# UNIT 1 ELECTRONIC AND PHOTONIC MATERIALS

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29 Οκτ 2013 (πριν από 4 χρόνια και 8 μήνες)

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1

UNIT 1 ELECTRONIC AND PHOTONIC MATERIALS

LECTURE

1

:

IMPORTANCE

OF

CLASSICAL

AND

QUANTUM

THEORY

OF

FREE

ELECTRONS
.

LECTURE

2

:

FERMI
-

DIRAC

STATISTICS

SEMICONDUCTORS,

FERMI

ENERGY

LEVEL

VARIATION
.

LECTURE

3

:

HALL

EFFECT

AND

ITS

APPLICATION,

DILUTE

MAGNETIC

SEMICONDUCTORS

AND

SUPERCONDUCTOR

AND

ITS

CHARACTERISTICS
.

LECTURE

4
:

APPLICATIONS

OF

SUPERCONDUCTOR

AND

PHOTONIC

MATERIALS

LECTURE

5

:

PHOTOCONDUCTING

MATERIALS

LECTURE
6

:

NON

LINEAR

OPTICAL

MATERIALS

AND

APPLICATIONS

2

LECTURE 1

CONTENTS

BASIC

DEFINITION

IN

CONDUCTORS

CLASSIFICATION

OF

CONDUCTORS

IMPORTANCE

OF

CLASSICAL

AND

QUANTUM

FREE

ELECTRON

THEORY

OF

METALS

SCHRODINGER

EQUATIONS

3

ELECTRONIC AND PHOTONIC MATERIALS

The

detailed

knowledge

with

the

properties

of

materials

like

electrical,

dielectric,

conduction,

semi

conduction,

magnetic,

superconductivity,

optical

etc
.
,

is

known

as

`
Materials

Science

.

In

terms

of

electrical

properties,

the

materials

can

be

divided

into

three

groups

(
1
)

c
onductors

,(
2
)

semi

conductors

and

(
3
)

dielectrics

(or)

insulators
.

4

Electric

current

The

rate

of

flow

of

charge

through

a

conductor

is

known

as

the

current
.

If

a

charge

dq

flows

through

the

conductor

for

dt

second

then

Ohm’s

law

At

constant

temperature,

the

potential

difference

between

the

two

ends

of

a

conductor

is

directly

proportional

to

the

current

that

passes

through

it
.

where

R

=

resistance

of

the

conductor

5

Resistance

of

a

conductor

The

resistance

(
R
)

of

a

conductor

is

the

ratio

of

the

potential

difference

(
V
)

applied

to

the

conductor

to

the

current

(
I
)

that

passes

through

it
.

The

specific

resistance

(or)

resistivity

of

a

conductor

The

resistance

(
R
)

of

conductor

depends

upon

its

length

(
L
)

and

cross

sectional

area

(
A
)

i
.
e
.
,

or

where

is a proportional constant and is known as the
specific resistance (or ) resistivity of the material.

6

The

electrical

conductivity

is

also

defined

as”

the

charge

that

flows

in

unit

time

per

unit

area

of

cross

section

of

the

conductor

per

unit

potential

.

The

resistivity

and

conductivity

of

materials

are

pictured

as

shown

below,

Conductivities and resistivities of materials

7

Conductors

The

materials

that

conduct

electricity

when

an

electrical

potential

difference

is

applied

across

them

are

conductors
.

The resistivity of the material of a
conductor

is defined as
the
resistance of the material having unit length and unit
cross sectional area.

8

The

electrical

conductivity

(

)

of

a

conductor

The

reciprocal

of

the

electrical

resistivity

is

known

as

electrical

conductivity

(σ)

and

is

expressed

in

ohm

1

metre

1
.

The

conductivity

(

)

We

Know

that,

R

=

V/I

9

The conducting materials based on their conductivity
can be classified into three categories

1. Zero resistivity materials

2. Low resistivity materials

3.

High resistivity materials

1) Zero Resistivity Materials

Superconductors

like

alloys

of

aluminium,

zinc,

gallium,

nichrome,

niobium

etc
.
,

are

a

special

class

of

materials

that

conduct

electricity

almost

with

zero

resistance

below

transition

temperature
.

These

materials

are

known

as

zero

resistivity

materials
.

USES

Energy saving in power systems, super conducting
magnets, memory storage elements

10

2
)
.

Low

Resistivity

Materials

The

metals

and

alloys

like

silver,

aluminium

have

very

high

electrical

conductivity
.

These

materials

are

known

as

low

resistivity

materials
.

