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ELECTRICAL PROPERTIES
The objective of this chapter is to explore the electrical properties of materials, i.e. their
responses to an applied electric field. We begin with the phenomenon of electrical
conduction: the parameters by which it is expressed, the mechanism of conduction by
electrons, and how the electron energy band structure of a materialinfluences its ability to
conduct. These principles are extended to metals, semiconductors and insulators. Particular
attention is given to the dielectric characteristics of insulating materials. The final sections are
devoted to the peculiar phenomena of Ferro electricity and piezoelectricity.
FREE ELECTRON THEORY
The electrical conductivity is one of the properties of materials that vary most widely and the
best conductors such as copper and silver have values of about 108Ω
-1m
-1 and a good
insulator such as polystyrene has conductivity 10-15
Ω-1
m-1
. In a metal, the current is carried
by conduction electrons, and hence the name electronic conduction.
In an isolated atom, the electrons are tightly bound to the nucleus. These electrons are known
as core electrons. But there are a few electrons which are loosely bound to the nucleus and
are called as valance or conduction electrons which are responsible to the properties of the
materials. When the crystal is formed, the valence electrons are treated as electrons of the
crystal and do not belong to the individual atom in that crystal. In a crystal, the valence
electrons experience an effective potential of all the positive ion cores and is less than that
when it experience an effective potential of all positive ion cores and is less than that when it
experiences in an isolated atom.
A useful assumption can be made here that the effective potential inside the metal is constant
and can be considered as zero as a first assumption. This is the fact that at the surface of a
metal, the atoms are not completely bonded and there appears the so called surface potential
which is much higher than the effective potential. Thus the valance electrons are freely
moving in a metal, and it requires certain amount of energy which is more than the surface
energy to leave the metal. Thus, the electrons are in potential energy well of depth W as
shown in figure 3.1. Remember that we have studied infinite potential well or particle in a
rigid box problem in quantum mechanics. The free electron theory of metals is a fundamental
theory for understanding the electrical properties.
Figure:Free electrons are in potential well of depth W.
Here are some important characteristics of metals and these will be discussed in detailed in
the classical free electron theory of metals proposed by Drude and Lorentz
Main characteristics of metals:
(1) Metals exhibit an excellent electrical and thermal conductivity.
(2) Metals obey Ohm`s law. In the steady state condition, the current density j is proportional
to the applied electric potential.
J α E or J = σ E
(3) Where, σ is electrical conductivity, which is about 107mhos m
-1.
(4) At low temperatures, the resistivity ρ is proportional to the fifth power of absolute
temperature. ρ α T5
(5) Above Debye temperature, the resistivity of metals is proportional to the absolute
temperature. Thus metals have positive temperature coefficient of resistivity.
(6) In the case if the metal contains small impurity, the resistivity can be written as
ρ = ρi +ρph (T)
(7) Where ρi is constant which increases with impurity content and ρph (T) is the temperature
dependent part, and this equation is known as Matthiessen`s rule.
(8) The ratio of thermal and electrical conductivity is a constant at constant temperature.
K/σ = LT.
(9) This is known as Wiedemann-Franz Law. The constant L is called Lorentz number
(10) Metals exhibit the phenomenon of superconductivity, its resistivity disappears in
temperature range 0 < T < Tc. The temperature Tc is called critical temperature and above this
temperature the resistivity is finite.
(11) For most metals, the electrical resistivity decreases with increase of pressure.
(12) Metal alloys exhibit order-disorder transitions and the resistivity shows pronounced
minimum corresponding to the ordered phase.
CLASSICAL FREE ELECTRON THEORY OF METALS: DRUDE – LORENTZ
THEORY:
The discovery of the electron by J. J. Thomson in 1897 enbaled P. Drude to produce a
very satisfactory theory of the electrical and thermal conductivities of metals in 1900. In
1904-1905, H. A. Lorentz gave an improved mathematical formulation of Drude theory
without essentially adding anything to its physical content. These two independent
models are now combined and called it as Drude-Lorentz classical free electron theory of
metals.
In order to understand this model, consider a sodium crystal. Each sodium atom contains
11 electrons and its electronic configuration is 1s22s
22p
63s
1. The inner shells are
completely filled and the outer shell is unfilled and contains only one electron. The inner
10 of the electrons are in filled shells which are tightly bound to the nucleus to form a net
positive charge. These electrons can be treated as core electrons. The remaining 11th
electron in the outer shell is loosely bound to the nucleus and moves around the positive
ion core. This electron is called as valence electron or conduction electron. In this
sodium crystal the atoms are close enough to form metallic bonds between them and the
valence electrons are not associated with the parent atom as illustrated in the figure 3.2.
These electrons may be considered as moving freely in between positive ions or the
crystal hence they constitute a free electron gas similar to the molecules in a gas
container. The theory connected to these electrons is known as free electron theory or
free electron gas theory of metals.
Figure: Valence electrons in sodium metal (left) and the free electron approximation
(right).
This model has been developed by the application of Kinetic theory of gases, Maxwell-
Boltzmann statistics and Equilibrium Theorem to the electrons in the metal. For the
purpose of understanding the electrical and thermal properties of metals, the following
basic assumptions have been made in the theory.
Basic Assumptions:
A metal consists of positive ion cores surrounded by delocalized electrons called
valence electrons which are freely moving between charged ions similar to the
molecules in a gas container. Hence they constitute a free electron gas.
The mutual repulsion between the electrons is neglected and this approximation is
called independent electron approximation (true).
Forces between conduction electrons and ion cores should be neglected and this
approximation is called free electro approximation (not true).
All collisions are assumed to be elastic in nature i.e., no energy is lost in the
interaction process.
The potential experienced by the electrons due to the positive ions is completely
uniform and without loss if generality, this potential can be considered as zero. As a
result, there is no change in the energy of the electron when it is moving inside the
metal or it strikes the surfaces.
The total energy of conduction electrons in a metal is all kinetic energy.
At room temperature, the conduction electrons cannot escape from the surface of the
metal because of the surface potential. Thus the electrons are confined within volume
of the metal. This means that the potential energy of the electron inside the metal must
be lower than that at the surface of the metal. Therefore, we assume that the edges of
the metal surfaces are the sites of sharp potential barrier and crystal acts as potential
energy box. In this energy box, the conduction electrons move freely like molecules
in a gas container.
Since the electron gas in a metal having properties similar to those of an ordinary
molecular gas, so one can approximate that the free electron gas also obeys the same
laws of Kinetic theory of gases.
The free electron gas is now treated as a perfect gas and at thermal equilibrium; the
electrons obey the Maxwell-Boltzmann statistics.
According to the equipartition theorem, in the absence of electric field, the energy
associated with each electron at a temperature T is 3/2KBT, where KB is the
Boltzmann constant.
Based on the above assumptions, a simple physical model has been developed to understand
the electrical conductivity in metals. According to the free electron theory, the free electrons
in a metal move randomly with an average speed ~ 105 m/s. Consider a metal consists of
valence electrons. These electrons are freely moving in all possible directions in the absence
of an electric field as shown in the figure 3.3. Hence, their average velocity becomes zero.
Let n be number of free electrons per cubic meter in the conductor and vibe the velocity of the
ith
electron, then the average velocity va of free electrons can be expressed as
Va = 1
𝑛 𝑣i𝑛
𝑖=1 = 0
At thermal equilibrium, there is an equal probability for number of electrons moving in one
direction to those moving in the opposite direction. Therefore the free electrons follow
Maxwell-Boltzmann distribution function. This fact explains that in the absence of an
external electric field, the contribution of electrons to electric current in a conductor is zero.
Figure:Random motion of electrons in the absence of electric potential.
SUCCESSESOF THE CLASSICAL FREE ELECTRON THEORY
This theory has been given an idea about the electrical conductivity which depends on
temperature.
This theory successfully explained the Ohm‘s law.
This theory successfully explained the conductivity of monovalent metals.
This theory successfully explained the skin depth.
