APPLIED PHYSICS

ENGINEERING PHYSICS
UNIT-I: BONDING IN SOLIDS
INTRODUCTION
.Matter can exist three ways such as 1).solid state2).liquid
State 3).gaseous state
.1)In gases, the atoms or molecules are free whereas in solids they are bound in a
particular form because of which, they possess different properties such as physical,
electrical, mechanical, chemical and optical properties
2) But in solid state, the constituent atoms or molecules that build the solid are confined
to a localized region due attractive force between atoms
BONDING
• 1 Bonding is the physical state of existence of two or more atoms together in a
bound form by forces of attraction
• 2 .The attractive force which hold the constitute of particles of a substance
together are called bonds.
• . 3. The supply of external energy is required to move an atom completely
from its equilibrium position or to break the bonds. This energy is called
dissociation (binding) or cohesive energy.
• Bonding occurs between similar or dissimilar atoms, when an electrostatic
interaction between them produces a resultant state whose energy is lesser than
the sum of the energies possessed by individual atoms when they are free.
TYPES OF BONDING IN SOLIDS
Bonds in solids are classified basically into two groups
1) Namely primary and secondary bonds. Primary bonds are inter atomic bonds i.e.
bonding between the atoms and secondary bonds are intermolecular bonds i.e.
between the molecules.
Primary bonds
1 The primary bonds are inter atomic bonds.(strong bonds)
2. DEF . The electrostatic forces hold the atoms together in solids known as
primary bond
3 Inter atomic distance between these bonds are ranges from 1 to 2A
(or)0.1 to 10eV/ bond(bond energy)
4. In this bonding interaction occurs only through the electrons in the
outermost orbit, i.e. the valence electrons. These are further classified into
three types
1. Ionic bonding
2. Covalent bonding
3. Metallic bonding
1. Ionic Bonding(Hetropolar bond).
DEF: Ionic bonding results due to transfer of electrons from an electropositive
elements(1st&2ndGROUP) to an electronegative elements(.6th&7thGROUP)
Example:
1) In Na Cl crystal, Na atom has only one electron in outer most shell and a Cl atom
needs one more electron to attain inert gas configuration.
2) During the formation of NaCl molecule, one electron from the Na atom is transferred
to the Cl atom as a result both Na and Cl ions attain filled- shell configuration.
Na + Cl 􀃆 Na+ + Cl
- 􀃆 NaCl
A strong electrostatic attraction is set up that bond the Na+ cation and the Cl
-
anion into a very stable molecule NaCl at the equilibrium spacing.
Since Cl exist as molecules, the chemical reaction must be written as
2Na + Cl2 􀃆 2Na+ + 2Cl
- 􀃆 2NaCl
Other examples of ionic crystals are
2Mg + O2 􀃆 2Mg++ + 2O
-- 􀃆 2MgO
Mg + Cl2 􀃆 Mg++ + 2Cl
- 􀃆 MgCl2
Properties:
1. As the ionic bonds are strong, the materials are hard and possess high melting
and boiling points.
2. They are good ionic conductors, but poor conductors of both heat and
electricity
3. They are transparent over wide range of electromagnetic spectrum
4. They are brittle. They possess neither ductility (ability to be made into sheets)
nor malleability (ability to be made into wires).
5. They are soluble in polar liquids such as water but not in non-polar liquids
such as ether.
2. Covalent Bonding(homopolar bond)
DEF : Covalent bond is formed by sharing of electrons between two atoms to form
molecule.
Example: 1) Covalent bonding is found in the H2 molecule.
2) Here the outer shell of each atom possesses 1 electron. Each H atom would like
to gain an electron, and thus form a stable configuration.
3) This can be done by sharing 2 electrons between pairs of H atoms, there by
producing stable diatomic molecules.
Thus covalent bonding is also known as shared electron pair bonding.
Other examples for covalant crystals:
Properties:
1. Covalent crystals are very hard since the bond is strong.
2. The best example is diamond which is the hardest naturally occurring
material and possess high melting and boiling points, but generally lower
than that for ionic crystals.
3. Their conductivity falls in the range between insulators and
semiconductors. For example, Si and Ge are semiconductors, where as
diamond as an insulator
.
4. They are transparent to electromagnetic waves in infrared region, but opaque
at shorter wavelengths.
5. They are brittle and hard.
6. They are not soluble in polar liquids, but they dissolve in non-polar liquids such
as ether, acetone, benzene etc.
7. The bonding is highly directional.
3. Metallic Bonding:
1)The valance electrons from all the atoms belonging to the crystal are free to move
throughout the crystal.
2) The crystal may be considered as an array of positive metal ions embedded in a
cloud of free electrons. This type of bonding is called metallic bonding.
3) In a solid even a tiny portion of it comprises of billions of atoms. Thus in a
metallic body, the no. of electrons that move freely will be so large that it is
considered as though there is an electron gas contained with in the metal.
4) The atoms may embedded in this gas but having lost the valence electrons, they
become positive ions.
5) The electrostatic interaction between these positive ions and the electron gas as a
whole is responsible for the metallic bonding.
Properties:
1Compared to ionic and covalent bonds, the metallic bonds are weaker. Their
melting and boiling points are also lower.
2) Because of the easy movement possible to them, the electrons can transport energy
efficiently. Hence all metals are excellent conductors of heat and electricity.
3)They are good reflectors and are opaque to E.M radiation.
4) They are ductile and malleable.
5) They exists in solid form only
6) They are neither soluble in water nor in benzene
Secondary Bonds
There are two types of secondary bonds. They are Vander Waal’s bonds and
Hydrogen bonds.
2. secondary bonding energy’s in the range 0.01-0.5eV/bond
Vander Waal’s bonding: Vander Waal’s bonding is due to Vander Waal’s forces.
These forces exist over a very short range. The force decreases as the 4th power of
the distance of separation between the constituent atoms or molecules when the
ambient temperature is low enough. These forces lead to condensation of gaseous
to liquid state and even from liquid to solid state though no other bonding
mechanism exists.( except He)
Fig
Properties:
The bonding is weak because of which they have low melting points.
They are insulators and transparent for visible and UV light.
They are brittle.
They are non-directional
Hydrogen bonding: Covalently bonded atoms often produce an electric dipole
configuration. With hydrogen atom as the positive end of the dipole if bonds arise
as a result of electrostatic attraction between atoms, it is known as hydrogen
bonding.
FIG………………………..
Properties:
1. The bonding is weak because of which they have low melting points.
2. They are insulators and transparent for visible and UV light.
3. They are brittle.
4. The hydrogen bonds are directional.
Forces between atoms:
In solid materials, the forces between the atoms are of two kinds.
1) Attractive force 2) Repulsive force
IMPORTANCE: To keep the atoms together in solids, these forces play an important
role.
1).When the atoms are infinitely far apart they do not interact with each other to form a
solid and the potential energy will be zero.
2)From this, it can be understood that the potential energy between two atoms is
inversely proportional to some power of the distance of separation.
3). The atoms attract each other when they come close to each other due to inter-atomic
attractive force which is responsible for bond formation.
.Suppose two atoms A and B experiences attractive and repulsive forces on each other,
and then the interatomic or bonding force ‘F(r)’ between them may be represented as
F(r) =A / rM – B / rN (N > M) -------------- (1)
Where ‘r ‘is the inter atomic distance
A, B, M, N are constants.
M=2,N=7 to 10 for metallic bonds,10 to 12 for ionic, covalent bonds
In eqn-1, the first term represents attractive force and the second the repulsive force.
At larger separation, the attractive force predominates. The two atoms approach until they
reach equilibrium spacing. If they continue to approach further, the repulsive force pre
dominates, tending to push them back to their equilibrium spacing.
Fig. Variaration of interatomic force with interatomic spacing
To calculate equilibrium spacing r0:
The general expression for bonding force between two atoms is
F(r)=A / rM – B / rN
At equilibrium spacing: r = r0, F(r) = 0 because both forces should be equal
Hence A / r0
M = B / r0
N
i.e. (r0)
N-M
= B / A
Or r0 = (B / A) 1/ (N-M)
Cohesive energy:
Def;
The amount of energy required to separate the atoms completely
from the structure is called cohesive energy. this energy also called energy of
dissociation. Or
The amount of energy evolved or released from crystalline solid when two
atoms are formed bond formation from infinite distance
Since this is the energy required to dissociate the atoms, this is also called the energy of
dissociation.
The potential energy or stored internal energy of a material is the sum of the individual
energies of the atoms plus their interaction energies.
Consider the atoms are in the ground state and are infinitely far apart. Hence they do not
interact with each other to form a solid. The potential energy, which is inversely
proportional to some power of the distance of separation, is nearly zero. The potential
energy varies greatly with inter-atomic separation. It is obtained by integrating the eqn –
(1)
U(r) = ∫F(r) dr
= ∫ [A/rM - B/rN] dr
= [(A/1-M) x r1-M – (B/1-N) x r1-N] + c
= [-(A/M-1) r – (M-1) + (B/N-1) r–(N-1) + c
= -a / rm + b / r n + c where a = A/M-1, b = B/N-1, m = M-1, n = N-1
At r = α , U(r) = 0, then c = 0
Therefore U(r) = -a / rm + b / r n
The condition under which the particles form a stable lattice is that the function U(r)
exhibits min. for a finite value of r i.e. r = r0 this spacing r0 is known as equilibrium
spacing of the system. This min. energy Umin at r = r0 is negative and hence the energy
needed to dissociate the molecule then equals the positive quantity of ( - U min ). U min
occurs only if m and n satisfy the condition n>m
When the system in equilibrium then r = r0 and U(r) = Umin
[du/ dr] r = ro = 0
=d / dr [-a / ro
m + b / ro
n ] = 0
or 0 = [a m ro–m-1] – [b n ro–n-1]
or 0 = [a m / ro
m+1 ] – [ b n / ro
n+1] -------􀃆 ( 2)
Solving for ro
ro=[( b / a ) ( n / m )]1/n-m
or ro n = ro
m[( b /a ) (n / m)]
at the same time , n>m to prove this,
[ d 2U / dr2 ]r = ro = - [a m( m+1) / ro
m+2 ] + [ b n(n+1) / ro
n+2] > 0
[ro
m+2 b n(n+1)] - [ro
n+2a m( m+1) ] > 0
ro
m b n (n+1) > ro
n a m ( m+1)
b n (n+1) > a m ( m+1) ro
n-m
b n (n+1) > a m ( m+1) ( b / a ) ( n / m )
i.e. n > m
Calculation of cohesive energy:
The energy corresponding to the equilibrium position r = r0 , denoted by U(r0 ) is
called boning energy or cohesive energy of the molecule.
Substituting ‘ro
n’ in expression for U min,
We get
U(min) = - a / ro
m + b / ro
n
= - a / ro m+ b (a / b) ( m/ n )1 / ro
m
= - a / ro
m + ( m/ n) ( a / ro
m)
= - a / ro m[1-m / n]
Umin = - a / ro
m [1-m / n]
Thus the min. value of energy of U min is negative. The positive quantity │U min│ is the
dissociation energy of the molecule, i.e. the energy required to separate the two atoms.
Calculation of cohesive energy of NaCl Crystal
Let Na and Cl atoms be free at infinite distance of separation. The energy required
removing the outer electron from Na atom (ionization energy of Na atom), leaving it a
Na+ ion is 5.1eV.
i.e. Na + 5.1eV 􀃆 Na+ + e-
The electron affinity of Cl is 3.6eV. Thus when the removed electron from Na atom is
added to Cl atom, 3.6eV of energy is released and the Cl atom becomes negatively
charged.
Hence Cl + e- 􀃆 Cl- +3.6eV
Net energy = 5.1 – 3.6 = 1.5 eV is spent in creating Na+ and Cl- ions at infinity.
Thus Na + Cl + 1.5 eV 􀃆 Na+ + Cl-
At equilibrium spacing r0= 0.24nm, the potential energy will be min. and the energy
released in the formation of NaCl molecule is called bond energy of the molecule and is
obtained as follows:
V = -e2 / 4Πε0 r0
= - [ (1.602x10-19)2 / 4Π(8.85x10-12 )( 2.4x10-10 ) ] joules
= - [ (1.602x10-19)2 / 4Π(8.85x10-22 x2.4 )( 1.602x10-19 ) ] eV
= -6 eV
Thus the energy released in the formation of NaCl molecule is ( 5.1 - 3.6 – 6 ) = - 4.5 eV
To dissociate Na Cl molecule into Na+ and Cl- ions, it requires energy of 4.5 eV.
CRYSTAL STRUCTURES
CRYSTALLOGRAPHY
The branch of science which deals with the study of geometric form and
other physical properties of the crystalline solids by using X-rays, electron beam, and
neutron beams etc is called crystallography or crystal physics.
The solids are classified into two types 1)crystalline and2) amorphous. A
substance is said to be crystalline, when the arrangement of atoms, molecules or ions
inside it is regular and periodic. Ex. NaCl, Quartz crystal. Though two crystals of same
substance may look different in external appearance, the angles between the
2)corresponding faces are always the same. In amorphous solids, there is no particular
order in the arrangement of their constituent particles. Ex. Glass.
CRYSTALLINE SOLIDS AMORPHOUS SOLIDS
1. Crystalline solids have regular periodic 1. Amorphous solids have no
Arrangement of particles (atoms, ions, regularity in the arrangement
Or molecules). Of particles.
2. They are un-isotropic i.e., they differ in 2. They are usually isotropic i.e.,
Properties with direction. They possess same properties in
different directions.
3. They have well defined melting and 3. They do not posses well defined
Freezing points. Melting and freezing points.
.
4. Crystalline solids may be made up of 4. Most important amorphous
Materials are metallic crystals or glasses, plastics and rubber.
Non-metallic crystals. Some of the
Metallic crystals are Copper, silver,
Aluminum, tungsten , and manganese.
Non-metallic crystals are crystalline
Carbon, crystallized polymers or plastics.
5. Metallic crystals have wide use in 5. An amorphous structure does not
engineering because of their favorable generally posses elasticity but only
Lattice points: They are the imaginary points in space about which the atoms are
located.
Lattice: The regular repetition of atomic, ionic or molecular units in 2-dimensional, 3-
dimensional space is called lattice.
Space lattice or Crystal lattice: The totality of all the lattice point in space is called
space lattice, the environment about any two points is same or An array of points in
space such that the environment about each point is the same.
Consider the case of a 2-dimensional array of points.
Let O be any arbitrary point as origin, r1, r2 are position vectors of any two lattice
points joining to O.
If T ( translational vector) is the difference of two vectors r1, r2 and if it satisfies the
condition
T= n1a + n2b where n1, n2 are integers
Then T represent 2-dimensional lattice.
For 3- dimensional lattice,
T= n1a + n2b + n3c, n1, n2 n3 no of trans lation vectors along X,Y,Z,
Note: crystal lattice is the geometry of set of points in space where as the structure of the
crystal is the actual ordering of the constituent ions, atoms, molecules in space
Basis and Crystal structure:
1) Basis or pattern is a group of atoms, molecule or ions identical in
composition, arrangement and orientation.
2) When the basis is repeated with correct periodicity in all directions, it
gives the actual crystal structure.
Crystal structure = Lattice + Basis
FIG………………………..
The crystal structure is real while the lattice is imaginary.
In crystalline solids like Cu and Na, the basis is a single atom
In NaCl and CsCl- basis is diatomic
In CaF2 – basis is triatomic
Unit cell and Lattice parameters:
1) Unit cell is the smallest portion of the space lattice or geometrical
figure which can generate the complete crystal by repeating its own dimensions in varies
directions.
\
2) In describing the crystal structure, it is convenient to subdivide the structure into
small repetitive entities called unit cells.
3) unit cell may contain one or more atoms
4) shape of unit cell gives shape of the entire crystal
5) shape of the wall depends shape of the brick, brick it is considered as a unit cell.
Unit cell is the parallelepiped or cubes having 3 sets of parallel faces. It is the basic
structural unit or the building block of the crystal.
A unit cell can be described by 3 vectors or intercepts a, b, c, the lengths of the vectors
and the interfacial angles α, β, γ between them. If the values of these intercepts and
interfacial angles are known, then the form and actual size of the unit cell can be
determined. They may or may not be equal. Based on these conditions, there are 7
different crystal systems.
Primitive Cell: A unit cell having only one lattice point at the corners is called the
primitive cell. The unit cell differs from the primitive cell in that it is not restricted to
being the equivalent of one lattice point.
2) In some cases, the two coincide. unit cells may be primitive cells, but all the primitive
cells need not be unit cells.
CRYSTAL SYSTEMS AND BRAVAIS LATTICES:
There are 7 basic crystal systems which are distinguished based on three
vectors or the intercepts and the 3 interfacial angles between the 3 axes of the crystal.
They are
1. Cubic
2. Tetragonal
3. Orthorhombic
4. Monoclinic
5. Triclinic
6. Trigonal (Rhombohedral)
7. Hexagonal
More space lattices can be constructed by atoms at the body centers of unit cells or at
the centers of the faces. Based on this property, bravais ,in 1880,classified the space
lattices into 14.in the 7 crystal systems
1. 1.primitive cell. in this lattice, the unit cell consists of eight corners,all
corners have one atom
2.Body centered lattice ; in addition to the eight corner atoms,it consists of one
compleate atomat the center
4. Fa ce centered lattice ; along with corner atoms, each face will have one
center atom
5. Base centered lattice; the base and opposite face will have center atoms along
with the corner atoms
1. Cubic crystal system
a = b = c, α = β = γ =900
The crystal axes are perpendicular to one another, and the repetitive interval in the same
along all the three axes. Cubic lattices may be simple, body centered or face-centered.
2. Tetragonal crystal system
a = b ≠ c, α = β = γ =900
The crystal axes are perpendicular to one another. The repetitive intervals along the two
axes are the same, but the interval along the third axes is different. Tetragonal lattices
may be simple or body-centered.
3. Orthorhombic crystal system.
a ≠ b ≠ c, α = β = γ =900
The crystal axes are perpendicular to one another but the repetitive intervals are different
along the three axes. Orthorhombic lattices may be simple, base centered, body centered
or face centered.
4.Monoclinic crystal system
a ≠ b ≠ c, α = β = 900 ≠γ
Two of the crystal axes are perpendicular to each other, but the third is obliquely
inclined. The repetitive intervals are different along all the three axes. Monoclinic lattices
may be simple or base centered.
5. Triclinic crystal system
a ≠ b ≠ c, α ≠ β ≠γ ≠ 900
None of the crystal axes is perpendicular to any of the others, and the repetitive intervals
are different along the three axes.
6. Trigonal(rhombohedral) crystal system
a = b = c, α = β = γ ≠ 900
The three axes are equal in length and are equally inclined to each other at an angle other
than 900
7. Hexagonal crystal system.
a = b ≠ c, α = β =γ = 900 , γ = 1200
Two of the crystal axes are 600 apart while the third is perpendicular to both of them. The
repetitive intervals are the same along the axes that are 600 apart, but the interval along
the third axis is different.
Basic Crystal Structures:
The important fundamental quantities which are used to study the different arrangements
of atoms to form different structure are
Nearest neighbouring distance ( 2r) : the distance between the centers of two
nearest neighboring atoms is called nearest neighboring distance. If r is the
radius of the atom, nearest neighboring distance= 2r.
Atomic radius ( r) : It is defined as of the distance between the nearest neighboring
atoms in a crystals.
Coordination number (N): It is defined as the number of equidistant nearest
neighbours that an atom as in a given structure. More closely packed structure as
greater coordination number.
Atomic packing factor or fraction: It is the ratio of the volume occupied by the
atoms in unit cell(v) to the total volume of the unit cell (V).
P.F. = v/ V
Simple cubic (SC) structure:
In the simple cubic lattice, there is one lattice point at each of the 8
corners of the unit cell. The atoms touch along cubic edges.
Fig. Simple Cubic Structure
Nearest neighbouring distance = 2r = a
Atomic radius = r = a / 2
Lattice constant = a = 2r
Coordination number = 6 (since each corner atom is surrounded by 6 equidistant
nearest neighbours)
Effective number of atoms belonging to the unit cell or no. of atoms per unit cell =
(⅛)x8 = 1 atom per unit cell.
Atomic packing factor = v/ V = volume of the all atoms in the unit cell
-----------------------------------------------
Volume of the unit cell.
= 1 x (4 / 3) Π r3 / a3 = 4Π r3 / 3(2 r )3
= Π / 6 = 0.52 = 52%
Void space =48%
This structure is loosely packed. In simple cubic atoms are occupied upto 52%,remaining
vacant space is 48%
EXAMPLE. Polonium
Body centered cube structure (BCC):
BCC structure has one atom at the centre of the cube and one atom at each corner. The
centre atom touches all the 8 corner atoms.
Fig. Body Centered Cubic Structure
Diagonal length = 4r
Body diagonal = (√ 3) a
i.e. 4r = (√ 3) a
Nearest neighbouring distance = 2r = (√ 3) a / 2
Atomic radius = r = (√ 3) a / 4
Lattice constant = a = 4r / √ 3
Coordination number = 8 (since the central atom touches all the corner 8 atoms)
Effective number of atoms belonging to the unit cell or no. of atoms per unit cell = (⅛)
x8 + 1 = 2 atom per unit cell.
I.e. each corner atom contributes ⅛th to the unit cell. In addition to it, there is a one
more atom at the center
Atomic packing factor = v/ V = volume of the all atoms in the unit cell
-----------------------------------------------
Volume of the unit cell.
= 2 x (4 / 3) Π r3 / a3 = 8Π r3 / 3(4r /√ 3 )3
= √ 3 Π / 8 = 0.68 = 68%
Void space : 32%
EXAMPLES : Tungsten, Na, Fe and Cr,Molybdenum, exhibits this type of structure.\
Face centered cubic (FCC) structure:
In FCC structure, ther is one lattice point at each of the 8 corners of the unit cell and 1
centre atom on each of the 6 faces of the cube.
Fig. Face Centered Cubic Structure
Face diagonal length = 4r = (√ 2) a
Nearest neighbouring distance = 2r = (√ 2)a / 2 = a / √ 2
Atomic radius = r = a / 2√ 2
Lattice constant = a = 2√ 2 r
Coordination number = 12 ( the centre of each face has one atom. This centre atom
touches 4 corner atoms in its plane, 4 face centered atoms in each of the 2 planes on
either side of its plane)
Effective number of atoms belonging to the unit cell or no. of atoms per unit cell = (⅛)
x8 + (1/2) x 6 = 1 + 3 = 4 atom per unit cell.
I.e. each corner atom contributes ⅛th to the unit cell. In addition to it, there is a centre
atom on each face of the cube.
Atomic packing factor = v/ V = volume of the all atoms in the unit cell
-----------------------------------------------
Volume of the unit cell.
= 4 * (4 / 3) Π r3 / a3 = 16Π r3 / 3(2√ 2 r )3
= Π / 3√ 2 = 0.74 = 74%
Void space is equal to 26%
EX : Cu, Al, Pb and Ag, NI,AU, have this structure. FCC has highest packing factor.



Directions & Planes in Crystals:
While dealing with the crystals, it is necessary to refer to crystal planes,
and directions of straight lines joining the lattice points in a space lattice. For this
purpose, an indexing system deviced by Miller known as Miller indices is widely used.
Directions in A Crystal:
Consider a cubic lattice in which a straight line is passing
through the lattice points A, B, C etc and 1 lattice point on the line such as point A is
chosen as the origin.
Then the vector R which joins A to any other point on the line such as B (position vector)
can be represented by the vector eqn.
R = n1 a + n2 b + n3 c------ (1) where a, b, c are basic vectors
The direction of the vector R depends on the integers n1, n2, n3 since a, b, c are
constants. The common multiple is removed and n1, n2, n are re-expressed as the
smallest integers bearing the same relative ratio. The direction is then specified as [n1 n2
n3].
R =a+b+c, which provides the value of 1 for each of n1, n2, n3
Thus the direction is denoted as [1 1 1].
All lines in the space lattice which are parallel to the line AB possess either same set of
values for n1, n2, n3 as that of AB, or its common multiples.
Ex: 1. The direction that connects the origin and (1/3, 1/3, 2/3) point is [1 1 2].
i.e, (1/3, 1/3, 2/3)
L.C.M = 3.
(1/3x3 1/3x3 2/3x3)= [1 1 2].
2. [2 1 1] is the direction that connects the origin (0, 0, 0) and point (1, 1/2, 1/2)
Largest Number = 2
(2/2, 1/2, 1/2) = (1, 1/2, 1/2)
Planes in Crystals (Miller Indices):
Crystal plane; .the plane passing through the latice points is known as crystal plane
Def: Miller indices is a set of three lowest possible integers whose ratio taken in order
is the same as that of the reciprocals of the intercepts of the planes on the corresponding
axes in the same order. OR
Miller indices are defined as reciprocal of the intercepts made by the plane on the
crystallographic axes which are reduced to smallest numbers
It is possible for defining a system of parallel and equidistant planes which can
be imagined to pass through the atoms in a space lattice, such that they include all the
atoms in the crystal. Such a system of planes is called crystal planes. Many different
systems of planes could be identified for a given space lattice.
The position of a crystal plane can be specified in terms of three
integers called Miller indices
Consider a crystal plane intersecting the crystal axes.
Procedure for finding Miller indices
1. Find the intercepts of the desired plane on the three coordinate axes.
Let these be ( pa, qb, rc).
2. Express the intercepts as multiples of unit cell dimensions or lattice parameters
i.e. (p, q, r)
3. Take the reciprocals of these numbers i.e. 1/p: 1/q: 1/r
4. Convert these reciprocals into whole numbers by multiplying each with their
LCM to get the smallest whole number.
This gives the Miller indices (h k l) of the plane.
Ex: (3a, 4b, α c)
(3, 4, α)
1/3 1/4 1/α
(4 3 0) = (h, k, l)
Important features of Miller indices:
1. When a plane is parallel to any axis, the intercept of the plane on that axis is
infinity. Hence its Miller index for that axis is zero.
2. When the intercept of a plane on any axis is negative, a bar is put on the
corresponding Miller index.
3. All equally spaced parallel planes have the same index number (h k l)
Ex: The planes ( 1 1 2) and (2 2 4) are parallel to each other.
Separation Between successive (h k l) Planes:
Let (h k l) be the Miller indices of the plane ABC.
Let OP=d h k l be the normal to the plane ABC passing through the origin O.
Let OP make angles α, β, γ with X, Y & Z axes respectively.
Then cos α =d / OA = d / x = d / (a / h)
Fig. Inter planar Spacing.
Cos β =d /OB = d / y = d / (b / k)
Cos γ =d /OC = d/z =d/(c/l)
(Since convention in designing Miller indices x=a/h, y=b/k, z=c/l)
Now cos2 α + cos2 β +cos2 γ = 1
Hence d2 / (a / h) 2 + d2 / (b/k) 2 + d2 / (c/l) 2 = 1
􀃎 (d h / a )2 + (d k / b)2 +(d l / c)2 =1
􀃎 d (h k l) = OP = 1 / √ (h2/a2 +k2/b2 + l2/c2).
Therefore for cubic structure, a=b=c,
d (h k l) = a / √ ( h2 + k2 + l2 )
Let the next plane be parallel to ABC be at a distance OQ from the origin. Then its
intercepts are 2a / h, 2a / k, 2a / l.
Therefore OQ = 2d = 2a / √ (h2 + K+ l2)
Hence the spacing between adjacent planes = OQ - OP = PQ.
i.e. d = a / √ (h2 + k2 + l2)
Expression for Space Lattice Constant ‘a’ For a Cubic Lattice:
Density р = (total mass of molecules belonging to unit cell) / (volume of unit cell)
Total mass of molecule belonging to unit cell = nM / NA
Where n-number of molecules belonging to unit cell
M-Molecular weight
NA -Avagadro Number
Volume of cube = a3
Therefore р= nM/ a3 NA
Or a3 = nM / р NA
Lattice Constant for Cubic Lattice a = ( n M / р NA )1/3.
X-RAY DIFFRACTION
Diffraction of X- Rays by Crystal Planes:
1)X-Rays are electromagnetic waves like ordinary light; therefore, they should exhibit
interference and diffraction.
2) Diffraction occurs when waves pass across an object whose dimensions are of the
order of their own wavelengths. The wavelength of X-rays is of the order of 0.1nm or
10 -8 cm so that ordinary devices such as ruled diffraction gratings do not produce
observable effects with X-rays.
3) Laue suggested that a crystal which consisted of a 3-dimensional array of regularly
spaced atoms could serve the purpose of a grating. The crystal differs from ordinary
grating in the sence that the diffracting centers in the crystal are not in one plane.
4) Hence the crystal acts as a space grating rather than a plane grating.
X-ray diffraction methods
(1) Laue Method – for single crystal
(2) Powder Method- for finely divided crystalline or polycrystalline powder
(3) Rotating crystal Method - for single crystal
Bragg’s Law:
DEF. It states that the X-rays reflected from different parallel planes of a crystal
interfere constructively when the path difference is integral multiple of wavelength of
X-rays.
Consider a crystal made up of equidistant parallel planes of atoms with the interplanar
spacing d.
Let wave front of a monochromatic X-ray beam of wavelength ג fall at an angle θ on
these atomic planes. Each atom scatters the X-rays in all directions.
In certain directions these scattered radiations are in phase ie they interfere
constructively while in all other directions, there is destructive interference.
Figure 1 X-Ray Scattering by Crystal.
Consider the X-rays PE and P’A are inclined at an angle θ with the top of the crystal
plane XY. They are scattered along AQ and EQ’ at an angle θ w.r.t plane XY.
Consider another incoming beam P’C is scattered along CQ”
Let normal EB & ED be drawn to AC &CF. if EB & ED are parallel incident and
reflected wave fronts then the path difference between the incident and reflected waves is
given by
Δ = BC + CD --------------- (1)
In Δ ABC , sin θ = BC / EC = BC / d
i e BC = d sin θ
Similarly, in Δ DEC, CD = d sin θ
Hence path difference Δ = 2d sin θ FIG-------------------
If the 2 consecutive planes scatter waves in phase with each other , then the path
difference must be an integral multiple of wavelength.
i e Δ = n λ where n = 0 ,1 , 2 , 3 ,…….is the order of reflection
Thus the condition for in phase scattering by the planes in a crystal is given by
2d sin θ = n λ …………….(2)
This condition is known as Bragg’s Law.
The maximum possible value for θ is 1.
n λ
v=2d, λ-<2d
LIMITATION; λ should not exceed twice the inter planar spacing for diffraction to
occur.
Laue Method : 1)S1 & S2 are 2 lead screens in which 2 pin holes act as slits .
2) X-ray beam from an X –ray tube is allowed to pass through these 2 slits S1 & S2 . the
beam transmitted through S2 will be a narrow pencil of X – rays . the beam proceeds
further to fall on a single crystal such that Zinc blended (ZnS) which is mounted suitably
on a support . the single crystal acts as a 3 – dimensional diffraction grating to the
incident beam.
3)Thus, the beam undergoes diffraction in the crystal and then falls on the photographic
film.
4)The diffracted waves undergo constructive interference in certain directions, and fall on
the photographic film with reinforced intensity.
5)In all other directions, the interference will be destructive and the photographic film
remains unaffected.
6)The resultant interference pattern due to diffraction through the crystal as a whole will
be recorded on the photographic film (which requires many hours of exposure to the
incident beam
7) When the film is developed, it reveals a pattern of fine spots, known as Laue spots.
8)The distribution spots follow a particular way of arrangement that is the characteristic
of the specimen used in the form of crystal to diffract the beam.
MERITS; The Laue spot photograph obtained by diffracting the beam at several
orientations of the crystal to the incident beam are used for determining the symmetry
and orientations of the internal arrangement of atoms, molecules in the crystal lattice . it
is also used to study the imperfections in the crystal .
DEMERITS this method is not convenient for actual crystal structure determination
because of several wavelengths X rays diffract in different order from same plane,
and they super impose single lave spot
POWDER METHOD ( Debye – Scherrer Method ):
1)This method is widely used for experimental determination of crystal structures.
2)A monochromatic X- ray beam is incident on randomly oriented crystals in powder
form.
3)In this we used a camera called Debye – Scherer camera.
4)It consists of a cylindrical cassette, with a strip of photographic film positioned around
the circular periphery of cassette.
5) The powder specimen is placed at the centre, either pasted on a thin fiber of glass or
filled in a capillary glass tube.
6)The X- ray beam enters through a small hole in the camera and falls on the powder
specimen.
7)Some part of the beam is diffracted by the powder while the remaining passes out
through the exit port.
8)Since large no. of crystals is randomly oriented in the powder, set of planes which
make an angle θ with the incident beam can have a no. of possible orientations.
9) Hence reflected radiation lies on the surface of a cone whose apex is at the point of
contact of the radiation with the specimen.
10)If all the crystal planes of interplanar spacing d reflect at the same bragg angle θ, all
reflections from a family lie on the same cone.
After taking n=1 in the Bragg’s law
2dsin θ = λ
11)There are still a no of combinations of d and θ, which satisfies Bragg’s law. Hence
many cones of reflection are emitted by the powder specimen. In the powder camera a
part of each cone is recorded by the film strip.
12)The full opening angle of the diffracted cone 4θ is determined by measuring the
distance S between two corresponding arcs on the photographic film about the exit point
direction beam. The distance S on the film between two diffraction lines corresponding to
a particular plane is related to Bragg’s angle by the equation
4θ = (S / R) radians (or)
4θ = (S / R) x (180 /Π ) degrees where R- radius of the camera
13)A list of θ values can be thus be obtained from measured values of S. Since the
wavelength ‘λ’ is known, substitution of λ gives a list of spacing‘d’.
Each spacing is the distance between neighbouring plane (h k l). From the ratio of
interplanar spacing, the type of lattice can be identified.
CRYSTAL DEFECTS
In real materials we find:
Crystalline Defects or lattice irregularity
Most real materials have one or more “errors in
perfection”
with dimensions on the order of an atomic diameter to
many lattice sites
Defects can be classification: crystalline
imperfection
zero-dimensional or point defects
one-dimensional or line defects (dislocation)
two-dimensional or planar defects
three-dimensional or volume defects
(1) point defects
vacancy – atom is missing, may be created
by
 local disturbances during the crystal
growth
 atomic arrangements in an existing crystal
 plastic defromation, rapid cooling
 bombardment with energetic particles
interstitialcy or self-interstitial – an atom in
a
crystal can occupy an interstitial site
between
surrounding atoms
can be introduced by irradiation
1. according to geometry
(point, line or plane)
2. dimensions of the defect
POINT DEFECTS
• The simplest of the point defect is a vacancy, or
vacant lattice site.
• All crystalline solids contain vacancies.
• Principles of thermodynamics is used explain the
necessity of the existence of vacancies in
crystalline solids.
• The presence of vacancies increases the entropy
(randomness) of the crystal.
• The equilibrium number of vacancies for a given
quantity of material depends on and increases
with temperature as follows: (an Arrhenius
model)
• n = N exp(-Ev/kT)
Schottky imperfection – two oppositely
charged
ions are missing form an ionic crystal
a cation-anion divacancy
impurity is also a type of point defect
(2) line defects (dislocations)
crystalline solids are defects that cause
lattice
distortion centered around a line
formed by plastic deformation, vacancy
condensation, and atomic mismatch
Frenkel imperfection – a cation moves into
an
interstitial site, and a cation vacancy is
created
vacancy-interstitialcy pair
the presence of these defects in ionic
crystals
increases their electrical conductivity
Linear Defects (Dislocations)
– Are one-dimensional defects around which
atoms are misaligned
• Edge dislocation:
– extra half-plane of atoms inserted in a crystal
structure
– b (the berger’s vector) is ⊥ (perpendicular) to
dislocation line
• Screw dislocation:
– spiral planar ramp resulting from shear
deformation
– b is || (parallel) to dislocation line
Burger’s vector, b: is a measure of lattice distortion and
is measured as a distance along the close packed
directions in the lattice
Definition of the Burgers vector, b, relative to an edge
dislocation. (a) In the perfect crystal, an m× n atomic step
loop closes at the starting point. (b) In the region of a
dislocation, the same loop does not close, and the closure
vector (b) represents the magnitude of the structural
defect. For the edge dislocation, the Burgers vector is
perpendicular to the dislocation line.
Screw dislocation. The spiral
stacking of crystal planes leads to the Burgers vector
being parallel to the dislocation
line.

Simple grainboundary
structure. This is termed a tilt boundary
because it is formed when two adjacent crystalline grains
are tilted relative to each other by a few degrees (θ). The
resulting structure is equivalent to isolated edge
dislocations separated by the distance b/θ, where b is the
length of the Burgers vector, b.
Dislocation Line:
A dislocation line is the boundary between slip and no slip regions of a crystal
Burgers vector:
The magnitude and the direction of the slip is represented by a vector b called the
Burgers vector
In general, there can be any angle between the Burgers vector b (magnitude and the
direction of slip) and the line vector t (unit vector tangent to the dislocation line
b ⊥ t ⇒ Edge dislocation
b ⎜⎜ t ⇒ Screw dislocation
grain boundaries – a narrow region
between
two grains of about 2~5 atomic diameters in
width and a region of atomic mismatch
between adjacent grains the higher energy of
grain boundaries and
more open structure make them more
favorable for nucleation and growth of
precipitates
twin or twin boundary – a region in which
a
mirror image of the structure exists across a
plane or a boundary
ex. twin boundaries in the grain structure of
brass
Grain Boundary: low and high angle
One grain orientation can be obtained by rotation of another grain across the grain
boundary about an axis through an angle
If the angle of rotation is high, it is called a high angle grain boundary
If the angle of rotation is low it is called a low angle grain boundary
Grain Boundary: tilt and twist
One grain orientation can be obtained by rotation of another grain across the grain
boundary about an axis through an angle
If the axis of rotation lies in the boundary plane it is called tilt boundary
If the angle of rotation is perpendicular to the boundary plane it is called a twist boundary
stacking faults or piling-up faults
one or more of the stacking planes may be
missing, give rise to another twodimensional
defect
ex. ABCABAACBABC in FCC crystal
ABAABBAB in HCP crystal
(4) volume defects
a cluster of point defects join to form a
three-dimensional void or a pore
a cluster of impurity atoms join to form a
three-dimensional precipitate
the size from a few nm to cm
⊥

Questions;
Explain the various types of bonding in crystals. Illustrate with examples.
Distinguish between ionic and covalent bonding in solids.
Obtain a relation between potential energy and inter atomic spacing of a molecule.
Derive an expression for cohesive energy of a solid.
Obtain an equation for total binding energy of sodium chloride crystal.
Differentiate between crystalline and amorphous solids.
Explain the terms i) Basis ii) Space lattice iii) Unit cell
Explain with neat diagram the following crystal structures.
simple cubic structure(SC)
body centered cubic structure(BCC)
face centered cubic structure (FCC)
What do you understand by packing density? Show that packing density for simple
lattice, body centered lattice and face centered lattice is Π/6 , √3Π/8 , √2Π/6
respectively
Show that FCC is the most closely packed of the three cubic structures.
a) For a crystal having a ≠ b ≠ c and α = β = γ = 900, what is the crystal system?
b) For a crystal having a ≠ b ≠ c and α ≠ β ≠ γ ≠ 900, what is the crystal system?
c) Can you specify the Bravais lattices for parts (a) and (b) explain.
Explain the special features of the three types of lattices of cubic crystals?
What are ionic crystals? Explain the formation of an ionic crystal and obtain an
expression for its cohesive energy?
What is a Bravais lattice? What are the different space lattices in the cubic system?
What are the miller indices? How they obtained?
Derive the expression for the interplannar spacing between two adjacent planes of miller
indices (h k l ) in a cubic lattice of edge length ‘a ‘.
Derive f rombragg’s law 2d sin θ= n λ
Describe Laue’s method of determination of crystal structu Explain the power method
of crystal structure analysis.
UNIT -II
PRINCIPLES OF QUANTUM MECHANICS
Introduction: Quantum mechanics is a new branch of study in physics which is
indispensable in understanding the mechanics of particles in the atomic and sub-atomic
scale.
The motion of macro particles can be observed either directly or
through microscope. Classical mechanics can be applied to explain their motion. But
classical mechanics failed to explain the motion of micro particles like electrons, protons
etc...
Max Plank proposed the Quantum theory to explain Blackbody
radiation. Einstein applied it to explain the Photo Electric Effect. In the mean time,
Einstein’s mass – energy relationship (E = mc2) had been verified in which the radiation
and mass were mutually convertible. Louis deBroglie extended the idea of dual nature of
radiation to matter, when he proposed that matter possesses wave as well as particle
characteristics.
The classical mechanics and the quantum mechanics have
fundamentally different approaches to solve problems. In the case of classical mechanics
it is unconditionally accepted that position, mass, velocity, acceleration etc of a particle
can be measured accurately, which, of course, true in day to day observations. In contrast,
the structure of quantum mechanics is built upon the foundation of principles which are
purely probabilistic in nature. As per the fundamental assumption of quantum mechanics,
it is impossible to measure simultaneously the position and momentum of a particle,
whereas in the case of classical mechanics, there is nothing which contradicts the
measurements of both of them accurately.
Plank’s Quantum Theory:
Max Plank, a German physicist derived an equation which successfully accounted for the
spectrum of the blackbody radiation. He incorporated a new idea in his deduction of
Plank eqn. that the probability of emission of radiation decreases as its frequency
increases so that, the curve slopes down in the high frequency region. The oscillators in
the blackbody can have only a discrete set of energy values. Such an assumption was
radically different from the basic principles of physics.
The assumption in the derivation of Plank’s law is that the wall of the
experimental blackbody consists of a very large number of electrical oscillators, with
each oscillator vibrating with a frequency of its own. Plank brought two special
conditions in his theory.They are
(1) Only an integral multiple of energies h ν where ‘h’ is Plank’s constant and ‘ν ‘ is
frequency of vibration i e, the allowed energy values are E = n h ν where n = 0 ,
1 , 2 , ………
(2) An oscillator may lose or gain energy by emitting or absorbing radiation of
frequency ν = (ΔE / h), where ΔE is the difference in the values of energies of the
oscillator before and the emission or absorption had taken place.
Based on the above ideas, he derived the law governing the entire spectrum of the
Blackbody radiation, given by
U λ d λ = (8Πhc / λ5) [1 / (e h ν /kT – 1)] d λ (since ν = c / λ)
This is called Plank’s radiation law.
Waves and Particles: deBroglie suggested that the radiation has dual nature i e
both particle as well as wave nature. The concept of particle is easy to grasp. It has
mass, velocity, momentum and energy. The concept of wave is a bit more difficult
than that of a particle. A wave is spread out over a relatively large region of space, it
cannot be said to be located just here and there, and it is hard to think of mass being
associated with a wave. A wave is specified by its frequency, wavelength, phase,
amplitude, intensity.
Considering the above facts, it appears difficult to accept the
conflicting ideas that radiation has dual nature. However this acceptance is essential
because radiation sometimes behaves as a wave and at other times as a particle.
(1) Radiations behaves as waves in experiments based on interference, diffraction,
polarization etc. this is due to the fact that these phenomena require the presence
of two waves at the same position and at the same time. Thus we conclude that
radiation behaves like wave.
(2) Plank’s quantum theory was successful in explaining blackbody radiation,
photoelectric effect, Compton Effect and had established that the radiant energy,
in its interaction with the matter, behaves as though it consists of corpuscles. Here
radiation interacts with matter in the form of photons or quanta. Thus radiation
behaves like particle.
Hence radiation cannot exhibit both particle and wave nature simultaneously.
deBroglie hypothesis :The dual nature of light possessing both wave and
particle properties was explained by combining Plank’s expression for the energy of a
photon E = h ν with Einstein’s mass energy relation E = m c2 (where c is velocity of
light , h is Plank’s constant , m is mass of particle )
􀃖 h ν = m c2
Introducing ν = c / λ, we get h c / λ = m c2
==> λ = h / mc = h / p where p is momentum of particle
λ is deBroglie wavelength associated with a photon.
deBroglie proposed the concept of matter waves , according to which a material particle
of mass ‘m’ moving with velocity ‘v’ should be associated with deBroglie wavelength ‘λ’
given by
λ = h / m v = h / p
The above eqn. represents deBroglie wave eqn.
Characteristics of Matter waves:
Since λ = h / m v
1. Lighter the particle, greater is the wavelength associated with it.
2. Lesser the velocity of the particle, longer the wavelength associated with it.
3. For v = 0, λ = ∞. This means that only with moving particle, matter waves is
associated.
4. Whether the particle is changed or not, matter waves is associated with it.
5. It can be proved that matter waves travel faster than light.
We know that E = h ν and E = m c2
􀃖 h ν = m c2 or ν = m c2 / h
Wave velocity (ω) is given by
ω = ν λ = m c2 λ / h = (m c2 / h) (h / m v)
􀃖 ω = c 2 / v
As the particle velocity ‘v’ cannot exceed velocity of light, ω is greater than
the velocity of light.
6. No single phenomena exhibit both particle nature and wave nature
simultaneously.
7. The wave nature of matter introduces an uncertainty in the location of the particle
& the momentum of the particle exists when both are determined simultaneously.
Davisson and Germer’s experiment:
C. J. Davisson and L. H. Germer were studying scattering of electrons by a metal target
and measuring the intensity of electrons scattered in different directions.
Experimental Arrangement:
An electron gun, which comprises of a tungsten filament is heated by a
low tension battery B1, produces electrons. These electrons are accelerated to desired
velocity by applying suitable potential from a high tension source B2. The accelerated
electrons are collimated into a fine beam by allowing them to pass through a system of
pin holes provided in the cylinder. The whole instrument is kept in an evacuated
chamber.
The past moving beam of electrons is made to strike the Nickel target
capable of rotating about an axis perpendicular to the plane. The electrons are now
scattered in all directions by the atomic planes of crystals. The intensity of the electron
beam scattered in a direction can be measured by the electron collector which can be
rotated about the same axis as the target. The collector is connected to a galvanometer
whose deflection is proportional to the intensity of the electron beam entering the
collector.
Fig. Davisson and Germer's Apparatus
The electron beam is accelerated by 54 V is made to strike the Nickel crystal
and a sharp maximum is occurred at angle of 50o with the incident beam. The incident
beam and the diffracted beam in this experiment make an angle of 65o with the family of
Bragg’s planes.
d = 0.091nm (for Ni crystals)
According to Bragg’s law for maxima in diffracted pattern,
2d sin θ = n λ
For n =1, λ = 2d sin θ
= 2 x0.91x 10-10x sin 65o
= 0.165 nm
For a 54 V electron, the deBrogllie wavelength associated with the electron is given by
= 12.25 / √ V = (12.25 / √54) oA
= 0.166 nm.
This value is in agreement with the experimental value. This experiment provides a direct
verification of deBroglie hypothesis of wave nature of moving particles.
G.P. Thomson’s Experiment:
1) G.P. Thomson investigated high speed electrons produced by applying
the high voltage ranges from10 to 50kV. The principle is similar to that of powdered
crystal method of X- ray diffraction.
In this experiment, an extremely thin ( 10-8m ) metallic film F of gold, aluminium etc.,
is used as a transmission grating to a narrow beam of high speed electrons emitted by a
cathode C and accelerated by anode A. A fluorescent screen S or a suitable photographic
plate P is used to observe the scattered electron beam. The whole arrangement is enclosed
in a vacuum chamber.
The electron beam transmitted through the metal foil gets scattered
producing diffraction pattern consisting of concentric circulars rings around a central
spot. The experimental results and its analysis are similar to those of powdered crystal
experiment of X- ray diffraction.
Fig. Thomson Apparatus
L: distance between the foil and screen
R(r): radius of the diffraction ring
θ: glancing angle
The method of interpretation of the experimental results is same as that of the Davisson
and Germer experiment. Here also we use Bragg’s law and deBroglie wavelength.
We have 2d sin θ = n λ
θ = n λ / 2d --􀃆 I since θ is small sin θ = θ
From fig.
R/L = tan2 θ ≈ 2 θ
R = L 2 θ -􀃆 II
From eqn I, II
R = L 2 n λ / 2d = L n λ / d
Since L and d are fixed in the experiment
R α n λ or D α n λ D is diameter of the ring
Combing with the de Broglie expression
λ = √ (150 / V (1+r))
Where (1+r) is the relativistic correction we notice that D√ (V (1+r)) must be constant for
a given order the experiment repeated with different voltages. The data shows that D√
(V(1+r)) is constant, thus supporting the deBroglie concept of matter waves.
Heisenberg Uncertainty Principle:
According to classical mechanics, a moving particle at any instant has a
fixed position in space and a definite momentum which can be determined
simultaneously with any desired accuracy. The classical point of view represents an
approximation which is adequate for the objects of appreciable size, but not for the
particles of atomic dimensions.
Since a moving particle has to be regarded as a deBroglie group,
there is a limit to the accuracy with which we can measure the particle properties. The
particle may be found anywhere within the wave group, moving with the group
velocity. If the group is narrow, it is easy to locate its position but the uncertainty in
calculating its velocity or momentum increases. On the other hand, if the group is
wide, its momentum can be estimated satisfactorily, but the uncertainty in finding the
location of the particle is great. Heisenberg stated that the simultaneous determination
of exact position and momentum of a moving particle is impossible.
If Δ x is Error in the measurement of position of the particle along X-axis
Δ p is Error in the measurement of momentum
Then Δ x. Δ p = h ---------- (1) where h is Plank’s constant
The above relation represents the uncertainty involved in measurement of both the
position and momentum of the particle.
To optimize the above error, lower limit is applied to the eqn. (1)
Then (Δ x). (Δ p) ≥ Ђ / 2 where ђ = h / 2 Π
A particle can be exactly located (Δ x → 0) only at the expense of an infinite
momentum (Δ p → ∞).
There are uncertainty relatio0ns between position and momentum, energy and time,
and angular momentum and angle.
If the time during which a system occupies a certain state is not greater than Δ t, then
the energy of the state cannot be known within Δ E,
i e (Δ E ) ( Δ t ) ≥ ђ / 2 .
Schrödinger’s Time Independent Wave Equation:
Schrödinger, in 1926, developed wave equation for the moving
particles. One of its forms can be derived by simply incorporating the deBroglie
wavelength expression into the classical wave eqn.
If a particle of mass ‘m’ moving with velocity ‘v’ is associated
with a group of waves.
Let ψ be the wave function of the particle. Also let us consider a simple form of
progressing wave like the one represented by the following equation,
Ψ = Ψ0 sin (ω t – k x) --------- (1)
Where Ψ = Ψ (x, t) and Ψ0 is the amplitude.
Differentiating Ψ partially w.r.to x,
∂ Ψ / ∂ x = Ψ0 cos (ω t – k x) (- k)
= -k Ψ0 cos (ω t – k x)
Once again differentiate w.r.to x
∂2 ψ / ∂ x2 = (- k) Ψ0 (- sin (ω t – k x)) (- k)
= - k 2 Ψ0 sin (ω t – k x)
∂2 ψ / ∂ x2 = - k 2 ψ (from eqn (1))
∂2 ψ / ∂ x2 + k 2 ψ = 0 ----------- (2)
∂2 ψ / ∂ x2 + (4 Π2 / λ2) ψ = 0 --------- (3) (since k = 2 Π / λ)
From eqn. (2) or eqn. (3) is the differential form of the classical wave eqn. now we
incorporate deBroglie wavelength expression λ = h / m v.
