)

20

30

40

50

60

1,5 1,6 1,7 1,8 1,9 2 2,1

Roughness analysis-single & poly crystals

tau rough/tau 0Tau rough/tau0

Temperature (K)

Nb Polycrystal

Annealed + chemically polished

(RRR 647)

Nb Single crystal

Chemically polished

(RRR 300)

• RK of ideal surface is >> RK of real surface

• Resonant scattering is ~40 times more effective

0,3

0,35

0,4

0,45

0,5

0,55

0,6

1,5 1,6 1,7 1,8 1,9 2 2,1

Polycryst sample 3_JLTP2000 _ also called Ktcbis

sigma/lamda(sigma/lamda)CP

T (K)

5.03.0~

l

s

Ratio of Surface Roughness to

phonon wavelength

Selective diffuse resonant scattering

enhances transmission

Re

son

ant

scat

teri

ng

: ac

ou

stic

imp

ed

ance


0,5

0,6

0,7

0,8

0,9

1

1,5 1,6 1,7 1,8 1,9 2 2,1

Polycryst sample 3_JLTP2000 _ also called Ktcbis

sigma_poly 3sigma (nm)_single CP

s (nm)

T(K)

Nb polycrystal

Nb single crystal

Su

rfa

ce

Rou

gh

ne

ss

Effective heat transfer

between Niobium and He II

occurs at scales less than

a nanometer

Impact of Nanoscale surface roughness on RK

(in progress)


Pressure dependency of Kapitza resistance Will cooling of SRF cavities be more efficient if the He II pressure is increased to 25 bars?

3000

3500

4000

4500

5000

5500

6000

6500

0 5 10 15 20 25

Impédance acoustique de l'HeII

en fonction de la pression

T = 0.10KT = 0.20KT = 0.30KT = 0.40KT = 0.50KT = 0.60KT = 0.70KT = 0.80KT = 0.90KT = 1.00KT = 1.10KT = 1.20KT = 1.30KT = 1.40KT = 1.50KT = 1.60KT = 1.70KT = 1.80KT = 1.90KT = 2.00K

P (atm)

LLHe cZ

changes by ~80%

Aco

ustic im

pe

da

nce

of L

iqu

id H

e

cL and L increases with pressure

Nb

He

HeNb

HeNbAM

Z

Z

ZZ

ZZ 4

)(

42

Transmission

Coefficient

Niobium

He II


Pressure dependency of Kapitza resistance Will cooling of SRF cavities be more efficient if the He II pressure is increased to 25 bars?

3000

3500

4000

4500

5000

5500

6000

6500

0 5 10 15 20 25

Impédance acoustique de l'HeII

en fonction de la pression

T = 0.10KT = 0.20KT = 0.30KT = 0.40KT = 0.50KT = 0.60KT = 0.70KT = 0.80KT = 0.90KT = 1.00KT = 1.10KT = 1.20KT = 1.30KT = 1.40KT = 1.50KT = 1.60KT = 1.70KT = 1.80KT = 1.90KT = 2.00K

P (atm)

LLHe cZ

changes by ~80%

Aco

ustic im

pe

da

nce

of L

iqu

id H

e

cL and L increases with pressure

Nb

He

HeNb

HeNbAM

Z

Z

ZZ

ZZ 4

)(

42

Transmission

Coefficient

0

1

2

3

4

5

6

7

150

200

250

0 5 10 15 20 25

P (bar)

~1.8K

2

l

s

Silicon crystal (111)

Niobium

He II

~80% change in acoustic impedance of He

NO change in transmission!


0,2

0,4

0,6

0,8

1

1,2

1,4

1,6 1,7 1,8 1,9 2 2,1 2,2

Gra

in-g

rain

RK(c

m2K

/W)

T(K)

RK

~2T-3

(cm2K/W)

RKy

RKx

Kapitza resistance RG-G at grain boundaries in polycrystalline Nb

O

GG

o

lpolycrysta

KRnd

n

KK

)1(1

GGR Kapitza resistance at

grain-grain boundaries

Thermal conductivity of Nb :

oK Casimir thermal conductivity

n Nb. of grains

Anisotropy in heat flow through cavity walls:

-grain size d

-grain distribution

In plane

Cross plane

Solid-solid model


KGG

acc

RRn

)1n(

d

e

1E

•Cooling of cavities is controlled by K and RK

•As purity of Nb improves, RK dominates

•Surface quality (chemical purity, structural order, surface roughness…) rather

than effective surface area

•Poly-crystals : Annealed + CP + s ~ 1.2µm is better than

Annealed + electro-polished

•Single crystals : Chemically polished (RRR 300) give better performance

•Thermo-mechanical history of sample : dislocations due to machining

•Equivalence in performance with poly-crystals and single crystals

• is important parameter

•Presence of dislocations and/or impurities increase RK

•Raising the pressure to 25 bars changes impedance by ~80%, but no effect on RK

•RK is anomalous – cannot be explained by acoustic mismatch theory

•New analysis : Resonant scattering of phonons from nanoscale roughnesses

Summary

KeRK /

1


Possible Future work related to Kapitza resistance

How does the properties of superfluid He affect RK?

…

3

34322 )(~ q

Ts

Aq

TsdT

s

nn

Viscous flow of non-turbulent He II Mutual friction for high heat fluxesbetween normal fluid and vortices


Kapitza resistance under High heat fluxes ?

32 )(25.0)()(5.11 TTTTTT

TTcorrect

32

0,

)(25.0)()(5.11 TTTTTT

R

q

TR

KcorrectK


Radiation model for heat flux :

)( 441 LTTq TTT L 1

with

For 5.0T

T

2

0KK

RR we have



-Convection in liquid

-Nucleate boiling

-Film boiling

- Development of Turbulence & attenuation

of second sound

Physics of quantum fluids



Will coating of surfaces modify van der Waals

forces on the Niobium surface ?

Niobium

He II


Thank you for your attention


Nb +1 µm Ti

Nb after removal of 30µm

Nb after removal of 5 µm

jay amrit kapitza resistance at niobiumsuperfluid he interfaces

Science