ELECTRICAL PROPERTIES
Department of Physics
K L University
1
Session 1
15-Sep-15
2
Contents
 Free Electron Model
 Bloch theorem, Kronig- Penny model, Brillouin Zones
 Energy band theory, Band structures in Conductors, Semi conductors and Insulators
 Electrical properties of conductors- Ohms, Mathiessen rule, conductivity, Mobility
 Electrical properties of Semi conductors, Factors effecting the carrier concentration,
Conductivity and Mobility of charge carriers
 Electric properties of Insulator-Dielectrics- Types of Dielectrics, Dielectric
Constant, Polarization, Types of Polarizations, Frequency Dependence of
Polarization, Ferro, Piezo Electrics.
Free Electron Model
 To explain the structure and properties of solid
 To explain bondings in solids, behavior of conductors,
semiconductors and insulators, electrical and thermal conductivities
of solids, magnetism, elasticity, through their electronic structure
 Development of Free Electron Theory:
1. The classical free electron theory (Drude and Lorentz Model)
2. The quantum free electron theory (Sommerfeld Model)
3. Band Theory (Brillouin Zone Theory)
The Classical Free Electron Theory
 Postulates:
1. The valance electrons (electron gas) are free to move about the whole
volume of the metal like the molecules of a perfect gas in a container.
2. Electrons suffer collisions among themselves, with ion core and with
boundaries of the specimen.
3. All these collisions are ELASTIC, i.e., there is no loss of energy.
Electrons obey classical kinetic theory of gases.
Postulates (Cont.)
4. Velocities of electrons in metal obey classical Maxwell-Boltzmann
distribution of velocities. Root mean square velocity of electron is
Vrms = (3KBT/m)1/2
Where, KB is Boltzmann constant, T is absolute temperature and m is
mass of the electron.
5. As Vrms is RANDOM, it does not contribute to any current. Only
directed motion of electrons, imparted by an external force causes
current.
The Classical Free Electron Theory (Cont.)
 Neglecting electron–electron interaction between collisions is “independent electron
approximation.”
 In the absence of external fields, random motion of electrons is observed.
 In the presence of external fields, electrons acquire some amount of energy from the
field and are directed to move towards higher potential. As a result, the electrons
acquire a constant velocity known as DRIFT VELOCITY (Vd).
Trajectory of a conduction electron
The Classical Free Electron Theory (Cont.)
 Time taken for the drift velocity to decay (1/e) of its initial value is
known as RELAXATION TIME (τ).
 The mean time between successive collisions is called MEAN
COLLISION TIME (τc).
 The average distance travelled by an electron between any two
successive collisions in the presence of external field is called
MEAN FREE PATH (λ).
 Mathematically, mean free path
λ = Vrms . τc.
Success of Classical Free Electron Theory
1. Explains the concept of resistance in metals
2.
Verifies Ohm‟s law
3. Explains high electrical and thermal conductivity of metals
4. Establishes relation between electrical and thermal conductivities
of metals (Wiedemann – Franz law)
k/σ = L.T; wher, k is thermal conductivity, σ is electrical
conductivity, L is Lorentz number, T is temperature (in K)
5. Explains optical properties of metals
Drawbacks of Classical Free Electron Theory
 Classical theory failed to explain:
1.
Many phenomenon observed in materials such as photoelectric effect,
Compton effect and black body radiation, etc.
2.
Electrical conductivity of semiconductors and insulators.
3.
Specific heat capacity of solids.
4.
The concept of ferromagnetism.
5.
The theoretical value of paramagnetic susceptibility is greater than
the experimental value.
Quantum Free Electron Theory
 Postulates:
 Sommerfeld retained the concept of free electrons moving in a uniform
potential within the metal.
 Treated electrons obeying laws of quantum mechanics instead of those of
classical mechanics.
 Electron within the boundaries of the metal is considered as electron trapped in
a potential well.
 Energy levels of electrons are explained by distribution functions besides the
laws of quantum mechanics.
 Fermi-Dirac statistics was used instead of Maxwell-Boltzmann statistics.
Schrödinger Time Dependent Wave Equation
The Schrödinger time dependent wave equation is
2


 2  V  i
2m
t

2
2


 V

2m

or


  i

t

H  E
 2 2
  V , Hamiltonian operator
where H =
2m
E =

i
t
, Energy operator
Success of Quantum Free Electron Theory
1.
According to classical theory, which follows Maxwell- Boltzmann
statistics, all the free electrons gain energy. So it leads to much larger
predicted quantities than that is actually observed.
