Electrical conduction Thermal properties: Thermal ... · PDF fileIt is simply the reciprocal...

Electrical Properties of Materials

Electrical conduction

Thermal expansion

Thermal properties:

1Functional MaterialsFunctional Materials Saarland UniversitySaarland University 1

Thermal conductivity

Thermal expansion

Heat capacity

Electrical Conduction

Ohm‘s Law

Ohm‘s law relates the current – or time rate of charge passage – to an applied voltage:

V = IR R: resistance of the material through which the current is passing

The resistivity ρρρρ is independent of the specimen geometry but related to R through


The resistivity ρρρρ is independent of the specimen geometry but related to R through

the expression::

l: distance between the 2 points at which the voltage is measured

A: cross-sectional area perpendicular to the direction of the current


Ohm‘s Law

Schematic representation of the apparatus used to measure electrical resistivity.



Ohm‘s Law


Ag/SnO2 contact materials with different oxide fraction.


Ohm‘s Law

Sometimes, electrical conductivity σσσσ is used to specify the electrical

character of a material. It is simply the reciprocal of the resistivity, or

Ohm’s law may be expressed as:


Ohm’s law may be expressed as:

J: current density: current per unit area I/A

E: electric field intensity, or the voltage difference between

two points divided by the distance separating them:

Room temperature conductivity of various materials


Electronic and ionic conduction

Valence e- in Metals Semiconductors and Insulators


Ionically bonded materials

Energy Band Structures in Solids


Schematic plot of electron energy versus interatomic separation for an aggregate of 12 atoms (N

12). Upon close approach, each of the 1s and 2s atomic states splits to form an electron energy

band consisting of 12 states.



Leiter Isolator Halbleiter

Cu Mg

Conduction in terms of band

and atomic bonding models

Metals:


For a metal, occupancy of electron states (a) before and

(b) after an electron excitation

Insulators and semiconductors:

Conduction in terms of band

and atomic bonding models


For an insulator or semiconductor, occupancy of electron states (a) before and

(b) after an electron excitation from the valence band into the conduction band, in

which both a free electron and a hole are generated.

Electron Mobility

Perfect crystal Crystal heated to high temperature


Crystal containing lattice defects

Schematic diagram showing

the path of an electron that is

deflected by scattering events.

Electron Mobility


Collisions of electrons with: - other electrons

- Metallic atoms

- Phonons

Electron Mobility: Ohm´s law

In an electrical field: F = e E = a · m*

become an electron an acceleration a = e E / m*

Energy lost by collisions with Phonons, Foreign Atoms, Crystal defects

Middle drift velocity: v = le / τ = τ a= τ e E / m* le = middle free path length

τ = relaxation time between collisions


Current density: j = n e v = n τ e2 E / m* n = concentration of conduction electrons

Specific electrical

conductivity:σσσσ = j / E = n τ e2 / m*

j = σ E = (1/ρ) EOhm´s law: ρ = specific electrical resistance

The electrical conductivity (or Resistance) of metals is independent of the field intensity

Electrical Resistivity of Metals


Room-Temperature Electrical Conductivities for

Nine Common Metals and Alloys


Mathiessen‘s Rule:


ρt: thermal resistivity contribution

ρi: impurity resistivity contribution

ρd: deformations resistivity

contribution


Influence of the Temperature


Low Temperatures (T<<Θ): R ~ T5

Scattering on lattice defects!

High Temperatures (T>Θ): R ~ TLineal relationship for high temperatures

Scattering of the conduction e- with the

lattice vibration (Phonons)

Influence of the temperature and impurities at very low temperatures



Influence of lattice Orientation

Conductivity

Dependent on crystallographic

orientation



For hexagonal Lattice:

ρα = ρ + (ρ - ρ ) · cos2α

Influence of impurities (alloying)


the impurity resistivity ρi is related

to the impurity concentration ci in

terms of the atom fraction (at%/

100) as follows:


where A is a composition-

independent constant that is a

function of both the

impurity and host metals

Isomorphous system (solid solution)



Room temperature electrical resistivity versus

composition for copper–nickel alloys.


Isomorphous system (solid solution)


Influence of order

3Cu.1Au 1Cu.1Au

disordered



CuAuCu3Au

ordered

Resistivity in inhomogeneous alloys



Eutectic system: Ag-Ni

AgNi40

(contact material)

a) Adition of resistances




a) Adition of resistances

RA RB RA RB RA RB

R = R1 + R2 + …+ Rn

b) Adition of conductivitiesRA

RA

RB

RB

R R1 R2 Rn

1 1 1 1= + + …+



c) Real system

Best fit à Adition of conductivities

RA

RA

RB


Addition Conductivity

Addition Resistance

RA

RB

Electrical Properties of Materials

Electrical conduction

Thermal expansion

Thermal properties:


Thermal conductivity

Thermal expansion

Heat capacity

Thermal properties: thermal expansion

Most solid materials expand upon heating and contract when cooled. The change

in length with temperature for a solid material may be expressed as follows:

where l0 and lf represent, respectively, initial and final lengths with the temperature


where l0 and lf represent, respectively, initial and final lengths with the temperature

change from T0 to Tf . The parameter αl is called the linear coefficient of thermal expansion.

Volume changes with temperature may be computed from

where ∆V and V0 are the volume change and the original volume, respectively,

and αv symbolizes the volume coefficient of thermal expansion.

