Transcript Lecture 1 - SRM University

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
LECTURE6 :
NON
LINEAR
APPLICATIONS
OPTICAL
MATERIALS
AND
1
LECTURE 1
CONTENTS
• BASIC DEFINITION IN CONDUCTORS
• CLASSIFICATION OF CONDUCTORS
• IMPORTANCE OF
CLASSICAL AND
QUANTUM FREE ELECTRON THEORY
OF METALS
• SCHRODINGER EQUATIONS
2
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) conductors ,(2) semi conductors and (3) dielectrics
(or) insulators.
3
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
Electric current (I)
Ohm’s law
dq
dt
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
V I (or)V  IR
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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.
Resistance ( R ) 
V
I
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.,
L
or R   L
R
A
A
where  is a proportional constant and is known as the
specific resistance (or ) resistivity of the material.
5
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,
Resistivity ( ohm metre )
10
5
10
12
10

10
Insulators
10
5
10
12

10

10
1
Semiconductors
10

10

10

( ohm1 metre

10

Metals
1
1
10

10

)
Conductivities and resistivities of materials
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Conductors
The materials that conduct electricity when an
electrical potential difference is applied across them are
conductors.
 R
A
ohm metre
L
The resistivity of the material of a conductor is defined as
the resistance of the material having unit length and unit
cross sectional area.
7
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,
1


1
RA

L


L
RA
R = V/I
Q
 t
L
I
 


V A V A V
I
L
L



A
     
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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
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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.
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USES:
Manufacturing of resistors,
resistance thermometers etc.,
heating
elements,
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.
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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.
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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.
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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.
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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.
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In a single isolated atom of the metallic element has a
nucleus of charge e Za as shown in Figure below.
Figure represents Arrangement of atoms in a metal
where Za - 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 Za electrons of the total charge –eZa.
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Some of these electrons ‘Z’, are the relatively weakly bound
valence electrons. The remaining (Za-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.
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The density of the electron gas is calculated as
follows. A metallic element contains 6.023X1023 atoms per
mole (Avogadro’s number) and ρm/A moles per m3
Here ρm 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
m Z
electrons per cubic metre.
N
.
n
V
A
The conduction electron densities are of the order of
1028 conduction electrons for cubic metre, varying from
0.91X1028 for cesium upto 24.7X1028 for beryllium.
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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.
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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”.
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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.
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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
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Success of classical free electron theory
It is used to verify ohm’s law.
It is used to explain
conductivities of metals.
the
electrical
and
thermal
It is used to explain the optical properties of metals.
Ductility and malleability of metals can be explained by this
model.
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Drawbacks of classical free electron theory
From the classical free electron theory the value of
specific heat of metals is given by 4.5R, where ‘R’ is called
the universal gas constant. But the experimental value of
specific heat is nearly equal to 3R.
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.
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At low temperature, the electrical conductivity and the
thermal conductivity vary in different ways. Therefore K/σT
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.
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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
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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.
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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.
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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
microscopic and macroscopic particles.
for both
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Schrödinger Time Independent Wave Equation
The Schrödinger's time independent wave equation is
given by
2
8

m
2
 
( E  V )  0
2
h
For one-dimensional motion, the above equation becomes
d 2
dx 2

8 m
2
h2
( E  V )  0
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Introducing,
h
 
2
In the above equation
d 2
2m

( E  V )  0
2
2
dx

For three dimension,
 
2
2m

2
( E  V )  0
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Schrödinger time dependent wave equation
The Schrödinger time dependent wave equation is
2

2

    V  i
2m
t

2
2


 V

2
m



  i

t

(or)
H  E
where H =
 2 2
 V
2m

E = i
t
= Hamiltonian operator
= Energy operator
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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.)
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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.
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Success of quantum free electron theory
According to classical theory, which follows MaxwellBoltzmann 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.
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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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