Thus we obtain
∂2 ψ / ∂ x2 + (4 Π2 / (h / m v) 2) ψ = 0
∂2 ψ / ∂ x2 + 4 Π2 m2 v 2 ψ / h2 = 0 -------------- (4)
The total energy E of the particle is the sum of its kinetic energy K and potential energy V
i e E = K + V -------------- (5)
And K = mv2 / 2 ---------- (6)
Therefore m2 v 2 = 2 m (E – V) ------------ (7)
From (4) and (7)
=> ∂2 ψ / ∂ x2 + [8Π2 m (E-V) / h2] ψ = 0 ------------ (8)
In quantum mechanics, the value h / 2 Π occurs more frequently. Hence we denote,
ђ = h / 2 Π
Using this notation, we have
∂2 ψ / ∂ x2 + [2 m (E – V) / ђ 2] ψ = 0 ------------ (9)
For simplicity, we considered only one – dimensional wave. Extending eqn. (9) for a three
– dimensional, we have
∂2 ψ / ∂ x2 + ∂2 ψ / ∂ y2 + ∂2 ψ / ∂ z2 + [2 m (E – V) / ђ 2] ψ = 0 ------------ (10)
Where Ψ = Ψ (x, y, z).
Here, we have considered only stationary states of ψ after separating the time dependence
of Ψ.
Using the Laplacian operator,
▼2 = ∂2 / ∂ x2 + ∂2 / ∂ y2 + ∂2 / ∂ z2 ------------- (11)
Eqn. (10) can be written as
▼2 Ψ + [2 m (E – V) / ђ 2] ψ = 0 --------------- (12)
This is the Schrödinger Time Independent Wave Equation.
Physical Significance of Wave Function:
Max Born in 1926 gave a satisfactory interpretation of the wave function ψ associated
with a moving particle. He postulated that the square of the magnitude of the wave
function |ψ|2 (or ψ ψ* it ψ is complex), evaluated at a particular point represents the
probability of finding the particle at the point. |ψ|2 is called the probability density and ψ
is the probability amplitude. Thus the probability of the particle within an element
volume dt is |ψ|2 dτ. Since the particle is certainly somewhere, the integral at |ψ|2 dτ
over all space must be unity i.e.
-∞∫∞ |ψ|2 .dτ = 1 ___________________ (28)
A wave function that obeys the above equations is said to be normalized. Energy
acceptable wave function must be normalizable besides being normalizable; an
acceptable wave function should fulfill the following requirements (limitations)
.
1. It must be finite everywhere.
2. It must be single valued.
3. It must be continuous and have a continuous first derivative everywhere.
Normalization Of a wave function:
Since |ψ(x, y, z) 2| .dν is the probability that the particle will be found in a volume
element dν. Surrounding the point at positron (x, y, z), the total probability that the
particle will be somewhere in space must be equal to 1. Thus, we have
-∞∫∞ |ψ(x, y, z) 2| .dν = 1
Where ψ is a function of the space coordinates (x, y, z) from this ‘normalization
condition’ we can find the value of the complaint and its sign. A wave function which
satisfies the above condition is said to be normalized (to unity).
The normalizing condition for the wave function for the motion of a particle in
one dimension is
-∞∫∞ |ψ(x) |2 .dx = 1
From these equations, we see that for one – dimensional case, the dimension of ψ(x) in L-
1/2 and for the three – dimensional case the dimension of ψ(x, y, z) in L-3/2.
Particle in One Dimensional Potential Box:
Consider a particle of mass ‘m’ placed inside a one-dimensional box of infinite
height and width L.
Fig. Particle in a potential well of infinite height.
Assume that the particle is freely moving inside the box. The motion of the particle is
restricted by the walls of the box. The particle is bouncing back and forth between the walls
of the box at x = 0 and x = a. For a freely moving particle at the bottom of the potential
well, the potential energy is very low. Since the potential energy is very low, moving
particle energy is assumed to be zero between x =0 and x = a.
The potential energy of the particle outside the walls is infinite due to the infinite P.E
outside the potential well.
The particle cannot escape from the box
i.e. V = 0 for 0 < x < a
V = ∞ for 0 ≥ x ≥ a
Since the particle cannot be present outside the box, its wave function is zero
i e │ψ│2 = 0 for 0 > x > a
│ψ│2 = 0 for x = a & x = 0
The Schrödinger one – dimensional time independent eqn. is
▼2 Ψ + [2 m (E – V) / ђ 2] ψ = 0 ----------(1)
For freely moving particle V = 0
▼2 Ψ + [ 2 m E / ђ 2 ] ψ = 0 ----------(2)
Taking 2 m E / ђ 2 = K2 ------------ (3)
Eqn.(1) becomes ∂2 ψ / ∂ x2 + k2 Ψ = 0 -------------(4)
Eqn. (1) is similar to eq. of harmonic motion and the solution of above eqn. is written as
Ψ = A sin kx + B cos kx -----------(5) where A, B and k are unknown
quantities and to calculate them it is necessary to construct boundary conditions.
Hence boundary conditions are
When x = 0, Ψ = 0 => from (5) 0 = 0 + B => B = 0 -------- (6)
When x = a, Ψ = 0 => from (5) 0 = A sin ka + B cos ka ---------- (7)
But from (6) B = 0 therefore eqn. (7) may turn as
A sin ka = 0
Since the electron is present in the box a ≠ 0
Sin ka = 0
Ka = n Π
k = n Π / a -------------- (8)
Substituting the value of k in eqn. (3)
2 m E / ђ 2 = (n Π / a )2
E = ( n Π / a )2 ( ђ 2 / 2 m ) = ( n Π / a )2 ( h2 / 8 m Π2 )
E = n2 h2 / 8 m a2
In general En = n2 h2 / 8 m a2 ------------(9)
The wave eqn. can be written as
Ψ = A sin (n Π x / a) ---------- (10)
Let us find the value of A, if an electron is definitely present inside the box, then
=>∫∞
-∞ │ψ│2 dx = 1
=>∫a
0 A2 sin2 ( n Π x / a ) dx = 1
=>∫a
0 sin2 (n Π x / a) dx = 1 / A2
=>∫a
0 [1 - cos (2 Π n (x / a)) / 2] dx = 1 / A2
A = √ 2 / a ---------- (11)
From eqn’s. (10) & (11)
Ψn = √ 2 / a sin ( n Π x / a ) ----------(12)
Eqn. (9) represents an energy level for each value of n. the wave function this energy
level is given in eqn. (12). Therefore the particle in the box can have discrete values of
energies. These values are quantized. Not that the particle cannot have zero energy .The
normalized wave functions Ψ1 , Ψ2, Ψ3 given by eqn (12) is plotted. the values
corresponding to each En value is known as Eigen value and the corresponding wave
function is known as Eigen function.
The wave function Ψ1, has two nodes at x = 0 & x = a
The wave function Ψ2, has three nodes at x = 0, x = a / 2 & x = a
The wave function Ψ3, has three nodes at x = 0, x = a / 3, x = 2 a / 3 & at x = a
The wave function Ψn, has (n + 1) nodes
Substituting the value of E in (3), we get
(2 m / ђ 2 ) ( p2 / 2 m) = k2
=> p2 / ђ 2 = k2
k = p / ђ = p / (h / 2 Π) = 2 Π p / h
k = 2 Π / λ where k is known as wave vector.
Questions:
1. What are the matter waves? Explain their properties.
2. Explain de Broglie hypothesis.
3. Explain the duality of matter waves
4. Describe Davisson and Germer’s experiment an explain how it enabled the
verification of the de Broglie equation.
5. Explain G.P. Thompson’s experiment in support of de Broglie hypothesis
6. Explain Heisenberg’s uncertainty principle. Give its physical significance.
7. Derive time independent one dimensional Schrödinger’s equation.
8. Explain the physical significance of wave function.
9Write down Schrödinger’s wave equation for a particle in one dimensional
potential box.
UNIT- III
ELECTRON THEORY OF METALS
INTRODUCTION:
The electrons in the outer most orbital of the atoms which consists the
solids determine its electrons properties .The electron theory of metals solids aims to
explain the structure and properties of solids through their electronic structure the
electron theory is applicable to the all solids both metals and non metals .
It explains the electrical thermal and magnetic properties of solids etc .The theory has
been developed in three main steps.
1) THE CLASSICAL FREE ELECTRON THEORY: Drude and Lorentz developed this
theory in 1900. According to this theory the metals containing free electrons obey the law
of classical mechanics.
2) THE QUANTUM FREE ELECTRON THEORY: Sommerfeld developed this
theory during 1928. According to this theory free electrons obey quantum laws.
3) THE ZONE THEORY (OR) BAND THEORY OF SOLIDS: Bloch developed 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”.
PHYSICAL PROPERTIES OF METALS
Metallic conductors obey ohm’s law which states that the current in the steady state is
proportional to the electric field strength.
Metals have high electrical and thermal conductivities.
At low temperatures, the resistivity is proportional to the fifth power of absolute
temperature.
i.e. ρ α T5
4. The resistivity of metals at room temperatures is of the order of 10-7 ohm- meter
and above Debye’s temperatures varies linearly with temperature.
i.e . ρ α T
5 For most metals, resistivity is inversely proportional to the pressure.
i.e. . ρ α 1 / P
The resistivity of an impure specimen is given by Mathiessen rule.
ρ = ρ0 + ρ(T)
6)Where ρ0 is a constant for impure specimen and ρ(T) is the temperature dependent
resistivity of pure specimen
7)Near absolute zero, the resistivity of certain metals tends towards zero. i.e it
exhibits the phenomena of super conductivity.
8)The conductivity varies in the presence of magnetic field this effect is known as
magneto resistance
9)The ratio of thermal to electrical conductivity is directly proportional to absolute
temperature. This is known as Wiedemann-Franz law.
Classical free electron theory of metals:
The classical free electron theory is based on following postulates:
1) In an atom, electrons revolve around the nucleus. A metal is composed of such
atoms.
2) The valence electrons of atoms are free to move about the whole volume of
metals like the molecules of perfect gas in a container. The collection of valence
electrons from all the atoms in a given piece of metal forms electron gas. It is free
to move through out the volume of metals.
3) These free electrons move in random direction and collide with either positive
ions fixed to the lattice or other free electrons. All the collision are elastic.
i.e. there is no loss of energy
4) The moments of free electrons obey the classical kinetic theory of gases.
5) The electron velocities in metals obey the classical Maxwell Boltzmann
distributions of velocities.
6) The free electrons move in a completely uniform potential field due to ions fixed
in the lattice.
7) When an electric field is applied to the metal, the free electrons are accelerated in
the direction opposite to the direction of the applied electric field.
Success of classical free electron theory:
1) It verifies ohm’s law
2) It explains the electrical and the thermal conductivities of metals.
3) It derives Wiedemann Franz law.
4) It explains the optical properties of metals.
Draw backs of classical free electron theory:
In spite of the success seen above, classical theory has the following draw backs
1)The phenomena such as photo electric effect, Compton effect and the black body
radiation could not be explained by classical free electron theory.
2) According to the classical free electron theory the values of specific heat of metals
is given by 4.5Ru where Ru is the universal gas constant where as the experimental
value is nearly equal to 3Ru .
1) Also according to this theory, the value of electronic specific heat is
equal to (3/2)Ru while the actual value is about 0.01Ru only.
5) Though K / σT is a constant (Wiedemann Franz law). According to the classical
free electron theory it is not a constant at low temperature.
6) Ferromagnetism could not be explained by this theory. The theoretical value of
para magnetic susceptibility is greater than experimental value.
Fermi – Dirac Distribution:
Permitted energy values for electrons in a material are represented by the energy
levels in the energy bands for that material. These energy levels are occupied by the
electrons in the material in a particular order. This is called distribution of electrons.
Fermi – Dirac distribution deals with the distribution of electrons among the
permitted energy levels.
According to Fermi – Dirac distribution, the probability of electron occupying energy
level E is given by
F(E) = 1/ (1 + exp [( E – EF) / kT] ----------------(1)
Where EF is called the Fermi energy and is constant for a given system. F(E) is called Fermi
function.
At T = 0K, for E < EF F(E) = 1
E < EF F(E) = 0 ------------- (2)
This means that at 0K, all quantum states with energy below EF are completely occupied
and those above EF are unoccupied. With increase of temperature, the Fermi function
plot shows deviation.
At any temperature other than 0K, If E = EF
F(E) = 1 / 2 ----------------(3)
Hence, the Fermi level is that state at which the probability of electron occupation
is ½ at any temperature above 0K and also it is the level of the max. energy of the
filled states at 0K. Fermi energy is the energy of the state at which the probability
of electron occupation is ½ at any temperature above 0K. it also the max.energy of
filled states at 0K.
BAND THEORY OF SOLIDS
Bloch Theorem: A crystalline solid consists of a lattice which is composed of a
large number of ionic cores at regular intervals, and, the conduction electrons can move
freely throughout the lattice.
Let the lattice is in only one-dimension ie only an array of ionic cores along x-axis is
considered. If we plot the potential energy V of a conduction electron as a function of
its position in the lattice, the variation of potential energy.
Since the potential energy of any body bound in a field of attraction is negative, and
since the conduction electron is bound to the solid, its potential energy V is negative.
Further, as it approaches the site of an ionic core V → - ∞. Since this occurs
symmetrically on either side of the core, it is referred to as potential well. The width of
the potential well b is not uniform, but has a tapering shape.
If V0 is the potential at a given depth of the well, then the variation is such that
b → 0 , as V0 → ∞ ., and ,
The product b V0 is a constant.
Now, since the lattice is a repetitive structure of the ion arrangement in a crystal, the
type of variation of V also repeats itself. If a is the interionic distance, then , as we
move in x-direction , the value of V will be same at all points which are separated by a
distance equal to a.
ie V(x) = V( x + a ) where, x is distance of the electron from the core.
Such a potential is said to be a periodic potential.
The Bloch’s theorem states that, for a particle moving in a periodic
potential, the Eigen functions for a conduction electron are of the form,
χ ( x ) = U (x) cos kx
Where U ( x ) = U ( x + a )
The Eigen functions are the plane waves modulated by the function U (x). The function
U (x) has the same periodicity as the potential energy of the electron, and is called the
modulating function.
In order to understand the physical properties of the system, it is required to
solve the Schrödinger’s equation. However, it is extremely difficult to solve the
Schrödinger’s equation with periodic potential described above. Hence the Kronig –
Penney Model is adopted for simplification.
THE KRONIG -PENNEY MODEL:
It is assumed in quantum free electron theory of metals that the free electrons in
a metal express a constant potential and is free to move in the metal. This theory
explains successfully most of the phenomena of solids. But it could not explain why
some solids are good conductors and some other are insulators and semi conductors. It
can be understood successfully using the band theory of solids.
According to this theory, the electrons move in a periodic potential
produced by the positive ion cores. The potential of electron varies periodically with
periodicity of ion core and potential energy of the electrons is zero near nucleus of the
positive ion core. It is maximum when it is lying between the adjacent nuclei which
are separated by interatomic spacing. The variation of potential of electrons while it is
moving through ion core is shown fig.
Fig. One dimensional periodic potential
V ( x ) = { 0 , for the region 0 < x < a
{ V0 for the region -b < x < a ---------------------(1)
Applying the time independent Schrödinger’s wave equation for above two regions
d2Ψ / dx2 + 2 m E Ψ / ħ2 = 0 for region 0 < x < a -----------(2)
and d2Ψ / dx2 + 2 m ( E – V ) Ψ / ħ2 = 0 for region -b < x < a -----------(3)
Substituting α2 = 2 m E / ħ2 ---------------(4)
β2 = 2 m ( E – V ) / ħ2 -----------(5)
d2Ψ / dx2 + α2 Ψ = 0 for region 0 < x < a -----------(6)
d2Ψ / dx2 + β2 Ψ = 0 for region -b < x < a -----------(7)
The solution for the eqn.s (6) and (7) can be written as
Ψ ( x ) = Uk ( x ) eikx ------------------(8)
The above solution consists of a plane wave eikx modulated by the periodic function.
Uk(x), where this Uk(x) has the periodicity of the ion such that
Uk(x) = Uk(x+a) ------------------(9)
and where k is propagating vector along x-direction and is given by k = 2 Π / λ . This
k is also known as wave vector.
Differentiating equation (8) twice with respect to x, and substituting in equation (6)
and (7), two independent second order linear differential equations can be obtained
for the regions 0 < x < a and -b < x < 0 .
Applying the boundary conditions to the solution of above equations, for linear
equations in terms of A,B,C and D it can be obtained (where A,B,C,D are constants )
the solution for these equations can be determined only if the determinant of the
coefficients of A , B , C , and D vanishes, on solving the determinant.
(β2 - α2 / 2 α β)sin hβb sin αa + cos hβb cos αa = cos k ( a + b ) ---------- ---- (10)
The above equation is complicated and Kronig and Penney could conclude with the
equation. Hence they tried to modify this equation as follows
Let Vo is tending to infinite and b is approaching to zero. Such that Vob remains
finite. Therefore sin hβb → βb and cos hβb→1
β2 - α2 = ( 2 m / ħ2 ) (Vo – E ) – ( 2 m E / ħ2 )
= ( 2 m / ħ2 ) (Vo – E - E ) = ( 2 m / ħ2 ) (Vo – 2 E )
= 2 m Vo / ħ2 ( since Vo >> E )
Substituting all these values in equation (10) it verities as
( 2 m Vo / 2 ħ2 α β ) β b . sin α a + cos α a == cos k a
( m Vo b a / ħ2 ) ( sin α a / α a ) + cos α a == cos k a
( P / α a ) sin α a + cos α a == cos k a -------------------(11)
Where P = [ m Vo b a / ħ2 ] ------------------(12)
and is a measure of potential barrier strength.
The left hand side of the equation (11) is plotted as a function of α for the value of P
= 3 Π / 2 which is shown in fig, the right hand side one takes values between -1 to +1
as indicated by the horizontal lines in fig. Therefore the equation (11) is satisfied
only for those values of ka for which left hand side between ± 1.
From fig , the following conclusions are drawn.
The energy spectrum of the electron consists of a number of allowed and forbidden
energy bands.
The width of the allowed energy band increases with increase of energy values ie
increasing the values of αa. This is because the first term of equation(11)
decreases with increase of αa.
( P / α a ) sin α a + cos α a == 3 Π / 2
Fig. a) P=6pi b) p--> infinity c) p--> 0
With increasing P, ie with increasing potential barrier, the width of an allowed band
decreases. As P→∞, the allowed energy becomes infinitely narrow and the
energy spectrum is a line spectrum as shown in fig.
If P→∞, then the equation (11) has solution ie
Sin αa = 0
αa = ± n Π
α = ± n Π / a
α2 = n2 Π2 / a2
But α2 = 2 m E / ħ2
Therefore 2 m E / ħ2 = n2 Π2 / a2
E = [ħ2 Π2 / 2 m a2] n2
E = n h2 / 8 m a2 ( since ħ = h / 2 Π )
This expression shows that the energy spectrum of the electron contains discrete energy
levels separated by forbidden regions.
4) When P→0 then
Cos αa = cos ka
α = k , α2 = k2
but α2 = 2 m E / ħ2
therefore k2 = (h2 / 2 m ) ( 1 / λ2 ) = (h2 / 2 m ) (P2 / h2 )
E = P2 / 2 m
E =1/2mv2 ---------------- (14)
The equation (11) shows all the electrons are completely free to move in the crystal
without any constraints. Hence, no energy level exists ie all the energies are allowed to
the electrons and shown in fig (5). This case supports the classical free electrons theory.
[ ( P / α a ) sin α a + cos α a ] , P → 0
Velocity of the electron in periodic potential.:
According to quantum theory, an electron moving with a velocity can be treated as a
wave packet moving with the group velocity vg
v = vg = dω / dk -----------------(1)
where ω is the angular frequency of deBroglie wave and k = 2 Π / λ is the wave vector.
The energy of an electron can be expressed as
E = ħ ω --------------------- (2)
Differentiating the equation (2) with respect to k
dE / dk = ħ dω / dk -------------(3)
from (1) & (3)
vg = 1 / ħ (dE / dk ) -----------------(4)
According to band theory of solids, the variations of E with k as shown in fig(1). Using
this graph and equation (4), the velocity of electron can be calculated. The variation of
velocity with k is shown in fig(2). From this fig, it is clear that the velocity of electron is
zero at the bottom of the energy band. As the value of k increases, the velocity of
electron increases and reaches to maximum at K=k.Further ,the increases of k, the
velocity of electron decreases and reaches to zero at K= Π / a at the top of energy band.
Origin Of Energy Bands In Solids:
Solids are usually moderately strong, slightly elastic structures. The
individual atoms are held together in solids by interatomic forces or bonds. In addition to
these attractive forces, repulsive forces also act and hence solids are not easily
compressed.
The attractive forces between the atoms are basically electrostatic
in origin. The bonding is strongly dependent on the electronic structure of the atoms. The
attraction between the atoms brings them closer until the individual electron clouds begin
to overlap. A strong repulsive force arises to comply with Pauli’s exclusion principle.
When the attractive force and the repulsive force between any two atoms occupy a stable
position with a minimum potential energy. The spacing between the atoms under this
condition is called equilibrium spacing.
In an isolated atom, the electrons are tightly bound and have
discrete, sharp energy levels. When two identical atoms are brought closer , the outermost
orbits of these atoms overlap and interact. When the wave functions of the electrons on
different atoms begin to overlap considerably, the energy levels corresponding to those
wave functions split. if more atoms are brought together more levels are formed and for a
solid of N atoms, each of the energy levels of an atom splits into N levels of energy. The
levels are so close together that they form an almost continuous band. The width of this
band depends on the degree of overlap of electrons of adjacent atoms and is largest for
the outermost atomic electrons. In a solid many atoms are brought together so that the
split energy levels form a set of bands of very closely spaced levels with forbidden
energy gaps between them.
Classification Of Materials:
The electrons first occupy the lower energy bands and are of no importance in
determining many of the electrical properties of solids. Instead, the electrons in the higher
energy bands of solids are important in determining many of the physical properties of
solids. Hence the two allowed energy bands called valence and conduction bands are
required. The gap between these two allowed bands is called forbidden energy gap or
band gap since electrons can’t have any energy values in the forbidden energy gap. The
valence band is occupied by valence electrons since they are responsible for electrical,
thermal and optical properties of solids. above the valence band we have the conduction
band which is vacant at 0K. According to the gap between the bands and band occupation
by electrons, all solids can be classified broadly into two groups.
In the first group of solids called metals there is a partially filled
band immediately above the uppermost filled band .this is possible when the valence
band is partially filled or a completely filled valence band overlaps with the partially
filled conduction band.
In the second group of solids , there is a gap called band gap between the
completely filled valence band and completely empty conduction band. Depending on the
magnitude of the gap we can classify insulators and semiconductors.
Insulators have relatively wide forbidden band gaps. For typical
insulators the band gap Eg > 3 eV. On the other hand , semiconductors have relatively
narrow forbidden bands. For typical semiconductors Eg ≤ 1 eV.
Effective mass of the electron: When an electron in a period potential is
accelerated by an electric field (or) magnetic field, then the mass of the electron is called
effective mass ( m*).
Let an electron of charge ‘e’ and mass ‘m’ moving inside a crystal lattice of electric field
E.
Acceleration a = eE / m is not a constant in the periodic lattice of the crystal. It can be
considered that its variation is caused by the variation of electron’s mass when it moves
in the crystal lattice.
Therefore Acceleration a = eE / m*
Electrical force on the electron F = m* a --------------(1)
Considering the free electron as a wave packet , the group velocity vg corresponding to
the particle’s velocity can be written as
vg = dw / dk = 2 Π dv/ dk = (2 Π / h ) dE / dk ------------------(2)
where the energy E = h υ and ħ = h / 2 Π.
Acceleration a = d vg / dt = ( 1 / ħ ) d2E / dk dt = ( 1 / ħ ) ( d2E / dk2 ) dk / dt
Since ħ k = p and dp / dt = F,
dk / dt = F / ħ
Therefore a = ( 1 / ħ2 ) ( d2E / dk2 ) F
Or F = ( ħ2 / ( d2E / dk2 ) ) a -----------------------(3)
Comparing eqns . (1) and (3) we get
m*= ħ2 / ( d2E / dk2 ) ---------------(4)
This eqn indicates that the effective mass is determined by d2E / dk2 .
Questions:
1. Explain classical free- electron theory of metals.
2. Define electrical resistance
3. Give the basic assumptions of the classical free electron theory.
4. Explain the following: Drift velocity, mobility, relaxation time
and mean free path.
5. Based on free electron theory derive an expression for electrical
conductivity of metals.
6. Explain the failures of classical free theory.
7. Explain the salient features of quantum free electron theory.
8. Explain the Fermi- Dirac distribution for free electrons in a
metal. Discuss its variation with temperature.
9. Explain the following i) Effective mass, ii) Bloch theorem.
10. Discuss the band theory of solids based on Kronig –Penney
model. Explain the important features of this model.
11. Explain the origin of energy bands in solids.
12. Distinguish between metals, semiconductors and insulators.
UNIT – IV
DIELECTRIC PROPERTIES
Introduction: Dielectrics are insulating materials. In dielectrics, all the electrons are bound
to their parent molecules and there are no free charges. Even with normal voltage or thermal
energy, electrons are not released.
Electric Dipole: A system consisting of two equal and opposite charges separated by a
distance is called electric dipole.
Dipole moment: The product of charge and distance between two charges is called dipole
moment.
i e, μ = q x dl
Permittivity: It is a quantity, which represents the dielectric property of a medium.
Permittivity of a medium indicates the easily polarisable nature of the material.
Units: Faraday / Meter or Coulomb / Newton-meter.
Dielectric constant: The dielectric characteristics are determined by the dielectric constant.
The dielectric constant or relative permittivity of a medium is defined as the ratio between
the permittivity of the medium to the permittivity of the free space.
ε r = ε / ε 0 = C / C 0 where
ε is permittivity of the medium
ε 0 is permittivity of the free space
C is the capacitance of the capacitor with dielectric
C 0 is the capacitance of the capacitor without dielectric
Units: No Units.
Capacitance: The property of a conductor or system of conductor that describes its ability to
store electric charge.
C = q / V = A ε / d where
C is capacitance of capacitor
q is charge on the capacitor plate
V is potential difference between plates
A is area of capacitor plate
ε is permittivity of medium
d is distance between capacitor plates
Units: Farad .
Polarizability (α ) : When the strength of the electric field E is increased the strength of
the induced dipole μ also increases . Thus the induced dipole moment is proportional to
the intensity of the electric field.
μ = α E where α the constant of proportionality is called
polarizability .It can be defined as induced dipole moment per unit electric field.
α = μ / E
Units: Farad – meter2
Polarization Vector ( P ) : The dipole moment per unit volume of the dielectric material
is called polarization vector P .if μ is the average dipole moment per molecule and N is
the number of molecules per unit volume then polarization vector
P = N μ
The dipole moment per unit volume of the solid is the sum of all the individual dipole
moments within that volume and is called the polarization of the solid.
Electric Flux Density or Electric Displacement (D): The Electric Flux Density or
Electric Displacement at a point in the material is given by
D = ε r ε 0 E -------------(1) where
E is electric field strength
ε r is relative permittivity of material
ε 0 is permittivity of free space
As polarization measures additional flux density arising from the presence of the material
as compared to free space, it has same units as D.
Hence D = ε 0 E + P -----------(2)
Since D = ε 0 ε r E
ε 0 ε r E = ε 0 E + P
P = ε 0 ε r E - ε 0 E
P = ε 0(ε r - 1 ) E.
Electric Susceptibility ( χe ) : The polarization P is proportional to the total electronic
flux density E and is in the same direction of E . Therefore, the polarization vector can be
written as
P = ε 0 χe E
Therefore χe = P / ε 0 E = ε 0(ε r - 1 ) E / ε 0 E
χe = (ε r - 1 )
Dielectric Strength: It can be defined as the minimum voltage required for producing
dielectric breakdown. Dielectric strength decreases with raising the temperature,
humidity and age of the material.
Various polarization Process: polarization occurs due to several atomic
mechanisms. When a specimen is placed in a d.c. electric field, polarization is due to four
types of processes. They are
(1) electronic polarization
(2) ionic polarization
(3) orientation polarization and
(4) space charge polarization
Electronic Polarization: the process of producing electric dipoles which are oriented
along the field direction is called polarization in dielectrics
Consider an atom placed inside an electric field. The centre of
positive charge is displaced along the applied field direction while the centre of negative
charge is displaced in the opposite direction .thus a dipole is produced.
The displacement of the positively charged nucleus and the negative electrons of an atom
in opposite directions, on application of an electric field, result in electronic polarization.
Induced dipole moment
μ α E or μ = αe E where αe is electronic polarizability
Electronic polarizability is independent of temperature.
Derivation: Consider the nucleus of charge Ze is surrounded by an electron cloud of charge
-Ze distributed in a sphere of radius R.
Charge density ρ is given by
ρ = -Ze / ( 4/3ΠR3 ) = - (3/4) (Ze / ΠR3 ) ----------(1)
When an external field of intensity E is applied, the nucleus and electrons experiences
Lorentz forces in opposite direction. Hence the nucleus and electron cloud are pulled apart.
Then Coulomb force develops between them, which tends to oppose the
displacement. When Lorentz and coulomb forces are equal and opposite, equilibrium is
reached.
Let x be the displacement
Lorentz force = -Ze E (since = charge x applied field )
Coulomb force = Ze x [ charge enclosed in sphere of radius ‘x’ / 4 Π ε 0 x2 ]
Charge enclosed = ( 4 / 3 ) Π x3 ρ
= ( 4 / 3 ) Π x3 [( - 3 / 4 ) ( Z e / Π R3 ) ]
= - Z e x3 / R3
Therefore Coulomb force = ( Ze )( - Z e x3 / R3 ) / 4 Π ε 0 x2 = - Z2 e2 x / 4 Π ε 0 R3
At equilibrium, Lorentz force = Coulomb force
􀃖 -Ze E = - Z2 e2 x / 4 Π ε 0 R3
􀃖 E = -Ze x / 4 Π ε 0 R3
􀃖 or x = 4 Π ε 0 R3 E / Ze
Thus displacement of electron cloud is proportional to applied field.
The two charges +Ze and -Ze are separated by a distance ‘x‘ under applied field constituting
induced electric dipoles .
Induced dipole moment μe = Ze x
Therefore μe = Ze (4 Π ε 0 R3 E / Ze ) = 4 Π ε 0 R3 E
Therefore μe α E , μe = αe E where αe = 4 Π ε 0 R3 is electronic polarizability
The dipole moment per unit volume is called electronic polarization. It is independent of
temperature.
P = N μe = N αe E where
N is Number of atoms / m3
Pe = N (4 Π ε 0 R3 E ) = 4 Π ε 0 R3 N E where
R is radius of atom
Electric Susceptibility χ = P / ε 0 E
Therefore P = ε 0 E χ
P = (4 Π R3 N) ε 0 E where χ = 4 Π R3 N
Also Pe = ε 0 E (ε r - 1) = N αe E
Or ε r - 1 = N αe / ε 0
Hence αe = ε 0 (ε r - 1 ) / N .
Ionic Polarization: It is due to the displacement of cat ions and anions in opposite
directions and occurs in an ionic solid .
Consider a NaCl molecule. Suppose an electric field is applied in the positive direction . The
positive ion moves by x1 and the negative ion moves by x2
Let M is mass of positive ion
M is mass of negative ion
x1 is displacement of positive ion
x2 is displacement of negative ion
Total displacement x = x1 + x2 --------------(1)
Lorentz force on positive ion = + e E ----------(2)
Lorentz force on negative ion = - e E ---------- (3)
Restoring force on positive ion = -k1 x1---------- (2 a)
Restoring force on negative ion = +k2 x2---------- (3 a) where k1, k2 Restoring force constants
At equilibrium, Lorentz force and restoring force are equal and opposite
For positive ion, e E = k1 x1
For negative ion, e E = k2 x2 ] ---------- (4)
Where k1 = M ω0
2 & K2 = m ω0
2 where ω0 is angular velocity of ions
Therefore x = x1 + x2 = ( e E / ω0
2 ) [ 1/M + 1/m ] ------------(5)
From definition of dipole moment
μ = charge x distance of separation
μ = e x = ( e2 E / ω0
2 ) [ 1/M + 1/m ] ------------(6)
But μ α E or μ = αi E
Therefore αi = (e2 / ω0
2) [1/M + 1/m]
This is ionic polarizability.
Orientational Polarization:
In methane molecule, the centre of negative and positive charges coincides, so that there
is no permanent dipole moment. On the other hand, in certain molecules such as Ch3Cl,
the positive and negative charges do not coincide .Even in the absence of an electric field,
this molecule carries a dipole moment, and they tend to align themselves in the direction
of applied field. The polarization due to such alignment is called orientation polarization.
It is dependent on temperature. With increase of temperature the thermal energy tends to
randomize the alignment.
Orientation polarization Po = Nμ = Nμ2E / 3kT
= N α0 E
Therefore Orientation polarizability α0 = Po / NE = μ2 / 3kT
Thus orintational polarizability α0 is inversely proportional t absolute temperature of
material.
Internal field or Local field or Lorentz field: Internal field is the total electric field at
atomic site.
Internal field A = E1 + E2 + E3 + E4 ------- (I) where
E1 is field intensity due to charge density on plates
E2 is charge density induced on two sides of dielectric
E3 is field intensity due to other atoms in cavity and
E4 is field intensity due to polarization charges on surface of cavity
Field E1 : E1 is field intensity due to charge density on plates
From the field theory
E1 = D / ε 0
D = P + ε 0 E
Therefore E1 = P + ε 0 E / ε 0 = E + P / ε 0 ---------- (1)
Field E2: E2 is the field intensity at A due to charge density induced on two sides of dielectric
Therefore E2 = - P / ε 0 -----------(2)
Field E3: E3 is field intensity at A due to other atoms contained in the cavity and for a cubic
structure,
E3 = 0 because of symmetry. ----------- (3)
Field E4: E4 is field intensity due to polarization charges on surface of cavity and was
calculated by Lorentz in the following way:
If dA is the surface area of the sphere of radius r lying between θ and θ + dθ, where θ is the
direction with reference to the direction of applied force.
Then dA = 2 Π (PQ) (QR)
But sin θ = PQ / r => PQ = r sin θ
And dθ = QR / r => QR = r dθ
Hence dA = 2 Π (r sin θ) (r dθ) = 2 Π r2 sin θ dθ
Charge on surface dA is dq = P cos θ dA (cos θ is normal component)
dq = P cos θ (2 Π r2 sin θ dθ) = P (2 Π r2 sin θ cos θ dθ )
The field due to the charge dq at A, is denoted by dE4 in direction θ = 0
dE4 = dq cos θ / 4 Π ε 0 r2 = P (2 Π r2 sin θ cos θ dθ) cos θ
dE4 = P sin θ cos2 θ dθ / 2 ε 0
∫ dE4 = P / 2 ε 0 ∫0
Π sin θ cos2 θ dθ = P / 2 ε 0 ∫0
Π cos2 θ d (- cos θ)
Let cos θ = x
∫ dE4 = - P / 2 ε 0 ∫0
Π x2 dx
Therefore E4= - P / 2 ε 0 [x3 / 3]0
Π
= - P / 2 ε 0 [cos3 θ/ 3] 0
Π = - P / 6 ε 0 [-1 – 1] = P / 3 ε 0 ------- (4)
Local field Ei = E1 + E2 + E3 + E4
= E + P / ε 0 - P / ε 0 + 0 + P / 3 ε 0
= E + P / 3 ε 0
Clausius – Mosotti Relation:
Let us consider the elemental dielectric having cubic structure. Since there are no ions
and perment dipoles in these materials, them ionic polarizability αi and orientational
polarizability α0 are zero.
i.e. αi = α0 = 0
Hence polarization P = N αe Ei
= N αe ( E + P / 3ε0)
i.e. P [ 1 - N αe / 3 ε0 ] = N αe E
P = N αe E / P [ 1 - N αe / 3 ε0 ] -------------􀃆 1
D = P + ε0E
P = D - ε0E
Dividing on both sides by E
P / E = D / E - ε0 = ε - ε0 = ε0 εr - ε0
P = E ε0 (εr - 1) ----------------------------􀃆 2
From eqn 1 and 2 , we get
P = E ε0 (εr - 1) = N αe E / [ 1 - N αe / 3 ε0 ]
[ 1 - N αe / 3 ε0 ] = N αe / ε0 (εr - 1)
1 = N αe / 3 ε0 + N αe / ε0 (εr - 1)
1 = (N αe / 3 ε0 ) ( 1 + 3 / (εr - 1))
1 = (N αe / 3 ε0 )[ (εr - 1 + 3) / (εr - 1) ]
1 = (N αe / 3 ε0 )[ (εr + 2) / (εr - 1) ]
(εr + 2) / (εr - 1) = N αe / 3 ε0 Where N – no of molecules per unit volume
This is Clausius – Mosotti Relation.
Dielectric Breakdown : The dielectric breakdown is the sudden change in state of a
dielectric material subjected to a very high electric field , under the influence of which , the
electrons are lifted into the conduction band causing a surge of current , and the ability of the
material to resist the current flow suffers a breakdown .
Or
When a dielectric material loses its resistivity and permits very large current to flow through it,
then the phenomenon is called dielectric breakdown.
There are many factors for dielectric breakdown which are (1) Intrinsic breakdown (2) Thermal
breakdown (3) Discharge breakdown (4) Electro Chemical breakdown (5) Defect breakdown.
(1) Intrinsic breakdown: The dielectric strength is defined as the breakdown voltage per unit
thickness of the material. When the applied electric field is large, some of the electrons in
the valence band cross over to the conduction band across the large forbidden energy gap
giving rise to large conduction currents. The liberation or movement of electrons from
valence band is called field emission of electrons and the breakdown is called the intrinsic
breakdown or zener breakdown.
The number of covalent bonds broken and the number of charge
carriers released increases enormously with time and finally dielectric breakdown occurs. This
type of breakdown is called Avalanche breakdown.
(2) Thermal breakdown: It occurs in a dielectric when the rate of heat generation is greater
than the rate of dissipation. Energy due to the dielectric loss appears as heat. If the rate of
generation of heat is larger than the heat dissipated to the surrounding, the temperature of
the dielectric increases which eventually results in local melting .once melting starts, that
particular region becomes highly conductive, enormous current flows through the material
and dielectric breakdown occurs.
Thus thermal breakdown occurs at very high temperatures. Since the
dielectric loss is directly proportional to the frequency, for a.c fields, breakdown occurs at
relatively lower field strengths.
(3) Discharge breakdown: Discharge breakdown is classified as external or internal. External
breakdown is generally caused by a glow or corona discharge .Such discharges are
normally observed at sharp edges of electrodes. It causes deterioration of the adjacent
dielectric medium. It is accompanied by the formation of carbon so that the damaged areas
become conducting leading to power arc and complete failure of the dielectric. Dust or
moisture on the surface of the dielectric may also cause external discharge breakdown.
Internal breakdown occurs when the insulator contains blocked gas bubbles .If large
number of gas bubbles is present, this can occur even at low voltages.
(4) Electro Chemical breakdown: Chemical and electro chemical breakdown are related to
thermal breakdown. When temperature rises, mobility of ions increases and hence
electrochemical reaction takes place. When ionic mobility increases leakage current also
increases and this may lead to dielectric breakdown. Field induced chemical reaction
gradually decreases the insulation resistance and finally results in breakdown.
(5) Defect breakdown: if the surface of the dielectric material has defects such as cracks and
porosity, then impurities such as dust or moisture collect at these discontinuities leading to
breakdown. Also if it has defect in the form of strain in the material, that region will also
break on application of electric field.
Frequency dependence of polarizability:
On application of an electric field, polarization process occurs as a
function of time. The polarization P(t) as a function of time. The polarization P(t) as a
function of time t is given by
P(t) = P[ 1- exp ( - t / tr )]
Where P – max. Polarization attained on prolonged application of static field.
tr - relaxation time for particular polarization process
The relaxation time tr is a measure of the time scale of polarization process. It is the time
taken for a polarization process to reach 0.63 of the max. value.
Electronic polarization is extremely rapid. Even when the frequency of the applied
voltage is very high in the optical range (≈1015 Hz), electronic polarization occurs during
every cycle of the applied voltage.
Ionic polarization is due to displacement of ions over a small distance due to the applied
field. Since ions are heavier than electron cloud, the time taken for displacement is larger.
The frequency with which ions are displaced is of the same order as the lattice vibration
frequency (≈1013Hz). Hence, at optical frequencies, there is no ionic polarization. If the
frequency of the applied voltage is less than 1013 Hz, the ions respond.
Orientation polarization is even slower than ionic polarization. The relaxation time for
orientation polarization in a liquid is less than that in a solid. Orientation polarization
occurs, when the frequency of applied voltage is in audio range (1010 Hz).
Space charge polarization is the slowest process, as it involves the diffusion of ions over
several interatomic distances. The relaxation time for this process is related to frequency
of ions under the influence of applied field. Space charge polarization occurs at power
frequencies (50-60 Hz).
Piezo – Electricity: These materials have the property of becoming electrically polarized
when mechanical stress is applied. This property is known as Piezo – electric effect has an
inverse .According to inverse piezo electric effect, when an electric stress or voltage is applied,
the material becomes strained. The strain is directly proportional to the applied field E.
When piezo electric crystals are subjected to compression or tension,
opposite kinds of charges are developed at the opposite faces perpendicular to the direction of
applied force. The charges produced are proportional to the applied force.
Piezo – Electric Materials and Their Applications: Single crystal of quartz is used for
filter, resonator and delay line applications. Natural quartz is now being replaced by synthetic
material.
Rochelle salt is used as transducer in gramophone pickups, ear phones,
hearing aids, microphones etc. the commercial ceramic materials are based on barium titanate,
lead zirconate and lead titanate. They are used for high voltage generation (gas lighters),
accelerometers, transducers etc.
Piezo electric semiconductors such as GaS, ZnO & CdS are used as
amplifiers of ultrasonic waves.
Ferro electricity: Ferro electric materials are an important group not only because of intrinsic
Ferro electric property, but because many possess useful piezo electric, birefringent and electro
optical properties.
The intrinsic Ferro electric property is the possibility of reversal or
change of orientation of the polarization direction by an electric field. This leads to hysteresis
in the polarization P, electric field E relation , similar to magnetic hysteresis. Above a critical
temperature, the Curie point Tc, the spontaneous polarization is destroyed by thermal disorder.
The permittivity shows a characteristic peak at Tc.
Pyroelectricity: It is the change in spontaneous polarization when the temperature of
specimen is changed.
Pyroelectric coefficient ‘λ’ is defined as the change in polarization per unit temperature
change of specimen.
λ= dP / dT
change in polarization results in change in external field and also changes the surface.
Required Qualities of Good Insulating Materials: The required qualities can be classified as
under electrical, mechanical, thermal and chemical applications.
i) Electrical: 1. electrically the insulating material should have high electrical resistivity and
high dielectric strength to withstand high voltage.
2 .The dielectric losses must be minimum.
3. Liquid and gaseous insulators are used as coolants. For example transformer oil, hydrogen
and helium are used both as insulators and coolant.
ii) Mechanical: 1. insulating materials should have certain mechanical properties depending on
the use to which they are put.
2. When used for electric machine insulation, the insulator should have sufficient mechanical
strength to withstand vibration.
iii) Thermal: Good heat conducting property is also desirable in such cases. The insulators
should have small thermal expansion and it should be non-ignitable.
iv) Chemical: 1. chemically, the insulators should be resistant to oils, liquids, gas fumes, acids
and alkali’s.
2. The insulators should be water proof since water lowers the insulation resistance and the
dielectric strength.
MAGNETIC PROPERTIES
Introduction : The basic aim in the study of the subject of magnetic materials is to
understand the effect of an external magnetic field on a bulk material ,and also to account for
its specific behavior. A dipole is an object that a magnetic pole is on one end and a equal and
opposite second magnetic dipole is on the other end.
A bar magnet can be considered as a dipole with a north pole at one end
and South Pole at the other. If a magnet is cut into two, two magnets or dipoles are created out
of one. This sectioning and creation of dipoles can continue to the atomic level. Therefore, the
source of magnetism lies in the basic building block of all the matter i.e. the atom.
Consider electric current flowing through a conductor. When the electrons
are flowing through the conductor, a magnetic field is forms around the conductor. A magnetic
field is produced whenever an electric charge is in motion. The strength of the field is called
the magnetic moment.
Magnetic materials are those which can be easily magnetized as they have
permanent magnetic moment in the presence of applied magnetic field. Magnetism arises from
the magnetic dipole moments. It is responsible for producing magnetic influence of attraction
or repulsion.
Magnetic dipole : it is a system consisting of two equal and opposite magnetic poles separated
by a small distance of ‘2l’metre.
Magnetic Moment ( μm ) :It is defined as the product of the pole strength (m) and the
distance between the two poles (2l) of the magnet.
i . e . . μm = (2l ) m
Units: Ampere – metre2
Magnetic Flux Density or Magnetic Induction (B): It is defined as the number of magnetic
lines of force passing perpendicularly through unit area.
i . e . . B = magnetic flux / area = Φ / A
Units: Weber / metre2 or Tesla.
Permeability:
Magnetic Field Intensity (H): The magnetic field intensity at any point in the magnetic field is
the force experienced by a unit north pole placed at that point.
Units: Ampere / meter
The magnetic induction B due to magnetic field intensity H applied in vacuum is related by
B = μ0 H where μ0 is permeability of free space = 4 Π x 10-7 H / m
If the field is applied in a medium, the magnetic induction in the solid is given by
B = μ H where μ is permeability of the material in the medium
μ = B / H
Hence magnetic Permeability μ of any material is the ratio of the magnetic induction to the
applied magnetic field intensity. The ratio of μ / μ0 is called the relative permeability (μr ).
μr = μ / μ0
Therefore B = μ0 μr H
Magnetization: It is the process of converting a non – magnetic material into a magnetic
material. The intensity of magnetization (M) of a material is the magnetic moment per unit
volume. The intensity of magnetization is directly related to the applied field H through the
susceptibility of the medium (χ) by
χ = M / H ------------(1)
The magnetic susceptibility of a material is the ratio of the intensity of magnetization produced
to the magnetic field intensity which produces the magnetization. It has no units.
We know
B = μ H
= μ0 μr H
i.e B = μ0 μr H + μ0 H - μ0 H
= μ0 H + μ0 H ( μr – 1 )
= μ0 H + μ0 M where M is magnetization = H ( μr – 1 )
i.e B = μ0 ( H + M ) ----------(2)
The first term on the right side of eqn (2) is due to external field. The second term is due to the
magnetization.
Hence μ0 = B / H + M
Relative Permeability ,
μr = μ / μ0 = ( B / H ) / ( B / H + M ) = H + M / H = 1 + M / H
μr = 1 + χ ---------(3)
The magnetic properties of all substances are associated with the orbital and spin motions of
the electrons in their atoms. Due to this motion, the electrons become elementary magnets of
the substance. In few materials these elementary magnets are able to strengthen the applied
magnetic field , while in few others , they orient themselves such that the applied magnetic
field is weakened.
Origin of Magnetic Moment : In atoms , the permanent magnetic moments can arise due to the
following :
1. the orbital magnetic moment of the electrons
2. the spin magnetic moment of the electrons
3. the spin magnetic moment of the nucleus.
Orbital magnetic moment of the electrons: In an atom, electrons revolve round the nucleus in
different circular orbits.
Let m be the mass of the electron and r be the radius of the orbit in which it moves with
angular velocity ω.
The electric current due to the moving electron I = - ( number of electrons flowing per second
x charge of an electron )
Therefore I = - e ω / 2 Π --------------(1)
The current flowing through a circular coil produces a magnetic field in a direction
perpendicular to the area of coil and it is identical to the magnetic dipole. the magnitude of the
magnetic moment produced by such a dipole is
μm = I .A
= ( - e ω / 2 Π ) ( Π r2 )
= - e ω r2 / 2 = ( - e / 2 m ) ( m ω r2 ) = - ( e / 2 m ) L -----------(2)
where L = m ω r2 is the orbital angular momentum of electron. The minus sign
indicates that the magnetic moment is anti – parallel to the angular momentum L. A substance
therefore possesses permanent magnetic dipoles if the electrons of its constituent atom have a
net non-vanishing angular momentum. The ratio of the magnetic dipole moment of the electron
due to its orbital motion and the angular momentum of the orbital motion is called orbital gyro
magnetic ratio , represented by γ.
Therefore γ = magnetic moment / angular momentum = e / 2m
The angular momentum of an electron is determined by the orbital quantum number ‘l’ given
by l = 0 , 1 , 2 , ……( n – 1 ) where n is principal quantum number n = 1 , 2 , 3 , 4 , ……
……corresponding to K , L , M , N……shells .
The angular momentum of the electrons associated with a particular value of l is given by l( h /
2 Π )
The strength of the permanent magnetic dipole is given by
μ el = - ( e / 2 m ) ( l h / 2 Π )
i.e μ el = - ( e h l / 4 Π m ) = - μB l ---------------(3)
The quantity μB = e h / 4 Π m is an atomic unit called Bohr Magneton and has a value 9.27
x 10 -24 ampere metre2
In an atom having many electrons, the total orbital magnetic moment is determined by taking
the algebraic sum of the magnetic moments of individual electrons. The moment of a
completely filled shell is zero. An atom with partially filled shells will have non zero orbital
magnetic moment.
Magnetic Moment Due to Electron Spin : The magnetic moment associated with spinning of
the electron is called spin magnetic moment μ es .Magnetic moment due to the rotation of the
electronic charge about one of the diameters of the electron is similar to the earth’s spinning
motion around it’s north – south axis.
An electronic charge being spread over a spherical volume ,the electron spin would cause
different charge elements of this sphere to form closed currents, resulting in a net spin
magnetic moment. This net magnetic moment would depend upon the structure of the electron
and its charge distribution.
μ es = γ ( e / 2 m ) S -------------------(1)where S= h / 4 Π is spin angular momentum
therefore μ es ≈ 9.4 x 10 -24 ampere metre2
Thus, the magnetic moments due to the spin and the orbital motions of an electron are of the
same order of magnitude. The spin and electron spin magnetic moment are intrinsic properties
of an electron and exist even for a stationary electron. Since the magnitude of spin magnetic
moment is always same, the external field can only influence its direction. If the electron spin
moments are free to orient themselves in the direction of the applied field B. In a varying field
,it experiences a force in the direction of the increasing magnetic field due to spin magnetic
moments of its various electrons.
Magnetic Moment due to Nuclear Spin : Another contribution may arise from the nuclear
magnetic moment. By analogy with Bohr Magneton, the nuclear magneton arises due to spin of
the nucleus. It is given by
μ ps = e h / 4 Π Mp
μ ps = 5.05 x 10 -27 ampere metre2 where Mp is mass of proton.
