Unit - IV
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Solid State Physics
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Que.1. Explain Opening of band gap due to
internal electron diffraction and hence
explain formation of bands in solids [6]
When an electron moves through a lattice of
positive ions, it experiences varying potentials.
The potential of an electron at the positive ion
site is zero and is, maximum in between two
ions as shown in figure below. The dotted line is representing the motion for a
free electron. As electron is moving in a
periodic potential, it appears that there are gaps
which open in energy spectrum at an integer of
These gaps correspond to energy bands inside
the solids. Thus, the energy level of electron in
the crystal lattice is discrete.
Formation of Bands:
1. In a single isolated atom, the electrons in
each orbit have definite energy associated with
it. But in case of solids all the atoms are close
to each other, so the energy levels of outermost
orbit electrons are affected by the neighboring
atoms.
2. Band formation is preferred in solids as
atomic spacing is very small. In liquids and
gases as atomic spacing is much larger as
Kronig and Penney approximated these compared to solids so band formation is not
potentials inside the crystal to the shape of preferred.
rectangular steps. 3. When two single or isolated atoms are bring
close to each other, then the outermost orbit
electrons of two atoms are interact or shared
with each other i.e. the electrons in the
outermost orbit of one atom experience an
attractive force from the nearest or neighboring
atomic nucleus. Due to this the energies of the
electrons will not be in same level, the energy
The energies of electrons can be known by levels of electrons are changed to a value which
solving Schrödinger’s wave equation in such a is higher or lower than that of the original
lattice. energy level of the electron.
By solving the Schrödinger’s wave equation 5. However, the energy levels of inner orbit
using Bloch theorem, we get Energy vs electrons are not much affected by the presence
K (momentum) diagram of neighboring atoms.
1 Sinhgad COE, Engineering Physics
, Que.2. Explain in brief how free electron theory it can be shown that 𝑘/σ𝑇=1.11x10-8
theory explains electrical conductivity and WΏ/K2.
thermal conductivity of solids, and relation
between electrical and thermal conductivity This number is also known as Lorentz’s
of solids (Wiedemann–Franz law). What are number. This number is based on
the limitations of this theory? [4] calculations as per Maxwell-Boltzman
distribution for electrons. If Fermi-Dirac
The electrons present in the outermost orbitals statistics is applied to electrons, this number
of the atom, determine the electrical properties is modified to 2.45x10-8 WΏ/K2., which
of the solid. The electron theory of solids agrees quite well with experimental
explains the structures and properties of solids findings.
through their electronic structure. The electron 6. Advantages (Success) of Free Electron
theory is applicable to all solids. The theory is Theory
developed in three phases: i) It verified Ohm’s law
ii) It explains electrical and thermal
1. This theory was developed by Drude and conductivities of metals.
Lorentz in 1900 and hence is also known as iii) It derived Wiedemann-Franz law (i.e.
Drude-Lorentz theory. According to this relation between electrical conductivity
theory, a metal consists of electrons which and thermal conductivity.
are free to move about in the crystal like iv) It explains optical properties of metals.
molecules of a gas in a container.
7. Disadvantages (Limitations) of Free
2. Neglecting the electron-electron interaction Electron Theory
and the electron-ion interaction, it appears
i) The phenomenon such as photoelectric
as if the electrons move in a confined box
effect, Compton effect, black body
with periodic collision with ions in the
radiations, etc could not be explained
lattice. This idea is utilized to explain many
properties of metals satisfactorily such as ii) The theoretical values of specific heat
electrical conductivity, Ohm’s law, thermal and electronic specific heat as calculated
conductivity etc. from this theory do not agree with
experimental values
3. Electrical conductivity of metal: It can be iii) Electrical conductivity of
shown that Electrical conductivity = semiconductors or insulators could not be
explained.
𝜎=𝑛𝑒𝜇, where n is electron density, e is
charge on electron and µ is the mobility of iv)Wiedemann-Franz law deviates at low
electrons. temperature.
v) Ferromagnetism could not be explained
4. Thermal Conductivity: Thermal by this theory.
conductivity is the property of material’s vi) The theoretical value of paramagnetic
ability to conduct the heat. From free susceptibility is greater than experimental
electron theory, it can be shown that value.
thermal conductivity𝐾=k𝐵𝑛𝑣λ/2, where kB
is Boltzmann constant. Que.3. Explain terms (a) valence band (b)
conduction band (c) band gap energy. [4]
5. Wiedemann-Franz Law:
The Wiedemann–Franz law states that the There are number of energy bands in solids but
ratio of the electronic contribution to the three of them are very important. These three
thermal conductivity (K) and the electrical energy bands are important to understand the
conductivity (σ) of a metal is proportional behavior of solids. These energy bands are -
to the temperature (T). From free electron
2 Sinhgad COE, Engineering Physics
---------------------------------------------------------
Solid State Physics
---------------------------------------------------------
Que.1. Explain Opening of band gap due to
internal electron diffraction and hence
explain formation of bands in solids [6]
When an electron moves through a lattice of
positive ions, it experiences varying potentials.
