1

Department of Physics

K L University

ELECTRICAL PROPERTIES

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

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.

The Classical Free Electron Theory (Cont.)

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.

tiV

m

22

2

tiV

m

22

2

EH

Vm

2

2

2

ti

Schrödinger Time Dependent Wave Equation

The Schrödinger time dependent wave equation is

or

where H = , Hamiltonian operator

E = , 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

Consider the available energies for electrons in the materials.

As two atoms are brought close

together, electrons must occupy different

energies due to Pauli Exclusion principle.

Instead of having discrete energies

as in the case of free atoms, the

available energy states form bands.

Conductors, Insulators, and Semiconductors

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)

vdI

E

∆x

Random motion of electrons

In the absence of external field

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

Si

+4

Si

+4

Si

+4

Si

+4

Si

+4

Si

+4

Si

+4

Si

+4

Si

+4

The diagram above shows the 2D structure

of the Si crystal.

The light green lines represent the electronic

bonds made when the valence electrons are

shared.

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.

The extra valence electrons are introduced by

putting impurities or dopants into the silicon.

The dopants used to create an n-type material are

Group V elements. The most commonly used

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.

+4+4

+5

+4

+4+4+4

+4+4

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

The free electrons in n type silicon support the flow of current

This crystal has been doped with a pentavalent impurity

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.

This creates a hole (h+), or a positive charge that can

move around in the material.

Commonly used Group III dopants are aluminum,

boron, and gallium.

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.

+4+4

+3

+4

+4+4+4

+4+4

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:

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)

Extrinsic semiconductors:

Conductivity is mainly due to free

electrons in n-type semiconductors

n >> p

σ = neµe

Conductivity is mainly due to free

holes in p-type semiconductors

p >> n

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

room temperature

Filled valance band and empty conduction

band with relatively large band gap, (more

than 2 eV)

At room temperature, only very few

electrons may be excited across the band

gap

Very small values of conductivity

Material Electrical Conductivity

(Ω-m)-1

Ceramics

Dry concrete 10-9

Soda lime glass 10-10 – 10-11

Borosilicate glass 10-13

Fused silica 10-18

Polymers

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 capacitorQ = charge stored on each either plate

V = applied potential

Capacitor with vacuum 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

+

Electric field

Dielectric atom

+

+

+

+

+

+

+

+

_

_

_

_

_

_

_

__

dipole

Fundamental definitions

Dielectric constant (ϵr):

Dielectric Constant is the ratio between the permittivity of the

medium to the permittivity of free space

ϵr = 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 = α E

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 = ϵ E

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 = N µ

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

FactorElectronic

Polarization

Ionic

Polarization

Oriantational

Polarization

Definition

Electron

cloud shift

wrt nucleus

Cations &

anions are

shifted

Arrangement

of random

dipoles

Examples Inert gases Ionic crystalsAlcohol,

methane

Temperature Independent Independent 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), Pottasium dihydrogen phosphate(NH4H2PO4),

Rochelle salt(NaKC4H4O6.4H2O)

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

Industry Application

AutomotiveAir bag sensor, air flow sensor, audible alarms, fuel atomiser, keyless door entry, seat

belt buzzers, knock sensors.

Computer Disc drives, inkjet printers.

ConsumerCigarette 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.

Applications of piezoelectric materials

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.

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

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.

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

Engineering applications

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.

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

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.

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.

Review questions on electrical properties

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.

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.

Problems on electrical properties