The nuclear magnetic moments are smaller than those associated with electrons.
Classification Of Magnetic Materials :All matter respond in one way or the other when
subjected to the influence of a magnetic field. The response could be strong or weak, but there
is none with zero response ie, there is no matter which is non magnetic in the absolute sense.
Depending upon the magnitude and sign of response to the applied field , and also on the basis
of effect of temperature on the magnetic properties, all materials are classified broadly under 3
categories.
1. Diamagnetic materials 2. Paramagnetic materials, 3. Ferromagnetic materials
two more classes of materials have structure very close to ferromagnetic
materials but possess quite different magnetic effects. They are i. Anti ferromagnetic
materials and ii . Ferri magnetic materials
1. Diamagnetic materials: Diamagnetic materials are those which experience a repelling
force when brought near the pole of a strong magnet. In a non uniform magnetic field they
are repelled away from stronger parts of the field.
In the absence of an external magnetic field , the net magnetic dipole
moment over each atom or molecule of a diamagnetic material is zero.
Ex: Cu, Bi , Pb .Zn and rare gases.
Paramagnetic materials: Paramagnetic materials are those which experience a feeble
attractive force when brought near the pole of a magnet. They are attracted towards the
stronger parts of magnetic field. Due to the spin and orbital motion of the electrons, the
atoms of paramagnetic material posses a net intrinsic permanent moment.
Susceptibility χ is positive and small for these materials. The susceptibility is inversely
proportional to the temperature T.
χ α 1/T
χ = C/T where C is Curie’s temperature.
Below superconducting transition temperatures, these materials exhibit the Para
magnetism.
Examples: Al, Mn, Pt, CuCl2.
Ferromagnetic Materials: Ferromagnetic materials are those which experience a very
strong attractive force when brought near the pole of a magnet. These materials, apart
from getting magnetized parallel to the direction of the applied field, will continue to
retain the magnetic property even after the magnetizing field removed. The atoms of
ferromagnetic materials also have a net intrinsic magnetic dipole moment which is due to
the spin of the electrons.
Susceptibility is always positive and large and it depends upon temperature.
χ = C / (T- θ) ( only in paramagnetic region i.e., T > θ)
θ is Curie’s temperature.
When the temperature of the material is greater than its Curie temperature then it converts
into paramagnetic material.
Examples: Fe, Ni, Co, MnO.
Antiferromagnetic matériels : These are the ferromagnetic materials in which equal
no of opposite spins with same magnitude such that the orientation of neighbouring spins
is in antiparallel manner are present.
Susceptibility is small and positive and it is inversely proportional to the temperature.
χ=C /(T+θ)
the temperature at which anti ferromagnetic material converts into paramagnetic material
is known as Neel’s temperature.
Examples: FeO, Cr2O3.
Ferrimagnetic materials: These are the ferromagnetic materials in which equal no of
opposite spins with different magnitudes such that the orientation of neighbouring spins
is in antiparallel manner are present.
Susceptibility positive and large, it is inversely proportional to temperature
χ=C /(T ± θ) T> TN ( Neel’s temperature)
Examples : ZnFe2O4, CuFe2O4
Domain theory of ferromagnetism: According to Weiss, a virgin specimen of
ferromagnetic material consists of a no of regions or domains (≈ 10-6 m or larger) which
are spontaneously magnetized. In each domain spontaneous magnetization is due to
parallel alignment of all magnetic dipoles. The direction of spontaneous magnetization
varies from domain to domain. The resultant magnetization may hence be zero or nearly
zero. When an external field is applied there are two possible ways of alignment fo a
random domain.
i). By motion of domain walls: The volume of the domains that are favourably oriented
with respect to the magnetizing field increases at the cost of those that are unfavourably
oriented
ii) By rotation of domains: When the applied magnetic field is strong, rotation of the
direction of magnetization occurs in the direction of the field.
Hysteresis curve (study of B-H curve): The hysteresis of ferromagnetic materials
refers to the lag of magnetization behind the magnetization field. when the temperature of
the ferromagnetic substance is less than the ferromagnetic Curie temperature ,the
substance exhibits hysteresis. The domain concept is well suited to explain the
phenomenon of hysteresis. The increase in the value of the resultant magnetic moment of
the specimen by the application of the applied field , it attributes to the 1. motion of the
domain walls and 2. rotation of domains.
When a weak magnetic field is applied, the domains that are
aligned parallel to the field and in the easy direction of magnetization , grow in size at the
expense of less favorably oriented ones. This results in Bloch wall movement and when
the weak field is removed, the domains reverse back to their original state. This reverse
wall displacement is indicated by OA of the magnetization curve. When the field
becomes stronger ,the Bloch wall movement continues and it is mostly irreversible
movement. This is indicated by the path AB of the graph. The phenomenon of hysteresis
is due to this irreversibility.
At the point B all domains have got magnetized along their easy directions. Application
of still higher fields rotates the domains into the field direction which may be away from
the easy direction. Once the domain rotation is complete the specimen is saturated
denoted by C. on removal of the field the specimen tends to attain the original
configuration by the movement of Bloch walls. But this movement is hampered by the
impurities, lattice imperfections etc, and so more energy must be supplied to overcome
the opposing forces. This means that a coercive field is required to reduce the
magnetization of the specimen to zero. The amount of energy spent in this regard is a
loss. Hysteresis loss is the loss of energy in taking a ferromagnetic body through a
complete cycle of magnetization and this loss is represented by the area enclosed by the
hysteresis loop.
A hysteresis curve shows the relationship between the magnetic
flux density B and applied magnetic field H. It is also referred to as the B-H curve(loop).
Hard and Soft Magnetic Materials:
Hysteresis loop of the ferromagnetic materials vary in size and shape. This variation in
hysteresis loops leads to a broad classification of all the magnetic materials into hard type
and soft type.
Hard Magnetic Materials:
Hard magnetic materials are those which are characterized by large
hysteresis loop because of which they retain a considerable amount of their magnetic
energy after the external magnetic field is switched off. These materials are subjected to a
magnetic field of increasing intensity, the domain walls movements are impeded due to
certain factors. The cause for such a nature is attributed to the presence of impurities or
non-magnetic materials, or the lattice imperfections. Such defects attract the domain
walls thereby reducing the wall energy. It results in a stable state for the domain walls
and gives mechanical hardness to the material which increases the electrical resistivity.
The increase in electrical resisitivity brings down the eddy current loss if used in a.c
conditions. The hard magnetic materials can neither be easily magnetized nor easily
demagnetized.
Properties:
High remanent magnetization
High coercivity
High saturation flux density
Low initial permeability
High hysteresis energy loss
High permeability
The eddy current loss is low for ceramic type and large for metallic type.
Examples of hard magnetic materials are, i) Iron- nickel- aluminum alloys with certain
amount of cobalt called Alnico alloy. ii) Copper nickel iron alloys. iii) Platinum cobalt
alloy.
Applications of hard magnetic materials: For production of permanent magnets, used in
magnetic detectors, microphones, flux meters, voltage regulators, damping devices and
magnetic separators.
Soft Magnetic Materials:
Soft magnetic materials are those for which the hysteresis loops enclose very small
area. They are the magnetic materials which cannot be permanently magnetized. In these
materials ,the domain walls motion occurs easily. Consequently, the coercive force
assumes a small value and makes the hysteresis loop a narrow one because of which, the
hysteresis loss becomes very small. For the sane reasons, the materials can be easily
magnetized and demagnetized.
Properties:
Low remanent magnetization
Low coercivity
Low hysteresis energy loss
Low eddy current loss
High permeability
High susceptibility
Examples of soft magnetic materials are
i) Permalloys ( alloys of Fe and Ni)
ii) Si – Fe alloy
iii) Amorphous ferrous alloys ( alloys of Fe, Si, and B)
iv) Pure Iron (BCC structure)
Applications of soft magnetic materials: Mainly used in electro- magnetic machinery and
transformer cores. They are also used in switching circuits, microwave isolators and
matrix storage of computers.
Questions:
Describe how polarization occurs in a dielectric material.
Define dielectric constant of a material.
Explain the origin of different kinds of polarization.
Describe in brief various types of polarization.
Obtain an expression for the internal field.
Derive Clausius - Mossotti equation.
Describe the frequency dependence of dielectric constant.
Write note on Dielectric loss.
Explain the properties of ferroelectric materials.
What is piezoelectricity?
Distinguish between dia, para, ferro, antiferro,and ferromagnetic materials.
what is meant by Neel temperature
Define magnetization and show that B = μ0 (H + M).
Explain the origin of magnetic moment.
Decribe the domain theory of ferromagnetism.
What is Bhor Magneton.
Draw and explain the hysteresis curve.
Discuss the characteristic features of soft and hard magnetic materials.
What are the applications of soft and hard magnetic materials?
UNIT - V
SEMICONDUCTORS
INTRODUCTION:
Semiconductors are materials whose electronic properties are intermediate between
those of conductors and insulators. These electrical properties of a solid depend on its
band structure. A semiconductor has two bands of importance (neglecting bound
electrons as they play no part in the conduction process) the valence and the conduction
bands. They are separated by a forbidden energy gap. At OK the valence band is full
and the conduction band is empty, the semiconductor behaves as an insulator.
Semiconductor has both positive (hole) and negative (electron) carriers of electricity
whose densities can be controlled by doping the pure semiconductor with chemical
impurities during crystal growth.
At higher temperatures, electrons are transferred across the gap into the conduction band
leaving vacant levels in the valence band. It is this property that makes the
semiconductor a material with special properties of electrical conduction.
Generally there are two types of semiconductors. Those in which electrons and holes are
produced by thermal activation in pure Ge and Si are called intrinsic semiconductors. In
other type the current carriers, holes or free electrons are produced by the addition of
small quantities of elements of group III or V of the periodic table, and are known as
extrinsic semiconductors. The elements added are called the impurities or dopants.
Intrinsic semiconductors:
A pure semiconductor which is not doped is termed as intrinsic semiconductor. In Si
crystal, the four valence electrons of each Si atom are shared by the four surrounding Si
atoms. An electron which may break away from the bond leaves deficiency of one
electron in the bond. The vacancy created in a bond due to the departure of an electron is
called a hole. The vacancy may get filled by an electron from the neighboring bond, but
the hole then shifts to the neighboring bond which in turn may get filled by electron from
another bond to whose place the hole shifts, and so on thus in effect the hole also
undergoes displacement inside a crystal. Since the hole is associated with deficiency of
one electron, it is equivalent for a positive charge of unit magnitude. Hence in a
semiconductor, both the electron and the hole act as charge carriers.
In an intrinsic semiconductor, for every electron freed from the bond, there will be one
hole created. It means that, the no of conduction electrons is equal to the no of holes at
any given temperature. Therefore there is no predominance of one over the other to be
particularly designated as charge carriers.
Carriers Concentration in intrinsic semiconductors:
A broken covalent bond creates an electron that is raised in energy, so as to
occupy the conduction bond, leaving a hole in the valence bond. Both electrons and holes
contribute to overall conduction process.
In an intrinsic semiconductor, electrons and holes are equal in numbers. Thus
n = p = ni
Where n is the number of electrons in the conduction band in a unit volume of the
material (concentration), p is the number of holes in the valance band in a unit volume of
the material. And ni, the number density of charge carriers in an intrinsic semi conductor.
It is called intrinsic density.
For convenience, the top of the valence bond is taken as a zero energy reference level
arbitrarily.
The number of electrons in the conduction bond is
n = N P(Eg)
Where P(Eg) is the probability of an electron having energy Eg. It is given by Fermi
Dirac function eqn., and N is the total number of electrons in both bands.
Thus,
N
n = -----------------------------------
1 + exp [(Eg – EF)/KT]
Where EF is the Fermi Level
The probability of an electron being in the valence bond is given by putting Eg = 0 in
eqn. Hence, the number of electrons in the valence bond is given by
N
nv = ------------------------------
1 + exp(-EF/KT)
The total number of electrons in the semiconductor. N is the sum of those in the
conduction band n and those in the valence bond nv. Thus,
N N
N = -------------------------------- + --------------------------
1 + exp [ (Eg – EF) / KT ] 1 + exp(-EF/KT)
For semiconductors at ordinary temperature, Eg >> KT as such in equation one may be
neglected when compared with exp Eg – EF Then
RT
1 1
1 = +
exp Eg – EF 1 + exp – EF
RT RT
Rearranging the terms, we get
- Eg + EF exp (-EF/RT)
exp =
RT 1 + exp (-EF/RT)
-EF
≈ exp
RT
2 EF - Eg
or exp = 1
RT
This leads to
EF = Eg/2
Thus in an intrinsic semiconductor, the Fermi level lies mid way between the conduction
and the valence bonds. The number of conduction electrons at any temperature T is
given by
N
n = (∴ EF = Eg/2)
1 + exp(Eg/2KT)
In eqn may be approximated as
n ≈ N exp(-Eg / 2RT)
From the above discussion, the following conclusions may be drawn.
a) The number of conduction electrons and hence the number of holes in an intrinsic
semiconductor, decreases exponentially with increasing gap energy Eg this accounts
for lack of charge carries in insulator of large forbidden energy gap.
b) The number of available charge carries increases exponentially with increasing
temperature.
The above treatment is only approximate as we have assumed that all states in a bond
have the same energy. Really it is not so. A more rigorous analysis must include
additions terms in eqn.
The no of conduction bond, in fact is given by
n = ∫ S(E) P(E) dE
Where S(E) is the density of available states in the energy range between E and E + dE,
and P(E) is the probability, that an electron can occupy a state of energy E.
S(E) 8√2 π m3/2 E1/2
n3
Inclusion of S(E) and integration over the conduction bond leads to
n = Ne exp [(-Eg-EF)/RT]
In a similar way, we arrive at
p = NV exp [ -EF/RT]
If we multiply eq: we get
np = ni
2 = Ne NV exp(-Eg/RT)
For the intrinsic material
Ni = 2 (2πRT)3/2 (me* mn*)3/4 exp(-Eg/2RT)
h2
Notice that this expression agress with the less rigorous one derived earlier since the
temperature dependence is largely controlled by the rapidly varying exponential term.
EXTRINSIC SEMICONDUCTORS:
Intrinsic Semiconductors are rarely used in semiconductor devices as their
conductivity is not sufficiently high. The electrical conductivity is extremely sensitive to
certain types of impurity. It is the ability to modify electrical characteristics of the
material by adding chosen impurities that make extrinsic semiconductors important and
interesting.
Addition of appropriate quantities of chosen impurities is called doping, usually, only
minute quantities of dopants (1 part in 103 to 1010) are required. Extrinsic or doped
semiconductors are classified into main two main types according to the type of charge
carries that predominate. They are the n-type and the p-type.
N-TYPE SEMICONDUCTORS:
Doping with a pentavalent impurity like phosphorous,
arsenic or antimony the semiconductor becomes rich in conduction electrons. It is called
n-type the bond structure of an n-type semiconductor is shown in Fig below.
Even at room temperature, nearby all impurity atoms lose an electron into the conduction
bond by thermal ionization. The additional electrons contribute to the conductivity in
the same way as those excited thermally from the valence bond. The essential difference
beam the two mechanisms is that ionized impurities remain fixed and no holes are
produced. Since penta valent impurities denote extra carries elections, they are called
donors.
P-TYPE SEMICONDUCTORS:-
p-type semiconductors have holes as majority charge carries. They are
produced by doping an intrinsic semiconductor with trivalent impurities.(e.g. boron,
aluminium, gallium, or indium). These dopants have three valence electrons in their
outer shell. Each impurity is short of one electron bar covalent bonding. The vacancy
thus created is bound to the atom at OK. It is not a hole. But at some higher temperature
an electron from a neighbouring atom can fill the vacancy leaving a hole in the valence
bond for conduction. It behaves as a positively charge particle of effective mass mh*.
The bond structure of a p-type semiconductor is shown in Fig below.
Dopants of the trivalent type are called acceptors, since they accept electrons to create
holes above the tope of the valence bond. The acceptor energy level is small compared
with thermal energy of an electron at room temperature. As such nearly all acceptor
levels are occupied and each acceptor atom creates a hole in the valence bond. In
extrinsic semiconductors, there are two types of charge carries. In n-type, electrons are
more than holes. Hence electrons are majority carriers and holes are minority carries.
Holes are majority carries in p-type semiconductors; electrons are minority carriers.
CARRIER CONCENTRATION IN EXTRINSIC SEMICONDUCTORS:
Equation gives the relation been electron and hole concentrations in a
semiconductor. Existence of charge neutrality in a crystal also relates n and p.
The charge neutrality may be stated as
ND + p = NA + n
Since donors atoms are all ionized, ND positive charge per cubic meter are
contributed by ND donor ions. Hence the total positive charge density = ND + p.
Similarly if NA is the concentration of the acceptor ions, they contribute NA
negative charge per cubic meter. The total negative charges density = NA + n.
Since the semiconductor is electrically neutral the magnitude of the positive
charge density must be equal to the magnitude of the total negative charge
density.
n-type material : NA = 0 Since n >> p, eqn reduces to n ≈ ND i.e., in an n-type
material the where subscript n indicates n-type material. The concentration pn of
holes in the n-type semiconductor is obtained from eqn i.e.,
nn pn = n2
i
Thus pn ≈ n2
i
ND
Similarly, for a p-type semiconductor pp ≈ NA and np ≈ n2
i
NA
Expression for electrical conductivity:
There are two types of carries in a Semiconductor electrons and holes. Both these
carries contribute to conduction. The general expression for conductivity can be written
down as
σ = e (nμe + ρ μh)
Where μe and μh are motilities of electrons and holes respectively.
Intrinsic Semiconductor; For an intrinsic Semiconductor
n = p = ni
eqn becomes
σi = eni (μn + μp)
If the scattering is predominantly due to lattice vibrations.
μe = AT3/2
μh = BT3/2
We may put μe + μh = (A + B)T3/2 = CT3/2
σi = ni CT3/2
Substituting for ni from eq we get
σi = 2 2μRT 3/2 CT3/2 (me* mh*)3/4 exp -Eg .
h2 2RT
log σi = log x - Eg
2RT
A graph of log σi Vs 1/T gives a straight line shown in fig below:
LIFE TIME OF MINORITY CARRIER:
In Semiconductor devices electron and hole concentrations are very often
disturbed from their equilibrium values. This may happen due to thermal agitation or
incidence of optical radiation. Even in a pure Semiconductor there will be a dynamic
equilibrium. In a pure Semiconductor the number of holes is equal to the number of free
electrons. Thermal agitation continuously produces of new EHP per unit volume per
second while other EHP disappear due to recombination. On the average a hole exists for
a time period of Tp while an electron exists for a time period Tn before recombination
take place. This time is called the mean life time. If we are dealing with holes in an ntype
Semiconductor, Tp is called the minority carrier life time. These parameters are
important in Semiconductor devices as they indicate the time required for electron and
hole concentration to return to their equilibrium values after they are disturbed.
Let the equilibrium concentration of electrons and holes in an n-type
Semiconductor be n0 and p0 respectively. If the specimen is illuminated at t = ti ,
additional EHPS are generated throughout the specimen. The existing equilibrium is
disturbed and the new equilibrium concentrations are p and n. The excess
concentration of holes = p – p0 – Excess concentration of electrons = n – n0 Since the
radiation creates EHPS.
p – p0 = n – n0
Due to incident radiation equal no of holes and electrons are created. How ever,
the percentage increase of minority carriers is much more than the percentage of majority
carries. In fact, the majority charge carrier change is negligibly small. Hence it is the
minority charge carrier density that is important. Hence, we shall discuss the behaviour
of minority carriers.
As indicated in radiation is removed at t = 0. Let us investigate how the minority
carrier density returns to its original equilibrium value.
The hole concentration decreases as a result of recombination. Decrease in hole
concentration per second due to recombination = p/Tp. But the increase in hole
concentration per second due to thermal generation = g Since charge can neither be
created nor destroyed
dp = g - p
dt Tp
When the radiation is with drawn, the hole concentration p reaches equilibrium
value p0 Hence g = p0/Tp. Then eqn may be rewritten as
dp = p0 – p
dt Tp
We define excess carrier concentration. Since p is a function of time.
p′ = p – p0 = p′(t)
from we may write dp = -p′
dt Tp
The solution to the above differential equation is given by
p′(t) = p (0) e-t/Tp
The excess concentration decreases exponentially to zero with a time constant Tp.
DRIFT CURRENT:
In an electric field E, the drift velocity Vd of carriers superposes on the thermal
velocity Vth. But the flow of charge carriers results in an electric current, known as the
drift current. Let a field E be applied, in the positive creating drifts currents Jnd and Jpd of
electrons and holes respectively.
Without E, the carriers move randomly with rms velocity Vth. Their mean
velocity is zero. The current density will be zero. But the field E applied, the electrons
have the velocity Vde and the holes Vdh.
Consider free electrons in a Semiconductor moving with uniform velocity Vde in the
negative x direction due to an electric field E. Consider a smaller rectangular block of
AB of length Vde inside the Semiconductor. Let the area of the side faces each be unity.
The total charge Q in the elements AB is
Q = Volume of the element x density of partially change on each particle
= (Vde x 1x 1) x n x –q
Thus Q = -qnVde
Where n is the number density of electrons. The entire charge of the block will
cross the face B, in unit time. Thus the drift current density Jnd due to free electrons at
the face B will be.
Jnd = - q nVde
Similarly for holes Jpd = q nVdh
but Vde = -μnE
and Vdh = μpE
hence Jnd = n qμnE
and Jpd = p qμnE
The total drift current due to both electrons and holes Jd is
Jd = Jnd + Jpd = (nqμn + pq μp)E
Even though electrons and holes move in opposite direction the effective direction of
current flow, is the same for both and hence they get added up. Ohm’s Law can be
written in terms of electrical conductivity, as
Jd = σE
Equating the RHS of eq we have
σ = nqμn + pqμp = σn + σp
For an intrinsic Semiconductor n = p = ni
σi = ni q(μn + μp)
DIFFUSION CURRENTS:
1. Diffusion Current: Electric current is Setup by the directed movement of
charge carriers. The movements of charge carriers could be due to either drift or
diffusion. Non- uniform concentration of carriers gives rise to diffusion. The first law of
diffusion by Fick States that the flux F, i.e., the particle current is proportional and is
directed to opposite to the concentration gradient of particles. It can be written
mathematically, in terms of concentration N, as
F = -D V N
Where D stands for diffusion constant.
In one dimension it is written as
F = -D ∂N
∂x
In terms of Je and Jp the flux densities of electrons holes and their densities n and p
respectively.
We get Je = -Dn ∂N
∂x
and Jn = -Dp ∂p
∂x
Where Dn and Dp are the electron and hole diffusion constant constants
respectively. Then the diffusion current densities become
Jn diff = q Dn ∂N
∂x
Jp diff = - q Dp ∂p
∂x
THE EINSTEIN RELATIONS:
When both the drift and the diffusion currents are present total electron and hole current
densities can be summed up as
Jn = Jnd + Jn diff
∂n
Jn = nqμn E + qDn -----
∂x
∂p
Jp = pqμp E - qDp -----
∂x
Now, let us consider a non uniformly doped n-type slab of the Semiconductor fig shown
below (9) under thermal equilibrium. Let the slab be intrinsic at x = 0 while the donor
concentration, increases gradually upto x = 1, beyond which it becomes a constant.
Assume that the Semiconductor is non-degenerate and that all the donors are ionized.
Due to the concentration gradient, electrons tend to diffuse to the left to x = l. This
diffusion leaves behind a positive charge of ionized donors beyond x = 1 and accumulates
electron near x = 0 plane. This charge imbalance. Sets up an electric field in which the
electrons experience fill towards x = 1.
Fig shows illustrates the equilibrium potential φ(x) fig shown refers to the bond diagram
of the Semiconductor.
Both E1 and EF coincide till x = 0 when n0 = ni that EF continues to be the same
throughout the slab. But since the bond structure is not changed due to doping, the bond
edges bond with equal separation all along. How ever, the level Ei continues to lie
midway between EV and E.
In thermal equilibrium, the electrons tend to diffuse down the concentration tending to
setup a current from the right to left. The presence of electric field tends to set up drift
current of electrons in the opposite direction. Both the currents add upto zero. Thus we
obtain
∂n
Jn = qDn ----- + nqμn E = 0
∂x
∂n
i.e., Dn ----- + nμn E = 0
∂x
For a non degenerate Semi conductor.
EF – Ei(x)
n(x) = ni exp -------------
RT
Thus relation is valid at all points in the Semiconductor further. The electronic
concentration is not influenced by the small in balance of charge. Energy is defined in
terms of φ(x) the potential.
E(x) = -qφ(x)
Then EF – Ei(x) = Ei(0)– Ei(x) = -q[φ(0) - φ(x)]
qφ(x)
Assuming φ(0) = 0 we get n(x) = ni exp ------
RT
dn -dφ
Substituting ------ from eqn along with E = ----- we get
dx dx
nq dφ dφ
Dn ------- ------- = μn n -----
RT dx dx
RT
Simplifying we obtain, Dn = ------ μn
q
RT
Simplifying for holes Dp = ----- μp
q
These are known as Einstein relations and the factor (RT/q) as thermal voltage. The
above relations hold good only for non degenerate Semiconductors. For the degenerate
case the Einstein’s relations are complex.
It is clear from the Einstein’s relation that Dp μp and Dn,μn are related and they are
functions of temperature also. The relation of diffusion constant D and the mobility μ
confirms the fact, that both the diffusion an drift processes arise due to thermal motion
and scattering of free electrons, even though they appear to be different.
EQUATION OF CONTINUITY:
If the equilibrium concentrations of carriers in a Semiconductor are disturbed, the
concentrations of electrons and holes vary with time. How ever the carrier concentration
in a Semiconductor is a function of both time and position.
The fundamental law governing the flow of charge is called the continuity equation. It is
arrived at by assuming law to conservation of charge provided drift diffusion and
recombination processes are taken into account.
Consider a small length Δx of a Semiconductor sample with area A in the Z plane
fig shown above. The hole current density leaving the volume (ΔxΔ) under consideration
is Jp ( x + Δx) and the current density entering the volume is Jp(x). Jp (x + Δx) may be
smaller or larger than Jp(x) depending upon the generation and recombination of carriers
in the element. The resulting change in hole concentration per unit time.
∂p = hole flux entering per unit time – hole flux leaving per unit
∂p Jp(x+Δx) δp
----- = Jp(x) - --------------- - ------
∂t x→ x+Δx q Δx Tp
Where Tp is the recombination life time. According to eqn, the rate of hole build up is
equla to the rate of increase of hole concentration remains the recombination rate. As Δx
approaches zero, we may write
∂p ∂ δp - 1 ∂ Jp δp
-----(x,t) = ------ = ----------- - ------
∂t ∂t q ∂x Tp
The above is called the continuity equation for holes for electrons
∂δn 1 ∂Jn δn
----- = -------- - -----
∂t q dx Tn
If there is no drift we may write
∂δ
Jn(diff) = qDn ----
∂x
Substituting the above eqn we get the following diffusion eqn for electrons.
∂δn ∂2δn δn
----- = Dn -------- - -----
∂t ∂x2 Tn
For holes we may write
∂δp ∂2δp δp
----- = Dp -------- - -----
∂t ∂x2 Tp
HALL EFFECT:
When a material carrying current is subjected to a magnetic field in a direction
perpendicular to the direction of current, an electric field is developed across the material
in a direction perpendicular to both the direction of the magnetic field and the current
direction. This phenomenon is called Hall Effect.
Hall Effect finds important application in studying the electron properties of semi
conductor, such as determination of carrier concentration and carrier mobility. It also
used to determine whether a semi conductor is n-type, or p- type.
THEORY:
Consider a rectangular slab of an n-type Semiconductor carrying current in
the positive x-direction. The magnetic field B is acting in the positive direction as
indicated in fig above. Under the influence of the magnetic field, electrons experience a
force FL given by
FL = - Bev --------------- (1)
Where e = magnitude of the charge of the electron
v = drift velocity
Appling the Fleming’s Left Hand Rule, it indicates a force FH acting on the electrons in
the negative y-direction and electron are deflected down wards. As a consequence the
lower face of the specimen gets negatively charged (due to increases of electrons) and the
upper face positively charged (due to loss of electrons). Hence a potential VH, called the
Hall voltage appears between the top and bottom faces of the specimen, which establishes
an electric field EH, called the Hall field across the conductor in negative y-direction. The
field EH exerts an upward force FH on the electrons. It is given by
FH = - eEH --------------------------------------(2)
FH acts on electrons in the upward direction. The two opposing forces FL and FH
establish an equilibrium under which
|FL = FH
using eqns 1 and 2 -Bev = -eEH
EH = Bv --------------(3)
If ‘d’ is the thickness of the Specimen
VH
EH = ------
d
VH = EH d = Bvd from eqn (3)------------------------- 4
If ω is the width of the specimen in z- direction.
The current density
I
J = -----
ωd
But J = nev = ρv -------------------- 5
Where n = electron concentration
And ρ = charge density
I
∴ ρv = ----
ωd
I
or v = ------ --------------------------------- 6
ρωd
Substitutinf for v, from eqns 6 and 4
VH = BI / ρω
BI
or ρ= --------
VH ω
Thus, by measuring VH, I, and ω and by knowing B, the charge density ρ can be
determined.
Hall Coefficient:
The Hall field EH, for a given material depends on the current density J, and
the applied field B
i.e., EH ∝ JB
EH= RH JB
Where RH is called the Hall Coefficient
BI
Since VH = --------
ρω
VH
EH = -----
Jωd
I
J = ------
ωd
BI I
-------- = RH ------- B
Jωd ωd
I
This leads to RH = -----
ρ
Mobility of charge carriers:
The mobility μ is given by μ = v
E
But J = σE = nev = ρv
∴ σE = ρv
ρv
or E = -----
σ
σ
⇒ μ = ---- = σRH (∴ 1/ρ = RH)
ρ
σ is the conductivity of the semi conductor.
(C) Applications
Determination of the type of Semiconductor: The Hall Coefficient RH is
negative for an n-type Semiconductor and positive for a p-type material.
Thus, the sign of the Hall coefficient can be utilized to determine whether a
given Semiconductor is n or p type.
Determination of Carrier Concentration: Equation relates the Hall Coefficient
RH and charge density is
1 - 1
RH = ------ = ------ ( for n-type
p ne
1
= ------ ( for p-type
pe
1
Thus n = ------
eRH
1
and -------
eRH
Determination of mobility: According to equation the mobility of charge carriers
is given by
μ = σ|RH|
Determination of σ and RH leads to a value of mobility of charge carriers.
Measurement of Magnetic Induction (B):- The Hall Voltage is proportional to
the flux density B. As such measurement of VH can be used to9 estimate B.
SUPERCONDUCTIVITY
Introduction : Certain metals and alloys exhibit almost zero resistivity( i.e. infinite
conductivity ) when they are cooled to sufficiently low temperatures. This phenomenon is
called superconductivity. This phenomenon was first observed by H.K. Onnes in 1911.
He found that when pure mercury was cooled down to below 4K, the resistivity suddenly
dropped to zero. Since then hundreds of superconductors have been discovered and
studied. Superconductivity is strictly a low temperature phenomenon. Few new oxides
exhibited superconductivity just below 125K itself. This interesting phenomena has many
important applications in many emerging fields.
General Properties: The temperature at which the transition from normal state to
superconducting state takes place on cooling in the absence of magnetic field is called the
critical temperature (Tc ) or the transition temperature.
The following are the general properties of the superconductors:
1. The transition temperature is different to different substances.
2. For a chemically pure and structurally perfect specimen, the superconducting
transition is very sharp.
3. Superconductivity is found to occur in metallic elements in which the number of
valence electrons lies between 2 and 8.
4. Transition metals having odd number of valence electrons are favourable to
exhibit superconductivity while metals having even number of valence electrons
are unfavourable.
5. Materials having high normal resistivities exhibit superconductivity.
6. Materials for which Zρ > 106 (where Z is the no. of valence electrons and ρ is
the resistivity) show superconductivity.
7. Ferromagnetic and antiferromagnetic materials are not superconductors.
8. The current in a superconducting ring persists for a very long time .
Effect of Magnetic Field: Superconducting state of metal depends on temperature
and strength of the magnetic field in which the metal is placed. Superconducting
disappears if the temperature of the specimen is raised above Tc or a strong enough
magnetic field applied. At temperatures below Tc, in the absence of any magnetic field,
the material is in superconducting state. When the strength of the magnetic field applied
reaches a critical value Hc the superconductivity disappears.
At T= Tc, Hc = 0. At temperatures below Tc, Hc increases. The dependence of the
critical field upon the temperature is given by
HC(T) = HC(0) [1 – (T/Tc)2]------------------------------(1)
Where Hc(0) is the critical field at 0K. Hc(0) and Tc are constants of the characteristics
of the material.
Meissner effect: When a weak magnetic field applied to super conducting
specimen at a temperature below transition temperature Tc , the magnetic flux lines are
expelled. This specimen acts as on ideal diamagnet. This effect is called meissner effect.
This effect is reversible, i.e. when the temperature is raised from below Tc , at T = Tc the
flux lines suddenly start penetrating and the specimen returns back to the normal state.
Under this condition, the magnetic induction inside the specimen is given by
B = μ0(H + M) -------------------------------------(2)
Where H is the external applied magnetic field and M is the magnetization produced
inside the specimen.
When the specimen is super conducting, according to meissner effect inside the bulk
semiconductor B= 0.
Hence μ0(H + M) = 0
Or M = - H ---------------------------------------------(3)
Thus the material is perfectly diamagnetic.
Magnetic susceptibility can be expressed as
χ=M/H = -1-----------------------------------------------------(4)
Consider a superconducting material under normal state. Let J be the current passing
through the material of resistivity ρ. From ohm’s law we know that the electric field
E = Jρ
On cooling the material to its transition temperature, ρ tends to zero. If J is held finite.
E must be zero. Form Maxwell’s eqn, we know
▼X E = - dB/ dt ----------------------------(5)
Under superconducting condition since E = 0, dB/dt = 0, or B= constant.
This means that the magnetic flux passing through the specimen should not change on
cooling to the transition temperature. The Meissner effect contradicts this result.
According to Meissner effect perfect diamagnetism is an essential property of defining
the superconducting state. Thus
From zero resistivity E = 0,
From Meissner effect B= 0.
Type- I , Type- II superconductors: Based on diamagnetic response
Superconductors are divided into two types, i.e type-I and type-II.
Superconductors exhibiting a complete Meissner effect are called type-1, also called Soft
Superconductors. When the magnetic field strength is gradually increased from its initial
value H< HC, at HC the diamagnetism abruptly disappear and the transition from
superconducting state to normal state is sharp. Example Zn, Hg, pure specimens of Al
and Sn.
In type-2 Superconductors, transition to the normal state takes place gradually. For fields
below HC1, the material is diamagnetic i.e., the field is completely excluded HC1 is called
the lower critical field. At HC1 the field begins to penetrate the specimen. Penetration
increases until HC2 is reached. At HC2, the magnetizations vanishes i.e., the material
becomes normal state. HC2 is the upper critical field. Between HC1 and HC2 the state of the
material is called the mixed or vortex state. They are also known as hard
superconductors. They have high current densities. Example Zr , Nb etc.
Penetration Depth : The penetration depth λ can be defined as the depth from the
surface at which the magnetic flux density falls to 1/e of its initial value at the surface.
Since it decreases exponentially the flux inside the bulk of superconductor is zero and
hence is in agreement with the Meissner effect. The penetration depth is found to depend
on temperature. its dependence is given by the relation
λ ( T ) = λ ( 0 ) ( 1 – T4 / Tc
4 )-1/2 ----------------(1)
where λ ( 0 ) is the penetration depth at T = 0K.
According to eqn.(1) , λ increases with the increase of T and at T= Tc
, it becomes
infinite . At T= Tc
, the substance changes from super conducting state to normal state
and hence the field can penetrate to the whole specimen , ie , the specimen has an infinite
depth of penetration.
BCS Theory : BCS theory of superconductivity was put forward by Bardeen, Cooper and
Schrieffer in 1957. This theory could explain many observed effects such as zero
resistivity , Meissner effect, isotope effect etc. The BCS theory is based on advanced
quantum concept.
1. Electron – electron interaction via lattice deformation: Consider an electron is
passing through the lattice of positive ions. The electron is attracted by the
neighbouring positive ion, forming a positive ion core and gets screened by them.
The screening greatly reduces the effective charge of this electron. Due to the
attraction between the electron and the positive ion core, the lattice gets deformed
on local scale. Now if another electron passes by its side of the assembly of the
electron and the ion core, it gets attracted towards it. The second electron interacts
with the first electron via lattice deformation. The interaction is said to be due to
the exchange of a virtual phonon ,q, between the two electrons. The interaction
process can be written in terms of wave vector k ,as
K1 – q = K1
1 and K2 + q = K2
1 -----------------(1)
This gives K1 – K2 = K1
1 + K2
1 , ie , the net wave vector of the pair is conserved.
The momentum is transferred between the electrons. These two electrons together form a
cooper pair and is known as cooper electron.
2. Cooper Pair: To understand the mechanism of cooper pair formation , consider
the distribution of electrons in metals as given by Fermi – Dirac distribution
function .
F ( E ) = 1 / [ exp ( E – EF / KT ) + 1 ]
At T = 0K, all the energy states below Fermi level EF are completely filled and all the
states above are completely empty.
Let us see what happens when two electrons are added to a metal at absolute zero.
Since all the quantum states with energies E ≤ EF are filled , they are forced to occupy
states having energies E > EF . Cooper showed that if there is an attraction between
the two electrons ,they are able to form a bound state so that their total energy is less
than 2 EF . These electrons are paired to form a single system. These two electrons
together form a cooper pair and is known as cooper electron. cooper pair and is
known as cooper electron. Their motions are correlated. The binding is strongest
when the electrons forming the pair have opposite momenta and opposite spins. All
electron pairs with attraction among them and lying in the neighbourhood of the
Fermi surface form cooper pairs. These are super electrons responsible for the
superconductivity.
In normal metals, the excited states lie just above the Fermi
surface. To excite an electron from the Fermi surface even an arbitrarily small
excitation energy is sufficient. In super conducting material , when a pair of electrons
lying just below the Fermi surface is taken just above it ,they form a cooper pair and
their total energy is reduced. This continues until the system can gain no additional
energy by pair formation. Thus the total energy of the system is further reduced.
At absolute zero in normal metals there is an abrupt
discontinuity in the distribution of electrons across the Fermi surface whereas such
discontinuity is not observed in superconductors. As super electrons are always
occupied in pairs and their spins are always in opposite directions.
Isotope Effect : in super conducting materials, the transition temperature varies with
the average isotopic mass , M , of their constituents. The variation is found to follow
the general form
Tc α M-α --------------------(1)
Or Mα Tc = constant
Where α is called the isotope effect coefficient .
Flux Quantization:
Consider a superconductor in the form of a ring. Let it be at temperature above
its TC because of which it will be in the normal state. When it is subjected to the
influence of a magnetic field, the flux lines pass through the body and also exist out side
and inside ring also.
If the body is cooled to a temperature below its TC, then as per Meissner effect, the flux
lines are expelled from the body i.e. the flux exists both outside the ring and in the hole
region but not in the body of the ring. But when the external field is switched off, the
magnetic flux lines continue to exist within the hole region, through the rest that
surrounded the ring from the exterior would vanish. This is known as flux trapping. It is
due to the large currents that are induced as per Faraday’s law during the flux decay when
the field switched off. Because of the zero resistance property that the superconductor
enjoys, these induced currents continue to circulate in the ring practically externally.
Thus the flux stand trapped in the loop forever.
It was F.London who gave the idea that the trapped magnetic flux is quantized, as
super conductivity is governed by the quantum phenomenon. At first he suggested the
quantization of Φ as
Φ = nh / e n = 1,2,3,………….
But experiments carried out carefully on very small superconducting hollow cylinders by
Deaver and Fairbank , that gave half the values of flux quanta given by London. Thus the
governing equation for flux quantization was changed to,
Φ = nh / 2e
It happened so because, London’s theory was based on supercurrents constituted by
electrons as individual entities. This demonstrates conclusively that superconducting
current carriers are pairs of electrons and not single ones.
Then the above equation is written as
Φ = n Φ0
Where Φ0 = (h/2e) is the quantum of flux and is called fluxoid.
Josephson Effect : Consider a thin insulating layer sandwiched between two metals.
This insulating layer acts as a potential barrier for flow of electrons from one metal to
another. Since the barrier is so thin , mechanically electrons can tunnel through from
a metal of higher chemical potential to the other having a lower chemical potential.
This continues until the chemical potential of electrons in both the metals become
equal.
Consider application of a potential difference across the
potential barrier. Now more electrons tunnel through the insulating layer from higher
potential side to lower potential side. The current – voltage relation across the
tunneling junction obeys the ohm’s law at low voltages.
Now consider another case that one of the metals is a
superconductor. On applying the potential, it can be observed that no current flows
across the junction until the potential reaches a threshold value. It has been found that
the threshold potential is nothing but half the energy gap in the superconducting state.
Hence the measurement of threshold potential under this condition helps one to
calculate the energy gap of superconductor. As the temperature is increased towards
Tc ,more thermally excited electrons are generated. Since they require less energy to
tunnel , the threshold voltage decreases. This results in decrease of energy gap itself.
Consider a thin insulating layer sandwiched between two superconductors. In
addition to normal tunneling of single electrons, the super electrons also tunnel
through the insulating layer from one superconductor to another without dissociation,
even at zero potential difference across the junction. Their wave functions on both
sides are highly correlated. This is known as Josephson effect.
The tunneling current across the junction is very less since the two
superconductors are only weakly coupled because of the presence of a thin insulating
layer in between.
D.C. Josephson Effect : According to Josephson, when tunneling occurs through the
insulator it introduces a phase difference ΔΦ between the two parts of the wave
function on the opposite sides of the junction.
The tunneling current is given by
I = I0 sin ( Φ0 ) --------------------(1)
Where I0 is the maximum current that flows through the junction without any
potential difference across the junction. I0 depends on the thickness of the junction
and the temperature.
When there is no applied voltage, a d.c. current flows across the
junction. The magnitude of the current varies between I0 and -I0 according to the
value of phase difference Φ0 = (Φ2 – Φ1 ). This is called d.c Josephson effect.
A.C. Josephson Effect : let a static potential V0 is applied across the junction. This
results in additional phase difference introduced by cooper pair during tunneling
across the junction. This additional phase difference ΔΦ at any time t can be
calculated using quantum mechanics
ΔΦ = E t / ħ ----------------------(1)
Where E is the total energy of the system.
In the present case E = (2 e ) V0 . since a cooper pair contains 2 electrons , the factor
2 appears in the above eqn.
Hence ΔΦ = 2 e V0 t / ħ .
The tunneling current can be written as
I = I0 sin ( Φ0 + ΔΦ ) = I0 sin ( Φ0 + 2 e V0 t / ħ )---------------(1)
This is of the form
I = I0 sin ( Φ0 + ω t ) --------------------(2)
Where ω = 2 e V0 / ħ .
This represents an alternating current with angular frequency ω. This is the a.c.
Josephson effect. when an electron pair crosses the junction a photon of energy ħω =
2eV0 is emitted or absorbed.
Current – voltage characteristics of a Josephson junction are :
When V0 = 0 , there is a constant flow of d.c current ic through the junction. This
current is called superconducting current and the effect is the d.c. Josephson
effect.
So long V0 < Vc , a constant d.c.current ic flows.
When V0 > Vc , the junction has a finite resistance and the current oscillates with a
frequency ω = 2 e V0 / ħ . This effect is the a.c Josephson effect.
Applications Of Josephson Effect:
1. It is used to generate microwaves with frequency ω = 2 e V0 / ħ .
2. A.C Josephson effect is used to define standard volt.
3. A.C Josephson effect is also used to measure very low temperatures based on the
variation of frequency of emitted radiation with temperature.
4. A Josephson junction is used for switching of signals from one circuit to another.
The switching time is of the order of 1ps and hence very useful in high speed
computers.
Applications Of Superconductors :
1. Electric generators : superconducting generators are very smaller in size and
weight when compared with conventional generators. The low loss
superconducting coil is rotated in an extremely strong magnetic field. Motors with
very high powers could be constructed at very low voltage as low as 450V. this is
the basis of new generation of energy saving power systems.
2. Low loss transmission lines and transformers : Since the resistance is almost zero
at superconducting phase, the power loss during transmission is negligible. Hence
electric cables are designed with superconducting wires. If superconductors are
used for winding of a transformer, the power losses will be very small.
3. Magnetic Levitation : Diamagnetic property of a superconductor ie , rejection of
magnetic flux lines is the basis of magnetic levitation. A superconducting material
can be suspended in air against the repulsive force from a permanent magnet. This
magnetic levitation effect can be used for high speed transportation.
4. Generation of high Magnetic fields : superconducting materials are used for
producing very high magnetic fields of the order of 50Tesla. To generate such a
high field, power consumed is only 10kW whereas in conventional method for
such a high field power generator consumption is about 3MW. Moreover in
conventional method ,cooling of copper solenoid by water circulation is required
to avoid burning of coil due to Joule heating.
5. Fast electrical switching :A superconductor possesses two states , the
superconducting and normal. The application of a magnetic field greater than Hc
can initiate a change from superconducting to normal and removal of field
reverses the process. This principle is applied in development of switching
element cryotron. Using such superconducting elements, one can develop
extremely fast large scale computers.
6. Logic and storage function in computers : they are used to perform logic and
storage functions in computers. The current – voltage characteristics associated
with Josephson junction are suitable for memory elements.
7. SQUIDS ( superconducting Quantum Interference Devices ) : It is a double
junction quantum interferometer. Two Josephson junctions mounted on a
superconducting ring forms this interferometer. The SQUIDS are based on the
flux quantization in a superconducting ring. Very minute magnetic signals are
detected by these SQUID sensors. These are used to study tiny magnetic signals
from the brain and heart. SQUID magnetometers are used to detect the
paramagnetic response in the liver. This gives the information of iron held in the
liver of the body accurately.
Questions:
UNIT – VI
LASERS
Introduction:
It is a device to produce a powerful monochromatic narrow beam of light in which the
waves are convergent. Laser is an acronym for light amplification by stimulated emission
of radiation.
Maser is an acronym of microwave amplification by stimulated emission of radiation.
The light emitted from the conventional light source (eg: sodium lamp, candle) is said to
be incoherent. Because the radiation emitted from different atom do not have any definite
phase relationship with each other. Lasers are much important because the light sources
having high monochromaticity, high intensity, high directionality and high coherence.
In the laser the principal of maser is employed in the frequency range of 1014to 1015Hz
and it is termed as optical maser. Laser principle now a day is extended up to γ-rays
hence Gamma ray lasers are called Grazers. The first two successful lasers developed
during 1960 were Ruby laser and He- Ne lasers. Some lasers emit light is pulses while
others emit as a continuous wave.
Characteristics of laser radiation:
The four characteristics of a laser radiation over conventional light sources are
(1) Laser is highly monochromatic
(2) Laser is highly directional
(3) Laser is highly coherent
(4) The intensity of laser is very high
HIGHLY MONOCHROMATIC:
The band width of ordinary light is about 1000A0 . The band width of laser light is about
10A0 . The narrow band width of a laser light is called on high monochromacity.
BAND WIDTH:- The spread of the wavelength (frequency ) about the wavelength of
maximum intensity is band width.
Laser light is more monochromatic than that of a conventional light source. Because of
this monochromaticity large energy can be concentrated in to an extremely. Small band
width.
For good laser dv=50Hz v=5×1014 Hz. The degree of non-monochromaticity
for a conventional sodium light.
HIGH DIREATIONALITY:
The conventional light sources like lamp, torch light, sodium lamp emits light in all
directions. This is called divergence. Laser in the other hand emits light only in one
direction. This is called directionality of laser light.
Light from ordinary light spreads in about few kilometers.
Light from laser spreads to a diameter less than 1 cm for many kilometers.
The directionality of laser beam is given by (or) expressed in divergence.
The divergence Δθ = (r2 –r1) /d2-d1
Where r2, r1 are the radius of laser beam spots
d2 ,d1 are distances respectively from the laser source. Hence for getting a high
directionality then should be low divergence.
HIGHLY COHERENT:
When two light rays are having same phase difference then they are said to be coherent.
It is expressed in terms of ordering of light field
Laser has high degree of ordering than other common sources. Due to its coherence only
it is possible to create high power(1012 watts) in space with laser beam of 1μm diameter.
There are two independent concepts of coherence.
1) Spatial coherence (2) Temporal coherence
SPATIAL COHERENCE: The two light fields at different point in space maintain a
constant phase difference over any time (t) they are said to be spatial coherence.
In He- Ne gas laser the coherence length( lc ) is about 600km.It means over the distance
the phase difference is maintained over any time .For sodium light it is about 3cm.
The coherence & monochoremacity is related by
ξ = (Δυ / υ) α 1/ lc
For the higher coherence length ξ is small hence it has high monochromacity
TEMPORAL COHERENCE: The correlation of phase between the light fields at a
point over a period of time. For He- Ne laser it is a about 10-3 second , for sodium it is
about 10-10 second only.
ξ = (Δυ / υ) α 1/ tc
Higher is the tc higher is the monochromacity.
HIGH INTENSITY:
Intensity of a wave is the energy per unit time flowing through a unit area.
The light from an ordinary source spreads out uniformly in all directions and from
spherical wave fronts around it.
Ex:- If you look a 100W bulb from a distance of 30cm the power entering the eye is
1 / 1000 of watt.
But in case of a laser light, energy is in small region of space and in a small wavelength
and hence is said to be of great intensity.
The power range of laser about 10-3W for gas laser and 109W for solid state laser
The number of photons coming out from a laser per second per unit area is given by
N1 = p / hυЛr2 ≈ 1022 to 1034 photons/ m2 – sec.
SPONTANEOUS AND STIMULATED (INDUCED) EMISSION:
Light is emitted or absorbed by particles during their transitions from one energy state to
another .the process of transferring a particle from ground state to higher energy state is
called excitation. Then the particle is said to be excited.
The particle in the excited state can remain for a short interval of time known as life time.
The life time is of the order of 10-8 sec, in the excited states in which the life time is
much greater than 10-8 sec are called meta stable states. The life time of the particle in the
Meta stable state is of the order 10-3 sec
The probability of transition to the ground state with emission of radiation is made
up of two factors one is constant and the other variable.
The constant factor of probability is known as spontaneous emission and the variable
factor is known as stimulated emission.
SPONTANEOUS EMISSION: The emission of particles from higher energy state to
lower energy state spontaneously by emitting a photon of energy hυ is known as
“spontaneous emission”
STIMULATED EMISSION: The emission of a particle from higher state to lower state
by stimulating it with another photon having energy equal to the energy difference
between transition energy levels called stimulated emission.