2.
But according to quantum mechanics only one percent of the free
electrons can absorb energy. So the resulting specific heat and
paramagnetic susceptibility values are in much better agreement with
experimental values.
3.
According to quantum free electron theory, both experimental and
theoretical values of Lorentz number (L = 2.44x10-8 WΩK-1) are in
good agreement with each other.
Drawbacks of Quantum Free Electron Theory
1. It is incapable of explaining why some crystals have metallic
properties and others do not have.
2. It fails to explain why the atomic arrays in crystals including
metals prefer certain structures and not others.
Bloch Functions
 The periodicity of the lattice is described by the potential
 V(x) = V(x+a); where, a is the periodicity of the lattice
 Schrodinger wave equation for motion of electron in periodic potential
is
 Statement of Bloch Theorem: The eigen functions of the wave equation
for a periodic potential are the product of a plane wave times a function
with the periodicity of the crystal lattice
 Ѱ (x) = uk(x)eikx
 Periodic function uk(x) = uk(x+a)
Kronig-Penney Model
 Periodic potential is described by:
V(x) = 0, for 0 < x < a (potential well)
V(x) = V0, for –b < x < 0 (potential barrier)
Schrodinger wave equations for motion of electron in
periodic potential (in wells and barriers) is
 The two equations can be solved by inclusion of Bloch function,
due to the periodicity of the potential, described by:
Ѱ (x) = uk(x)eikx, where, uk(x) = uk(x+a)
(P/αa)sin αa + cosαa = coska
Where, P = mV0ab/ħ is called scattering power
V0b = barrier strength
α2 = 2mE/ħ2 and
k = 2π/λ is the wave vector
Brillouin Zones
 Electrons in solids are permitted to be in allowed energy bands
separated by forbidden energy gaps.
 The allowed energy band width increases with αa.
 The electron has allowed energy values in the region or zone
extending from k = (-π/a) to (π/a), called the first Brillouin zone.
 Similarly, second Brillouin zone extends from
k = (-π/a) to (-2π/a) and
k = (π/a) to (2π/a).
Band Theory of Solids
In isolated atoms the
electrons are arranged in
energy levels
Conductors, Insulators, and Semiconductors
Consider the available energies for electrons in the materials.
As two atoms are brought close
Instead of having discrete energies
together, electrons must occupy different
as in the case of free atoms, the
energies due to Pauli Exclusion principle.
available energy states form bands.
In solids the outer electron energy levels become
smeared out to form bands
The highest occupied band is called the VALENCE band. This
is full.
For conduction of electrical energy there must be electrons in
the conduction band. Electrons are free to move in this band.
Insulators : There is a big energy gap between the valence and
conduction band. Examples are plastics, paper …
Conductors : There is an overlap between the valence and
conduction band hence electrons are free to move about. Examples
are copper, lead …
Semiconductors : There is a small energy gap between the two
bands. Thermal excitation is sufficient to move electrons from the
valence to conduction band. Examples are silicon, germanium…
Session 2
15-Sep-15
23
Electrical conductivity (σ)
Electrical conductivity (σ)
 It is the ability of a substance to conduct an electric current
 It is the inverse of the resistivity (ρ)
 Metals: σ > 105 (Ω.m)-1
 Semiconductors: 10-6<σ < 105 (Ω.m)-1
 Insulators: 10-6 (Ω.m)-1<σ<10-20(Ω.m)-1
Electrical Conductivity in Metals
 One of the best materials for electrical conduction (low resistivity) is silver
but its use is restricted due to the high cost
 Most widely used conductor is copper: inexpensive, abundant, high σ
Ohm‟s law
 The voltage applied to a conductor which is equal to the product of current passing
through the conductor times its resistance
V= IR (Macroscopic form)
 Ohm‟s law in this form is independent of size and shape of the conductor under
consideration
 However, it can also be expressed in terms of current density J and electric field E
J = σ E (Microscopic form)
 It follows that the electric current density is proportional to the applied electric field
 Where, proportionality constant σ is called electrical conductivity
Matthiessen‟s rule
 The resistivity ρ is defined by scattering events due to the imperfections
and thermal vibrations
 Total resistivity ρtot can be described by the Matthiessen‟s rule:
 Where
 ρthermal - resistivity due to thermal vibrations
 ρimpurity - due to impurities
 ρdeformation - due to deformation-induced defects
 resistivity increases with temperature, with deformation, and with alloying
Graphical verification of Mathiessen‟s rule
Drift velocity of electrons (vd)
∆x
vd
I
Random motion of electrons
In the absence of external field
E
Directed motion of electrons
In the presence of external field
Relaxation time (τ)
 The conduction electrons acquire random motion after collision with
impurities and lattice imperfections
 Electrons are not accelerated indefinitely because during their motion
the electrons collide with impurities and lattice imperfections in the
crystal
 These collisions are considered similar to the collision process of an
ideal gas in a container
 The average time between successive collisions of electron with the
lattice imperfection is called relaxation time and is denoted by τ
Drift velocity of electrons (vd)
 When an electric field E is applied to a conductor, the electrons
modify their random motion and move with an average drift
velocity vd in a direction opposite to that of the electric field
vd =
 The drift velocity is proportional to applied electric field and the
proportionality constant is called the electron mobility µ
Electron Mobility (µ)
 The electron mobility is defined as the drift velocity vd per unit
applied electric field E.
