Physics devices function property MATERIALS.pdf

Physics of Functional Materials & Devices
Prof. Amreesh Chandra
Department of Physics, IIT KHARAGPUR
Module 03: Introduction to theory of solids
Lecture 14 : Theory of Solids
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 Definitions and Classifications of Solids
 Drude-Lorentz’s Classical Theory
 Sommerfeld’s Quantum Theory
Applications of Free Electron Gas Model
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Definition of solids:
 A solid is a state of matter which has a fixed shape, mass, and volume. It shows very small changes in volume by
changing the temperature.
 According to the nature of band occupation by electrons, all solids can be broadly classified into three types.
Metals Semiconductors Insulators
Solids
 The first group includes a completely filled valance band overlapping with a partially filled
conduction band (Metal).
 Depending on the width of the forbidden band, the second group can be divided into:
 Insulator (band gap 𝑬𝑬𝒈𝒈 > 𝟑𝟑 eV)
 Semiconductor (band gap 𝑬𝑬𝒈𝒈 ≤1.5 eV)
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Difference between metal, insulator, and semiconductor in terms of energy band
 In general, metal is any class of substances characterized by high electrical and thermal
conductivity as well as by malleability, ductility, and high reflectivity of light.
 The theory of metals are postulated by Drude in 1900.
 The theory was named as free electron gas model.
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Postulates of free electron gas model:
 The metals consist of positive ion cores with the valance electrons moving freely among these cores.
 The electrons are bound to move within the metal, due to electrostatic attraction between the positive ion cores and
electrons.
 The potential field of these ion cores, which is responsible for such an attraction, is assumed to be constant throughout the
metal.
 The mutual repulsion, among the electrons, is neglected.
 The behavior of free electrons inside the metals is considered similar to that of atoms or molecules in a perfect gas.
Hence, it is called free electron gas.
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Free Electron Gas Model
Definitely, there are some basic differences
between ordinary gas and free electron gas.
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Basic differences between ordinary gas and free electron gas
 The free electron gas is negatively charged, whereas the molecules of an ordinary gas are mostly neutral.
 The concentration of electrons in an electron gas is quite large compared to the concentration of molecules in an ordinary
gas.
 The valance electrons are responsible for the conduction of electricity through metals, hence called conduction electrons.
 Since the conduction electrons move in a uniform electrostatic field of ion cores, their potential energy remains constant
and is normally taken as zero, ignoring the existence of ion cores.
 As the movement of electrons is restricted to within the crystal only, the potential energy of a stationary electron inside a
metal is less than the potential energy of an identical electron just outside it.
 The energy difference V0 serves as a potential barrier and stops the inner electrons from
leaving the surface of the metal.
Some important points about free electron gas
 Thus, the movement of a free electron in metal is similar to the movement of a free electron
gas inside a “potential energy box”.
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Fig. Metallic surface bounded by a potential barrier V0
Free Electron
Gas Model
Successfully
explained
 Lorentz in 1909, postulated that the electrons constituting the electrons
constituting the electron gas obey Maxwell-Boltzman statistics.
 The Drude-Lorentz theory of the electron gas is called classical theory.
 The model is successful in explaining various properties of metals.
Failure of Free
Electron Gas
Model Can not
explain
 Temperature dependence resistivity
 Heat capacity
 Paramagnetic susceptibility, etc.
 Electrical conductivity
 Thermal conductivity
 Thermionic emission
 Thermoelectric effect
 Galvanomagnetic effect
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 To overcome the failure of the Free electron gas model, Sommerfeld’s quantum theory was adopted.
 Sommerfeld treated the problem quantum mechanically using the Fermi-Dirac statistics.
Free Electron Gas in One-Dimensional box
Potential
Fig. one dimensional potential box
V0 (for x ≤ 0 and x ≥ L
V(x) =
0 (for 0 ˂ x ˂L)
𝒅𝒅𝟐𝟐
𝝍𝝍(𝒙𝒙)
𝒅𝒅𝒅𝒅𝟐𝟐
+
𝟐𝟐𝒎𝒎
𝒉𝒉𝟐𝟐
𝑬𝑬𝝍𝝍 𝒙𝒙 = 𝟎𝟎
The Schrödinger takes the following form inside the crystal:
By solving the equation with the proper boundary conditions: (i) ψ 𝑥𝑥 = 0 = 0
(ii) ψ 𝑥𝑥 = 𝐿𝐿 = 0
The general solution to this equations is:
ψ 𝑥𝑥 = 𝐴𝐴 𝑠𝑠𝑠𝑠𝑠𝑠(𝑘𝑘𝑘𝑘) + 𝐵𝐵 cos(𝑘𝑘𝑘𝑘)
(1)
(2)
The solution of eq. 1 in the region 0 ˂ x ˂a
becomes,
𝝍𝝍𝒏𝒏 = 𝑨𝑨 𝒔𝒔𝒔𝒔𝒔𝒔
𝒏𝒏𝝅𝝅𝒙𝒙
𝑳𝑳
(3)
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Sommerfeld’s Quantum Theory
For each value of ‘n’, there is a corresponding quantum state 𝝍𝝍𝒏𝒏; whose energy 𝑬𝑬𝒏𝒏 can be obtained as:
𝑬𝑬𝒏𝒏 =
ћ𝟐𝟐
𝟐𝟐𝟐𝟐
(
𝒏𝒏𝝅𝝅
𝑳𝑳
)𝟐𝟐
 The bound electron can have only discrete energy values corresponding to 𝒏𝒏 = 𝟏𝟏, 𝟐𝟐, 𝟑𝟑, …
 The lowest energy of the particle is obtained as 𝑬𝑬𝟏𝟏 =
𝒉𝒉𝟐𝟐
𝟖𝟖𝟖𝟖𝑳𝑳𝟐𝟐 (for 𝒏𝒏 = 𝟏𝟏) which is not equals to zero.
 The spacing between two consecutive levels increases as: (𝒏𝒏 + 𝟏𝟏)𝟐𝟐
𝑬𝑬𝟏𝟏 − 𝒏𝒏𝟐𝟐
𝑬𝑬𝟏𝟏 = (𝟐𝟐𝟐𝟐 + 𝟏𝟏)𝑬𝑬𝟏𝟏
Substituting equation 3 in equation 1, the energy value can be calculated.
(4)
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Fig. First three energy levels and wave function of a
free electron
o The quantum mechanical energy levels are discrete.
o Electrons are accommodated according to Pauli’s exclusion principle.
o More than one orbital may have the same energy, and that number is called the
degeneracy.
𝑬𝑬𝑭𝑭 =
ћ𝟐𝟐
𝟐𝟐𝟐𝟐
(
𝒏𝒏𝑭𝑭𝝅𝝅
𝟐𝟐𝑳𝑳
)𝟐𝟐
The fermi energy is given as:
The topmost filled energy level at 0 K is known as the Fermi level and the energy
corresponding to this level is called the Fermi energy EF.
Where nF represents the principal quantum number of
the Fermi level.
(5)
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 According to FD statistics, one energy state can be occupied by a maximum of 2 electrons.
