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, Orsay
jay.amrit@limsi.fr
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 state
Poly-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

Part I : Introduction on KTBR Why is it important?/Model predictionsPart II : Experiments with Nb: bulk purity & surface statePoly-crystalsSingle crystalsComparison and impact on SRF cavitiesPart III : New analysis & ongoing work Importance of nanoscale surface roughness Kapitza resistance at grain boundaries in NbPressure dependency of KTBR : What would happen to cavity performance if we increased the pressure to 25 bars? Summary & possible future studies… OutlineThin Films and New Ideas for RF Superconductivity, Padova, 2014

Part I : Introduction on KTBR Why is it important?/Model predictionsPart II : Experiments with Nb: bulk purity & surface statePoly-crystalsSingle crystalsComparison and impact on SRF cavitiesPart III : New analysis & ongoing work Importance of nanoscale surface roughness Kapitza resistance at grain boundaries in NbPressure dependency of KTBR : What would happen to cavity performance if we increased the pressure to 25 bars? Summary & possible future studies…
OutlineThin Films and New Ideas for RF Superconductivity, Padova, 2014

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 conductivity
Copper
Superfluid
4He
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
3  10
S
K
T
T


8  10
L
K
T
T


Typical temperature gradient with temperature jump over atomic distances
1 mm 1 mm
TL
TS
Solid (Cu) Superfluid He
TK
x
Kcu ~1 W/(mK) KHe ~ 800 x KCu
K L T  T 8 10
K S T  T 3 10
For Nb : K Nb T  (100 1000)T
Kapitza length K K L  K  R
L km K HeII 8 , 
L cm K Cu 10 , 
Temperature

Introduction: Kapitza resistance in SRF cavities
Electrical surface resistance :
 
T / 2
RF 0
acc
RF
E(z, t)dt
T
2
Accelerating field : E (MV/m)
Quality factor QO :
E dV
2
1
U v
2
 o  (stored energy)
Power dissipated
  Energy stored per sec
q
ωU
Qo 
(3-4 nano ohms)
Heat dissipation in inner walls : B// penetrates ~50 nm into walls
Joule effect
c residual
2
Rs  A( / T )exp( 1.76T / T ) R

e
Nb He II
Tin
THe
Kapitza T
q
B 
Tin Tbath
Tout
RK K(T)
e
q
K
s
o
acc
R
K
R e
T
E

 


 

 

1
8
2 1/ 2 
Cavités ellipsoïdales : ] [ ] [ / 4 mT acc MV m B  E
2
// 2
1
q R B s
o
 


 




Power dissipation : 
R q
K
e
T T T in He K  



     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
 240 L c m.s-1
 = cLk
Dispersion relation of Helium

Heat transport in superfluid Helium
 240 L c 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  n s
Superfluid component : s  s   0 s s
s 
n 
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
s s n n J       
n n s s   0    
Normal component = source & Superfluid component = sink
s n    

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 the
Nb/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


 c cos
4
d
N1,b  n( ,T )g( k )dk 1,b
  k T 1 n( ,T ) e B 1

    
3
b
2
3
3
c
4 d
( 2 / L )
d k
g( k )dk
 

 

N° of modes with wavevector k for a given branch
  N1,b
3
1 2T
A
RK



Thermal boundary resistance: formal approach
   






b 
b
K
d c d
T
n T
g
T
q
R
 
q
   q q q


0
/ 2
0
0 2 cos sin
( , )
( )
4
1 1


Specific heat
  1 1 2
1 2
4
1 1

 


   D
K
C
T
q
R

      

 
b
b q Q Q n T g d c d
 
q
     q q q
0
/ 2
0
1 2 2 1 0 ( , ) ( ) cos sin
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 p p
 

L S  
Snell’s law
S
S
L
L
c
sin
c
sinq q

  S
L
L S
L S
AM Z
Z
Z Z
4Z Z 4
( ) 2 

 q 
Transmission coefficient is due to
discontinuity in sound velocity & density
  L L L, Z  c
  S S S , Z  c
L q
L L p k




S S p k




k k sin i k cos j S S S S S
  
 q  q
k k i k j L L L L L
  
 sinq  cosq
S q
 240 L c m.s-1
 3 5 c
L q
Critical cone in superfluid limits transmission of phonons !!
5068 ,  Nb L c m.s-1 2092 m.s-1 ,  Nb T 0.002 c
4
2 ( )
0


 

  q
q d
c
AM AM
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
t ( ) 1 
t ( ) 2 
r ( ) 2 
r ( ) 1 
t ( ) r ( ) 1 1 1    
t ( ) r ( ) 1 2 2    
1 1 2 2 t ()q  t ()q
~ 0.5
q
q
t ( ) 1
1
2
1
1   


 


 




r ( ) t ( ) 1 2   
He II
Nb
DMM phonon transmission is increased by two orders of magnitude
How 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
Experiments
Nb/HeII
The Kapitza resistance at a solid/ superfluid He II interface is anomalous !

