CH.8 Electrical and Thermal Properties of Materials

About This Presentation

Title:

CH.8 Electrical and Thermal Properties of Materials

Description:

CH.8 Electrical and Thermal Properties of Materials 8.1 MACROSCOPIC ELECTRICAL PROPERTIES 8.1.1 Generalized Ohm s law conductivity J : current density ... – PowerPoint PPT presentation

Number of Views:221

Avg rating:3.0/5.0

Slides: 84

Provided by: STFL

Category:

more less

Transcript and Presenter's Notes

Title: CH.8 Electrical and Thermal Properties of Materials

1
CH.8 Electrical and Thermal Properties of
Materials
2

8.1 MACROSCOPIC ELECTRICAL PROPERTIES
8.1.1 Generalized Ohms law conductivity
J current density, conductivity,
electric field.
N number density of electrons, e charge, v
average drift velocity
mobility

In metals the charge carriers are electrons
and so we are concerned in this case only with
the number density, charge and mobility of
electrons. In semiconductors both electrons and
holes contribute to the conduction, so that using
a similar equation for the current density in
terms of both contributions from holes and
electrons leads to

8.1.2 Temperature dependence of conductivity in
metals
Assume one type of charge carrier for
simplicity.
The temperature dependence of the conductivity
is dependent on the temperature dependence of N
and µ
In a metal N is the density of valence
(conduction) electrons. This has a value of
typically N 1028m-3 in a metal, and is largely
temperature independent. Therefore the
temperature dependence of conductivity should be
due to a temperature dependence of mobility

The mobility of electrons in metals is of the
order of µ 10-3 10-1 m2 (Vs)-1 and so this
leads to a conductivity
of typically 106 108 (ohm m)-1. In fact
all of the observed temperature dependence of
in metals arises from the temperature
dependence of the electron mobility µ which is
affected by phonon scattering and impurity
scattering of electrons in the metal

6
ELECTRON MOBILITY

vd drift velocity (average electron velocity)
µe electron mobility (the frequency of
scattering events)

7
FREE ELECTRON THEORY OF CONDUCTION

charge? mass m? free electron?
electric field E? ????.
electron? ??? force F?
but, electrons periodically collide with the
atom in the lattice and loose their kinetic
energy. ?Scattering

v? ??? linear? ??
v electronic draft velocity
8
average
mean free path
quantum theory
thermal velocity of electrons at Fermi level
9
METALS RESISTIVITY VS T, IMPURITIES

assumption scattering simply involves the
collision of an electron with an ion core.
Thermal component of ? ?th
interaction between moving electrons and atomic
vibration atomic vibration ? with Temp.?
? mean free path?

mean free path due to thermal vibration mean free
path due to impurities
Mathiessens rule
of phonon
proportionality constant
10

Impurity component of ? ?I
at fixed impurity concentration ?I constant
Impurity? ???? scattering ?? ? lI ?? ? ?I ??

11
(No Transcript)
12

8.1.3 Temperature dependence of conductivity in
semiconductors
In intrinsic semiconductors, the number
density of charge carriers increases with
temperature according to the equation
Where Eg is the band gap and the above
equation assumes that the Fermi level is in the
middle of the band gap. This equation show that
there is an increase in the number density of
conduction electrons with temperature.

8.1.4 Temperature dependence of mobility
If the mean free time between collisions is
, the charge e and the mass m, then the
mobility is given by,
And it can be seen that it is the temperature
dependence of which determines the mobility,
or alternatively we can view this as the
temperature dependence of the resistive
coefficient in the equation of motion of the
electrons. In a metal increases with
temperature,
? reduction in mobility ,decrease in
conductivity

8.1.5 Different types of mobility
Four different kinds of mobility of electrons
Microscopic mobility
This is defined for a particular electron
moving with drift velocity v in an electric field
2. Conductivity mobility
This is the macroscopic or average mobility
which is determined from measurement of
electrical conductivity
assuming N and e are both known.

3. Hall mobility
is the mobility of charge carriers as
determined from a Hall effect measurement.
4. Drift mobility
This is determined from measurement of the
time t required for carriers to travel a distance
d in the material under the action of an electric
field

8.2 QUANTUM MECHANICAL DESCRIPTION OF CONDUCTION
ELECTRON BEHAVIOUR
In the absence of an electric field, the
valence electrons in a metal have no net or
preferential velocity in any direction. If we
plot the vectors of these electrons in velocity
space, then for a free-electron metal we obtain a
velocity sphere, the surface of which corresponds
to the Fermi velocity. All points inside the
Fermi sphere are occupied. Integrating over the
entire sphere we obtain zero drift velocity

17
Thermal and Drift Velocities

The carrier-gas model assumes that the kinetic
energy of any single carrier is given by
(3.18)
The kinetic energy of the carriers is related to
the crystal temperature T
(3.19)