SPONTANEOUS EMISSION STIMULATED EMISSION
1) Incoherent radiation 1) coherent radiation
2) Less Intensity 2) high intensity
3) Poly chromatic 3) mono chromatic
4) One photon released 4) two photons released
5) Less directionality 5) high directionality
6) More angular spread during propagation 6) less angular spread during
Propagation
Ex:-Light from sodium ex: - light from a laser source
Mercury vapour lamp ruby laser, He-He gas laser gas
Laser
EINSTEINS EQUATIONS (OR) EINSTAINS CO- EFFICIENTS:
Based on Einstein’s theory of radiation one can get the expression for probability for
stimulated emission of radiation to the probability for spontaneous emission of radiation
under thermal equilibrium.
E1, E2 be the energy states
N1, N2 be the no of atoms per unit volume
ABSORPTION: If ρ(υ)dυ is the radiation energy per unit volume between the
frequency range of υ and υ+dυ
The number of atoms under going absorption per unit volume per second from level
E1 to E2 = N1 ρ(υ )B12------􀃆 1
B12 represents probability of absorption per unit time
STIMILATED EMISSION: When an atom makes transition E2 toE1 in the presence of
external photon whose energy equal to (E2-E1) stimulated emission takes place thus the
number of stimulated emission per unit volume per second from levels.
E2 to E1 = N2 ρ(υ) B21------􀃆 2
B21 is represents probability of stimulated emission per unit time.
SPONSTANEOUS EMISSION: An atom in the level E2 can also make a
spontaneous emission by jumping in to lower energy level E1.
E2 to E1=N2A21---􀃆 3
A21 represents probability of spontaneous emission per unit time.
Under steady state
(dN / dt) = 0
No of atoms under going absorption per second = no of atoms under going emission per
second
1 =2+3
N1 ρ(υ )B12 = N2 ρ(υ )B21 + N2 A21
N2 A21 = N1 ρ(υ )B12 - N2 ρ(υ )B21
= ρ(υ )(N1B12 - N2B21)
ρ (υ) = N2 A21 / (N1B12 - N2B21)
= A21 / [(N1 / N2 )B12 - B21)] --------􀃆 4
From distribution law we know that
N1 / N2 = e(E
2
- E
1
) / k
B
T
= ehυ / k
B
T --------------􀃆 5
Substituting N1 / N2 in eqn (4) we get
ρ (υ) = A21 / B21 (ehυ / k
B
T - 1)---------􀃆 6
From Planck’s radiation
ρ (υ) = 8Πhn3 / λ3 x [ 1 / (ehυ / k
B
T - 1)] -------􀃆 7
n – refractive index of the medium
λ - wave length of the light in air.
λm= λ / n wavelength of light in medium --------􀃆 8
ρ (υ) = 8Πh / λm
3 x [ 1 / (ehυ / k
B
T - 1)] ------􀃆 9
Comparing eqn 6 and 9
A21 / B21 = 8Π h / λm
3
Where A21 , B21 are Einstein’s co-efficient of spontaneous emission probability per unit
time and stimulated emission probability per unit time respectively.
For stimulated emission to be predominant, we have
A21 / B21 << 1
POPULATION INVERSION:
The no of atoms in higher energy level is less than the no of atoms in lowest
energy level. The process of making of higher population in higher energy level than the
population in lower energy level is known as population inversion.
Population inversion is achieved by pumping the atoms from the ground level to the
higher energy level through optical (or) electrical pumping. It is easily achieved at the
matastable state, where the life time of the atoms is higher than that in other higher
energy levels.
The states of system, in which the population of higher energy state is more in
comparision with the population of lower energy state, are called “Negative temperature
state”.
A system in which population inversion is achieved is called as an active system. The
method of raising the particles from lower energy state to higher energy state is called
”Pumping”.
DIFFERENMT TYPES OF LASES:
1. Solid state laser - Ruby laser, Nd-YAG laser
2. Gas laser - He-Ne laser, CO2 laser
3. Semi conductor laser - GaAs laser
RUBY LASER:
Ruby laser is a three level solid state laser having very high power of hundreds of
mega watt in a single pulse it is a pulsed laser.
The system consists of mainly two parts
1) ACTIVE MATERIAL: Ruby crystal in the form of rod.
2) RESONANT CAVITY: A fully reflecting plate at the left end of the ruby crystal
and partially reflecting end at the right side of the ruby crystal both the surfaces
are optically flat and exactly parallel to each other.
3) EXCITING SYSTEM: A helical xenon flash tube with power supply source.
CONSTRUCTION: In ruby laser, ruby rod is a mixture of Al2O3+Cr2O3. It is a
mixture of Aluminum oxide in which some of ions Al+3 ions concentration doping of Cr+3
is about 0.05% , then the colour of rod becomes pink. The active medium in ruby rod is
Cr+3 ions. The length of the ruby rod is 4cm and diameter 5mm and both the ends of the
ruby rod are silvered such that one end is fully reflecting and the other end is partially
reflecting. The ruby rod is surrounded by helical xenon flash lamp tube which provides
the optical pumping to raise Cr+3 ions to upper energy level.
The chromium atom has been excited to an upper energy level by absorbing photons of
wave length 5600A0 from the flash lamp. Initially the chromium ions (Cr+3) are excited to
the energy levels E1 to E3 , the population in E3 increases. Since the life time of E3 level is
very less (10_8Sec). The Cr+3 ions drop to the level E2 which is matastable of life time 10-
3Sec. The transition from E3 to E2 is non-radiative.
Since the life time of metastable state is much longer, the no of ions in this state goes on
increasing hence population inversion achieved between the excited metastable state E2
and the ground state E1.
When an excited ion passes spontaneously from the metastable state E2 to the ground
state E1. It emit a photon of wave length 6993A0 this photon travel through the ruby rod
and if it is moving parallel to the axis of the crystal and it is reflected back and forth by
the silver ends until it stimulates an excited ion in E2. The emitted photon and stimulated
photon are in phase the process is repeated again and again finally the photon beam
becomes intense; it emerges out through partially silvered ends.
Since the emitted photon and stimulating photon in phase, and have same frequency and
are traveling in the same direction, the laser beam has directionality along with spatial
and temporal coherence.
IMPORTANCE OF RESONATOR CAVITY: To make the beams parallel to each
other curved mirrors are used in the resonator cavity. Resonator mirrors are coated with
multi layer dielectric materials to reduce the absorption loss in the mirrors. Resonators act
as frequency selectors and also give rise to directionality to the out put beam. The
resonator mirror provides partial feedback to the protons.
He- Ne Laser
CONSTRUTION:
He - Ne gas laser consists of a gas discharge tube of length 80cm
and diameter of 1cm. The tube is made up of quartz and is filled with a mixture of Neon
under a pressure of 0.1mm of Hg. The Helium under the pressure of 1mm of Hg the ratio
of He-Ne mixture of about 10:1, hence the no of helium atoms are greater than neon
atoms. The mixtures is enclosed between a set of parallel mirrors forming a resonating
cavity, one of the mirrors is completely reflecting and the other partially reflecting in
order to amplify the output laser beam.
WORKING:
When a discharge is passed through the gaseous mixture electrons are
accelerated down the tube these accelerated electrons collide with the helium atoms and
excite them to higher energy levels since the levels are meta stable energy levels he
atoms spend sufficiently long time and collide with neon atoms in the ground level E1 .
Then neon atoms are excited to the higher energy levels E4 & E6 and helium atoms are de
excited to the ground state E1
Since E6 & E4 are meta stable states, population inversion takes place at there levels. The
stimulated emission takes place between E6 to E3 gives a laser light of wave length
6328A0 and the stimulated emission between E6 and E5 gives a laser light wave length of
3.39μm. Another stimulated emission between E4 to E3 gives a laser light wave length of
1.15μm. The neon atoms undergo spontaneous emission from E3 toE2 and E5 to E2.
Finally the neon atoms are returned to the ground state E1 from E2 by non-radeative
diffusion and collision process.
After arriving the ground state, once again the neon atoms are raised to E6 & E4 by
excited helium atoms thus we can get continuous out put from He-Ne laser.
But some optical elements placed insides the laser system are used to absorb the infrared
laser wave lengths 3.39μm and1.15μm. hence the output of He-Ne laser contains only a
single wave length of 6328A0 and the output power is about few milliwatts .
CO2 LASER
Construction and working:
We know that a molecule is made up of two or more atoms bound together. In molecule
in addition to electronic motion, the constituent atoms in a molecule can vibrate in
relation to each other and the molecule as a whole can rotate. Thus the molecule is not
only characterized by electronic levels but also by vibration and rotational levels. The
fundamental modes of vibrations of CO2 molecule shown I fig.
CO2 Laser is a gas discharge, which is air cooled. The filling gas within the discharge
tube consists primarily of, CO2 gas with 10 – 20%, Nitrogen around 10 – 20 %
H2 or Xenon ( Xe ) a few percent usually only in a sealed tube.
The specific proportions may vary according to the particular application. The population
inversion in the laser is achieved by following sequence:
1. Electron impact excites vibration motion of the N2. Because N2 is a homo nuclear
molecule, it cannot lose this energy by photon emission and it is excited vibration
levels are therefore metestable and live for long time.
2. Collision energy transfer between the N2 and the CO2 molecule causes vibration
excitation of the CO2, with sufficient efficiency to lead to the desired population
inversion necessary for laser operation.
Because CO2 lasers operate in the infrared, special materials are necessary for their
construction. Typically the mirrors are made of coated silicon, molybdenum or gold,
while windows and lenses are made of either germanium or zinc sulenide. For high
power application gold mirrors and zinc selenide windows and lenses are preferred.
Usually lenses and windows are made out of salt NaCl or KCl ). While the material is
inexpensive, the lenses windows degraded slowly with exposure to atmosphere moisture.
The most basic form of a CO2 laser consist of a gas discharge ( with a mix close to that
specified above) with a total reflector at one end and an output coupler ( usually a semi
reflective coated zinc selendine mirror ) at the output end. The reflectivity of the out put
coupler is typically around 5 – 15 %. The laser output may be edge coupled in higher
power systems to reduce optical heating problems.
CO2 lasers output power is very high compared to Ruby laser or He – Ne lasers. All CO2
lasers are rated in Watts or kilowatts where the output power of Ruby laser or He – Ne
laser is rated in mill watts. The CO2 laser can be constructed to have CW powers between
mill watts and hundreds of kilowatts.
SEMICONDIRCTOR LASER (Gallium Arsenide Diode Laser or
Homojunction Laser):
Semiconductor laser is also known as diode laser
PRINCIPLE:
In the case direct band gap semiconductors there is a large possibility for direct
recombination of hole and electron emitting a photon. GaAs is a direct band gap
semiconductor and hence it is used to make lasers and light emitting diodes. The wave
lengths of the emitted light depend on the band hap of the material.
CONSTRUCTIONS: The P-region and N-region in the diode are obtained by
heavily doping germanium and tellurium respectively in GaAs. The thickness of the P-N
junction layer is very narrow at the junction, the sides are well polished and parallel to
each other. Since the refractive index of GaAs is high the reflectance at the material air
interface is sufficiently large so that the external mirror are not necessary to produce
multiple reflections. The P-N junction is forward biased by connection positive terminal
to P-type and negative terminal to N type.
WORKING: When the junction is forward biased a large current of order104
amp/cm2 is passed through the narrow junction to provide excitation. Thus the electrons
and holes injected from N side and P side respectively. The continuous injection of
charge carries the population inversion of minority carriers in N and P sides respectively.
The excess minority electrons in the conduction band of P – layers recombine with the
majority holes in the valence band of P-layer emitting light photons similarly the excess
minority holes in the valence band of N- layers recombine with the majority electrons in
the conduction band of N- layer emitting light photons.
The emitted photons increase the rate of recombination of injected electrons from the Nregion
and holes in P- region. Thus more no of photons are produced hence the
stimulated emission take place, light is amplified.
The wave length of emitted radiation depends upon the concentration of donor and
acceptor atoms in GaAs the efficiency of laser emission increases, when we cool the
GaAs diode.
DRAWBACKS:-
1. Only pulsed laser output is obtained
2. Laser output has large divergence
3. Poor coherence
APPLICATION OF LASERS:
Lasers in scientific research
1) Lasers are used to clean delicate pieces of art, develop hidden finger prints
2) Laser are used in the fields of 3D photography called holography
3) Using lasers the internal structure of micro organisms and cells are studied very
accurately
4) Lasers are used to produce certain chemical reactions.
Laser in Medicine:
1) The heating action of a laser bean used to remove diseased body tissue
2) Lasers are used for elimination of moles and tumours, which are developing in the skin
tissue.
3) Argon and CO2 lasers are used in the treatment liver and lungs
4) Laser beam is used to correct the retinal detachment by eye specialist
Lasers in Communication:
1) More amount of data can be sent due to the large band with of semiconductor
lasers
2) More channels can be simultaneously transmitted
3) Signals cannot tapped
4) Atmospheric pollutants concentration, ozone concentration and water vapour
concentration can be measured
Lasers in Industry: Lasers are used
1) To blast holes in diamonds and hard steel
2) To cut, drill, welding and remove metal from surfaces
3) To measure distance to making maps by surveyors
4) For cutting and drilling of metals and non metals such a ceramics plastics glass
Questions:
1. Explain the terms i) Absorption, ii) spontaneous emission, iii) Stimulated
emission, iv) pumping mechanism , v) Population inversion, vi) Optical cavity
2. What is population inversion? How it is achieved?.
3. Explain the characteristics of a laser beam.
4. Distinguish between spontaneous and stimulated emission.
5. Derive the Einstein’s coefficients.
6. With neat diagrams, describe the construction and action of ruby laser.
7. Explain the working of He – Ne laser
8. Describe the CO2 laser
9. Explain the semiconductor laser.
10. Mentions some applications of laser in different fields.
UNIT - VII
FIBER OPTICS AND HOLOGRAPHY
Introduction: In 1870 John Tyndall demonstrated that light follows the curve of a
stream of water pouring from a container; it was this simple principle that led to the study
and development of application of the fiber optics. The transmission of information over
fibers has much lower losses than compared to that of cables. The optical fibers are most
commonly used in telecommunication, medicine, military, automotive and in the area of
industry. In fibers, the information is transmitted in the form of light from one end of the
fiber to the other end with min.losses.
Advantages of optical fibers:
• Higher information carrying capacity.
• Light in weight and small in size.
• No possibility of internal noise and cross talk generation.
• No hazards of short circuits as in case of metals.
• Can be used safely in explosive environment.
• Low cost of cable per unit length compared to copper or G.I cables.
• No need of additional equipment to protect against grounding and voltage
problems.
• Nominal installation cost.
• Using a pair of copper wires only 48 independent speech signals can be sent
simultaneously whereas using an optical fiber 15000 independent speeches
can be sent simultaneously.
Basic principle of Optical fiber:
The mechanism of light propagation along fibers can be understood using the
principle of geometrical optics. The transmission of light in optical fiber is based on
the phenomenon of total internal reflection.
Let n1 and n2 be the refractive indices of core and cladding
respectively such that n1>n2.Let a light ray traveling from the medium of refractive
index n1 to the refractive index n2 be incident with an angle of incidence “i” and the
angle of refraction “r”. By Snell’s law
n1sin i = n2sin r…………………(1)
The refractive ray bends towards the normal as the ray travels from rarer medium to
denser medium .On the other hand ,the refracted ray bends away from normal as it
travel from denser medium to rarer medium. In the later case , there is a possibility to
occur total internal reflection provided, the angle of incidence is greater than critical
angle(θ c ). This can be understood as follows.
1. When i < θ c ,then the ray refracted is into the second medium as shown in
below fig1.
2. When i= θc, then the ray travels along the interface of two media as shown in fig2.
3. When i> θc then the ray totally reflects back into the same medium as shown in
fig3.
Suppose if i= θc then r =90o ,hence
n1sin θc = n2sin 90o
sin θc = n2/ n1 (since sin 90o=1)
θc=sin-1(n2/ n1)………………………….(2)
thus any ray whose angle of incidence is greater than the critical angle ,total internal
reflection occurs ,when a ray is traveling from a medium of high refractive index to
low refractive index.
Construction of optical fiber:
The optical fiber mainly consists of the following parts.
i .Core ii .Cladding iii .Silicon coating iv .Buffer jacket
v .Strength material vi . Outer jacket
􀂾 A typical glass fiber consists of a central core of thickness 50μm surrounded by
cladding.
􀂾 Cladding is made up of glass of slightly lower refractive index than core’s
refractive index, whose over all diameter is 125μm to 200μm.
Of course both core and cladding are made of same glass and to put refractive index
of cladding lower than the refractive index of core, some impurities like Boron,
Phosphorous or Germanium are doped.
􀂾 Silicon coating is provided between buffer jacket and cladding in order to
improve the quality of transmission of light.
􀂾 Buffer jacket over the optical fiber is made of plastic and it protects the fiber from
moisture and abrasion.
􀂾 In order to provide necessary toughness and tensile strength, a layer of strength
material is arranged surrounding the buffer jacket.
􀂾 Finally the fiber cable is covered by black polyurethane outer jacket. Because of
this arrangement fiber cable will not be damaged during hard pulling ,bending
,stretching or rolling, though the fiber is of brittle glass.
Acceptance angle and Numerical aperture of optical fiber:
When the light beam is launched into a fiber, the entire light may not pass
through the core and propagate. Only the rays which make the angle of incidence greater
than critical angle at the core –cladding interface undergoes total internal reflection. The
other rays are refracted to the cladding and is lost. Hence the angle we have to launch the
beam at its end is essential to enable the entire light to pass through the core .This
maximum angle of launch is called acceptance angle.
Consider an optical fiber of cross sectional view as shown in figure no, n1and n2 are
refractive indices of air ,core and cladding respectively such that n1>n2>no.let light ray is
incidenting on interface of air and core medium with an angle of incidence α .This
particular ray enters the core at the axis point A and proceeds after refraction at an angle
αr from the axis .It then undergoes total internal reflection at B on core at an internal
incidence angle θ.
To find α at A:-
In triangle ABC, α r =90- θ ……………………. (1)
From snell’ s law ,
n0 sin α = n1sinαr…………….(2)
sin α= n1/n0 sin αr …………………….(3)
From equations 1, 3
sin α= n1/n0 sin(90- θ ) => sin α= n1/n0 cos θ………(4)
If θ < θc, the ray will be the lost by refraction. Therefore limiting value for the beam to
be inside the core, by total internal reflection is θc. Let α (max) be the maximum possible
angle of incident at the fiber end face at A for which θ= θc. If for a ray α exceeds α (max),
then θ< θc and hence at B the ray will be refracted.
Hence equation 4 can be written as
sin α(max)= n1/n0 cos θc…………………..(5)
We know that
cos θc=√(1-sin2 θc) = √ (1- n2
2/n1
2)
=√ ( n1
2- n2
2)/ n1……………………(6)
Therefore
sin α(max) =√ ( n1
2- n2
2)/no
α(max)=sin-1√ ( n1
2- n2
2)/no
This maximum angle is called acceptance angle or acceptance cone angle. Rotating the
acceptance angle about the fiber axis gives the acceptance cone of the fiber. Light
launched at the fiber end within this acceptance cone alone will be accepted and
propagated to the other end of the fiber by total internal reflection. Larger acceptance
angles make launching easier.
Numerical aperture:-
Light collecting capacity of the fiber is expressed in terms of
acceptance angle using numerical aperture. Sine of the maximum acceptance angle is
called the numerical aperture of the fiber.
Numerical aperture =NA = Sin α (max) =√ ( n1
2- n2
2)/no…………….(7)
Let Δ = (n1
2- n2
2)/2n1
2…………… (8)
For most fiber n1 n2
Hence Δ = (n1+ n2) (n1-n2)/2n1
2 = 2n1 (n1-n2)/ 2n1
2
Δ = (n1-n2)/ n1 (fractional difference in refractive indices)……………(9)
From equation (8) n1
2- n2
2 = Δ 2n1
2
Taking under root on both sides
Hence =√ (n1
2- n2
2) = √2 Δ n1
Substituting this in equation (7) we get
NA ≈ √2 Δ n1/no ………(10)
For air no= 1, then the above equation can changed as
NA ≈ √2 Δ n1
Numerical aperture of the fiber is dependent only on refractive indices of
the core and cladding materials and is not a function of fiber dimensions.
Types or classification of optical fibers:
Optical fibers are classified as follows:
Depending upon the refractive index profile of the core, optical fibers are classified into
two categories
_______________
Step index Graded index
Depending upon the number of modes of propagation, optical fibers are classified into
two categories, they are
__________________
Single mode Multi mode
Based on the nature of the material used, optical fibers are classified into four categories.
􀂾 Glass fiber
􀂾 Plastic fiber
􀂾 Glass Core with plastic cladding
􀂾 PCS Fibers(Polymer-Clad Silica fiber)
Step index fibers:-
In step index fibers the refractive index of the core is uniform through out the medium
and undergoes an abrupt change at the interface of core and cladding.The diameter of the
core is about 50-200μm and in case of multi mode fiber.And 10 μm in the case of single
mode fiber.The transmitted optical signals travel through core medium in the form of
meridonal rays, which will cross the fiber axis during every reflection at the corecladding
interface. The shape of the propagation appears in a zig-zac manner.
Graded index fiber:-
In these fibers the refractive index of the core varies radially.As the radius increases in
the core medium the refractive index decreases. The diameter of the core is about
50μm.The transmitted optical signals travel through core medium in the form of helical
rays, which will not cross the fiber axis at any time.
Attenuations in optical fiber (or losses):
While transmitting the signals through optical fiber some energy is lost due to few
reasons. The major losses in fibers are 1.Distotrion losses 2.Transmission losses
3.Bending losses.
1. Distortion losses:-
When a pulse is launched at one end of the fiber and collected at the other end of the
fiber, the shape and size of the pulse should not be changed. Distortion of signals in
optical fiber is an undesired feature. If the out put pulse is not same as the input pulse,
then it is said that the pulse is distorted .If the refractive index of the core is not uniform
most of the rays will travel through the medium of lower refractive index region. Due to
this the rays which are travel in fiber will become broadened. Because of this the out put
pulses will no longer matches with the input pulses.
The distortion takes place due to the presence of imperfections, impurities
and doping concentrations in fiber crystals.Disperion is large in multi mode than in
single mode fiber.
2. Transmission losses (attenuation):-
The attenuation or transmission losses may be classified into two
categories i ) Absorption losses ii) scattering losses
i) Absorption losses:-
Absorption is a characteristic possessed by all materials every material in universe
absorb suitable wavelengths as they incident or passed through the material. In the
same way core material of the fiber absorbs wavelengths as the optical pulses or
wavelengths pass through it.
ii) Scattering losses:-
The core medium of the fiber is made of glass or silica .In
the passage of optical signals in the core medium if crystal defects are encountered,
they deviate from the path and total internal reflection is discontinued, hence such
signals will be destroyed by entering into the cladding however attenuation is
minimum in optical fibers compared to other cables.
ii) Bending losses:-
The distortion of the fiber from the ideal strait line
configuration may also result in fiber .Let us consider a way front that travels
perpendicular to the direction of propagation. In order to maintain this, the part of the
mode which is on the out side of the bend has to travel faster than that on the inside.
As per the theory each mode extends an infinite distance into the cladding though the
intensity falls exponentially. Since the refractive index of cladding is less than that of
the core (n1>n2), the part of the mode traveling in the cladding will attempt to travel
faster. As per Einistein’s theory of relativity since the part of the mode cannot travel
faster the energy associated with this particular part of the mode is lost by radiation.
Attenuation loss is generally measured in terms of decibels (dB),
which is a logarithmic unit.
Loss of optical power = -10 log (Pout/ Pin ) dB
Where Pout is the power emerging out of the fiber
Pin is the power launched into the fiber.
Optical fiber in communication systems as a wave guide:
An efficient optical fiber communication system requires high information carrying
capacity fast operating speed over long distances with minimum number of repeaters
.An optical fiber communication system mainly contains
1. Encoder
2. Transmitter
3. Wave guide
4. Receiver
5. Decoder.
Encoder:-
It is an electronic system that convert the analog information like voice,
figure, objects etc into binary data. This binary data contain a series of electrical
pulses.
Transmitter:-
It consist of two parts namely drive circuit and light source .Drive circuit
supplies electric signals to the light source from the encoder in the required form. In
the place of light source either an LED or A diode Laser can be used ,which converts
electric signals into optical signals. With the help of specially made connecter optical
signal will be injected into wave guide from the transmitter.
Wave guide:-
It is an optical fiber which carries information in the form of optical
signals over long distances with the help of repeaters. with the help of specially made
connector optical signal will be received by the receiver from the wave guide.
Receiver:-
It consists of three parts namely photo detector, amplifier and signal
restorer. The photo detector converts the optical signals into equivalent electric
signals and supply them to amplifier. The amplifier amplifies the electric signals as
they become weak during the journey trough the wave guide over a long distances.
The signal restorer keeps all the electric signals in sequential form and supplies to
decoder in suitable way.
Decoder:-
It convert the received electric signals into the analog information.
Advantages of optical fiber communication:
􀂾 Enormous Band width:-
In the coaxial cable transmission the band width is up to around
5000MHz only where as in fiber optical communication it is as large as 105 GHz.
Thus the information carrying capacity of optical fiber system is far superior to the
copper cable system.
􀂾 Electrical isolation:-
Since fiber optic material are insulators unlike there
metallic counterpart, they do not exhibit earth loop and interface problems. Hence
communication through fiber even in electrically hazarders environment do not cause
any fear of spark hazards.
􀂾 Immunity to interference and cross talk:-
Since optical fibers are dielectric waveguides, they are free
from any electromagnetic interference (EMI) and radio frequency interference (RIF).
Hence fiber cable do not require special shielding from EMI. Since optical
interference among different fiber is not possible ,unlike communication using
electrical conductors cross talk is negligible even when many fibers are cabled
together.
􀂾 Signal security:-
Unlike the situation with copper cables a transmitted optical
signal can not be drawn fiber with out tampering it. Such an attempt will affect the
original signal and hence can easily detected.
􀂾 Small size and weight:-
Since fibers are very small in diameter the space occupied the
fiber cable is negligibly small compare to metallic cables. Optical cables are light in
weight , these merits make them more useful in air crafts and satellites.
􀂾 Low transmission loss:-
Since the loss in fibers is as low as 0.2dB/Km , transmission loss
is very less compare to copper conductors. Hence for long distance communication,
fibers are preferred. Number of repeaters required is reduced.
􀂾 Low cost:-
Since fibers are made up of silica which is available in nature ,optical
fibers are less expensive.
Applications of optical fibers:
􀂾 Optical fibers are used as sensors
􀂾 These are used in Endoscopy
􀂾 These are used in communication system
􀂾 For decorative peaces in home needs.
􀂾 These are used in defence areas for the sake of high security.
􀂾 These are used in electrical engineering.
HOLOGRAPHY
Holography was invented in 1948 by Dennis Gabor. The word ‘holo’ means
entire, complete, full: ‘graphy’ means recording. Hence holography means recording of
the complete information of an object. Recording of the complete information of an
object: i.e., its amplitude and phase is called holography. It carries information of light
intensity distribution on the object only.
Differences between holography and photography:
Photography
Holography
1. Ordinary light
2. Amplitude recording
3. Two dimensional recording
4. Lens is used
5. Ordinary film
6. No need for vibration isolation table
7. No special optics
8. When cut into pieces information is
lost
Laser
Amplitude and phase recording
Three dimensional recording
Generally no lens is used
Very high resolution film
Vibration isolation table is required
Special optics
When cut into pieces, each bit carries full
information
Basic principle of holography:
Light from a coherent source such as laser is split
into two parts. One beam called object beam is used to illuminate the object to be
recorded kept infront of a photo sensitive recording material. Scattered light from the
object falls on the recording material. The other beam called reference beam is expanded
to falls on the recording material. At the recording plane, interference of the object beam
and reference beam takes place. This interference pattern is recorded in the photo
sensitive material as bright and dark fringes on developing and fixing the recorded
material. Such a recorded material is called hologram.
When such a hologram is illuminated with reconstructing beam which is identical to the
reference beam used at the time of recording, light passes through the fine fringes pattern
recorded resulting in diffraction. The diffracted beam is identical to the object wave
which recording material received at the time of recording. Hence when the diffracted
beam is viewed it forms virtual image of the object.
Recording (Construction) of a hologram:
The laser beam is split into two parts, one is called the object beam is used to illuminate
the object. The beam falling on the object after scattering reaches the recording material.
The other one is called reference beam after expansion reaches the recording material. At
the recording plane, interference of the object beam and the reference beam takes place.
On developing and processing we get ‘hologram’ which is nothing but the recorded
interference pattern, say, grating. Since the fringe spacing of the interference pattern is
less than microns, the grains in the photosensitive recording material must be much
smaller in size to record it. This makes the recoding film very in slow, hence slightly
longer exposure is required. Since the manual seismic vibrations of the earth is always of
the order of few microns, the recoding of fine fringes giving longer exposure to the
recording medium is not possible unless we isolate the systems used for recording from
the vibration of the earth. Hence vibration isolation table is required. This table is heavy
one floating on compressed air bed/pillars so that earth vibrations are arrested by this
arrangement.
Fig. Recording (Construction) of Hologram
Steps involved in recording a hologram:
1. Check that vibration isolation table is floated.
2. The laser beam has to be split into two, one to act as object beam and another to
act as reference beam.
3. Front coated mirrors and beam expanders are to be arranged as given in the
diagram. At the recording plane the scattered waves from the object and reference
waves are made to interfere.
4. Object beam and reference beam should travel almost equal distances. The path
difference should not exceed the coherent length of the source.
5. All the components are to be rigidly fixed to the table.
6. In total darkness the recording material is loaded such that emulsion side faces the
object.
7. Without disturbing the table, the laser has to be switched on and the exposure
given.
8. The exposed plate is to be removed and processed in the developer and fixer.
Then it has to be washed well and dried. The dried plate is called hologram.
Reconstruction of a hologram:
The beam splitter is removed and the reference beam is used for reconstruction.
The hologram diffracts the light and the diffracted light received forms virtual image. It is
identical to the object and hence it appears as if the object is really present. By moving
the head side wise the observer can get three dimensional effects. Since while recording,
the secondary waves from each and every point on the object participates in this process
to regenerate both a real and virtual image of the object point by point. By seeing through
the hologram from the transmission side, it appears as though the original object is lying
on the other side of it at the same place. This is the virtual image due to regeneration. By
switching the direction of view, different set of points which correspond to constructive
interference are observed, which regenerate a different perspective of the object. Thus the
sense of depth is felt.
Fig. Reconstruction of a hologram
The other wave stream starting from the various zone plates is convergent as though
coming from a convex lens and gives rise to the real image of the object. The image can
be photographed by keeping a photographic plate in the plane of the image formation of
the convergent beam.
Application:
1. Holographic Interferometry: This method is used to study minute distortions of
an object that take place such as due to stress or vibration. When a defective is
deformed it deforms more at the defective region. This can be holographically
recorded. First the object is recorded giving an exposure to the recording material.
Then the object is stressed and second exposure is given on the same recording
material. After this double exposure, the recording material is processed. When
reconstructed, two image waves are formed one corresponding to the normal
position of the object and another corresponding to the deformed position of the
object. These two image waves interfere to produce interference pattern on the
object. These fringes are contours of locations of same displacement. Hence when
recorded holographicallly by double exposure method and re constructed, one can
observe variation in fringe patter at defective region. The fringe pattern at the
defect point will exhibit discontinuity if there is a crack on the object. Crowding
of fringes can be seen if there is a weak point which has deformed more when
compared with normal region. Thus double exposure holographic interferometry
is very useful in non destructive study of objects.
2. Holographic Diffraction Gratings: The holographic gratings are obtained by
interference of two laser beams each one with plane wavefronts, on a hologram.
This method produces the rulings much more uniformly than the ones made by a
ruling engine on a conventional grating.
3. Acoustical Holography: Using some intricate techniques, procedures have
been formulated for hologram recording with coherent ultrasonic waves. The
hologram could be made for scanning of the human body with ultrasound. When
the image is reconstructed using laser light, the hologram can provide three
dimensional view of internal structures of the human body.
4. Thick Holograms: Thick holograms are the ones in which the interference
pattern is recorded in 3- dimension. Such holograms are in use on credit cards in
which improvement of security is the criteria.
5. Infornation Coding: In the process of recording a hologram, the reference
wave is passed through an encoding mask which generates a special wavefront.
Then in order to get back the information recorded, the hologram has to be
illuminated with the same kind of wavefront only. Since the masking details are
kept secret. The information in the hologram is also guarded.
Questions:
1. Explain the principle behind the functioning of an optical fiber.
2. With the help of a suitable diagram explain the principle construction and
working of optical fiber.
3. Derive an expression for acceptance angle.
4. Explain numerical aperture and derive an expression for it.
5. Write short note on i). Step index fiber, ii) Graded index fiber.
6. Describe the structure of different types of Optical fibers with ray paths.
7. What are the different losses (attenuation) in an optical fiber? Write brief note on
each.
8. Explain the advantages of optical fiber in communication.
9. Explain the basic principle of holography.
UNIT – VIII
SCIENCE AND TECHNOLOGY OF NANOMATERIALS
Introduction
In 1959, Richard Feynman made a statement ‘there is plenty of room at the bottom’.
Based on his study he manipulated smaller units of matter. He prophesied that “we can
arrange the atoms the way we want, the very atoms, all the way down”. The term
‘nanotechnology’ was coined by Norio Taniguchi at the University of Tokyo. Nano
means 10-9. A nano metre is one thousand millionth of a metre (i.e. 10-9 m).
Nanomaterials could be defined as those materials which have structured components
with size less than 100nm at least in one dimension. Any bulk material we take, its size
can express in 3-dimensions. Any planer material, its area can be expressed in 2-
dimension. Any linear material, its length can be expressed in 1-dimension.
Materials that are nano scale in 1-dimension or layers, such as thin films or surface
coatings.
Materials that are nano scale in 2-dimensions include nanowires and nanotubes.
Materials that are nano scale in 3- dimensions are particles like precipitates, colloids and
quantum dots.
Nanoscience: it can be defined as the study of phenomena and manipulation of
materials at atomic, molecular and macromolecular scales, where properties differ
significantly from those at a larger scale.
Nanotechnology: It can be defined as the design, characterization, production and
application of structures, devices and systems by controlling shape and size at the nano
metre scale. It is also defined as “A branch of engineering that deals with the design and
manufacture of extremely small electronic circuits and mechanical devices built at
molecular level of matter. Now nanotechnology crosses and unites academic fields such
as Physics, Chemistry and Computer science.
Properties of nano particles:
The properties of nano scale materials are very much different from
those at a larger scale. Two principal factors that cause that the properties to differ
significantly are increased relative surface area and quantum effects. These can enhance
or change properties such as reactivity, strength and electrical characteristics.
1. Increase in surface area to volume ratio
Nano materials have relatively larger surface area when compared to the volume of
the bulk material.
Consider a sphere of radius r
Its surface area =4Πr2
Its volume = 4Πr3 /3
Surface area = 4Πr2 = 3
Volume 4Πr3 /3 r
Thus when the radius of sphere decreases, its surface area to volume ratio increases.
EX: For a cubic volume,
Surface area = 6x1m2 =6m2
When it is divided it 8 pieces
It surface area = 6x (1/2m) 2x8=12m2
When the same volume is divided into 27 pieces,
It surface area = 6x (1/3m) 2x27=18m2
Therefore, when the given volume is divided into smaller pieces, the surface area
increases. Hence as particle size decreases, greater proportions of atoms are found at the
surface compared to those inside. Thus nano particles have much greater surface to
volume ratio. It makes material more chemically reactive.
As growth and catalytic chemical reaction occur at surfaces, then given mass of material
in nano particulate form will be much more reactive than the same mass of bulk material.
This affects there strength or electrical properties.
2. Quantum confinement effects
When atoms are isolated, energy levels are discrete or discontinuous. When
very large number of atoms is closely packed to form a solid, the energy levels split and
form bands. Nan materials represent intermediate stage.
When dimensions of potential well and potential box are of the order of deBroglie wave
length of electrons or mean free path of electrons, then energy levels of electrons
changes. This effect is called Quantum confinement effect.
When the material is in sufficiently small size, organization of energy levels into
which electrons can climb of or fall changes. Specifically, the phenomenon results
from electrons and holes being squeezed into a dimension that approaches a critical
quantum measurement called the exciton Bohr radius. These can affect the optical,
electrical and magnetic behaviour of materials.
Variations of properties of nano materials
The physical, electronic, magnetic and chemical properties of materials depend on
size. Small particles behave differently from those of individual atoms or bulk.
Physical properties: The effect of reducing the bulk into particle size is to create
more surface sites i.e. to increase the surface to volume ratio. This changes the surface
pressure and results in a change in the inter particle spacing. Thus the inter atomic
spacing decreases with size.
The change in the inter particle spacing and the large surface to volume ratio in
particle have a combined effect on material properties. Variation in the surface free
energy changes the chemical potential. This affects the thermodynamic properties like
melting point. The melting point decreases with size and at very small sizes the decrease
is faster.
Chemical properties: the large surface to volume ratio, the variations in geometry
and electronic structure has a strong effect on catalytic properties. The reactivity of small
clusters increases rapidly even when the magnitude of the cluster size is changed only by
a few atoms.
Another important application is hydrogen storage in metals. Most metals do not absorb,
hydrogen is typically absorbed dissociatively on surfaces with hydrogen- to- metal atom
ratio of one. This limit is significantly enhanced in small sizes. The small positively
charged clusters of Ni, Pd and Pt and containing between 2 and 60 atoms decreases with
increasing cluster size. This shows that small particles may be very useful in hydrogen
storage devices in metals.
Electrical properties: The ionization potential at small sizes is higher than that for
the bulk and show marked fluctuations as function of size. Due to quantum confinement
the electronic bands in metals become narrower. The delocalized electronic states are
transformed to more localized molecular bands and these bands can be altered by the
passage of current through these materials or by the application of an electric field.
In nano ceramics and magnetic nano composites the electrical conductivity
increases with reduction in particle size where as in metals, electrical conductivity
decreases.
Optical properties: Depending on the particle size, different colours are same.
Gold nano spheres of 100nm appear orange in colour while 50nm nano spheres appear
green in colour. If semiconductor particles are made small enough, quantum effects come
into play, which limits the energies at which electrons and holes can exist in the particles.
As energy is related to wavelength or colour, the optical properties of the particles can be
finely tuned depending on its size. Thus particles can be made to emit or absorb specific
wavelength of light, merely by controlling their size.
An electro chromic device consist of materials in which an optical absorption band
can be introduced or existing band can be altered by the passage of current through the
materials, or by the application of an electric field. They are similar to liquid crystal
displays (LCD) commonly used in calculator and watches. The resolution, brightness and
contrast of these devices depend on tungstic acid gel’s grain size.
Magnetic properties: The strength of a magnet is measured in terms of coercivity
and saturation magnetization values. These values increase with a decrease in the grain
size and an increase in the specific surface area (surface area per unit volume) of the
grains.
In small particle a large number or fraction of the atoms reside at the surface. These
atoms have lower coordination number than the interior atoms. As the coordination
number decreases, the moment increases towards the atomic value there is small particles
are more magnetic than the bulk material.
Nano particle of even non magnetic solids are found to be magnetic. It has been found
theoretically and experimentally that the magnetism special to small sizes and disappears
in clusters. At small sizes, the clusters become spontaneously magnetic.
Mechanical properties: If the grains are nano scale in size, the interface area with in
the material greatly increases, which enhances its strength. Because of the nano size
many mechanical properties like hardness, elastic modulus, fracture toughness, scratch
resistance, fatigue strength are modified.
The presence of extrinsic defects such as pores and cracks may be responsible for low
values of E (young’s modulus) in nano crystalline materials. The intrinsic elastic modulli
of nano structured materials are essentially the same as those for conventional grain size
material until the grain size becomes very small. At lower grain size, the no. of atoms
associated with the grain boundaries and triple junctions become very large. The
hardness, strength and deformation behaviour of nano crystalline materials is unique and
not yet well understood.
Super plasticity is the capability of some polycrystalline materials to exhibit very
large texture deformations without fracture. Super plasticity has been observed occurs at
somewhat low temperatures and at higher strain rates in nano crystalline material.
PRODUCTION OF NANOMATETIALS:
Material can be produced that are nanoscale in one dimension like
thin surface coatings in two dimensions like nanowires and nanotubes or in 3 dimensions
like nanoparticles
Nano materials can be synthesized by’ top down’ techniques producing very small
structures from larger pieces of material. One way is to mechanical crushing of solid into
fine nano powder by ball milling.
Nanomaterials may also be synthesized by ‘bottom up’ techniques, atom by atom or
molecule by molecule. One way of doing this is to allow the atoms or molecules arranges
themselves into a structure due to their natural properties
Ex: - Crystals growth
PERPARATION:
There are many methods to produce nanomaterials. They are
1). PLASMA ARCING:
Plasma is an ionized gas. To produce plasma, potential difference is
applied across two electrodes. The gas yields up its electrons and gets ionized .Ionized
gas (plasma) conducts electricity. A plasma arcing device consists of two electrodes. An
arc passes from one electrode to the other. From the anode electrode due to the potential
difference electrons are emitted. Positively charged ions pass to the other electrode
(cathode), pick up the electron and are deposited to form nanoparticles. As a surface
deposit the depth of the coating must be only a few atoms. Each particle must be
nanosized and independent. The interaction among them must be by hydrogen bonding or
Vander Waals forces. Plasma arcing is used to produced carbon nanotubes.
2). CHEMICAL VAPOUR DEPOSITION:
In this method, nanoparticles are deposited from the gas phase.
Material is heated to from a gas and then allowed to deposit on solid surface, usually
under vacuum condition. The deposition may be either physical or chemical. In
deposition by chemical reaction new product is formed. Nanopowder or oxides and
carbides of metals can be formed, if vapours of carbon or oxygen are present with the
metal.
Production of pure metal powders is also possible using this method.
The metal is meted exciting with microwave frequency and vapourised to produce
plasma at 15000c . This plasma then enters the reaction column cooled by water where
nanosized particles are formed.
CVD can also be used to grow surfaces. If the object to be coated is
introduced inside the chemical vapour, the atoms/molecules coated may react with the
substrate atoms/molecules. The way the atoms /molecules grow on the surface of the
substrate depends on the alignment of the atoms /molecules of the substrate. Surfaces
with unique characteristics can be grown with these techniques.
3. Sol – Gels:
Sol: - A material which when reacts with liquid converts in to a gelly or viscous fluid.
Colloid:- A substance which converts liquid to semisolid or viscous or cloudy.
Gel : Amore thicky substance.
Soot :- When a compound is brunt, it given black fumes called soot.
In solutions molecules of nanometer size are dispersed and move
around randomly and hence the solutions are clear. In colloids, the molecules of size
ranging from 20μm to100μm are suspended in a solvent. When mixed with a liquid
colloids look cloudy or even milky. A colloid that is suspended in a liquid is called as sol.
A suspension that keeps its shape is called a gel. Thus sol-gels are suspensions of colloids
in liquids that keep their shape. Sol -gels formation occurs in different stages.
Hydrolysis
Condensation and polymerization of monomers to form particles
Agglomeration of particles. This is followed by formation of networks which extends
throughout the liquid medium and forms a gel.
The rate of hydrolysis and condensation reactions are governed by various factors
such as PH, temperature, H2O/Si molar ratio, nature and concentration of catalyst and
process of drying. Under proper conditions spherical nanoparticles are produced.
3. ELECTRODEPOSITION:
This method is used to electroplate a material. In many liquids called
electrolytes (aqueous solutions of salts , acids etc) when current is passed through
two electrodes immersed inside the electrolyte, certain mass of the substance liberated
at one electrode gets deposited on the surface of the other. By controlling the current
and other parameters, it is possible to deposit even a single layer of atoms. The films
thus obtained are mechanically robust, highly flat and uniform. These films have very
wide range of application like in batteries, fuel cells, solar cells, magnetic read heads
etc.
5. BALL MILLING (MECHANICAL CRUSHING):
In this method, small balls are allowed to rotate around the inside
of a drum and then fall on a solid with gravity force and crush the solid into
nanocrystallites. Ball milling can be used to prepare a wide range of elemental and
oxide powders. Ball milling is the preferred method for preparing metal oxides.
Materials referred to as nano materials fall under two categories:
Fullerenes :they are a class of allotropes of carbon which conceptually a re graphene
sheets rolled into tubes or spheres .A common method used to produce fullerness is
to send a large current between two near by graphite electrodes in an inert
atmosphere .The resulting carbon plasma arc between the electrodes cools into
sooty residue from which many fullerenes can be isolated .
Eg : carbon nano tubes, buckyball(buckminsterfullerene C-60),cnt…,
Bucky ball: C-60 molecules &buckminister fullerene made up of 60 carbon atoms
arranged in a series of inter locking hexagon(20) and pentagons(12) forming a
structure that looks like soccer ball c60 actually is a truncated icosahedron.
CARBON NANOTUBES (CNT’S):
We know three forms of carbon namely diamond graphite and amorphous
carbon. There is a whole family of other forms of carbon known as carbon nanotubes,
which are related to graphite. In conventional graphite, the sheets of carbon are
stacked on top of one another .They can easily slide over each other. That’s why
graphite is not hard and can be used as a lubricant. When graphite sheets are rolled
into a cylinder and their edges joined, they form carbon nanotubes i.e. carbon
nanotubes are extended tubes of rolled graphite sheets.
TYPES OF CNT’S: A nanotube may consist of one tube of graphite, a one atom
thick single wall nanotube or number of concentric tubes called multiwalled
nanotubes.
There are different types of CNT’S because the graphite sheets can be rolled in
different ways .The 3 types of CNT’S are ZigZag, Armchair and chiral. It is possible
to recognize type by analyzing their cross sectional structures.
Multiwalled nanotubes come in even more complex array of forms.
Each concentric single – walled nanotube can have different structures, and hence
there are a variety of sequential arrangements. There can have either regular layering
or random layering .The structure of the nanotubes influences its properties .Both
type and diameter are important .The wider the diameter of the nanotube, the more it
behaves like graphite. The narrower the diameters of nanotube, the more its intrinsic
properties depends upon its specific type. Nanotubes are mechanically very strong,
flexible and can conduct electricity extremely well.
The helicity of the graphite sheet determines whether the CNT is a
semiconductor of metallic.
PRODUCTION OF CNT’S: There are a number of methods of making CNT’S
few method adopted for the production of CNT’S.
ARC MRTHOD: This method creates CNT’S through arc- vapourisation of two
carbon rods placed end to end, separated by 1mm , in an enclosure filled with inert
gas at low pressure .It is also possible to create CNT’S with arc method in liquid
nitrogen. A direct current of 50-100A, driven by a potential difference of 20V apprx,
creates a high temperature discharge between the two electrodes .The discharges
vapourizes the surface of one of the carbon electrodes, and forms a small rod shaped
deposit on the other electrode. Producing CNT’S in high yield depends on the
uniformity of the plasma arc, and the temperatures of deposits forming on the carbon
electrode.
LASER METHOD: CNT’S were first synthesized using a dual-pulsed laser.
Samples were prepared by laser vapourizations of graphite rods with a 50:50 catalyst
mixture of Cobalt & Nickel at 12000c in flowing argon. The initial layer
vapourization pulse was followed by a second pulse, to vapourize the target more
uniformly. The use of two successive laser pulses minimizes the amount of carbon
deposited as soot. The second laser pulse breaks up the larger particles ablated by the
first one and feeds then into growing nanotube structure. The CNT’S produced by
this method are 10-20nm in diameter and upto 100m or more in length. By varying
the growth temperatures, the catalyst composition and other process parameters the
average nanotube diameter and size distribution can be varied.
CHEMICAL VAPOUR DEPOSITION (CVD): Large amount of CNT”S can be
formed by catalytic CVD of acetylene over Cobalt and Iron catalysts supported on
silica or zeolite. The carbon deposition activity seems to relate to the cobalt content of
the catalyst; where as the CNT’S selectivity seems to be a function of the PH in
catalyst preparation. CNT’S can be formed from ethylene. Supported catalysts such
as iron cobalt and Nickel containing either a single metal or a mixture of metals, seem
to induce the growth of isolated single walled nanotubes or single walled nanotubes,
bundles in the ethylene atmosphere. The production of single walled nanotubes as
well as double walled CNT’S, molybdenum and molylodenum-iron catalysts has also
been demonstrated.
PROPERTIES OF CNT;S : Few unique properties of CNT”S are
1) ELECTRICAL CONDUCTIVITY: CNT’S can be highly conducting , and
hence can be said to be metallic. Their conductivity will be a function of chirality, the
degree of twist and diameter. CNT’S can be either metallic or semi conducting in
their electrical behaviour. Conductivity in multi walled CNT’S is more complex .The
resistivity of single walled nanotubes ropes is of the order of 10-4 ohm –cm at 270c
.This means that single walled nanotube ropes are most conductive carbon fibers.
Individual single walled nanotubes may contain defects. These defects allow the
single walled nanotubes to act as transistors. Similarly by joining CNT”S together
forms transistor - like devices. A nanotube with natural junctions behaves as a
rectifying diode.
2) Strength and elasticity: Because of the strong carbon bonds, the basal plane
elastic modules of graphite, it is one of the largest of any known material. For this
reason, CNT”S are the ultimate high strength fibers. Single walled nanotubes are
stiffer than Steel, and are very resistant to damage from physical forces.
3)THERMAL CONDUCTIVITY AND EXPANSION: The strong in- plane
graphite carbon- carbon bonds make them exceptionally strong and stiff against axial
strains. The almost zero- in -plane thermal expansion but large inter - plane expansion
of single walled nanotubes implies strong in plane coupling and high flexibility
against non- axial strains. CNT’S show very high thermal conductivity. The nanotube
reinforcements in polymeric materials may also significantly improve the thermal and
thermo mechanical properties of composites.
4) HIGHLY ABSORBENT: The large surface area and high absorbency of
CNT”S make them ideal for use in air, gas and water filteration. A lot of research is
being done in replacing activated charcoal with CNT’S in certain ultra high purity
application.
APPLICATION OF NANOMATERIALS:
Engineering: i).Wear protection for tools and machines (anti blocking coatings,
scratch resistant coatings on plastic parts). ii) Lubricant – free bearings.
Electronic industry: Data memory(MRAM,GMR-HD), Displays(OLED,FED),
Laser diodes, Glass fibres
Automotive industry: Light weight construction, Painting (fillers, base coat, clear
coat), Sensors, Coating for wind screen and car bodies.
Construction: Construction materials, Thermal insulation, Flame retardants.
Chemical industry: Fillers for painting systems, Coating systems based on nano
composites. Impregnation of papers, Magnetic Fluids.
Medicine: Drug delivery systems, Agents in cancer therapy, Anti microbial agents
and coatings, Medical rapid tests Active agents.
Energy: Fuel cells, Solar cells, batteries, Capacitors.
Cosmetics: Sun protection, Skin creams, Tooth paste, Lipsticks.
Questions:
What are Nanomaterials? Why do they exhibit different properties?
How are optical, physical and chemical properties of nano particles vary with their
size.
How are electrical, magnetic and mechanical properties of nano particles vary with
their size?
How are nano materials produced?
What are carbon nano tubes? How are they produced?
What are the different types of carbon nano tubes? What are their properties?