 Hence total number of accommodated electrons, upto Fermi level is: N = 2nF
 Hence equation 5 becomes:
For example: If we accommodate 𝟓𝟓 × 𝟏𝟏𝟏𝟏−𝟏𝟏𝟏𝟏
electrons on one centimetre length of line the fermi
energy of the topmost electron would be
𝑬𝑬𝑭𝑭 = 𝟑𝟑. 𝟕𝟕 × 𝟏𝟏𝟏𝟏−𝟏𝟏𝟏𝟏 erg = 2.4 eV
Thus the Fermi energy depends on the length of the box and the number of electrons in the box.
𝑬𝑬𝑭𝑭 =
ћ𝟐𝟐
𝟐𝟐𝟐𝟐
(
𝑵𝑵𝝅𝝅
𝟐𝟐𝑳𝑳
)𝟐𝟐
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Free Electron Gas in Three Dimensions
12
The free particle Schrödinger equation in three dimensional cube of edge L is:
Fig. Three dimensional potential box
−
ћ𝟐𝟐
𝟐𝟐𝟐𝟐
(
𝝏𝝏𝟐𝟐
𝝏𝝏𝒙𝒙𝟐𝟐
+
𝝏𝝏𝟐𝟐
𝝏𝝏𝒚𝒚𝟐𝟐
+
𝝏𝝏𝟐𝟐
𝝏𝝏𝒛𝒛𝟐𝟐
)𝝍𝝍𝒌𝒌(𝒓𝒓) = 𝑬𝑬𝒌𝒌𝝍𝝍𝒌𝒌
Similar to the 1D potential well case, here the wavefunction can be expressed as:
𝝍𝝍𝒏𝒏(𝒓𝒓) = 𝑨𝑨 𝒔𝒔𝒔𝒔𝒔𝒔
𝝅𝝅𝒏𝒏𝒙𝒙𝒙𝒙
𝑳𝑳
𝒔𝒔𝒔𝒔𝒔𝒔
𝝅𝝅𝒏𝒏𝒚𝒚𝒚𝒚
𝑳𝑳
𝒔𝒔𝒔𝒔𝒔𝒔
𝝅𝝅𝒏𝒏𝒛𝒛𝒛𝒛
𝑳𝑳
where, 𝑛𝑛𝑥𝑥, 𝑛𝑛𝑦𝑦 and 𝑛𝑛𝑧𝑧 are positive integers
Proceeding further, we have the energy value 𝐸𝐸𝑘𝑘 as:
𝑬𝑬𝒌𝒌 =
ћ𝟐𝟐
𝟐𝟐𝟐𝟐
𝒌𝒌𝟐𝟐
=
ћ𝟐𝟐
𝟐𝟐𝟐𝟐
(𝒌𝒌𝒙𝒙
𝟐𝟐
+ 𝒌𝒌𝒚𝒚
𝟐𝟐
+ 𝒌𝒌𝒛𝒛
𝟐𝟐
) where; 𝑘𝑘𝑥𝑥 =
𝜋𝜋𝑛𝑛𝑥𝑥
𝐿𝐿
, 𝑘𝑘𝑦𝑦 =
𝜋𝜋𝑛𝑛𝑦𝑦
𝐿𝐿
and 𝑘𝑘𝑧𝑧 =
𝜋𝜋𝑛𝑛𝑧𝑧
𝐿𝐿
 Several combinations can yield the same value of energy.
 Each combination of the quantum numbers is called a quantum state, and
several states having the same energy are termed degenerate.
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Filling of Energy Levels
Applications of the Free Electron Gas Model
o The distribution of electrons follows Pauli’s exclusion principle.
o Available set of quantum numbers: kx, ky, kz and ms �
𝟏𝟏
𝟐𝟐
− �
𝟏𝟏
𝟐𝟐
or
Hence each level can accommodate two electrons with set of (kx, ky, kz, ±1/2)
Or
Each energy level is doubly degenerate
𝑭𝑭𝑭𝑭𝑭𝑭 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝑵𝑵
𝒏𝒏𝒏𝒏𝒏𝒏 − 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊
𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆
It needs 𝑵𝑵/𝟐𝟐
energy levels to fill up
𝒂𝒂𝒂𝒂 𝟎𝟎 𝑲𝑲
Energy
(E)
Surface
energy
(E
s
)
Vacuum
EF0
𝑵𝑵/𝟐𝟐𝟐𝟐𝟐𝟐 level
𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆 level
 Unlike classical theory, Sommerfeld’s theory does not allow condensation of all
electrons into the zero energy level even at 0.
 Electrons are distributed among the discrete energy levels having energies ranging
from 0 to EF0.
Fig. Sommerfeld’s free electron model at 0 K.
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The density of states (D(E)) can be also found by applying Sommerfeld’s quantum theory.
The density of states (D(E)) is the total number of available electronic states (or orbitals) per
unit energy range at energy E.
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Derivation of the Density of States
Linear momentum is represented by quantum mechanical operator P ⇒
Now, 𝑷𝑷𝝍𝝍𝒌𝒌 𝒓𝒓 = −𝒊𝒊ћ ▽ 𝝍𝝍𝒌𝒌 𝒓𝒓 = ћ𝒌𝒌𝝍𝝍𝒌𝒌 𝒓𝒓
From eigen value, particle velocity in the orbital 𝒌𝒌 is given by:
−𝒊𝒊ћ ▽
𝒗𝒗 = �
ћ𝒌𝒌
𝒎𝒎
Fig. Schematic representation of electron in 𝒌𝒌 space
𝒌𝒌𝒙𝒙
𝒌𝒌𝒚𝒚
𝒌𝒌𝒛𝒛
Fermi surface
at energy 𝝐𝝐𝑭𝑭
o In the ground state of a system of N free electrons, the occupied orbitals of the system fill a
sphere of radius 𝑘𝑘𝐹𝐹, in the 𝒌𝒌 space.
o The Fermi energy at the surface of the sphere is given by:
𝝐𝝐𝑭𝑭 =
ћ𝟐𝟐
𝟐𝟐𝟐𝟐
𝒌𝒌𝑭𝑭
𝟐𝟐
o The allowed wave vectors, i.e., 𝑘𝑘𝑥𝑥, 𝑘𝑘𝑦𝑦 and 𝑘𝑘𝑧𝑧 occupy the volume element ( ⁄
𝟐𝟐𝝅𝝅
𝑳𝑳)𝟑𝟑
of 𝑘𝑘 space.
So, the total number of orbital in the Fermi sphere of volume �
𝟒𝟒𝝅𝝅𝒌𝒌𝑭𝑭
𝟑𝟑
𝟑𝟑 is:
2.
�
4𝜋𝜋𝑘𝑘𝐹𝐹
3
3
( ⁄
2𝜋𝜋
𝐿𝐿)3 =
𝑉𝑉
3𝜋𝜋2 𝑘𝑘𝐹𝐹
3
= 𝑁𝑁 𝒌𝒌𝑭𝑭= (
𝟑𝟑𝝅𝝅𝟐𝟐𝑵𝑵
𝑽𝑽
) �
𝟏𝟏
𝟑𝟑
(1)
(2)
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Density of States
16
16
From (1) and (2) ⇒ 𝝐𝝐𝑭𝑭 =
ћ𝟐𝟐
𝟐𝟐𝟐𝟐
(
𝟑𝟑𝝅𝝅𝟐𝟐𝑵𝑵
𝑽𝑽
) �
𝟐𝟐
𝟑𝟑
⇒ 𝑵𝑵 =
𝑽𝑽
𝟑𝟑𝝅𝝅𝟐𝟐 (
𝟐𝟐𝟐𝟐𝝐𝝐
ћ𝟐𝟐 ) �
𝟑𝟑
𝟐𝟐
Density of states D(E) is the total number of available electronic
states (or orbitals) per unit energy range at energy E.