USES

Resistors,

conductors

in

electrical

devices

and

in

electrical

power

transmission

and

distribution,

winding

wires

in

motors

and

transformers
.

3
)

High

Resistivity

Materials

The

materials

like

tungsten,

platinum,

nichrome

etc
.
,

have

high

resistivity

and

low

temperature

co
-
efficient

of

resistance
.

These

materials

are

known

as

high

resistivity

materials
.

11

USES
:

Manufacturing

of

resistors,

heating

elements,

resistance

thermometers

etc
.
,

The

conducting

properties

of

a

solid

are

not

a

function

of

the

total

number

of

the

electrons

in

the

metal

as

only

the

valence

electrons

of

the

atoms

can

take

part

in

conduction
.

These

valence

electrons

are

called

free

electrons
.

Conduction

electrons

and

in

a

metal

the

number

of

free

electrons

available

is

proportional

to

its

electrical

conductivity
.

Hence

the

electronic

structure

of

a

metal

determines

its

electrical

conductivity
.

12

Free

Electron

Theory

The

electron

theory

explain

the

structure

and

properties

of

solids

through

their

electronic

structure
.

It

explains

the

binding

in

solids,

behaviour

of

conductors

and

insulators,

ferromagnetism,

electrical

and

thermal

conductivities

of

solids,

elasticity,

cohesive

and

repulsive

forces

in

solids

etc
.

Development

of

Free

Electron

Theory

The

classical

free

electron

theory

[Drude

and

Lorentz]

It

is

a

macroscopic

theory,

through

which

free

electrons

in

lattice

and

it

obeys

the

laws

of

classical

mechanics
.

Here

the

electrons

are

assumed

to

move

in

a

constant

potential
.

13

The quantum free electron theory

[Sommerfeld

Theory]

It is a microscopic theory, according to this theory the
electrons
in lattice moves in a constant potential and it obeys law of
quantum mechanics
.

Brillouin Zone Theory [Band Theory]

Bloch

developed this theory in which the
electrons move in a
periodic potential provided by periodicity of crystal lattice
.It
explains the mechanisms of conductivity, semiconductivity on
the basis of energy bands and hence band theory.

The Classical Free Electron Theory

According to kinetic theory of gases in a metal ,Drude
assumed free electrons are
as a gas of electrons.

14

Kinetic

theory

treats

the

molecules

of

a

gas

as

identical

solid

spheres,

which

move

in

straight

lines

until

they

collide

with

one

another
.

The

time

taken

for

single

collision

is

assumed

to

be

negligible,

and

except

for

the

forces

coming

momentarily

into

play

each

collision,

no

other

forces

are

assumed

to

act

between

the

particles
.

There

is

only

one

kind

of

particle

present

in

the

simplest

gases
.

However,

in

a

metal,

there

must

be

at

least

two

types

of

particles,

for

the

electrons

are

negatively

charged

and

the

metal

is

electrically

neutral
.

15

Drude

assumed

that

the

compensating

positive

charge

was

attached

to

much

heavier

particles,

so

it

is

immobile
.

In

Drude

model,

when

atoms

of

a

metallic

element

are

brought

together

to

form

a

metal,

the

valence

electrons

from

each

atom

become

detached

and

wander

freely

through

the

metal,

while

the

metallic

ions

remain

intact

and

play

the

role

of

the

immobile

positive

particles
.

16

In

a

single

isolated

atom

of

the

metallic

element

has

a

nucleus

of

charge

e

Z
a

as

shown

in

Figure

below
.

Figure represents Arrangement of atoms in a metal

where

Z
a

-

is

the

atomic

number

and

e

-

is

the

magnitude

of

the

electronic

charge

[e

=

1
.
6

X

10
-
19

coulomb]

surrounding

the

nucleus,

there

are

Z
a

electrons

of

the

total

charge

eZ
a
.

17

Some

of

these

electrons

Z
’,

are

the

relatively

weakly

bound

valence

electrons
.

The

remaining

(Z
a
-
Z)

electrons

are

relatively

tightly

bound

to

the

nucleus

and

are

known

as

the

core

electrons
.

These

isolated

atoms

condense

to

form

the

metallic

ion,

and

the

valence

electrons

are

allowed

to

wander

far

away

from

their

parent

atoms
.

They

are

called

`
conduction

electron

gas

or

`
conduction

electron

cloud

.

Due

to

kinetic

theory

of

gas

Drude

assumed,

conduction

electrons

of

mass

m

move

against

a

background

of

heavy

immobile

ions
.

