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
gradient”
.
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
(Avogadro’s
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
radiation
cannot
be
explained
by
the
classical
free
electron
theory
.
26
Quantum
free
electron
theory
deBroglie
wave
concepts
The
universe
is
made
of
Radiation(light)
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
addition
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
addition
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
leads
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
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