This theory is used to verify the Wiedmann-Franz law, which relates the electric
conductivity and thermal conductivity K in metals.
𝐾
𝜍𝑇= 𝐿
Where, L is called the Lorentz number and it is a constant value.
FAILURES OF CLASSICAL FREE ELECTRON THEORY
This theory fails to explain the temperature variation of electrical conductivity in metals.
Experimentally the conductivity σ 𝛼 T-1
.
According to the free electron theory, the conductivity is proportional to the electron
concentration. It is supersizing that the divalent metals and even trivalent metals exhibit less
conductivity than monovalent metals despite the fact that the former have higher electron
concentration compared to the later one.
Wiedmann-Franz law is constant for all the temperatures in the free electron theory. But this
is not true at low temperatures is ten times larger than the value at room temperature.
This theory is unable to explain the negative and positive hall coefficients in semiconductor.
This theory could not predict the different colors of metals.
QUANTUM FREE ELECTRON THEORY OF METALS: SOMMMERFELD
THEORY
The study of metallic properties in terms of the motion of free electrons has been started
before the invention of quantum mechanics. Drude and Lorentz made first attempts to
understand the properties of metals using the laws of classical mechanics. This classical
theory has several remarkable successes like the verification of ohm‘s law and the
Wiedmann-Franz law but it could not reconcile with all the experimental observations. The
failures of classical free electron theory leads to the invention of quantum free electron theory
which was introduced by sommerfeld. In this theory, he applied quantum laws to the
electrons in a metal in addition to the Fermi-Dirac statistics and Pauli Exclusion Principle. He
developed a model based on the assumptions which are discussed as below.
Basic assumptions of the theory:
Sommerfeld free electron theory of metals is simple theory in which the electron can be
treated as a quantum particle.
The classical free electron model is modified by taking into account quantum mechanics
and the Pauli Exclusion Principle.
Free electrons are valence electrons of the atoms just like in the classical theory.
A valence electron in a metal finds itself in the field of all ions and that of the other
electrons. The mutual repulsion between the electrons is neglected and the potential
representing the attractive interaction of ions is assumed to be completely uniform
everywhere inside the solid.
The independent electron approximation and the free electron approximations are retained
in the sommerfeld theory also.
Distribution of electrons in the energy states in an electron gas obeys the Fermi-Dirac
quantum statistics.
Only electrons close to the Fermi energy are scattered that is only a fraction of valence
electrons contribute to the electrical conductivity and such a fraction of electrons is called
effective number of electrons or conduction electrons.
Each energy level corresponding to the principal quantum number splits up into two very
close energy levels representing the electron spin-up and spin-down states. According to
Pauli Exclusion Principle only one electron can occupy one energy level. Thus an energy
level in this model can accommodate at the most two electrons.
The potential energy of the electrons within the metal is constant and might be taken as
zero everywhere except at the ends of the box. Hence, the electrons are moving freely in
the metal and can be treated as free electron gas in a potential energy box. Thus it is
called sommerfeld quantum free electron theory of metals.
Sommerfeld applied these assumptions and the first to find the possible energy levels of
electrons in the potential energy box by solving the solution of the wave equation. Then he
applied the Pauli Exclusion Principle for the distribution of a large number of electrons
among these energy levels in the metal. The expression for the energy of a free electron in a
potential energy box and is given by,
𝐸 =ħ2𝜋2
2𝑚𝐿2𝑛2
n=1,2,3…
From the above equation it is clear that the energy eigen values depend on the values of n, the
principal quantum number. For a given value of n, there must be one energy level. So we
have a set of energy levels constituting a discrete energy spectrum.
QUESTIONS ON FREE ELECTRON THEORY:
1. List out basic postulates of classical free electron theory.
2. Explain the failures of classical free electron theory.
3. List out basic postulates of quantum free electron theory.
4. Explain the failures of quantum free electron theory.
PARTICLE IN A PERIODIC POTENTIAL:
BLOCH THEOREM
One of the first great successes of quantum theory in applied physics is band theory, which
eventually resolved the puzzle of electrical transport in metals, semiconductors, and
insulators. The core of modern understanding of electronic and optical properties of solid-
state materials is based on the band theory, which describes the properties of electrons in a
periodic potential. The periodic potential is due to the periodic arrangement of atoms in a
crystal.
The description of the electron in the periodic material has to be via the Schrodinger equation
−ħ𝟐
𝟐𝒎𝟎𝜵𝟐 + 𝑼 𝒓 𝝍 𝒓 = 𝑬𝝍(𝒓)
where U(r) is the background potential seen by the electrons. Due to the crystalline nature of
the material, the potential U(r) has the same periodicity, R, as the lattice U(r) = U ( r + R)
If the background potential is zero, the electronic function in a volume V is
Ψ(r)=𝑒 𝑖𝑘 .𝑟
𝑉
and the electron momentum and energy are
p= ħk
𝑬 =ħ𝟐𝒌𝟐
𝟐𝒎𝟎
The wave function is spread in the entire sample and has equal probability (ψ*ψ) at every
point in space. Let us examine the periodic crystal. We expect the electron probability to be
same in all unit cells ofthe crystal because each cell isidentical. If the potential was random,
this would not be the case, as shown schematically in Fig. 3.7a. This expectation is, indeed,
correct and is put in a mathematical form by Bloch‘s theorem. Bloch‘s theorem states that the
eigen functions of the Schrodinger equation for a periodic potential are the product of a plane
wave eik.r
and a function Uk(r), which has the same periodicity as theperiodic potential.
Thus Ψk r = 𝑒𝑖𝑘 .𝑟𝑈𝑘(𝑟)
Figure: (a) Potential and electron probability value of a typical electronic wave-functionin a
random material. (b) The effect of a periodic background potential on an electronic wave-
function.
In the case of the periodic potential, 𝜓 2has the same spatial periodicity as the potential. This
puts a special constraint on ψ(r) according to Bloch's theorem is the form of the electronic
function. The periodic part Uk(r) has the same periodicity as the crystal; i.e.,
𝑼𝒌(r)=𝑼𝒌(r+R)
The wave-function has the property
Ψk r + R = 𝑒𝑖𝑘 . 𝑟+𝑅 𝑈𝑘 𝑟 + 𝑅 = 𝑒𝑖𝑘 . 𝑟 𝑈𝑘 𝑟 𝑒𝑖𝑘 .𝑅 = 𝑒𝑖𝑘 .𝑅Ψk r
The wave-function is illustrated in Fig. 3.7b. Before discussing the solutions of the periodic
potential problem, let us take a look at some of the important properties of crystalline
materials.
KRONIG-PENNEY MODEL
The Kronig-Penney model is represented by the one-dimensional periodic potential shown in
figure 3.8.
Even though the model is one-dimensional, it is the periodicity of the potential that is the
crucial property that yields electronic band structure. The mathematical form of the repeating
unit of the potential is
Figure: Kronig-penney potential.
V(x)=Vo -b<x<0
=0 0<x<a
As shown in the above figure the potential has a period of c=a+b. The schrodinger equation
for the model is,
The coefficients in both of these equations are constants, so the solutions are similar to that of
a particle in a box. The solutions to Equations 2 are
Figure: A plot of energy against wave number k showing the band gaps that appear when k
A(1+R)= Nπ. The dashed line is the parabolic result for free electrons.
After solving the above equations,
𝐸 =ħ2𝑘2
2𝑚
and
which represent the allowed energies of a particle in a box.
BRILLOUIN ZONES:
The Brillouin zones highly depend on momentum of electrons (k-space). The energy
spectrum of an electron moving in the presence of periodic potential field is divided into
allowed and forbidden zones. When a parabola representing the energy of a free electron is
compared with the energy of an electron in a periodic field, it is clear that the discontinuities
in the parabola occur at values of k given by,
K = nπ/a
where n = ±1, ±2, ±3. … since k is the wave vector.
k = 2π/λ = n π/a
2a = n λ
From the parabola it can be observed that the electron has allowed energy values in the
region (zone) extending from k = -π/a to +π/a. This zone is called first Brillouin zone. After
the forbidden gap, where there is a discontinuity in energy. Another zone of energy
extending fromk = -π/a to -2π/a and k = π/a to 2π/a. This zone is known as second Brillouin
zone. Similarly, other higher order Brillouin zones can be defined.
