Materials & Properties II:
Thermal & Electrical Characteristics
Sergio Calatroni - CERN
Outline (we will discuss mostly metals)
•
Electrical properties
Electrical conductivity
o
Temperature dependence
o
Limiting factors
Surface resistance
-
•
o
Relevance for accelerators
o
Heat exchange by radiation (emissivity)
Thermal properties
Thermal conductivity
o
Temperature dependence, electron & phonons
o
Limiting factors
Properties II: Thermal & Electrical
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The electrical resistivity of metals changes with temperature
Copper
T
Constant
T-5
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All pure metals…
100
Electrical resistivity
of Be
Electrical resistivity
of Al
10
10
1
1
Electrical resistivity  [µ.cm]
Electrical resistivity  [µ.cm]
Electrical resistivity  [µ.cm]
10
10-1
10-1
1
10-1
10-2
Electrical resistivity
of Ag
10-2
10-3
10-2
10-3
1
10
100
Temperature [K]
1000
1
10
100
1000
1
10
Temperature [K]
Properties II: Thermal & Electrical
100
1000
Temperature [K]
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Alloys?
Resistivity of Fe Alloys
90.00
80.00
Resistivity [µOhm.cm]
70.00
60.00
AISI 304 L
50.00
AISI 316 L
40.00
Invar 36
30.00
20.00
10.00
0.00
0.00
50.00
100.00
150.00
200.00
250.00
300.00
Temperature [K]
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Some resistivity values (in µ.cm) (pure metals)
Variation of a factor ~70
for pure metals at room
temperature
Even alloys have seldom more than a few 100s of µcm
We will not discuss semiconductors (or in general effects not due to electron transport)
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Definition of electrical resistivity 
R

  length
The electrical resistance of a real object
(for example, a cable)
section
1
The electrical resistivity is measured in Ohm.m
Its inverse is the conductivity measured in S/m

mv
m
  e 2 F  2e
ne  ne 
Changes with: temperature, impurities, crystal defects
Electron relaxation time
Electron mean free path
Constant for a given material
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Basics (simplified free electron Drude model)
Electrical current = movement of conduction electrons
-
Properties II: Thermal & Electrical
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Defects
Defects in metals result in electron-defect collisions
They lead to a reduction in mean free path ℓ,
or equivalently in a reduced relaxation time .
They are at the origin of electrical resistivity 
-
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Possible defects: phonons
Crystal lattice vibrations: phonons
Temperature dependent
-
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Possible defects: phonons
Crystal lattice vibrations: phonons
Temperature dependent
-
Properties II: Thermal & Electrical
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Possible defects: impurities
Can be inclusions of foreign atoms, lattice defects, dislocations
Not dependent on temperature
-
Properties II: Thermal & Electrical
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Possible defects: grain boundaries
Grain boundaries, internal or external surfaces
Not dependent on temperature
-
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The two components of electrical resistivity
Temperature
dependent part
It is characteristic
of each metal, and
can be calculated
Proportional to:
- Impurity content
- Crystal defects
- Grain boundaries
Does not depend on
temperature
Total resistivity
Properties II: Thermal & Electrical
Varies of several
orders of
magnitude
between room
temperature and
“low” temperature
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Temperature dependence: Bloch-Grüneisen function
 d 
 ph (T )  

 T 
13
d 
hvs  2 N 
 6

2k B 
V
 T
T   d
  T 5 T   d
5
d
T
0

x5
dx
x
x
(e  1)(1  e )
Debye temperature:
~ maximum frequency of
crystal lattice vibrations
(phonons)
Given by total number of
high-energy phonons
proportional ~T
Given by total number of
phonons at low energy ~T3
and their scattering
efficiency T2
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Low-temperature limits: Matthiessen’s rule
 total (T )   phonons (T )   impurities   grain boundaries  ...
Or in other terms
 1

1
1

 total (T ) 


 ... 




