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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.
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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
LECTURE 5 :PHOTOCONDUCTING MATERIALS
LECTURE6 :NON LINEAR OPTICAL MATERIALS AND APPLICATIONS
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) resistivityof a conductor
The resistance (R) of conductor depends upon its length (L) and cross sectional area (A) i.e.,
where is a proportional constant and is known as the specific resistance (or ) resistivity of the material.
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 asshown below,
Conductivities and resistivities of materials
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.
The reciprocal of the electrical resistivity is known as electrical conductivity (σ) and is expressed in ohm1 metre1.
The conductivity ()
We Know that, R = V/I
The conducting materials based on their conductivity can be classified into three categories
The metals and alloys like silver, aluminium have very high electrical conductivity. These materials are known as low resistivity materials.
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.
Kinetic theory treats the molecules of a gas as identical solid spheres, which move in straight lines until they collide with one another.
Drude assumed that the compensating positive charge was attached to much heavier particles, so it is immobile.
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.
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.
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
These densities are typically a thousand times greater than those of a classical gas at normal temperature and pressures.
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.
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.
Trajectory of a conduction electron
At low temperature, the electrical conductivity and the thermal conductivity vary in different ways. Therefore K/σT
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.
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.
where h - is the planck’s constant.
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.
The Schrödinger\'s time independent wave equation is given by
For one-dimensional motion, the above equation becomes
In the above equation
For three dimension,
The Schrödinger time dependent wave equation is
= Hamiltonian operator
where H =
= Energy operator
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.)
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.
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.
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