Wave Optics:
Wave nature of light, including interference, diffraction, and polarization.
Double-slit experiment and Young's modulus.
Fresnel and Fraunhofer diffraction.
Quantum Mechanics:
Wave-particle duality and the uncertainty principle.
Schrödinger equation and its solutions.
Quantum s...
Band Theory of Solids is a quantum mechanical model of electrons in solids which
envisages certain constrained ranges or the bands, for the energies of the electrons.
It is also called as energy band theory of solids.
The electron band theory of solids basically describes the quantum states that electrons
can take inside metal solids.
The overlap of the electron probability distributions of all the individual atoms in the
metal solid leads to a creation of a continuous band of energies.
The ranges of allowed energies of electrons in a solid are called allowed bands.
Bands of energies between two such allowed bands are called forbidden bands which
implies that electrons within the solid cannot be allowed to possess these energies. Band
theory accounts for many of the electrical and thermal properties of solids. Solids can be
categorised into conductors, semiconductors or insulators by their ability to conduct
electricity.
Electron band theory explains differences in conductivity of these solids.
Band theory was also successful in giving us an insight into theoretical understanding of
semiconductors and their physical properties.
FORMATION OF ENERGY BANDS IN SOLIDS
A single isolated atom has discrete energy levels.
When two identical atoms are considered to be far apart, the electron energy levels in an
individual atom are not affected by the presence of the other.
As long as the atoms are widely separated, they have identical energy levels; electrons fill
the levels in each atom independently.
But when the atoms are brought closer, they begin to interact strongly and as a result,
each isolated energy level will be transformed into two energy levels of similar energies.
Transformation of single energy level into two or more separate energy levels is defined
as the energy level splitting.
Similarly, if we consider N atoms, their isolated energy levels will be split into N energy
levels. These N energy levels are so close to each other that form a near continuum.
Therefore, when atoms are brought together to form a solid, their energy levels split up
and form a group of closely space allowed energy levels of same energy value. This
group of closely spaced energy levels of same energy is called Energy band.
The concept of energy level splitting and formation of valence band and conduction band
is illustrated in Fig.3.1
1
, Engineering Physics
Fig.3.1 Energy level splitting and formation of valence band and conduction band.
When atoms are brought together, application of Pauli’s exclusion principle becomes
important.
It states that no two electrons can have their entire quantum numbers same. Hence an
energy level can accommodate at the most two electrons of opposite spin.
The degree of splitting of energy levels depends on the depth in an atom.
The energy levels of core electrons belonging to inner shells split to a lesser degree and
hence they form a narrow core band. They are always full and do not take part in the
conduction process.
The energy levels occupied by valence electrons split more and form wider bands.
Energy levels above the valence levels also split though they are not occupied.
While occupying a band, electron starts from lowest energy level and fill the levels in the
ascending order of energy (Aufbau’s Principle).
QUE: Explain formation of energy bands (in solids) on the basis of band theory of solids.
(4)[Summer-05, 07]
VALENCE BAND, CONDUCTION BAND AND ENERGY GAP
VALENCE BAND:
The Energy band occupied by valence electrons that are involved in covalent bonding, is
called as valence band.
Depending upon the number of valence electrons this band may get partially or
completely filled.
At absolute zero, covalent bonds are complete, therefore valence band is completely
filled.
CONDUCTION BAND:
The energy band above the valence band, having free electrons responsible for electrical
conduction, is called as conduction band.
At absolute zero, this energy band is empty.
ENERGY GAP:
The energy interval between top of the valence band and bottom of the conduction band
which is empty and forbidden is called energy gap or band gap.
2
, Engineering Physics
It is the special characteristic of semiconductor material.
It is the minimum amount of energy required for breaking a covalent bond and to
excite an electron from valence band to conduction band.
CLASSIFICATION OF SOLIDS BASED ON BAND THEORY
Solids can be classified into conductors, semiconductors or insulators depending upon
width of Energy gap.
Completely filled bands contain large number of electrons but do not contribute to the
conductivity of the material.
Partially filled bands are necessary for electrical conduction.
The energy band diagram of conductors, semiconductors or insulators is shown in figure
3.2.
Conductors:
The solids in which conduction and valence band overlap each other are called
conductors. Therefore, the energy gap between valence band and conduction band is zero.
Electrons can easily jump from lower energy band to higher one and become available for
conduction.
An application of a small amount of voltage leads to generation of large amount of
current.
Hence these solids are good electrical conductors. For e.g. Lithium, Berylium and
sodium.
Semiconductors:
The solids in which the conduction and valence bands are separated by a small energy gap
of less than 2eV are called semiconductors.
For e.g. Semiconductors like Silicon has Bandgap of 1.12 eV and Germanium has
bandgap of 0.72 eV.
A small energy gap means that a small amount of energy is required to free the electrons
and move them from the valence band to the conduction band.
The semiconductors behave like insulators at 0K, because valence electrons do not have
required energy to jump to the conduction band.
If the temperature is increased, valence electrons acquire sufficient energy to jump into
the conduction band.
Therefore, the conductivity of semiconductors increases with the increase in temperature.
Insulators:
The solids in which the conduction band and valence bands are separated by a large
energy gap of ≥ 3 eV are called insulators.
At room temperature, the valence electrons do not have enough energy to jump into the
conduction band, therefore insulator do not conduct current.
Thus, insulators have very high resistivity and extremely low conductivity at room
temperatures. For e.g. Diamond and glass.
3
, Engineering Physics
Figure3.2: Energy band diagram of conductor, semiconductor and insulator
QUE: Explain classification of solids on the basis of energy band diagrams.
(3) [Summer-17]
QUE: Discuss classification of solids on the basis of forbidden energy gaps.
(3) [Summer-18]
FERMI LEVEL
Fermi level is defined as the highest filled energy level in a conductor at 0K.
At 0K, all the levels below Fermi level are completely filled with electrons and all the
levels above Fermi levels are completely empty.
But at high temperature there is a possibility that some of the electrons from levels below
Fermi level then gets transferred (jump) to the levels above Fermi level.
Fermi Energy: Fermi energy is the maximum energy that a free electron can have in a
conductor at 0K.It is the energy associated with the Fermi level.
FERMI DIRAC DISTRIBUTION FUNCTION F(E)
Fermi Dirac distribution function gives the probability that any energy level ‘E’ at given
temperature T is occupied or not.
1
𝑓(𝐸) = (𝐸−𝐸𝐹 )
1 + exp [ ]
𝑘𝑇
E – Energy level for which occupancy is to be determined.
EF – Fermi level.
k – Boltzmann constant.
T – Temperature at which occupancy is to be determined.
When f (E ) = 0; it indicates that energy level E is completely empty.
When f (E ) = 1; it indicates that energy level E is completely filled.
QUE: What is meant by Femi-Dirac Distribution function? Define Fermi level at 0K.
QUE: What is fermi function?
VARIATION OF FERMI FUNCTION F(E) WITH TEMPERATURE
The Fermi function is given by
4
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