µ=
 Substituting the value of drift velocity in the above equation we get
µ=
 The above equation shows that mobility of an electron depends on
the relaxation time and hence it depends on temperature
Relation between mobility and conductivity
 The conductivity of materials is expressed as
σ = neµe
 where,
 n is the number of free electrons per unit volume
 e is the absolute magnitude of the electrical charge on
an electron
 µe is electron mobility
Session 3
15-Sep-15
35
Semiconductors
 Features:
 Conductivity lies between that of conductors and insulators
 The electrical properties of semiconductors are extremely
sensitive to the presence of even minute concentrations of
impurities
 Classification:
 Intrinsic: Materials in their pure form
 Extrinsic: Materials doped with impure atoms
Semiconducting materials for devices
 Silicon (Si) and Germanium (Ge)
 Gallium Arsenide (GaAs) is commonly used compound, especially in the case of LEDs
because of its large bandgap
 Silicon and Germanium are both group 4 elements, having 4 valence electrons
 Their structure allows them to grow in a shape called the diamond lattice
 Gallium is a group 3 element while Arsenide is a group 5 element. When put together as a
compound, GaAs creates a zincblend lattice structure
 In both the diamond lattice and zincblend lattice, each atom shares its valence electrons with
its four closest neighbors. This sharing of electrons is what ultimately allows diodes to be
build
 When dopants from groups 3 or 5 (in most cases) are added to Si, Ge or GaAs it changes the
properties of the material so we are able to make the P- and N-type materials that become the
diode
Semiconducting materials for devices
The diagram above shows the 2D structure
Si
+4
Si
+4
Si
+4
of the Si crystal.
The light green lines represent the electronic
Si
+4
Si
+4
Si
+4
bonds made when the valence electrons are
shared.
Si
+4
Si
+4
Si
+4
 Each Si atom shares one electron with each
of its four closest neighbors so that its valence
band will have a full 8 electrons.
N-type material
When extra valence electrons are introduced into a
material such as silicon an n-type material is
produced.
+4
+4
+4
The extra valence electrons are introduced by
putting impurities or dopants into the silicon.
+4
+5
+4
The dopants used to create an n-type material are
Group V elements. The most commonly used
+4
+4
+4
dopants from Group V are arsenic, antimony and
phosphorus.
The 2D diagram to the left shows the extra electron
that will be present when a Group V dopant is
introduced to a material such as silicon. This extra
electron is very mobile.
The Phosphorus atom
Phosphorus is number 15
in the periodic table
It has 15 protons and 15
electrons – 5 of these
electrons are in its outer shell
This crystal has been doped with a pentavalent impurity
The free electrons in n type silicon support the flow of current
P-type material
P-type material is produced when the dopant that is
introduced is from Group III.
Group III elements have only 3 valence electrons.
+4
+4
+4
This creates a hole (h+), or a positive charge that can
move around in the material.
+4
+3
+4
 Commonly used Group III dopants are aluminum,
boron, and gallium.
+4
+4
+4
The 2D diagram to the left shows the hole that will
be present when a Group III dopant is introduced to a
material such as silicon. This hole is quite mobile in
the same way the extra electron is mobile in a n-type
material.