phonons
impurities
grain
boundaries


1
Every contribution is additive.
Physically, it means that the different sources of scattering for the
electrons are independent
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Effect of added impurities (copper)
(Cu)(300K)=1.65 µ.cm
Note: alloys behave as
having a very large amount
of impurities embedded in
the material
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An useful quantity: RRR
 total (300 K )   phonons (300 K )   impurities   grain boundaries
 total (4.2 K )   impurities   grain boundaries
Fixed number
 total (300 K )  phonons (300 K )   0
RRR 

 total (4.2 K )
0
0 
 phonons (300 K )
Practical formula
( RRR  1)
Experimentally, we have a very neat feature remembering that
R(300 K )  total (300 K )
RRR 

R(4.2 K )  total (4.2 K )
Depends only on
“impurities”
Dominant in alloys
R
  length
section
Independent of the geometry of the sample.
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Final example: copper RRR 100
Copper
𝑅𝑅𝑅 =
𝜌 300 𝐾
𝜌 <10 𝐾
= 100
300K = phonon. = 1.55x10-8 m
 10K = 1.55x10-10 m
If this is due only to oxygen:
imp. = 5.3 x10-8 m / at% of O
1.55𝑥10−10
= 0.003 at % of O
5.3𝑥10−8
 30 ppm atomic !
This is Cu-OFE
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Estimates of mean free path

•
•
•
•
me vF
m

ne 2 ne 2 
Typical values? Example of Cu at room temperature
Let’s assume one conduction electron per atom.
 = 1.55 x 10-8 m.
density = 89400 kg/m3
m = 9.11 x 10-31 kg, e = 1.6 x 10-19 C, A = 63.5, NA = 6.022 x 1023
Exercise ! Solution:
•   2.5 x 10-14 s. Knowing that vF = 1.6 x 106 m/s we have
• ℓ  4 x 10-8 m at room temperature. It can be x100  x1000 larger at low
temperature
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Interlude: LHC
8.33 T dipoles (nominal field) @ 1.9 K
Beam screen operating from 4 K to 20 K
SS + Cu colaminated, RRR ≈ 60
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Magnetoresistance
The LHC
Electron trajectories are bent
due to the magnetic field
Cyclotron radius: r 
mvF
eB
B x RRR [T]
 B-field
ℓ𝑒𝑓𝑓
ℓ
 eff
 eff
ℓ⟶ℓ 4
𝜏⟶𝜏 4
   
   
 r sin    r sin  
  r 
  2 
2
r 

Properties II: Thermal & Electrical
ℓ ℓ𝑒𝑓𝑓 ≈ ℓ
   
 eff  r sin  
  8 
 2
 eff  
 4
8
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Fermi sphere
•
The real picture: the whole Fermi sphere is displaced from
equilibrium under the electric field E, the force F acting on each
electron being –eE
•
This displacement in steady state results in a net momentum per
electron 𝛿𝒌 = 𝑭𝜏/ℏ thus a net speed increment 𝛿𝒗 = 𝑭𝜏/𝑚 =
− 𝑒𝑬𝜏/𝑚
•
𝒋 = 𝑛𝑒𝛿𝒗 = 𝑛𝑒 2 𝑬𝜏/𝑚 and from the definition of Ohm’s law 𝒋 = 𝜎𝑬
we have 𝜎 =
𝑛𝑒 2 𝜏
𝑚
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The speed of conduction electrons
•
Fermi velocity vF = 1.6 x 106 m/s
𝜎𝑬
𝑛𝑒
•
𝛿𝒗 = 𝒋/𝑛𝑒 thus 𝛿𝒗 =

As an order-of-magnitude, in a common conductor, we
may have a potential drop of ~1V over ~1m
𝑉
𝑑
=
𝑒𝜏𝑬
𝑚
•
𝑬=
≈1 V/m and as a consequence 𝛿𝒗 ≈4 x 10-3 m/s
•
The drift velocity of the conduction electrons is orders of
magnitude smaller than the Fermi velocity

(Repeat the same exercise with 1 A of current, in a
copper conductor of 1 cm2 cross section)
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Square resistance and surface resistance
Consider a square sheet of metal and calculate its resistance to a
transverse current flow:
a
current
a
a

R

da
d
d
This is the so-called square resistance often indicated as 𝑅∎
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Square resistance and surface resistance
And now imagine that instead of DC we have RF, and the RF current is
2
confined in a skin depth:  
 0
a
current
a

