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CHAPTER
1
INTRODUCTORY
CONCEPTS
Electronics is that branch of science and technology which deals with the theory
and application of a class of devices in which transport of electrons occurs through
a vacuum, gas or semiconductor. Vacuum and gas-filled tubes, transistors, etc. are
examples of such devices, known as electron devices. Motion of electrons in these
devices is ordinarily controlled by the application of electric fields.
During the last few decades, the growth of electronics has been tremendous,
and, at present, it may be classified into two broad branches: The branch of
electronics which is concerned with the flow of electrons in a vacuum, gas or
solid is called physical electronics and the branch which deals with the design,
development, and application of electron devices is termed as electronic
engineering.
Electronics, in recent years, has found widespread application in many fields
of science and engineering, and has become indispensable to modern living.
1.1
THE ELECTRON
The electron is one of the fundamental particles of which an atom is composed.
The charge on an electron is negative, and has the smallest possible value. All
conceivable charges are multiples of the electronic charge. The magnitude of the
electronic charge is usually denoted by e; its numerical value is
e = 1.601 × 10–19 coulomb.
The mass of an electron depends on its velocity. When the electron velocity is
much smaller than the velocity of light, the mass of an electron is called its rest
mass. The rest mass of an electron is
m0 = 9.107 × 10–31 kg.
If the electron moves with a velocity v, its mass is given by
mv =
m0
1 − v 2 /c 2
,
where c is the velocity of light: c = 3 × 1010cm/s = 3 × 108 m/s.
1
(1.1)
2
FOUNDATIONS
OF
ELECTRONICS
From Eq. (1.1), we find that the mass of an electron increases with its velocity
and approaches infinity as the electron velocity approaches the velocity of light.
However, the mass of an electron increases by only one per cent when its velocity
is 15 per cent of the velocity of light. Such a high speed is achieved by an electron
when it passes through a potential difference of 6000 volts. Hence, the variation
of electron mass with velocity may be neglected below 6000 volts.
The most widely used unit of the energy of an electron is the electron volt,
abbreviated eV. It is defined as the kinetic energy acquired by an initially stationary
electron in moving through a potential difference of one volt. Since the charge on
an electron is 1.601 × 10–19 coulomb, one electron volt is 1.601 × 10–19 coulomb
× 1 volt = 1.601 × 10–19 joule.
1.2
THE ATOM
An atom of an element consists in general of electrons, protons, and neutrons. The
only exception is the atom of hydrogen which contains only one electron and one
proton but no neutron. An electron is known as a negatively charged particle. A
proton is a positively charged particle; the magnitude of this charge is equal to that
of an electron. The mass of the proton is 1837 times that of an electron. A neutron
is a neutral particle with a mass approximately equal to that of a proton. In an
atom, the protons and neutrons exist together in the nucleus while electrons revolve
in some definite orbits round the nucleus. The number of electrons in an atom is
equal to the number of protons so that the atom is, as a whole, electrically neutral.
The distinction between different atoms arises from the varying number of protons,
neutrons, and electrons in the atom.
Different models have been proposed for the structure of an atom. In the Bohr
model, electrons are assumed to revolve round the nucleus, without radiating any
energy, in certain discrete circular orbits. The orbital radius is such that the angular
momentum of the electron is equal to an integral multiple of the reduced Planck’s
constant* . For different orbits this integral number n takes different values, i.e.,
n = 1, 2, 3, ..., etc. The higher the value of n, the greater the radius of the orbit.
As discrete values of n are only possible, all values of energy for the orbital
electrons are not allowed. The electron energy is quantised: only certain values of
energy corresponding to different integral values of n are allowed. These values of
energy are diagrammatically represented by horizontal lines, as shown in Fig. 1.1.
This diagram is known as the energy-level diagram of the atom.
* The reduced Planck’s constant is Planck’s constant h divided by 2π. Thus =
h 6.626 × 10−34
=
joule-second = 1.055 × 10–34 joule-second.
2π
2 × 3.14
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INTRODUCTORY CONCEPTS
Chapter 1
Energy (eV)
n=5
n=4
n=3
n=2
n=1
Fig. 1.1 Energy-level diagram of an atom.