What are the important applications of nano materials?ENGINEERING PHYSICS
UNIT-I: BONDING IN SOLIDS
INTRODUCTION
.Matter can exist three ways such as 1).solid state2).liquid
State 3).gaseous state
.1)In gases, the atoms or molecules are free whereas in solids they are bound in a
particular form because of which, they possess different properties such as physical,
electrical, mechanical, chemical and optical properties
2) But in solid state, the constituent atoms or molecules that build the solid are confined
to a localized region due attractive force between atoms
BONDING
• 1 Bonding is the physical state of existence of two or more atoms together in a
bound form by forces of attraction
• 2 .The attractive force which hold the constitute of particles of a substance
together are called bonds.
• . 3. The supply of external energy is required to move an atom completely
from its equilibrium position or to break the bonds. This energy is called
dissociation (binding) or cohesive energy.
• Bonding occurs between similar or dissimilar atoms, when an electrostatic
interaction between them produces a resultant state whose energy is lesser than
the sum of the energies possessed by individual atoms when they are free.
TYPES OF BONDING IN SOLIDS
Bonds in solids are classified basically into two groups
1) Namely primary and secondary bonds. Primary bonds are inter atomic bonds i.e.
bonding between the atoms and secondary bonds are intermolecular bonds i.e.
between the molecules.
Primary bonds
1 The primary bonds are inter atomic bonds.(strong bonds)
2. DEF . The electrostatic forces hold the atoms together in solids known as
primary bond
3 Inter atomic distance between these bonds are ranges from 1 to 2A
(or)0.1 to 10eV/ bond(bond energy)
4. In this bonding interaction occurs only through the electrons in the
outermost orbit, i.e. the valence electrons. These are further classified into
three types
1. Ionic bonding
2. Covalent bonding
3. Metallic bonding
1. Ionic Bonding(Hetropolar bond).
DEF: Ionic bonding results due to transfer of electrons from an electropositive
elements(1st&2ndGROUP) to an electronegative elements(.6th&7thGROUP)
Example:
1) In Na Cl crystal, Na atom has only one electron in outer most shell and a Cl atom
needs one more electron to attain inert gas configuration.
2) During the formation of NaCl molecule, one electron from the Na atom is transferred
to the Cl atom as a result both Na and Cl ions attain filled- shell configuration.
Na + Cl 􀃆 Na+ + Cl
- 􀃆 NaCl
A strong electrostatic attraction is set up that bond the Na+ cation and the Cl
-
anion into a very stable molecule NaCl at the equilibrium spacing.
Since Cl exist as molecules, the chemical reaction must be written as
2Na + Cl2 􀃆 2Na+ + 2Cl
- 􀃆 2NaCl
Other examples of ionic crystals are
2Mg + O2 􀃆 2Mg++ + 2O
-- 􀃆 2MgO
Mg + Cl2 􀃆 Mg++ + 2Cl
- 􀃆 MgCl2
Properties:
1. As the ionic bonds are strong, the materials are hard and possess high melting
and boiling points.
2. They are good ionic conductors, but poor conductors of both heat and
electricity
3. They are transparent over wide range of electromagnetic spectrum
4. They are brittle. They possess neither ductility (ability to be made into sheets)
nor malleability (ability to be made into wires).
5. They are soluble in polar liquids such as water but not in non-polar liquids
such as ether.
2. Covalent Bonding(homopolar bond)
DEF : Covalent bond is formed by sharing of electrons between two atoms to form
molecule.
Example: 1) Covalent bonding is found in the H2 molecule.
2) Here the outer shell of each atom possesses 1 electron. Each H atom would like
to gain an electron, and thus form a stable configuration.
3) This can be done by sharing 2 electrons between pairs of H atoms, there by
producing stable diatomic molecules.
Thus covalent bonding is also known as shared electron pair bonding.
Other examples for covalant crystals:
Properties:
1. Covalent crystals are very hard since the bond is strong.
2. The best example is diamond which is the hardest naturally occurring
material and possess high melting and boiling points, but generally lower
than that for ionic crystals.
3. Their conductivity falls in the range between insulators and
semiconductors. For example, Si and Ge are semiconductors, where as
diamond as an insulator
.
4. They are transparent to electromagnetic waves in infrared region, but opaque
at shorter wavelengths.
5. They are brittle and hard.
6. They are not soluble in polar liquids, but they dissolve in non-polar liquids such
as ether, acetone, benzene etc.
7. The bonding is highly directional.
3. Metallic Bonding:
1)The valance electrons from all the atoms belonging to the crystal are free to move
throughout the crystal.
2) The crystal may be considered as an array of positive metal ions embedded in a
cloud of free electrons. This type of bonding is called metallic bonding.
3) In a solid even a tiny portion of it comprises of billions of atoms. Thus in a
metallic body, the no. of electrons that move freely will be so large that it is
considered as though there is an electron gas contained with in the metal.
4) The atoms may embedded in this gas but having lost the valence electrons, they
become positive ions.
5) The electrostatic interaction between these positive ions and the electron gas as a
whole is responsible for the metallic bonding.
Properties:
1Compared to ionic and covalent bonds, the metallic bonds are weaker. Their
melting and boiling points are also lower.
2) Because of the easy movement possible to them, the electrons can transport energy
efficiently. Hence all metals are excellent conductors of heat and electricity.
3)They are good reflectors and are opaque to E.M radiation.
4) They are ductile and malleable.
5) They exists in solid form only
6) They are neither soluble in water nor in benzene
Secondary Bonds
There are two types of secondary bonds. They are Vander Waal’s bonds and
Hydrogen bonds.
2. secondary bonding energy’s in the range 0.01-0.5eV/bond
Vander Waal’s bonding: Vander Waal’s bonding is due to Vander Waal’s forces.
These forces exist over a very short range. The force decreases as the 4th power of
the distance of separation between the constituent atoms or molecules when the
ambient temperature is low enough. These forces lead to condensation of gaseous
to liquid state and even from liquid to solid state though no other bonding
mechanism exists.( except He)
Fig
Properties:
The bonding is weak because of which they have low melting points.
They are insulators and transparent for visible and UV light.
They are brittle.
They are non-directional
Hydrogen bonding: Covalently bonded atoms often produce an electric dipole
configuration. With hydrogen atom as the positive end of the dipole if bonds arise
as a result of electrostatic attraction between atoms, it is known as hydrogen
bonding.
FIG………………………..
Properties:
1. The bonding is weak because of which they have low melting points.
2. They are insulators and transparent for visible and UV light.
3. They are brittle.
4. The hydrogen bonds are directional.
Forces between atoms:
In solid materials, the forces between the atoms are of two kinds.
1) Attractive force 2) Repulsive force
IMPORTANCE: To keep the atoms together in solids, these forces play an important
role.
1).When the atoms are infinitely far apart they do not interact with each other to form a
solid and the potential energy will be zero.
2)From this, it can be understood that the potential energy between two atoms is
inversely proportional to some power of the distance of separation.
3). The atoms attract each other when they come close to each other due to inter-atomic
attractive force which is responsible for bond formation.
.Suppose two atoms A and B experiences attractive and repulsive forces on each other,
and then the interatomic or bonding force ‘F(r)’ between them may be represented as
F(r) =A / rM – B / rN (N > M) -------------- (1)
Where ‘r ‘is the inter atomic distance
A, B, M, N are constants.
M=2,N=7 to 10 for metallic bonds,10 to 12 for ionic, covalent bonds
In eqn-1, the first term represents attractive force and the second the repulsive force.
At larger separation, the attractive force predominates. The two atoms approach until they
reach equilibrium spacing. If they continue to approach further, the repulsive force pre
dominates, tending to push them back to their equilibrium spacing.
Fig. Variaration of interatomic force with interatomic spacing
To calculate equilibrium spacing r0:
The general expression for bonding force between two atoms is
F(r)=A / rM – B / rN
At equilibrium spacing: r = r0, F(r) = 0 because both forces should be equal
Hence A / r0
M = B / r0
N
i.e. (r0)
N-M
= B / A
Or r0 = (B / A) 1/ (N-M)
Cohesive energy:
Def;
The amount of energy required to separate the atoms completely
from the structure is called cohesive energy. this energy also called energy of
dissociation. Or
The amount of energy evolved or released from crystalline solid when two
atoms are formed bond formation from infinite distance
Since this is the energy required to dissociate the atoms, this is also called the energy of
dissociation.
The potential energy or stored internal energy of a material is the sum of the individual
energies of the atoms plus their interaction energies.
Consider the atoms are in the ground state and are infinitely far apart. Hence they do not
interact with each other to form a solid. The potential energy, which is inversely
proportional to some power of the distance of separation, is nearly zero. The potential
energy varies greatly with inter-atomic separation. It is obtained by integrating the eqn –
(1)
U(r) = ∫F(r) dr
= ∫ [A/rM - B/rN] dr
= [(A/1-M) x r1-M – (B/1-N) x r1-N] + c
= [-(A/M-1) r – (M-1) + (B/N-1) r–(N-1) + c
= -a / rm + b / r n + c where a = A/M-1, b = B/N-1, m = M-1, n = N-1
At r = α , U(r) = 0, then c = 0
Therefore U(r) = -a / rm + b / r n
The condition under which the particles form a stable lattice is that the function U(r)
exhibits min. for a finite value of r i.e. r = r0 this spacing r0 is known as equilibrium
spacing of the system. This min. energy Umin at r = r0 is negative and hence the energy
needed to dissociate the molecule then equals the positive quantity of ( - U min ). U min
occurs only if m and n satisfy the condition n>m
When the system in equilibrium then r = r0 and U(r) = Umin
[du/ dr] r = ro = 0
=d / dr [-a / ro
m + b / ro
n ] = 0
or 0 = [a m ro–m-1] – [b n ro–n-1]
or 0 = [a m / ro
m+1 ] – [ b n / ro
n+1] -------􀃆 ( 2)
Solving for ro
ro=[( b / a ) ( n / m )]1/n-m
or ro n = ro
m[( b /a ) (n / m)]
at the same time , n>m to prove this,
[ d 2U / dr2 ]r = ro = - [a m( m+1) / ro
m+2 ] + [ b n(n+1) / ro
n+2] > 0
[ro
m+2 b n(n+1)] - [ro
n+2a m( m+1) ] > 0
ro
m b n (n+1) > ro
n a m ( m+1)
b n (n+1) > a m ( m+1) ro
n-m
b n (n+1) > a m ( m+1) ( b / a ) ( n / m )
i.e. n > m
Calculation of cohesive energy:
The energy corresponding to the equilibrium position r = r0 , denoted by U(r0 ) is
called boning energy or cohesive energy of the molecule.
Substituting ‘ro
n’ in expression for U min,
We get
U(min) = - a / ro
m + b / ro
n
= - a / ro m+ b (a / b) ( m/ n )1 / ro
m
= - a / ro
m + ( m/ n) ( a / ro
m)
= - a / ro m[1-m / n]
Umin = - a / ro
m [1-m / n]
Thus the min. value of energy of U min is negative. The positive quantity │U min│ is the
dissociation energy of the molecule, i.e. the energy required to separate the two atoms.
Calculation of cohesive energy of NaCl Crystal
Let Na and Cl atoms be free at infinite distance of separation. The energy required
removing the outer electron from Na atom (ionization energy of Na atom), leaving it a
Na+ ion is 5.1eV.
i.e. Na + 5.1eV 􀃆 Na+ + e-
The electron affinity of Cl is 3.6eV. Thus when the removed electron from Na atom is
added to Cl atom, 3.6eV of energy is released and the Cl atom becomes negatively
charged.
Hence Cl + e- 􀃆 Cl- +3.6eV
Net energy = 5.1 – 3.6 = 1.5 eV is spent in creating Na+ and Cl- ions at infinity.
Thus Na + Cl + 1.5 eV 􀃆 Na+ + Cl-
At equilibrium spacing r0= 0.24nm, the potential energy will be min. and the energy
released in the formation of NaCl molecule is called bond energy of the molecule and is
obtained as follows:
V = -e2 / 4Πε0 r0
= - [ (1.602x10-19)2 / 4Π(8.85x10-12 )( 2.4x10-10 ) ] joules
= - [ (1.602x10-19)2 / 4Π(8.85x10-22 x2.4 )( 1.602x10-19 ) ] eV
= -6 eV
Thus the energy released in the formation of NaCl molecule is ( 5.1 - 3.6 – 6 ) = - 4.5 eV
To dissociate Na Cl molecule into Na+ and Cl- ions, it requires energy of 4.5 eV.
CRYSTAL STRUCTURES
CRYSTALLOGRAPHY
The branch of science which deals with the study of geometric form and
other physical properties of the crystalline solids by using X-rays, electron beam, and
neutron beams etc is called crystallography or crystal physics.
The solids are classified into two types 1)crystalline and2) amorphous. A
substance is said to be crystalline, when the arrangement of atoms, molecules or ions
inside it is regular and periodic. Ex. NaCl, Quartz crystal. Though two crystals of same
substance may look different in external appearance, the angles between the
2)corresponding faces are always the same. In amorphous solids, there is no particular
order in the arrangement of their constituent particles. Ex. Glass.
CRYSTALLINE SOLIDS AMORPHOUS SOLIDS
1. Crystalline solids have regular periodic 1. Amorphous solids have no
Arrangement of particles (atoms, ions, regularity in the arrangement
Or molecules). Of particles.
2. They are un-isotropic i.e., they differ in 2. They are usually isotropic i.e.,
Properties with direction. They possess same properties in
different directions.
3. They have well defined melting and 3. They do not posses well defined
Freezing points. Melting and freezing points.
.
4. Crystalline solids may be made up of 4. Most important amorphous
Materials are metallic crystals or glasses, plastics and rubber.
Non-metallic crystals. Some of the
Metallic crystals are Copper, silver,
Aluminum, tungsten , and manganese.
Non-metallic crystals are crystalline
Carbon, crystallized polymers or plastics.
5. Metallic crystals have wide use in 5. An amorphous structure does not
engineering because of their favorable generally posses elasticity but only
Lattice points: They are the imaginary points in space about which the atoms are
located.
Lattice: The regular repetition of atomic, ionic or molecular units in 2-dimensional, 3-
dimensional space is called lattice.
Space lattice or Crystal lattice: The totality of all the lattice point in space is called
space lattice, the environment about any two points is same or An array of points in
space such that the environment about each point is the same.
Consider the case of a 2-dimensional array of points.
Let O be any arbitrary point as origin, r1, r2 are position vectors of any two lattice
points joining to O.
If T ( translational vector) is the difference of two vectors r1, r2 and if it satisfies the
condition
T= n1a + n2b where n1, n2 are integers
Then T represent 2-dimensional lattice.
For 3- dimensional lattice,
T= n1a + n2b + n3c, n1, n2 n3 no of trans lation vectors along X,Y,Z,
Note: crystal lattice is the geometry of set of points in space where as the structure of the
crystal is the actual ordering of the constituent ions, atoms, molecules in space
Basis and Crystal structure:
1) Basis or pattern is a group of atoms, molecule or ions identical in
composition, arrangement and orientation.
2) When the basis is repeated with correct periodicity in all directions, it
gives the actual crystal structure.
Crystal structure = Lattice + Basis
FIG………………………..
The crystal structure is real while the lattice is imaginary.
In crystalline solids like Cu and Na, the basis is a single atom
In NaCl and CsCl- basis is diatomic
In CaF2 – basis is triatomic
Unit cell and Lattice parameters:
1) Unit cell is the smallest portion of the space lattice or geometrical
figure which can generate the complete crystal by repeating its own dimensions in varies
directions.
\
2) In describing the crystal structure, it is convenient to subdivide the structure into
small repetitive entities called unit cells.
3) unit cell may contain one or more atoms
4) shape of unit cell gives shape of the entire crystal
5) shape of the wall depends shape of the brick, brick it is considered as a unit cell.
Unit cell is the parallelepiped or cubes having 3 sets of parallel faces. It is the basic
structural unit or the building block of the crystal.
A unit cell can be described by 3 vectors or intercepts a, b, c, the lengths of the vectors
and the interfacial angles α, β, γ between them. If the values of these intercepts and
interfacial angles are known, then the form and actual size of the unit cell can be
determined. They may or may not be equal. Based on these conditions, there are 7
different crystal systems.
Primitive Cell: A unit cell having only one lattice point at the corners is called the
primitive cell. The unit cell differs from the primitive cell in that it is not restricted to
being the equivalent of one lattice point.
2) In some cases, the two coincide. unit cells may be primitive cells, but all the primitive
cells need not be unit cells.
CRYSTAL SYSTEMS AND BRAVAIS LATTICES:
There are 7 basic crystal systems which are distinguished based on three
vectors or the intercepts and the 3 interfacial angles between the 3 axes of the crystal.
They are
1. Cubic
2. Tetragonal
3. Orthorhombic
4. Monoclinic
5. Triclinic
6. Trigonal (Rhombohedral)
7. Hexagonal
More space lattices can be constructed by atoms at the body centers of unit cells or at
the centers of the faces. Based on this property, bravais ,in 1880,classified the space
lattices into 14.in the 7 crystal systems
1. 1.primitive cell. in this lattice, the unit cell consists of eight corners,all
corners have one atom
2.Body centered lattice ; in addition to the eight corner atoms,it consists of one
compleate atomat the center
4. Fa ce centered lattice ; along with corner atoms, each face will have one
center atom
5. Base centered lattice; the base and opposite face will have center atoms along
with the corner atoms
1. Cubic crystal system
a = b = c, α = β = γ =900
The crystal axes are perpendicular to one another, and the repetitive interval in the same
along all the three axes. Cubic lattices may be simple, body centered or face-centered.
2. Tetragonal crystal system
a = b ≠ c, α = β = γ =900
The crystal axes are perpendicular to one another. The repetitive intervals along the two
axes are the same, but the interval along the third axes is different. Tetragonal lattices
may be simple or body-centered.
3. Orthorhombic crystal system.
a ≠ b ≠ c, α = β = γ =900
The crystal axes are perpendicular to one another but the repetitive intervals are different
along the three axes. Orthorhombic lattices may be simple, base centered, body centered
or face centered.
4.Monoclinic crystal system
a ≠ b ≠ c, α = β = 900 ≠γ
Two of the crystal axes are perpendicular to each other, but the third is obliquely
inclined. The repetitive intervals are different along all the three axes. Monoclinic lattices
may be simple or base centered.
5. Triclinic crystal system
a ≠ b ≠ c, α ≠ β ≠γ ≠ 900
None of the crystal axes is perpendicular to any of the others, and the repetitive intervals
are different along the three axes.
6. Trigonal(rhombohedral) crystal system
a = b = c, α = β = γ ≠ 900
The three axes are equal in length and are equally inclined to each other at an angle other
than 900
7. Hexagonal crystal system.
a = b ≠ c, α = β =γ = 900 , γ = 1200
Two of the crystal axes are 600 apart while the third is perpendicular to both of them. The
repetitive intervals are the same along the axes that are 600 apart, but the interval along
the third axis is different.
Basic Crystal Structures:
The important fundamental quantities which are used to study the different arrangements
of atoms to form different structure are
Nearest neighbouring distance ( 2r) : the distance between the centers of two
nearest neighboring atoms is called nearest neighboring distance. If r is the
radius of the atom, nearest neighboring distance= 2r.
Atomic radius ( r) : It is defined as of the distance between the nearest neighboring
atoms in a crystals.
Coordination number (N): It is defined as the number of equidistant nearest
neighbours that an atom as in a given structure. More closely packed structure as
greater coordination number.
Atomic packing factor or fraction: It is the ratio of the volume occupied by the
atoms in unit cell(v) to the total volume of the unit cell (V).
P.F. = v/ V
Simple cubic (SC) structure:
In the simple cubic lattice, there is one lattice point at each of the 8
corners of the unit cell. The atoms touch along cubic edges.
Fig. Simple Cubic Structure
Nearest neighbouring distance = 2r = a
Atomic radius = r = a / 2
Lattice constant = a = 2r
Coordination number = 6 (since each corner atom is surrounded by 6 equidistant
nearest neighbours)
Effective number of atoms belonging to the unit cell or no. of atoms per unit cell =
(⅛)x8 = 1 atom per unit cell.
Atomic packing factor = v/ V = volume of the all atoms in the unit cell
-----------------------------------------------
Volume of the unit cell.
= 1 x (4 / 3) Π r3 / a3 = 4Π r3 / 3(2 r )3
= Π / 6 = 0.52 = 52%
Void space =48%
This structure is loosely packed. In simple cubic atoms are occupied upto 52%,remaining
vacant space is 48%
EXAMPLE. Polonium
Body centered cube structure (BCC):
BCC structure has one atom at the centre of the cube and one atom at each corner. The
centre atom touches all the 8 corner atoms.
Fig. Body Centered Cubic Structure
Diagonal length = 4r
Body diagonal = (√ 3) a
i.e. 4r = (√ 3) a
Nearest neighbouring distance = 2r = (√ 3) a / 2
Atomic radius = r = (√ 3) a / 4
Lattice constant = a = 4r / √ 3
Coordination number = 8 (since the central atom touches all the corner 8 atoms)
Effective number of atoms belonging to the unit cell or no. of atoms per unit cell = (⅛)
x8 + 1 = 2 atom per unit cell.
I.e. each corner atom contributes ⅛th to the unit cell. In addition to it, there is a one
more atom at the center
Atomic packing factor = v/ V = volume of the all atoms in the unit cell
-----------------------------------------------
Volume of the unit cell.
= 2 x (4 / 3) Π r3 / a3 = 8Π r3 / 3(4r /√ 3 )3
= √ 3 Π / 8 = 0.68 = 68%
Void space : 32%
EXAMPLES : Tungsten, Na, Fe and Cr,Molybdenum, exhibits this type of structure.\
Face centered cubic (FCC) structure:
In FCC structure, ther is one lattice point at each of the 8 corners of the unit cell and 1
centre atom on each of the 6 faces of the cube.
Fig. Face Centered Cubic Structure
Face diagonal length = 4r = (√ 2) a
Nearest neighbouring distance = 2r = (√ 2)a / 2 = a / √ 2
Atomic radius = r = a / 2√ 2
Lattice constant = a = 2√ 2 r
Coordination number = 12 ( the centre of each face has one atom. This centre atom
touches 4 corner atoms in its plane, 4 face centered atoms in each of the 2 planes on
either side of its plane)
Effective number of atoms belonging to the unit cell or no. of atoms per unit cell = (⅛)
x8 + (1/2) x 6 = 1 + 3 = 4 atom per unit cell.
I.e. each corner atom contributes ⅛th to the unit cell. In addition to it, there is a centre
atom on each face of the cube.
Atomic packing factor = v/ V = volume of the all atoms in the unit cell
-----------------------------------------------
Volume of the unit cell.
= 4 * (4 / 3) Π r3 / a3 = 16Π r3 / 3(2√ 2 r )3
= Π / 3√ 2 = 0.74 = 74%
Void space is equal to 26%
EX : Cu, Al, Pb and Ag, NI,AU, have this structure. FCC has highest packing factor.



Directions & Planes in Crystals:
While dealing with the crystals, it is necessary to refer to crystal planes,
and directions of straight lines joining the lattice points in a space lattice. For this
purpose, an indexing system deviced by Miller known as Miller indices is widely used.
Directions in A Crystal:
Consider a cubic lattice in which a straight line is passing
through the lattice points A, B, C etc and 1 lattice point on the line such as point A is
chosen as the origin.
Then the vector R which joins A to any other point on the line such as B (position vector)
can be represented by the vector eqn.
R = n1 a + n2 b + n3 c------ (1) where a, b, c are basic vectors
The direction of the vector R depends on the integers n1, n2, n3 since a, b, c are
constants. The common multiple is removed and n1, n2, n are re-expressed as the
smallest integers bearing the same relative ratio. The direction is then specified as [n1 n2
n3].
R =a+b+c, which provides the value of 1 for each of n1, n2, n3
Thus the direction is denoted as [1 1 1].
All lines in the space lattice which are parallel to the line AB possess either same set of
values for n1, n2, n3 as that of AB, or its common multiples.
Ex: 1. The direction that connects the origin and (1/3, 1/3, 2/3) point is [1 1 2].
i.e, (1/3, 1/3, 2/3)
L.C.M = 3.
(1/3x3 1/3x3 2/3x3)= [1 1 2].
2. [2 1 1] is the direction that connects the origin (0, 0, 0) and point (1, 1/2, 1/2)
Largest Number = 2
(2/2, 1/2, 1/2) = (1, 1/2, 1/2)
Planes in Crystals (Miller Indices):
Crystal plane; .the plane passing through the latice points is known as crystal plane
Def: Miller indices is a set of three lowest possible integers whose ratio taken in order
is the same as that of the reciprocals of the intercepts of the planes on the corresponding
axes in the same order. OR
Miller indices are defined as reciprocal of the intercepts made by the plane on the
crystallographic axes which are reduced to smallest numbers
It is possible for defining a system of parallel and equidistant planes which can
be imagined to pass through the atoms in a space lattice, such that they include all the
atoms in the crystal. Such a system of planes is called crystal planes. Many different
systems of planes could be identified for a given space lattice.
The position of a crystal plane can be specified in terms of three
integers called Miller indices
Consider a crystal plane intersecting the crystal axes.
Procedure for finding Miller indices
1. Find the intercepts of the desired plane on the three coordinate axes.
Let these be ( pa, qb, rc).
2. Express the intercepts as multiples of unit cell dimensions or lattice parameters
i.e. (p, q, r)
3. Take the reciprocals of these numbers i.e. 1/p: 1/q: 1/r
4. Convert these reciprocals into whole numbers by multiplying each with their
LCM to get the smallest whole number.
This gives the Miller indices (h k l) of the plane.
Ex: (3a, 4b, α c)
(3, 4, α)
1/3 1/4 1/α
(4 3 0) = (h, k, l)
Important features of Miller indices:
1. When a plane is parallel to any axis, the intercept of the plane on that axis is
infinity. Hence its Miller index for that axis is zero.
2. When the intercept of a plane on any axis is negative, a bar is put on the
corresponding Miller index.
3. All equally spaced parallel planes have the same index number (h k l)
Ex: The planes ( 1 1 2) and (2 2 4) are parallel to each other.
Separation Between successive (h k l) Planes:
Let (h k l) be the Miller indices of the plane ABC.
Let OP=d h k l be the normal to the plane ABC passing through the origin O.
Let OP make angles α, β, γ with X, Y & Z axes respectively.
Then cos α =d / OA = d / x = d / (a / h)
Fig. Inter planar Spacing.
Cos β =d /OB = d / y = d / (b / k)
Cos γ =d /OC = d/z =d/(c/l)
(Since convention in designing Miller indices x=a/h, y=b/k, z=c/l)
Now cos2 α + cos2 β +cos2 γ = 1
Hence d2 / (a / h) 2 + d2 / (b/k) 2 + d2 / (c/l) 2 = 1
􀃎 (d h / a )2 + (d k / b)2 +(d l / c)2 =1
􀃎 d (h k l) = OP = 1 / √ (h2/a2 +k2/b2 + l2/c2).
Therefore for cubic structure, a=b=c,
d (h k l) = a / √ ( h2 + k2 + l2 )
Let the next plane be parallel to ABC be at a distance OQ from the origin. Then its
intercepts are 2a / h, 2a / k, 2a / l.
Therefore OQ = 2d = 2a / √ (h2 + K+ l2)
Hence the spacing between adjacent planes = OQ - OP = PQ.
i.e. d = a / √ (h2 + k2 + l2)
Expression for Space Lattice Constant ‘a’ For a Cubic Lattice:
Density р = (total mass of molecules belonging to unit cell) / (volume of unit cell)
Total mass of molecule belonging to unit cell = nM / NA
Where n-number of molecules belonging to unit cell
M-Molecular weight
NA -Avagadro Number
Volume of cube = a3
Therefore р= nM/ a3 NA
Or a3 = nM / р NA
Lattice Constant for Cubic Lattice a = ( n M / р NA )1/3.
X-RAY DIFFRACTION
Diffraction of X- Rays by Crystal Planes:
1)X-Rays are electromagnetic waves like ordinary light; therefore, they should exhibit
interference and diffraction.
2) Diffraction occurs when waves pass across an object whose dimensions are of the
order of their own wavelengths. The wavelength of X-rays is of the order of 0.1nm or
10 -8 cm so that ordinary devices such as ruled diffraction gratings do not produce
observable effects with X-rays.
3) Laue suggested that a crystal which consisted of a 3-dimensional array of regularly
spaced atoms could serve the purpose of a grating. The crystal differs from ordinary
grating in the sence that the diffracting centers in the crystal are not in one plane.
4) Hence the crystal acts as a space grating rather than a plane grating.
X-ray diffraction methods
(1) Laue Method – for single crystal
(2) Powder Method- for finely divided crystalline or polycrystalline powder
(3) Rotating crystal Method - for single crystal
Bragg’s Law:
DEF. It states that the X-rays reflected from different parallel planes of a crystal
interfere constructively when the path difference is integral multiple of wavelength of
X-rays.
Consider a crystal made up of equidistant parallel planes of atoms with the interplanar
spacing d.
Let wave front of a monochromatic X-ray beam of wavelength ג fall at an angle θ on
these atomic planes. Each atom scatters the X-rays in all directions.
In certain directions these scattered radiations are in phase ie they interfere
constructively while in all other directions, there is destructive interference.
Figure 1 X-Ray Scattering by Crystal.
Consider the X-rays PE and P’A are inclined at an angle θ with the top of the crystal
plane XY. They are scattered along AQ and EQ’ at an angle θ w.r.t plane XY.
Consider another incoming beam P’C is scattered along CQ”
Let normal EB & ED be drawn to AC &CF. if EB & ED are parallel incident and
reflected wave fronts then the path difference between the incident and reflected waves is
given by
Δ = BC + CD --------------- (1)
In Δ ABC , sin θ = BC / EC = BC / d
i e BC = d sin θ
Similarly, in Δ DEC, CD = d sin θ
Hence path difference Δ = 2d sin θ FIG-------------------
If the 2 consecutive planes scatter waves in phase with each other , then the path
difference must be an integral multiple of wavelength.
i e Δ = n λ where n = 0 ,1 , 2 , 3 ,…….is the order of reflection
Thus the condition for in phase scattering by the planes in a crystal is given by
2d sin θ = n λ …………….(2)
This condition is known as Bragg’s Law.
The maximum possible value for θ is 1.
n λ
v=2d, λ-<2d
LIMITATION; λ should not exceed twice the inter planar spacing for diffraction to
occur.
Laue Method : 1)S1 & S2 are 2 lead screens in which 2 pin holes act as slits .
2) X-ray beam from an X –ray tube is allowed to pass through these 2 slits S1 & S2 . the
beam transmitted through S2 will be a narrow pencil of X – rays . the beam proceeds
further to fall on a single crystal such that Zinc blended (ZnS) which is mounted suitably
on a support . the single crystal acts as a 3 – dimensional diffraction grating to the
incident beam.
3)Thus, the beam undergoes diffraction in the crystal and then falls on the photographic
film.
4)The diffracted waves undergo constructive interference in certain directions, and fall on
the photographic film with reinforced intensity.
5)In all other directions, the interference will be destructive and the photographic film
remains unaffected.
6)The resultant interference pattern due to diffraction through the crystal as a whole will
be recorded on the photographic film (which requires many hours of exposure to the
incident beam
7) When the film is developed, it reveals a pattern of fine spots, known as Laue spots.
8)The distribution spots follow a particular way of arrangement that is the characteristic
of the specimen used in the form of crystal to diffract the beam.
MERITS; The Laue spot photograph obtained by diffracting the beam at several
orientations of the crystal to the incident beam are used for determining the symmetry
and orientations of the internal arrangement of atoms, molecules in the crystal lattice . it
is also used to study the imperfections in the crystal .
DEMERITS this method is not convenient for actual crystal structure determination
because of several wavelengths X rays diffract in different order from same plane,
and they super impose single lave spot
POWDER METHOD ( Debye – Scherrer Method ):
1)This method is widely used for experimental determination of crystal structures.
2)A monochromatic X- ray beam is incident on randomly oriented crystals in powder
form.
3)In this we used a camera called Debye – Scherer camera.
4)It consists of a cylindrical cassette, with a strip of photographic film positioned around
the circular periphery of cassette.
5) The powder specimen is placed at the centre, either pasted on a thin fiber of glass or
filled in a capillary glass tube.
6)The X- ray beam enters through a small hole in the camera and falls on the powder
specimen.
7)Some part of the beam is diffracted by the powder while the remaining passes out
through the exit port.
8)Since large no. of crystals is randomly oriented in the powder, set of planes which
make an angle θ with the incident beam can have a no. of possible orientations.
9) Hence reflected radiation lies on the surface of a cone whose apex is at the point of
contact of the radiation with the specimen.
10)If all the crystal planes of interplanar spacing d reflect at the same bragg angle θ, all
reflections from a family lie on the same cone.
After taking n=1 in the Bragg’s law
2dsin θ = λ
11)There are still a no of combinations of d and θ, which satisfies Bragg’s law. Hence
many cones of reflection are emitted by the powder specimen. In the powder camera a
part of each cone is recorded by the film strip.
12)The full opening angle of the diffracted cone 4θ is determined by measuring the
distance S between two corresponding arcs on the photographic film about the exit point
direction beam. The distance S on the film between two diffraction lines corresponding to
a particular plane is related to Bragg’s angle by the equation
4θ = (S / R) radians (or)
4θ = (S / R) x (180 /Π ) degrees where R- radius of the camera
13)A list of θ values can be thus be obtained from measured values of S. Since the
wavelength ‘λ’ is known, substitution of λ gives a list of spacing‘d’.
Each spacing is the distance between neighbouring plane (h k l). From the ratio of
interplanar spacing, the type of lattice can be identified.
CRYSTAL DEFECTS
In real materials we find:
Crystalline Defects or lattice irregularity
Most real materials have one or more “errors in
perfection”
with dimensions on the order of an atomic diameter to
many lattice sites
Defects can be classification: crystalline
imperfection
zero-dimensional or point defects
one-dimensional or line defects (dislocation)
two-dimensional or planar defects
three-dimensional or volume defects
(1) point defects
vacancy – atom is missing, may be created
by
 local disturbances during the crystal
growth
 atomic arrangements in an existing crystal
 plastic defromation, rapid cooling
 bombardment with energetic particles
interstitialcy or self-interstitial – an atom in
a
crystal can occupy an interstitial site
between
surrounding atoms
can be introduced by irradiation
1. according to geometry
(point, line or plane)
2. dimensions of the defect
POINT DEFECTS
• The simplest of the point defect is a vacancy, or
vacant lattice site.
• All crystalline solids contain vacancies.
• Principles of thermodynamics is used explain the
necessity of the existence of vacancies in
crystalline solids.
• The presence of vacancies increases the entropy
(randomness) of the crystal.
• The equilibrium number of vacancies for a given
quantity of material depends on and increases
with temperature as follows: (an Arrhenius
model)
• n = N exp(-Ev/kT)
Schottky imperfection – two oppositely
charged
ions are missing form an ionic crystal
a cation-anion divacancy
impurity is also a type of point defect
(2) line defects (dislocations)
crystalline solids are defects that cause
lattice
distortion centered around a line
formed by plastic deformation, vacancy
condensation, and atomic mismatch
Frenkel imperfection – a cation moves into
an
interstitial site, and a cation vacancy is
created
vacancy-interstitialcy pair
the presence of these defects in ionic
crystals
increases their electrical conductivity
Linear Defects (Dislocations)
– Are one-dimensional defects around which
atoms are misaligned
• Edge dislocation:
– extra half-plane of atoms inserted in a crystal
structure
– b (the berger’s vector) is ⊥ (perpendicular) to
dislocation line
• Screw dislocation:
– spiral planar ramp resulting from shear
deformation
– b is || (parallel) to dislocation line
Burger’s vector, b: is a measure of lattice distortion and
is measured as a distance along the close packed
directions in the lattice
Definition of the Burgers vector, b, relative to an edge
dislocation. (a) In the perfect crystal, an m× n atomic step
loop closes at the starting point. (b) In the region of a
dislocation, the same loop does not close, and the closure
vector (b) represents the magnitude of the structural
defect. For the edge dislocation, the Burgers vector is
perpendicular to the dislocation line.
Screw dislocation. The spiral
stacking of crystal planes leads to the Burgers vector
being parallel to the dislocation
line.

Simple grainboundary
structure. This is termed a tilt boundary
because it is formed when two adjacent crystalline grains
are tilted relative to each other by a few degrees (θ). The
resulting structure is equivalent to isolated edge
dislocations separated by the distance b/θ, where b is the
length of the Burgers vector, b.
Dislocation Line:
A dislocation line is the boundary between slip and no slip regions of a crystal
Burgers vector:
The magnitude and the direction of the slip is represented by a vector b called the
Burgers vector
In general, there can be any angle between the Burgers vector b (magnitude and the
direction of slip) and the line vector t (unit vector tangent to the dislocation line
b ⊥ t ⇒ Edge dislocation
b ⎜⎜ t ⇒ Screw dislocation
grain boundaries – a narrow region
between
two grains of about 2~5 atomic diameters in
width and a region of atomic mismatch
between adjacent grains the higher energy of
grain boundaries and
more open structure make them more
favorable for nucleation and growth of
precipitates
twin or twin boundary – a region in which
a
mirror image of the structure exists across a
plane or a boundary
ex. twin boundaries in the grain structure of
brass
Grain Boundary: low and high angle
One grain orientation can be obtained by rotation of another grain across the grain
boundary about an axis through an angle
If the angle of rotation is high, it is called a high angle grain boundary
If the angle of rotation is low it is called a low angle grain boundary
Grain Boundary: tilt and twist
One grain orientation can be obtained by rotation of another grain across the grain
boundary about an axis through an angle
If the axis of rotation lies in the boundary plane it is called tilt boundary
If the angle of rotation is perpendicular to the boundary plane it is called a twist boundary
stacking faults or piling-up faults
one or more of the stacking planes may be
missing, give rise to another twodimensional
defect
ex. ABCABAACBABC in FCC crystal
ABAABBAB in HCP crystal
(4) volume defects
a cluster of point defects join to form a
three-dimensional void or a pore
a cluster of impurity atoms join to form a
three-dimensional precipitate
the size from a few nm to cm
⊥

Questions;
Explain the various types of bonding in crystals. Illustrate with examples.
Distinguish between ionic and covalent bonding in solids.
Obtain a relation between potential energy and inter atomic spacing of a molecule.
Derive an expression for cohesive energy of a solid.
Obtain an equation for total binding energy of sodium chloride crystal.
Differentiate between crystalline and amorphous solids.
Explain the terms i) Basis ii) Space lattice iii) Unit cell
Explain with neat diagram the following crystal structures.
simple cubic structure(SC)
body centered cubic structure(BCC)
face centered cubic structure (FCC)
What do you understand by packing density? Show that packing density for simple
lattice, body centered lattice and face centered lattice is Π/6 , √3Π/8 , √2Π/6
respectively
Show that FCC is the most closely packed of the three cubic structures.
a) For a crystal having a ≠ b ≠ c and α = β = γ = 900, what is the crystal system?
b) For a crystal having a ≠ b ≠ c and α ≠ β ≠ γ ≠ 900, what is the crystal system?
c) Can you specify the Bravais lattices for parts (a) and (b) explain.
Explain the special features of the three types of lattices of cubic crystals?
What are ionic crystals? Explain the formation of an ionic crystal and obtain an
expression for its cohesive energy?
What is a Bravais lattice? What are the different space lattices in the cubic system?
What are the miller indices? How they obtained?
Derive the expression for the interplannar spacing between two adjacent planes of miller
indices (h k l ) in a cubic lattice of edge length ‘a ‘.
Derive f rombragg’s law 2d sin θ= n λ
Describe Laue’s method of determination of crystal structu Explain the power method
of crystal structure analysis.
UNIT -II
PRINCIPLES OF QUANTUM MECHANICS
Introduction: Quantum mechanics is a new branch of study in physics which is
indispensable in understanding the mechanics of particles in the atomic and sub-atomic
scale.
The motion of macro particles can be observed either directly or
through microscope. Classical mechanics can be applied to explain their motion. But
classical mechanics failed to explain the motion of micro particles like electrons, protons
etc...
Max Plank proposed the Quantum theory to explain Blackbody
radiation. Einstein applied it to explain the Photo Electric Effect. In the mean time,
Einstein’s mass – energy relationship (E = mc2) had been verified in which the radiation
and mass were mutually convertible. Louis deBroglie extended the idea of dual nature of
radiation to matter, when he proposed that matter possesses wave as well as particle
characteristics.
The classical mechanics and the quantum mechanics have
fundamentally different approaches to solve problems. In the case of classical mechanics
it is unconditionally accepted that position, mass, velocity, acceleration etc of a particle
can be measured accurately, which, of course, true in day to day observations. In contrast,
the structure of quantum mechanics is built upon the foundation of principles which are
purely probabilistic in nature. As per the fundamental assumption of quantum mechanics,
it is impossible to measure simultaneously the position and momentum of a particle,
whereas in the case of classical mechanics, there is nothing which contradicts the
measurements of both of them accurately.
Plank’s Quantum Theory:
Max Plank, a German physicist derived an equation which successfully accounted for the
spectrum of the blackbody radiation. He incorporated a new idea in his deduction of
Plank eqn. that the probability of emission of radiation decreases as its frequency
increases so that, the curve slopes down in the high frequency region. The oscillators in
the blackbody can have only a discrete set of energy values. Such an assumption was
radically different from the basic principles of physics.
The assumption in the derivation of Plank’s law is that the wall of the
experimental blackbody consists of a very large number of electrical oscillators, with
each oscillator vibrating with a frequency of its own. Plank brought two special
conditions in his theory.They are
(1) Only an integral multiple of energies h ν where ‘h’ is Plank’s constant and ‘ν ‘ is
frequency of vibration i e, the allowed energy values are E = n h ν where n = 0 ,
1 , 2 , ………
(2) An oscillator may lose or gain energy by emitting or absorbing radiation of
frequency ν = (ΔE / h), where ΔE is the difference in the values of energies of the
oscillator before and the emission or absorption had taken place.
Based on the above ideas, he derived the law governing the entire spectrum of the
Blackbody radiation, given by
U λ d λ = (8Πhc / λ5) [1 / (e h ν /kT – 1)] d λ (since ν = c / λ)
This is called Plank’s radiation law.
Waves and Particles: deBroglie suggested that the radiation has dual nature i e
both particle as well as wave nature. The concept of particle is easy to grasp. It has
mass, velocity, momentum and energy. The concept of wave is a bit more difficult
than that of a particle. A wave is spread out over a relatively large region of space, it
cannot be said to be located just here and there, and it is hard to think of mass being
associated with a wave. A wave is specified by its frequency, wavelength, phase,
amplitude, intensity.
Considering the above facts, it appears difficult to accept the
conflicting ideas that radiation has dual nature. However this acceptance is essential
because radiation sometimes behaves as a wave and at other times as a particle.
(1) Radiations behaves as waves in experiments based on interference, diffraction,
polarization etc. this is due to the fact that these phenomena require the presence
of two waves at the same position and at the same time. Thus we conclude that
radiation behaves like wave.
(2) Plank’s quantum theory was successful in explaining blackbody radiation,
photoelectric effect, Compton Effect and had established that the radiant energy,
in its interaction with the matter, behaves as though it consists of corpuscles. Here
radiation interacts with matter in the form of photons or quanta. Thus radiation
behaves like particle.
Hence radiation cannot exhibit both particle and wave nature simultaneously.
deBroglie hypothesis :The dual nature of light possessing both wave and
particle properties was explained by combining Plank’s expression for the energy of a
photon E = h ν with Einstein’s mass energy relation E = m c2 (where c is velocity of
light , h is Plank’s constant , m is mass of particle )
􀃖 h ν = m c2
Introducing ν = c / λ, we get h c / λ = m c2
==> λ = h / mc = h / p where p is momentum of particle
λ is deBroglie wavelength associated with a photon.
deBroglie proposed the concept of matter waves , according to which a material particle
of mass ‘m’ moving with velocity ‘v’ should be associated with deBroglie wavelength ‘λ’
given by
λ = h / m v = h / p
The above eqn. represents deBroglie wave eqn.
Characteristics of Matter waves:
Since λ = h / m v
1. Lighter the particle, greater is the wavelength associated with it.
2. Lesser the velocity of the particle, longer the wavelength associated with it.
3. For v = 0, λ = ∞. This means that only with moving particle, matter waves is
associated.
4. Whether the particle is changed or not, matter waves is associated with it.
5. It can be proved that matter waves travel faster than light.
We know that E = h ν and E = m c2
􀃖 h ν = m c2 or ν = m c2 / h
Wave velocity (ω) is given by
ω = ν λ = m c2 λ / h = (m c2 / h) (h / m v)
􀃖 ω = c 2 / v
As the particle velocity ‘v’ cannot exceed velocity of light, ω is greater than
the velocity of light.
6. No single phenomena exhibit both particle nature and wave nature
simultaneously.
7. The wave nature of matter introduces an uncertainty in the location of the particle
& the momentum of the particle exists when both are determined simultaneously.
Davisson and Germer’s experiment:
C. J. Davisson and L. H. Germer were studying scattering of electrons by a metal target
and measuring the intensity of electrons scattered in different directions.
Experimental Arrangement:
An electron gun, which comprises of a tungsten filament is heated by a
low tension battery B1, produces electrons. These electrons are accelerated to desired
velocity by applying suitable potential from a high tension source B2. The accelerated
electrons are collimated into a fine beam by allowing them to pass through a system of
pin holes provided in the cylinder. The whole instrument is kept in an evacuated
chamber.
The past moving beam of electrons is made to strike the Nickel target
capable of rotating about an axis perpendicular to the plane. The electrons are now
scattered in all directions by the atomic planes of crystals. The intensity of the electron
beam scattered in a direction can be measured by the electron collector which can be
rotated about the same axis as the target. The collector is connected to a galvanometer
whose deflection is proportional to the intensity of the electron beam entering the
collector.
Fig. Davisson and Germer's Apparatus
The electron beam is accelerated by 54 V is made to strike the Nickel crystal
and a sharp maximum is occurred at angle of 50o with the incident beam. The incident
beam and the diffracted beam in this experiment make an angle of 65o with the family of
Bragg’s planes.
d = 0.091nm (for Ni crystals)
According to Bragg’s law for maxima in diffracted pattern,
2d sin θ = n λ
For n =1, λ = 2d sin θ
= 2 x0.91x 10-10x sin 65o
= 0.165 nm
For a 54 V electron, the deBrogllie wavelength associated with the electron is given by
= 12.25 / √ V = (12.25 / √54) oA
= 0.166 nm.
This value is in agreement with the experimental value. This experiment provides a direct
verification of deBroglie hypothesis of wave nature of moving particles.
G.P. Thomson’s Experiment:
1) G.P. Thomson investigated high speed electrons produced by applying
the high voltage ranges from10 to 50kV. The principle is similar to that of powdered
crystal method of X- ray diffraction.
In this experiment, an extremely thin ( 10-8m ) metallic film F of gold, aluminium etc.,
is used as a transmission grating to a narrow beam of high speed electrons emitted by a
cathode C and accelerated by anode A. A fluorescent screen S or a suitable photographic
plate P is used to observe the scattered electron beam. The whole arrangement is enclosed
in a vacuum chamber.
The electron beam transmitted through the metal foil gets scattered
producing diffraction pattern consisting of concentric circulars rings around a central
spot. The experimental results and its analysis are similar to those of powdered crystal
experiment of X- ray diffraction.
Fig. Thomson Apparatus
L: distance between the foil and screen
R(r): radius of the diffraction ring
θ: glancing angle
The method of interpretation of the experimental results is same as that of the Davisson
and Germer experiment. Here also we use Bragg’s law and deBroglie wavelength.
We have 2d sin θ = n λ
θ = n λ / 2d --􀃆 I since θ is small sin θ = θ
From fig.
R/L = tan2 θ ≈ 2 θ
R = L 2 θ -􀃆 II
From eqn I, II
R = L 2 n λ / 2d = L n λ / d
Since L and d are fixed in the experiment
R α n λ or D α n λ D is diameter of the ring
Combing with the de Broglie expression
λ = √ (150 / V (1+r))
Where (1+r) is the relativistic correction we notice that D√ (V (1+r)) must be constant for
a given order the experiment repeated with different voltages. The data shows that D√
(V(1+r)) is constant, thus supporting the deBroglie concept of matter waves.
Heisenberg Uncertainty Principle:
According to classical mechanics, a moving particle at any instant has a
fixed position in space and a definite momentum which can be determined
simultaneously with any desired accuracy. The classical point of view represents an
approximation which is adequate for the objects of appreciable size, but not for the
particles of atomic dimensions.
Since a moving particle has to be regarded as a deBroglie group,
there is a limit to the accuracy with which we can measure the particle properties. The
particle may be found anywhere within the wave group, moving with the group
velocity. If the group is narrow, it is easy to locate its position but the uncertainty in
calculating its velocity or momentum increases. On the other hand, if the group is
wide, its momentum can be estimated satisfactorily, but the uncertainty in finding the
location of the particle is great. Heisenberg stated that the simultaneous determination
of exact position and momentum of a moving particle is impossible.
If Δ x is Error in the measurement of position of the particle along X-axis
Δ p is Error in the measurement of momentum
Then Δ x. Δ p = h ---------- (1) where h is Plank’s constant
The above relation represents the uncertainty involved in measurement of both the
position and momentum of the particle.
To optimize the above error, lower limit is applied to the eqn. (1)
Then (Δ x). (Δ p) ≥ Ђ / 2 where ђ = h / 2 Π
A particle can be exactly located (Δ x → 0) only at the expense of an infinite
momentum (Δ p → ∞).
There are uncertainty relatio0ns between position and momentum, energy and time,
and angular momentum and angle.
If the time during which a system occupies a certain state is not greater than Δ t, then
the energy of the state cannot be known within Δ E,
i e (Δ E ) ( Δ t ) ≥ ђ / 2 .
Schrödinger’s Time Independent Wave Equation:
Schrödinger, in 1926, developed wave equation for the moving
particles. One of its forms can be derived by simply incorporating the deBroglie
wavelength expression into the classical wave eqn.
If a particle of mass ‘m’ moving with velocity ‘v’ is associated
with a group of waves.
Let ψ be the wave function of the particle. Also let us consider a simple form of
progressing wave like the one represented by the following equation,
Ψ = Ψ0 sin (ω t – k x) --------- (1)
Where Ψ = Ψ (x, t) and Ψ0 is the amplitude.
Differentiating Ψ partially w.r.to x,
∂ Ψ / ∂ x = Ψ0 cos (ω t – k x) (- k)
= -k Ψ0 cos (ω t – k x)
Once again differentiate w.r.to x
∂2 ψ / ∂ x2 = (- k) Ψ0 (- sin (ω t – k x)) (- k)
= - k 2 Ψ0 sin (ω t – k x)
∂2 ψ / ∂ x2 = - k 2 ψ (from eqn (1))
∂2 ψ / ∂ x2 + k 2 ψ = 0 ----------- (2)
∂2 ψ / ∂ x2 + (4 Π2 / λ2) ψ = 0 --------- (3) (since k = 2 Π / λ)
From eqn. (2) or eqn. (3) is the differential form of the classical wave eqn. now we
incorporate deBroglie wavelength expression λ = h / m v.
