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UNIT 1 ELECTRONIC AND PHOTONIC MATERIALS LECTURE 1 : IMPORTANCE OF CLASSICAL AND QUANTUM

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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.

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 OPTICAL MATERIALS AND APPLICATIONS

LECTURE 1

CONTENTS

- BASIC DEFINITION IN CONDUCTORS
- CLASSIFICATION OF CONDUCTORS
- IMPORTANCE OF CLASSICAL AND QUANTUM FREE ELECTRON THEORY OF METALS
- SCHRODINGER EQUATIONS

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.

- 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

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.,

or

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 electrical conductivity () of a conductor

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

- 1. Zero resistivity materials
- 2. Low resistivity materials
- High resistivity materials
- 1) Zero Resistivity Materials
- Superconductorslike 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

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.

- 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 atomscan 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.

- 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.

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.

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.

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.

Figure represents Arrangement of atoms in a metal

- In a single isolated atom of the metallic element has a nucleus of charge e Zaas shown in Figure below.

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.

- 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.

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 electrons per cubic metre.
- 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.

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.

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”.

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.

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

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.

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.

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.

- 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

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.

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.

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.

Schrödinger Time Independent WaveEquation

The Schrödinger\'s time independent wave equation is given by

For one-dimensional motion, the above equation becomes

Schrödinger time dependent wave equation

The Schrödinger time dependent wave equation is

(or)

= Hamiltonian operator

where H =

= Energy operator

E =

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.)

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.

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.

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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