Thermal velocity
18
Figure 3.5 The concept of drift velocity (a) No
electric field is applied. (b) A small electric
field is applied. (c) A larger electric field is
applied.
19

Fig. 3.5a An electron that returns to the
initial position after a number of scattering
events, performing therefore no effective motion
in any particular direction in the crystal.
? no drift current ? drift velocity 0
Fig. 3.5b If an electric field is applied to
the electron gas, the electric-field will deviate
slightly the electron paths between collisions,
producing an effective shift of the electrons in
the direction opposite to the direction of the
electric field.
? drift velocity gt 0
Fig. 3.5c An increase in the electric field
increases the drift velocity.

20
(No Transcript)
21

When an electric field is applied
The majority of electron velocities cancel,
but now some are uncompensated and it is these
electrons which cause the electric current.
The important result that only certain
specific electrons which are close to the Fermi
surface can contribute to the conduction
mechanism .
Note that a similar effect was found for heat
capacity where only those electrons within kBT of
the Fermi level could contribute to the heat
capacity

8.2.1 Quantum corrections to the conductivity in
Ohms law
The highest energy that electrons can take in
a metal in its ground state is the Fermi energy
EF.
Density of states
This means that only a small change of energy
is needed to raise a large number of
electrons above the Fermi level. The velocity of
the uncompensated electrons under the action of
the field is close to the Fermi velocity.

Our Ohms law equation of section 8.1.1.
needs to be slightly modified to take into
account the fact that not all free electrons
contribute to the conductivity.
vF velocity of electrons at the Fermi
level, N number of displaced electrons

24
(No Transcript)
25

8.2.2 Number of conduction electrons
contributing to conduction

For free electron, we have
and hence

8.2.3 displacement of the Fermi sphere under the
action of an electric field
the mean free time of the electrons
between collisions.
With this expression for

Only the projections of vF along the
direction of the electric field , that is
vFcos?, contribute to the current
For a spherical Fermi surface there is a slight
correction which gives.

And finally, the conductivity is given by
, so that
This quantum mechanical statement of
conductivity shows that not all conduction
electrons can contribute to the conductivity, but
only those close to the Fermi surface. In
addition, the conductivity is determined by the
density conduction electron per atom, this
density is high, leading to high conductivity.

30
8.3 DIELECTRIC PROPERTIES capacitance

A dielectric material is one that is electrically
insulating (nonmetallic) and exhibits or may be
made to exhibit an electric dipole structure
that is, there is a separation of positive and
negative electrically charged entities on a
molecular or atomic level.
As a result of dipole interactions with electric
fields, dielectric materials are utilized in
capacitors.
the capacitance
the quantity of charge stored on either plate
the voltage applied across the capacitor

31
the area of the plates the distance between
the plates the permittivity of
vacuum 8.8510-12 F/m
the permittivity of the dielectric medium
the dielectric constant (the relative
permittivity)
32
(No Transcript)
33

Most electronic applications involve the use
of alternation electric fields or currents. In
these cases the atoms in insulators oscillate
under the action of the applied electric field,
and these oscillations can be expressed in terms
of the dielectric constant, e.
This dielectric constant is actually
dependent on the frequency of the applied
electric field. When considering its dependence
on the frequency of electromagnetic radiation it
is often represented as e(?).

8.3.1 Polarization
The polarization can result from the relative
displacement of the electrons and ionic cores or
alternatively from the relative displacement of
positive and negative ionic cores
The force F on a charge e under the action
of an electric field is,
And it is this force which causes
polarization of a material by displacing the
positive and negative charges within an atom in
opposite directions, or by displacing the ionic
cores within the lattice

Electric polarization of the material (P) An
electric dipole moment per unit volume, which is
measured in coulomb metres per cubic metre (or
effectively coulombs per square metre).
P Np
Where p is the dipole moment of an individual
atom and N is the number of atoms per unit
volume. P can also be defined as the surface
density of charge which appears on the faces of
the specimen when placed in a field. The
polarization can be expressed in terms of the
electric field by the equation.
Where e is the permittivity or dielectric
constant.