The different Brillouin zones correspond to primitive cells of a different type that come up in
the theory of electronic levels in a periodic potential. The first Brillouin zone is considered as
the Wigner-Seitz (WS) primitive cell in the reciprocal lattice. In other words, the first
Brillouin zone is a geometrical construction to the WS primitive cell in the k-space.
In a direct lattice, the procedure of drawing a WS cell is as follows:
i) Draw lines to connect a given lattice points to all nearby lattice points.
ii) Draw new lines or plane at the mid point and normal to the lines in (i).
iii) The smallest volume enclosed in this way is the WS primitive cell. [See figure 3.10].
Figure: Construction of primitive Wigner-Sietz cell in 2-D direct space lattice.
Construction of a Wigner-Seitz cell in the reciprocal lattice (called first Brillouin zone):
To construct the first Brillouin zone, we need to find the link between the incident beam (like
electron or neutron or phonon beam) of wave vector 𝑘 and the reciprocal lattice vector
𝐺 .
This relation may be found as , [for example, an x-ray beam in the crystal will be
diffracted if its wave vector 𝑘 has the magnitude and direction required by this latter
relation]. Thus the procedure to build up the first Brillouin zone is as follows (see figure
3.11):
i) Select a vector 𝐺 from the origin to a reciprocal lattice point. Figure 3.11: Construction of
primitive Wigner-Sietz cell in 2-D direct space lattice.
ii) Construct a plane normal to the vector 𝐺 at its mid point. This plane forms a part of the
zone boundary.
iii) The diffracted beam will be in the direction
iv) Thus the Brillouin construction exhibits all the wave vectors k which can be Bragg-
reflected by the crystal.
Important note:
A wave whose wave vector drawn from the origin terminates on any of the planes will satisfy
the condition of diffraction. Such planes are the perpendicular bisectors of the reciprocal
vectors.
Remarks:
The planes divide the Fourier space of the crystal into fragments as shown for a square
lattice. - The central square is a primitive cell of the reciprocal lattice. It is a Wigner-Seitz
cell of the reciprocal lattice (called the first Brillouin zone). (See figure 3.12). - The first
Brillouin zone is the smallest volume entirely enclosed by the planes.
Figure: Construction of first Brillouin Zone
Conclusion:
Wigner-Seitz cell: smallest possible primitive cell, which consist of one lattice point and all
the surrounding space closer to it than to any other point. The construction of the W-S cell in
the reciprocal lattice delivers the first Brillouin zone (important for diffraction).
Figure: Smallest volume formed by the first Brillouin zone
Importance of Brillouin zone:
The Brillouin zones are used to describe and analyze the electron energy in the band energy
structure of crystals.
ENERGY BAND STRUCTURES IN SOLIDS:
In an isolated atom electron occupy well defined energy states, when atoms come together to
form a solid, their valence electrons interact with each other and with nuclei due to Coulomb
forces. Inter atomic spacing is small in solids compared with liquids and gases. Due to this
closeness of atoms, energy levels of the atoms are disturbed. Energy levels of the valence
electrons are most affected. Each energy level splits up into a number of closely spaced
levels. These split energy levels form a continuous band called energy band as shown in fig
3.13.
Energy level is the discrete energy of an electron in an atom. But energy band is the energy of
the same electron when it is in a solid. At large separation distances, each atom is
independent of all the others, so they have energy levels. But the atoms come close to each
other, electrons respond to the influence of other nuclei and electrons. So each atomic level
splits into N levels each of which may be occupied by pairs of electrons of opposite spins.
The extent of splitting depends on inter atomic separation. Firstly outer most electron shells
are disturbed.
In each energy band, energy levels are discrete; the difference between adjacent states is
infinitesimally small. At the equilibrium spacing, band formation may not occur for electron
sub shells nearestthe nucleus as shown in fig 3.13. Further gaps exist between adjacent bands.
The electron does not have any energy with in energy gap.
Figure: Electron energy versus inter atomic separation for an aggregate of atoms
ENERGY BAND STRUCTURES AND CONDUCTIVITY:
There are four types of band structures which are possible at 0 K. The highest filled state at 0
K is called Fermi Energy (EF). The two important energy bands are:
Valence band: Valence band is the wide range of energies possessed by the valence
electrons.The highest filled energy band where the electrons are present at 0 K.
Conduction band: Conduction band is the wide range of energies possessed by the
conduction electrons. A partially filled or empty energy band where the electrons can
increase their energies by going to higher energy levels within the band when an electric field
is applied.
CLASSIFICATION OF SOLIDS INTO CONDUCTORS, SEMICONDUCTORS &
INSULATORS:
ENERGY BAND STRUCTURES AND CONDUCTIVITY FOR METALS
In metals (conductors), highest occupied band is partially filled or bands overlap. Conduction
occurs by promoting electrons into conducting states that starts right above the Fermi level.
The conducting states are separated from the valence band by an infinitesimal amount.
Energy provided by an electric field is sufficient to excite many electrons into conducting
states.
Ex: Copper atomic number is 29, its electronic configuration is - 1s22s
22p
63s
23p
63d
104s
1.
Each copper atom has one 4s electron. For a solid consisting of N atoms, the 4s band can
accommodate 2N electrons. Thus only half of the 4s band states are filled. This is shown in
fig 3.14(a).
For the second band structure, there is an overlap of filled band (valence band) and empty
band (conduction band). Ex: Magnesium atomic number is 12, its electronic configuration is
1s22s
22p
63s
2. Each magnesium atom has two 3s electrons. In a solid of magnesium, the 3s
and 3p bands are overlapped. This is shown in fig 3.14(b).
Figure: The various possible electron band structures in solids at 0 K. (a) The electron band
structure found in metals such as copper, in which there are available electron states above
and adjacent to filled states, in the same band. (b) The electron band structure of metals such
as magnesium, wherein there is an overlap of filled and empty outer bands. (c) The band
structures of insulators. (d) The band structure of semiconductors.
Energy Band Structures and Conductivity for Semiconductors and insulators:
As shown in fig 3.14, (c) and (d) band structures are similar. The valence band is completely
filled with electrons. There is large energy gap between filled valence band and empty
conduction band. Ex: Insulators. If the energy gap between valence and conduction band is
smaller than that of insulator, it is semiconductor. Electrical conduction requires that
electrons be able to gain energy in an electric field. To become free, electrons must be
promoted (excited) across the band gap. The excitation energy can be provided by heat or
light. In semiconductors and insulators, electrons have to jump across the band gap into
conduction band to find conducting states above Ef. The energy needed for the jump may
come from heat or from irradiation at sufficiently small wavelength (photo excitation).
The difference between semiconductors and insulators is that in semiconductors electrons can
reach the conduction band at ordinary temperatures, where in insulators they cannot. An
electron promoted into the conduction band leaves a hole(positive charge) in the valence
band, that can also participate in conduction. Holes exist in metals as well, but are more
important in semiconductors and insulators.
REVIEW QUESTIONS:
1.1. What are the salient features of the free electron gas model? Obtain the Ohm‘s law based
on it.
1.2. Assuming the electron-lattice interaction to be responsible for scattering of conduction in
a metal, obtain an expression for conductivity in terms of relaxation time and explain any
three drawbacks of classical theory of free electrons.
1.3. Discuss the various drawbacks of classical free electron theory of metals and explain the
assumptions made in quantum theory to overcome the drawbacks.
1.4. Explain the salient features of quantum free electron theory.
1.5. In terms of electron energy band structure, discuss reasons for the difference in electrical
conductivity between metals, semiconductors, and insulators.
PROBLEMS:
1.1. An electron is bound by a potential which closely approaches an infinite square well of
width of width 0.2nm. Calculate the lowest three permissible quantum energies the electron
can have.