Fig. Density of states as a function of electron
energy
D(E) is given as:
𝑫𝑫 𝝐𝝐 =
𝒅𝒅𝒅𝒅
𝒅𝒅𝝐𝝐
=
𝑽𝑽
𝟐𝟐𝝅𝝅𝟐𝟐
(
𝟐𝟐𝟐𝟐
ћ𝟐𝟐
) �
𝟑𝟑
𝟐𝟐. 𝝐𝝐 �
𝟏𝟏
𝟐𝟐 =
𝟑𝟑𝟑𝟑
𝟐𝟐𝝐𝝐
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 The basic classical (Drude-Lorentz) and quantum (Sommerfeld) theories of metals were
discussed.
 The density of states of the electrons in metals was derived from Sommerfeld’s quantum
theory.
 Some drawbacks of the classical theory could be solved by the quantum theory.
NPTEL

• Physics of Functional Materials by Hasse Fredriksson & Ulla Akerlind
• Introduction to Nanotechnology, Charles P. Poole, Jr. and Frank J. Owens, wiley-
interscience.
• Shriver And Atkins Inorganic Chemistry by Peter Atkins Tina Overton, Jonathon Rourke.
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Physics of Functional Materials & Devices
Module 03: Introduction to Theory of Solids
Lecture 15: Nearly free electron model
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Nearly free electron model
Bloch’s Theorem
Kronig Penney Model
Extended and Reduced Zone Scheme
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In continuation of the free electron model of solids, it can be inferred that
Drawbacks of the free electron model:
The free electron model can not explain some phenomena.
1. It could not account for the difference in conductors and insulators, i.e. why the conductors have plenty of
free electrons but the insulators do not have any.
2. It can not explain the variation of resistivity with temperature, for the insulators.
3. The various properties of semiconductors could not be explained with the basic model.
To overcome the deficiencies, the electrons are considered to move in
a periodic potential with a period equal to the lattice constant, which
is known as the “Nearly free electron model”.
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Figure: Periodic lattice in one dimension
According to the nearly free electron model, it is assumed that the ion cores are at rest and the potential experienced by
the electrons in a crystal is periodic with a period equal to the lattice constant.
Nearly free electron model:
 The assumption of the model is based on the fact that the
ion cores in the crystals are distributed periodically on the
lattice sites.
 The potential contribution due to all other free electrons are
taken as a constant.
 This type of periodic potential extends up to infinity in all
directions except at the surface of the crystal, as there
remains interruption in the periodicity.
Important facts of nearly free electron model:
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 In the case of free electron theory, there is no upper limit of energy. According to that:
𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬, 𝑬𝑬 =
ћ𝟐𝟐
𝒌𝒌𝟐𝟐
𝟐𝟐𝟐𝟐
 k can have discrete values in any range which means that the energy levels are discrete and may have any spacing
depending on the dimensions of the box.
 According to the free electron model, the one-dimensional Schrodinger equation for an electron in a constant potential
V0 is:
𝒅𝒅𝟐𝟐
𝝍𝝍
+
𝟐𝟐𝟐𝟐
ћ𝟐𝟐
(𝑬𝑬 − 𝑽𝑽𝟎𝟎)𝝍𝝍 = 𝟎𝟎 (1)
𝝍𝝍 𝒙𝒙 = 𝒆𝒆±𝒊𝒊𝒊𝒊𝒊𝒊 (2)
 The solution to equation 1, is a type of plane wave as given:
𝑾𝑾𝑾𝑾𝑾𝑾𝑾𝑾𝑾𝑾, 𝑬𝑬 − 𝑽𝑽𝟎𝟎 =
ћ𝟐𝟐 𝒌𝒌𝟐𝟐
𝟐𝟐𝟐𝟐
=
𝑷𝑷𝟐𝟐
𝟐𝟐𝟐𝟐
= 𝑬𝑬𝒌𝒌𝒌𝒌𝒌𝒌 (3)
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 When we consider, periodic potential V(x), the Schrodinger equation is written as:
 The potential V(x) is periodic with the lattice constant a. i.e. V (x + a) = V (x)
 The solution of equation 4 is governed by a famous theorem, known as
Bloch’s theorem.
𝒅𝒅𝟐𝟐
𝝍𝝍
+
𝟐𝟐𝟐𝟐
ћ𝟐𝟐
(𝑬𝑬 − 𝑽𝑽 𝒙𝒙 )𝝍𝝍 = 𝟎𝟎 (4)
Bloch’s Theorem:
Where, 𝑼𝑼𝒌𝒌 𝒙𝒙 has the same periodicity of the crystal lattice. i.e., 𝑼𝑼𝒌𝒌(𝒙𝒙 + 𝒂𝒂) = 𝑼𝑼𝒌𝒌(𝒙𝒙)
The Schrödinger equation in a periodic potential possesses the form: 𝝍𝝍 𝒙𝒙 = 𝑼𝑼𝒌𝒌(𝒙𝒙)𝒆𝒆±𝒊𝒊𝒊𝒊𝒊𝒊
(5)
 The solution of equation 4 takes the form of equation 5 according to Bloch’s theorem.
 In three dimensions, Bloch’s theorem is expressed as: 𝝍𝝍 𝒓𝒓 = 𝑼𝑼𝒌𝒌(𝒓𝒓)𝒆𝒆𝒊𝒊𝒌𝒌.𝒓𝒓
Figure: Periodic lattice in one dimension
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Figure: Ideal periodic square well potential suggested by Kronig and Penney
Kronig Penney model
 The potential energy of an electron in a linear array of positive nuclei is assumed to have the form of a periodic array
of square wells with a period of (a+b).
 The model illustrates the behavior of electrons in a periodic potential by assuming a relatively simple one-dimensional
model of a periodic potential.
 For, 𝟎𝟎 < 𝒙𝒙 < 𝒂𝒂 (Region - I), at the bottom of a well, V = 0, the electron is assumed to be in the vicinity of the nucleus.
 Whereas outside a well, i.e., for −𝒃𝒃 < 𝒙𝒙 < 𝟎𝟎, the potential energy is assumed to be V0.
