This document discusses band theory and its application to understanding the properties of materials. It begins by introducing classical and quantum free electron theories, which treat electrons in solids as free particles. The behavior of electrons in periodic potentials is then described, leading to the development of band theory. Band theory explains that the discrete energy levels of isolated atoms merge into continuous energy bands as atoms form solids. This allows classification of materials as conductors, semiconductors, or insulators based on whether their energy bands are fully filled, partially filled, or fully empty. Band formation in silicon is provided as an example. The document concludes that band theory determines a material's ability to conduct electricity based on its energy band structure.

1. BAND THEORY
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2. Content
• Introduction
• Free Electron Theory
• Behaviour of an Electron in Periodic Potential
• Kronig-Penney Potential
• Band Theory
• Band Formation in Silicon
• Classification of Materials based on Energy Band
• Conclusion
• References
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3. Introduction
• Why Band Theory?
• Conductors/ Semiconductors/
Insulators
• Physical properties of materials
(Electrical resistivity, Optical
absorption)
• Foundation in understanding all solid-
state devices (Transistors, Solar cells)
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4. Free Electron Theory
• Classical free electron theory
(Macroscopic)
• Proposed by Paul Drude in 1900 and
elaborated (after the discovery of electron by
J J Thomson) by Lorentz in 1909.
• Drude and Lorentz theory.
• Metals contains free electrons which are
responsible for the electrical conductivity in
metals and obeys the laws of classical
mechanics.
Paul Drude
Hendrik Lorentz
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5. • boundaries of the neighboring atoms
slightly overlap with each other.
• Due to this overlapping, the valance
electrons of all the atoms are free to move
(Free electron) within the metal lattice,
randomly in all direction with an average
speed of the order 106 m/s.
• This is similar to the motion of gas
molecules confined in a vessel.
• The free electrons are responsible for
electrical and thermal conduction in metals,
they are also called as conduction electrons.
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6. • Assumptions
• All metals contain large number of free electrons.
• The free electrons are treated as equivalent to gas molecules; the laws of
classical kinetic theory of gases can be applied to them. Therefore these
electrons have mean free path (λ), mean collision time (T), average speed (v).
• Since the motion of the electrons is random, the net current is zero in the
absence of electric field. But when an electric field is applied, current is
produced due to the drift velocity of the electrons.
• The electric field (or Potential) due to positive ionic cores is considered to be
uniform throughout the metal and hence neglected.
• The force of attraction between the electrons & lattice ions and the force of
repulsion between the electrons themselves are considered to be negligible.
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7. • Success of classical free electron theory
• It verifies Ohm’s law
• It derives Wiedemann-Frenz theory
• It explains Electrical and Thermal conductivity of metals
• It explains optical properties of metals
• Drawbacks
• According to classical free electron theory Cv is independent of temperature, but the
experimental value of Cv is directly proportional to temperature
• Experimental value of mean free path (λ) is found to be 0.285 nm, which is 10 times
less than the value obtained from classical free electron theory
• The actual relationship between Temperature and thermal conductivity deviates from
classical free electron theory
• According to classical free electron theory; bivalent & trivalent metals should posses
much higher electrical conductivity than monovalent metals. This is contrary to the
experimental observations that the monovalent element metals such as silver is more
conducting than Zinc (bivalent) & aluminum (trivalent)
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8. Free Electron Theory
• Quantum free electron theory
(microscopic)
• Developed by Sommerfeld in 1928.
• Free electrons move with a constant potential
obeys quantum laws
• Electron as a quantum particle
• Retains the vital features of classical free
electron theory and included the Pauli
Exclusion Principle & Fermi-Dirac statistics
Arnold Sommerfeld
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9. • The free electrons in a metal can have only discrete energy values. Thus the
energies are quantized
• The electrons obey Pauli’s Exclusion Principle, which states that there cannot be
more than two electrons in any energy level
• The distribution of electrons in various energy levels obey the Fermi-Dirac
quantum statistics
• Free electrons have the same potential energy everywhere within the metal,
because the potential due to ionic cores is uniform throughout the metal
• The force of attraction between electrons & lattice ions and the force of repulsion
between electrons can be neglected
• Electrons are treated as wave-like particles
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10. • Success of Quantum free electron theory
• The predicted value of specific heat capacity agrees well with the experimental
results
• Thus quantum free electron theory properly explains the dependence of
Thermal Conductivity on Temperature
• Electrical conductivity depends on electron concentration, mean free path and
Fermi velocity, Explains why the conductivity of Copper is higher than
Aluminum
• Drawback
• In real case, electron moves as a periodic potential function.
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11. Behaviour of an Electron in Periodic
Potential
• Nearly Free electron theory
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1D periodic crystal potential