The Boron atom
Boron is number 5 in the
periodic table
It has 5 protons and 5
electrons – 3 of these
electrons are in its outer
shell
Extrinsic conduction – p-type silicon
This crystal has been doped with a trivalent impurity
The holes in p type silicon contribute to the current
Note that the hole current direction is opposite to electron current
so the electrical current is in the same direction
Intrinsic Semiconductors
 Group IV elements (C, Si, Ge, Sn, Pb, Fl)
 relatively narrow forbidden band gap, generally less than 2 eV
 Si (1.1 eV) and Ge (0.7 eV) are widely used for device applications
 At 0 K, valence band is completely filled, conduction band is empty
 Equal carrier concentration (electron and hole)
 ni = n = p
 The magnitude of hole mobility is always less than electron mobility for
semiconductors
 Conductivity is given by
 σ = neµe+peµh = nie(µe+µh)
Electron bonding in intrinsic silicon
 Before excitation
 After excitation
Extrinsic Semiconductors
 All commercial semiconductors are extrinsic
 Impurity concentration of one atom in 1012 is sufficient to render silicon
extrinsic at room temperature
 n-type semiconductors are obtained by adding impure atoms of group V (P,As,
Sb…)
 These impure atoms are called donor atoms
 Conductivity is mainly due to free electrons
 σ = neµe
 At room temperature, the thermal energy available is sufficient to excite large
numbers of electrons from donor states
Extrinsic Semiconductors
 p-type semiconductors are obtained by adding impure
atoms of group III (Al, B, Ga…)
 These impure atoms are called acceptor atoms
 Conductivity is mainly due to free holes
 σ = peµh
 Energy level within the band gap introduced by this
type of impurity is called an acceptor state
Band diagram of semiconductors
n-type semiconductors
Band diagram of semiconductors
p-type semiconductors
Role of temperature in conductivity
 In extrinsic semiconductors, large numbers of charge carriers
are created at room temperature by the available thermal
energy
 As a consequence, relatively high room-temperature electrical
conductivities are obtained in extrinsic semiconductors
 Most of these materials are designed for use in electronic
devices to be operated at ambient conditions
Factors effecting conductivity
Influence of Dopant Content:
 Dependence of electron and hole
mobilities in silicon plotted as a function
of dopant concentration, at room
temperature
 Both the axes on the plot are scaled
logarithmically
 At dopant concentrations less than about
1020 m-3, mobilities are at their maximum
levels and independent of the doping
concentration
 Both mobilities decrease with increasing
impurity content
Factors effecting conductivity
 Influence of Temperature:
 The temperature dependences of electron and hole mobilities for silicon are presented in
plots
 Both the axes of the plot are scaled logarithmically
 For dopant concentrations of 1024 m-3, and below, both mobilities decrease in magnitude
with rising temperature
 It is due to enhanced thermal scattering of the carriers
 For both electrons and holes, and dopant levels less than 1020 m-3, the dependence of
mobility on temperature is independent of acceptor/donor concentration
 For concentrations greater than 1020 m-3, curves in are shifted to progressively lower
mobility values with increasing dopant level
Influence of temperature
Conductivity and mobility of charge
carriers in semi-conductors
 Intrinsic semiconductors:
 Extrinsic semiconductors:
 Equal carrier concentration
 Conductivity is mainly due to free
(electron and hole)
electrons in n-type semiconductors
 ni = n = p
 n >> p
 The magnitude of hole mobility is
 σ = neµe
always less than electron mobility
for semiconductors
 Conductivity is mainly due to free
holes in p-type semiconductors
 Conductivity is given by
 p >> n
 σ = neµe+peµh = nie(µe+µh)
 σ = peµh
Session 4
15-Sep-15
58
Energy band diagram
 The magnitude of the band gap is
the key parameter to understand
the electrical properties of
insulators, semiconductors and
metals
 Even very high electric fields is
also unable to promote electrons
across the band gap in an insulator
Introduction to insulators
 Internal electric charges do not flow freely
 Not possible to conduct an electric current under the influence of an
electric field
 Higher resistivity than semiconductors or conductors
 A perfect insulator does not exist, because even insulators contain small
numbers of mobile charges
 All insulators become electrically conductive when a sufficiently large
voltage is applied
 This is known as the breakdown voltage of an insulator
 Ex: glass, paper and teflon
Conductivity of various insulators
Various insulators
 Many polymers and ionic ceramics at
Material
Electrical Conductivity
(Ω-m)-1
room temperature
 Filled valance band and empty conduction
Ceramics
band with relatively large band gap, (more
Dry concrete
10-9
than 2 eV)
Soda lime glass
10-10 – 10-11
Borosilicate glass
10-13
Fused silica
10-18
 At room temperature, only very few
electrons may be excited across the band
Polymers
gap
 Very small values of conductivity
Nylon 6,6
10-12 – 10-13
Polystyrene
< 10-14
Polyethylene
10-15 – 10-17
Polytetrafluoroethylene
< 10-17
Properties of insulators
 Used in electrical system to prevent unwanted flow of current to
the earth from its supporting points
 Insulator is a very high resistive path through which practically no
current can flow
 In transmission and distribution system, there must be insulator
between tower and current carrying conductors to prevent the flow
of current from conductor to earth through the towers
Properties of insulators
cont.