1.2. Assuming the electrons to be free, calculate the total number of states below E=5eV in a
4.1. The resistivity of aluminium at room temperature is 2.62x10-8
ohm-m. Calculate (i) the
drift velocity (ii) mobility (iii) relaxation time and (iv) mean free path on the basis of classical
free electron theory.
1.3. Using the Fermi function, evaluate the temperature at which there is 1% probability that
an electron in a solid will have an energy 0.5eV above EF of 5eV.
1.4. Use the Fermi distribution function to obtain the value of F(E) for (E-EF) = 0.01eV at
200K.
OHM’S LAW:
It is defined as the voltage apply to a conductor which is equal to the product of current
passing through the conductor times its resistance. This law is independent of size and shape
of the conductor under consideration. However, it can alsobe expressed in terms of current
density J and electric field E. The current density in the direction of electric field is defined as
the amount of charge flowing across a unit area of cross section per unit time in a conductor.
If n is a number of electrons crossing a unit area in unit time then the amount of charge is
equal to ―– ne‖. Average velocity of electrons is the drift velocity. The flow of charge per
unit time through unit area of the conductor is equal to the product of electon charge times
the drift velocity of the electrons which is current density
J = -nevd.
Substituting the value of vd in the above equation
J = 𝑛𝑒2𝜏
𝑚 E
J α E
Or J = σ E
It follows that the electric current density is proportional to the applied electric field. This
equation is known as Ohm‘s law and proportionality constant is called electrical conductivity
σ.
ELECTRICAL CONDUCTIVITY (MICROSCOPIC FORM OF OHM’S LAW):
In general the electrical conductivity is a property of a conducting material which opposes the
flow of current.
σ = 𝑛𝑒2𝜏
𝑚
This is one of the central results of the classical free electron theory of metals. This formula is
not useful for calculating the conductivity of a given metal, since it contains an unknown
term called relaxation time. The theory itself has important implications. This picture of
electron gas undergoing constant scattering and is still used. For a particular metal n, e and m
are constants. Therefore, the electrical conductivity depends on the relaxation time ‗τ‘. Thus
in the absence of collision process the relaxation time is infinite and hence the conductivity
becomes infinite. The electrical conductivity can also be expresses interms of mobility.
σ = neµ
Thus ingeneral the electrical conductivity of a metal depends on two factors. They are n and
µ.
Electrical Resistivity (ρ)
Electrical resistivity is a measure of how strongly a materialopposes the flow of electric
current. A low resistivity indicates a material that readily allows the movement of electric
charge. The S.I. unit of electrical resistivity is Ω-m. It is commonly represented by a Greek
letter ρ.
(vi) Electrical conductivity (σ)
It is the ability of a substance to conduct an electric current. It is the inverse of the resistivity.
𝜍 = 1
𝜌
Figure: Schematic representation of the apparatus used to measure electrical resistivity.
Ohm's law can be rewritten in terms of the current density J = I/A as: J = σ E. Ohm‘s law can
be expressed in both microscopic and macroscopic forms.
𝑉 = 𝐼𝑅 (Macroscopic form)
𝐽 = 𝜍𝐸 (Microscopic form)
Electrical conductivity varies between different materials by over 27 orders of magnitude.
Metals: σ > 105 (Ω.m)
-1
Semiconductors: 10-6
<σ < 105 (Ω.m)-1
Insulators: 10-6
(Ω.m)-1
<σ<10-20
(Ω.m)-1
ELECTRICAL RESISTIVITY OF METALS (MATTHIESSEN’S RULE):
The resistivity ρ is defined by scattering events due to the imperfections and thermal
vibrations. Total resistivity ρtot can be described by the Matthiessen‘s rule:
𝜌𝑡𝑜𝑡𝑎𝑙 = 𝜌𝑡ℎ𝑒𝑟𝑚𝑎𝑙 + 𝜌𝑖𝑚𝑝𝑢𝑟𝑖𝑡𝑦 + 𝜌𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛
Whereρthermal - from thermal vibrations, ρimpurity - from impurities, ρdeformation - from
deformation-induced defects resistivity increases with temperature, with deformation, and
with alloying.
APPLICATIONS OF ELECTRICAL CONDUCTING MATERIALS
One of the best materials for electrical conduction (low resistivity) is silver, but its use is
restricted due to the high cost. Most widely used conductor is copper: inexpensive, abundant,
high σ, but rather soft – cannot be used in applications where mechanical strength is
important.
Table: Room temperature electrical conductivities for eight common metals and alloys
Solid solution alloying and cold working improve strength but decrease conductivity. If
hardeningis preferred, Cu-Be alloy is used. When weight is important one uses aluminum,
which is half as good as Cu and more resistant to corrosion. Heating elements require low
(high R), and resistance to high temperature oxidation: nickel chromium alloy.
QUESTIONS:
2.1. Verify Ohm‘s law based on quantum free electron theory.
2.2. Discuss the origin of electrical resistance in metals.
2.3. Demonstrate that the two Ohm‘s law expressions, V = I R and J = σ E, are equivalent.
2.4. How does the electrical resistance of a metal change with temperature?
ELECTRON MOBILITY AND DRIFT VELOCITY:
When an electric field E is applied to a conductor, the electrons modify their random motion
and move with an average drift velocity vd in a direction opposite to that of the electric field
as illustrated in figure 3.4 and an electric current is established in the conductor. The flow of
current in a conductor is an indication that the electron in it moves in a specific direction. As
a result, Maxwell-Boltzmann distribution function of the electrons in the conductor
undergoes a change. The directional motion of the electrons is due to the applied electric
field. The group of electrons within the conductor drifts in the direction opposite to the
applied electric field. This motion is called a drift motion and the associated velocity of the
electron is called drift velocity.
Figure: Electrons are moving opposite to the applied electric field E.
Due to the electric field E an electron inside the metal experiences a force and this force F is
equal to the product of electric charge and electric field.
F=-e * E
where‗–e‘ is the charge of the electron. As a result, the electron acquires acceleration.
According to newtons second law of motion, the force experienced by the electron is given
by
F=m * a
where‗a‘ is the acceleration of the electron and m is the mass of the electron. Eliminating F
from the above equations gives the acceleration of the electron under the influence of electric
field –eE=ma.
a = −𝑒𝐸
𝑚
It is clear from the above equation that the electrons are accelerated indefinitely because of
growth of their velocity due to electric field i.e. its drift velocity vd increases with time.
However the electrons are not accelerated indefinitely because during their motion the
electrons collide with impurities and lattice imperfections in the crystal. Here Drude assumed
that the conduction electron acquires random motion after collision with impurities and lattice
imperfections. In the collision process the electron loses the energy and after collision it again
gains energy from the electric field. The velocity of the electrons after collision does not
depend upon its velocity before collision. When the electron collides with the lattice
imperfection it stops momentarily and moves in another direction with a new velocity. Thus
electrons acquire random motion even in the presence of electric field and this is shown in
figure 3.5. This problem can be solved by considering these collisions are similar to the
collision process of an ideal gas in a container. The collision process introduces the concept
of relaxation time, which influences the conductivity in the metals. The average time
between successive collisions of electron with the lattice imperfection is called relaxation
time and is denoted by τ.
Figure: Random velocity of an electron in the applied electric field E.
The acceleration of the electron due to collision with impurities is given by –vd/ τ (negative
sign is due to the acceleration is opposite to the direction of electric field). Now, the
acceleration is equal to sum of the acceleration due to electric field and acceleration due to
collision with impurities.
a = −𝑒𝐸
𝑚–𝑣𝑑
𝜏
After sometime the acceleration of the electrons due to the electric field and acceleration due
to collision should compensate with each other but leads to steady state condition of the
electron gas. As a result the acceleration of electron becomes zero
−𝑒𝐸
𝑚 –
𝑣𝑑
𝜏 = 0
The drift velocity of the electron is then obtained by arranging the above equation.
vd= −𝑒𝜏𝐸
𝑚
Thus the electrons in a conductor attain a constant drift velocity under the application of the
electric field. The drift velocity is proportional to applied electric field and the
proportionality constant is called the electron mobility µ.