12. • To find out the motion of electron in
crystalline solid, Bloch has solved the
Schrodinger equation for periodic
boundary condition
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Felix
Bloch

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Travelling Part
Periodic Function

14. Kronig-Penney Potential
• Rectangular array of potentials
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Ralph Kronig William Penney
Braggs Law
nλ = 2dSinθ
1D crystal θ=900, λ=2π/a
n π/a = d

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Allowed and forbidden
energy band in E-k Diagram
No solution for
Schrodinger
equation

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17. • Free electron theory vs Nearly free electron theory
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• Constant Potential
• Continuous Parabolic Curve
• Periodic Potential
• Discontinuous Curve

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Variation of potential energy of a
conduction electron in the field of a ion
cores of a linear lattice
Distribution of probability density
of standing wave
+
_
Travelling wave

20. Band Theory
• In 1928-1931, BLOCH/ WILSON/ PEIERLS move
on to solids and invent band theory. they gave
convincing explanation of metallic insulating and
semiconducting behaviour of solids.
• A solid is made up of enormous number of closely
packed atoms. When these atoms are isolated they
have discrete set of energy levels as 1s,2s,2p etc.
• To form a solid many isolated atoms are brought
together, then a continuously increasing interactions
occurs between them so that the split energy levels
form essentially continuous bands of energies.
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Rudolf Peierls
Alan Wilson

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22. Band Formation in Silicon
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23. Classification of Materials based on Energy
Band
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24. Properties Conductor Semi-conductor Insulator
Definition A conductor is a material
that allows the flow of
charge when applied with
a voltage
A semiconductor is a
material whose
conductivity lies between
conductor & insulator
An insulator is a material
that does not allow the
flow of current
Temperature Dependence The resistance of a
conductor increases with
an increase in
temperature
The resistance of a
semiconductor decrease
with increases in
temperature. Thus it acts
as an insulator at absolute
zero
Insulator has very high
resistance but it still
decreases with
temperature
Conductivity Conductors have
very high conductivity
(10-7 Ʊ/m), thus they can
conduct electrical current
easily
They have intermediate
conductivity
(10-7 Ʊ /m to 10-13 Ʊ /m),
thus they can acts as
insulator & conductor at
different conditions
They have very low
conductivity
(10-13 Ʊ /m), thus they do
not allow current flow
Resistivity Low
(10-5 Ω/m)
Normal
(10-5Ω/m to 105 Ω/m)
Very High
(105 Ω/m)
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25. Conclusion
• Band theory specifies the types of solid’s, i.e., we have the insulator, the
conductors and the semi conductors.
• If a material takes more energy to conduct electricity through it then it is a
Insulator, it also says that a material which takes very less or no energy
(theoretically) is a conductor and material which takes more energy than a
conductor and less energy than the Insulator is categorized as semi-conductor.
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26. THANK YOU
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27. • Fermi-Dirac Statistics
• It deals with the system constituted by identical , non identifiable particles
having odd half integer spins. These particles are called Fermions.
• The angular momentum of fermions are ħ/2, 3ħ/2, 5ħ/2,…….
• The spins of Fermions are 1/2, 3/2, 5/2,….
• These particles obeys the Pauli exclusion principle.
• The wave function associated with Fermions are anti-symmetric.
• Pauli's Exclusion Principle
• No two electrons in the same atom can have identical values for all four of
their quantum numbers.
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