Several improvements of the simple Bohr model have been made to explain
the fine structure of the spectral lines of some elements. Four quantum-numbers,
denoted by n, l, ml, and ms have been introduced. The energy of the orbital electron
is primarily determined by the quantum number n, referred to as the principal
quantum number. The quantum number l is a measure of the angular momentum
of the electron, and is known as the orbital angular momentum quantum number.
The quantity ml, referred to as the magnetic quantum number, specifies the splitting
of energies with a given n and l in a magnetic field. The number ms is called the
spin quantum number; it shows that the spin of the electron about its own axis is
quantised. The values of the four quantum numbers are:
n = 1, 2, 3, ......;
l = 0, 1, 2, 3, ..., (n – 1);
ml = 0, ± 1, ± 2, ..., ± l;
and
ms = ±
1
.
2
The state of an electron in an atom is uniquely determined by all the four
quantum numbers taken together. This follows from Pauli’s exclusion principle
which states that in a given system more than one electron can occupy a quantum
state described by the four quantum numbers n, l, ml and ms.
1.3
ENERGY BANDS IN CRYSTALLINE SOLIDS
A solid in which the atoms are arranged in a regular and repetitive geometrical
pattern is known as a crystal. The positions of the atoms in a crystal are represented
by an array points in space. This array of points is called a crystal lattice. The
spacing between the atoms in a crystal is fixed and is called the lattice constant of
the crystal.
We shall discuss here the behaviour of electrons in a crystalline solid. Let us
first consider an isolated atom of the crystal. If Z be atomic number, the nucleus of
the atom has a positive charge equal to Ze. The electrostatic potential at a distance
r from the nucleus due to positive charge is given by
V(r) =
Ze
,
Cr
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(1.2)
4
FOUNDATIONS
OF
ELECTRONICS
where C is constant. The value of this constant depends on the chosen system of
units. In SI units C = 4πε0; ε0 is termed the permittivity of free space. Since the
charge on an electron is negative, the potential energy of an electron at a distance
r from the nucleus is
Epot(r) = – eV(r) = −
Ze2
.
Cr
(1.3)
The variation of the potential V(r) with r is shown in Fig. 1.2(a) while that
of the potential energy of an electron Epot(r) with r is shown in Fig. 1.2(b). Note
that V(r) is positive but Epot(r) is negative. The electrostatic potential and the
potential energy of an electron are taken to be zero at infinite distances from the
nucleus.
V(r)
r
+Ze
r
(a)
+Ze
r
Zero level of
potential energy
Epot (r)
r
–
(b)
Fig. 1.2 (a) Variation of potential in the field of a nucleus with distance.
(b) Variation of potential energy of an electron with distance from the nucleus.
Let us now consider two similar atoms close together. The resultant potential
energy of an electron will now be the sum of the potential energies due to the two
individual nuclei. As a result, the resultant potential energy in a region between
the two nuclei will be smaller than the potential energy due to an isolated nucleus.
This is represented in Fig. 1.3.
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INTRODUCTORY CONCEPTS
Chapter 1
Potential energy
Zero level of potential energy
–
+Ze
Distance
+Ze
Fig. 1.3 Potential energy of an electron due to two similar nuclei close together.
In a crystalline solid, the atoms are arranged in a three-dimensional crystal
lattice. Hence, the potential energy of an electron at every point in the solid cannot
be represented by a two dimensional diagram. However, the potential energy along
a line through a row of nuclei of the atoms can be represented in a two dimensional
diagram, as shown in Fig. 1.4. It is seen from this figure that the distribution of
potential energy between the nuclei is a series of humps. The potential energy
increases at the boundary of the solid and becomes zero at infinity.
–
+Ze
+Ze
+Ze
+Ze
Boundary of
the solid surface
Potential energy
Zero level of
potential energy
Distance through crystal
Fig. 1.4
Potential enegy of an electron along a line through a row of atoms
in a crystalline solid.
