Materials & Properties II: Thermal & Electrical ...€¦ · Outline (we will discuss mostly metals) • Electrical properties-Electrical conductivityo Temperature dependence o Limiting

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


Copper

T

ConstantT-5

10

10-1

10-2

1

10-3

All pure metals…


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?


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)


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


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)


+-

Electrical current = movement of conduction electrons

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

Possible defects: phonons


+-

Crystal lattice vibrations: phonons

Temperature dependent

Possible defects: phonons


+-

Crystal lattice vibrations: phonons

Temperature dependent

Possible defects: impurities


+-

Can be inclusions of foreign atoms, lattice defects, dislocations

Not dependent on temperature

Possible defects: grain boundaries


+-

Grain boundaries, internal or external surfaces

Not dependent on temperature

The two components of electrical resistivity


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


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


...)()( 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)


(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


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


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


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


8.33 T dipoles (nominal field) @ 1.9 K

Beam screen operating from 4 K to 20 K

SS + Cu colaminated, RRR ≈ 60

Magnetoresistance


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


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)


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∎


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)


2

0

sR

current a d

a R

d

a

a

d

0

2

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)

• 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:


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.


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:


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


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


r 11

7.376

2

0

0

0

vacuum

s

R

R

vacuum

s

R

Rr 41

Practical case: 316 LN


Thermal conductivity of metals


Copper

peak

constant

T-1

Thermal conductivity: insulators


Determined by phonons (lattice vibrations). Phonons behave like a “gas”

peak

constant

T-3

Thermal conductivity: insulators


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


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


Copper

Wiedemann-Franz


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


Contact resistance (both electrical and thermal)


• 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”


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


)(

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


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


)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


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


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


Skin depth

Mean free path

0

2

1~

.2

0

const

Fe

eff

Fe vm

ne

vm

ne

22

0

0nneff

Anomalous skin effect


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


Heat capacity of solids: Dulong-Petit law


Low-temperature heat capacity of phonon gas


(simplified plot in 2D)

Phonon spectrum and Debye temperature


Density of states :

How many elemental

oscillators of frequency

Assuming constant

speed of sound

D

Materials & Properties II: Thermal & Electrical ...€¦ · Outline (we will discuss mostly metals) • Electrical properties-Electrical conductivityo Temperature dependence o Limiting

Documents