Electrical Properties of Materials
Electrical conduction
Thermal properties:
Thermal expansion
Heat capacity
Thermal conductivity
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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 ρ 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
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Electrical Conduction
Ohm‘s Law
Schematic representation of the apparatus used to measure electrical resistivity.
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Electrical Conduction
Ohm‘s Law
Ag/SnO2 contact materials with different oxide fraction.
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Electrical Conduction
Ohm‘s Law
Sometimes, electrical conductivity σ 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:
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:
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Room temperature conductivity of various materials
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Electronic and ionic conduction
Valence e- in Metals
Semiconductors and Insulators
Ionically bonded materials
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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.
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Energy Band Structures in Solids
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Energy Band Structures in Solids
Cu
Mg
Leiter
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Isolator
Halbleiter
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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
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Conduction in terms of band
and atomic bonding models
Insulators and semiconductors:
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.
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Electron Mobility
Perfect crystal
Crystal heated to high temperature
Crystal containing lattice defects
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Electron Mobility
Schematic diagram showing
the path of an electron that is
deflected by scattering events.
Collisions of electrons with: - other electrons
- Metallic atoms
- Phonons
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Electron Mobility: Ohm´s law
F = e E = a · m*
In an electrical field:
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*
Specific electrical
conductivity:
σ = j / E = n τ e2 / m*
Ohm´s law:
j = σ E = (1/ρ) E
n = concentration of conduction electrons
ρ = specific electrical resistance
The electrical conductivity (or Resistance) of metals is independent of the field intensity
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Electrical Resistivity of Metals
Room-Temperature Electrical Conductivities for
Nine Common Metals and Alloys
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Electrical Resistivity of Metals
Mathiessen‘s Rule:
ρt: thermal resistivity contribution
ρi: impurity resistivity contribution
ρd: deformations resistivity
contribution
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Electrical Resistivity of Metals
Influence of the Temperature
Lineal relationship for high temperatures
Scattering of the conduction e- with the
lattice vibration (Phonons)
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High Temperatures (T>Θ): R ~ T
Low Temperatures (T<<Θ): R ~ T5
Scattering on lattice defects!
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Electrical Resistivity of Metals
Influence of the temperature and impurities at very low temperatures
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Electrical Resistivity of Metals
Influence of lattice Orientation
Conductivity
Dependent on
crystallographic
orientation
For hexagonal Lattice:
ρα = ρ + (ρ - ρ ) · cos2α
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Electrical Resistivity of Metals
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 compositionindependent constant that is a
function of both the
impurity and host metals
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Electrical Resistivity of Metals
Isomorphous system (solid solution)
Room temperature electrical resistivity versus
composition for copper–nickel alloys.
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Electrical Resistivity of Metals
Isomorphous system (solid solution)
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Electrical Resistivity of Metals
Influence of order
disordered
3Cu.1Au
1Cu.1Au
Cu3Au
CuAu
ordered
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Electrical Resistivity of Metals
Resistivity in inhomogeneous alloys
AgNi40
(contact material)
Eutectic system: Ag-Ni
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Electrical Resistivity of Metals
Resistivity in inhomogeneous alloys
a) Adition of resistances
RA
RB
RA
RB
RA
RB
R = R1 + R2 + …+ Rn
b) Adition of conductivities
1 1
1
1
=
+
+ …+
R R1 R2
Rn
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RA
RB
RA
RB
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Electrical Resistivity of Metals
Resistivity in inhomogeneous alloys
c) Real system
Best fit
Adition of conductivities
RA
RB
RA
RB
Addition Conductivity
Addition Resistance
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Electrical Resistivity of Metals
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Electrical Properties of Materials
Electrical conduction
Thermal properties:
Thermal expansion
Heat capacity
Thermal conductivity
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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
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.
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Thermal properties: thermal expansion
potential energy versus
interatomic spacing curve
Morse-Potential
(exponential approximation)
(
Φ = D ⋅ e [−2α (r − r0 )] − 2e
2e [−α (r − r0 )]
Lennard-Jones-Potential
(potential law approximation)
Φ=
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A
r
12
−
B
r6
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)
Thermal properties: thermal expansion
T
E
at 0 K
no thermal expansion
(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 ).
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Thermal properties: thermal expansion
Für T > Θ /2
ist α Konstant
Θ: Debye
temperature
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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).
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Thermal properties: heat capacity
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.
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Thermal properties: heat capacity
Temperature dependence of the heat capacity
High T
U = 3·k·T
Low T
U∝T
(klassischen Gesetzt von Dulong-Petit)
1. Einstein Model:
Atome ~ Oszillatoren
(Quantenmechanik)
Phonen = Quanten der
Gitterschwingungen
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Thermal properties: heat capacity
Temperature dependence of the heat capacity
Assumption: Atoms oscillate independent from each other and with the same
frequency
(
Energy values of the Oscillators:
(ν = oscillation frequency; n = quantum number)
E n = hν n + 1
Frequency distribution of quantum
states n (Bose-Einstein-Distribution):
< n >=
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)
1
e
hν
kT
−1
1

U = 3 ⋅ N ⋅  < n > + h ⋅ν
2

For N atoms and oscillations
in all 3 space direction:
the specific heat results
2
cV =
dU
dT
 hν 
exp

kT
 hν 


= 3 Nk 
 ⋅
2
 kT  
 hν  
 exp
 − 1
 kT  

2
V
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Thermal properties: heat capacity
Temperature dependence of the heat capacity
2. Debye model: Atoms oscillate like coupled oscillators with different frequencies
Total Energy through integration
of frequencies:
νD
1

U = ∫ D(ν ) ⋅  n(ν ) + hν ⋅ dν
2

0
υD: the maximal possible frequency
(Debye-Frequency)
Dυ: number of oscillation states in an
interval between υ and dυ in a cube with
sides length L
2ν 2 ⋅ L3
D (ν ) =
VS3
VS: sound velocity
Debye-Temperature:
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ΘD =
hν D
k
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Thermal properties: heat capacity
Temperature dependence of the heat capacity
For T << ΘD:
 T 

cV ≅ 234 Nk 
 ΘD 
3
For T >> ΘD:
cV ≅ 3 Nk
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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
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Thermal properties: thermal conductivity
λ or k: thermal conduction coefficient
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Thermal properties: thermal conductivity
Heat conduction mechanisms
Electron thermal conductivity (λe)
Lattice vibration conductivity (λG)
crystal lattice
free electrons
electrons become energy
from much excited atoms and
deliver it to less excited atoms
collisions between phonons
• Metals: λe>> λG
• Ceramics: λe<< λG
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thermal conductivity: crystal lattice vibration
From the kinetic theory of gases:
1
λ = C ⋅v ⋅l
3
C: specific heat per volume unit
v: mean particle velocity
l: mean free path
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
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thermal conductivity: crystal lattice vibration
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thermal conductivity: free electrons
1
λ = C ⋅v ⋅l
3
I: „contaminated“ Na
II: pure Na
For electrons:
Ce: specific heat of the ev: velocity of electrons
l: mean free path between collisions
• Ce ~ T
• v:
Fermi-Energie: ε F
=
T-independent
•T
1
m ⋅ v2
2
⇒l
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thermal conductivity: free electrons
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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:
λ
= L ⋅T
σ
Lorenz-Zahl: L = 2.443·10-8 WΩ/K2
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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