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8
Kronig Penney model
Schrödinger equation for two region can be written as:
𝒅𝒅𝟐𝟐
𝝍𝝍
+
𝟐𝟐𝟐𝟐
ћ𝟐𝟐
𝑬𝑬𝝍𝝍 = 𝟎𝟎
𝒅𝒅𝟐𝟐𝝍𝝍
𝒅𝒅𝒅𝒅𝟐𝟐 +
𝟐𝟐𝟐𝟐
ћ𝟐𝟐 (𝑬𝑬 − 𝑽𝑽𝟎𝟎)𝝍𝝍 = 𝟎𝟎
For, 𝟎𝟎 < 𝒙𝒙 < 𝒂𝒂
For, 0> 𝒙𝒙 > −𝒃𝒃
Region-I
Region-II
Region-I
Region-II
(6)
(7)
Assuming that the energy E of the electrons is less than V0, we can define two real quantities 𝐾𝐾 and 𝑄𝑄, as
𝑲𝑲𝟐𝟐 =
𝟐𝟐𝟐𝟐𝟐𝟐
ћ𝟐𝟐
𝑸𝑸𝟐𝟐
=
𝟐𝟐𝟐𝟐
ћ𝟐𝟐
(𝑽𝑽𝟎𝟎 − 𝑬𝑬)
and
So, the equations 6 and 7 become 𝒅𝒅𝟐𝟐
𝝍𝝍
𝒅𝒅𝒅𝒅𝟐𝟐 + 𝑲𝑲𝟐𝟐
𝝍𝝍 = 𝟎𝟎
𝒅𝒅𝟐𝟐𝝍𝝍
𝒅𝒅𝒅𝒅𝟐𝟐 − 𝑸𝑸𝟐𝟐
𝝍𝝍 = 𝟎𝟎
(8)
(9)
Figure: Ideal periodic square well potential suggested by Kronig and
Penney
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Since the above two Schrödinger equations obey Bloch’s theorem, the wave function takes the following form:
𝝍𝝍 𝒙𝒙 = 𝑼𝑼𝒌𝒌(𝒙𝒙)𝒆𝒆𝒊𝒊𝒊𝒊𝒊𝒊
𝒅𝒅𝟐𝟐
𝑼𝑼𝟐𝟐
+ 𝟐𝟐𝟐𝟐𝟐𝟐
𝒅𝒅𝒅𝒅
− 𝑸𝑸𝟐𝟐 + 𝒌𝒌𝟐𝟐 𝑼𝑼𝟐𝟐 = 𝟎𝟎
𝐾𝐾2 =
2𝑚𝑚𝑚𝑚
ћ2
where,
𝑄𝑄2 =
2𝑚𝑚
ћ2 (𝐸𝐸0 − 𝐸𝐸)
where,
(9)
Where, 𝑈𝑈𝑘𝑘 𝑥𝑥 is the periodic function in x with periodicity of (a+b), i.e., 𝑼𝑼𝒌𝒌(𝒙𝒙) = 𝑼𝑼𝒌𝒌(𝒙𝒙 + 𝒂𝒂 + 𝒃𝒃)
From equation 9, we have 𝝍𝝍
𝒆𝒆𝒊𝒊𝒊𝒊𝒊𝒊𝑼𝑼𝒌𝒌(𝒙𝒙) 𝒆𝒆𝒊𝒊𝒊𝒊𝒊𝒊
and
𝝍𝝍
= - 𝒆𝒆𝒊𝒊𝒊𝒊𝒊𝒊𝑼𝑼𝒌𝒌 𝒙𝒙 + 𝟐𝟐𝟐𝟐𝟐𝟐𝒆𝒆𝒊𝒊𝒊𝒊𝒊𝒊 𝒆𝒆𝒊𝒊𝒊𝒊𝒊𝒊
(10)
Substituting equations 9 and 10 in equations 8 and 9, we get two equations for two regions (I) and (II).
(11)
(12)
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Boundary conditions
𝑈𝑈1(𝑥𝑥) 𝑥𝑥=0 = 𝑈𝑈2(𝑥𝑥) 𝑥𝑥=0
𝑑𝑑𝑑𝑑1
𝑑𝑑𝑑𝑑 𝑥𝑥=0
=
𝑑𝑑𝑈𝑈2
𝑑𝑑𝑑𝑑 𝑥𝑥=0
𝑈𝑈1(𝑥𝑥) 𝑥𝑥=𝑎𝑎 = 𝑈𝑈2(𝑥𝑥) 𝑥𝑥=−𝑏𝑏
𝑑𝑑𝑑𝑑1
𝑑𝑑𝑑𝑑 𝑥𝑥=𝑎𝑎
=
𝑑𝑑𝑈𝑈2
𝑑𝑑𝑑𝑑 𝑥𝑥=−𝑏𝑏
(ii)
(i) (iii)
(iv)
𝐴𝐴 + 𝐵𝐵 = 𝐶𝐶 + 𝐷𝐷
𝑖𝑖𝑖𝑖 𝐴𝐴 − 𝐵𝐵 = 𝑄𝑄(𝐶𝐶 − 𝐷𝐷)
𝐴𝐴𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖
+ 𝐵𝐵𝑒𝑒−𝑖𝑖𝑖𝑖𝑖𝑖
= 𝐶𝐶𝑒𝑒−𝑄𝑄𝑄𝑄
+ 𝐷𝐷𝑒𝑒𝑄𝑄𝑄𝑄
𝑒𝑒𝑖𝑖𝑖𝑖(𝑎𝑎+𝑏𝑏)
𝑖𝑖𝑖𝑖 𝐴𝐴𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖 + 𝐵𝐵𝑒𝑒−𝑖𝑖𝑖𝑖𝑖𝑖 = 𝑄𝑄 𝐶𝐶𝑒𝑒−𝑄𝑄𝑄𝑄 − 𝐷𝐷𝑒𝑒𝑄𝑄𝑄𝑄 𝑒𝑒𝑖𝑖𝑖𝑖(𝑎𝑎+𝑏𝑏)
(i)
(ii)
(iii)
(iv)
General Solutions of
Equations 11 and 12
𝑼𝑼𝟏𝟏(𝒙𝒙) = 𝐀𝐀𝒆𝒆𝒊𝒊(𝑲𝑲−𝒌𝒌)𝒙𝒙
+ 𝑩𝑩𝒆𝒆−𝒊𝒊(𝑲𝑲+𝒌𝒌)𝒙𝒙
𝑼𝑼𝟐𝟐(𝒙𝒙) = 𝐂𝐂𝒆𝒆𝒊𝒊(𝑸𝑸−𝒊𝒊𝒊𝒊)𝒙𝒙 + 𝑫𝑫𝒆𝒆−𝒊𝒊(𝑸𝑸+𝒊𝒊𝒊𝒊)𝒙𝒙
Obtained modified
equations after applying
all the boundary
conditions
Now, simplifying the determinant of these equations, we can obtain:
𝑄𝑄2+ 𝐾𝐾2
2𝑄𝑄𝑄𝑄
sin ℎ𝑄𝑄𝑄𝑄 sin 𝐾𝐾𝐾𝐾 + cos ℎ𝑄𝑄𝑄𝑄 cos 𝐾𝐾𝐾𝐾 = cos 𝑘𝑘(𝑎𝑎 + 𝑏𝑏)
(13)
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(i) Potential barrier 𝑉𝑉0 → ∞
(ii) Barrier width b → 0
Such a way that (𝑽𝑽𝟎𝟎𝒃𝒃)
remains finite
New modified equation: 𝑄𝑄2
+ 𝐾𝐾2
2𝑄𝑄𝑄𝑄
𝑄𝑄𝑄𝑄 sin 𝐾𝐾𝐾𝐾 + cos 𝐾𝐾𝐾𝐾 = cos 𝑘𝑘𝑘𝑘
mV0b
ħ2K
sin 𝐾𝐾𝐾𝐾 + cos 𝐾𝐾𝐾𝐾 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
[where, 𝑷𝑷 =
𝐦𝐦𝑽𝑽𝟎𝟎𝐛𝐛𝒂𝒂
ħ𝟐𝟐 ]
As, b → 0, sin ℎ𝑄𝑄𝑄𝑄 → 𝑄𝑄𝑄𝑄 And, cos ℎ𝑄𝑄𝑄𝑄 → 1
Also,
𝑄𝑄2+ 𝐾𝐾2
2𝑄𝑄𝑄𝑄
=
𝑚𝑚𝑉𝑉0
𝑄𝑄𝑄𝑄ħ2
(14)
𝑃𝑃
sin 𝐾𝐾𝐾𝐾
𝐾𝐾𝐾𝐾
+ cos 𝐾𝐾𝐾𝐾 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 (15)
Assumptions for
simplification
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12
Fig. Plot of ⁄
𝑷𝑷
𝑲𝑲𝑲𝑲 𝒔𝒔𝒔𝒔𝒔𝒔 𝑲𝑲𝑲𝑲 + 𝒄𝒄𝒄𝒄𝒄𝒄 𝑲𝑲𝑲𝑲 vs 𝒌𝒌𝒌𝒌
 The above condition must be satisfied for the
solutions to the wave equations to exist.