The dielectric constant is a measure of the
amount of electric polarization induced by unit
field strength. A high dielectric constant means
that a material is easily polarized in an
electric field. Typical values of the relative
dielectric permittivity er are in the range 1.0
10 (dimensionless), although its value can be
much higher in some special materials, for
example er is 94 in titanium dioxide(TiO2)

8.3.2 Dielectric field strength
The dielectric field strength is a measure of
the largest electric field strength that an
insulation material can sustain before the
electrostatic forces holding the atoms in place
are overcome. Once this happens the material
suffers electrical breakdown and suddenly becomes
an electrical conductor. Typical values of the
dielectric strength are in the range of
megavolts per metre.
The breakdown strength often increases with
frequency, and in particular for most materials
breakdown is somewhat inhibited above 108Hz

8.3.4 Polymers
Most polymers are insulators of course, but
conduction polymers exist which have electrical
properties resembling those of conventional
metals or semiconductors. Polyacetylene contains
a high degree of crystallinity and a relatively
high conductivity compared with other polymers.
Trans-polyacetylene has a conductivity that is
comparable to silicon. The electron band
structure of this polymer has even been
calculated and it has been found that when all
carbon lengths are equal, this material has a
band structure which is reminiscent of a metal.
When the carbon bonds alternate in length it is
found that band gaps appear in the structure.

39
(No Transcript)
40

8.4 OTHER EFFECTS CAUSED BY ELECTRIC FIELDS,
MAGNETIC FIELDS AND THERMAL GRADIENTS
8.4.1 Magnetoresistance
?The change in electrical conductivity
associated with an applied magnetic field.
?Hall effect
Under equilibrium conditions the motion of
the change carriers is identical in the presence
or absence of a magnetic field.
? magnetoresistance 0
If not all the charge carriers have the same
properties, the current flow is disturbed by the
presence of a magnetic field and some of the
charge carriers travel a longer distance.

? drift velocity ?
? mobility ?
?

42
(No Transcript)
43
8.4.2 Thermoelectric power (Seeback effect)
SEEBACK EFFECT
Metal
Cold
Hot

Hot region? electrons? cold region? electrons?? ?
kinetic energy? ???.
Electrons? hot ? cold? ??? average K.E.? ???.
?potential gradient
?Seeback voltage

Hot
Cold
Seeback potential thermal driving force
equilibrium
44

If a material is subjected to a temperature
gradient, the energy of the carriers at the hot
end is greater than at the cold end and this
leads to a carrier concentration gradient along
the material. Displaced charge resulting from
this concentration gradient generates a
counteraction electric field until the total
current becomes zero. The magnitude of this
electric field in terms of the voltage per degree
difference is known as the thermoelectric power
, In a metal,
EF The Fermi energy

In a metal is typically a few microvolts
per degree Kelvin
In a semiconductor, for an n type material
Here A is a constant which depends on the
specific scattering mechanism, A 2 for lattice
scattering and A 4 for charged impurity
scattering

The Seeback effect is utilized in the
thermocouple which is used for measuring
temperature. The thermoelectric power is
determined from the open circuit electric field
caused by a temperature gradient dT/dx

47
THOMSON EFFECT

Thomson effect the effect of heat production
and/or absorption when an electric current flows
in a temperature gradient
Electrons move from cold to hot end ? electrons
absorb heat from hot region ? thermoelectric
cooling
Electron moves from hot to cold end ? electrons
give up heat to the rod

hot
cold
hot
cold
48
8.4.3 PELTIER EFFECT

? The evolution or absorption of heat that is
created when an electric current flows across a
junction between two dissimilar materials.
Metal-Semiconductor Ohmic contact

Q
electron flow
metal n-type
electron flow
Q
49
(No Transcript)
50

When a current flows in a material, a
temperature gradient is developed. This of course
is the inverse of the Seeback effect and is used
in some cases for temperature control. The
Peltier coefficient is simply the ratio of
the electrical current density J to the thermal
current density JQ

8.4.4 Nernst effect
When a magnetic field is applied at right
angles to a temperature gradient, the diffusing
charge carriers are deflected in the same way as
when the magnetic field is applied at right
angles to a conventional electric current, the
result is a Nernst voltage. However, since charge
carriers of both signs diffuse in the same
direction the polarity of the Nernst voltage is
not dependent on the sign of the charge carrier.

8.4.5 Ettingshausen effect
In the Hall effect, the application of a
magnetic field normal to the passage of an
electric current leads not only to a transverse
voltage but also to a transverse temperature
gradient. The appearance of this temperature
gradient is known as the Ettingshausen effect.
This arises because charge carriers with
different energies (velocities) are deflected
differently by the magnetic field. This is a
small effect which adds to the Hall voltage

8.5 THERMAL PROPERTIES OF MATERIALS
8.5.1 Thermal conductivity
In the case of metals, the thermal conduction
mechanism is similar to the electrical conduction
mechanism and proceeds via the free electrons
which migrate throughout the material. In
semiconductors, conduction can take place by the
electrons which are thermally stimulated into the
conduction band.
In insulators another mechanism must be
involved and in this case the thermal conduction
is due to phonons which are created at the hot
part of a solid and destroyed at the cold part.
These phonons provide the mechanism by which
energy is transferred through the material.
Thermal conductivity electron phonon

In metals the phonon contribution to thermal
conductivity is also present, but the electronic
contribution is so much greater that in these
cases the phonon contribution is neglected.
Units W -1m -1k -1

8.5.2 Mechanism of thermal conduction
If we begin from the assumption that thermal
conduction can arise from both the motion of free
electrons and phonons, we can derive a theory of
the thermal conductivity. Again as in electrical
conductivity, only those electrons close to the
Fermi surface can contribute to the thermal
conductivity.