18

The

density

of

the

electron

gas

is

calculated

as

follows
.

A

metallic

element

contains

6
.
023
X
10
23

atoms

per

mole

number)

and

ρ
m
/A

moles

per

m
3

Here

ρ
m

is

the

mass

density

(in

kg

per

cubic

metre)

and

A

is

the

atomic

mass

of

the

element
.

Each atom contributes ‘
Z
’ electrons, the number of
electrons per cubic metre.

The

conduction

electron

densities

are

of

the

order

of

10
28

conduction

electrons

for

cubic

metre,

varying

from

0
.
91
X
10
28

for

cesium

upto

24
.
7
X
10
28

for

beryllium
.

19

These

densities

are

typically

a

thousand

times

greater

than

those

of

a

classical

gas

at

normal

temperature

and

pressures
.

Due

to

strong

electron
-
electron

and

electron
-
ion

electromagnetic

interactions,

the

Drude

model

boldly

treats

the

dense

metallic

electron

gas

by

the

methods

of

the

kinetic

theory

of

a

neutral

dilute

gas
.

20

BASIC ASSUMPTION FOR KINETIC THEORY OF

A NEUTRAL DILUTE GAS

In the
absence of an externally applied electromagnetic
fields
, each electron is taken to move freely here and there
and it collides with other free electrons or positive ion cores.
This collision is known as elastic collision.

The neglect of
electron

electron interaction

between
collisions is known as the

independent electron
approximation

.

21

In

the

presence

of

externally

applied

electromagnetic

fields,

the

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
.

In

Drude

model,

due

to

kinetic

theory

of

collision,

that

abruptly

alter

the

velocity

of

an

electron
.

Drude

attributed

the

electrons

bouncing

off

the

impenetrable

ion

cores
.

Let

us

assume

an

electron

experiences

a

collision

with

a

probability

per

unit

time

1
/
τ

.

That

means

the

probability

of

an

electron

undergoing

collision

in

any

infinitesimal

time

interval

of

length

ds

is

just

ds/
τ
.

22

The

time

is

known

as

the

relaxation

time

and

it

is

defined

as

the

time

taken

by

an

electron

between

two

successive

collisions
.

That

relaxation

time

is

also

called

mean

free

time

[or]

collision

time
.

Electrons

are

assumed

to

achieve

thermal

equilibrium

with

their

surroundings

only

through

collision
.

These

collisions

are

assumed

to

maintain

local

thermodynamic

equilibrium

in

a

particularly

simple

way
.

Trajectory of a conduction electron

23

Success

of

classical

free

electron

theory

It

is

used

to

verify

ohm’s

law
.

It

is

used

to

explain

the

electrical

and

thermal

conductivities

of

metals
.

It

is

used

to

explain

the

optical

properties

of

metals
.

Ductility

and

malleability

of

metals

can

be

explained

by

this

model
.

24

Drawbacks

of

classical

free

electron

theory

From

the

classical

free

electron

theory

the

value

of

specific

heat

of

metals

is

given

by

4
.
5
R
,

where

R

is

called

the

universal

gas

constant
.

But

the

experimental

value

of

specific

heat

is

nearly

equal

to

3
R
.

With

help

of

this

model

we

can’t

explain

the

electrical

conductivity

of

semiconductors

or

insulators
.

The

theoretical

value

of

paramagnetic

susceptibility

is

greater

than

the

experimental

value
.

Ferromagnetism

cannot

be

explained

by

this

theory
.

25

At

low

temperature,

the

electrical

conductivity

and

the

thermal

conductivity

vary

in

different

ways
.

Therefore

K/
σ
T

is

not

a

constant
.

But

in

classical

free

electron

theory,

it

is

a

constant

in

all

temperature
.

The

photoelectric

effect,

Compton

effect

and

the

black

body

cannot

be

explained

by

the

classical

free

electron

theory
.

26

Quantum

free

electron

theory

deBroglie

wave

concepts

The

universe

is

of

and

matter(Particles)
.
The

light

exhibits

the

dual

nature(i
.
e
.
,)

it

can

behave

s

both

as

a

wave

[interference,

diffraction

phenomenon]

and

as

a

particle[Compton

effect,

photo
-
electric

effect

etc
.
,]
.

Since

the

nature

loves

symmetry

was

suggested

by

Louis

deBroglie
.

He

also

suggests

an

electron

or

any

other

material

particle

must

exhibit

wave

like

properties

in

to

particle

nature

27

In

mechanics,

the

principle

of

least

action

states”

that

a

moving

particle

always

chooses

its

path

for

which

the

action

is

a

minimum”
.