R

da
d
a

d
Rs 
0



2
This is a (simplified) definition of surface resistance Rs
(We will discuss this in more details at the tutorials)
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Surface impedance in normal metals
•
The Surface Impedance 𝑍𝑠 is a complex number defined at the interface
between two media.
•
The real part 𝑅𝑠 contains all information about power losses (per unit
surface)
1
Rs I 2
1
P  2 2  Rs H 02
d
2
•
The imaginary part 𝑋𝑠 contains all information about the field penetration in
the material

2
 0
Xs
•
For copper ( = 1.75x10-8 µ.cm) at 350 MHz:
•
𝑅𝑠 = 𝑋𝑠 = 5 m and  = 3.5 µm
Properties II: Thermal & Electrical
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Why the surface resistance (impedance)?
•
It is used for all interactions between E.M. fields and
materials
Q0 

Rs
•
In RF cavities: quality factor
•
In beam dynamics (more at the tutorials):
-
Longitudinal impedance and power dissipation from wakes is
Ploss  MI b2 Re Z seff where Z seff is a summation of 2R 2b  Z s over
the bunch frequency spectrum
-
Transverse impedance:
ZT 
Properties II: Thermal & Electrical
2R c
Zs
3
b 
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From RF to infrared: the blackbody
Thermal exchanges by radiation are mediated by EM waves in the
infrared regime.
Peak ≈ 3000 µm x K
Schematization of
a blackbody
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Blackbody radiation
•
A blackbody is an idealized perfectly emitting and absorbing body
(a cavity with a tiny hole)
•
Stefan-Boltzmann law of radiated power density:
P
 T4
A
σ ≈ 5.67 × 10−8 W/(m2K4)
 
•
At thermal equilibrium:
•
 is the emissivity (blackbody=1)
•
•
A “grey” body will obey: 1  r   (t )
Thus for a grey body: P   T 4
A
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From RF to infrared in metals
•
Thermal exchanges by radiation are mediated by EM waves in the
infrared regime.
•
At 300 K, peak ≈ 10 µm of wavelength -> ≈ 1013 Hz or RF ≈ 10-13 s
•
The theory of normal skin effect is usually applied for: RF  1
•
But it can be applied also for: RF  1
•
In the latter case it means:    RF
•
For metals at moderate T we can then use the standard skin effect
theory to calculate emissivity
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Emissivity of metals
1       1 r
•
From:
•
Thus we can calculate emissivity from reflectivity:
  1 r  4
Rs
Rvacuum
Rs 
 0
Rvacuum 
2

0
 376.7
0
•
The emissivity of metals is small
•
The emissivity of metals depends on resistivity
•
Thus, the emissivity of metals depends on temperature
and on frequency
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Practical case: 316 LN
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Thermal conductivity of metals
Copper
peak
T-1
constant
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Thermal conductivity: insulators
Determined by phonons (lattice vibrations). Phonons behave like a “gas”
peak
T-3
constant
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Thermal conductivity: insulators
Thermal conductivity K ph from heat capacity C ph (as in thermodynamics of gases)
1
1
K ph  C ph vs   C ph vs2
3
3
C ph  3 Nk BT  3RT
C ph
T   d
 T 
12 4


Nk B 
5
 d 
1
T
  const.

3
T   d
K ph  const
 T 

K ph  
 d 
T   d
3
Properties II: Thermal & Electrical
T   d
T   d
T   d
for ultra-pure crystals
1
K peak  C ph vs 
3
 = max dimension
of specimen
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Thermal conductivity: metals
Determined by both electrons and phonons.
Thermal conductivity K el from heat capacity Cel
1
 2 nk B2T
 2 nk B2T
K el  Cel vF  
vF  

3
3mvF2
3m
1
T
  const.