 Only those values of 𝑲𝑲𝑲𝑲 are allowed for which the
left-hand side of the equation lies between +1 and -1,
as the values of 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 lie between those values.
 The other values of 𝑲𝑲𝑲𝑲 are not allowed and they
reside in the forbidden energy bands.
𝑷𝑷
𝒔𝒔𝒔𝒔𝒔𝒔 𝑲𝑲𝑲𝑲
𝑲𝑲𝑲𝑲
+ 𝒄𝒄𝒄𝒄𝒄𝒄 𝑲𝑲𝑲𝑲 = 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄
 The energy spectrum of the electron consists of alternate regions of allowed energy bands
(solid lines on the abscissa) and forbidden energy bands (broken lines).
 The width of the allowed energy band increases with 𝑲𝑲𝑲𝑲 or the energy.
 The width of a particular energy band decreases with an increase in the value of P, i.e.,
with the increase in the binding energy of electrons.
Important points
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The width of allowed and forbidden energy band can be changed on basis of 𝑷𝑷 value, and various band structures can be formed within the lattice
crystal.
If 𝑃𝑃 → ∞,
sin 𝐾𝐾𝐾𝐾 = 0
𝐾𝐾𝐾𝐾 = 𝑛𝑛𝜋𝜋
𝐾𝐾 =
𝑛𝑛𝜋𝜋
𝑎𝑎
𝐸𝐸 =
ћ2
2𝑚𝑚
𝑛𝑛𝑛𝑛
𝑎𝑎
2
If 𝑃𝑃 → 0,
cos 𝐾𝐾𝐾𝐾 = cos 𝑘𝑘𝑘𝑘
𝐾𝐾2
= 𝑘𝑘2
2𝑚𝑚𝑚𝑚
ћ2 =
2𝜋𝜋
λ
2
𝐸𝐸 =
1
2𝑚𝑚
ℎ
λ
2
=
𝑝𝑝2
2𝑚𝑚
Fig. Band structure for 𝑷𝑷 → ∞
Fig. Band structure for 𝑷𝑷 → ∞
Fig. Band structure for 𝑷𝑷 = 𝟔𝟔𝝅𝝅
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14
𝑷𝑷
𝒔𝒔𝒔𝒔𝒔𝒔 𝑲𝑲𝑲𝑲
𝑲𝑲𝑲𝑲
+ 𝒄𝒄𝒄𝒄𝒄𝒄 𝑲𝑲𝑲𝑲 = 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄
 The relation infers that 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 takes a specific value for each of the
allowed energy value of E.
 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 is an even periodic function, with period
𝟐𝟐𝝅𝝅𝝅𝝅
𝒂𝒂
, where n is any
integer. Hence, E is also an even periodic function off k with period of
𝟐𝟐𝝅𝝅
𝒂𝒂
.
There exists a general convention to express the relationship between the energy E and k. The general convenient two
schemes are:
(a) The extended zone scheme
(b) The reduced zone scheme
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15
Fig. Energy vs wave vector for one dimensional lattice
(extended zone scheme)
 The solid lines in the adjacent figure represent the E-k
relationship in the extended zone scheme.
 The corresponding dotted parabolic curve for free electrons in
the constant potential is also shown for comparison.
 The E-k curve of the extended zone scheme is not continuous and
has discontinuities at
𝒌𝒌 = ±
𝒏𝒏𝒏𝒏
𝒂𝒂
, where n = 1, 2, 3 ….
 The allowed values of k define the boundaries of the Brillouin
zones.
 The first Brillouin zone extends from -
𝝅𝝅
𝒂𝒂
to +
𝝅𝝅
𝒂𝒂
; the second one
extends from
𝝅𝝅
𝒂𝒂
to
𝟐𝟐𝝅𝝅
𝒂𝒂
and from -
𝝅𝝅
𝒂𝒂
to -
𝟐𝟐𝟐𝟐
𝒂𝒂
.
Notable points regarding extended zone scheme:
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16
 The E-k curve in the reduced zone is shown in the adjacent figure, whose x-axis
is limited from -
𝝅𝝅
𝒂𝒂
to +
𝝅𝝅
𝒂𝒂
.
 The scheme is obtained by reducing the contents of the other zones as to
correspond, in general, to the first zone, i.e. to the region -
𝝅𝝅
𝒂𝒂
to +
𝝅𝝅
𝒂𝒂
.
 The wave vector 𝒌𝒌 belonging to this region is called the reduced wave vector.
Notable points regarding the reduced zone scheme:
Fig. Reduced zone scheme
Major applications of the Nearly free electron model
and band theory of solids:
(a) To calculate the band gap of solids.
(b) To differentiate among the insulators,
semiconductors, and metals.
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 The nearly free electron model considers that the electrons are not totally free but it
encounters the potential due to the presence of positive ion cores.
 The nearly free electron model is successful in explaining the differences in the band gap
among metals, semiconductors, and insulators.
 Extended and reduced zone schemes of band theory have been discussed.
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• Physics of Functional Materials by Hasse Fredriksson & Ulla Akerlind
• Solid State Physics by Charles Kittel.
• Shriver And Atkins Inorganic Chemistry by Peter Atkins Tina Overton, Jonathon Rourke.
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Physics of Functional Materials and Devices
Module 03: Introduction to Theory of Solids
Lecture 16: Bonds in Molecules and Solids
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 Introduction to Bonds in Molecules and Solids
 Atomic energies: Binding, Dissociation, Ionization, Electron Affinity,
Sublimation, Condensation, Cohesive and Lattice.
 Bonds in Molecules: Molecular Bonds, Ionic Bonds, and Covalent Bonds.
 Metallic Bonds
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Bonds in Molecules and Solids
 What are bonds?
A bond is the attractive force that connects distinct constituents (atoms, ions, etc.) of various
types of molecules together.
 Why do bonds form?
 Atoms tend to stay stable, which is why they create bonds with one another. Atoms attain this
stability by either sharing electrons or donating electrons to other atoms.
Sharing of electrons Donation of electrons
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 Getting familiar with general terms:
 Potential energy: For particles, the concept of potential energy is often used to describe the
energy associated with the interactions between particles.
 Particles can interact with each other through various fundamental forces, such as the
electromagnetic force, the strong nuclear force, and the weak nuclear force.
 Potential Energy in terms of force can be written as:
𝐸𝐸𝑃𝑃𝑃𝑃𝑃𝑃 = �
𝑎𝑎
𝑥𝑥
−𝐹𝐹𝐹𝐹𝐹𝐹
a: Zero level of the potential energy
 The above concept is used to understand the interaction between two
particles.