RELATION BETWEEN ELECTRON MOBILITY (µ) AND CONDUCTIVITY:
The electron mobility is defined as the drift velocity vdper unit applied electric field E.
µ = 𝑣𝑑
𝐸
Substituting the value of dirft velocity in the above equation we get
µ = −𝑒𝜏
𝑚
The above equation shows that mobility of an electron depends on the relaxation time and
hence it depends on temperature.
PROBLEMS:
1. Calculate the drift velocity and mean free path of copper when it carries a steady current oh
10 amp and whose radius is 0.08cm.Assume that the mean thermal velocity=1.6*106 m/s and
resistivity of copper is 2x108 ohm-m.
SEMI CONDUCTORS
INTRODUCTION:
The electrical conductivity of the semiconducting materials is not as high as that of the
metals. Nevertheless they have some unique electrical characteristics that render them
especially useful. The electrical properties of these materials are extremely sensitive to the
presence of even minute concentrations of impurities.
Energy Band Structure for Intrinsic and Extrinsic Semiconductors:
The band gap of a semiconductor is the minimum energy required to excite an electron that is
stuck in its bound state into a free state where it can participate in conduction. The band
structure of a semiconductor gives the energy of the electrons on the y-axis and is called a
"band diagram". The lower energy level of a semiconductor is called the "valence band" (EV)
and the energy level at which an electron can be considered free is called the "conduction
band" (EC). The band gap (EG) is the gap in energy between the bound state and the free state,
between the valence band and conduction band. Therefore, the band gap is the minimum
change in energy required to excite the electron so that it can participate in conduction.
Once the electron becomes excited into the conduction band, it is free to move about the
semiconductor and participate in conduction. However, the excitation of an electron to the
conduction band will also allow an additional conduction process to take place. The
excitation of an electron to the conduction band leaves behind an empty space for an electron.
An electron from a neighboring atom can move into this empty space. When this electron
moves, it leaves behind another space. The continual movement of the space for an electron,
called a "hole", can be illustrated as the movement of a positively charged particle through
the crystal structure. Consequently, the excitation of an electron into the conduction band
results in not only an electron in the conduction band but also a hole in the valence band.
Thus, both the electron and hole can participate in conduction and are called "carriers".
The concept of a moving "hole" is analogous to that of a bubble in a liquid. Although it is
actually the liquid that moves, it is easier to describe the motion of the bubble going in the
opposite direction.
Intrinsic semiconductors - electrical conductivity is defined by the electronic structure of
pure material.
Extrinsic semiconductors - electrical conductivity is defined by impurity atoms.
Intrinsic semiconductors
Highly pure semiconductors with no impurities are called intrinsic semiconductors. In such a
material there are no charge carriers at 0K. Since the valence band is filled and the
conduction band is empty. At higher temperatures, the electrons reaching the conduction
band due to thermal excitation leave equal number of holes in valence band. In intrinsic
semiconductor, the number of free electrons is equal to the number of holes. The bond
structure and band structures of intrinsic semiconductors are shown in below fig 3.15.
Examples: Si, Ge, GaP, GaAs, InSb, CdS, ZnTe
(a) (b)
Figure: (a) For a semiconductor, occupancy of electron states after an electron excitation
from the valence band into the conduction band in which both a free electron and a hole are
generated. (b) Electron bonding model of electrical conduction in intrinsic silicon after
excitation.
Number of electrons in the conduction band increases exponentially with temperature.Eg is
the band-gap width. In an electric field, electrons and holes move in opposite direction and
participate in conduction.Since both electrons and holes are charge carriers in an intrinsic
semiconductor, the conductivity is
𝜍 = 𝑛 𝑒 𝜇𝑒 + 𝑝 𝑒 𝜇ℎ
where p is the hole concentration and μh the hole mobility.
Electrons are more mobile than holes, μe>μhIn an intrinsic semiconductor, a hole is produced
by thepromotion of each electron to the conduction band. Therefore, n = pand (only for
intrinsic semiconductors)
𝜍 = 𝑛 𝑒 𝜇𝑒 + 𝜇ℎ = 𝑝 𝑒 𝜇𝑒 + 𝜇ℎ
n (and p) increase exponentially with temperature, whereas μe and μh decrease (about
linearly) with temperature.The conductivity of intrinsic semiconductors is increasing with
temperature (different from metals)
𝜍 = 𝑛 𝑒 𝜇𝑒 + 𝜇ℎ
FERMI LEVEL AND DOPING:
The application of intrinsic semiconductors is restricted due to its low conductivity. In
electronic devices, high conducting semiconductors are more essential. The concentration of
either electrons or holes in a semiconductor should be increased depending upon the
requirements in the electronic devices.
This can be carried out simply by adding impurities (one atom in 107
host atoms) to the
intrinsic semiconductors. The process of adding impurity to the intrinsic semiconductors is
called doping. The doped semiconductor is called as extrinsic semiconductor. The
concentration of electrons and holes are not equal in an extrinsic semiconductor.
Usually the doping material is either penta-valent atoms (P, AS, Sb, Bi) or trivalent atoms (B,
Al, Ga, In). The penta-valent atom is called donor atom because it donates one electron to the
conduction band of pure semiconductor. The trivalent doping atom is called acceptor atom
because it accepts one electron from semiconductor atom. The added impurity is called
dopant.
In Extrinsic semiconductors, electrical conductivity is defined by impurity atoms.
Example: Si is considered to be extrinsic at room T if impurity concentration is one impurity
per 1012
lattice sites (remember our estimation of the number of electrons promoted to the
conduction band by thermal fluctuations at 300 K) unlike intrinsic semiconductors, an
extrinsic semiconductor may have different concentrations of holes and electrons. It is called
p-type if p > n and n-type if n > p.
Thus there are two types of extrinsic semiconductors depending on the type of impurity
added.
1. N-type extrinsic semiconductors
2. P-type extrinsic semiconductors
N-type extrinsic semiconductors:
Excess electron carriers are produced by substituting impurities that have more valence
electron per atom than the semiconductor matrix.
Example: phosphorus (or As, Sb) with 5 valence electrons, is an electron donor in Si since
only 4 electrons are used to bond to the Si lattice when it substitutes for a Si atom. Fifth outer
electron of P atom is weakly bound in a donor state(~ 0.01 eV) and can be easily promoted to
the conduction band.Impurities which produce extra conduction electrons are called
donors,Elements in columns V and VI of the periodic table are donors for semiconductors in
the IV column, Si and Ge.
ENERGY BAND DIAGRAM FOR P-TYPE AND N-TYPE SEMI-CONDUCTORS:
The hole created in donor state is far from the valence band and is immobile. Conduction
occurs mainly by the donated electrons thus n-type.
Figure: Extrinsic n-type semi conduction model (electron bonding). An impurity atom such
as phosphorus, having five valence electrons, may substitute for a silicon atom.
(a) (b)
Figure: (a) Electron energy band scheme for a donor impurity level located within the band
gap and just below the bottom of the conduction band. (b) Excitation from a donor state in
which a free electron is generated in the conduction band.
P-type extrinsic semiconductors:
Excess holes are produced by substituting impurities that have fewer valence electrons per
atom than the matrix. A bond with the neighbors is incomplete and can be viewed as a
holeweakly bound to the impurity atom. Elements in columns III of the periodic table (B, Al,
Ga) are donors for semiconductors in the IV column, Si and Ge. Impurities of this type are
called acceptors.
(a) (b)
Figure: (a) Energy band scheme for an acceptor impurity level located within the band gap
and just above the top of the valence band. (b) Excitation of an electron into the acceptor
level leaving behind a hole in the valence band.
Figure: Extrinsic p-type semi conduction model (electron bonding). An impurity atom such
as boron having three valence electrons, may substitute for a silicon atom.