Electron energy
The electrons in an atom passess through both potential and kinetic energies,
and the total energy of the electrons is known to have negative discrete values or
levels [Fig. 1.5(a)]. When several atoms are brought together to form a solid crystal,
each energy level of the individual atoms splits into as many energy levels as
there are atoms in the solid, so that Pauli’s exclusion principle is not violated for
the atoms in the solid. These energy levels are situated very close together and
form an energy bond, as shown in Fig. 1.5(b).
Zero level
Potential
energy
Energy
level
+Ze
(a) Isolated atom
Fig. 1.5
Distance
Energy
bands
Solid surface
(b) Crystalline solid
(a) Energy-level in an isolated atom.
(b) Formation of energy bands in a crystal.
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FOUNDATIONS
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ELECTRONICS
The width of a band depends on the corresponding energy level in the isolated
atom and the spacing between the atoms in the solid crystal. The lower energy
levels are slightly affected by the interactions between the neighbouring atoms
and thus become very narrow bands. On the other hand, the higher energy levels
are affected considerably by the interaction between the neighbouring atoms and
expansion into a wide band. The spacing between the atoms in a crystalline solid
varies from solid to solid and is fixed for a particular solid. Hence, the width of a
band depends on the type of the solid crystal and is greater for the solid for which
the interatomic distance is less.
Again, the number of energy levels in a band is equal to the number of atoms
in the solid crystal but the width of the band is independent of the number of
atoms. Hence, as the latter increases, the spacing between the levels in a band
decreases. Practically, the spacing between the levels in a band is so small that the
band may be regarded as continuous.
As electrons always prefer states with a minimum energy, the lower energy
bands are usually completely filled or occupied by electrons. The higher energy
bands may be completely empty, i.e., unoccupied by electrons; or, they may be
partly filled, i.e., contain a number of electron less than the number they can
accommodate. It may be noted that Pauli’s exclusion principle restricts the number
of electrons that can be accommodated in a band.
A partly-filled band arises either when a partly-filled level expands into a
band, or when a completely filled band and a completely empty band overlap.
As in a single atom, the electrons cannot have any energy value intermediate
between the permitted levels, the electrons in a solid crystal also cannot have
energy values in the region between the successive bands. That is, energy bands in
a solid are separated by regions of forbidden energy.
The average energy of the electrons in the topmost band of the crystal is
normally much less than the zero energy level shown in Fig. 1.5(b). The rise in
potential energy near the boundary of the solid, as shown in this figure, acts as a
barrier preventing the electrons from coming out of the solid. If sufficient energy
is delivered to the electrons by external means, they may overcome the surface
potential energy barrier and escape from the solid surface.
1.4
METALS, INSULATORS AND SEMICONDUCTORS
The band structure of solids provides a basis which enables us to classify solids
into three distinct groups: (i) metals (or conductors), (ii) insulators, and (iii)
semiconductors.
Metals
A metal is a crystalline solid which contains a partly-filled band. Electrons within
this band are able to move about freely since the band is not filled. When an
electric field is applied, electrons acquire additional energy from the applied field
and move into the empty higher energy states lying adjacent to the filled levels in
the partly-filled band. Hence, conduction of electrons occurs easily in a metal
which is therefore a good conductor. The partly-filled band of the solid, which is
responsible for the conduction of electrons, is called the conduction band of the
solid. The electrons within this band are called free electrons.
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INTRODUCTORY CONCEPTS
If, in a crystal of a solid, the forbidden energy gap between the topmost filled band
(called the valence band) and the lowermost empty band (called the conduction
band) is so wide that at room temperature only a negligible fraction of the thermally
excited electrons can jump from the valence band to the conduction band, the
solid is classed as an insulator. As only a few free electrons are available in the
conduction band, the material is a bad conductor of electricity.
Semiconductors
A substance for which the width of the forbidden energy gap between the valence
band and the conduction band is relatively small is called a semiconductor. As the
forbidden gap (also called the band gap) is narrow, at room temperature some of
the valence electrons acquire sufficient thermal energy to move into the condition
band. The semiconductor exhibits an electrical conductivity larger than that of an
insulator but less than of a metal.