Experimental Cell
Nb disc sample
Heater
Carbon
Thermometers
Q 
superfluid
To
T1 4He Filling line
Temperature profile
across a sample
K : thermal conductivity
e : sample thickness
K
e
Q
S(T T )
R K . 2
1 0 


To
T1
RK
Q 
To
RK
T
Experiments : cell configuration & method

Polycrystalline Niobium: 4 different bulk & surface treatments 100 μmchemically etched250 μmannealed and chemicallyetchedLightHeavy250 μm
electropolished50 μmannealed, chemically etched & electropolishedMicrographics of surfaces1234RRR 178
RRR 178
RRR 647
RRR 6474.08.1σ 25.085.0σ 4.03.1σ 1.02.0σ )mμ(σK2.4K300ρρRRR

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
R
K
(cm2K.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
e
R
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
e
R
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
1/ 2
1






K
e
R
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
3.05 10 9
T
N
R d
dp
  
K DL K CP dp d R  R  R  Analysis : , ,
~1 d 
Plausible dislocation density which explains results
12 ~ 910 d N 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
R
K
/[R
K
+(d/K)]
T(K)
(a)
(b)
(c)
(d)
polycrystalline
RRR = 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 Layer
Nb crystal _chemically polished
CP polycrystalline Nb (sample 3)
Temperature (K)
1/ 2
1






K
e
R
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 interface
I. N. Adamenko & I. M. Fuks, JETP, 32, 1123 (1971)

s
He II
Nb surface
ζ(r) 
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

2σ
γ 
(roughness)
ideal interface
l = phonon wavelength
Phonon wavelength =
Transmission coefficient =
 


 






 
2
1 231
l
s
   AM
( )
3
3.8
( )
T K
nm
k T
hc
nm
B
l  L 
2
169 



 



l
s
l
s
f








 
l
s
  Q Q f o
2
2
1
  1

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 0
Tau 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
1,5 1,6 1,7 1,8 1,9 2 2,1
T (K)
5 . 0 3 . 0 ~  



l
s
Ratio of Surface Roughness to
phonon wavelength
Selective diffuse resonant scattering
enhances transmission
Resonant scattering : acoustic impedance

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 3
sigma (nm)_single CP
s (nm)
T(K)
Nb polycrystal
Nb single crystal
Surface Roughness
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 = 0.10K
T = 0.20K
T = 0.30K
T = 0.40K
T = 0.50K
T = 0.60K
T = 0.70K
T = 0.80K
T = 0.90K
T = 1.00K
T = 1.10K
T = 1.20K
T = 1.30K
T = 1.40K
T = 1.50K
T = 1.60K
T = 1.70K
T = 1.80K
T = 1.90K
T = 2.00K
P (atm)
He L L Z   c
changes by ~80%
Acoustic impedance
of Liquid He
cL and L increases with pressure
 


Nb
He
Nb He
Nb He
AM Z
Z
Z Z
Z Z 4
( )
4
2  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 = 0.10K
T = 0.20K
T = 0.30K
T = 0.40K
T = 0.50K
T = 0.60K
T = 0.70K
T = 0.80K
T = 0.90K
T = 1.00K
T = 1.10K
T = 1.20K
T = 1.30K
T = 1.40K
T = 1.50K
T = 1.60K
T = 1.70K
T = 1.80K
T = 1.90K
T = 2.00K
P (atm)
He L L Z   c
changes by ~80%
Acoustic impedance
of Liquid He
cL and L increases with pressure
 


Nb
He
Nb He
Nb He
AM Z
Z
Z Z
Z Z 4
( )
4
2  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
Grain-grain R
K
(cm2K/W)
T(K)
R
K
~2T-3 (cm2K/W)
R
Ky
R
Kx
Kapitza resistance RG-G at grain boundaries in polycrystalline Nb



 



O
G G
o
polycrystal
R K
nd
n
K
K
( 1)
1
G G R  Kapitza resistance at
grain-grain boundaries
Thermal conductivity of Nb :
o K 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
G G K
acc
R R
n
(n 1)
d
e
1
E





•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
R e K K /
1


Possible Future work related to Kapitza resistance
How does the properties of superfluid He affect RK?
…
3
2 2 3 4 3 ( )
~ q
s T
A
q
d s T
T
s
n  n 




 
Viscous flow of non-turbulent He II Mutual friction for high heat fluxes
between normal fluid and vortices

Kapitza resistance under High heat fluxes ?
 2 3  1 1.5( T T) ( T T) 0.25( T T)
T
Tcorrect
  


  

 2 3 
,0
1 1.5( T T) ( T T) 0.25( T T)
R
q
T
R correct K
K
  

  
 

Radiation model for heat flux :
( ) 4 4
1 L q  T T T T T L   1 with
For  0.5
T
T
2
K0
K
R
we have R 

-Convection in liquid
-Nucleate boiling
-Film boiling
-Development of Turbulence & attenuation of second sound
Physics of quantum fluidsThin Films and New Ideas for RF Superconductivity, Padova, 2014

Possible Future work related to Kapitza resistanceWill coating of surfaces modify van der Waals forces on the Niobium surface ? NiobiumHe IIThin Films and New Ideas for RF Superconductivity, Padova, 2014

Thin Films and New Ideas for RF Superconductivity, Padova, 2014Nb +1 μm TiNb after removal of 30μm Nb after removal of 5 μm

Jay amrit kapitza resistance at niobiumsuperfluid he interfaces

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