The energy state that corresponds to the hole (acceptor state) is close to the top of the valence
band. An electronmay easily jump from the valence band to complete the bondleaving a hole
behind. Conduction occurs mainly by theholes (thus p-type). For extrinsic p-type
semiconductors
𝜍 = 𝑝 𝑒 𝜇𝑝
CONCLUSION AND SUMMARY:
1. Review of conductivity and resistivity of different types of semiconductors.
2. Recapturing the temperature dependence of the conductivity for metals and intrinsic
semiconductors.
FACTORS EFFECTING CARRIER CONCENTRATION IN SEMI-CONDUCTORS:
Temperature effect on intrinsic carrier concentration
When we speak of an intrinsic semiconductor several factors come to mind:
1. It is extremely pure, containing an insignificant amount of impurities.
2. The properties of the material depend only on the element(s) the semiconductor is made of.
3. For every electron created, a hole is created also, no = po = ni.
For an electron-hole pair to be created in an intrinsic semiconductor, a bond must be broken
in the lattice, and this requires energy. An electron in the valence band must gain enough
energy to jump to the conduction band and leave a hole behind. ni represents the intrinsic
carrier concentration, or we can see it as the number of bonds broken in an intrinsic
semiconductor.
As the temperature is increased, the number of broken bonds (carriers) increases because
there is more thermal energy available so more and more electrons gain enough energy to
break free. Each electron that makes it to the conduction band leaves behind a hole in the
valence band and there is an increase in both the electron and hole concentration. As the
temperature is decreased, electrons do not receive enough energy to break a bond and remain
in the valence band. If electrons are in the conduction band they will quickly lose energy and
fall back to the valence band, annihilating a hole. Therefore, lowering the temperature causes
a decrease in the intrinsic carrier concentration, while raising the temperature causes an
increase in intrinsic carrier concentration.
QUESTIONS:
1. Review of energy band diagram for p-type and n-type semi-conductors.
2. Recapture of factors effecting carrier concentration.
CONDUCTIVITY AND MOBILITY OF CHARGE CARRIERS IN SEMI-
CONDUCTORS:
When an electric field E is applied across a piece of material, the electrons respond by
moving with an average velocity called the drift velocity. Then the electron mobility μ is
defined as
where is the drift velocity (m/s)
is the magnitude of the applied electric field (V/m)
is the mobility (m2/(V.s))
In other words, the electrical mobility of the particle is defined as the ratio of the drift
velocity to the magnitude of the electric field:
Electrical mobility is proportional to the net charge of the particle. This was the basis
for Robert Millikan's demonstration that electrical charges occur in discrete units, whose
magnitude is the charge of the electron.
Electrical mobility of spherical particles much larger than the mean free path of the molecules
of the medium is inversely proportional to the diameter of the particles; for spherical particles
much smaller than the mean free path, the electrical mobility is inversely proportional to the
square of the particle diameter.
Derivation for Electric current and conductivity
Let L be the length of the conductor
A be the area of cross-section
N be number of electrons in the conductor
T be the time taken by electron to travel a distance L
Vd be the average drift velocity
Current ( I )= Total charge Q
Time
Total charge = number of electrons in the conductor * charge = Nq
I = Nq
T
Speed = distance / time
Average drift speed = L / T = V
T = L / V
Substituting T in current equation
I = NqV / L
Current density ( J ) = I / A
= NqV / LA
in the above equation N / LA is electron concentration,let it be n
J = n q v = ρv
= n q µ E
=σ E
Where σ is called conducty of metal and its units are ( ohm – meter )-1
INTRODUCTION TO INSULATORS:
An electrical insulator is a material whose internal electric charges do not flow freely, and
therefore make it impossible to conduct an electric current under the influence of an electric
field. This contrasts with other materials, semiconductors and conductors, which conduct
electric current more easily. The property that distinguishes an insulator is its resistivity;
insulators have higher resistivity than semiconductors or conductors. A perfect insulator does
not exist, because even insulators contain small numbers of mobile charges (charge carriers)
which can carry current. In addition, all insulators become electrically conductive when a
sufficiently large voltage is applied that the electric field tears electrons away from the atoms.
This is known as the breakdown voltage of an insulator. Some materials such as glass, paper
and Teflon, which have high resistivity, are very good electrical insulators. A much larger
class of materials, even though they may have lower bulk resistivity, are still good enough to
prevent significant current from flowing at normally used voltages, and thus are employed as
insulation for electrical wiring and cables. Examples include rubber-like polymers and most
plastics.
ENERGY BAND DIAGRAM AND BASIC REQUIREMENTS OF INSULATORS:
Energy Band Diagram :-
The range of energies that an electron may possess in an atom is known as the energy band.
There Important energy bands are,
Valence Band
Conduction Band
Forbidden Band
Valence Band :-
The Range of Energy possessed by valence electrons is known as valence Bands.
Conduction Band :-
The valence electrons are less tightly bound with the nucleus. So that even an appication of
small elctric field some of the valence electrons detached from the nucleus and it becomes
free electrons. These fee elctrons are responsible for the conduction of current in good
conductors. The electrons are also called conduction electrons. The Range of energy possed
by these elctrons is known as conduction band.
Forbidden Band (or) Energy Gap :-
The energy band in between the condition band and the valence band is called forbidden
Band.
Classification of Materials (or) Solids According to Energy Bands :-
Solids are classified in to there types
1. Insulators
2. Conudctors
3. Semi conductors
Insulators :-
The materials in which the condition band and valence bands are separeated by a wide energy
gap (≈ 15 eV) as shown in figure. A wide energy gap means that a large amount of energy is
required, to free the electrons, by moving them from the valence band into the conduction
band.
Since at room temperature, the valence electrons of an insulator do not have enough energy to
jump in to the condition, therefore insulator do not have an ability to conduct current. Thus
insulators have very high resistively (or extremely low conductivity) at room temperatures.
However if the temperature is raised, some of the valence electrons may acquire energy and
jump in to the conduction band. It causes the resistively of insulators to decrease. Therefore
an insulator have negative temperature co-efficient of resistance.
Conductors :-
The materilas in which conduction and valence bands overlap as shown in figure are called
conductors. The overlapping indicates a large number of electrons available for conduction.
Hence the application of a small amount of voltage results a large amount of current.
Semiconductors :-
The materials, in which the conduction and valence bands are separeated by a small energy
gap (1eV) as shown in figure are called semiconductors. Silicon and germanium are the
commonly used semiconductors.
A small energy gap means that a small amount of energy is required to free the elctrons by
moving them from the valence band in to the conduction band. The semiconductors behave4
like insulators at 0K, because no electrons are available in the conduction band. If the
temperature is further increased, more valence elctrons will acquire energy to jump into the
conduction band.
Thus like insulators, semiconductors also have negative temperature co-efficient of
resistance. It means that conductivity of semiconductors increases with the increases
tempertature.
VARIOUS EXAMPLES OF INSULATORS:
A very flexible coating of an insulator is often applied to electric wire and cable, this is called
insulated wire. Since air is an insulator, in principle no other substance is needed to keep
power where it should be. High-voltage power lines commonly use just air, since a solid (e.g.
plastic) coating is impractical. However, wires that touch each other produce cross
connections, short circuits, and fire hazards. In coaxial cable the center conductor must be
supported exactly in the middle of the hollow shield in order to prevent EM wave reflections.
Finally, wires that expose voltages higher than 60 V can cause human shock and
electrocution hazards. Insulating coatings help to prevent all of these problems. Some wires
have a mechanical covering with no voltage rating—e.g.: service-drop, welding, doorbell,
thermostat wire. An insulated wire or cable has a voltage rating and a maximum conductor
temperature rating. It may not have an ampacity (current-carrying capacity) rating, since this
is dependent upon the surrounding environment (e.g. ambient temperature).
PROPERTIES OF VARIOUS INSULATORS:
Electrical Insulator must be used in electrical system to prevent unwanted flow of current to
the earth from its supporting points. The insulator plays a vital role in electrical system.
Electrical Insulator is a very high resistive path through which practically no current can
flow. In transmission and distribution system, the overhead conductors are generally
supported by supporting towers or poles. The towers and poles both are properly grounded.