1.
Mechanically strong enough to carry tension and weight of conductors
2.
Very high dielectric strength to withstand the voltage stresses in High
Voltage system
3.
Must be free from unwanted impurities
4.
It should not be porous -
5.
Entrance on the surface of electrical insulator so that the moisture or gases
can enter in it
6.
Physical as well as electrical properties must be less affected by changing
temperature
Dielectrics Vs insulators
 The insulating materials are used to resist the flow of current
through it, when a potential difference is applied across its ends
 The distinction between a dielectric material and an insulator lies
in the application
 For instance, vacuum is an insulator but it is not dielectric
 All dielectric materials are electrical insulators but all electric
insulators need not be dielectrics
Introduction to dielectrics
 High electrical resistivities
 Can store energy/charge
 No free electrons
 Band gap larger than 3eV
 No excitation of electron from valance band to conduction band with normal
voltage or thermal energy
 The electrons are strongly bound to the atoms or molecules
 A steady flow of electrons cannot flow through it
 Net separation of positive and negative charges is observed at molecular or
atomic level
Types of dielectrics
 Polar Dielectrics: dielectrics in which centers of the positive as well as
negative charges do not coincide with each other
 Ex: NH3, HCl, H2O
 They are of asymmetric shape
Types of dielectrics cont.

Non Polar dielectrics: dielectrics in which the centers of both positive as
well as negative charges coincide with each other

Ex: methane, benzene, CO2 etc.

Molecules of this category are symmetric in nature
Dielectric materials in capacitors
Capacitance of a parallel plate capacitor
 Capacitor with vacuum
Q = charge stored on each either plate
V = applied potential
 Capacitor with dielectric
A is area of the plates, separated by a distance l
 permittivity of vacuum, ϵ0 = 8.85x10-12 Fm-1
 ϵ = permittivity of dielectric medium
ϵ > ϵ0, represents the increase in charge storing capacity
by insertion of the dielectric medium between the plates
dipole
_
+
Electric field
+
+
+
_
_
_
_
+
+
+
_
_
+
_
+
_
Dielectric atom
Fundamental definitions
 Dielectric constant (ϵr):
 Dielectric Constant is the ratio between the permittivity of the
medium to the permittivity of free space
 ϵ r = ϵ / ϵ0
 The characteristics of a dielectric material are determined by the
dielectric constant
 It has no units
 It is a measure of polarization in the dielectric material
Electric polarization
 Formation of dipoles:
 When a dielectric material is placed inside an electric field,
net separation of positive and negative charges is observed,
which is called a dipole
 Such dipoles are created in all the atoms of the dielectric material
 The process of producing electric dipoles by an electric field is called polarization
in dielectrics
(or)
 In the presence of an electric field, the dipoles experience a force to orient in the
field direction. This process of dipole alignment is termed polarization
Polarization vector
 Electric dipole moment of an electric dipole generated by two electric
charges, each of magnitude q, separated by the distance d is given by
µ = qd
 The dipole moment per unit volume of the dielectric material is called
polarization vector
P = Nµ
where, μ is the average dipole moment per molecule,
N is the no. of molecules per unit volume
Polarizability
 The induced dipole moment per unit electric field is called
Polarizability
 The induced dipole moment is proportional to the intensity
of the electric field
μ∝E
μ=αE
 α is the constant of proportionality, called the polarizibility
Electric flux Density
 Electric flux density is defined as charge per unit area and it has same units of dielectric
polarization.
 Electric flux density D at a point in a free space or air in terms of electric field strength is
D0 = ϵ0 E
 At the same point in a medium is given by
D=ϵE
 As the polarization measures the additional flux density arising from the presence of
material as compared to free space
i.e, D = ϵ0 E + P
D = ϵ E = ϵ0 E + P
(ϵ - ϵ0) E = P (or ) (ϵr.ϵ0 - ε0) E = P
(ϵr−1) ϵ0 . E = P
Electric susceptibility
 The polarization vector P is proportional to the total
electric flux density and direction of electric field.
