Materials & Properties II: Thermal & Electrical ...€¦ · Outline (we will discuss mostly metals) • Electrical properties-Electrical conductivityo Temperature dependence o Limiting
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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 CAS Vacuum 2017 - S.C. 3
The electrical resistivity of metals changes with temperature
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 4
Copper
T
ConstantT-5
10
10-1
10-2
1
10-3
All pure metals…
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 5
10
10-1
100
10-2
1
Ele
ctr
ical re
sis
tivity
[µ
.cm
]
Ele
ctr
ical re
sis
tivity
[µ
.cm
]
10
10-1
10-2
1
Ele
ctr
ical re
sis
tivity
[µ
.cm
]
10-3
Electrical resistivity
of BeElectrical resistivity
of Al
Temperature [K]
1 10 100 1000
Temperature [K]
1 10 100 1000
Temperature [K]
1 10 100 1000
Electrical resistivity
of Ag
Alloys?
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 6
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
90.00
0.00 50.00 100.00 150.00 200.00 250.00 300.00
Re
sist
ivit
y [µ
Oh
m.c
m]
Temperature [K]
Resistivity of Fe Alloys
AISI 304 L
AISI 316 L
Invar 36
Some resistivity values (in µ.cm) (pure metals)
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 7
Variation of a factor ~70
for pure metals at room
temperature
Even alloys have seldom more than a few 100s of µcm
We will not discuss semiconductors (or in general effects not due to electron transport)
Definition of electrical resistivity
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 8
section
lengthR
The electrical resistance of a real object
(for example, a cable)
1 The electrical resistivity is measured in Ohm.m
Its inverse is the conductivity measured in S/m
22 ne
m
ne
vm eFe
Constant for a given material
Changes with: temperature, impurities, crystal defects
Electron relaxation time
Electron mean free path
Basics (simplified free electron Drude model)
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 9
+-
Electrical current = movement of conduction electrons
Defects
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 10
+-
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
Possible defects: phonons
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 11
+-
Crystal lattice vibrations: phonons
Temperature dependent
Possible defects: phonons
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 12
+-
Crystal lattice vibrations: phonons
Temperature dependent
Possible defects: impurities
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 13
+-
Can be inclusions of foreign atoms, lattice defects, dislocations
Not dependent on temperature
Possible defects: grain boundaries
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 14
+-
Grain boundaries, internal or external surfaces
Not dependent on temperature
The two components of electrical resistivity
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 15
Temperature
dependent part
It is characteristic
of each metal, and
can be calculated
Varies of several
orders of
magnitude
between room
temperature and
“low” temperature
Proportional to:
- Impurity content
- Crystal defects
- Grain boundaries
Does not depend on
temperatureTotal resistivity
Temperature dependence: Bloch-Grüneisen function
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 16
T
xx
dph
d
dxee
x
TT
0
55
)1)(1()(
31
262
V
N
k
hv
B
sd
Debye temperature:
~ maximum frequency of
crystal lattice vibrations
(phonons)
d
d
TT
TT
5
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
Low-temperature limits: Matthiessen’s rule
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 17
...)()( boundariesgrainimpuritiesphononstotal TT
Or in other terms
1
...111
)(
boundariesgrainimpuritiesphonons
total T
Every contribution is additive.
Physically, it means that the different sources of scattering for the
electrons are independent
Effect of added impurities (copper)
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 18
(Cu)(300K)=1.65 µ.cm
Note: alloys behave as
having a very large amount
of impurities embedded in
the material
An useful quantity: RRR
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 19
boundariesgrainimpuritiestotal
boundariesgrainimpuritiesphononstotal
K
KK
)2.4(
)300()300(
0
0)300(
)2.4(
)300(
K
K
KRRR
phonons
total
total
Fixed number
Depends only on
“impurities”
Dominant in alloys
)1(
)300(0
RRR
Kphonons Practical formula
Experimentally, we have a very neat feature remembering thatsection
lengthR
)2.4(
)300(
)2.4(
)300(
K
K
KR
KRRRR
total
total
Independent of the geometry of the sample.
Final example: copper RRR 100
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 20
Copper𝑅𝑅𝑅 =
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
5.3𝑥10−8= 0.003 at % of O
30 ppm atomic !