Thus we obtain
∂2 ψ / ∂ x2 + (4 Π2 / (h / m v) 2) ψ = 0
∂2 ψ / ∂ x2 + 4 Π2 m2 v 2 ψ / h2 = 0 -------------- (4)
The total energy E of the particle is the sum of its kinetic energy K and potential energy V
i e E = K + V -------------- (5)
And K = mv2 / 2 ---------- (6)
Therefore m2 v 2 = 2 m (E – V) ------------ (7)
From (4) and (7)
=> ∂2 ψ / ∂ x2 + [8Π2 m (E-V) / h2] ψ = 0 ------------ (8)
In quantum mechanics, the value h / 2 Π occurs more frequently. Hence we denote,
ђ = h / 2 Π
Using this notation, we have
∂2 ψ / ∂ x2 + [2 m (E – V) / ђ 2] ψ = 0 ------------ (9)
For simplicity, we considered only one – dimensional wave. Extending eqn. (9) for a three
– dimensional, we have
∂2 ψ / ∂ x2 + ∂2 ψ / ∂ y2 + ∂2 ψ / ∂ z2 + [2 m (E – V) / ђ 2] ψ = 0 ------------ (10)
Where Ψ = Ψ (x, y, z).
Here, we have considered only stationary states of ψ after separating the time dependence
of Ψ.
Using the Laplacian operator,
▼2 = ∂2 / ∂ x2 + ∂2 / ∂ y2 + ∂2 / ∂ z2 ------------- (11)
Eqn. (10) can be written as
▼2 Ψ + [2 m (E – V) / ђ 2] ψ = 0 --------------- (12)
This is the Schrödinger Time Independent Wave Equation.
Physical Significance of Wave Function:
Max Born in 1926 gave a satisfactory interpretation of the wave function ψ associated
with a moving particle. He postulated that the square of the magnitude of the wave
function |ψ|2 (or ψ ψ* it ψ is complex), evaluated at a particular point represents the
probability of finding the particle at the point. |ψ|2 is called the probability density and ψ
is the probability amplitude. Thus the probability of the particle within an element
volume dt is |ψ|2 dτ. Since the particle is certainly somewhere, the integral at |ψ|2 dτ
over all space must be unity i.e.
-∞∫∞ |ψ|2 .dτ = 1 ___________________ (28)
A wave function that obeys the above equations is said to be normalized. Energy
acceptable wave function must be normalizable besides being normalizable; an
acceptable wave function should fulfill the following requirements (limitations)
.
1. It must be finite everywhere.
2. It must be single valued.
3. It must be continuous and have a continuous first derivative everywhere.
Normalization Of a wave function:
Since |ψ(x, y, z) 2| .dν is the probability that the particle will be found in a volume
element dν. Surrounding the point at positron (x, y, z), the total probability that the
particle will be somewhere in space must be equal to 1. Thus, we have
-∞∫∞ |ψ(x, y, z) 2| .dν = 1
Where ψ is a function of the space coordinates (x, y, z) from this ‘normalization
condition’ we can find the value of the complaint and its sign. A wave function which
satisfies the above condition is said to be normalized (to unity).
The normalizing condition for the wave function for the motion of a particle in
one dimension is
-∞∫∞ |ψ(x) |2 .dx = 1
From these equations, we see that for one – dimensional case, the dimension of ψ(x) in L-
1/2 and for the three – dimensional case the dimension of ψ(x, y, z) in L-3/2.
Particle in One Dimensional Potential Box:
Consider a particle of mass ‘m’ placed inside a one-dimensional box of infinite
height and width L.
Fig. Particle in a potential well of infinite height.
Assume that the particle is freely moving inside the box. The motion of the particle is
restricted by the walls of the box. The particle is bouncing back and forth between the walls
of the box at x = 0 and x = a. For a freely moving particle at the bottom of the potential
well, the potential energy is very low. Since the potential energy is very low, moving
particle energy is assumed to be zero between x =0 and x = a.
The potential energy of the particle outside the walls is infinite due to the infinite P.E
outside the potential well.
The particle cannot escape from the box
i.e. V = 0 for 0 < x < a
V = ∞ for 0 ≥ x ≥ a
Since the particle cannot be present outside the box, its wave function is zero
i e │ψ│2 = 0 for 0 > x > a
│ψ│2 = 0 for x = a & x = 0
The Schrödinger one – dimensional time independent eqn. is
▼2 Ψ + [2 m (E – V) / ђ 2] ψ = 0 ----------(1)
For freely moving particle V = 0
▼2 Ψ + [ 2 m E / ђ 2 ] ψ = 0 ----------(2)
Taking 2 m E / ђ 2 = K2 ------------ (3)
Eqn.(1) becomes ∂2 ψ / ∂ x2 + k2 Ψ = 0 -------------(4)
Eqn. (1) is similar to eq. of harmonic motion and the solution of above eqn. is written as
Ψ = A sin kx + B cos kx -----------(5) where A, B and k are unknown
quantities and to calculate them it is necessary to construct boundary conditions.
Hence boundary conditions are
When x = 0, Ψ = 0 => from (5) 0 = 0 + B => B = 0 -------- (6)
When x = a, Ψ = 0 => from (5) 0 = A sin ka + B cos ka ---------- (7)
But from (6) B = 0 therefore eqn. (7) may turn as
A sin ka = 0
Since the electron is present in the box a ≠ 0
Sin ka = 0
Ka = n Π
k = n Π / a -------------- (8)
Substituting the value of k in eqn. (3)
2 m E / ђ 2 = (n Π / a )2
E = ( n Π / a )2 ( ђ 2 / 2 m ) = ( n Π / a )2 ( h2 / 8 m Π2 )
E = n2 h2 / 8 m a2
In general En = n2 h2 / 8 m a2 ------------(9)
The wave eqn. can be written as
Ψ = A sin (n Π x / a) ---------- (10)
Let us find the value of A, if an electron is definitely present inside the box, then
=>∫∞
-∞ │ψ│2 dx = 1
=>∫a
0 A2 sin2 ( n Π x / a ) dx = 1
=>∫a
0 sin2 (n Π x / a) dx = 1 / A2
=>∫a
0 [1 - cos (2 Π n (x / a)) / 2] dx = 1 / A2
A = √ 2 / a ---------- (11)
From eqn’s. (10) & (11)
Ψn = √ 2 / a sin ( n Π x / a ) ----------(12)
Eqn. (9) represents an energy level for each value of n. the wave function this energy
level is given in eqn. (12). Therefore the particle in the box can have discrete values of
energies. These values are quantized. Not that the particle cannot have zero energy .The
normalized wave functions Ψ1 , Ψ2, Ψ3 given by eqn (12) is plotted. the values
corresponding to each En value is known as Eigen value and the corresponding wave
function is known as Eigen function.
The wave function Ψ1, has two nodes at x = 0 & x = a
The wave function Ψ2, has three nodes at x = 0, x = a / 2 & x = a
The wave function Ψ3, has three nodes at x = 0, x = a / 3, x = 2 a / 3 & at x = a
The wave function Ψn, has (n + 1) nodes
Substituting the value of E in (3), we get
(2 m / ђ 2 ) ( p2 / 2 m) = k2
=> p2 / ђ 2 = k2
k = p / ђ = p / (h / 2 Π) = 2 Π p / h
k = 2 Π / λ where k is known as wave vector.
Questions:
1. What are the matter waves? Explain their properties.
2. Explain de Broglie hypothesis.
3. Explain the duality of matter waves
4. Describe Davisson and Germer’s experiment an explain how it enabled the
verification of the de Broglie equation.
5. Explain G.P. Thompson’s experiment in support of de Broglie hypothesis
6. Explain Heisenberg’s uncertainty principle. Give its physical significance.
7. Derive time independent one dimensional Schrödinger’s equation.
8. Explain the physical significance of wave function.
9Write down Schrödinger’s wave equation for a particle in one dimensional
potential box.
UNIT- III
ELECTRON THEORY OF METALS
INTRODUCTION:
The electrons in the outer most orbital of the atoms which consists the
solids determine its electrons properties .The electron theory of metals solids aims to
explain the structure and properties of solids through their electronic structure the
electron theory is applicable to the all solids both metals and non metals .
It explains the electrical thermal and magnetic properties of solids etc .The theory has
been developed in three main steps.
1) THE CLASSICAL FREE ELECTRON THEORY: Drude and Lorentz developed this
theory in 1900. According to this theory the metals containing free electrons obey the law
of classical mechanics.
2) THE QUANTUM FREE ELECTRON THEORY: Sommerfeld developed this
theory during 1928. According to this theory free electrons obey quantum laws.
3) THE ZONE THEORY (OR) BAND THEORY OF SOLIDS: Bloch developed 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”.
PHYSICAL PROPERTIES OF METALS
Metallic conductors obey ohm’s law which states that the current in the steady state is
proportional to the electric field strength.
Metals have high electrical and thermal conductivities.
At low temperatures, the resistivity is proportional to the fifth power of absolute
temperature.
i.e. ρ α T5
4. The resistivity of metals at room temperatures is of the order of 10-7 ohm- meter
and above Debye’s temperatures varies linearly with temperature.
i.e . ρ α T
5 For most metals, resistivity is inversely proportional to the pressure.
i.e. . ρ α 1 / P
The resistivity of an impure specimen is given by Mathiessen rule.
ρ = ρ0 + ρ(T)
6)Where ρ0 is a constant for impure specimen and ρ(T) is the temperature dependent
resistivity of pure specimen
7)Near absolute zero, the resistivity of certain metals tends towards zero. i.e it
exhibits the phenomena of super conductivity.
8)The conductivity varies in the presence of magnetic field this effect is known as
magneto resistance
9)The ratio of thermal to electrical conductivity is directly proportional to absolute
temperature. This is known as Wiedemann-Franz law.
Classical free electron theory of metals:
The classical free electron theory is based on following postulates:
1) In an atom, electrons revolve around the nucleus. A metal is composed of such
atoms.
2) The valence electrons of atoms are free to move about the whole volume of
metals like the molecules of perfect gas in a container. The collection of valence
electrons from all the atoms in a given piece of metal forms electron gas. It is free
to move through out the volume of metals.
3) These free electrons move in random direction and collide with either positive
ions fixed to the lattice or other free electrons. All the collision are elastic.
i.e. there is no loss of energy
4) The moments of free electrons obey the classical kinetic theory of gases.
5) The electron velocities in metals obey the classical Maxwell Boltzmann
distributions of velocities.
6) The free electrons move in a completely uniform potential field due to ions fixed
in the lattice.
7) When an electric field is applied to the metal, the free electrons are accelerated in
the direction opposite to the direction of the applied electric field.
Success of classical free electron theory:
1) It verifies ohm’s law
2) It explains the electrical and the thermal conductivities of metals.
3) It derives Wiedemann Franz law.
4) It explains the optical properties of metals.
Draw backs of classical free electron theory:
In spite of the success seen above, classical theory has the following draw backs
1)The phenomena such as photo electric effect, Compton effect and the black body
radiation could not be explained by classical free electron theory.
2) According to the classical free electron theory the values of specific heat of metals
is given by 4.5Ru where Ru is the universal gas constant where as the experimental
value is nearly equal to 3Ru .
1) Also according to this theory, the value of electronic specific heat is
equal to (3/2)Ru while the actual value is about 0.01Ru only.
5) Though K / σT is a constant (Wiedemann Franz law). According to the classical
free electron theory it is not a constant at low temperature.
6) Ferromagnetism could not be explained by this theory. The theoretical value of
para magnetic susceptibility is greater than experimental value.
Fermi – Dirac Distribution:
Permitted energy values for electrons in a material are represented by the energy
levels in the energy bands for that material. These energy levels are occupied by the
electrons in the material in a particular order. This is called distribution of electrons.
Fermi – Dirac distribution deals with the distribution of electrons among the
permitted energy levels.
According to Fermi – Dirac distribution, the probability of electron occupying energy
level E is given by
F(E) = 1/ (1 + exp [( E – EF) / kT] ----------------(1)
Where EF is called the Fermi energy and is constant for a given system. F(E) is called Fermi
function.
At T = 0K, for E < EF F(E) = 1
E < EF F(E) = 0 ------------- (2)
This means that at 0K, all quantum states with energy below EF are completely occupied
and those above EF are unoccupied. With increase of temperature, the Fermi function
plot shows deviation.
At any temperature other than 0K, If E = EF
F(E) = 1 / 2 ----------------(3)
Hence, the Fermi level is that state at which the probability of electron occupation
is ½ at any temperature above 0K and also it is the level of the max. energy of the
filled states at 0K. Fermi energy is the energy of the state at which the probability
of electron occupation is ½ at any temperature above 0K. it also the max.energy of
filled states at 0K.
BAND THEORY OF SOLIDS
Bloch Theorem: A crystalline solid consists of a lattice which is composed of a
large number of ionic cores at regular intervals, and, the conduction electrons can move
freely throughout the lattice.
Let the lattice is in only one-dimension ie only an array of ionic cores along x-axis is
considered. If we plot the potential energy V of a conduction electron as a function of
its position in the lattice, the variation of potential energy.
Since the potential energy of any body bound in a field of attraction is negative, and
since the conduction electron is bound to the solid, its potential energy V is negative.
Further, as it approaches the site of an ionic core V → - ∞. Since this occurs
symmetrically on either side of the core, it is referred to as potential well. The width of
the potential well b is not uniform, but has a tapering shape.
If V0 is the potential at a given depth of the well, then the variation is such that
b → 0 , as V0 → ∞ ., and ,
The product b V0 is a constant.
Now, since the lattice is a repetitive structure of the ion arrangement in a crystal, the
type of variation of V also repeats itself. If a is the interionic distance, then , as we
move in x-direction , the value of V will be same at all points which are separated by a
distance equal to a.
ie V(x) = V( x + a ) where, x is distance of the electron from the core.
Such a potential is said to be a periodic potential.
The Bloch’s theorem states that, for a particle moving in a periodic
potential, the Eigen functions for a conduction electron are of the form,
χ ( x ) = U (x) cos kx
Where U ( x ) = U ( x + a )
The Eigen functions are the plane waves modulated by the function U (x). The function
U (x) has the same periodicity as the potential energy of the electron, and is called the
modulating function.
In order to understand the physical properties of the system, it is required to
solve the Schrödinger’s equation. However, it is extremely difficult to solve the
Schrödinger’s equation with periodic potential described above. Hence the Kronig –
Penney Model is adopted for simplification.
THE KRONIG -PENNEY MODEL:
It is assumed in quantum free electron theory of metals that the free electrons in
a metal express a constant potential and is free to move in the metal. This theory
explains successfully most of the phenomena of solids. But it could not explain why
some solids are good conductors and some other are insulators and semi conductors. It
can be understood successfully using the band theory of solids.
According to this theory, the electrons move in a periodic potential
produced by the positive ion cores. The potential of electron varies periodically with
periodicity of ion core and potential energy of the electrons is zero near nucleus of the
positive ion core. It is maximum when it is lying between the adjacent nuclei which
are separated by interatomic spacing. The variation of potential of electrons while it is
moving through ion core is shown fig.
Fig. One dimensional periodic potential
V ( x ) = { 0 , for the region 0 < x < a
{ V0 for the region -b < x < a ---------------------(1)
Applying the time independent Schrödinger’s wave equation for above two regions
d2Ψ / dx2 + 2 m E Ψ / ħ2 = 0 for region 0 < x < a -----------(2)
and d2Ψ / dx2 + 2 m ( E – V ) Ψ / ħ2 = 0 for region -b < x < a -----------(3)
Substituting α2 = 2 m E / ħ2 ---------------(4)
β2 = 2 m ( E – V ) / ħ2 -----------(5)
d2Ψ / dx2 + α2 Ψ = 0 for region 0 < x < a -----------(6)
d2Ψ / dx2 + β2 Ψ = 0 for region -b < x < a -----------(7)
The solution for the eqn.s (6) and (7) can be written as
Ψ ( x ) = Uk ( x ) eikx ------------------(8)
The above solution consists of a plane wave eikx modulated by the periodic function.
Uk(x), where this Uk(x) has the periodicity of the ion such that
Uk(x) = Uk(x+a) ------------------(9)
and where k is propagating vector along x-direction and is given by k = 2 Π / λ . This
k is also known as wave vector.
Differentiating equation (8) twice with respect to x, and substituting in equation (6)
and (7), two independent second order linear differential equations can be obtained
for the regions 0 < x < a and -b < x < 0 .
Applying the boundary conditions to the solution of above equations, for linear
equations in terms of A,B,C and D it can be obtained (where A,B,C,D are constants )
the solution for these equations can be determined only if the determinant of the
coefficients of A , B , C , and D vanishes, on solving the determinant.
(β2 - α2 / 2 α β)sin hβb sin αa + cos hβb cos αa = cos k ( a + b ) ---------- ---- (10)
The above equation is complicated and Kronig and Penney could conclude with the
equation. Hence they tried to modify this equation as follows
Let Vo is tending to infinite and b is approaching to zero. Such that Vob remains
finite. Therefore sin hβb → βb and cos hβb→1
β2 - α2 = ( 2 m / ħ2 ) (Vo – E ) – ( 2 m E / ħ2 )
= ( 2 m / ħ2 ) (Vo – E - E ) = ( 2 m / ħ2 ) (Vo – 2 E )
= 2 m Vo / ħ2 ( since Vo >> E )
Substituting all these values in equation (10) it verities as
( 2 m Vo / 2 ħ2 α β ) β b . sin α a + cos α a == cos k a
( m Vo b a / ħ2 ) ( sin α a / α a ) + cos α a == cos k a
( P / α a ) sin α a + cos α a == cos k a -------------------(11)
Where P = [ m Vo b a / ħ2 ] ------------------(12)
and is a measure of potential barrier strength.
The left hand side of the equation (11) is plotted as a function of α for the value of P
= 3 Π / 2 which is shown in fig, the right hand side one takes values between -1 to +1
as indicated by the horizontal lines in fig. Therefore the equation (11) is satisfied
only for those values of ka for which left hand side between ± 1.
From fig , the following conclusions are drawn.
The energy spectrum of the electron consists of a number of allowed and forbidden
energy bands.
The width of the allowed energy band increases with increase of energy values ie
increasing the values of αa. This is because the first term of equation(11)
decreases with increase of αa.
( P / α a ) sin α a + cos α a == 3 Π / 2
Fig. a) P=6pi b) p--> infinity c) p--> 0
With increasing P, ie with increasing potential barrier, the width of an allowed band
decreases. As P→∞, the allowed energy becomes infinitely narrow and the
energy spectrum is a line spectrum as shown in fig.
If P→∞, then the equation (11) has solution ie
Sin αa = 0
αa = ± n Π
α = ± n Π / a
α2 = n2 Π2 / a2
But α2 = 2 m E / ħ2
Therefore 2 m E / ħ2 = n2 Π2 / a2
E = [ħ2 Π2 / 2 m a2] n2
E = n h2 / 8 m a2 ( since ħ = h / 2 Π )
This expression shows that the energy spectrum of the electron contains discrete energy
levels separated by forbidden regions.
4) When P→0 then
Cos αa = cos ka
α = k , α2 = k2
but α2 = 2 m E / ħ2
therefore k2 = (h2 / 2 m ) ( 1 / λ2 ) = (h2 / 2 m ) (P2 / h2 )
E = P2 / 2 m
E =1/2mv2 ---------------- (14)
The equation (11) shows all the electrons are completely free to move in the crystal
without any constraints. Hence, no energy level exists ie all the energies are allowed to
the electrons and shown in fig (5). This case supports the classical free electrons theory.
[ ( P / α a ) sin α a + cos α a ] , P → 0
Velocity of the electron in periodic potential.:
According to quantum theory, an electron moving with a velocity can be treated as a
wave packet moving with the group velocity vg
v = vg = dω / dk -----------------(1)
where ω is the angular frequency of deBroglie wave and k = 2 Π / λ is the wave vector.
The energy of an electron can be expressed as
E = ħ ω --------------------- (2)
Differentiating the equation (2) with respect to k
dE / dk = ħ dω / dk -------------(3)
from (1) & (3)
vg = 1 / ħ (dE / dk ) -----------------(4)
According to band theory of solids, the variations of E with k as shown in fig(1). Using
this graph and equation (4), the velocity of electron can be calculated. The variation of
velocity with k is shown in fig(2). From this fig, it is clear that the velocity of electron is
zero at the bottom of the energy band. As the value of k increases, the velocity of
electron increases and reaches to maximum at K=k.Further ,the increases of k, the
velocity of electron decreases and reaches to zero at K= Π / a at the top of energy band.
Origin Of Energy Bands In Solids:
Solids are usually moderately strong, slightly elastic structures. The
individual atoms are held together in solids by interatomic forces or bonds. In addition to
these attractive forces, repulsive forces also act and hence solids are not easily
compressed.
The attractive forces between the atoms are basically electrostatic
in origin. The bonding is strongly dependent on the electronic structure of the atoms. The
attraction between the atoms brings them closer until the individual electron clouds begin
to overlap. A strong repulsive force arises to comply with Pauli’s exclusion principle.
When the attractive force and the repulsive force between any two atoms occupy a stable
position with a minimum potential energy. The spacing between the atoms under this
condition is called equilibrium spacing.
In an isolated atom, the electrons are tightly bound and have
discrete, sharp energy levels. When two identical atoms are brought closer , the outermost
orbits of these atoms overlap and interact. When the wave functions of the electrons on
different atoms begin to overlap considerably, the energy levels corresponding to those
wave functions split. if more atoms are brought together more levels are formed and for a
solid of N atoms, each of the energy levels of an atom splits into N levels of energy. The
levels are so close together that they form an almost continuous band. The width of this
band depends on the degree of overlap of electrons of adjacent atoms and is largest for
the outermost atomic electrons. In a solid many atoms are brought together so that the
split energy levels form a set of bands of very closely spaced levels with forbidden
energy gaps between them.
Classification Of Materials:
The electrons first occupy the lower energy bands and are of no importance in
determining many of the electrical properties of solids. Instead, the electrons in the higher
energy bands of solids are important in determining many of the physical properties of
solids. Hence the two allowed energy bands called valence and conduction bands are
required. The gap between these two allowed bands is called forbidden energy gap or
band gap since electrons can’t have any energy values in the forbidden energy gap. The
valence band is occupied by valence electrons since they are responsible for electrical,
thermal and optical properties of solids. above the valence band we have the conduction
band which is vacant at 0K. According to the gap between the bands and band occupation
by electrons, all solids can be classified broadly into two groups.
In the first group of solids called metals there is a partially filled
band immediately above the uppermost filled band .this is possible when the valence
band is partially filled or a completely filled valence band overlaps with the partially
filled conduction band.
In the second group of solids , there is a gap called band gap between the
completely filled valence band and completely empty conduction band. Depending on the
magnitude of the gap we can classify insulators and semiconductors.
Insulators have relatively wide forbidden band gaps. For typical
insulators the band gap Eg > 3 eV. On the other hand , semiconductors have relatively
narrow forbidden bands. For typical semiconductors Eg ≤ 1 eV.
Effective mass of the electron: When an electron in a period potential is
accelerated by an electric field (or) magnetic field, then the mass of the electron is called
effective mass ( m*).
Let an electron of charge ‘e’ and mass ‘m’ moving inside a crystal lattice of electric field
E.
Acceleration a = eE / m is not a constant in the periodic lattice of the crystal. It can be
considered that its variation is caused by the variation of electron’s mass when it moves
in the crystal lattice.
Therefore Acceleration a = eE / m*
Electrical force on the electron F = m* a --------------(1)
Considering the free electron as a wave packet , the group velocity vg corresponding to
the particle’s velocity can be written as
vg = dw / dk = 2 Π dv/ dk = (2 Π / h ) dE / dk ------------------(2)
where the energy E = h υ and ħ = h / 2 Π.
Acceleration a = d vg / dt = ( 1 / ħ ) d2E / dk dt = ( 1 / ħ ) ( d2E / dk2 ) dk / dt
Since ħ k = p and dp / dt = F,
dk / dt = F / ħ
Therefore a = ( 1 / ħ2 ) ( d2E / dk2 ) F
Or F = ( ħ2 / ( d2E / dk2 ) ) a -----------------------(3)
Comparing eqns . (1) and (3) we get
m*= ħ2 / ( d2E / dk2 ) ---------------(4)
This eqn indicates that the effective mass is determined by d2E / dk2 .
Questions:
1. Explain classical free- electron theory of metals.
2. Define electrical resistance
3. Give the basic assumptions of the classical free electron theory.
4. Explain the following: Drift velocity, mobility, relaxation time
and mean free path.
5. Based on free electron theory derive an expression for electrical
conductivity of metals.
6. Explain the failures of classical free theory.
7. Explain the salient features of quantum free electron theory.
8. Explain the Fermi- Dirac distribution for free electrons in a
metal. Discuss its variation with temperature.
9. Explain the following i) Effective mass, ii) Bloch theorem.
10. Discuss the band theory of solids based on Kronig –Penney
model. Explain the important features of this model.
11. Explain the origin of energy bands in solids.
12. Distinguish between metals, semiconductors and insulators.
UNIT – IV
DIELECTRIC PROPERTIES
Introduction: Dielectrics are insulating materials. In dielectrics, all the electrons are bound
to their parent molecules and there are no free charges. Even with normal voltage or thermal
energy, electrons are not released.
Electric Dipole: A system consisting of two equal and opposite charges separated by a
distance is called electric dipole.
Dipole moment: The product of charge and distance between two charges is called dipole
moment.
i e, μ = q x dl
Permittivity: It is a quantity, which represents the dielectric property of a medium.
Permittivity of a medium indicates the easily polarisable nature of the material.
Units: Faraday / Meter or Coulomb / Newton-meter.
Dielectric constant: The dielectric characteristics are determined by the dielectric constant.
The dielectric constant or relative permittivity of a medium is defined as the ratio between
the permittivity of the medium to the permittivity of the free space.
ε r = ε / ε 0 = C / C 0 where
ε is permittivity of the medium
ε 0 is permittivity of the free space
C is the capacitance of the capacitor with dielectric
C 0 is the capacitance of the capacitor without dielectric
Units: No Units.
Capacitance: The property of a conductor or system of conductor that describes its ability to
store electric charge.
C = q / V = A ε / d where
C is capacitance of capacitor
q is charge on the capacitor plate
V is potential difference between plates
A is area of capacitor plate
ε is permittivity of medium
d is distance between capacitor plates
Units: Farad .
Polarizability (α ) : When the strength of the electric field E is increased the strength of
the induced dipole μ also increases . Thus the induced dipole moment is proportional to
the intensity of the electric field.
μ = α E where α the constant of proportionality is called
polarizability .It can be defined as induced dipole moment per unit electric field.
α = μ / E
Units: Farad – meter2
Polarization Vector ( P ) : The dipole moment per unit volume of the dielectric material
is called polarization vector P .if μ is the average dipole moment per molecule and N is
the number of molecules per unit volume then polarization vector
P = N μ
The dipole moment per unit volume of the solid is the sum of all the individual dipole
moments within that volume and is called the polarization of the solid.
Electric Flux Density or Electric Displacement (D): The Electric Flux Density or
Electric Displacement at a point in the material is given by
D = ε r ε 0 E -------------(1) where
E is electric field strength
ε r is relative permittivity of material
ε 0 is permittivity of free space
As polarization measures additional flux density arising from the presence of the material
as compared to free space, it has same units as D.
Hence D = ε 0 E + P -----------(2)
Since D = ε 0 ε r E
ε 0 ε r E = ε 0 E + P
P = ε 0 ε r E - ε 0 E
P = ε 0(ε r - 1 ) E.
Electric Susceptibility ( χe ) : The polarization P is proportional to the total electronic
flux density E and is in the same direction of E . Therefore, the polarization vector can be
written as
P = ε 0 χe E
Therefore χe = P / ε 0 E = ε 0(ε r - 1 ) E / ε 0 E
χe = (ε r - 1 )
Dielectric Strength: It can be defined as the minimum voltage required for producing
dielectric breakdown. Dielectric strength decreases with raising the temperature,
humidity and age of the material.
Various polarization Process: polarization occurs due to several atomic
mechanisms. When a specimen is placed in a d.c. electric field, polarization is due to four
types of processes. They are
(1) electronic polarization
(2) ionic polarization
(3) orientation polarization and
(4) space charge polarization
Electronic Polarization: the process of producing electric dipoles which are oriented
along the field direction is called polarization in dielectrics
Consider an atom placed inside an electric field. The centre of
positive charge is displaced along the applied field direction while the centre of negative
charge is displaced in the opposite direction .thus a dipole is produced.
The displacement of the positively charged nucleus and the negative electrons of an atom
in opposite directions, on application of an electric field, result in electronic polarization.
Induced dipole moment
μ α E or μ = αe E where αe is electronic polarizability
Electronic polarizability is independent of temperature.
Derivation: Consider the nucleus of charge Ze is surrounded by an electron cloud of charge
-Ze distributed in a sphere of radius R.
Charge density ρ is given by
ρ = -Ze / ( 4/3ΠR3 ) = - (3/4) (Ze / ΠR3 ) ----------(1)
When an external field of intensity E is applied, the nucleus and electrons experiences
Lorentz forces in opposite direction. Hence the nucleus and electron cloud are pulled apart.
Then Coulomb force develops between them, which tends to oppose the
displacement. When Lorentz and coulomb forces are equal and opposite, equilibrium is
reached.
Let x be the displacement
Lorentz force = -Ze E (since = charge x applied field )
Coulomb force = Ze x [ charge enclosed in sphere of radius ‘x’ / 4 Π ε 0 x2 ]
Charge enclosed = ( 4 / 3 ) Π x3 ρ
= ( 4 / 3 ) Π x3 [( - 3 / 4 ) ( Z e / Π R3 ) ]
= - Z e x3 / R3
Therefore Coulomb force = ( Ze )( - Z e x3 / R3 ) / 4 Π ε 0 x2 = - Z2 e2 x / 4 Π ε 0 R3
At equilibrium, Lorentz force = Coulomb force
􀃖 -Ze E = - Z2 e2 x / 4 Π ε 0 R3
􀃖 E = -Ze x / 4 Π ε 0 R3
􀃖 or x = 4 Π ε 0 R3 E / Ze
Thus displacement of electron cloud is proportional to applied field.
The two charges +Ze and -Ze are separated by a distance ‘x‘ under applied field constituting
induced electric dipoles .
Induced dipole moment μe = Ze x
Therefore μe = Ze (4 Π ε 0 R3 E / Ze ) = 4 Π ε 0 R3 E
Therefore μe α E , μe = αe E where αe = 4 Π ε 0 R3 is electronic polarizability
The dipole moment per unit volume is called electronic polarization. It is independent of
temperature.
P = N μe = N αe E where
N is Number of atoms / m3
Pe = N (4 Π ε 0 R3 E ) = 4 Π ε 0 R3 N E where
R is radius of atom
Electric Susceptibility χ = P / ε 0 E
Therefore P = ε 0 E χ
P = (4 Π R3 N) ε 0 E where χ = 4 Π R3 N
Also Pe = ε 0 E (ε r - 1) = N αe E
Or ε r - 1 = N αe / ε 0
Hence αe = ε 0 (ε r - 1 ) / N .
Ionic Polarization: It is due to the displacement of cat ions and anions in opposite
directions and occurs in an ionic solid .
Consider a NaCl molecule. Suppose an electric field is applied in the positive direction . The
positive ion moves by x1 and the negative ion moves by x2
Let M is mass of positive ion
M is mass of negative ion
x1 is displacement of positive ion
x2 is displacement of negative ion
Total displacement x = x1 + x2 --------------(1)
Lorentz force on positive ion = + e E ----------(2)
Lorentz force on negative ion = - e E ---------- (3)
Restoring force on positive ion = -k1 x1---------- (2 a)
Restoring force on negative ion = +k2 x2---------- (3 a) where k1, k2 Restoring force constants
At equilibrium, Lorentz force and restoring force are equal and opposite
For positive ion, e E = k1 x1
For negative ion, e E = k2 x2 ] ---------- (4)
Where k1 = M ω0
2 & K2 = m ω0
2 where ω0 is angular velocity of ions
Therefore x = x1 + x2 = ( e E / ω0
2 ) [ 1/M + 1/m ] ------------(5)
From definition of dipole moment
μ = charge x distance of separation
μ = e x = ( e2 E / ω0
2 ) [ 1/M + 1/m ] ------------(6)
But μ α E or μ = αi E
Therefore αi = (e2 / ω0
2) [1/M + 1/m]
This is ionic polarizability.
Orientational Polarization:
In methane molecule, the centre of negative and positive charges coincides, so that there
is no permanent dipole moment. On the other hand, in certain molecules such as Ch3Cl,
the positive and negative charges do not coincide .Even in the absence of an electric field,
this molecule carries a dipole moment, and they tend to align themselves in the direction
of applied field. The polarization due to such alignment is called orientation polarization.
It is dependent on temperature. With increase of temperature the thermal energy tends to
randomize the alignment.
Orientation polarization Po = Nμ = Nμ2E / 3kT
= N α0 E
Therefore Orientation polarizability α0 = Po / NE = μ2 / 3kT
Thus orintational polarizability α0 is inversely proportional t absolute temperature of
material.
Internal field or Local field or Lorentz field: Internal field is the total electric field at
atomic site.
Internal field A = E1 + E2 + E3 + E4 ------- (I) where
E1 is field intensity due to charge density on plates
E2 is charge density induced on two sides of dielectric
E3 is field intensity due to other atoms in cavity and
E4 is field intensity due to polarization charges on surface of cavity
Field E1 : E1 is field intensity due to charge density on plates
From the field theory
E1 = D / ε 0
D = P + ε 0 E
Therefore E1 = P + ε 0 E / ε 0 = E + P / ε 0 ---------- (1)
Field E2: E2 is the field intensity at A due to charge density induced on two sides of dielectric
Therefore E2 = - P / ε 0 -----------(2)
Field E3: E3 is field intensity at A due to other atoms contained in the cavity and for a cubic
structure,
E3 = 0 because of symmetry. ----------- (3)
Field E4: E4 is field intensity due to polarization charges on surface of cavity and was
calculated by Lorentz in the following way:
If dA is the surface area of the sphere of radius r lying between θ and θ + dθ, where θ is the
direction with reference to the direction of applied force.
Then dA = 2 Π (PQ) (QR)
But sin θ = PQ / r => PQ = r sin θ
And dθ = QR / r => QR = r dθ
Hence dA = 2 Π (r sin θ) (r dθ) = 2 Π r2 sin θ dθ
Charge on surface dA is dq = P cos θ dA (cos θ is normal component)
dq = P cos θ (2 Π r2 sin θ dθ) = P (2 Π r2 sin θ cos θ dθ )
The field due to the charge dq at A, is denoted by dE4 in direction θ = 0
dE4 = dq cos θ / 4 Π ε 0 r2 = P (2 Π r2 sin θ cos θ dθ) cos θ
dE4 = P sin θ cos2 θ dθ / 2 ε 0
∫ dE4 = P / 2 ε 0 ∫0
Π sin θ cos2 θ dθ = P / 2 ε 0 ∫0
Π cos2 θ d (- cos θ)
Let cos θ = x
∫ dE4 = - P / 2 ε 0 ∫0
Π x2 dx
Therefore E4= - P / 2 ε 0 [x3 / 3]0
Π
= - P / 2 ε 0 [cos3 θ/ 3] 0
Π = - P / 6 ε 0 [-1 – 1] = P / 3 ε 0 ------- (4)
Local field Ei = E1 + E2 + E3 + E4
= E + P / ε 0 - P / ε 0 + 0 + P / 3 ε 0
= E + P / 3 ε 0
Clausius – Mosotti Relation:
Let us consider the elemental dielectric having cubic structure. Since there are no ions
and perment dipoles in these materials, them ionic polarizability αi and orientational
polarizability α0 are zero.
i.e. αi = α0 = 0
Hence polarization P = N αe Ei
= N αe ( E + P / 3ε0)
i.e. P [ 1 - N αe / 3 ε0 ] = N αe E
P = N αe E / P [ 1 - N αe / 3 ε0 ] -------------􀃆 1
D = P + ε0E
P = D - ε0E
Dividing on both sides by E
P / E = D / E - ε0 = ε - ε0 = ε0 εr - ε0
P = E ε0 (εr - 1) ----------------------------􀃆 2
From eqn 1 and 2 , we get
P = E ε0 (εr - 1) = N αe E / [ 1 - N αe / 3 ε0 ]
[ 1 - N αe / 3 ε0 ] = N αe / ε0 (εr - 1)
1 = N αe / 3 ε0 + N αe / ε0 (εr - 1)
1 = (N αe / 3 ε0 ) ( 1 + 3 / (εr - 1))
1 = (N αe / 3 ε0 )[ (εr - 1 + 3) / (εr - 1) ]
1 = (N αe / 3 ε0 )[ (εr + 2) / (εr - 1) ]
(εr + 2) / (εr - 1) = N αe / 3 ε0 Where N – no of molecules per unit volume
This is Clausius – Mosotti Relation.
Dielectric Breakdown : The dielectric breakdown is the sudden change in state of a
dielectric material subjected to a very high electric field , under the influence of which , the
electrons are lifted into the conduction band causing a surge of current , and the ability of the
material to resist the current flow suffers a breakdown .
Or
When a dielectric material loses its resistivity and permits very large current to flow through it,
then the phenomenon is called dielectric breakdown.
There are many factors for dielectric breakdown which are (1) Intrinsic breakdown (2) Thermal
breakdown (3) Discharge breakdown (4) Electro Chemical breakdown (5) Defect breakdown.
(1) Intrinsic breakdown: The dielectric strength is defined as the breakdown voltage per unit
thickness of the material. When the applied electric field is large, some of the electrons in
the valence band cross over to the conduction band across the large forbidden energy gap
giving rise to large conduction currents. The liberation or movement of electrons from
valence band is called field emission of electrons and the breakdown is called the intrinsic
breakdown or zener breakdown.
The number of covalent bonds broken and the number of charge
carriers released increases enormously with time and finally dielectric breakdown occurs. This
type of breakdown is called Avalanche breakdown.
(2) Thermal breakdown: It occurs in a dielectric when the rate of heat generation is greater
than the rate of dissipation. Energy due to the dielectric loss appears as heat. If the rate of
generation of heat is larger than the heat dissipated to the surrounding, the temperature of
the dielectric increases which eventually results in local melting .once melting starts, that
particular region becomes highly conductive, enormous current flows through the material
and dielectric breakdown occurs.
Thus thermal breakdown occurs at very high temperatures. Since the
dielectric loss is directly proportional to the frequency, for a.c fields, breakdown occurs at
relatively lower field strengths.
(3) Discharge breakdown: Discharge breakdown is classified as external or internal. External
breakdown is generally caused by a glow or corona discharge .Such discharges are
normally observed at sharp edges of electrodes. It causes deterioration of the adjacent
dielectric medium. It is accompanied by the formation of carbon so that the damaged areas
become conducting leading to power arc and complete failure of the dielectric. Dust or
moisture on the surface of the dielectric may also cause external discharge breakdown.
Internal breakdown occurs when the insulator contains blocked gas bubbles .If large
number of gas bubbles is present, this can occur even at low voltages.
(4) Electro Chemical breakdown: Chemical and electro chemical breakdown are related to
thermal breakdown. When temperature rises, mobility of ions increases and hence
electrochemical reaction takes place. When ionic mobility increases leakage current also
increases and this may lead to dielectric breakdown. Field induced chemical reaction
gradually decreases the insulation resistance and finally results in breakdown.
(5) Defect breakdown: if the surface of the dielectric material has defects such as cracks and
porosity, then impurities such as dust or moisture collect at these discontinuities leading to
breakdown. Also if it has defect in the form of strain in the material, that region will also
break on application of electric field.
Frequency dependence of polarizability:
On application of an electric field, polarization process occurs as a
function of time. The polarization P(t) as a function of time. The polarization P(t) as a
function of time t is given by
P(t) = P[ 1- exp ( - t / tr )]
Where P – max. Polarization attained on prolonged application of static field.
tr - relaxation time for particular polarization process
The relaxation time tr is a measure of the time scale of polarization process. It is the time
taken for a polarization process to reach 0.63 of the max. value.
Electronic polarization is extremely rapid. Even when the frequency of the applied
voltage is very high in the optical range (≈1015 Hz), electronic polarization occurs during
every cycle of the applied voltage.
Ionic polarization is due to displacement of ions over a small distance due to the applied
field. Since ions are heavier than electron cloud, the time taken for displacement is larger.
The frequency with which ions are displaced is of the same order as the lattice vibration
frequency (≈1013Hz). Hence, at optical frequencies, there is no ionic polarization. If the
frequency of the applied voltage is less than 1013 Hz, the ions respond.
Orientation polarization is even slower than ionic polarization. The relaxation time for
orientation polarization in a liquid is less than that in a solid. Orientation polarization
occurs, when the frequency of applied voltage is in audio range (1010 Hz).
Space charge polarization is the slowest process, as it involves the diffusion of ions over
several interatomic distances. The relaxation time for this process is related to frequency
of ions under the influence of applied field. Space charge polarization occurs at power
frequencies (50-60 Hz).
Piezo – Electricity: These materials have the property of becoming electrically polarized
when mechanical stress is applied. This property is known as Piezo – electric effect has an
inverse .According to inverse piezo electric effect, when an electric stress or voltage is applied,
the material becomes strained. The strain is directly proportional to the applied field E.
When piezo electric crystals are subjected to compression or tension,
opposite kinds of charges are developed at the opposite faces perpendicular to the direction of
applied force. The charges produced are proportional to the applied force.
Piezo – Electric Materials and Their Applications: Single crystal of quartz is used for
filter, resonator and delay line applications. Natural quartz is now being replaced by synthetic
material.
Rochelle salt is used as transducer in gramophone pickups, ear phones,
hearing aids, microphones etc. the commercial ceramic materials are based on barium titanate,
lead zirconate and lead titanate. They are used for high voltage generation (gas lighters),
accelerometers, transducers etc.
Piezo electric semiconductors such as GaS, ZnO & CdS are used as
amplifiers of ultrasonic waves.
Ferro electricity: Ferro electric materials are an important group not only because of intrinsic
Ferro electric property, but because many possess useful piezo electric, birefringent and electro
optical properties.
The intrinsic Ferro electric property is the possibility of reversal or
change of orientation of the polarization direction by an electric field. This leads to hysteresis
in the polarization P, electric field E relation , similar to magnetic hysteresis. Above a critical
temperature, the Curie point Tc, the spontaneous polarization is destroyed by thermal disorder.
The permittivity shows a characteristic peak at Tc.
Pyroelectricity: It is the change in spontaneous polarization when the temperature of
specimen is changed.
Pyroelectric coefficient ‘λ’ is defined as the change in polarization per unit temperature
change of specimen.
λ= dP / dT
change in polarization results in change in external field and also changes the surface.
Required Qualities of Good Insulating Materials: The required qualities can be classified as
under electrical, mechanical, thermal and chemical applications.
i) Electrical: 1. electrically the insulating material should have high electrical resistivity and
high dielectric strength to withstand high voltage.
2 .The dielectric losses must be minimum.
3. Liquid and gaseous insulators are used as coolants. For example transformer oil, hydrogen
and helium are used both as insulators and coolant.
ii) Mechanical: 1. insulating materials should have certain mechanical properties depending on
the use to which they are put.
2. When used for electric machine insulation, the insulator should have sufficient mechanical
strength to withstand vibration.
iii) Thermal: Good heat conducting property is also desirable in such cases. The insulators
should have small thermal expansion and it should be non-ignitable.
iv) Chemical: 1. chemically, the insulators should be resistant to oils, liquids, gas fumes, acids
and alkali’s.
2. The insulators should be water proof since water lowers the insulation resistance and the
dielectric strength.
MAGNETIC PROPERTIES
Introduction : The basic aim in the study of the subject of magnetic materials is to
understand the effect of an external magnetic field on a bulk material ,and also to account for
its specific behavior. A dipole is an object that a magnetic pole is on one end and a equal and
opposite second magnetic dipole is on the other end.
A bar magnet can be considered as a dipole with a north pole at one end
and South Pole at the other. If a magnet is cut into two, two magnets or dipoles are created out
of one. This sectioning and creation of dipoles can continue to the atomic level. Therefore, the
source of magnetism lies in the basic building block of all the matter i.e. the atom.
Consider electric current flowing through a conductor. When the electrons
are flowing through the conductor, a magnetic field is forms around the conductor. A magnetic
field is produced whenever an electric charge is in motion. The strength of the field is called
the magnetic moment.
Magnetic materials are those which can be easily magnetized as they have
permanent magnetic moment in the presence of applied magnetic field. Magnetism arises from
the magnetic dipole moments. It is responsible for producing magnetic influence of attraction
or repulsion.
Magnetic dipole : it is a system consisting of two equal and opposite magnetic poles separated
by a small distance of ‘2l’metre.
Magnetic Moment ( μm ) :It is defined as the product of the pole strength (m) and the
distance between the two poles (2l) of the magnet.
i . e . . μm = (2l ) m
Units: Ampere – metre2
Magnetic Flux Density or Magnetic Induction (B): It is defined as the number of magnetic
lines of force passing perpendicularly through unit area.
i . e . . B = magnetic flux / area = Φ / A
Units: Weber / metre2 or Tesla.
Permeability:
Magnetic Field Intensity (H): The magnetic field intensity at any point in the magnetic field is
the force experienced by a unit north pole placed at that point.
Units: Ampere / meter
The magnetic induction B due to magnetic field intensity H applied in vacuum is related by
B = μ0 H where μ0 is permeability of free space = 4 Π x 10-7 H / m
If the field is applied in a medium, the magnetic induction in the solid is given by
B = μ H where μ is permeability of the material in the medium
μ = B / H
Hence magnetic Permeability μ of any material is the ratio of the magnetic induction to the
applied magnetic field intensity. The ratio of μ / μ0 is called the relative permeability (μr ).
μr = μ / μ0
Therefore B = μ0 μr H
Magnetization: It is the process of converting a non – magnetic material into a magnetic
material. The intensity of magnetization (M) of a material is the magnetic moment per unit
volume. The intensity of magnetization is directly related to the applied field H through the
susceptibility of the medium (χ) by
χ = M / H ------------(1)
The magnetic susceptibility of a material is the ratio of the intensity of magnetization produced
to the magnetic field intensity which produces the magnetization. It has no units.
We know
B = μ H
= μ0 μr H
i.e B = μ0 μr H + μ0 H - μ0 H
= μ0 H + μ0 H ( μr – 1 )
= μ0 H + μ0 M where M is magnetization = H ( μr – 1 )
i.e B = μ0 ( H + M ) ----------(2)
The first term on the right side of eqn (2) is due to external field. The second term is due to the
magnetization.
Hence μ0 = B / H + M
Relative Permeability ,
μr = μ / μ0 = ( B / H ) / ( B / H + M ) = H + M / H = 1 + M / H
μr = 1 + χ ---------(3)
The magnetic properties of all substances are associated with the orbital and spin motions of
the electrons in their atoms. Due to this motion, the electrons become elementary magnets of
the substance. In few materials these elementary magnets are able to strengthen the applied
magnetic field , while in few others , they orient themselves such that the applied magnetic
field is weakened.
Origin of Magnetic Moment : In atoms , the permanent magnetic moments can arise due to the
following :
1. the orbital magnetic moment of the electrons
2. the spin magnetic moment of the electrons
3. the spin magnetic moment of the nucleus.
Orbital magnetic moment of the electrons: In an atom, electrons revolve round the nucleus in
different circular orbits.
Let m be the mass of the electron and r be the radius of the orbit in which it moves with
angular velocity ω.
The electric current due to the moving electron I = - ( number of electrons flowing per second
x charge of an electron )
Therefore I = - e ω / 2 Π --------------(1)
The current flowing through a circular coil produces a magnetic field in a direction
perpendicular to the area of coil and it is identical to the magnetic dipole. the magnitude of the
magnetic moment produced by such a dipole is
μm = I .A
= ( - e ω / 2 Π ) ( Π r2 )
= - e ω r2 / 2 = ( - e / 2 m ) ( m ω r2 ) = - ( e / 2 m ) L -----------(2)
where L = m ω r2 is the orbital angular momentum of electron. The minus sign
indicates that the magnetic moment is anti – parallel to the angular momentum L. A substance
therefore possesses permanent magnetic dipoles if the electrons of its constituent atom have a
net non-vanishing angular momentum. The ratio of the magnetic dipole moment of the electron
due to its orbital motion and the angular momentum of the orbital motion is called orbital gyro
magnetic ratio , represented by γ.
Therefore γ = magnetic moment / angular momentum = e / 2m
The angular momentum of an electron is determined by the orbital quantum number ‘l’ given
by l = 0 , 1 , 2 , ……( n – 1 ) where n is principal quantum number n = 1 , 2 , 3 , 4 , ……
……corresponding to K , L , M , N……shells .
The angular momentum of the electrons associated with a particular value of l is given by l( h /
2 Π )
The strength of the permanent magnetic dipole is given by
μ el = - ( e / 2 m ) ( l h / 2 Π )
i.e μ el = - ( e h l / 4 Π m ) = - μB l ---------------(3)
The quantity μB = e h / 4 Π m is an atomic unit called Bohr Magneton and has a value 9.27
x 10 -24 ampere metre2
In an atom having many electrons, the total orbital magnetic moment is determined by taking
the algebraic sum of the magnetic moments of individual electrons. The moment of a
completely filled shell is zero. An atom with partially filled shells will have non zero orbital
magnetic moment.
Magnetic Moment Due to Electron Spin : The magnetic moment associated with spinning of
the electron is called spin magnetic moment μ es .Magnetic moment due to the rotation of the
electronic charge about one of the diameters of the electron is similar to the earth’s spinning
motion around it’s north – south axis.
An electronic charge being spread over a spherical volume ,the electron spin would cause
different charge elements of this sphere to form closed currents, resulting in a net spin
magnetic moment. This net magnetic moment would depend upon the structure of the electron
and its charge distribution.
μ es = γ ( e / 2 m ) S -------------------(1)where S= h / 4 Π is spin angular momentum
therefore μ es ≈ 9.4 x 10 -24 ampere metre2
Thus, the magnetic moments due to the spin and the orbital motions of an electron are of the
same order of magnitude. The spin and electron spin magnetic moment are intrinsic properties
of an electron and exist even for a stationary electron. Since the magnitude of spin magnetic
moment is always same, the external field can only influence its direction. If the electron spin
moments are free to orient themselves in the direction of the applied field B. In a varying field
,it experiences a force in the direction of the increasing magnetic field due to spin magnetic
moments of its various electrons.
Magnetic Moment due to Nuclear Spin : Another contribution may arise from the nuclear
magnetic moment. By analogy with Bohr Magneton, the nuclear magneton arises due to spin of
the nucleus. It is given by
μ ps = e h / 4 Π Mp
μ ps = 5.05 x 10 -27 ampere metre2 where Mp is mass of proton.
