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Electrical Properties of Materials
Electrical conduction
Thermal expansion
Thermal properties:
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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
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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
Electrical Conduction
Ohm‘s Law
Schematic representation of the apparatus used to measure electrical resistivity.
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Electrical Conduction
Ohm‘s Law
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Ag/SnO2 contact materials with different oxide fraction.
Electrical Conduction
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:
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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
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Electronic and ionic conduction
Valence e- in Metals Semiconductors and Insulators
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Ionically bonded materials
Energy Band Structures in Solids
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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.
Energy Band Structures in Solids
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Energy Band Structures in Solids
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Leiter Isolator Halbleiter
Cu Mg
Conduction in terms of band
and atomic bonding models
Metals:
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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
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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
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Crystal containing lattice defects
Schematic diagram showing
the path of an electron that is
deflected by scattering events.
Electron Mobility
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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
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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
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Room-Temperature Electrical Conductivities for
Nine Common Metals and Alloys
Electrical Resistivity of Metals
Mathiessen‘s Rule:
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ρt: thermal resistivity contribution
ρi: impurity resistivity contribution
ρd: deformations resistivity
contribution
Electrical Resistivity of Metals
Influence of the Temperature
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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
Electrical Resistivity of Metals
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Influence of lattice Orientation
Conductivity
Dependent on crystallographic
orientation
Electrical Resistivity of Metals
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For hexagonal Lattice:
ρα = ρ + (ρ - ρ ) · cos2α
Influence of impurities (alloying)
Electrical Resistivity of Metals
the impurity resistivity ρi is related
to the impurity concentration ci in
terms of the atom fraction (at%/
100) as follows:
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where A is a composition-
independent constant that is a
function of both the
impurity and host metals
Isomorphous system (solid solution)
Electrical Resistivity of Metals
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Room temperature electrical resistivity versus
composition for copper–nickel alloys.
Electrical Resistivity of Metals
Isomorphous system (solid solution)
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Influence of order
3Cu.1Au 1Cu.1Au
disordered
Electrical Resistivity of Metals
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CuAuCu3Au
ordered
Resistivity in inhomogeneous alloys
Electrical Resistivity of Metals
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Eutectic system: Ag-Ni
AgNi40
(contact material)
a) Adition of resistances
Resistivity in inhomogeneous alloys
Electrical Resistivity of Metals
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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= + + …+
Resistivity in inhomogeneous alloys
Electrical Resistivity of Metals
c) Real system
Best fit à Adition of conductivities
RA
RA
RB
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Addition Conductivity
Addition Resistance
RA
RB
Electrical Resistivity of Metals
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Electrical Properties of Materials
Electrical conduction
Thermal expansion
Thermal properties:
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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
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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)
Thermal properties: thermal expansion
potential energy versus
interatomic spacing curve
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( )2eeD −⋅=Φ
612r
B
r
A−=Φ
Lennard-Jones-Potential
(potential law approximation)
Thermal properties: thermal expansion
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(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
Thermal properties: thermal expansion
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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:
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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.
Thermal properties: heat capacity
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High T U = 3·k·T (klassischen Gesetzt von Dulong-Petit)
Low T U ∝ T
Thermal properties: heat capacity
Temperature dependence of the heat capacity
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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
Thermal properties: heat capacity
Temperature dependence of the heat capacity
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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)
Thermal properties: heat capacity
Temperature dependence of the heat capacity
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(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:
Thermal properties: heat capacity
Temperature dependence of the heat capacity
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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:
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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
Thermal properties: thermal conductivity
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Heat conduction mechanisms
crystal latticefree electrons
Thermal properties: thermal conductivity
Electron thermal conductivity (λe) Lattice vibration conductivity (λG)
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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
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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
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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
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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
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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:
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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