This is Cu-OFE
Estimates of mean free path
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 21
22 ne
vm
ne
m Fe
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
Interlude: LHC
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 22
8.33 T dipoles (nominal field) @ 1.9 K
Beam screen operating from 4 K to 20 K
SS + Cu colaminated, RRR ≈ 60
Magnetoresistance
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 23
ℓ ℓ𝑒𝑓𝑓 ℓ ℓ𝑒𝑓𝑓 ≈ ℓ
B-field
2 reff
2sinsin
r
rreff
8sin
reff
42
8
eff
ℓ ⟶ Τℓ 4𝜏 ⟶ Τ𝜏 4
B x RRR [T]
Cyclotron radius:eB
mvr F
The LHCElectron trajectories are bent
due to the magnetic field
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 = 𝑛𝑒2𝑬𝜏/𝑚 and from the definition of Ohm’s law 𝒋 = 𝜎𝑬
we have 𝜎 =𝑛𝑒2𝜏
𝑚
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 24
The speed of conduction electrons
• Fermi velocity vF = 1.6 x 106 m/s
• 𝛿𝒗 = 𝒋/𝑛𝑒 thus 𝛿𝒗 =𝜎𝑬
𝑛𝑒=
𝑒𝜏𝑬
𝑚
As an order-of-magnitude, in a common conductor, we may have a potential drop of ~1V over ~1m
• 𝑬 =𝑉
𝑑≈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)
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 25
Square resistance and surface resistance
Consider a square sheet of metal and calculate its resistance to a
transverse current flow:
This is the so-called square resistance often indicated as 𝑅∎
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 27
current a d
a R
d
a
a
d
Square resistance and surface resistance
And now imagine that instead of DC we have RF, and the RF current is
confined in a skin depth:
This is a (simplified) definition of surface resistance Rs
(We will discuss this in more details at the tutorials)
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 28
2
0
sR
current a d
a R
d
a
a
d
0
2
Surface impedance in normal metals
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 29
• 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)
• The imaginary part 𝑋𝑠 contains all information about the field penetration in
the material
• For copper ( = 1.75x10-8 µ.cm) at 350 MHz:
• 𝑅𝑠 = 𝑋𝑠 = 5 m and = 3.5 µm
2
02
2
H2
12
1
s
s
Rd
IR
P
sX
0
2
Why the surface resistance (impedance)?
• It is used for all interactions between E.M. fields and materials
• In RF cavities: quality factor
• In beam dynamics (more at the tutorials):
- Longitudinal impedance and power dissipation from wakes is
where is a summation of over
the bunch frequency spectrum
- Transverse impedance:
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 30
sRQ
0
sT Zb
cRZ
3
2
eff
sbloss ZMIP Re2 sZbR 22eff
sZ
From RF to infrared: the blackbody
Thermal exchanges by radiation are mediated by EM waves in the
infrared regime.
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 31
Schematization of
a blackbody
Peak ≈ 3000 µm x K
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:
• At thermal equilibrium:
• is the emissivity (blackbody=1)
• A “grey” body will obey:
• Thus for a grey body:
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 32
4TA
P σ ≈ 5.67 × 10−8 W/(m2K4)
)(1 tr
4TA
P
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:
• But it can be applied also for:
• In the latter case it means:
• For metals at moderate T we can then use the standard skin effect
theory to calculate emissivity
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 33
1RF
1RF
RF
Emissivity of metals
• From:
• Thus we can calculate emissivity from reflectivity:
• The emissivity of metals is small
• The emissivity of metals depends on resistivity
• Thus, the emissivity of metals depends on temperature
and on frequency
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 34
r 11
7.376
2
0
0
0
vacuum
s
R
R
vacuum
s
R
Rr 41
Practical case: 316 LN
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 35
Thermal conductivity of metals
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 36
Copper
peak
constant
T-1
Thermal conductivity: insulators
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 37
Determined by phonons (lattice vibrations). Phonons behave like a “gas”
peak
constant
T-3
Thermal conductivity: insulators
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 38
Thermal conductivity from heat capacity (as in thermodynamics of gases)
2
3
1
3
1sphsphph vCvCK
phK phC
d
d
Bph
dBph
TT
NkC
TRTTNkC
34
5
12
33
d
d
Tconst
TT
.