The potential of an electron at the positive ion
site is zero and is, maximum in between two
ions as shown in figure below. The dotted line is representing the motion for a
free electron. As electron is moving in a
periodic potential, it appears that there are gaps
which open in energy spectrum at an integer of
These gaps correspond to energy bands inside
the solids. Thus, the energy level of electron in
the crystal lattice is discrete.
Formation of Bands:
1. In a single isolated atom, the electrons in
each orbit have definite energy associated with
it. But in case of solids all the atoms are close
to each other, so the energy levels of outermost
orbit electrons are affected by the neighboring
atoms.
2. Band formation is preferred in solids as
atomic spacing is very small. In liquids and
gases as atomic spacing is much larger as
Kronig and Penney approximated these compared to solids so band formation is not
potentials inside the crystal to the shape of preferred.
rectangular steps. 3. When two single or isolated atoms are bring
close to each other, then the outermost orbit
electrons of two atoms are interact or shared
with each other i.e. the electrons in the
outermost orbit of one atom experience an
attractive force from the nearest or neighboring
atomic nucleus. Due to this the energies of the
electrons will not be in same level, the energy
The energies of electrons can be known by levels of electrons are changed to a value which
solving Schrödinger’s wave equation in such a is higher or lower than that of the original
lattice. energy level of the electron.
By solving the Schrödinger’s wave equation 5. However, the energy levels of inner orbit
using Bloch theorem, we get Energy vs electrons are not much affected by the presence
K (momentum) diagram of neighboring atoms.
1 Sinhgad COE, Engineering Physics
, Que.2. Explain in brief how free electron theory it can be shown that 𝑘/σ𝑇=1.11x10-8
theory explains electrical conductivity and WΏ/K2.
thermal conductivity of solids, and relation
between electrical and thermal conductivity This number is also known as Lorentz’s
of solids (Wiedemann–Franz law). What are number. This number is based on
the limitations of this theory? [4] calculations as per Maxwell-Boltzman
distribution for electrons. If Fermi-Dirac
The electrons present in the outermost orbitals statistics is applied to electrons, this number
of the atom, determine the electrical properties is modified to 2.45x10-8 WΏ/K2., which
of the solid. The electron theory of solids agrees quite well with experimental
explains the structures and properties of solids findings.
through their electronic structure. The electron 6. Advantages (Success) of Free Electron
theory is applicable to all solids. The theory is Theory
developed in three phases: i) It verified Ohm’s law
ii) It explains electrical and thermal
1. This theory was developed by Drude and conductivities of metals.
Lorentz in 1900 and hence is also known as iii) It derived Wiedemann-Franz law (i.e.
Drude-Lorentz theory. According to this relation between electrical conductivity
theory, a metal consists of electrons which and thermal conductivity.
are free to move about in the crystal like iv) It explains optical properties of metals.
molecules of a gas in a container.
7. Disadvantages (Limitations) of Free
2. Neglecting the electron-electron interaction Electron Theory
and the electron-ion interaction, it appears
i) The phenomenon such as photoelectric
as if the electrons move in a confined box
effect, Compton effect, black body
with periodic collision with ions in the
radiations, etc could not be explained
lattice. This idea is utilized to explain many
properties of metals satisfactorily such as ii) The theoretical values of specific heat
electrical conductivity, Ohm’s law, thermal and electronic specific heat as calculated
conductivity etc. from this theory do not agree with
experimental values
3. Electrical conductivity of metal: It can be iii) Electrical conductivity of
shown that Electrical conductivity = semiconductors or insulators could not be
explained.
𝜎=𝑛𝑒𝜇, where n is electron density, e is
charge on electron and µ is the mobility of iv)Wiedemann-Franz law deviates at low
electrons. temperature.
v) Ferromagnetism could not be explained
4. Thermal Conductivity: Thermal by this theory.
conductivity is the property of material’s vi) The theoretical value of paramagnetic
ability to conduct the heat. From free susceptibility is greater than experimental
electron theory, it can be shown that value.
thermal conductivity𝐾=k𝐵𝑛𝑣λ/2, where kB
is Boltzmann constant. Que.3. Explain terms (a) valence band (b)
conduction band (c) band gap energy. [4]
5. Wiedemann-Franz Law:
The Wiedemann–Franz law states that the There are number of energy bands in solids but
ratio of the electronic contribution to the three of them are very important. These three
thermal conductivity (K) and the electrical energy bands are important to understand the
conductivity (σ) of a metal is proportional behavior of solids. These energy bands are -
to the temperature (T). From free electron
2 Sinhgad COE, Engineering Physics