The nuclear magnetic moments are smaller than those associated with electrons.
Classification Of Magnetic Materials :All matter respond in one way or the other when
subjected to the influence of a magnetic field. The response could be strong or weak, but there
is none with zero response ie, there is no matter which is non magnetic in the absolute sense.
Depending upon the magnitude and sign of response to the applied field , and also on the basis
of effect of temperature on the magnetic properties, all materials are classified broadly under 3
categories.
1. Diamagnetic materials 2. Paramagnetic materials, 3. Ferromagnetic materials
two more classes of materials have structure very close to ferromagnetic
materials but possess quite different magnetic effects. They are i. Anti ferromagnetic
materials and ii . Ferri magnetic materials
1. Diamagnetic materials: Diamagnetic materials are those which experience a repelling
force when brought near the pole of a strong magnet. In a non uniform magnetic field they
are repelled away from stronger parts of the field.
In the absence of an external magnetic field , the net magnetic dipole
moment over each atom or molecule of a diamagnetic material is zero.
Ex: Cu, Bi , Pb .Zn and rare gases.
Paramagnetic materials: Paramagnetic materials are those which experience a feeble
attractive force when brought near the pole of a magnet. They are attracted towards the
stronger parts of magnetic field. Due to the spin and orbital motion of the electrons, the
atoms of paramagnetic material posses a net intrinsic permanent moment.
Susceptibility χ is positive and small for these materials. The susceptibility is inversely
proportional to the temperature T.
χ α 1/T
χ = C/T where C is Curie’s temperature.
Below superconducting transition temperatures, these materials exhibit the Para
magnetism.
Examples: Al, Mn, Pt, CuCl2.
Ferromagnetic Materials: Ferromagnetic materials are those which experience a very
strong attractive force when brought near the pole of a magnet. These materials, apart
from getting magnetized parallel to the direction of the applied field, will continue to
retain the magnetic property even after the magnetizing field removed. The atoms of
ferromagnetic materials also have a net intrinsic magnetic dipole moment which is due to
the spin of the electrons.
Susceptibility is always positive and large and it depends upon temperature.
χ = C / (T- θ) ( only in paramagnetic region i.e., T > θ)
θ is Curie’s temperature.
When the temperature of the material is greater than its Curie temperature then it converts
into paramagnetic material.
Examples: Fe, Ni, Co, MnO.
Antiferromagnetic matériels : These are the ferromagnetic materials in which equal
no of opposite spins with same magnitude such that the orientation of neighbouring spins
is in antiparallel manner are present.
Susceptibility is small and positive and it is inversely proportional to the temperature.
χ=C /(T+θ)
the temperature at which anti ferromagnetic material converts into paramagnetic material
is known as Neel’s temperature.
Examples: FeO, Cr2O3.
Ferrimagnetic materials: These are the ferromagnetic materials in which equal no of
opposite spins with different magnitudes such that the orientation of neighbouring spins
is in antiparallel manner are present.
Susceptibility positive and large, it is inversely proportional to temperature
χ=C /(T ± θ) T> TN ( Neel’s temperature)
Examples : ZnFe2O4, CuFe2O4
Domain theory of ferromagnetism: According to Weiss, a virgin specimen of
ferromagnetic material consists of a no of regions or domains (≈ 10-6 m or larger) which
are spontaneously magnetized. In each domain spontaneous magnetization is due to
parallel alignment of all magnetic dipoles. The direction of spontaneous magnetization
varies from domain to domain. The resultant magnetization may hence be zero or nearly
zero. When an external field is applied there are two possible ways of alignment fo a
random domain.
i). By motion of domain walls: The volume of the domains that are favourably oriented
with respect to the magnetizing field increases at the cost of those that are unfavourably
oriented
ii) By rotation of domains: When the applied magnetic field is strong, rotation of the
direction of magnetization occurs in the direction of the field.
Hysteresis curve (study of B-H curve): The hysteresis of ferromagnetic materials
refers to the lag of magnetization behind the magnetization field. when the temperature of
the ferromagnetic substance is less than the ferromagnetic Curie temperature ,the
substance exhibits hysteresis. The domain concept is well suited to explain the
phenomenon of hysteresis. The increase in the value of the resultant magnetic moment of
the specimen by the application of the applied field , it attributes to the 1. motion of the
domain walls and 2. rotation of domains.
When a weak magnetic field is applied, the domains that are
aligned parallel to the field and in the easy direction of magnetization , grow in size at the
expense of less favorably oriented ones. This results in Bloch wall movement and when
the weak field is removed, the domains reverse back to their original state. This reverse
wall displacement is indicated by OA of the magnetization curve. When the field
becomes stronger ,the Bloch wall movement continues and it is mostly irreversible
movement. This is indicated by the path AB of the graph. The phenomenon of hysteresis
is due to this irreversibility.
At the point B all domains have got magnetized along their easy directions. Application
of still higher fields rotates the domains into the field direction which may be away from
the easy direction. Once the domain rotation is complete the specimen is saturated
denoted by C. on removal of the field the specimen tends to attain the original
configuration by the movement of Bloch walls. But this movement is hampered by the
impurities, lattice imperfections etc, and so more energy must be supplied to overcome
the opposing forces. This means that a coercive field is required to reduce the
magnetization of the specimen to zero. The amount of energy spent in this regard is a
loss. Hysteresis loss is the loss of energy in taking a ferromagnetic body through a
complete cycle of magnetization and this loss is represented by the area enclosed by the
hysteresis loop.
A hysteresis curve shows the relationship between the magnetic
flux density B and applied magnetic field H. It is also referred to as the B-H curve(loop).
Hard and Soft Magnetic Materials:
Hysteresis loop of the ferromagnetic materials vary in size and shape. This variation in
hysteresis loops leads to a broad classification of all the magnetic materials into hard type
and soft type.
Hard Magnetic Materials:
Hard magnetic materials are those which are characterized by large
hysteresis loop because of which they retain a considerable amount of their magnetic
energy after the external magnetic field is switched off. These materials are subjected to a
magnetic field of increasing intensity, the domain walls movements are impeded due to
certain factors. The cause for such a nature is attributed to the presence of impurities or
non-magnetic materials, or the lattice imperfections. Such defects attract the domain
walls thereby reducing the wall energy. It results in a stable state for the domain walls
and gives mechanical hardness to the material which increases the electrical resistivity.
The increase in electrical resisitivity brings down the eddy current loss if used in a.c
conditions. The hard magnetic materials can neither be easily magnetized nor easily
demagnetized.
Properties:
High remanent magnetization
High coercivity
High saturation flux density
Low initial permeability
High hysteresis energy loss
High permeability
The eddy current loss is low for ceramic type and large for metallic type.
Examples of hard magnetic materials are, i) Iron- nickel- aluminum alloys with certain
amount of cobalt called Alnico alloy. ii) Copper nickel iron alloys. iii) Platinum cobalt
alloy.
Applications of hard magnetic materials: For production of permanent magnets, used in
magnetic detectors, microphones, flux meters, voltage regulators, damping devices and
magnetic separators.
Soft Magnetic Materials:
Soft magnetic materials are those for which the hysteresis loops enclose very small
area. They are the magnetic materials which cannot be permanently magnetized. In these
materials ,the domain walls motion occurs easily. Consequently, the coercive force
assumes a small value and makes the hysteresis loop a narrow one because of which, the
hysteresis loss becomes very small. For the sane reasons, the materials can be easily
magnetized and demagnetized.
Properties:
Low remanent magnetization
Low coercivity
Low hysteresis energy loss
Low eddy current loss
High permeability
High susceptibility
Examples of soft magnetic materials are
i) Permalloys ( alloys of Fe and Ni)
ii) Si – Fe alloy
iii) Amorphous ferrous alloys ( alloys of Fe, Si, and B)
iv) Pure Iron (BCC structure)
Applications of soft magnetic materials: Mainly used in electro- magnetic machinery and
transformer cores. They are also used in switching circuits, microwave isolators and
matrix storage of computers.
Questions:
Describe how polarization occurs in a dielectric material.
Define dielectric constant of a material.
Explain the origin of different kinds of polarization.
Describe in brief various types of polarization.
Obtain an expression for the internal field.
Derive Clausius - Mossotti equation.
Describe the frequency dependence of dielectric constant.
Write note on Dielectric loss.
Explain the properties of ferroelectric materials.
What is piezoelectricity?
Distinguish between dia, para, ferro, antiferro,and ferromagnetic materials.
what is meant by Neel temperature
Define magnetization and show that B = μ0 (H + M).
Explain the origin of magnetic moment.
Decribe the domain theory of ferromagnetism.
What is Bhor Magneton.
Draw and explain the hysteresis curve.
Discuss the characteristic features of soft and hard magnetic materials.
What are the applications of soft and hard magnetic materials?
UNIT - V
SEMICONDUCTORS
INTRODUCTION:
Semiconductors are materials whose electronic properties are intermediate between
those of conductors and insulators. These electrical properties of a solid depend on its
band structure. A semiconductor has two bands of importance (neglecting bound
electrons as they play no part in the conduction process) the valence and the conduction
bands. They are separated by a forbidden energy gap. At OK the valence band is full
and the conduction band is empty, the semiconductor behaves as an insulator.
Semiconductor has both positive (hole) and negative (electron) carriers of electricity
whose densities can be controlled by doping the pure semiconductor with chemical
impurities during crystal growth.
At higher temperatures, electrons are transferred across the gap into the conduction band
leaving vacant levels in the valence band. It is this property that makes the
semiconductor a material with special properties of electrical conduction.
Generally there are two types of semiconductors. Those in which electrons and holes are
produced by thermal activation in pure Ge and Si are called intrinsic semiconductors. In
other type the current carriers, holes or free electrons are produced by the addition of
small quantities of elements of group III or V of the periodic table, and are known as
extrinsic semiconductors. The elements added are called the impurities or dopants.
Intrinsic semiconductors:
A pure semiconductor which is not doped is termed as intrinsic semiconductor. In Si
crystal, the four valence electrons of each Si atom are shared by the four surrounding Si
atoms. An electron which may break away from the bond leaves deficiency of one
electron in the bond. The vacancy created in a bond due to the departure of an electron is
called a hole. The vacancy may get filled by an electron from the neighboring bond, but
the hole then shifts to the neighboring bond which in turn may get filled by electron from
another bond to whose place the hole shifts, and so on thus in effect the hole also
undergoes displacement inside a crystal. Since the hole is associated with deficiency of
one electron, it is equivalent for a positive charge of unit magnitude. Hence in a
semiconductor, both the electron and the hole act as charge carriers.
In an intrinsic semiconductor, for every electron freed from the bond, there will be one
hole created. It means that, the no of conduction electrons is equal to the no of holes at
any given temperature. Therefore there is no predominance of one over the other to be
particularly designated as charge carriers.
Carriers Concentration in intrinsic semiconductors:
A broken covalent bond creates an electron that is raised in energy, so as to
occupy the conduction bond, leaving a hole in the valence bond. Both electrons and holes
contribute to overall conduction process.
In an intrinsic semiconductor, electrons and holes are equal in numbers. Thus
n = p = ni
Where n is the number of electrons in the conduction band in a unit volume of the
material (concentration), p is the number of holes in the valance band in a unit volume of
the material. And ni, the number density of charge carriers in an intrinsic semi conductor.
It is called intrinsic density.
For convenience, the top of the valence bond is taken as a zero energy reference level
arbitrarily.
The number of electrons in the conduction bond is
n = N P(Eg)
Where P(Eg) is the probability of an electron having energy Eg. It is given by Fermi
Dirac function eqn., and N is the total number of electrons in both bands.
Thus,
N
n = -----------------------------------
1 + exp [(Eg – EF)/KT]
Where EF is the Fermi Level
The probability of an electron being in the valence bond is given by putting Eg = 0 in
eqn. Hence, the number of electrons in the valence bond is given by
N
nv = ------------------------------
1 + exp(-EF/KT)
The total number of electrons in the semiconductor. N is the sum of those in the
conduction band n and those in the valence bond nv. Thus,
N N
N = -------------------------------- + --------------------------
1 + exp [ (Eg – EF) / KT ] 1 + exp(-EF/KT)
For semiconductors at ordinary temperature, Eg >> KT as such in equation one may be
neglected when compared with exp Eg – EF Then
RT
1 1
1 = +
exp Eg – EF 1 + exp – EF
RT RT
Rearranging the terms, we get
- Eg + EF exp (-EF/RT)
exp =
RT 1 + exp (-EF/RT)
-EF
≈ exp
RT
2 EF - Eg
or exp = 1
RT
This leads to
EF = Eg/2
Thus in an intrinsic semiconductor, the Fermi level lies mid way between the conduction
and the valence bonds. The number of conduction electrons at any temperature T is
given by
N
n = (∴ EF = Eg/2)
1 + exp(Eg/2KT)
In eqn may be approximated as
n ≈ N exp(-Eg / 2RT)
From the above discussion, the following conclusions may be drawn.
a) The number of conduction electrons and hence the number of holes in an intrinsic
semiconductor, decreases exponentially with increasing gap energy Eg this accounts
for lack of charge carries in insulator of large forbidden energy gap.
b) The number of available charge carries increases exponentially with increasing
temperature.
The above treatment is only approximate as we have assumed that all states in a bond
have the same energy. Really it is not so. A more rigorous analysis must include
additions terms in eqn.
The no of conduction bond, in fact is given by
n = ∫ S(E) P(E) dE
Where S(E) is the density of available states in the energy range between E and E + dE,
and P(E) is the probability, that an electron can occupy a state of energy E.
S(E) 8√2 π m3/2 E1/2
n3
Inclusion of S(E) and integration over the conduction bond leads to
n = Ne exp [(-Eg-EF)/RT]
In a similar way, we arrive at
p = NV exp [ -EF/RT]
If we multiply eq: we get
np = ni
2 = Ne NV exp(-Eg/RT)
For the intrinsic material
Ni = 2 (2πRT)3/2 (me* mn*)3/4 exp(-Eg/2RT)
h2
Notice that this expression agress with the less rigorous one derived earlier since the
temperature dependence is largely controlled by the rapidly varying exponential term.
EXTRINSIC SEMICONDUCTORS:
Intrinsic Semiconductors are rarely used in semiconductor devices as their
conductivity is not sufficiently high. The electrical conductivity is extremely sensitive to
certain types of impurity. It is the ability to modify electrical characteristics of the
material by adding chosen impurities that make extrinsic semiconductors important and
interesting.
Addition of appropriate quantities of chosen impurities is called doping, usually, only
minute quantities of dopants (1 part in 103 to 1010) are required. Extrinsic or doped
semiconductors are classified into main two main types according to the type of charge
carries that predominate. They are the n-type and the p-type.
N-TYPE SEMICONDUCTORS:
Doping with a pentavalent impurity like phosphorous,
arsenic or antimony the semiconductor becomes rich in conduction electrons. It is called
n-type the bond structure of an n-type semiconductor is shown in Fig below.
Even at room temperature, nearby all impurity atoms lose an electron into the conduction
bond by thermal ionization. The additional electrons contribute to the conductivity in
the same way as those excited thermally from the valence bond. The essential difference
beam the two mechanisms is that ionized impurities remain fixed and no holes are
produced. Since penta valent impurities denote extra carries elections, they are called
donors.
P-TYPE SEMICONDUCTORS:-
p-type semiconductors have holes as majority charge carries. They are
produced by doping an intrinsic semiconductor with trivalent impurities.(e.g. boron,
aluminium, gallium, or indium). These dopants have three valence electrons in their
outer shell. Each impurity is short of one electron bar covalent bonding. The vacancy
thus created is bound to the atom at OK. It is not a hole. But at some higher temperature
an electron from a neighbouring atom can fill the vacancy leaving a hole in the valence
bond for conduction. It behaves as a positively charge particle of effective mass mh*.
The bond structure of a p-type semiconductor is shown in Fig below.
Dopants of the trivalent type are called acceptors, since they accept electrons to create
holes above the tope of the valence bond. The acceptor energy level is small compared
with thermal energy of an electron at room temperature. As such nearly all acceptor
levels are occupied and each acceptor atom creates a hole in the valence bond. In
extrinsic semiconductors, there are two types of charge carries. In n-type, electrons are
more than holes. Hence electrons are majority carriers and holes are minority carries.
Holes are majority carries in p-type semiconductors; electrons are minority carriers.
CARRIER CONCENTRATION IN EXTRINSIC SEMICONDUCTORS:
Equation gives the relation been electron and hole concentrations in a
semiconductor. Existence of charge neutrality in a crystal also relates n and p.
The charge neutrality may be stated as
ND + p = NA + n
Since donors atoms are all ionized, ND positive charge per cubic meter are
contributed by ND donor ions. Hence the total positive charge density = ND + p.
Similarly if NA is the concentration of the acceptor ions, they contribute NA
negative charge per cubic meter. The total negative charges density = NA + n.
Since the semiconductor is electrically neutral the magnitude of the positive
charge density must be equal to the magnitude of the total negative charge
density.
n-type material : NA = 0 Since n >> p, eqn reduces to n ≈ ND i.e., in an n-type
material the where subscript n indicates n-type material. The concentration pn of
holes in the n-type semiconductor is obtained from eqn i.e.,
nn pn = n2
i
Thus pn ≈ n2
i
ND
Similarly, for a p-type semiconductor pp ≈ NA and np ≈ n2
i
NA
Expression for electrical conductivity:
There are two types of carries in a Semiconductor electrons and holes. Both these
carries contribute to conduction. The general expression for conductivity can be written
down as
σ = e (nμe + ρ μh)
Where μe and μh are motilities of electrons and holes respectively.
Intrinsic Semiconductor; For an intrinsic Semiconductor
n = p = ni
eqn becomes
σi = eni (μn + μp)
If the scattering is predominantly due to lattice vibrations.
μe = AT3/2
μh = BT3/2
We may put μe + μh = (A + B)T3/2 = CT3/2
σi = ni CT3/2
Substituting for ni from eq we get
σi = 2 2μRT 3/2 CT3/2 (me* mh*)3/4 exp -Eg .
h2 2RT
log σi = log x - Eg
2RT
A graph of log σi Vs 1/T gives a straight line shown in fig below:
LIFE TIME OF MINORITY CARRIER:
In Semiconductor devices electron and hole concentrations are very often
disturbed from their equilibrium values. This may happen due to thermal agitation or
incidence of optical radiation. Even in a pure Semiconductor there will be a dynamic
equilibrium. In a pure Semiconductor the number of holes is equal to the number of free
electrons. Thermal agitation continuously produces of new EHP per unit volume per
second while other EHP disappear due to recombination. On the average a hole exists for
a time period of Tp while an electron exists for a time period Tn before recombination
take place. This time is called the mean life time. If we are dealing with holes in an ntype
Semiconductor, Tp is called the minority carrier life time. These parameters are
important in Semiconductor devices as they indicate the time required for electron and
hole concentration to return to their equilibrium values after they are disturbed.
Let the equilibrium concentration of electrons and holes in an n-type
Semiconductor be n0 and p0 respectively. If the specimen is illuminated at t = ti ,
additional EHPS are generated throughout the specimen. The existing equilibrium is
disturbed and the new equilibrium concentrations are p and n. The excess
concentration of holes = p – p0 – Excess concentration of electrons = n – n0 Since the
radiation creates EHPS.
p – p0 = n – n0
Due to incident radiation equal no of holes and electrons are created. How ever,
the percentage increase of minority carriers is much more than the percentage of majority
carries. In fact, the majority charge carrier change is negligibly small. Hence it is the
minority charge carrier density that is important. Hence, we shall discuss the behaviour
of minority carriers.
As indicated in radiation is removed at t = 0. Let us investigate how the minority
carrier density returns to its original equilibrium value.
The hole concentration decreases as a result of recombination. Decrease in hole
concentration per second due to recombination = p/Tp. But the increase in hole
concentration per second due to thermal generation = g Since charge can neither be
created nor destroyed
dp = g - p
dt Tp
When the radiation is with drawn, the hole concentration p reaches equilibrium
value p0 Hence g = p0/Tp. Then eqn may be rewritten as
dp = p0 – p
dt Tp
We define excess carrier concentration. Since p is a function of time.
p′ = p – p0 = p′(t)
from we may write dp = -p′
dt Tp
The solution to the above differential equation is given by
p′(t) = p (0) e-t/Tp
The excess concentration decreases exponentially to zero with a time constant Tp.
DRIFT CURRENT:
In an electric field E, the drift velocity Vd of carriers superposes on the thermal
velocity Vth. But the flow of charge carriers results in an electric current, known as the
drift current. Let a field E be applied, in the positive creating drifts currents Jnd and Jpd of
electrons and holes respectively.
Without E, the carriers move randomly with rms velocity Vth. Their mean
velocity is zero. The current density will be zero. But the field E applied, the electrons
have the velocity Vde and the holes Vdh.
Consider free electrons in a Semiconductor moving with uniform velocity Vde in the
negative x direction due to an electric field E. Consider a smaller rectangular block of
AB of length Vde inside the Semiconductor. Let the area of the side faces each be unity.
The total charge Q in the elements AB is
Q = Volume of the element x density of partially change on each particle
= (Vde x 1x 1) x n x –q
Thus Q = -qnVde
Where n is the number density of electrons. The entire charge of the block will
cross the face B, in unit time. Thus the drift current density Jnd due to free electrons at
the face B will be.
Jnd = - q nVde
Similarly for holes Jpd = q nVdh
but Vde = -μnE
and Vdh = μpE
hence Jnd = n qμnE
and Jpd = p qμnE
The total drift current due to both electrons and holes Jd is
Jd = Jnd + Jpd = (nqμn + pq μp)E
Even though electrons and holes move in opposite direction the effective direction of
current flow, is the same for both and hence they get added up. Ohm’s Law can be
written in terms of electrical conductivity, as
Jd = σE
Equating the RHS of eq we have
σ = nqμn + pqμp = σn + σp
For an intrinsic Semiconductor n = p = ni
σi = ni q(μn + μp)
DIFFUSION CURRENTS:
1. Diffusion Current: Electric current is Setup by the directed movement of
charge carriers. The movements of charge carriers could be due to either drift or
diffusion. Non- uniform concentration of carriers gives rise to diffusion. The first law of
diffusion by Fick States that the flux F, i.e., the particle current is proportional and is
directed to opposite to the concentration gradient of particles. It can be written
mathematically, in terms of concentration N, as
F = -D V N
Where D stands for diffusion constant.
In one dimension it is written as
F = -D ∂N
∂x
In terms of Je and Jp the flux densities of electrons holes and their densities n and p
respectively.
We get Je = -Dn ∂N
∂x
and Jn = -Dp ∂p
∂x
Where Dn and Dp are the electron and hole diffusion constant constants
respectively. Then the diffusion current densities become
Jn diff = q Dn ∂N
∂x
Jp diff = - q Dp ∂p
∂x
THE EINSTEIN RELATIONS:
When both the drift and the diffusion currents are present total electron and hole current
densities can be summed up as
Jn = Jnd + Jn diff
∂n
Jn = nqμn E + qDn -----
∂x
∂p
Jp = pqμp E - qDp -----
∂x
Now, let us consider a non uniformly doped n-type slab of the Semiconductor fig shown
below (9) under thermal equilibrium. Let the slab be intrinsic at x = 0 while the donor
concentration, increases gradually upto x = 1, beyond which it becomes a constant.
Assume that the Semiconductor is non-degenerate and that all the donors are ionized.
Due to the concentration gradient, electrons tend to diffuse to the left to x = l. This
diffusion leaves behind a positive charge of ionized donors beyond x = 1 and accumulates
electron near x = 0 plane. This charge imbalance. Sets up an electric field in which the
electrons experience fill towards x = 1.
Fig shows illustrates the equilibrium potential φ(x) fig shown refers to the bond diagram
of the Semiconductor.
Both E1 and EF coincide till x = 0 when n0 = ni that EF continues to be the same
throughout the slab. But since the bond structure is not changed due to doping, the bond
edges bond with equal separation all along. How ever, the level Ei continues to lie
midway between EV and E.
In thermal equilibrium, the electrons tend to diffuse down the concentration tending to
setup a current from the right to left. The presence of electric field tends to set up drift
current of electrons in the opposite direction. Both the currents add upto zero. Thus we
obtain
∂n
Jn = qDn ----- + nqμn E = 0
∂x
∂n
i.e., Dn ----- + nμn E = 0
∂x
For a non degenerate Semi conductor.
EF – Ei(x)
n(x) = ni exp -------------
RT
Thus relation is valid at all points in the Semiconductor further. The electronic
concentration is not influenced by the small in balance of charge. Energy is defined in
terms of φ(x) the potential.
E(x) = -qφ(x)
Then EF – Ei(x) = Ei(0)– Ei(x) = -q[φ(0) - φ(x)]
qφ(x)
Assuming φ(0) = 0 we get n(x) = ni exp ------
RT
dn -dφ
Substituting ------ from eqn along with E = ----- we get
dx dx
nq dφ dφ
Dn ------- ------- = μn n -----
RT dx dx
RT
Simplifying we obtain, Dn = ------ μn
q
RT
Simplifying for holes Dp = ----- μp
q
These are known as Einstein relations and the factor (RT/q) as thermal voltage. The
above relations hold good only for non degenerate Semiconductors. For the degenerate
case the Einstein’s relations are complex.
It is clear from the Einstein’s relation that Dp μp and Dn,μn are related and they are
functions of temperature also. The relation of diffusion constant D and the mobility μ
confirms the fact, that both the diffusion an drift processes arise due to thermal motion
and scattering of free electrons, even though they appear to be different.
EQUATION OF CONTINUITY:
If the equilibrium concentrations of carriers in a Semiconductor are disturbed, the
concentrations of electrons and holes vary with time. How ever the carrier concentration
in a Semiconductor is a function of both time and position.
The fundamental law governing the flow of charge is called the continuity equation. It is
arrived at by assuming law to conservation of charge provided drift diffusion and
recombination processes are taken into account.
Consider a small length Δx of a Semiconductor sample with area A in the Z plane
fig shown above. The hole current density leaving the volume (ΔxΔ) under consideration
is Jp ( x + Δx) and the current density entering the volume is Jp(x). Jp (x + Δx) may be
smaller or larger than Jp(x) depending upon the generation and recombination of carriers
in the element. The resulting change in hole concentration per unit time.
∂p = hole flux entering per unit time – hole flux leaving per unit
∂p Jp(x+Δx) δp
----- = Jp(x) - --------------- - ------
∂t x→ x+Δx q Δx Tp
Where Tp is the recombination life time. According to eqn, the rate of hole build up is
equla to the rate of increase of hole concentration remains the recombination rate. As Δx
approaches zero, we may write
∂p ∂ δp - 1 ∂ Jp δp
-----(x,t) = ------ = ----------- - ------
∂t ∂t q ∂x Tp
The above is called the continuity equation for holes for electrons
∂δn 1 ∂Jn δn
----- = -------- - -----
∂t q dx Tn
If there is no drift we may write
∂δ
Jn(diff) = qDn ----
∂x
Substituting the above eqn we get the following diffusion eqn for electrons.
∂δn ∂2δn δn
----- = Dn -------- - -----
∂t ∂x2 Tn
For holes we may write
∂δp ∂2δp δp
----- = Dp -------- - -----
∂t ∂x2 Tp
HALL EFFECT:
When a material carrying current is subjected to a magnetic field in a direction
perpendicular to the direction of current, an electric field is developed across the material
in a direction perpendicular to both the direction of the magnetic field and the current
direction. This phenomenon is called Hall Effect.
Hall Effect finds important application in studying the electron properties of semi
conductor, such as determination of carrier concentration and carrier mobility. It also
used to determine whether a semi conductor is n-type, or p- type.
THEORY:
Consider a rectangular slab of an n-type Semiconductor carrying current in
the positive x-direction. The magnetic field B is acting in the positive direction as
indicated in fig above. Under the influence of the magnetic field, electrons experience a
force FL given by
FL = - Bev --------------- (1)
Where e = magnitude of the charge of the electron
v = drift velocity
Appling the Fleming’s Left Hand Rule, it indicates a force FH acting on the electrons in
the negative y-direction and electron are deflected down wards. As a consequence the
lower face of the specimen gets negatively charged (due to increases of electrons) and the
upper face positively charged (due to loss of electrons). Hence a potential VH, called the
Hall voltage appears between the top and bottom faces of the specimen, which establishes
an electric field EH, called the Hall field across the conductor in negative y-direction. The
field EH exerts an upward force FH on the electrons. It is given by
FH = - eEH --------------------------------------(2)
FH acts on electrons in the upward direction. The two opposing forces FL and FH
establish an equilibrium under which
|FL = FH
using eqns 1 and 2 -Bev = -eEH
EH = Bv --------------(3)
If ‘d’ is the thickness of the Specimen
VH
EH = ------
d
VH = EH d = Bvd from eqn (3)------------------------- 4
If ω is the width of the specimen in z- direction.
The current density
I
J = -----
ωd
But J = nev = ρv -------------------- 5
Where n = electron concentration
And ρ = charge density
I
∴ ρv = ----
ωd
I
or v = ------ --------------------------------- 6
ρωd
Substitutinf for v, from eqns 6 and 4
VH = BI / ρω
BI
or ρ= --------
VH ω
Thus, by measuring VH, I, and ω and by knowing B, the charge density ρ can be
determined.
Hall Coefficient:
The Hall field EH, for a given material depends on the current density J, and
the applied field B
i.e., EH ∝ JB
EH= RH JB
Where RH is called the Hall Coefficient
BI
Since VH = --------
ρω
VH
EH = -----
Jωd
I
J = ------
ωd
BI I
-------- = RH ------- B
Jωd ωd
I
This leads to RH = -----
ρ
Mobility of charge carriers:
The mobility μ is given by μ = v
E
But J = σE = nev = ρv
∴ σE = ρv
ρv
or E = -----
σ
σ
⇒ μ = ---- = σRH (∴ 1/ρ = RH)
ρ
σ is the conductivity of the semi conductor.
(C) Applications
Determination of the type of Semiconductor: The Hall Coefficient RH is
negative for an n-type Semiconductor and positive for a p-type material.
Thus, the sign of the Hall coefficient can be utilized to determine whether a
given Semiconductor is n or p type.
Determination of Carrier Concentration: Equation relates the Hall Coefficient
RH and charge density is
1 - 1
RH = ------ = ------ ( for n-type
p ne
1
= ------ ( for p-type
pe
1
Thus n = ------
eRH
1
and -------
eRH
Determination of mobility: According to equation the mobility of charge carriers
is given by
μ = σ|RH|
Determination of σ and RH leads to a value of mobility of charge carriers.
Measurement of Magnetic Induction (B):- The Hall Voltage is proportional to
the flux density B. As such measurement of VH can be used to9 estimate B.
SUPERCONDUCTIVITY
Introduction : Certain metals and alloys exhibit almost zero resistivity( i.e. infinite
conductivity ) when they are cooled to sufficiently low temperatures. This phenomenon is
called superconductivity. This phenomenon was first observed by H.K. Onnes in 1911.
He found that when pure mercury was cooled down to below 4K, the resistivity suddenly
dropped to zero. Since then hundreds of superconductors have been discovered and
studied. Superconductivity is strictly a low temperature phenomenon. Few new oxides
exhibited superconductivity just below 125K itself. This interesting phenomena has many
important applications in many emerging fields.
General Properties: The temperature at which the transition from normal state to
superconducting state takes place on cooling in the absence of magnetic field is called the
critical temperature (Tc ) or the transition temperature.
The following are the general properties of the superconductors:
1. The transition temperature is different to different substances.
2. For a chemically pure and structurally perfect specimen, the superconducting
transition is very sharp.
3. Superconductivity is found to occur in metallic elements in which the number of
valence electrons lies between 2 and 8.
4. Transition metals having odd number of valence electrons are favourable to
exhibit superconductivity while metals having even number of valence electrons
are unfavourable.
5. Materials having high normal resistivities exhibit superconductivity.
6. Materials for which Zρ > 106 (where Z is the no. of valence electrons and ρ is
the resistivity) show superconductivity.
7. Ferromagnetic and antiferromagnetic materials are not superconductors.
8. The current in a superconducting ring persists for a very long time .
Effect of Magnetic Field: Superconducting state of metal depends on temperature
and strength of the magnetic field in which the metal is placed. Superconducting
disappears if the temperature of the specimen is raised above Tc or a strong enough
magnetic field applied. At temperatures below Tc, in the absence of any magnetic field,
the material is in superconducting state. When the strength of the magnetic field applied
reaches a critical value Hc the superconductivity disappears.
At T= Tc, Hc = 0. At temperatures below Tc, Hc increases. The dependence of the
critical field upon the temperature is given by
HC(T) = HC(0) [1 – (T/Tc)2]------------------------------(1)
Where Hc(0) is the critical field at 0K. Hc(0) and Tc are constants of the characteristics
of the material.
Meissner effect: When a weak magnetic field applied to super conducting
specimen at a temperature below transition temperature Tc , the magnetic flux lines are
expelled. This specimen acts as on ideal diamagnet. This effect is called meissner effect.
This effect is reversible, i.e. when the temperature is raised from below Tc , at T = Tc the
flux lines suddenly start penetrating and the specimen returns back to the normal state.
Under this condition, the magnetic induction inside the specimen is given by
B = μ0(H + M) -------------------------------------(2)
Where H is the external applied magnetic field and M is the magnetization produced
inside the specimen.
When the specimen is super conducting, according to meissner effect inside the bulk
semiconductor B= 0.
Hence μ0(H + M) = 0
Or M = - H ---------------------------------------------(3)
Thus the material is perfectly diamagnetic.
Magnetic susceptibility can be expressed as
χ=M/H = -1-----------------------------------------------------(4)
Consider a superconducting material under normal state. Let J be the current passing
through the material of resistivity ρ. From ohm’s law we know that the electric field
E = Jρ
On cooling the material to its transition temperature, ρ tends to zero. If J is held finite.
E must be zero. Form Maxwell’s eqn, we know
▼X E = - dB/ dt ----------------------------(5)
Under superconducting condition since E = 0, dB/dt = 0, or B= constant.
This means that the magnetic flux passing through the specimen should not change on
cooling to the transition temperature. The Meissner effect contradicts this result.
According to Meissner effect perfect diamagnetism is an essential property of defining
the superconducting state. Thus
From zero resistivity E = 0,
From Meissner effect B= 0.
Type- I , Type- II superconductors: Based on diamagnetic response
Superconductors are divided into two types, i.e type-I and type-II.
Superconductors exhibiting a complete Meissner effect are called type-1, also called Soft
Superconductors. When the magnetic field strength is gradually increased from its initial
value H< HC, at HC the diamagnetism abruptly disappear and the transition from
superconducting state to normal state is sharp. Example Zn, Hg, pure specimens of Al
and Sn.
In type-2 Superconductors, transition to the normal state takes place gradually. For fields
below HC1, the material is diamagnetic i.e., the field is completely excluded HC1 is called
the lower critical field. At HC1 the field begins to penetrate the specimen. Penetration
increases until HC2 is reached. At HC2, the magnetizations vanishes i.e., the material
becomes normal state. HC2 is the upper critical field. Between HC1 and HC2 the state of the
material is called the mixed or vortex state. They are also known as hard
superconductors. They have high current densities. Example Zr , Nb etc.
Penetration Depth : The penetration depth λ can be defined as the depth from the
surface at which the magnetic flux density falls to 1/e of its initial value at the surface.
Since it decreases exponentially the flux inside the bulk of superconductor is zero and
hence is in agreement with the Meissner effect. The penetration depth is found to depend
on temperature. its dependence is given by the relation
λ ( T ) = λ ( 0 ) ( 1 – T4 / Tc
4 )-1/2 ----------------(1)
where λ ( 0 ) is the penetration depth at T = 0K.
According to eqn.(1) , λ increases with the increase of T and at T= Tc
, it becomes
infinite . At T= Tc
, the substance changes from super conducting state to normal state
and hence the field can penetrate to the whole specimen , ie , the specimen has an infinite
depth of penetration.
BCS Theory : BCS theory of superconductivity was put forward by Bardeen, Cooper and
Schrieffer in 1957. This theory could explain many observed effects such as zero
resistivity , Meissner effect, isotope effect etc. The BCS theory is based on advanced
quantum concept.
1. Electron – electron interaction via lattice deformation: Consider an electron is
passing through the lattice of positive ions. The electron is attracted by the
neighbouring positive ion, forming a positive ion core and gets screened by them.
The screening greatly reduces the effective charge of this electron. Due to the
attraction between the electron and the positive ion core, the lattice gets deformed
on local scale. Now if another electron passes by its side of the assembly of the
electron and the ion core, it gets attracted towards it. The second electron interacts
with the first electron via lattice deformation. The interaction is said to be due to
the exchange of a virtual phonon ,q, between the two electrons. The interaction
process can be written in terms of wave vector k ,as
K1 – q = K1
1 and K2 + q = K2
1 -----------------(1)
This gives K1 – K2 = K1
1 + K2
1 , ie , the net wave vector of the pair is conserved.
The momentum is transferred between the electrons. These two electrons together form a
cooper pair and is known as cooper electron.
2. Cooper Pair: To understand the mechanism of cooper pair formation , consider
the distribution of electrons in metals as given by Fermi – Dirac distribution
function .
F ( E ) = 1 / [ exp ( E – EF / KT ) + 1 ]
At T = 0K, all the energy states below Fermi level EF are completely filled and all the
states above are completely empty.
Let us see what happens when two electrons are added to a metal at absolute zero.
Since all the quantum states with energies E ≤ EF are filled , they are forced to occupy
states having energies E > EF . Cooper showed that if there is an attraction between
the two electrons ,they are able to form a bound state so that their total energy is less
than 2 EF . These electrons are paired to form a single system. These two electrons
together form a cooper pair and is known as cooper electron. cooper pair and is
known as cooper electron. Their motions are correlated. The binding is strongest
when the electrons forming the pair have opposite momenta and opposite spins. All
electron pairs with attraction among them and lying in the neighbourhood of the
Fermi surface form cooper pairs. These are super electrons responsible for the
superconductivity.
In normal metals, the excited states lie just above the Fermi
surface. To excite an electron from the Fermi surface even an arbitrarily small
excitation energy is sufficient. In super conducting material , when a pair of electrons
lying just below the Fermi surface is taken just above it ,they form a cooper pair and
their total energy is reduced. This continues until the system can gain no additional
energy by pair formation. Thus the total energy of the system is further reduced.
At absolute zero in normal metals there is an abrupt
discontinuity in the distribution of electrons across the Fermi surface whereas such
discontinuity is not observed in superconductors. As super electrons are always
occupied in pairs and their spins are always in opposite directions.
Isotope Effect : in super conducting materials, the transition temperature varies with
the average isotopic mass , M , of their constituents. The variation is found to follow
the general form
Tc α M-α --------------------(1)
Or Mα Tc = constant
Where α is called the isotope effect coefficient .
Flux Quantization:
Consider a superconductor in the form of a ring. Let it be at temperature above
its TC because of which it will be in the normal state. When it is subjected to the
influence of a magnetic field, the flux lines pass through the body and also exist out side
and inside ring also.
If the body is cooled to a temperature below its TC, then as per Meissner effect, the flux
lines are expelled from the body i.e. the flux exists both outside the ring and in the hole
region but not in the body of the ring. But when the external field is switched off, the
magnetic flux lines continue to exist within the hole region, through the rest that
surrounded the ring from the exterior would vanish. This is known as flux trapping. It is
due to the large currents that are induced as per Faraday’s law during the flux decay when
the field switched off. Because of the zero resistance property that the superconductor
enjoys, these induced currents continue to circulate in the ring practically externally.
Thus the flux stand trapped in the loop forever.
It was F.London who gave the idea that the trapped magnetic flux is quantized, as
super conductivity is governed by the quantum phenomenon. At first he suggested the
quantization of Φ as
Φ = nh / e n = 1,2,3,………….
But experiments carried out carefully on very small superconducting hollow cylinders by
Deaver and Fairbank , that gave half the values of flux quanta given by London. Thus the
governing equation for flux quantization was changed to,
Φ = nh / 2e
It happened so because, London’s theory was based on supercurrents constituted by
electrons as individual entities. This demonstrates conclusively that superconducting
current carriers are pairs of electrons and not single ones.
Then the above equation is written as
Φ = n Φ0
Where Φ0 = (h/2e) is the quantum of flux and is called fluxoid.
Josephson Effect : Consider a thin insulating layer sandwiched between two metals.
This insulating layer acts as a potential barrier for flow of electrons from one metal to
another. Since the barrier is so thin , mechanically electrons can tunnel through from
a metal of higher chemical potential to the other having a lower chemical potential.
This continues until the chemical potential of electrons in both the metals become
equal.
Consider application of a potential difference across the
potential barrier. Now more electrons tunnel through the insulating layer from higher
potential side to lower potential side. The current – voltage relation across the
tunneling junction obeys the ohm’s law at low voltages.
Now consider another case that one of the metals is a
superconductor. On applying the potential, it can be observed that no current flows
across the junction until the potential reaches a threshold value. It has been found that
the threshold potential is nothing but half the energy gap in the superconducting state.
Hence the measurement of threshold potential under this condition helps one to
calculate the energy gap of superconductor. As the temperature is increased towards
Tc ,more thermally excited electrons are generated. Since they require less energy to
tunnel , the threshold voltage decreases. This results in decrease of energy gap itself.
Consider a thin insulating layer sandwiched between two superconductors. In
addition to normal tunneling of single electrons, the super electrons also tunnel
through the insulating layer from one superconductor to another without dissociation,
even at zero potential difference across the junction. Their wave functions on both
sides are highly correlated. This is known as Josephson effect.
The tunneling current across the junction is very less since the two
superconductors are only weakly coupled because of the presence of a thin insulating
layer in between.
D.C. Josephson Effect : According to Josephson, when tunneling occurs through the
insulator it introduces a phase difference ΔΦ between the two parts of the wave
function on the opposite sides of the junction.
The tunneling current is given by
I = I0 sin ( Φ0 ) --------------------(1)
Where I0 is the maximum current that flows through the junction without any
potential difference across the junction. I0 depends on the thickness of the junction
and the temperature.
When there is no applied voltage, a d.c. current flows across the
junction. The magnitude of the current varies between I0 and -I0 according to the
value of phase difference Φ0 = (Φ2 – Φ1 ). This is called d.c Josephson effect.
A.C. Josephson Effect : let a static potential V0 is applied across the junction. This
results in additional phase difference introduced by cooper pair during tunneling
across the junction. This additional phase difference ΔΦ at any time t can be
calculated using quantum mechanics
ΔΦ = E t / ħ ----------------------(1)
Where E is the total energy of the system.
In the present case E = (2 e ) V0 . since a cooper pair contains 2 electrons , the factor
2 appears in the above eqn.
Hence ΔΦ = 2 e V0 t / ħ .
The tunneling current can be written as
I = I0 sin ( Φ0 + ΔΦ ) = I0 sin ( Φ0 + 2 e V0 t / ħ )---------------(1)
This is of the form
I = I0 sin ( Φ0 + ω t ) --------------------(2)
Where ω = 2 e V0 / ħ .
This represents an alternating current with angular frequency ω. This is the a.c.
Josephson effect. when an electron pair crosses the junction a photon of energy ħω =
2eV0 is emitted or absorbed.
Current – voltage characteristics of a Josephson junction are :
When V0 = 0 , there is a constant flow of d.c current ic through the junction. This
current is called superconducting current and the effect is the d.c. Josephson
effect.
So long V0 < Vc , a constant d.c.current ic flows.
When V0 > Vc , the junction has a finite resistance and the current oscillates with a
frequency ω = 2 e V0 / ħ . This effect is the a.c Josephson effect.
Applications Of Josephson Effect:
1. It is used to generate microwaves with frequency ω = 2 e V0 / ħ .
2. A.C Josephson effect is used to define standard volt.
3. A.C Josephson effect is also used to measure very low temperatures based on the
variation of frequency of emitted radiation with temperature.
4. A Josephson junction is used for switching of signals from one circuit to another.
The switching time is of the order of 1ps and hence very useful in high speed
computers.
Applications Of Superconductors :
1. Electric generators : superconducting generators are very smaller in size and
weight when compared with conventional generators. The low loss
superconducting coil is rotated in an extremely strong magnetic field. Motors with
very high powers could be constructed at very low voltage as low as 450V. this is
the basis of new generation of energy saving power systems.
2. Low loss transmission lines and transformers : Since the resistance is almost zero
at superconducting phase, the power loss during transmission is negligible. Hence
electric cables are designed with superconducting wires. If superconductors are
used for winding of a transformer, the power losses will be very small.
3. Magnetic Levitation : Diamagnetic property of a superconductor ie , rejection of
magnetic flux lines is the basis of magnetic levitation. A superconducting material
can be suspended in air against the repulsive force from a permanent magnet. This
magnetic levitation effect can be used for high speed transportation.
4. Generation of high Magnetic fields : superconducting materials are used for
producing very high magnetic fields of the order of 50Tesla. To generate such a
high field, power consumed is only 10kW whereas in conventional method for
such a high field power generator consumption is about 3MW. Moreover in
conventional method ,cooling of copper solenoid by water circulation is required
to avoid burning of coil due to Joule heating.
5. Fast electrical switching :A superconductor possesses two states , the
superconducting and normal. The application of a magnetic field greater than Hc
can initiate a change from superconducting to normal and removal of field
reverses the process. This principle is applied in development of switching
element cryotron. Using such superconducting elements, one can develop
extremely fast large scale computers.
6. Logic and storage function in computers : they are used to perform logic and
storage functions in computers. The current – voltage characteristics associated
with Josephson junction are suitable for memory elements.
7. SQUIDS ( superconducting Quantum Interference Devices ) : It is a double
junction quantum interferometer. Two Josephson junctions mounted on a
superconducting ring forms this interferometer. The SQUIDS are based on the
flux quantization in a superconducting ring. Very minute magnetic signals are
detected by these SQUID sensors. These are used to study tiny magnetic signals
from the brain and heart. SQUID magnetometers are used to detect the
paramagnetic response in the liver. This gives the information of iron held in the
liver of the body accurately.
Questions:
UNIT – VI
LASERS
Introduction:
It is a device to produce a powerful monochromatic narrow beam of light in which the
waves are convergent. Laser is an acronym for light amplification by stimulated emission
of radiation.
Maser is an acronym of microwave amplification by stimulated emission of radiation.
The light emitted from the conventional light source (eg: sodium lamp, candle) is said to
be incoherent. Because the radiation emitted from different atom do not have any definite
phase relationship with each other. Lasers are much important because the light sources
having high monochromaticity, high intensity, high directionality and high coherence.
In the laser the principal of maser is employed in the frequency range of 1014to 1015Hz
and it is termed as optical maser. Laser principle now a day is extended up to γ-rays
hence Gamma ray lasers are called Grazers. The first two successful lasers developed
during 1960 were Ruby laser and He- Ne lasers. Some lasers emit light is pulses while
others emit as a continuous wave.
Characteristics of laser radiation:
The four characteristics of a laser radiation over conventional light sources are
(1) Laser is highly monochromatic
(2) Laser is highly directional
(3) Laser is highly coherent
(4) The intensity of laser is very high
HIGHLY MONOCHROMATIC:
The band width of ordinary light is about 1000A0 . The band width of laser light is about
10A0 . The narrow band width of a laser light is called on high monochromacity.
BAND WIDTH:- The spread of the wavelength (frequency ) about the wavelength of
maximum intensity is band width.
Laser light is more monochromatic than that of a conventional light source. Because of
this monochromaticity large energy can be concentrated in to an extremely. Small band
width.
For good laser dv=50Hz v=5×1014 Hz. The degree of non-monochromaticity
for a conventional sodium light.
HIGH DIREATIONALITY:
The conventional light sources like lamp, torch light, sodium lamp emits light in all
directions. This is called divergence. Laser in the other hand emits light only in one
direction. This is called directionality of laser light.
Light from ordinary light spreads in about few kilometers.
Light from laser spreads to a diameter less than 1 cm for many kilometers.
The directionality of laser beam is given by (or) expressed in divergence.
The divergence Δθ = (r2 –r1) /d2-d1
Where r2, r1 are the radius of laser beam spots
d2 ,d1 are distances respectively from the laser source. Hence for getting a high
directionality then should be low divergence.
HIGHLY COHERENT:
When two light rays are having same phase difference then they are said to be coherent.
It is expressed in terms of ordering of light field
Laser has high degree of ordering than other common sources. Due to its coherence only
it is possible to create high power(1012 watts) in space with laser beam of 1μm diameter.
There are two independent concepts of coherence.
1) Spatial coherence (2) Temporal coherence
SPATIAL COHERENCE: The two light fields at different point in space maintain a
constant phase difference over any time (t) they are said to be spatial coherence.
In He- Ne gas laser the coherence length( lc ) is about 600km.It means over the distance
the phase difference is maintained over any time .For sodium light it is about 3cm.
The coherence & monochoremacity is related by
ξ = (Δυ / υ) α 1/ lc
For the higher coherence length ξ is small hence it has high monochromacity
TEMPORAL COHERENCE: The correlation of phase between the light fields at a
point over a period of time. For He- Ne laser it is a about 10-3 second , for sodium it is
about 10-10 second only.
ξ = (Δυ / υ) α 1/ tc
Higher is the tc higher is the monochromacity.
HIGH INTENSITY:
Intensity of a wave is the energy per unit time flowing through a unit area.
The light from an ordinary source spreads out uniformly in all directions and from
spherical wave fronts around it.
Ex:- If you look a 100W bulb from a distance of 30cm the power entering the eye is
1 / 1000 of watt.
But in case of a laser light, energy is in small region of space and in a small wavelength
and hence is said to be of great intensity.
The power range of laser about 10-3W for gas laser and 109W for solid state laser
The number of photons coming out from a laser per second per unit area is given by
N1 = p / hυЛr2 ≈ 1022 to 1034 photons/ m2 – sec.
SPONTANEOUS AND STIMULATED (INDUCED) EMISSION:
Light is emitted or absorbed by particles during their transitions from one energy state to
another .the process of transferring a particle from ground state to higher energy state is
called excitation. Then the particle is said to be excited.
The particle in the excited state can remain for a short interval of time known as life time.
The life time is of the order of 10-8 sec, in the excited states in which the life time is
much greater than 10-8 sec are called meta stable states. The life time of the particle in the
Meta stable state is of the order 10-3 sec
The probability of transition to the ground state with emission of radiation is made
up of two factors one is constant and the other variable.
The constant factor of probability is known as spontaneous emission and the variable
factor is known as stimulated emission.
SPONTANEOUS EMISSION: The emission of particles from higher energy state to
lower energy state spontaneously by emitting a photon of energy hυ is known as
“spontaneous emission”
STIMULATED EMISSION: The emission of a particle from higher state to lower state
by stimulating it with another photon having energy equal to the energy difference
between transition energy levels called stimulated emission.