1
d
d
ph
dph
TT
K
TconstK
3 sphpeak vCK3
1
= max dimension
of specimen
for ultra-pure crystals
Thermal conductivity: metals
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 39
Thermal conductivity from heat capacity
m
Tnkv
mv
TnkvCK B
F
F
BFelel
333
1 22
2
22
elK elC
Determined by both electrons and phonons.
d
d
Tconst
TT
.
1
elK
d
T
phK
impurities
Thermal conductivity of metals: total
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 40
Copper
Wiedemann-Franz
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 41
TLTe
kK Bel
22
3
L = 2.45x10-8 WK-2
(Lorentz number)
Proportionality between thermal conductivity and electrical conductivity
Useful for simple estimations, if one or the other quantity are known
Useful also (very very approximately) to estimate contact resistances
dT
The LHC collimator
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 42
Contact resistance (both electrical and thermal)
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 43
• Complicated… and no time left
• Contacts depend also on oxidation, material(s) properties, temperature…
Example for electric contacts:
• Theoretically:
- RP-1/3 in elastic regime
- RP-1/2 in plastic regime
• Experimentally:
- RP-1-1/2 (same as for thermal contacts)
Contact area: )1(~ OnPA n
n depends on:
Plastic deformation
Elastic deformation
Roughness “height” and “shape”
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”
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 44
The end. Questions?
45
Plane waves in vacuum
Plane wave solution of Maxwell’s equations in vacuum:
Where (in vacuum):
So that:
The ratio is often called impedance of the free
space and the above equations are valid in a continuous medium
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 46
)(
0
00
)(0 ee tkzitkzi
μ
εEHEE
7.3760
0
H
EZ
)(0
)(0 ee tkzitkzi HHEE
0
000
2
ck
)(
00 e tkzik
EH
Plane waves in normal metals
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 47
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 metals
and the wave equations become:
z)(
0
)(
0 eee tzitkzi EEE
),(),( txtx
EJ eFe m
ne
vm
ne
22
0
kZ
H
E
iikik 22
More generally, in metals:
Surface impedance
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 48
)0(
)0(
)0(
)0()0(
0
x
z
z
z
z
zsss
H
E
J
E
dyyJ
EiXRZ
I
VZ s
0
;)0( dyyJdIxdEV zz;
0
)0(
1dyyJ
Jz
z
S=d2
V~
)0(zE
)(yJ z
y
z
x
222
2
1
2
1)( xsstot HdRIRtP
2
2
2
1
2
1/
o
rf
srfsrf
BRHRPSP
)0(zH
Normal metals in the local limit
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 49
y
zz eJyJ
)0(no
2
)1(2
)1(1
)0(
)0(ii
J
EiXRZ
n
o
nz
znnn
no
n
on
n
nn XR
22
1 )( nR
tieJtJ )0(
Limits for conductivity and skin effect
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 50
1. Normal skin effect if: e.g.: high temperature, low frequency
0
2
1~
2. Anomalous skin effect if: e.g.: low temperature, high frequency
Note: 1 & 2 valid under the implicit assumption 1
1 & 2 can also be rewritten (in advanced theory) as:
43221
It derives that 1 can be true for and also for1 1
Mean free path and skin depth
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 51
Skin depth
Mean free path
0
2
1~
.2
0
const
Fe
eff
Fe vm
ne
vm
ne
22
0
0nneff
Anomalous skin effect
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 52
Understood by Pippard, Proc. Roy. Soc. A191 (1947) 370Exact calculations Reuter, Sondheimer, Proc. Roy. Soc. A195 (1948) 336
Normal skin effect
Anomalous skin effect
Asymptotic value
Debye temperatures
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 53
Heat capacity of solids: Dulong-Petit law
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 54
Low-temperature heat capacity of phonon gas
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 55
(simplified plot in 2D)
Phonon spectrum and Debye temperature
Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 56
Density of states :
How many elemental
oscillators of frequency
Assuming constant
speed of sound
D
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