( )[ ] ( )[ ]( )00 22 rrrr

eeD−−−− −⋅=Φ αα

Morse-Potential

(exponential approximation)


potential energy versus

interatomic spacing curve


( )2eeD −⋅=Φ

612r

B

r

A−=Φ

Lennard-Jones-Potential

(potential law approximation)



(a) Plot of potential energy versus interatomic distance, demonstrating the increase in interatomic

separation with rising temperature. With heating, the interatomic separation increases from r0 to

r1 to r2 , and so on. (b) For a symmetric potential energy-versus-interatomic distance curve, there

is no increase in interatomic separation with rising temperature (i.e., r1, r2, r3 ).

at 0 K

Tä à Eä

à no thermal expansion

Für T > Θ /2

ist α Konstant



ist α Konstant

Θ: Debye

temperature

Thermal properties: heat capacity

Heat capacity is a property that is indicative of a material’s ability to absorb

heat from the external surroundings; it represents the amount of energy

required to produce a unit temperature rise. In mathematical terms, the heat

capacity C is expressed as follows:


where dQ is the energy required to produce a dT temperature change.

Ordinarily, heat capacity is specified per mole of material (e.g., J/mol-K, or

cal/mol-K). Specific heat (often denoted by a lowercase c) is sometimes

used; this represents the heat capacity per unit mass and has various units

(J/kg-K, cal/g-K, Btu/lbm-F).

VIBRATIONAL HEAT CAPACITY

In most solids the principal mode of thermal energy assimilation is by the increase

in vibrational energy of the atoms. Again, atoms in solid materials are constantly

vibrating at very high frequencies and with relatively small amplitudes. Rather than

being independent of one another, the vibrations of adjacent atoms are coupled by

virtue of the atomic bonding.



High T U = 3·k·T (klassischen Gesetzt von Dulong-Petit)

Low T U ∝ T


Temperature dependence of the heat capacity


1. Einstein Model:

Atome ~ Oszillatoren

(Quantenmechanik)

Phonen = Quanten der

Gitterschwingungen

Assumption: Atoms oscillate independent from each other and with the same

frequency

( )2

1+= nhEn

νEnergy values of the Oscillators:

(ν = oscillation frequency; n = quantum number)

Frequency distribution of quantum

states n (Bose-Einstein-Distribution):1

1

−

>=<kT

h

e

nν




states n (Bose-Einstein-Distribution):1−kTe

For N atoms and oscillations

in all 3 space direction:ν⋅

+><⋅⋅= hnNU

2

13

the specific heat results2

2

1exp

exp

3

−

⋅

==

kT

h

kT

h

kT

hNk

dT

dUc

V

V

ν

ν

ν

2. Debye model: Atoms oscillate like coupled oscillators with different frequencies

Total Energy through integration

of frequencies: ∫ ⋅

+⋅=

D

dhnDU

ν

νννν0

2

1)()(

υD: the maximal possible frequency

(Debye-Frequency)




(Debye-Frequency)

Dυ: number of oscillation states in an

interval between υ and dυ in a cube with

sides length L3

322

)(S

V

LD

⋅=

νν

VS: sound velocity

Debye-Temperature:k

hD

D

ν=Θ

3

234

Θ≅

D

V

TNkc

For T << ΘD:




For T >> ΘD:

NkcV

3≅

Thermal properties: thermal conductivity

Thermal conduction is the phenomenon by which heat is transported from high

to low-temperature regions of a substance. The property that characterizes the

ability of a material to transfer heat is the thermal conductivity. It is best

defined in terms of the expression:


where q denotes the heat flux, or heat flow, per unit time per unit area A (area

being taken as that perpendicular to the flow direction), k is the thermal

conductivity, and dT/dx is the temperature gradient through the conducting

medium. The unit of k is W/m.K

λ or k: thermal conduction coefficient



Heat conduction mechanisms

crystal latticefree electrons


Electron thermal conductivity (λe) Lattice vibration conductivity (λG)


collisions between phononselectrons become energy

from much excited atoms and

deliver it to less excited atoms

• Metals: λe>> λG

• Ceramics: λe<< λG

From the kinetic theory of gases:

C: specific heat per volume unit

v: mean particle velocity

l: mean free path

For phonons:

lvC ⋅⋅=3

1λ

thermal conductivity: crystal lattice vibration


For phonons:

Cph: specific heat of the lattice

v: sound velocity

l: mean free path

T C: const.

l decreases like 1/T

T l increases up to the sample size

then λ~C, which decreases like T3

thermal conductivity: crystal lattice vibration


For electrons:

Ce: specific heat of the e-

v: velocity of electrons

l: mean free path between collisions

lvC ⋅⋅=3

1λ I: „contaminated“ Na

II: pure Na

thermal conductivity: free electrons


l: mean free path between collisions

• Ce ~ T

• v: à Fermi-Energie:

à T-independent

2

2

1vm

F⋅=ε

• T ä ⇒ l æ

thermal conductivity: free electrons


Lorenz-Zahl: L = 2.443·10-8 WΩ/K2TL ⋅=σ

λ

Thermal conductivity in metals

Since free electrons are responsible for both electrical and thermal

conduction in pure metals, theoretical treatments suggest that the two

conductivities should be related according to the Wiedemann–Franz law:


σ

The theoretical value of L, 2.443.10-8 WΩ/K2, should be independent of

temperature and the same for all metals if the heat energy is transported

entirely by free electrons. Experiments for most metals confirm this

number quite well

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