So there must be insulator between tower or pole body and current carrying conductors to
prevent the flow of current from conductor to earth through the grounded supporting towers
or poles.
The materials generally used for insulating purpose is called insulating material. For
successful utilization, this material should have some specific properties as listed below- 1. It
must be mechanically strong enough to carry tension and weight of conductors. 2. It must
have very high dielectric strength to withstand the voltage stresses in High Voltage system. 3.
It must possess high Insulation Resistance to prevent leakage current to the earth. 4. The
insulating material must be free from unwanted impurities. 5. It should not be porous. 6.
There must not be any entrance on the surface of electrical insulator so that the moisture or
gases can enter in it. 7. There physical as well as electrical properties must be less affected by
changing temperature.
Now a days glass insulator has become popular in transmission and distribution system.
Annealed tough glass is used for insulating purpose. Glass insulator has numbers of
advantages over conventional porcelain insulator.
DIELECTRICS
INTRODUCTION:
All dielectric materials are electrical insulators. But all electric insulators need not be
dielectrics. Ex: Vacuum is an insulator but it is not dielectric. The distinction between a
dielectric material and an insulator lies in the application. The insulating materials are used to
resist the flow of current through it, when a potential difference is applied across its ends. On
the other hand the dielectric materials are used to store electric energy.
Dielectrics are the materials having electric dipole moment permanently or temporarily by
applying the electric field. In dielectrics all the electrons are bound to their parent molecules
and there are no free charges. Further at the ordinary temperatures, the electrons cannot be
dislocated either by thermal vibrations or by ordinary applied voltages. Generally the
dielectrics are non-metallic materials of high resistivity and have negative temperature
coefficient of resistance.
A dielectric material is an insulator in which electric dipoles can be induced by the electric
field (or permanent dipoles can exist even without electric field), that is where positive and
negative charge are separated on an atomic or molecular level with εr = 81 for water, 20 for
acetone,12 for silicon, 3 for ice, etc.
TYPES OF DIELECTRICS:
Those materials which have the ability to transfer the electric effects without conducting are
known as dielectrics. Dielectrics exist basically in two types.
Polar Dielectrics: Polar dielectrics are those in which the possibility of center coinciding of
the positive as well as negative charge is almost zero i.e. they don‘t coincide with each other.
The reason behind this is their shape. They all are of asymmetric shape. Some of the
examples of the polar dielectrics is NH3, HCl, water etc.
Non Polar dielectrics: In case of non polar dielectrics the centres of both positive as well as
negative charges coincide. Dipole moment of each molecule in non polar system is zero. All
those molecules which belong to this category are symmetric in nature. Examples of non
polar dielectrics are: methane, benzene etc.
DIELECTRIC MATERIALS IN CAPACITORS:
Dipole formation and/or orientation along the external electric field in the capacitor causes a
charge redistribution so that the surface nearest to the positive capacitor plate is negatively
charged and vice versa. The process of dipole formation/alignment in electric field is called
polarization and is described by P = Q1/A. Dipole formation induces additional charge Q
1 on
plates: total plate charge Qt = |Q+ Q1|.
Therefore, 𝐶 = 𝑄𝑡
𝑉 has increased and dielectric constant of the material εr = C / Cvac> 1.
In the capacitor surface charge density (also called dielectric displacement) is
D = Q/A = εrεoE = εoE +P
Polarization is responsible for the increase in charge density above that for vacuum.
FUNDAMENTAL DEFINITIONS:
Dielectric constant (€r)
The dielectric characteristic of a material are determined by the dielectric constant or relative
permittivity ‗€r ‗of that material. The dielectric constant determines the share of the electric
stress which is absorbed by the material. It is the ratio between the permittivity of the
medium and the permittivity of free space.
𝜀𝑟 = 𝜀
𝜖𝑓𝑟𝑒𝑒
The dielectric constant ‗€r‘ has no unit and it is a measure of the electric polarization in the
dielectric material.
Electric polarization
When an electric field is applied to a crystal or a glass containing positive and negative
charges, the positive charges are displayed in the direction of field, while the negative
charges are displaced in the opposite direction. This displacement produces local dipoles
throughout the solid. Thus the process of producing dipoles by an electric field is called
polarization in dielectrics.
Polarization vector (P)
It is found that the strength of the induced electric dipole moment is proportional to the
applied electric field.
𝜇 = 𝛼𝐸
Where α is the constant of proportionality, called the polarizability. If µ-is the average dipole
moment per molecule and N is the number of molecules per unit volume then the polarization
the solid is given by the polarization vector ‘P’ and it can be written as
𝑃 = 𝑁𝜇
So the polarization vector ‗P‘ is the dipole moment per unit volume of the dielectric material.
Electric displacement vector ’D’:
The electric displacement vector (or) electric induction ‗D‘ is a quantity which is a very
convenient function for analyzing the electrostatic fields in the dielectric and is given by
D = Ɛ E = Ɛ€o E + P
From this one can get
𝑃
𝐸= 𝜀 − 𝜀𝑜 = 𝜀𝑜 𝜀𝑟 − 1
This quantity ‗D‘ is similarly to the magnetic induction‘B‘ in the magnetism. Further the
electric displacement vector ‗D‘ has the remarkable property that its sources are only the free
electric charges.
The dielectric constant of vacuum is 1 and is close to 1 for air and many other gases. But
when a piece of a dielectric material is placed between the two plates in capacitor the
capacitance can increase significantly.
𝐶 = 𝜀𝑟𝜀𝑜
𝐴
𝐿
+ _
+ + _
Figure: Polarization Mechanism
Magnitude of electric dipole moment is µ = q d.
DIELECTRIC STRENGTH (OR) DIELECTRIC BREAKDOWN
Very high electric fields (>108 V/m) can excite electrons to the conduction band and
accelerate them to such high energies that they can, in turn, free other electrons, in an
avalanche process (or electrical discharge). The field necessary to start the avalanche process
is called dielectric strength or breakdown strength.
DIELECTRIC LOSS
When an ac electric field is applied to a dielectric material, some amount of electrical energy
is absorbed by the dielectric material and is wasted in the form of heat. This loss of energy is
called dielectric loss. This loss is at the electric field frequencies in the vicinity of the
relaxation frequency for each of the operative dipole types for a specific material. A low
dielectric loss is desired at the frequency of utilization.
QUESTIONS:
1. Define and explain (i) dielectric constant (ii) electric susceptibility (iii) electric polarization
and (iv) polarizability.
2. Explain local field in a dielectric material? Explain how the local field could be calculated
for a cubic dielectric crystal.
3. Explain the polarization mechanism in dielectric materials.
4. Establish a relation between the electronic polarizability (άe) and relative permittivity (έr)
using the concept of internal field.
ELECTRIC POLARIZATION:
When an electric field is applied to a crystal or a glass containing positive and negative
charges, the positive charges are displayed in the direction of field, while the negative
charges are displaced in the opposite direction. This displacement produces local dipoles
throughout the solid. Thus the process of producing dipoles by an electric field is called
polarization in dielectrics.
Polarization vector (P)
It is found that the strength of the induced electric dipole moment is proportional to the
applied electric field.
𝜇 = 𝛼𝐸
Where α is the constant of proportionality, called the polarizability. If µ-is the average dipole
moment per molecule and N is the number of molecules per unit volume then the polarization
the solid is given by the polarization vector ‘P’ and it can be written as
𝑃 = 𝑁𝜇
So the polarization vector ‗P‘ is the dipole moment per unit volume of the dielectric material.
TYPES OF POLARIZATION:
Polarization is the alignment of permanent or induced atomic or molecular dipole moments
with an external applied electric field. There are three types of polarization (fig. 3.21).
1. Electronic polarization
2. Ionic polarization
3. Molecular (orientation) polarization
(c)
Figure: (a) Electronic polarization that results from the distortion of an atomic electron cloud
by an electric field. (b) Ionic polarization that results from the relative displacements of
electrically charged ions in response to an electric field. (c) Response of permanent electric
dipoles (arrows) to an applied electric field, producing orientation polarization.