 Therefore the polarization vector can be written
P = ϵ0 χe E
χe = P / ϵ0 E
= ϵ0 (ϵr−1 ) E / ϵ0 E
χe = (ϵr−1)
Dielectric Strength
 The dielectric strength of a material measures the ability of that material to
withstand voltage differences
 When very high electric fields (>108 V/m) are applied across dielectric
materials, large numbers of electrons may suddenly be excited to higher energy
levels within the conduction band
 As a result, the current through the dielectric by the motion of these electrons
increases dramatically
 Then the voltage across a dielectric exceeds the breakdown potential, the
dielectric will break down and begin to conduct charge between the plates
 Real-life dielectrics enable a capacitor provide a given capacitance and hold
the required voltage without breaking down
Session 5
15-Sep-15
78
Electric polarization
 The process of producing electric dipoles by an electric field is
called polarization in dielectrics
 The dipole moment per unit volume of the dielectric material is
called polarization vector
P=Nµ
where, μ is the average dipole moment per molecule,
N is the no. of molecules per unit volume
Various polarization processes
 When the specimen is placed inside a d.c. electric field,
polarization occurs due to four types of processes..
1.Electronic polarization
2.Ionic polarization
3.Orientation polarization
Electronic polarization
 When an electric field is applied to an atom, positively charged nucleus
displaces in the direction of field and electron could in opposite direction
 This kind of displacement will produce an electric dipole with in the
atom.
 i.e, dipole moment is proportional to the magnitude of field strength and
is given by
μe ∝ E or
μe= αeE
 where „αe‟ is called electronic Polarizability constant
Electronic polarization cont.
It increases with increase of volume of the
atom
This kind of polarization is mostly exhibited in
monatomic gases
It occurs only at optical frequencies (1015Hz)
It is independent of temperature
Ionic polarization
 The ionic polarization occurs, when atoms form molecules and it is mainly due to a
relative displacement of the atomic components of the molecule in the presence of an
electric field
 When an electric field is applied to the molecule, the positive ions displaced by X1 to the
negative side of electric field and
 Negative ions displaced by X2 to the positive side of field
 The resultant dipole moment μ = e ( X1 + X2)
 Ionic polarization occurs in all ionic solids: NaCl, MgO…
Ionic polarization cont.
 Restoring force constant depend upon the mass of the ion and
natural frequency and is given by
F = eE = m.w02 x or
x = eE / m.w02
where „M‟ mass of anion and „m‟ is mass of cation
Ionic polarization cont.
This polarization occurs at frequency 1013 Hz (IR).
It is a slower process compared to electronic polarization.
It is independent of temperature
Orientational Polarization
 It is also called dipolar or molecular polarization
 The molecules such as H2 , N2,O2,Cl2 ,CH4,CCl4 etc., does not carry any
dipole because centre of positive charge and centre of negative charge
coincides
 On the other hand, molecules like CH3Cl, H2O, HCl, ethyl acetate ( polar
molecules) carries dipoles even in the absence of electric field
 However, the net dipole moment is negligibly small since all the molecular
dipoles are oriented randomly when there is no electric field
 In the presence of the electric field these all dipoles orient themselves in the
direction of field as a result the net dipole moment becomes enormous
Orientational Polarization cont.
 It occurs at a frequency 106 Hz to 1010Hz.
 It is slow process compare to ionic polarization.
 It greatly depends on temperature
 Expression for orientation polarization:
 This is called Langevin – Debye equation for total Polaris ability in dielectrics
Frequency dependence of the dielectric constant
 If a dielectric material that is
subject to polarization by an
ac (alternating current)
electric field
 With each direction reversal,
the dipoles attempt to reorient
with the field, in a process
requiring some finite time
Frequency dependence
(graphical)
Comparison of polarizations
Electronic
Ionic
Oriantational
Polarization
Polarization
Polarization
Electron
Cations &
Arrangement
cloud shift
anions are
of random
wrt nucleus
shifted
dipoles
Factor
Definition
Alcohol,
Examples
Inert gases
Ionic crystals
Temperature
Independent
Independent
methane
Dependent
Temperature dependence of polarization
(additional)
 Electronic and ionic polarizations  temperature independent
 Orientation polarization and Space charge polarization  temperature
dependent
 Sum of dipoles in presence of electric field is opposed by thermal vibrations of
atoms
 Polarization decreases with increasing temperature
 Normal temperatures will oppose the permanent dipoles to align in the field
direction
 Higher temperatures facilitate the movement of dipoles
 Polarization increases with increasing temperature
Ferro electric materials
(Ferro electricity)
 Exhibit electric polarization even in the absence of electric field, called Spontaneous
Polarization
 Analogous to ferromagnetic materials in magnetism
 Presence of permanent electric dipoles
 Ferro electric crystals possess high dielectric constant
 Ferroelectricity refers to creation of enormous value of induced dipole moment in a
weak electric field as well as existence of electric polarization even in the absence of
applied electric field
 Examples:
Barium Titanate (Ba Ti O3),
Rochelle salt(NaKC4H4O6.4H2O)
Pottasium dihydrogen phosphate(NH4H2PO4),
Properties of Ferro electric materials
 Easily polarized even for small electric fields
 Exhibits dielectric hysteresis
 Possess spontaneous polarization
 Possess permanent electric dipole
 Exhibit domain structure like ferromagnetic material
 All ferroelectric materials are piezoelectric but all
piezoelectric are not ferroelectric
Hysteresis loop
 Spontaneous polarization
without external field or stress
 Very similar to ferromagnetism
in many aspects:
 Alignment of dipoles, domains,
ferroelectric Curie temperature,
“paraelectric” above the Curie
temperature....