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Fig: Potential Energy of a free molecule as a function of interionic distance r.
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 Different types of particle interactions/energies
 Binding and Dissociation Energy
 Ionization Energy
 Electron Affinity
 Sublimation Energy and Condensation Energy
 Cohesive Energy
 Lattice Energy
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Bonds in Molecules and Solids: Binding Energy and dissociation Energy
 Binding Energy: It is the energy that holds the nucleus of an atom together, and is required to
overcome the attractive forces that exist between protons and neutrons. Further, we can also say
that the binding energy of an atom is a measure of the stability of the nucleus. The more stable
the nucleus, the greater the binding energy.
 Let us understand with an example:
 Considering two atoms A and B.
 Atoms A and B are at an infinite distance (∞) such that no interaction is present between the
atoms.
 Since the atoms are apart they are highly unstable.
 So, the atoms gradually comes close to each other.
 The energy released when the two atoms A and B are moved from
infinity to their equilibrium distance and form a stable molecule AB.
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 Binding Energy:
EB~ Binding Energy
Energy released when the two atoms A and B are moved from infinity to their
equilibrium distance
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 Dissociation Energy: It is the amount of energy required to break a bond between two atoms in a
molecule and separate them completely. In other words, it is the minimum energy needed to break
the bond and convert the molecule into its constituent atoms.
 Let us understand with an example:
 Consider a stable diatomic molecule AB.
 Atoms A and B are at an equilibrium distance such that the molecule is under its minimum
energy condition.
 The dissociation energy is equal to the binding energy, D0, of the
molecule or the energy required to separate the atoms A and B and
move them to infinite distance (∞) from each other.
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 Dissociation Energy Schematic:
D0~Dissociation Energy
Energy required to separate the atoms A and B
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Bonds in Molecules and Solids: Ionization Energy
 Ionization Energy: The energy needed to remove an electron from an atom or molecule is known as
ionization energy. Depending on the type of atom or molecule and the electron being extracted,
different amounts of energy are required. For instance, some atoms take more energy than
others to remove an electron.
Na
Na atom
The first ionization
energy of sodium
~5.14eV
Na
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Bonds in Molecules and Solids: Electron Affinity
 Electron Affinity: The energy required to transfer an electron from infinity to the lowest
feasible orbit in an atom or molecule. In other words, it is the energy change associated with
adding an electron to a neutral atom or molecule.
Atom
Nucleus
Cl atom
For chlorine gaining an electron, energy
is released ~ -3.62 eV
Atom
Nucleus
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Bonds in Molecules and Solids: Sublimation Energy and Condensation Energy
 Sublimation Energy: The amount of energy required to change a substance from its solid to its
gaseous state without first going through the liquid state is known as sublimation energy. To put
it simply, it is the energy needed to release the bonds binding together the molecules of a solid
and turn it into a gas. Atomic sublimation energy is defined as:
𝐸𝐸𝑆𝑆 = 𝑡𝑡𝑡𝑡𝑡 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑖𝑖𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
 Condensation Energy: The reverse process of sublimation i.e. the release of energy when vapour
changes directly into a solid. The energy released per atom is given as follows:
𝐸𝐸𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 𝐸𝐸𝑆𝑆
 Sublimation and condensation energy is frequently expressed in joules
per mole (J/mol) or kilojoules per mole (kJ/mol).
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Bonds in Molecules and Solids: Cohesive Energy
 Cohesive Energy: The amount of energy needed, starting from the substance's most stable form,
to completely separate a substance into its individual atoms or molecules is known as cohesive
energy. The energy needed to break every intermolecular bond holding the atoms or molecules in
the substance together.
 These forces may include hydrogen bonds, other kinds of chemical bonds, dipole-dipole
interactions, and London dispersion forces.
Energy required
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Bonds in Molecules and Solids: Lattice Energy
 Lattice Energy: The amount of energy needed to separate a solid ionic compound, which is made up
of charged particles known as ions, into its individual gaseous ions is known as lattice energy. It is
a way to estimate how strongly the ions in a solid's crystal lattice attract one another.
 The energy which has to be added to one stoichiometric unit of a crystal to separate its
component ions into free ions.
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Bonds in Molecules and Solids: Bonds in Molecules and Nonmetallic Solids
 What are the different types of bonds?
Types
of
Bonds
Molecular
Ionic
Covalent
Metallic
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 Molecular Bonds: Dipole interaction in a molecule acts as the source of origin of molecular bonds.
The unequal distribution of electrons among the atoms in a polar covalent connection results in
the formation of a dipole moment within a molecule. This dipole moment can interact with
adjacent molecules' dipoles to produce dipole-dipole interactions.
 Dipole Interaction Between Molecules
 Dipole moment of a dipole is defined as:
𝑝𝑝 = 𝑞𝑞𝑞𝑞
where q is the electrical charge and r a vector directed from the negative towards the positive
charge
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 Ionic Bonds: The electrostatic attraction between ions with opposing charges is what creates an ionic bond.
Ions are atoms or molecules with one or more electrons added or removed, creating a net electrical charge.
 When two atoms form an ionic bond, one of them gives one or more of its electrons to the other, resulting in
the formation of a positively charged cation and a negatively charged anion.
 As an illustration, sodium and chlorine can unite to form sodium chloride (NaCl), usually referred to as table
salt. In order to generate a sodium cation (Na+), one electron is lost, whereas, in a chlorine anion (Cl-), an
electron is gained. The ionic bond in NaCl is produced by electrostatic interaction between the ions of Na+
and Cl-.
 Now, one can imagine a Sodium ion and a chlorine ion forming a bond. But one can
ask:
 How does the bond attain stability?
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 As the electron clouds of the two subshells start to overlap, a strong repulsive force appears,
which causes the distance between the ions to shrink until it equals the equilibrium distance for
the Na+Cl- molecule.
 At this point, the electrostatic attraction force balances the strong repulsive force.
 Owing to the Pauli exclusion principle, no electrons can have four equal quantum numbers. Hence
some electrons must be excited to higher energy levels.
 This process requires a lot of energy and results in a steep energy curve.
The equilibrium corresponds to the lowest possible total energy of the
system and results in a stable ionic molecule.
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 IONIC BOND SCHEMATIC
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 Covalent Bonds: When two atoms share electrons, it forms a covalent bond, a type of chemical
bond. When two atoms with some electrons in their outer shells combine, you can imagine this.
Both of these atoms can have a more stable electron configuration by sharing electrons with one
another to form a bond.
 Covalent bonds tend to occur between nonmetal atoms, such as hydrogen, oxygen, and carbon.
They can be found in many molecules, including water, methane, and glucose.
 Let's take the example of water (H2O) to explain covalent bonding. Water is a molecule that
consists of two hydrogen atoms and one oxygen atom.
 In a covalent bond, the two hydrogen atoms and the oxygen atom
share electrons to form a stable molecule. Each hydrogen atom has
one electron in its outer shell, while the oxygen atom has six
electrons in its outer shell.
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 To complete their outer shells, the hydrogen atoms share their electrons with the oxygen atom, while
the oxygen atom shares its electrons with the hydrogen atoms.
 Some more examples of molecules having covalent bonds are Cl2, H2, CH4,
Graphite etc.