Electronic (induced) polarization: Applied electric field displaces negative electron
―clouds‖ with respect to positive nucleus. It occurs in all materials.
Ionic (induced) polarization: In ionic materials, applied electric field displaces cations and
anions in opposite directions
Molecular (orientation) polarization: Some materials possess permanent electric dipoles
(e.g. H2O). In absence of electric field, dipoles are randomly oriented. Applying electric field
aligns these dipoles, causing net (large) dipole moment.
Ptotal = Pe + Pi + Po
FREQUENCY DEPENDENCE OF POLARIZATION:
For alternating currents, an applied voltage or electric field changes direction with time. Now,
consider a dielectric material which is connected to ac supply. With each direction reversal,
the dipoles attempt to reorient with the field as shown in below fig 3.22.
Figure: Dipole orientations for (a) one polarity of an alternating electric field and (b) for the
reversed polarity.
As shown in above fig.in a process requires some finite time. For each type of polarization
some minimum reorientation time exists, which depends on the ease with which the particular
dipoles are capable of realignment. A relaxation frequency is taken as a reciprocal of this
minimum reorientation time.
Electrons have much smaller mass than ions, so they respond more rapidly to a changing
electric field. For electric field that oscillates at very high frequencies (1015
Hz) only
electronic polarization can occur. Compared with electrons, ions have large mass so ionic
polarization can‘t occur at optical frequencies. It requires large time to displace ions. At
smaller frequencies (1013
Hz) ionic polarization also occur in addition to electronic
polarization.
Figure: Variation of dielectric constant with frequency of an alternating electric field.
Electronic, ionic and orientation polarization contributions to the dielectric constant are
indicated.
Now third type is orientation polarization. Orientation of permanent dipoles which require the
rotation of a molecule can occur only if the oscillation is relatively slow. So at low
frequencies (106 – 10
10 Hz) the orientation polarization occurs in addition to electronic and
ionic polarization. Space charge polarization is the slowest process and occurs only at power
frequencies (102 Hz).
Thus at low frequencies the value of total polarization is very high and at high frequencies the
value of total polarization is very small. If polarization is large, dielectric constant is also
large from formulae D=ε0E+P and D=εE. Dielectric constant versus frequency graph is as
shown in fig 3.23.
TEMPERATURE DEPENDENCE OF POLARIZATION (Additional):
Electronic and ionic polarizations are practically temperature independent at normal
temperatures. But orientation polarization and space charge polarization are affected by
temperature.In orientation polarization, the sum of dipoles in presence of electric field is
opposed by thermal vibrations of atoms. So polarization decreases with increasing
temperature. In contrast to this in some polar substances, normal temperatures will oppose the
permanent dipoles to align in the field direction. But higher temperatures facilitate the
movement of dipoles. So polarization increases with increasing temperature.
For example:
1. Dielectric constant of solid HCl decreases from 19 to14 when temperature is increased
from 100 to 160 K. This shows that the number of molecules per unit volume decreases due
to the expansion on melting.
2. Solid nitrobenzene is a polar molecule, but it does not exhibit orientation polarization in its
solid state because the rotation of molecules is prevented under an applied field. But in liquid
state, the molecules have sufficient energy to reorient themselves in the applied field. The
relative dielectric constant increases from 3 to 37 on melting. This is in contrast with the
results obtained for solid HCl.
FERRO ELECTRICITY:
Ferro electricity is defined as the spontaneous alignment of electric dipoles by the mutual
interaction in the absence of an applied electric field. It arises from the fact that the local field
increases in proportion to the polarization. Thus, ferro-electric materials must possess
permanent dipoles.Ex.:BaTiO3, Rochelle salt(NaKC4H4O6.4H2O), potassium di hydrogen
phosphate(KH2PO4),Potassium niobate (KNbO3).These materials have extremely high
dielectric constants at relatively low applied field frequencies. Thus, capacitors made from
ferro electric materials are smaller than capacitors made of other dielectric materials.
Properties of Ferro electric Materials:
1. Ferro electric materials can be easily polarized even by very weak electric fields.
2. They exhibit dielectric hysteresis. Lagging of polarization behind the applied electric field
is called dielectric hysteresis. Ferro electricity is a result of dielectric hysteresis.
3. Ferro electric materials possess spontaneous polarization, which is a polarization that
persists when the applied field is zero.
4. Ferro electric materials exhibit ferro-electricity when the temperature T<Tc where Tc is
ferroelectric curie temperature when T>Tc, they are converted into para electric materials.
5. They exhibit domain structures as in the case of ferromagnetic materials.
6. Ferro electric materials exhibit piezo electricity and pyro electricity. Piezo electricity
means the creation of electric polarization by mechanical stress. Pyro electricity means
the creation of electric polarization by thermal stress.
Spontaneous polarization and domains in BaTiO3:
The structural changes in BaTiO3 crystal due to lattice variation give rise to ferro electricity.
Above 1200C BaTiO3 has a cubic crystal structure with the titanium ions exactly at the body
centre, Barium ions are at the body corners and oxygen ions are at the face centers as shown
in fig 3.25. At those temperatures, there is no spontaneous dipole moment.
If the crystal is cooled below 1200C the titanium ions shift to one side of the body centre.
This displaces the neighboring ions along one of the crystal directions which become
elongated. This gives the tetragonal structure due to the shifting of the titanium ion. This
displacement of the titanium ion creates electric dipoles and all the dipoles of the adjacent
unit cells get aligned in
Figure: A barium titanate unit cell in an isometric projection which shows the displacements
of Ti4+
and O2-
ions from the center of the face.
one direction. Below 1200C temperatures there is an enormous value for dielectric constant.
The dielectric constants of these materials are some three orders of magnitude larger than that
in ordinary dielectrics. Since ferro electric materials are similar to ferromagnetic materials.
They exhibit domain structure.
Applications of Ferro electric Materials:
1. In optical communication, the ferro electric crystals are used for optical modulation.
2. The high dielectric constant of ferro electric crystals is also useful for storing energy in
small sized capacitors in electrical circuits.
3. These are used in electro acoustic transducers such as microphone.
HYSTERESIS LOOP:
When most materials are polarized, the polarization induced, P, is almost exactly proportional
to the applied external electric field E; so the polarization is a linear function. This is called
dielectric polarization (see figure).
Figure: Hysteresis of Ferroelectrics
Some materials, known as paraelectric materials, show a more enhanced nonlinear
polarization (see figure). The electric permittivity, corresponding to the slope of the
polarization curve, is not constant as in dielectrics but is a function of the external electric
field. In addition to being nonlinear, ferroelectric materials demonstrate a spontaneous
nonzero polarization (after entrainment, see figure) even when the applied field E is zero. The
distinguishing feature of ferroelectrics is that the spontaneous polarization can be reversed by
a suite P Paraelectric polarization P E Ferroelectric polarization ably strong applied electric
field in the opposite direction; the polarization is therefore dependent not only on the current
electric field but also on its history, yielding a hysteresis loop. They are called ferroelectrics
by analogy to ferromagnetic materials, which have spontaneous magnetization and exhibit
similar hysteresis loops.
PIEZO ELECTRICITY:
In some ceramic materials, application of external forces produces an electric (polarization)
field and vice-versa. Applications of piezoelectricmaterials is based on conversion of
mechanical strain into electricity (microphones, strain gauges, sonar detectors) stress-free
with applied stress
Examples: barium titanate BaTiO3 leadzirconate PbZrO3 quartz.
Figure: (a) Dipoles within a piezoelectric material. (b) A voltage is generated when the
material is subjected to a compressive stress.
QUESTIONS:
1. Explain ferro electricity (write a note on ferro electricity).
2. What are the important characteristics of ferro electric materials?
3. Explain clearly the phenomenon of ferro electricity.
4. Briefly explain why the ferroelectric behavior of BaTiO3 ceases above its ferroelectric
Curie temperature.