This

is

very

much

analogous

to

Fermat’s

principle

of

optics,

which

states

that

light

always

chooses

a

path

for

which

the

time

of

transit

is

a

minimum
.

de

Broglie

suggested

that

an

electron

or

any

other

material

particle

must

exhibit

wave

like

properties

in

to

particle

nature
.

The

waves

associated

with

a

moving

material

particle

are

called

matter

waves,

pilot

waves

or

de

Broglie

waves
.

28

Wave

function

A

variable

quantity

which

characterizes

de
-
Broglie

waves

is

known

as

Wave

function

and

is

denoted

by

the

symbol

.

The

value

of

the

wave

function

associated

with

a

moving

particle

at

a

point

(x,

y,

z)

and

at

a

time

‘t’

gives

the

probability

of

finding

the

particle

at

that

time

and

at

that

point
.

de

Broglie

wavelength

deBroglie

formulated

an

equation

relating

the

momentum

(p)

of

the

electron

and

the

wavelength

(

)

associated

with

it,

called

de
-
Broglie

wave

equation
.

h

p

where

h

-

is

the

planck’s

constant
.

29

Schrödinger

Wave

Equation

Schrödinger

describes

the

wave

nature

of

a

particle

in

mathematical

form

and

is

known

as

Schrödinger

wave

equation
.

They

are

,

1
.

Time

dependent

wave

equation

and

2
.

Time

independent

wave

equation
.

To

obtain

these

two

equations,

Schrödinger

connected

the

expression

of

deBroglie

wavelength

into

classical

wave

equation

for

a

moving

particle
.

The

obtained

equations

are

applicable

for

both

microscopic

and

macroscopic

particles
.

30

Schrödinger Time Independent Wave

Equation

The
Schrödinger's time independent wave equation
is
given by

For one
-
dimensional motion, the above equation becomes

31

Introducing,

In the above equation

For

three

dimension,

32

Schrödinger time dependent wave equation

The
Schrödinger time dependent wave equation
is

(or)

where

H

=

=

Hamiltonian

operator

=
Energy operator

E =

33

The salient features of quantum free electron theory

Sommerfeld

proposed

this

theory

in

1928

retaining

the

concept

of

free

electrons

moving

in

a

uniform

potential

within

the

metal

as

in

the

classical

theory,

but

treated

the

electrons

as

obeying

the

laws

of

quantum

mechanics
.

Based

on

the

deBroglie

wave

concept
,

he

assumed

that

a

moving

electron

behaves

as

if

it

were

a

system

of

waves
.

(called

matter

waves
-
waves

associated

with

a

moving

particle)
.

According

to

quantum

mechanics,

the

energy

of

an

electron

in

a

metal

is

quantized
.
The

electrons

are

filled

in

a

given

energy

level

according

to

Pauli’s

exclusion

principle
.

(i
.
e
.

No

two

electrons

will

have

the

same

set

of

four

quantum

numbers
.
)

34

Each

Energy

level

can

provide

only

two

states

namely,

one

with

spin

up

and

other

with

spin

down

and

hence

only

two

electrons

can

be

occupied

in

a

given

energy

level
.

So,

it

is

assumed

that

the

permissible

energy

levels

of

a

free

electron

are

determined
.

It

is

assumed

that

the

valance

electrons

travel

in

constant

potential

inside

the

metal

but

they

are

prevented

from

escaping

the

crystal

by

very

high

potential

barriers

at

the

ends

of

the

crystal
.

In

this

theory,

though

the

energy

levels

of

the

electrons

are

discrete,

the

spacing

between

consecutive

energy

levels

is

very

less

and

thus

the

distribution

of

energy

levels

seems

to

be

continuous
.

35

Success of quantum free electron theory

According

to

classical

theory,

which

follows

Maxwell
-

Boltzmann

statistics,

all

the

free

electrons

gain

energy
.

So

it

to

much

larger

predicted

quantities

than

that

is

actually

observed
.

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
.

According

to

quantum

free

electron

theory,

both

experimental

and

theoretical

values

of

Lorentz

number

are

in

good

agreement

with

each

other
.

36

Drawbacks of quantum free electron theory

It

is

incapable

of

explaining

why

some

crystals

have

metallic

properties

and

others

do

not

have
.

It

fails

to

explain

why

the

atomic

arrays

in

crystals

including

metals

should

prefer

certain

structures

and

not

others