The energy band diagrams of a metal, an insulator, and a semiconductor are
shown in Fig. 1.6(a), (b) and (c), respectively.
Empty
(conduction band)
Partly full
(conduction band)
Forbidden gap
Forbidden gap
Full
(valence
band)
Full
(valence
band)
Full
Full
Full
(a) Metal
Fig. 1.6
1.5
Almost empty
(conduction band)
Full
(b) Insulator
(c) Semiconductor
Energy band diagrams for (a) Metal, (b) Insulator, and (c) Semiconductor.
ELECTRON GAS IN METALS
The free electrons in the conduction band of a metal move about at random just
like gas particles. These electrons are hence said to constitute an electron gas. As
the number of free electrons in a metal is quite large (about 1022 cm–3), statistical
methods are employed to find their average behaviour. An important concept in
the discussion of such behaviour is the distribution function which gives the
probability of occupancy of a given state by the electrons. The distribution function
due to Maxwell and Boltzmann is derived from classical concepts and does not
take into account Pauli’s exclusion principle. This function is applicable to the
cases where the number of particles in a system is less than the number of allowed
quantum states, and cannot, therefore, explain the behaviour of free electrons in a
metal. The defect of the Maxwell-Boltzmann function was removed in the
distribution function due to Fermi and Dirac. This function is applied with a good
approximation to find the energy distribution of free electrons in a metal.
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Chapter 1
Insulators
8
FOUNDATIONS
OF
ELECTRONICS
The Fermi-Dirac distribution function
f(E) =
1
,
1 + exp[( E − EF )/kT ]
(1.4)
where f(E) is the probability of occupancy of a given state with energy E; EF is an
energy level characteristic of a particular solid and is called the Fermi level; T is
the absolute temperature; and k is the Boltzmann constant: k = 1.38 × 10–23 J/K.
It is seen from Eq. (1.4) that at T = 0K the exponential term is zero for E less
than EF and infinite for E greater than EF, thus, when E is less than EF, f(E) = 1.
This means that the probability that all the quantum states below EF may be
occupied by the electrons is unity. All these states will, therefore, be filled by
electrons. On the other hand, when E is greater than EF, f(E) = 0, all the quantum
states above EF will therefor be unfilled. The Fermi energy EF is, therefore, the
maximum energy that can be occupied by electrons at 0K.
When T > 0K, f(E) may have a value greater than zero at energies greater than
the Fermi-level energy (see Fig. 1.7), this, in turn, signifies that at a finite
temperature, some of the electrons in the quantum states just below EF may
acquire extra energy and occupy the states above EF .
f (E)
T = 0K
1
1/2
T > 0K
0
1
E/EF
Fig. 1.7 Variation of f(E) with energy for T = 0K (solid curve) and for T > 0K
(broken curve).
Equation (1.4) shows that E = EF , f (E) = 1/2 for T > 0. Therefore, the Fermilevel may be defined as the energy level for which the probability of occupation is
equal to 1/2 for any temperature greater 0K.
1.6
WORKED-OUT PROBLEMS
1. Find the kinetic energy, in joules and eV, acquired by an initially stationary
electron after it has moved through a potential difference of 50 volts. Also find its
velocity.
Ans. The kinetic energy acquired by the electron is
KE = e × 50 joule
= 1.601 × 10 –19 × 50 joule
= 80.05 × 10 –19 joule
= 50 eV.
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If v be velocity of the electron after it has moved through a potential difference
of 50 volts, then
1
KE = m0v2
2
2.( KE )
or
v=
m0
=
2 × 80.05 × 10−19
9.107 × 10−31
= 4.193 × 106 m/s.
2. A positively charged particle has a mass equal to 7348 times that of an
electron and charge twice that of an electron. Find the velocity and kinetic energy
of this charged particle after it has been accelerated from rest through a potential
difference of 6000 volts.
Ans. If m and q be respectively the mass and charge of the charged particle
and if v be its velocity after is has moved through a potential difference of V volts,
we can write
1
mv2 = qV.
2
Here
m = 7348 m0 = 7348 × 9.107 × 10–31 kg
q = 2e = 2 × 1.601 × 10–19 = 3.202 × 10–19 coulomb
V = 6000 volts.