Applications of ferroelectric materials
 In optical communication, the ferroelectric crystals
are used for optical modulation.
 Useful for storing energy in small sized capacitors in
electrical circuits.
 In electro acoustic transducers such as microphone
Piezoelectric materials (Piezoelectricity)
 The process of creating electric polarization by mechanical stress is called piezo
electric effect
 This process is used in conversion of mechanical energy into electrical energy and
also electrical energy into mechanical energy
 According to inverse piezo electric effect, when an electric stress is applied, the
material becomes strained. This strain is directly proportional to the applied field
 Examples: quartz crystal , Rochelle salt etc
Piezoelectricity
cont.
 The (a) direct and (b) converse piezoelectric effect
 In the direct piezoelectric effect (a), applied stress causes a voltage to appear
 In the converse effect (b), an applied voltage leads to development of strain
Applications of piezoelectric materials
Industry
Application
Air bag sensor, air flow sensor, audible alarms, fuel atomiser, keyless door entry, seat
Automotive
Computer
Consumer
belt buzzers, knock sensors.
Disc drives, inkjet printers.
Cigarette lighters, depth finders, fish finders, humidifiers, jewellery cleaners, musical
instruments, speakers, telephones.
Medical
Disposable patient monitors, foetal heart monitors, ultrasonic imaging.
Military
Depth sounders, guidance systems, hydrophones, sonar.
Session 6
15-Sep-15
99
Engineering applications (Additional)
 Role of band theory in explaining photo-excitation and thermal excitation:
 Band theory successfully explains the process of photoexcitation, which is the principle
in many devices such as photovoltaic devices, photochemistry, luminiscence, and
optically pumped lasers.
 Photo-excitation is the photo-electrochemical process of electron excitation by photon
absorption. This absorption of photon is supported by Plank‟s quantum theory.
 Band theory also successfully explains the process of thermal excitation, which is a key
factor to be considered in fabrication any semiconductor device.
 Within a semiconductor crystal lattice, thermal excitation is a process where lattice
vibrations provide enough energy to transfer electrons to a higher energy band.
Engineering applications
 Semiconductor devices:
 Transistor is one of the most widely used semiconductor devices
 Primarily used to amplify an electrical signal and also to serve as
switching devices in computers for the processing and storage of
information
Engineering applications
 Semiconductor devices:
 In making high speed computer chips, calculators, telephones and
other variety of things like medical equipments and robotics
 Power semiconductor consisting of devices which have integrated
circuits
 In manufacturing computers, communication, space research,
medical sciences etc.
Engineering applications
 Applications of dielectrics:
 Major application is power line and electrical insulation
 Other applications include use in capacitors and transformers, motors and
generators
 A number of ceramics and polymers are utilized for this purpose
 Many of the ceramics, including glass, porcelain, steatite, and mica, have
dielectric constants within the range of 6 to 10
 These materials also exhibit a high degree of dimensional stability and mechanical
strength
 Titania (TiO2) and titanate ceramics, such as barium titanate (BaTiO3), having
extremely high dielectric constants, are specially useful for capacitor applications
Engineering applications
 Real time piezoelectric applications:
 Piezoelectric materials are mainly utilized in transducers
 Transducer is the devices that converts electrical energy into mechanical strains,
or vice versa
 other familiar applications that employ piezoelectrics include phonograph
cartridges, microphones, speakers, audible alarms, and ultrasonic imaging
 In a phonograph cartridge, a pressure variation is imposed on a piezoelectric
material located in the cartridge, which is then transformed into an electric
signal and is amplified before going to the speaker
 Piezoelectric materials include titanates of barium and lead, lead zirconate
ammonium dihydrogen phosphate and quartz
Review questions on electrical properties
I.