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Bonds in Molecules and Solids: Covalent Bonds
Covalent Bond Properties:
 The bonds are very strong and covalent solids are therefore characterized by high melting points
and high mechanical strength.
 They are poor conductors of heat and electricity because there are no non-localized electrons
which can carry energy or charge from one place to another.
 The electron excitation energies of covalent solids are high, of the
magnitude of several eV.
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Bonds in Molecules and Solids: Metallic Bonds
Metallic Bonds: The valence electrons of metal atoms in a metallic bond are delocalized, which
means they are not bound to any particular atom and are therefore free to move about the entire
metal lattice.
An atom is not tightly bound to any of the delocalized electrons in a metallic bond.
Rather, they move unrestrictedly within the metal lattice, giving metals their special ability to
conduct electricity.
Metals can conduct electricity and heat effectively due to the mobility of these
delocalized electrons.
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Bonds in Molecules and Solids: Metallic Bonds
Delocalised electrons Sea of electrons
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• The three main types of chemical bonds are ionic bonds, covalent bonds, and metallic bonds.
• Further, we have six types of bonding energies depending on the particle-particle
interaction namely:
• Binding and Dissociation Energy
• Ionization Energy
• Electron Affinity
• Sublimation Energy and Condensation Energy
• Cohesive Energy
• Lattice Energy
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Physics of Functional Materials and Devices
Module 03: Theory of solids
Lecture 17 : Introduction to transformation kinetics and reaction rates
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 Transformation or reaction
 Different classification of transformation
 Endothermic and Exothermic reactions
 Homogeneous and Heterogeneous reactions
 Reaction rate
 Factors influencing reaction rate
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Transformation kinetics
 In physics, a reaction or transformation refers to a process in which energy or matter is
changed from one form to another.
 In chemistry, a reaction or transformation in chemistry refers to a process in which one or
more substances, called reactants, are converted into one or more different substances, called
products.
 From material science aspects, transformation occurs when changes in the structure of the
materials in the form of either composition change or change in grain sizes in a crystal
structure.
The structure changes occur as a result of rearrangements of atoms in the material through:
 Chemical reactions
 Phase transformations
 Diffusion
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Chemical reactions
 Temperature: In general, increment in the temperature activates more number of reactant
molecules to collide and initiate a reaction.
 Pressure: Increase in the pressure increase the frequency of collision between reactant
molecules to initiate a reaction.
Factors that influences the chemical reactions?
What are chemical reactions?
 Process of interactions of different atoms, molecules or ions to form a new compounds with
different chemical and physical properties.
 Deformation and reformation of bonds between atoms or molecules occurs to form new
compounds.
 Formation of final product marked by release or absorption of energy, change in
temperature, color etc.
 Surface area: Increase in surface area provides more number of active
sites for reactants to collide and interact to initiate a reaction.
 Concentration: Higher concentrations increase the chances of reactant
molecules to collide with each other.
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Classifications on the basis of energy changes
Based of the variation of internal energy during a chemical reaction, they are be divided into two
categories:
Endothermic reactions Exothermic reactions
 Energy required to
break the bonds of
reactants is higher
than the energy
released by the
reaction.
 Energy is acquired
from its surroundings
as heat.
 Marked by drop in
temperature of overall
system.
Examples: Vaporization of water,
Photosynthesis, Melting of ice.
 Energy required to break the
bonds of reactants is lower
than the energy released by
the reaction.
 Energy is released in its
surroundings as heat.
 Marked by rise in temperature
of overall system.
Examples: Combustion reactions,
Rusting of iron, Respiration.
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Classifications on the basis of uniformity
Based of the uniformity of reactants or products in a reaction mixture, chemical reactions are divided
into two categories:
Homogeneous reactions Heterogeneous reactions
 Homogeneous reactions, by
definition, refers to a process where
the reactants and products are in the
same chemical phase.
 Reactants and products that are
uniformly distributed throughout the
reaction mixture.
 Heterogeneous reactions, by definition, refers to
a process where the reactants and products are
in the different chemical phase.
 Reactants and products that are NOT uniformly
distributed throughout the reaction mixture.
Examples: Depletion of Ozone layer,
Photochemical smog formation.
Examples: Haber’s process, Combustion reactions,
Corrosion reactions
Solid Solid Solid
Liquid Liquid Liquid Solid
Liquid Liquid
Liquid
Gas Liquid
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Classifications on the basis of spontaneity
 Gibbs free energy is a thermodynamic function that represents the maximum amount of energy
available to do useful work in a chemical system.
 Change in Gibbs free energy (ΔG) of a reaction influence its spontaneity.
 Most transformation chemical reactions occurs at constant temperature and pressure.
 At constant temperature and pressure processes always occur spontaneously in the direction of
decreasing Gibbs free energy.
 In other words, system is in equilibrium when the Gibbs free energy of the system is a minimum.
Consider the equilibrium reaction.
A B
For this,
 ΔG < 0, forward reaction is spontaneous.
 ΔG = 0, reaction is at equilibrium.
 ΔG > 0, backward reaction is spontaneous.
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Driving force for a chemical reaction
 We describe that a reaction or transformation is spontaneous when change in Gibbs free
energy (ΔG) of a reaction is negative or minimal.
 To know the probability of occurring a transformation or reaction, the concept of driving force
is used.
Driving force = −∆𝑮𝑮 = − ∫
𝒊𝒊𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏
𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇
𝒅𝒅𝒅𝒅 = −(𝑮𝑮𝒇𝒇 − 𝑮𝑮𝒊𝒊)
 The driving force of a spontaneous process is always a positive quantity. The larger the driving
force is, the more likely will be the transformation or reaction.
What about the energy barrier for
driving a chemical reaction ?
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Activation energy: the energy barrier
Consider the equilibrium reaction
 Activation energy is the minimum kinetic energy that
reactants must have in order to form products.
 In other words, the height of the barrier between the
reactants and products is the activation energy of the
reaction.
 For instance, there are numerous collisions of
reactants occurs each second, but only few collision
that have energy greater than the activation energy
result into final product.
 After additional energy, all the reactant molecules
have enough kinetic energy to pass the energy
barrier to form products.
Ea
Heat
A
B
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Activation energy: influencing the reaction rates
 Two necessary conditions:
(i) ∆𝑮𝑮 < 𝟎𝟎
(ii) 𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬 𝒐𝒐𝒐𝒐 𝒂𝒂 𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓 > 𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨 𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆
How activation energy influences the
rate of reaction?
Let us understand what is rate of reaction…….
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Reaction rate
𝒌𝒌 =
𝒅𝒅𝒅𝒅𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕/𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓 𝒕𝒕
𝒅𝒅𝒅𝒅
The reaction rate at the time t of a transformation is defined as:
where, the fractional transformation 𝒇𝒇𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕/𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓 𝒕𝒕 is defined as:
𝒇𝒇𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕/𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓 𝒕𝒕 =
𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏 𝒐𝒐𝒐𝒐 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂 𝒑𝒑𝒑𝒑𝒑𝒑 𝒖𝒖𝒖𝒖𝒖𝒖𝒖𝒖 𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗 𝒊𝒊𝒊𝒊 𝒕𝒕𝒕𝒕𝒕𝒕 𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔 𝒂𝒂𝒂𝒂 𝒕𝒕𝒕𝒕𝒕𝒕 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒕𝒕
𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏 𝒐𝒐𝒐𝒐 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂 𝒑𝒑𝒑𝒑𝒑𝒑 𝒖𝒖𝒖𝒖𝒖𝒖𝒖𝒖 𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂 𝒇𝒇𝒇𝒇𝒇𝒇 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒐𝒐𝒐𝒐 𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓
 The reaction rate k can also be expressed as the fraction of the total
number of particles which reach the final state per unit time.