2qV
2 × 3.202 × 10−19 × 6000
=
m
7348 × 9.107 × 10−31
= 7.577 × 105 m/s.
Again, the kinetic energy of the charged particle is
KE = qV = 2 × 1.601 × 10–19 × 6000 = 1.9212 × 10–15 J
= 12000 eV.
∴
v=
N.B. In the two problems given above, the velocity of the charged particle is
much less than the velocity of light. Hence, the variation of mass with velocity
was not taken into account.
QUESTIONS
1. What is meant by the rest mass of an electron? How does the mass of an electron
vary with its velocity?
2. Define an electron volt. How does it relate with joule?
3. Explain what is meant by energy level diagram of an atom . State and explain
Pauli’s exclusion principle.
4. What is a crystal? How does the potential energy of an electron vary along a
one-dimensional array of atomic nuclei?
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Chapter 1
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INTRODUCTORY CONCEPTS
10
FOUNDATIONS
OF
ELECTRONICS
5. How is the potential energy of an electron in the field of a nucleus modified by
the presence of another nucleus of the same type?
6. Explain what is meant by potential-energy barrier at the surface of a solid.
7. How are energy bands formed in a crystalline solid? What are the factors on
which the width of a band depends?
8. Describe the basis of principles for the band structure of electronic states.
(C.U. 1982)
9. Distinguish between a metal, an insulator, and a semiconductor
10. What is meant by electron gas in a metal? Define Fermi-energy level in a solid.
11. A stationary electron acquires a velocity of 6 × 106 m/s after moving through a
potential difference of V volts. Find V. Also, calculate the kinetic energy of the
electron in joules and eV. [Ans. 102.3 volts; 163.9 × 10–19 joule, 102.3 eV]
12. A positively charged particle with mass equal to 1837 times that of an electron
and charge the same as that of an electron, is accelerated from rest through a
potential difference of 200 volts. Find the kinetic energy of the particle in joule
and electron volt. Also, find its velocity in kilometer/second.
[Ans. 3.202 × 10–17 joule, 200 eV ; 195.5 km/s]
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CHAPTER
2.1
2
ELECTRON EMISSION
WORK FUNCTION
In Sec. 1.3, we have seen that the potential-energy hill or barrier existing at the
surface of a metal prevents free electrons from escaping from the metal at ordinary
temperatures. In order that these free electrons may be liberated from the metal,
they must be supplied with an energy sufficient to enable them to overcome this
potential-energy barrier. The least amount of this energy to be supplied at absolute
zero temperature is expressed quantitatively by the work function.
The potential-energy barrier occurs because of the asymmetry in the
arrangement of the positively charged nuclei about the surface of the metal. When
an electron is well inside the metal, it is surrounded symmetrically by positive
charges. As a result, the electron, on the average, experiences no net force in any
direction. However, when an electron approaches the boundary of the metal, the
positive charges behind the electron pull it back but none pulls it onward. Hence,
the electrons cannot ordinarily come out of the metal surface although they can
move freely in all directions inside the metal.
In order that an electron may escape from the metal, its kinetic energy
associated with the component of velocity normal to the metal surface must be
equal to or greater than the surface potential-energy barrier. At absolute zero of
temperature, the maximum value of this kinetic energy is equal to the Fermi energy.
In Fig. 2.1, let EF be the Fermi energy and EP be the height of the surface potentialenergy barrier with reference to an arbitrarily chosen zero level of energy. As EF is
less than EP, an amount of energy at least equal to EW = EP – EF must be supplied
to the electrons for their emission from the metal. This energy EW is called the
work function of the metal. It is, thus, the minimum energy to be supplied to an
electron at 0K to enable it to escape from the metal. At other temperatures, the
work function is not the minimum energy required to be supplied.
The work function is defined as the energy that an electron at the Fermi level
must acquire in order to be liberated from the metal. It is the work done in liberating
an electron at the Fermi level from the metal.
If φ is the voltage equivalent of the work function, then
(2.1)
EW = φe = EP – EF .
11