56
1
0
57
(No Transcript)
58
(No Transcript)
59
(No Transcript)
60
(No Transcript)
61
Neglect Too small
62

8.5.3 Thermal conductivity of metals
The number of participating electrons N is
determined by the population density at the Fermi
energy N(EF). To a first approximation, this is
about 1 of the number of free electrons per unit
volume.

Ein the heat energy flowing in per unit time
per unit area at the left end
Eout the heat energy flowing out per unit time
per unit area at the right end.

Z the number of electrons per unit time per
unit area impinging on the end face
The number density of free electrons (N) is
similar to the number contributing to the thermal
heat capacity
since in both
cases the electrons must be able to absorb heat
energy, Substituting z,

And from

The thermal conductivity increases with mean
free path
Number of electrons per unit volume at the
Fermi surface NF, and velocity of electrons at
the Fermi surface vF, Remembering that
, and that
The thermal conductivity increases with T and
and decreases with m

8.5.4 Thermal conductivity of insulators
The thermal conductivity K is related to the
heat capacity by the expression

8.6 OTHER THERMAL PROPERTIES
8.6.1 Thermoluminescence
Thermoluminescence is the emission of
electromagnetic radiation, in the visible
spectrum, when certain materials are heated.
These materials must be either insulators or
semiconductors, and they must have a large number
of electrons trapped in impurity states in the
band gap.

69
(No Transcript)
70

8.6.2 Mechanism of thermoluminescence

71
(No Transcript)
72

The requirements for a material to be able to
exhibit thermoluminescence are
Presence of a band gap
Presence of impurity energy states in the band
gap
Long lifetime of electrons in traps
Material must have been subjected to radiation to
excite electrons from valence band before
becoming trapped
Material must not have been inadvertently heated,
which could empty electrons from traps.

8.6.3 Theory of thermoluminescence
Electrons are thermally stimulated from the
traps into the conduction band ? fall back into
the valence band, emitting a photon

If we have an electron located in a trapped
state at an energy below the conduction
band, then the probability of the electron being
thermally stimulated into the conduction band in
unit time is given by the Arrhenius equation,
Where s is a constant, with dimensions time-1
and typically of magnitude 1011 - 1017s-1. this
means that there is a time frame associated with
the occupancy of the electron trap once the
electron is there

8.6.4 Occupation and vacation of trapped states
by electrons
The probability of filling any state in the
band gap will also be dependent on time. If dN/dt
is the rate of stimulation of electrons from
traps into the conduction band, then
N the number of electrons in traps , p the
probability of escape in unit time.

This is the Randall-Wilkins equation which
describes the number of electrons remaining in
traps as a function of both time t and
temperature T.

8.6.5 Lifetime of electrons in traps
The lifetime of occupancy of an electron
state is inversely proportional to the
probability p of a transition in unit time.
Lifetime as a function of temperature T
Raising the temperature T ? decreases the
expected lifetime of the electrons in the traps.
More thermal energy increases the probability
of the electron escaping by thermal stimulation

From the exponential decay equation N
N0exp(-pt) it is possible to define a half-life
for the occupancy of the electron traps. Simply,
when the number of traps remaining occupied has
declined to half, N N0/2, we have the half-life
of the occupancy

8.6.6 Intensity of light emitted during
thermoluminescence
The intensity of light emitted during
thermoluminescence is dependent on the rate of
emptying of the electron traps dN/dt. If we
assume that every electron from a trap enters the
bottom of the conduction band and then
instantaneously falls back to the top of the
valence band with emission of a photon of energy
equal to the band gap energy, then light of a
single frequency (Eg/h) will be emitted. The
intensity of the light will be equal to the rate
of emptying of electron traps,

8.6.7 Emission of light on heating
Suppose then the temperature of the specimen
is raised at a constant rate,
Then the fractional change in occupancy (dN/N) is

This emission assumes a single type of trap
at an energy
below the conduction band, a constant
rate of change of temperature and a constant
value of s for all traps of the given type

8.6.8 Location of the peaks in thermoluminescent
intensity
When intensity of emission I is measured as a
function of temperature T as the temperature is
swept at a fixed rate, peaks in the intensity
will correspond to the depth of electron traps
below the conduction band.
An empirical relationship has been given
between the depth in electron volts (eV)
and the peak temperature T in Kelvin (K) by
Urbach

83
(No Transcript)

Write a Comment

User Comments (0)