INTRODUCTION

Lecture-1

The electron theory of solids aims to explain the structures and properties of solids through their

electronic structure.

The electron theory of solids has been developed in three main

stages.www.calicutbtech.in

(i). The classical free electron theory:

Drude and Lorentz developed this theory in

1900. According to this theory, the metals

containing free electrons obey the laws of

classical mechanics.

(ii). The Quantum free electron theory:

Sommerfeld developed this theory during

1928. According to this theory, the free

electrons obey quantum laws.-

(iii). The Zone theory:

Bloch stated this theory in 1928.

According to this theory, the free

electrons move in a periodic field

provided by the lattice. This theory

is also called “Band theory of solids”.

The classical Free Electron Theory of

Metals (Drude - Lorentz theory of metals

postulates :(a). In an atom electrons revolue around the nucleus and

a metal is composed of such atoms.

(b). The valence electrons of atoms are free to move about the whole volume of the metals like the molecules of a perfect gas in a container. The collection of valence electrons

from all the atoms in a given piece of metal forms electrons gas. It is free to move throughout the volume of the metal

Lecture-2

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(c) These free electrons move in random directions and

collide with either positive ions fixed to the lattice or

other free electrons. All the collisions are elastic i.e.,

there is no loss of energy.

(d). The movements of free electrons obey the laws of

the classical kinetic theory of gases.

(e). The electron velocities in a metal obey the classical

Maxwell – Boltzmann distribution of velocities.

(f). The electrons move in a completely

uniform potential field due to ions fixed in

the lattice.

(g). When an electric field is applied to the

metal, the free electrons are accelerated in

the direction opposite to the direction of

applied electric field.

Success of classical free electron

theory:

(1). It verifies Ohm’s law.

(2). It explains the electrical and thermal conductivities of metals.

(3). It derives Wiedemann – Franz law. (i.e., the relation between electrical conductivity and thermal conductivity)

(4). It explains optical properties of metalsl.

Drawbacks of classical free electron

theory:

1. The phenomena such a photoelectric effect, Compton effect and the black body radiation couldn’t be explained by classical free electron theory.

2. According to the classical free electron theory the value of specific heat of metals is given by 4.5Ru is the Universal gas constant whereas the experimental value is nearly equal to 3Ru. Also according to this theory the value of electronic specific heat is equal to 3/2Ru while the actual value is about 0.01Ru

only.

3.Electrical conductivity of semiconductor or insulators couldn’t be explained using this model.

4. Though K/σT is a constant (Wiedemann –Franz Law) according to the Classical free electron theory, it is not a constant at low temperature.

5. Ferromagnetism couldn’t be explained by this theory. The theoretical value of paramagnetic susceptibility is greater than the experimental value. www.calicutbtech.in

Mean free path Lecture-3

The average distance traveled by an electron

between two successive collisions inside a

metal in the presence of applied field is known

as mean free path.

Relaxation Time

The time taken by the electron to

reach equilibrium position from

its disturbed position in the

presence of an electric field is

called relaxation time.

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Drift velocity

• In the presence of electric field, in

addition to random velocity there is an

additional net velocity associated with

electrons called drift velocity.

• Due to drift velocity, the electrons with

negative charge move opposie to the

field direction.

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Quantum free electron TheoryLecture-4

• According to quantum theory of free electrons energy of a free electron is given by

• En = n2h2/8mL2

• According to quantum theory of free electrons the electrical conductivity is given by

• σ = ne2T/m

Fermi LevelLecture-5

• “The highest energy level that can be

occupied at 0K” is called Fermi level.

• At 0K, when the metal is not under the

influence of an external field, all the levels

above the Fermi level are empty, those

lying below Fermi level are completely

filled.

• Fermi energy is the energy state at which the

probability of electron occupation is ½ at any

temperature above 0k.

Fermi-Dirac statisticsLecture-6

According to Fermi Dirac statistics, the

probability of electron occupation an

energy level E is given by

F(E) = 1/ 1+exp (E-EF/kT)

Electrical Resistivity Lecture-7

• The main factors affecting the electrical conductivity of solids are i) temperature and ii) defects (i.e. impurities).

• According to Matthiesens’s rule, the resistivity of a solid is given by

ρpure= ρpure+ ρimpurity

where ρpure is temperature dependent resistivity due to thermal vibrations of the lattice and ρimpurity is resistivity due to scattering of electrons by impurity atoms.

CLASSIFICATION OF MATERIALSLecture-8

• Based on „band theory‟, solids

can be classified into three

categories, namely,

1. insulators,

2. semiconductors &

3. conductors.

INSULATORS

• Bad conductors of electricity

• Conduction band is empty and

valence band is full, and these band

are separated by a large forbidden

energy gap.

• The best example is Diamond with

Eg=7ev.

SEMI CONDUCTORS

• Forbidden gap is less

• Conduction band an d valence band

are partially filled at room

temperature.

• Conductivity increases with

temperature as more and more

electrons cross over the small energy

gap.

• Examples Si(1.2ev) & Ge(0.7ev)

CONDUCTORS

• Conduction and valence bands are overlapped

• Abundant free electrons already exist in the conduction band at room temperature hence conductivity is high.

• The resistively increases with temperature as the mobility of already existing electrons will be reduced due to collisions.

• Metals are best examples.

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EFFECTIVE MASSLecture-9

• Def : When an electron in a periodic

potential of lattice is accelerated by

an electric field or magnetic field, then

the mass of the electron is called

effective mass.

• It is denoted by m*

m* = ћ2/(d2E/dk2)

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