 Unit of reaction rate is normally a function of time.
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Factors influencing the reaction rate
Three important factors that influence the rate of reaction:
• Direct correlation from Arrhenius equation
Activation Energy
• Direct correlation from Maxwell Distribution Law
Distribution of particles in energy states
• Arrhenius theory
• Transition state theory
Temperature
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Activation energy: influencing the reaction rates
The relation of reaction rate and activation energy was given by:
Arrhenius Equation
𝒌𝒌 = 𝑨𝑨𝒆𝒆−𝑬𝑬𝑬𝑬/𝑹𝑹𝑹𝑹
where,
k = reaction rate
A = pre-exponential factor
Ea = Activation energy (kJ mol-1)
R = Universal Gas constant (J K mol-1)
T = Temperature (K)
 If the activation energy is comparatively low and the temperature is
high, many atoms have kinetic energies high enough to overcome the
energy barrier and the transformation/reactions occurs readily.
 If the activation energy is high compared with the thermal energies of
the atoms/molecules, few of them have energies high enough to
overcome the barrier. The reaction rate becomes very low and the
transformation or reaction will be prohibited in practice.
Reactants
Products
Activation energy (Ea)
Transition state
Potential
Energy
Reaction Coordinate
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Distribution of particles: Influencing reaction rates
 The distribution of particles in different available energy states influences the reaction rates
of thermally activated reactions and transformations.
 The fraction of particles with energies above a given energy is described by Maxwell-
Boltzmann Distribution Law:
 The Maxwell–Boltzmann distribution law in its general form can be written as:
𝑵𝑵𝒊𝒊 =
𝑵𝑵𝑵𝑵
𝒁𝒁
𝒈𝒈𝒊𝒊𝒆𝒆−𝒖𝒖𝒊𝒊/𝑲𝑲𝑩𝑩𝑻𝑻
 Z is called the partition function and is given by
𝒁𝒁 = 𝒈𝒈𝟏𝟏𝒆𝒆
−
𝒖𝒖𝟏𝟏
𝑲𝑲𝑩𝑩𝑻𝑻 +𝒈𝒈𝟐𝟐𝒆𝒆
−
𝒖𝒖𝟐𝟐
𝑲𝑲𝑩𝑩𝑻𝑻 +𝒈𝒈𝟑𝟑𝒆𝒆−𝒖𝒖𝟑𝟑/𝑲𝑲𝑩𝑩𝑻𝑻 … . . = ∑𝒊𝒊 𝒈𝒈𝒊𝒊𝒆𝒆
−
𝒖𝒖𝒊𝒊
𝑲𝑲𝑩𝑩𝑻𝑻
Where,
𝒖𝒖𝒊𝒊= energy of particle i
𝑵𝑵𝒊𝒊= number of particles which have the energy 𝒖𝒖𝒊𝒊
𝑵𝑵𝑵𝑵 = total number of particles
𝒈𝒈𝒊𝒊= statistical weight of energy level 𝒖𝒖𝒊𝒊
𝑲𝑲𝑩𝑩= Boltzmann’s constant
𝑻𝑻 = absolute temperature of the system.
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Distribution of particles: Influencing reaction rates
From Maxwell-Boltzmann Law fraction of particles with energies equal to or greater
than a given energy can be calculated as:
where, 𝒇𝒇𝒊𝒊 = the fraction of the No particles which have the thermal energy 𝒖𝒖𝒊𝒊 per particle.
𝒇𝒇𝒊𝒊 =
𝑵𝑵𝒊𝒊
𝑵𝑵𝑵𝑵
=
𝒆𝒆−𝒖𝒖𝒊𝒊/𝑲𝑲𝑩𝑩𝑻𝑻
∑ 𝒆𝒆−𝒖𝒖𝒊𝒊/𝑲𝑲𝑩𝑩𝑻𝑻
From here, we can calculate the fraction of molecules which have enough thermal energy to
overcome the energy barrier, i.e. the activation energy
𝒇𝒇∗ =
𝑵𝑵𝒊𝒊
𝑵𝑵𝑵𝑵
=
𝒆𝒆−𝑼𝑼𝒂𝒂𝒂𝒂𝒂𝒂/𝑲𝑲𝑩𝑩𝑻𝑻
𝒁𝒁
Z = partition function
 If 𝑼𝑼𝒂𝒂𝒂𝒂𝒂𝒂>> 𝑲𝑲𝑩𝑩𝑻𝑻 the fraction is very SMALL and the transformation rate
will be very LOW.
 If 𝑼𝑼𝒂𝒂𝒂𝒂𝒂𝒂 < 𝑲𝑲𝑩𝑩𝑻𝑻 the fraction is very HIGH and the transformation rate
will be very HIGH.
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Temperature of reaction: Influencing reaction rates
From Arrhenius theory:
𝒌𝒌 = 𝑨𝑨𝒆𝒆−𝑬𝑬𝑬𝑬/𝑹𝑹𝑹𝑹
 According to the Arrhenius theory, the rate of a chemical reaction is proportional to the number
of collisions between reactant molecules, which in turn is dependent on temperature.
 Herein,
 For endothermic reactions, i.e heat absorb during reaction, T increases, rate of reaction
increases.
 For exothermic reactions, i.e heat released during reaction, T increases, rate of reaction
decreases. NPTEL

Temperature of reaction: Influencing reaction rates
From Transition state theory:
𝒌𝒌 =
𝑲𝑲𝑩𝑩𝑻𝑻
𝒉𝒉
𝒆𝒆−∆𝑮𝑮∗/𝑹𝑹𝑹𝑹
 According to the transition state theory (TST), the reaction
rate is proportional to the probability of forming the activated
complex, which is dependent on the activation energy and
temperature.
 As the temperature increases, the probability of forming the
activated complex increases, leading to an increase in the
reaction rate.
Where, 𝒌𝒌 = reaction rate, 𝑲𝑲𝑩𝑩= Boltzmann’s constant, 𝑻𝑻 = absolute temperature of the system, h =
Planck constant, ∆𝑮𝑮∗
= activation energy or the free energy difference between the reactants and
the activated complex, R= universal gas constant.
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 Transformation occurs through chemical reactions, phase transformations, and diffusion.
 At constant temperature and pressure, transformation always occur spontaneously in the
direction of decreasing Gibbs free energy.
 Driving force, a positive quantity used for knowing the probability of occurring a
transformation or reaction.
 Activation energy is the minimum kinetic energy that reactants must have in order to
form products.
 Three important factors that influence the rate of reaction are activation energy,
temperature, and molecular distribution of particles in different energy states.
NPTEL

⮚ PhysicsofFunctional M
aterialsbyH
asseFredriksson&U
llaA
kerlind
⮚ IntroductiontoSolidStatePhysicsbyC
harlesK
ittle
⮚ A
tkins’sPhysicalC
hem
istrybyPeterA
tkins,andJuliodePaula.
⮚ ATextbookofN
anoscienceandN
anotechnology, P.I.VargheseandThalappil, M
cG
rawH
ill
E
ducation, 2017.
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