Free electron models:
1.
List out the postulates of classical free electron theory along with its merits.
2.
Discuss the origin of electrical resistance in metals.
3.
Define relaxation time, mean free path and establish the relation between them.
4.
Explain Wiedemann – Franz law and mention its importance.
5.
Explain the any two drawbacks of classical free electron theory and explain the
assumptions made in quantum theory to overcome the drawbacks.
6.
Compare and contrast the postulates of classical and quantum free electron
theories.
7.
Explain the salient features of quantum free electron theory along with its
merits and demerits.
Review questions on electrical properties
I.
Free electron models (cont.):
8.
State and explain Bloch‟s theorem along with its significance.
9.
Illustrate the periodic potentials described by Kronig-Penney model
and draw conclusions in support of band theory of solids from the
model.
10. Explain the formation of Brillouin zones and list out the values of k,
for which second Brillouin zone exists.
11. Classify materials based on band theory of solids.
Review questions on electrical properties
II.
Conductors:
12. Explain macroscopic and microscopic forms of Ohm‟s law.
13. Explain Matthissen‟s rule with a supporting illustration.
14. Differentiate between random and drift velocity of electron in metals.
15. Define drift velocity and mobility of electron and also find the
relation between them.
16. Define conductivity and mobility of electron and also find the
relation between them.
Review questions on electrical properties
III. Semiconductors:
17. Illustrate the band structure of intrinsic and extrinsic
semiconductors.
18. Explain the key factors effecting carrier concentration.
19. Compare and contrast conductivity and mobility of charge
carriers in intrinsic and extrinsic semiconductors.
Review questions on electrical properties
IV.
Insulators and Dielectrics:
20.
List out various insulators and compare the band structure of insulators with
semiconductors and conductors.
21.
Recall important properties of insulators.
22.
Differentiate between insulators and dielectrics.
23.
Explain polar and non-polar dielectrics with an example for each type.
24.
Explain how dielectric materials can improve the charge storing capacity of a
parallel plate capacitor.
25.
Define dielectric constant and dipole moment.
26.
Define electric polarization and polarization vector.
27.
Define electric flux density (D), electric field strength (E) and polarization vector (P)
and establish relation among D,E and P.
Review questions on electrical properties
IV.
Insulators and Dielectrics (cont.):
28.
Define and explain dielectric strength and break down potential of dielectric
materials.
29.
Write a brief description of various types of polarization.
30.
Explain electronic polarization and write the expression for electronic polarization.
31.
Explain ionic polarization and write the expression for ionic polarization.
32.
Explain orientation polarization and write the expression for orientation
polarization.
33.
Discuss the frequency dependence of dielectric constant with the help of a neat
diagram.
Review questions on electrical properties
V. Ferroelectricity and Piezoelectricity:
34. Explain the concept of ferroelectricity along with its key
features.
35. Illustrate the hysteresis of ferroelectric materials along with
their applications.
36. Explain the concept of piezoelectricity with a neat diagram.
37. List out various industrial applications of piezoelectric
materials.
Problems on electrical properties
1. The resistivity of copper at 200C is 1.69x10-8 Ω-m and the
concentration of free electrons in copper is 8.5x1028m-3.
Calculate the relaxation time of electrons.
2. The collision time and the root mean square velocity of the
electron at room temperature are 2.5x10-14s and 1x105ms-1
respectively. Calculate the mean free path of the electron.
3. A copper wire of length 0.5m and diameter 0.3mm has a
resistance 0.12Ω at 200C. If the thermal conductivity of copper
at 200C is 390Wm-1K-1, calculate Lorentz number.
Problems on electrical properties
4.
Compute the electrical resistivity of sodium at 00C, if the mean free time at this
temperature is 3.1x10-14s. Furthermore, sodium builds a BCC lattice with two
atoms per unit cell, and the side of the unit cell is 0.429 nm.
5.
For intrinsic gallium arsenide, the room-temperature electrical conductivity is
10-6Ω-1m-1 the electron and hole mobilities are, respectively, 0.85 and 0.04m2/V-
s. Compute the intrinsic carrier concentration ni at room temperature.
6.
Consider a parallel-plate capacitor having an area of 6.45x10-4m2 and a plate
separation of 2x10-3m across which a potential of 10V is applied. If a material
having a dielectric constant of 6.0 is positioned within the region between the
plates, compute the capacitance and the magnitude of the charge stored on each
plate.