SPONTANEOUS EMISSION STIMULATED EMISSION
1) Incoherent radiation 1) coherent radiation
2) Less Intensity 2) high intensity
3) Poly chromatic 3) mono chromatic
4) One photon released 4) two photons released
5) Less directionality 5) high directionality
6) More angular spread during propagation 6) less angular spread during
Propagation
Ex:-Light from sodium ex: - light from a laser source
Mercury vapour lamp ruby laser, He-He gas laser gas
Laser
EINSTEINS EQUATIONS (OR) EINSTAINS CO- EFFICIENTS:
Based on Einstein’s theory of radiation one can get the expression for probability for
stimulated emission of radiation to the probability for spontaneous emission of radiation
under thermal equilibrium.
E1, E2 be the energy states
N1, N2 be the no of atoms per unit volume
ABSORPTION: If ρ(υ)dυ is the radiation energy per unit volume between the
frequency range of υ and υ+dυ
The number of atoms under going absorption per unit volume per second from level
E1 to E2 = N1 ρ(υ )B12------􀃆 1
B12 represents probability of absorption per unit time
STIMILATED EMISSION: When an atom makes transition E2 toE1 in the presence of
external photon whose energy equal to (E2-E1) stimulated emission takes place thus the
number of stimulated emission per unit volume per second from levels.
E2 to E1 = N2 ρ(υ) B21------􀃆 2
B21 is represents probability of stimulated emission per unit time.
SPONSTANEOUS EMISSION: An atom in the level E2 can also make a
spontaneous emission by jumping in to lower energy level E1.
E2 to E1=N2A21---􀃆 3
A21 represents probability of spontaneous emission per unit time.
Under steady state
(dN / dt) = 0
No of atoms under going absorption per second = no of atoms under going emission per
second
1 =2+3
N1 ρ(υ )B12 = N2 ρ(υ )B21 + N2 A21
N2 A21 = N1 ρ(υ )B12 - N2 ρ(υ )B21
= ρ(υ )(N1B12 - N2B21)
ρ (υ) = N2 A21 / (N1B12 - N2B21)
= A21 / [(N1 / N2 )B12 - B21)] --------􀃆 4
From distribution law we know that
N1 / N2 = e(E
2
- E
1
) / k
B
T
= ehυ / k
B
T --------------􀃆 5
Substituting N1 / N2 in eqn (4) we get
ρ (υ) = A21 / B21 (ehυ / k
B
T - 1)---------􀃆 6
From Planck’s radiation
ρ (υ) = 8Πhn3 / λ3 x [ 1 / (ehυ / k
B
T - 1)] -------􀃆 7
n – refractive index of the medium
λ - wave length of the light in air.
λm= λ / n wavelength of light in medium --------􀃆 8
ρ (υ) = 8Πh / λm
3 x [ 1 / (ehυ / k
B
T - 1)] ------􀃆 9
Comparing eqn 6 and 9
A21 / B21 = 8Π h / λm
3
Where A21 , B21 are Einstein’s co-efficient of spontaneous emission probability per unit
time and stimulated emission probability per unit time respectively.
For stimulated emission to be predominant, we have
A21 / B21 << 1
POPULATION INVERSION:
The no of atoms in higher energy level is less than the no of atoms in lowest
energy level. The process of making of higher population in higher energy level than the
population in lower energy level is known as population inversion.
Population inversion is achieved by pumping the atoms from the ground level to the
higher energy level through optical (or) electrical pumping. It is easily achieved at the
matastable state, where the life time of the atoms is higher than that in other higher
energy levels.
The states of system, in which the population of higher energy state is more in
comparision with the population of lower energy state, are called “Negative temperature
state”.
A system in which population inversion is achieved is called as an active system. The
method of raising the particles from lower energy state to higher energy state is called
”Pumping”.
DIFFERENMT TYPES OF LASES:
1. Solid state laser - Ruby laser, Nd-YAG laser
2. Gas laser - He-Ne laser, CO2 laser
3. Semi conductor laser - GaAs laser
RUBY LASER:
Ruby laser is a three level solid state laser having very high power of hundreds of
mega watt in a single pulse it is a pulsed laser.
The system consists of mainly two parts
1) ACTIVE MATERIAL: Ruby crystal in the form of rod.
2) RESONANT CAVITY: A fully reflecting plate at the left end of the ruby crystal
and partially reflecting end at the right side of the ruby crystal both the surfaces
are optically flat and exactly parallel to each other.
3) EXCITING SYSTEM: A helical xenon flash tube with power supply source.
CONSTRUCTION: In ruby laser, ruby rod is a mixture of Al2O3+Cr2O3. It is a
mixture of Aluminum oxide in which some of ions Al+3 ions concentration doping of Cr+3
is about 0.05% , then the colour of rod becomes pink. The active medium in ruby rod is
Cr+3 ions. The length of the ruby rod is 4cm and diameter 5mm and both the ends of the
ruby rod are silvered such that one end is fully reflecting and the other end is partially
reflecting. The ruby rod is surrounded by helical xenon flash lamp tube which provides
the optical pumping to raise Cr+3 ions to upper energy level.
The chromium atom has been excited to an upper energy level by absorbing photons of
wave length 5600A0 from the flash lamp. Initially the chromium ions (Cr+3) are excited to
the energy levels E1 to E3 , the population in E3 increases. Since the life time of E3 level is
very less (10_8Sec). The Cr+3 ions drop to the level E2 which is matastable of life time 10-
3Sec. The transition from E3 to E2 is non-radiative.
Since the life time of metastable state is much longer, the no of ions in this state goes on
increasing hence population inversion achieved between the excited metastable state E2
and the ground state E1.
When an excited ion passes spontaneously from the metastable state E2 to the ground
state E1. It emit a photon of wave length 6993A0 this photon travel through the ruby rod
and if it is moving parallel to the axis of the crystal and it is reflected back and forth by
the silver ends until it stimulates an excited ion in E2. The emitted photon and stimulated
photon are in phase the process is repeated again and again finally the photon beam
becomes intense; it emerges out through partially silvered ends.
Since the emitted photon and stimulating photon in phase, and have same frequency and
are traveling in the same direction, the laser beam has directionality along with spatial
and temporal coherence.
IMPORTANCE OF RESONATOR CAVITY: To make the beams parallel to each
other curved mirrors are used in the resonator cavity. Resonator mirrors are coated with
multi layer dielectric materials to reduce the absorption loss in the mirrors. Resonators act
as frequency selectors and also give rise to directionality to the out put beam. The
resonator mirror provides partial feedback to the protons.
He- Ne Laser
CONSTRUTION:
He - Ne gas laser consists of a gas discharge tube of length 80cm
and diameter of 1cm. The tube is made up of quartz and is filled with a mixture of Neon
under a pressure of 0.1mm of Hg. The Helium under the pressure of 1mm of Hg the ratio
of He-Ne mixture of about 10:1, hence the no of helium atoms are greater than neon
atoms. The mixtures is enclosed between a set of parallel mirrors forming a resonating
cavity, one of the mirrors is completely reflecting and the other partially reflecting in
order to amplify the output laser beam.
WORKING:
When a discharge is passed through the gaseous mixture electrons are
accelerated down the tube these accelerated electrons collide with the helium atoms and
excite them to higher energy levels since the levels are meta stable energy levels he
atoms spend sufficiently long time and collide with neon atoms in the ground level E1 .
Then neon atoms are excited to the higher energy levels E4 & E6 and helium atoms are de
excited to the ground state E1
Since E6 & E4 are meta stable states, population inversion takes place at there levels. The
stimulated emission takes place between E6 to E3 gives a laser light of wave length
6328A0 and the stimulated emission between E6 and E5 gives a laser light wave length of
3.39μm. Another stimulated emission between E4 to E3 gives a laser light wave length of
1.15μm. The neon atoms undergo spontaneous emission from E3 toE2 and E5 to E2.
Finally the neon atoms are returned to the ground state E1 from E2 by non-radeative
diffusion and collision process.
After arriving the ground state, once again the neon atoms are raised to E6 & E4 by
excited helium atoms thus we can get continuous out put from He-Ne laser.
But some optical elements placed insides the laser system are used to absorb the infrared
laser wave lengths 3.39μm and1.15μm. hence the output of He-Ne laser contains only a
single wave length of 6328A0 and the output power is about few milliwatts .
CO2 LASER
Construction and working:
We know that a molecule is made up of two or more atoms bound together. In molecule
in addition to electronic motion, the constituent atoms in a molecule can vibrate in
relation to each other and the molecule as a whole can rotate. Thus the molecule is not
only characterized by electronic levels but also by vibration and rotational levels. The
fundamental modes of vibrations of CO2 molecule shown I fig.
CO2 Laser is a gas discharge, which is air cooled. The filling gas within the discharge
tube consists primarily of, CO2 gas with 10 – 20%, Nitrogen around 10 – 20 %
H2 or Xenon ( Xe ) a few percent usually only in a sealed tube.
The specific proportions may vary according to the particular application. The population
inversion in the laser is achieved by following sequence:
1. Electron impact excites vibration motion of the N2. Because N2 is a homo nuclear
molecule, it cannot lose this energy by photon emission and it is excited vibration
levels are therefore metestable and live for long time.
2. Collision energy transfer between the N2 and the CO2 molecule causes vibration
excitation of the CO2, with sufficient efficiency to lead to the desired population
inversion necessary for laser operation.
Because CO2 lasers operate in the infrared, special materials are necessary for their
construction. Typically the mirrors are made of coated silicon, molybdenum or gold,
while windows and lenses are made of either germanium or zinc sulenide. For high
power application gold mirrors and zinc selenide windows and lenses are preferred.
Usually lenses and windows are made out of salt NaCl or KCl ). While the material is
inexpensive, the lenses windows degraded slowly with exposure to atmosphere moisture.
The most basic form of a CO2 laser consist of a gas discharge ( with a mix close to that
specified above) with a total reflector at one end and an output coupler ( usually a semi
reflective coated zinc selendine mirror ) at the output end. The reflectivity of the out put
coupler is typically around 5 – 15 %. The laser output may be edge coupled in higher
power systems to reduce optical heating problems.
CO2 lasers output power is very high compared to Ruby laser or He – Ne lasers. All CO2
lasers are rated in Watts or kilowatts where the output power of Ruby laser or He – Ne
laser is rated in mill watts. The CO2 laser can be constructed to have CW powers between
mill watts and hundreds of kilowatts.
SEMICONDIRCTOR LASER (Gallium Arsenide Diode Laser or
Homojunction Laser):
Semiconductor laser is also known as diode laser
PRINCIPLE:
In the case direct band gap semiconductors there is a large possibility for direct
recombination of hole and electron emitting a photon. GaAs is a direct band gap
semiconductor and hence it is used to make lasers and light emitting diodes. The wave
lengths of the emitted light depend on the band hap of the material.
CONSTRUCTIONS: The P-region and N-region in the diode are obtained by
heavily doping germanium and tellurium respectively in GaAs. The thickness of the P-N
junction layer is very narrow at the junction, the sides are well polished and parallel to
each other. Since the refractive index of GaAs is high the reflectance at the material air
interface is sufficiently large so that the external mirror are not necessary to produce
multiple reflections. The P-N junction is forward biased by connection positive terminal
to P-type and negative terminal to N type.
WORKING: When the junction is forward biased a large current of order104
amp/cm2 is passed through the narrow junction to provide excitation. Thus the electrons
and holes injected from N side and P side respectively. The continuous injection of
charge carries the population inversion of minority carriers in N and P sides respectively.
The excess minority electrons in the conduction band of P – layers recombine with the
majority holes in the valence band of P-layer emitting light photons similarly the excess
minority holes in the valence band of N- layers recombine with the majority electrons in
the conduction band of N- layer emitting light photons.
The emitted photons increase the rate of recombination of injected electrons from the Nregion
and holes in P- region. Thus more no of photons are produced hence the
stimulated emission take place, light is amplified.
The wave length of emitted radiation depends upon the concentration of donor and
acceptor atoms in GaAs the efficiency of laser emission increases, when we cool the
GaAs diode.
DRAWBACKS:-
1. Only pulsed laser output is obtained
2. Laser output has large divergence
3. Poor coherence
APPLICATION OF LASERS:
Lasers in scientific research
1) Lasers are used to clean delicate pieces of art, develop hidden finger prints
2) Laser are used in the fields of 3D photography called holography
3) Using lasers the internal structure of micro organisms and cells are studied very
accurately
4) Lasers are used to produce certain chemical reactions.
Laser in Medicine:
1) The heating action of a laser bean used to remove diseased body tissue
2) Lasers are used for elimination of moles and tumours, which are developing in the skin
tissue.
3) Argon and CO2 lasers are used in the treatment liver and lungs
4) Laser beam is used to correct the retinal detachment by eye specialist
Lasers in Communication:
1) More amount of data can be sent due to the large band with of semiconductor
lasers
2) More channels can be simultaneously transmitted
3) Signals cannot tapped
4) Atmospheric pollutants concentration, ozone concentration and water vapour
concentration can be measured
Lasers in Industry: Lasers are used
1) To blast holes in diamonds and hard steel
2) To cut, drill, welding and remove metal from surfaces
3) To measure distance to making maps by surveyors
4) For cutting and drilling of metals and non metals such a ceramics plastics glass
Questions:
1. Explain the terms i) Absorption, ii) spontaneous emission, iii) Stimulated
emission, iv) pumping mechanism , v) Population inversion, vi) Optical cavity
2. What is population inversion? How it is achieved?.
3. Explain the characteristics of a laser beam.
4. Distinguish between spontaneous and stimulated emission.
5. Derive the Einstein’s coefficients.
6. With neat diagrams, describe the construction and action of ruby laser.
7. Explain the working of He – Ne laser
8. Describe the CO2 laser
9. Explain the semiconductor laser.
10. Mentions some applications of laser in different fields.
UNIT - VII
FIBER OPTICS AND HOLOGRAPHY
Introduction: In 1870 John Tyndall demonstrated that light follows the curve of a
stream of water pouring from a container; it was this simple principle that led to the study
and development of application of the fiber optics. The transmission of information over
fibers has much lower losses than compared to that of cables. The optical fibers are most
commonly used in telecommunication, medicine, military, automotive and in the area of
industry. In fibers, the information is transmitted in the form of light from one end of the
fiber to the other end with min.losses.
Advantages of optical fibers:
• Higher information carrying capacity.
• Light in weight and small in size.
• No possibility of internal noise and cross talk generation.
• No hazards of short circuits as in case of metals.
• Can be used safely in explosive environment.
• Low cost of cable per unit length compared to copper or G.I cables.
• No need of additional equipment to protect against grounding and voltage
problems.
• Nominal installation cost.
• Using a pair of copper wires only 48 independent speech signals can be sent
simultaneously whereas using an optical fiber 15000 independent speeches
can be sent simultaneously.
Basic principle of Optical fiber:
The mechanism of light propagation along fibers can be understood using the
principle of geometrical optics. The transmission of light in optical fiber is based on
the phenomenon of total internal reflection.
Let n1 and n2 be the refractive indices of core and cladding
respectively such that n1>n2.Let a light ray traveling from the medium of refractive
index n1 to the refractive index n2 be incident with an angle of incidence “i” and the
angle of refraction “r”. By Snell’s law
n1sin i = n2sin r…………………(1)
The refractive ray bends towards the normal as the ray travels from rarer medium to
denser medium .On the other hand ,the refracted ray bends away from normal as it
travel from denser medium to rarer medium. In the later case , there is a possibility to
occur total internal reflection provided, the angle of incidence is greater than critical
angle(θ c ). This can be understood as follows.
1. When i < θ c ,then the ray refracted is into the second medium as shown in
below fig1.
2. When i= θc, then the ray travels along the interface of two media as shown in fig2.
3. When i> θc then the ray totally reflects back into the same medium as shown in
fig3.
Suppose if i= θc then r =90o ,hence
n1sin θc = n2sin 90o
sin θc = n2/ n1 (since sin 90o=1)
θc=sin-1(n2/ n1)………………………….(2)
thus any ray whose angle of incidence is greater than the critical angle ,total internal
reflection occurs ,when a ray is traveling from a medium of high refractive index to
low refractive index.
Construction of optical fiber:
The optical fiber mainly consists of the following parts.
i .Core ii .Cladding iii .Silicon coating iv .Buffer jacket
v .Strength material vi . Outer jacket
􀂾 A typical glass fiber consists of a central core of thickness 50μm surrounded by
cladding.
􀂾 Cladding is made up of glass of slightly lower refractive index than core’s
refractive index, whose over all diameter is 125μm to 200μm.
Of course both core and cladding are made of same glass and to put refractive index
of cladding lower than the refractive index of core, some impurities like Boron,
Phosphorous or Germanium are doped.
􀂾 Silicon coating is provided between buffer jacket and cladding in order to
improve the quality of transmission of light.
􀂾 Buffer jacket over the optical fiber is made of plastic and it protects the fiber from
moisture and abrasion.
􀂾 In order to provide necessary toughness and tensile strength, a layer of strength
material is arranged surrounding the buffer jacket.
􀂾 Finally the fiber cable is covered by black polyurethane outer jacket. Because of
this arrangement fiber cable will not be damaged during hard pulling ,bending
,stretching or rolling, though the fiber is of brittle glass.
Acceptance angle and Numerical aperture of optical fiber:
When the light beam is launched into a fiber, the entire light may not pass
through the core and propagate. Only the rays which make the angle of incidence greater
than critical angle at the core –cladding interface undergoes total internal reflection. The
other rays are refracted to the cladding and is lost. Hence the angle we have to launch the
beam at its end is essential to enable the entire light to pass through the core .This
maximum angle of launch is called acceptance angle.
Consider an optical fiber of cross sectional view as shown in figure no, n1and n2 are
refractive indices of air ,core and cladding respectively such that n1>n2>no.let light ray is
incidenting on interface of air and core medium with an angle of incidence α .This
particular ray enters the core at the axis point A and proceeds after refraction at an angle
αr from the axis .It then undergoes total internal reflection at B on core at an internal
incidence angle θ.
To find α at A:-
In triangle ABC, α r =90- θ ……………………. (1)
From snell’ s law ,
n0 sin α = n1sinαr…………….(2)
sin α= n1/n0 sin αr …………………….(3)
From equations 1, 3
sin α= n1/n0 sin(90- θ ) => sin α= n1/n0 cos θ………(4)
If θ < θc, the ray will be the lost by refraction. Therefore limiting value for the beam to
be inside the core, by total internal reflection is θc. Let α (max) be the maximum possible
angle of incident at the fiber end face at A for which θ= θc. If for a ray α exceeds α (max),
then θ< θc and hence at B the ray will be refracted.
Hence equation 4 can be written as
sin α(max)= n1/n0 cos θc…………………..(5)
We know that
cos θc=√(1-sin2 θc) = √ (1- n2
2/n1
2)
=√ ( n1
2- n2
2)/ n1……………………(6)
Therefore
sin α(max) =√ ( n1
2- n2
2)/no
α(max)=sin-1√ ( n1
2- n2
2)/no
This maximum angle is called acceptance angle or acceptance cone angle. Rotating the
acceptance angle about the fiber axis gives the acceptance cone of the fiber. Light
launched at the fiber end within this acceptance cone alone will be accepted and
propagated to the other end of the fiber by total internal reflection. Larger acceptance
angles make launching easier.
Numerical aperture:-
Light collecting capacity of the fiber is expressed in terms of
acceptance angle using numerical aperture. Sine of the maximum acceptance angle is
called the numerical aperture of the fiber.
Numerical aperture =NA = Sin α (max) =√ ( n1
2- n2
2)/no…………….(7)
Let Δ = (n1
2- n2
2)/2n1
2…………… (8)
For most fiber n1 n2
Hence Δ = (n1+ n2) (n1-n2)/2n1
2 = 2n1 (n1-n2)/ 2n1
2
Δ = (n1-n2)/ n1 (fractional difference in refractive indices)……………(9)
From equation (8) n1
2- n2
2 = Δ 2n1
2
Taking under root on both sides
Hence =√ (n1
2- n2
2) = √2 Δ n1
Substituting this in equation (7) we get
NA ≈ √2 Δ n1/no ………(10)
For air no= 1, then the above equation can changed as
NA ≈ √2 Δ n1
Numerical aperture of the fiber is dependent only on refractive indices of
the core and cladding materials and is not a function of fiber dimensions.
Types or classification of optical fibers:
Optical fibers are classified as follows:
Depending upon the refractive index profile of the core, optical fibers are classified into
two categories
_______________
Step index Graded index
Depending upon the number of modes of propagation, optical fibers are classified into
two categories, they are
__________________
Single mode Multi mode
Based on the nature of the material used, optical fibers are classified into four categories.
􀂾 Glass fiber
􀂾 Plastic fiber
􀂾 Glass Core with plastic cladding
􀂾 PCS Fibers(Polymer-Clad Silica fiber)
Step index fibers:-
In step index fibers the refractive index of the core is uniform through out the medium
and undergoes an abrupt change at the interface of core and cladding.The diameter of the
core is about 50-200μm and in case of multi mode fiber.And 10 μm in the case of single
mode fiber.The transmitted optical signals travel through core medium in the form of
meridonal rays, which will cross the fiber axis during every reflection at the corecladding
interface. The shape of the propagation appears in a zig-zac manner.
Graded index fiber:-
In these fibers the refractive index of the core varies radially.As the radius increases in
the core medium the refractive index decreases. The diameter of the core is about
50μm.The transmitted optical signals travel through core medium in the form of helical
rays, which will not cross the fiber axis at any time.
Attenuations in optical fiber (or losses):
While transmitting the signals through optical fiber some energy is lost due to few
reasons. The major losses in fibers are 1.Distotrion losses 2.Transmission losses
3.Bending losses.
1. Distortion losses:-
When a pulse is launched at one end of the fiber and collected at the other end of the
fiber, the shape and size of the pulse should not be changed. Distortion of signals in
optical fiber is an undesired feature. If the out put pulse is not same as the input pulse,
then it is said that the pulse is distorted .If the refractive index of the core is not uniform
most of the rays will travel through the medium of lower refractive index region. Due to
this the rays which are travel in fiber will become broadened. Because of this the out put
pulses will no longer matches with the input pulses.
The distortion takes place due to the presence of imperfections, impurities
and doping concentrations in fiber crystals.Disperion is large in multi mode than in
single mode fiber.
2. Transmission losses (attenuation):-
The attenuation or transmission losses may be classified into two
categories i ) Absorption losses ii) scattering losses
i) Absorption losses:-
Absorption is a characteristic possessed by all materials every material in universe
absorb suitable wavelengths as they incident or passed through the material. In the
same way core material of the fiber absorbs wavelengths as the optical pulses or
wavelengths pass through it.
ii) Scattering losses:-
The core medium of the fiber is made of glass or silica .In
the passage of optical signals in the core medium if crystal defects are encountered,
they deviate from the path and total internal reflection is discontinued, hence such
signals will be destroyed by entering into the cladding however attenuation is
minimum in optical fibers compared to other cables.
ii) Bending losses:-
The distortion of the fiber from the ideal strait line
configuration may also result in fiber .Let us consider a way front that travels
perpendicular to the direction of propagation. In order to maintain this, the part of the
mode which is on the out side of the bend has to travel faster than that on the inside.
As per the theory each mode extends an infinite distance into the cladding though the
intensity falls exponentially. Since the refractive index of cladding is less than that of
the core (n1>n2), the part of the mode traveling in the cladding will attempt to travel
faster. As per Einistein’s theory of relativity since the part of the mode cannot travel
faster the energy associated with this particular part of the mode is lost by radiation.
Attenuation loss is generally measured in terms of decibels (dB),
which is a logarithmic unit.
Loss of optical power = -10 log (Pout/ Pin ) dB
Where Pout is the power emerging out of the fiber
Pin is the power launched into the fiber.
Optical fiber in communication systems as a wave guide:
An efficient optical fiber communication system requires high information carrying
capacity fast operating speed over long distances with minimum number of repeaters
.An optical fiber communication system mainly contains
1. Encoder
2. Transmitter
3. Wave guide
4. Receiver
5. Decoder.
Encoder:-
It is an electronic system that convert the analog information like voice,
figure, objects etc into binary data. This binary data contain a series of electrical
pulses.
Transmitter:-
It consist of two parts namely drive circuit and light source .Drive circuit
supplies electric signals to the light source from the encoder in the required form. In
the place of light source either an LED or A diode Laser can be used ,which converts
electric signals into optical signals. With the help of specially made connecter optical
signal will be injected into wave guide from the transmitter.
Wave guide:-
It is an optical fiber which carries information in the form of optical
signals over long distances with the help of repeaters. with the help of specially made
connector optical signal will be received by the receiver from the wave guide.
Receiver:-
It consists of three parts namely photo detector, amplifier and signal
restorer. The photo detector converts the optical signals into equivalent electric
signals and supply them to amplifier. The amplifier amplifies the electric signals as
they become weak during the journey trough the wave guide over a long distances.
The signal restorer keeps all the electric signals in sequential form and supplies to
decoder in suitable way.
Decoder:-
It convert the received electric signals into the analog information.
Advantages of optical fiber communication:
􀂾 Enormous Band width:-
In the coaxial cable transmission the band width is up to around
5000MHz only where as in fiber optical communication it is as large as 105 GHz.
Thus the information carrying capacity of optical fiber system is far superior to the
copper cable system.
􀂾 Electrical isolation:-
Since fiber optic material are insulators unlike there
metallic counterpart, they do not exhibit earth loop and interface problems. Hence
communication through fiber even in electrically hazarders environment do not cause
any fear of spark hazards.
􀂾 Immunity to interference and cross talk:-
Since optical fibers are dielectric waveguides, they are free
from any electromagnetic interference (EMI) and radio frequency interference (RIF).
Hence fiber cable do not require special shielding from EMI. Since optical
interference among different fiber is not possible ,unlike communication using
electrical conductors cross talk is negligible even when many fibers are cabled
together.
􀂾 Signal security:-
Unlike the situation with copper cables a transmitted optical
signal can not be drawn fiber with out tampering it. Such an attempt will affect the
original signal and hence can easily detected.
􀂾 Small size and weight:-
Since fibers are very small in diameter the space occupied the
fiber cable is negligibly small compare to metallic cables. Optical cables are light in
weight , these merits make them more useful in air crafts and satellites.
􀂾 Low transmission loss:-
Since the loss in fibers is as low as 0.2dB/Km , transmission loss
is very less compare to copper conductors. Hence for long distance communication,
fibers are preferred. Number of repeaters required is reduced.
􀂾 Low cost:-
Since fibers are made up of silica which is available in nature ,optical
fibers are less expensive.
Applications of optical fibers:
􀂾 Optical fibers are used as sensors
􀂾 These are used in Endoscopy
􀂾 These are used in communication system
􀂾 For decorative peaces in home needs.
􀂾 These are used in defence areas for the sake of high security.
􀂾 These are used in electrical engineering.
HOLOGRAPHY
Holography was invented in 1948 by Dennis Gabor. The word ‘holo’ means
entire, complete, full: ‘graphy’ means recording. Hence holography means recording of
the complete information of an object. Recording of the complete information of an
object: i.e., its amplitude and phase is called holography. It carries information of light
intensity distribution on the object only.
Differences between holography and photography:
Photography
Holography
1. Ordinary light
2. Amplitude recording
3. Two dimensional recording
4. Lens is used
5. Ordinary film
6. No need for vibration isolation table
7. No special optics
8. When cut into pieces information is
lost
Laser
Amplitude and phase recording
Three dimensional recording
Generally no lens is used
Very high resolution film
Vibration isolation table is required
Special optics
When cut into pieces, each bit carries full
information
Basic principle of holography:
Light from a coherent source such as laser is split
into two parts. One beam called object beam is used to illuminate the object to be
recorded kept infront of a photo sensitive recording material. Scattered light from the
object falls on the recording material. The other beam called reference beam is expanded
to falls on the recording material. At the recording plane, interference of the object beam
and reference beam takes place. This interference pattern is recorded in the photo
sensitive material as bright and dark fringes on developing and fixing the recorded
material. Such a recorded material is called hologram.
When such a hologram is illuminated with reconstructing beam which is identical to the
reference beam used at the time of recording, light passes through the fine fringes pattern
recorded resulting in diffraction. The diffracted beam is identical to the object wave
which recording material received at the time of recording. Hence when the diffracted
beam is viewed it forms virtual image of the object.
Recording (Construction) of a hologram:
The laser beam is split into two parts, one is called the object beam is used to illuminate
the object. The beam falling on the object after scattering reaches the recording material.
The other one is called reference beam after expansion reaches the recording material. At
the recording plane, interference of the object beam and the reference beam takes place.
On developing and processing we get ‘hologram’ which is nothing but the recorded
interference pattern, say, grating. Since the fringe spacing of the interference pattern is
less than microns, the grains in the photosensitive recording material must be much
smaller in size to record it. This makes the recoding film very in slow, hence slightly
longer exposure is required. Since the manual seismic vibrations of the earth is always of
the order of few microns, the recoding of fine fringes giving longer exposure to the
recording medium is not possible unless we isolate the systems used for recording from
the vibration of the earth. Hence vibration isolation table is required. This table is heavy
one floating on compressed air bed/pillars so that earth vibrations are arrested by this
arrangement.
Fig. Recording (Construction) of Hologram
Steps involved in recording a hologram:
1. Check that vibration isolation table is floated.
2. The laser beam has to be split into two, one to act as object beam and another to
act as reference beam.
3. Front coated mirrors and beam expanders are to be arranged as given in the
diagram. At the recording plane the scattered waves from the object and reference
waves are made to interfere.
4. Object beam and reference beam should travel almost equal distances. The path
difference should not exceed the coherent length of the source.
5. All the components are to be rigidly fixed to the table.
6. In total darkness the recording material is loaded such that emulsion side faces the
object.
7. Without disturbing the table, the laser has to be switched on and the exposure
given.
8. The exposed plate is to be removed and processed in the developer and fixer.
Then it has to be washed well and dried. The dried plate is called hologram.
Reconstruction of a hologram:
The beam splitter is removed and the reference beam is used for reconstruction.
The hologram diffracts the light and the diffracted light received forms virtual image. It is
identical to the object and hence it appears as if the object is really present. By moving
the head side wise the observer can get three dimensional effects. Since while recording,
the secondary waves from each and every point on the object participates in this process
to regenerate both a real and virtual image of the object point by point. By seeing through
the hologram from the transmission side, it appears as though the original object is lying
on the other side of it at the same place. This is the virtual image due to regeneration. By
switching the direction of view, different set of points which correspond to constructive
interference are observed, which regenerate a different perspective of the object. Thus the
sense of depth is felt.
Fig. Reconstruction of a hologram
The other wave stream starting from the various zone plates is convergent as though
coming from a convex lens and gives rise to the real image of the object. The image can
be photographed by keeping a photographic plate in the plane of the image formation of
the convergent beam.
Application:
1. Holographic Interferometry: This method is used to study minute distortions of
an object that take place such as due to stress or vibration. When a defective is
deformed it deforms more at the defective region. This can be holographically
recorded. First the object is recorded giving an exposure to the recording material.
Then the object is stressed and second exposure is given on the same recording
material. After this double exposure, the recording material is processed. When
reconstructed, two image waves are formed one corresponding to the normal
position of the object and another corresponding to the deformed position of the
object. These two image waves interfere to produce interference pattern on the
object. These fringes are contours of locations of same displacement. Hence when
recorded holographicallly by double exposure method and re constructed, one can
observe variation in fringe patter at defective region. The fringe pattern at the
defect point will exhibit discontinuity if there is a crack on the object. Crowding
of fringes can be seen if there is a weak point which has deformed more when
compared with normal region. Thus double exposure holographic interferometry
is very useful in non destructive study of objects.
2. Holographic Diffraction Gratings: The holographic gratings are obtained by
interference of two laser beams each one with plane wavefronts, on a hologram.
This method produces the rulings much more uniformly than the ones made by a
ruling engine on a conventional grating.
3. Acoustical Holography: Using some intricate techniques, procedures have
been formulated for hologram recording with coherent ultrasonic waves. The
hologram could be made for scanning of the human body with ultrasound. When
the image is reconstructed using laser light, the hologram can provide three
dimensional view of internal structures of the human body.
4. Thick Holograms: Thick holograms are the ones in which the interference
pattern is recorded in 3- dimension. Such holograms are in use on credit cards in
which improvement of security is the criteria.
5. Infornation Coding: In the process of recording a hologram, the reference
wave is passed through an encoding mask which generates a special wavefront.
Then in order to get back the information recorded, the hologram has to be
illuminated with the same kind of wavefront only. Since the masking details are
kept secret. The information in the hologram is also guarded.
Questions:
1. Explain the principle behind the functioning of an optical fiber.
2. With the help of a suitable diagram explain the principle construction and
working of optical fiber.
3. Derive an expression for acceptance angle.
4. Explain numerical aperture and derive an expression for it.
5. Write short note on i). Step index fiber, ii) Graded index fiber.
6. Describe the structure of different types of Optical fibers with ray paths.
7. What are the different losses (attenuation) in an optical fiber? Write brief note on
each.
8. Explain the advantages of optical fiber in communication.
9. Explain the basic principle of holography.
UNIT – VIII
SCIENCE AND TECHNOLOGY OF NANOMATERIALS
Introduction
In 1959, Richard Feynman made a statement ‘there is plenty of room at the bottom’.
Based on his study he manipulated smaller units of matter. He prophesied that “we can
arrange the atoms the way we want, the very atoms, all the way down”. The term
‘nanotechnology’ was coined by Norio Taniguchi at the University of Tokyo. Nano
means 10-9. A nano metre is one thousand millionth of a metre (i.e. 10-9 m).
Nanomaterials could be defined as those materials which have structured components
with size less than 100nm at least in one dimension. Any bulk material we take, its size
can express in 3-dimensions. Any planer material, its area can be expressed in 2-
dimension. Any linear material, its length can be expressed in 1-dimension.
Materials that are nano scale in 1-dimension or layers, such as thin films or surface
coatings.
Materials that are nano scale in 2-dimensions include nanowires and nanotubes.
Materials that are nano scale in 3- dimensions are particles like precipitates, colloids and
quantum dots.
Nanoscience: it can be defined as the study of phenomena and manipulation of
materials at atomic, molecular and macromolecular scales, where properties differ
significantly from those at a larger scale.
Nanotechnology: It can be defined as the design, characterization, production and
application of structures, devices and systems by controlling shape and size at the nano
metre scale. It is also defined as “A branch of engineering that deals with the design and
manufacture of extremely small electronic circuits and mechanical devices built at
molecular level of matter. Now nanotechnology crosses and unites academic fields such
as Physics, Chemistry and Computer science.
Properties of nano particles:
The properties of nano scale materials are very much different from
those at a larger scale. Two principal factors that cause that the properties to differ
significantly are increased relative surface area and quantum effects. These can enhance
or change properties such as reactivity, strength and electrical characteristics.
1. Increase in surface area to volume ratio
Nano materials have relatively larger surface area when compared to the volume of
the bulk material.
Consider a sphere of radius r
Its surface area =4Πr2
Its volume = 4Πr3 /3
Surface area = 4Πr2 = 3
Volume 4Πr3 /3 r
Thus when the radius of sphere decreases, its surface area to volume ratio increases.
EX: For a cubic volume,
Surface area = 6x1m2 =6m2
When it is divided it 8 pieces
It surface area = 6x (1/2m) 2x8=12m2
When the same volume is divided into 27 pieces,
It surface area = 6x (1/3m) 2x27=18m2
Therefore, when the given volume is divided into smaller pieces, the surface area
increases. Hence as particle size decreases, greater proportions of atoms are found at the
surface compared to those inside. Thus nano particles have much greater surface to
volume ratio. It makes material more chemically reactive.
As growth and catalytic chemical reaction occur at surfaces, then given mass of material
in nano particulate form will be much more reactive than the same mass of bulk material.
This affects there strength or electrical properties.
2. Quantum confinement effects
When atoms are isolated, energy levels are discrete or discontinuous. When
very large number of atoms is closely packed to form a solid, the energy levels split and
form bands. Nan materials represent intermediate stage.
When dimensions of potential well and potential box are of the order of deBroglie wave
length of electrons or mean free path of electrons, then energy levels of electrons
changes. This effect is called Quantum confinement effect.
When the material is in sufficiently small size, organization of energy levels into
which electrons can climb of or fall changes. Specifically, the phenomenon results
from electrons and holes being squeezed into a dimension that approaches a critical
quantum measurement called the exciton Bohr radius. These can affect the optical,
electrical and magnetic behaviour of materials.
Variations of properties of nano materials
The physical, electronic, magnetic and chemical properties of materials depend on
size. Small particles behave differently from those of individual atoms or bulk.
Physical properties: The effect of reducing the bulk into particle size is to create
more surface sites i.e. to increase the surface to volume ratio. This changes the surface
pressure and results in a change in the inter particle spacing. Thus the inter atomic
spacing decreases with size.
The change in the inter particle spacing and the large surface to volume ratio in
particle have a combined effect on material properties. Variation in the surface free
energy changes the chemical potential. This affects the thermodynamic properties like
melting point. The melting point decreases with size and at very small sizes the decrease
is faster.
Chemical properties: the large surface to volume ratio, the variations in geometry
and electronic structure has a strong effect on catalytic properties. The reactivity of small
clusters increases rapidly even when the magnitude of the cluster size is changed only by
a few atoms.
Another important application is hydrogen storage in metals. Most metals do not absorb,
hydrogen is typically absorbed dissociatively on surfaces with hydrogen- to- metal atom
ratio of one. This limit is significantly enhanced in small sizes. The small positively
charged clusters of Ni, Pd and Pt and containing between 2 and 60 atoms decreases with
increasing cluster size. This shows that small particles may be very useful in hydrogen
storage devices in metals.
Electrical properties: The ionization potential at small sizes is higher than that for
the bulk and show marked fluctuations as function of size. Due to quantum confinement
the electronic bands in metals become narrower. The delocalized electronic states are
transformed to more localized molecular bands and these bands can be altered by the
passage of current through these materials or by the application of an electric field.
In nano ceramics and magnetic nano composites the electrical conductivity
increases with reduction in particle size where as in metals, electrical conductivity
decreases.
Optical properties: Depending on the particle size, different colours are same.
Gold nano spheres of 100nm appear orange in colour while 50nm nano spheres appear
green in colour. If semiconductor particles are made small enough, quantum effects come
into play, which limits the energies at which electrons and holes can exist in the particles.
As energy is related to wavelength or colour, the optical properties of the particles can be
finely tuned depending on its size. Thus particles can be made to emit or absorb specific
wavelength of light, merely by controlling their size.
An electro chromic device consist of materials in which an optical absorption band
can be introduced or existing band can be altered by the passage of current through the
materials, or by the application of an electric field. They are similar to liquid crystal
displays (LCD) commonly used in calculator and watches. The resolution, brightness and
contrast of these devices depend on tungstic acid gel’s grain size.
Magnetic properties: The strength of a magnet is measured in terms of coercivity
and saturation magnetization values. These values increase with a decrease in the grain
size and an increase in the specific surface area (surface area per unit volume) of the
grains.
In small particle a large number or fraction of the atoms reside at the surface. These
atoms have lower coordination number than the interior atoms. As the coordination
number decreases, the moment increases towards the atomic value there is small particles
are more magnetic than the bulk material.
Nano particle of even non magnetic solids are found to be magnetic. It has been found
theoretically and experimentally that the magnetism special to small sizes and disappears
in clusters. At small sizes, the clusters become spontaneously magnetic.
Mechanical properties: If the grains are nano scale in size, the interface area with in
the material greatly increases, which enhances its strength. Because of the nano size
many mechanical properties like hardness, elastic modulus, fracture toughness, scratch
resistance, fatigue strength are modified.
The presence of extrinsic defects such as pores and cracks may be responsible for low
values of E (young’s modulus) in nano crystalline materials. The intrinsic elastic modulli
of nano structured materials are essentially the same as those for conventional grain size
material until the grain size becomes very small. At lower grain size, the no. of atoms
associated with the grain boundaries and triple junctions become very large. The
hardness, strength and deformation behaviour of nano crystalline materials is unique and
not yet well understood.
Super plasticity is the capability of some polycrystalline materials to exhibit very
large texture deformations without fracture. Super plasticity has been observed occurs at
somewhat low temperatures and at higher strain rates in nano crystalline material.
PRODUCTION OF NANOMATETIALS:
Material can be produced that are nanoscale in one dimension like
thin surface coatings in two dimensions like nanowires and nanotubes or in 3 dimensions
like nanoparticles
Nano materials can be synthesized by’ top down’ techniques producing very small
structures from larger pieces of material. One way is to mechanical crushing of solid into
fine nano powder by ball milling.
Nanomaterials may also be synthesized by ‘bottom up’ techniques, atom by atom or
molecule by molecule. One way of doing this is to allow the atoms or molecules arranges
themselves into a structure due to their natural properties
Ex: - Crystals growth
PERPARATION:
There are many methods to produce nanomaterials. They are
1). PLASMA ARCING:
Plasma is an ionized gas. To produce plasma, potential difference is
applied across two electrodes. The gas yields up its electrons and gets ionized .Ionized
gas (plasma) conducts electricity. A plasma arcing device consists of two electrodes. An
arc passes from one electrode to the other. From the anode electrode due to the potential
difference electrons are emitted. Positively charged ions pass to the other electrode
(cathode), pick up the electron and are deposited to form nanoparticles. As a surface
deposit the depth of the coating must be only a few atoms. Each particle must be
nanosized and independent. The interaction among them must be by hydrogen bonding or
Vander Waals forces. Plasma arcing is used to produced carbon nanotubes.
2). CHEMICAL VAPOUR DEPOSITION:
In this method, nanoparticles are deposited from the gas phase.
Material is heated to from a gas and then allowed to deposit on solid surface, usually
under vacuum condition. The deposition may be either physical or chemical. In
deposition by chemical reaction new product is formed. Nanopowder or oxides and
carbides of metals can be formed, if vapours of carbon or oxygen are present with the
metal.
Production of pure metal powders is also possible using this method.
The metal is meted exciting with microwave frequency and vapourised to produce
plasma at 15000c . This plasma then enters the reaction column cooled by water where
nanosized particles are formed.
CVD can also be used to grow surfaces. If the object to be coated is
introduced inside the chemical vapour, the atoms/molecules coated may react with the
substrate atoms/molecules. The way the atoms /molecules grow on the surface of the
substrate depends on the alignment of the atoms /molecules of the substrate. Surfaces
with unique characteristics can be grown with these techniques.
3. Sol – Gels:
Sol: - A material which when reacts with liquid converts in to a gelly or viscous fluid.
Colloid:- A substance which converts liquid to semisolid or viscous or cloudy.
Gel : Amore thicky substance.
Soot :- When a compound is brunt, it given black fumes called soot.
In solutions molecules of nanometer size are dispersed and move
around randomly and hence the solutions are clear. In colloids, the molecules of size
ranging from 20μm to100μm are suspended in a solvent. When mixed with a liquid
colloids look cloudy or even milky. A colloid that is suspended in a liquid is called as sol.
A suspension that keeps its shape is called a gel. Thus sol-gels are suspensions of colloids
in liquids that keep their shape. Sol -gels formation occurs in different stages.
Hydrolysis
Condensation and polymerization of monomers to form particles
Agglomeration of particles. This is followed by formation of networks which extends
throughout the liquid medium and forms a gel.
The rate of hydrolysis and condensation reactions are governed by various factors
such as PH, temperature, H2O/Si molar ratio, nature and concentration of catalyst and
process of drying. Under proper conditions spherical nanoparticles are produced.
3. ELECTRODEPOSITION:
This method is used to electroplate a material. In many liquids called
electrolytes (aqueous solutions of salts , acids etc) when current is passed through
two electrodes immersed inside the electrolyte, certain mass of the substance liberated
at one electrode gets deposited on the surface of the other. By controlling the current
and other parameters, it is possible to deposit even a single layer of atoms. The films
thus obtained are mechanically robust, highly flat and uniform. These films have very
wide range of application like in batteries, fuel cells, solar cells, magnetic read heads
etc.
5. BALL MILLING (MECHANICAL CRUSHING):
In this method, small balls are allowed to rotate around the inside
of a drum and then fall on a solid with gravity force and crush the solid into
nanocrystallites. Ball milling can be used to prepare a wide range of elemental and
oxide powders. Ball milling is the preferred method for preparing metal oxides.
Materials referred to as nano materials fall under two categories:
Fullerenes :they are a class of allotropes of carbon which conceptually a re graphene
sheets rolled into tubes or spheres .A common method used to produce fullerness is
to send a large current between two near by graphite electrodes in an inert
atmosphere .The resulting carbon plasma arc between the electrodes cools into
sooty residue from which many fullerenes can be isolated .
Eg : carbon nano tubes, buckyball(buckminsterfullerene C-60),cnt…,
Bucky ball: C-60 molecules &buckminister fullerene made up of 60 carbon atoms
arranged in a series of inter locking hexagon(20) and pentagons(12) forming a
structure that looks like soccer ball c60 actually is a truncated icosahedron.
CARBON NANOTUBES (CNT’S):
We know three forms of carbon namely diamond graphite and amorphous
carbon. There is a whole family of other forms of carbon known as carbon nanotubes,
which are related to graphite. In conventional graphite, the sheets of carbon are
stacked on top of one another .They can easily slide over each other. That’s why
graphite is not hard and can be used as a lubricant. When graphite sheets are rolled
into a cylinder and their edges joined, they form carbon nanotubes i.e. carbon
nanotubes are extended tubes of rolled graphite sheets.
TYPES OF CNT’S: A nanotube may consist of one tube of graphite, a one atom
thick single wall nanotube or number of concentric tubes called multiwalled
nanotubes.
There are different types of CNT’S because the graphite sheets can be rolled in
different ways .The 3 types of CNT’S are ZigZag, Armchair and chiral. It is possible
to recognize type by analyzing their cross sectional structures.
Multiwalled nanotubes come in even more complex array of forms.
Each concentric single – walled nanotube can have different structures, and hence
there are a variety of sequential arrangements. There can have either regular layering
or random layering .The structure of the nanotubes influences its properties .Both
type and diameter are important .The wider the diameter of the nanotube, the more it
behaves like graphite. The narrower the diameters of nanotube, the more its intrinsic
properties depends upon its specific type. Nanotubes are mechanically very strong,
flexible and can conduct electricity extremely well.
The helicity of the graphite sheet determines whether the CNT is a
semiconductor of metallic.
PRODUCTION OF CNT’S: There are a number of methods of making CNT’S
few method adopted for the production of CNT’S.
ARC MRTHOD: This method creates CNT’S through arc- vapourisation of two
carbon rods placed end to end, separated by 1mm , in an enclosure filled with inert
gas at low pressure .It is also possible to create CNT’S with arc method in liquid
nitrogen. A direct current of 50-100A, driven by a potential difference of 20V apprx,
creates a high temperature discharge between the two electrodes .The discharges
vapourizes the surface of one of the carbon electrodes, and forms a small rod shaped
deposit on the other electrode. Producing CNT’S in high yield depends on the
uniformity of the plasma arc, and the temperatures of deposits forming on the carbon
electrode.
LASER METHOD: CNT’S were first synthesized using a dual-pulsed laser.
Samples were prepared by laser vapourizations of graphite rods with a 50:50 catalyst
mixture of Cobalt & Nickel at 12000c in flowing argon. The initial layer
vapourization pulse was followed by a second pulse, to vapourize the target more
uniformly. The use of two successive laser pulses minimizes the amount of carbon
deposited as soot. The second laser pulse breaks up the larger particles ablated by the
first one and feeds then into growing nanotube structure. The CNT’S produced by
this method are 10-20nm in diameter and upto 100m or more in length. By varying
the growth temperatures, the catalyst composition and other process parameters the
average nanotube diameter and size distribution can be varied.
CHEMICAL VAPOUR DEPOSITION (CVD): Large amount of CNT”S can be
formed by catalytic CVD of acetylene over Cobalt and Iron catalysts supported on
silica or zeolite. The carbon deposition activity seems to relate to the cobalt content of
the catalyst; where as the CNT’S selectivity seems to be a function of the PH in
catalyst preparation. CNT’S can be formed from ethylene. Supported catalysts such
as iron cobalt and Nickel containing either a single metal or a mixture of metals, seem
to induce the growth of isolated single walled nanotubes or single walled nanotubes,
bundles in the ethylene atmosphere. The production of single walled nanotubes as
well as double walled CNT’S, molybdenum and molylodenum-iron catalysts has also
been demonstrated.
PROPERTIES OF CNT;S : Few unique properties of CNT”S are
1) ELECTRICAL CONDUCTIVITY: CNT’S can be highly conducting , and
hence can be said to be metallic. Their conductivity will be a function of chirality, the
degree of twist and diameter. CNT’S can be either metallic or semi conducting in
their electrical behaviour. Conductivity in multi walled CNT’S is more complex .The
resistivity of single walled nanotubes ropes is of the order of 10-4 ohm –cm at 270c
.This means that single walled nanotube ropes are most conductive carbon fibers.
Individual single walled nanotubes may contain defects. These defects allow the
single walled nanotubes to act as transistors. Similarly by joining CNT”S together
forms transistor - like devices. A nanotube with natural junctions behaves as a
rectifying diode.
2) Strength and elasticity: Because of the strong carbon bonds, the basal plane
elastic modules of graphite, it is one of the largest of any known material. For this
reason, CNT”S are the ultimate high strength fibers. Single walled nanotubes are
stiffer than Steel, and are very resistant to damage from physical forces.
3)THERMAL CONDUCTIVITY AND EXPANSION: The strong in- plane
graphite carbon- carbon bonds make them exceptionally strong and stiff against axial
strains. The almost zero- in -plane thermal expansion but large inter - plane expansion
of single walled nanotubes implies strong in plane coupling and high flexibility
against non- axial strains. CNT’S show very high thermal conductivity. The nanotube
reinforcements in polymeric materials may also significantly improve the thermal and
thermo mechanical properties of composites.
4) HIGHLY ABSORBENT: The large surface area and high absorbency of
CNT”S make them ideal for use in air, gas and water filteration. A lot of research is
being done in replacing activated charcoal with CNT’S in certain ultra high purity
application.
APPLICATION OF NANOMATERIALS:
Engineering: i).Wear protection for tools and machines (anti blocking coatings,
scratch resistant coatings on plastic parts). ii) Lubricant – free bearings.
Electronic industry: Data memory(MRAM,GMR-HD), Displays(OLED,FED),
Laser diodes, Glass fibres
Automotive industry: Light weight construction, Painting (fillers, base coat, clear
coat), Sensors, Coating for wind screen and car bodies.
Construction: Construction materials, Thermal insulation, Flame retardants.
Chemical industry: Fillers for painting systems, Coating systems based on nano
composites. Impregnation of papers, Magnetic Fluids.
Medicine: Drug delivery systems, Agents in cancer therapy, Anti microbial agents
and coatings, Medical rapid tests Active agents.
Energy: Fuel cells, Solar cells, batteries, Capacitors.
Cosmetics: Sun protection, Skin creams, Tooth paste, Lipsticks.
Questions:
What are Nanomaterials? Why do they exhibit different properties?
How are optical, physical and chemical properties of nano particles vary with their
size.
How are electrical, magnetic and mechanical properties of nano particles vary with
their size?
How are nano materials produced?
What are carbon nano tubes? How are they produced?
What are the different types of carbon nano tubes? What are their properties?
What are the important applications of nano materials?