K el
K ph
T   d
T   d
impurities
T
d
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Thermal conductivity of metals: total
Copper
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Wiedemann-Franz
Proportionality between thermal conductivity and electrical conductivity
K el


 2  kB 
2
  T  LT
3  e 
T  d
L = 2.45x10-8 WK-2
(Lorentz number)
Useful for simple estimations, if one or the other quantity are known
Useful also (very very approximately) to estimate contact resistances
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The LHC collimator
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Contact resistance (both electrical and thermal)
•
Complicated… and no time left 
n depends on:
Plastic deformation
Elastic deformation
Contact area: A ~ P
•
n
n  O(1)
Roughness “height” and “shape”
Contacts depend also on oxidation, material(s) properties, temperature…
Example for electric contacts:
•
•
Theoretically:
-
RP-1/3 in elastic regime
-
RP-1/2 in plastic regime
Experimentally:
-
RP-1-1/2 (same as for thermal contacts)
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References
•
Charles Kittel, “Introduction to solid state physics”
•
Ashcroft & Mermin, “Solid State Physics”
•
S. W. Van Sciver, “Helium Cryogenics”
•
M. Hein, “HTS thin films at µ-wave frequencies”
•
J.A. Stratton, “Electromagnetic Theory”
•
Touloukian & DeWitt, “Thermophysical Properties of
Matter”
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The end. Questions?
45
Plane waves in vacuum
Plane wave solution of Maxwell’s equations in vacuum:
E  E0 e
i ( kz t )
H  H0 e
Where (in vacuum):
So that:
k
2

E  E0 e i ( kz t )
The ratio Z 
E
H


i ( kz t )

c
H  E0
   0 0 
H  E0
k
 0
e i ( kz t )
0
0
ε0 i ( kz t )
e
μ0
0
 376.7 is often called impedance of the free
0
space and the above equations are valid in a continuous medium
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Plane waves in normal metals
More generally, in metals:
Z
k     i
2
2
E
H

 


k
This results from taking the full Maxwell’s equations, plus a supplementary
equation which relates locally current density and field:


J ( x , t )   E( x , t )
ne 2 ne 2
0 

me vF
me
2
2
In metals      k  i
and the wave equations become:
k  i    i
E  E0 ei ( kz t )  E0 ei (z t ) e   z
With

1


is the damping coefficient of the wave inside a

2
metal, and  is also called the field penetration depth.
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V~
z
x
y
S=d2
E z (0)
Surface impedance
Zs 
H z (0)
V
I

V  dEz (0) x;; I  d  J z  y dy
J z ( y)

Z s  Rs  iX s 

E z (0)
 J  y dy

0
1
J z  y dy

J z (0) 0
E z ( 0)
E z ( 0)


J z (0) H x (0)
z
0
1
1
2
Ptot (t )  Rs I  Rs d 2 H x2
2
2
1
1  Brf
2
P / S  Prf  Rs H rf  Rs 
2
2  o
Properties II: Thermal & Electrical
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


2
48
Normal metals in the local limit
J z  y   J z (0)e

y




i t 


J
t

J
(
0
)
e




2
o n
E (0)
1
Z n  Rn  iX n  z

(1  i) 
J z (0)  n
Rn  X n 
1
 n

n
 o
 o


n

2 n
2
Properties II: Thermal & Electrical
o
(1  i)
2 n
( Rn   )
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Limits for conductivity and skin effect

2
 0
~
1. Normal skin effect if:
  
e.g.: high temperature, low frequency
2. Anomalous skin effect if:
  
e.g.: low temperature, high frequency
Note: 1 & 2 valid under the implicit assumption
1

  1
1 & 2 can also be rewritten (in advanced theory) as:


It derives that 1 can be true for

 1   2 2

34
  1 and also for   1
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Mean free path and skin depth
Mean free path  ~
1

neff  n0
Skin depth  
2
 0

ne 2 
ne 2
0 
  eff 
me vF
me vF

Properties II: Thermal & Electrical

2
0
  
 const.

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Anomalous skin effect
Anomalous skin effect
Asymptotic value
Normal skin effect
Understood by Pippard, Proc. Roy. Soc. A191 (1947) 370
Exact calculations Reuter, Sondheimer, Proc. Roy. Soc. A195 (1948) 336
Properties II: Thermal & Electrical
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Debye temperatures
Properties II: Thermal & Electrical
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Heat capacity of solids: Dulong-Petit law
Properties II: Thermal & Electrical
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Low-temperature heat capacity of phonon gas
(simplified plot in 2D)
Properties II: Thermal & Electrical
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Phonon spectrum and Debye temperature
Density of states D  :
How many elemental
oscillators of frequency 
Assuming constant
speed of sound
Properties II: Thermal & Electrical
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