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CHAPTER
1
Semiconductors and
Junction Diodes
1.1 Introduction
Semiconductors constitute a large class of substances which have resistivities lying between those of insulators and conductors. The resistivity of semiconductors varies within wide limits, i.e., 10–4 to 104 Ω-m and is
reduced to a very great extent with an increase in temperature (according to an exponential law) as shown in
Fig. 1.1.
In Mendeleev’s periodic table, semiconductors form the group of
elements shown in Fig. 1.2. The most typical and extensively employed semiconductors whose electrical properties have been well
investigated, are Germanium (Ge), Silicon (Si) and Tellurium (Te).
The study of their electrical properties reveals that
(i) Semiconductors have negative temperature Coefficient of
resistance, i.e., the resistance of semiconductors decreases
with increase in temperature and vice versa as shown in
Fig. 1.1. For example, Germanium is actually an insulator at
Fig. 1.1 The Temperature Dependence of the
low temperatures but it becomes good conductor at high
Resistance in Semiconductors.
temperature.
(ii) The resistivity of semiconductors lies between that of a good insulator and of a metal conductor, i.e.,
lies within the range 10–4 to l04 Ω-m.
(iii) The electrical conductivity of a semiconductors is very much affected when a suitable metallic impurity, e.g., Arsenic, Gallium, etc. is added to it. This property of semiconductors is most important.
Group
↓
→
Period
II
III
IV
B
C
V
VI
III
Si
P
S
IV
Ge
As
Se
V
Sn
Sb
Te
VII
I
Fig. 1.2 Position of Semiconductor Elements in the Mendeleev Table.
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Electronics—Theory and Applications
Today no society is considered modern or developed unless it has sizable electronics industry. We know
that there can be no electronics industry without the semiconductors and related technologies. Obviously,
semiconductors form the backbone of electronics. We find that semiconductors affect all walks of life whether
it is communications, computers, biomedical, power, aviation, defence, entertainment etc. The transistors, integrated circuits (IC), lasers and detectors, sensors and other semiconductor devices through the items of daily
use affecting our daily life.
In recent years a number of devices based on semiconductors have been developed that are of great practical applications in electronics. Most important of these are semiconductor diodes, transistors and ICs. A transistor, which is a three terminal device having properties similar to those of a vacuum tube (triode), but
requires no cathode power, and at the same time offers the hope of practically long life. In modern electronic
systems, the complete electronic circuit, containing many transistors, resistors, diodes, capacitors etc. is fabricated on a single chip is called an Integrated Circuit (IC). Recently, a new type of transistor which is called
metal oxide semiconductor (MOS) transistor has become very useful in microelectronic circuits. This MOS
transistor can be used also as a capacitor. In recent years, taking advantage of the silicon integrated circuit
technology, a new field has emerged which attempts to combine the sensor, aculator and the control circuit on
as one integrated unit. In this sense it emulates a biological system. These are known as smart sensors,
microsystem technology (MST) or microelectromechanical systems (MEMS). It has been possible to fashion
miniature mechanical devices such as gears, motors, springs etc. Their combination with sensing and actuating functions has given researchers and engineers the tools to build microsystems that could not be imagined
earlier. This technology is doing the mechanical system what the invention of transistors and ICs did to the
electronic systems. MEMS devices are small in size, light weight, low cost, reliable, with large batch fabrication technology. The MEMS technology involves a large number of materials. Silicon forms the backbone of
this systems also due to its excellent mechanical properties as well as mature micro-fabcrication technology
including lithography, etching and bonding. As compared to electronic valves, semiconductor devices offer
the following advantages: (i) low weight and small size (ii) no power for the filament (iii) long service life
(thousands of hours) (iv) mechanical ruggedness (v) low power losses and (vi) low supply voltages. At the
same time semiconductor devices suffer from a number of disadvantages: (i) marked spread in parameters
between devices within the same type (ii) deterioration in performance with time (ageing); higher noise level
than in electronic valves (iii) unsuitability of most transistors for use at frequencies over tens of megahertz;
(iv) low input resistance as compared with vacuum triodes; (v) inability to handle large power (vi) deterioration in performance after exposure to radioactive emissions.
Continuing efforts in research and development, however, are eliminating or minimizing many of the
demerits of semiconductor devices. There are semiconductor rectifiers for currents of thousands of amperes.
Replacement of germanium with silicon makes crystal diodes and transistors suitable for operation at temperatures upto 125°C. There are transistors for operation at hundreds of megahertz and more, and also microwave
devices such as gunn diode, tunnel diode, etc. The field of semiconductors is rapidly changing. This is
expected to continue in the next decade with some of the changes that can be foreseen now. We expect organic
semiconductors to play prominent role during this decade. Diamond as semiconductor will also be important.
Optoelectronic devices will provide three-dimensional integration of circuits, and optical computing.
Semiconductor devices impose a very small drain on power sources and make it possible to miniaturize or
even micro-miniaturize components and whole circuits. The minimum power for an electronic valve is 0.1
watt, while for a transistor it may be one microwatt, or one hundred thousandth of the former.
Of course, semiconductor devices will not replace electronic valves in each and all applications, for the
valves are also being continually improved. Simply, crystal diodes and transistors may be advantageous in
some uses and electronic valves in others.
Rapid progress in the fabrication of semiconductor structures has resulted into the reduction of three
dimensional bulk systems to two-dimensional, one dimensional and finally to zero-dimensional systems.
Semiconductors and Junction Diodes
3
These reduced dimensional systems are used in future applications like improved semiconductor lasers and
microelectronics.
Quantum Dots (QDs) represent ultimate reduction in the dimensionality of semiconductor devices. These are
3-dimensional semiconductor structures only nanometers in size confining electrons and holes. QDs operate
at the level of a single electron which is certainly the ultimate limit for an electronic device and are used as gain
material in lasers. QDs are used in quantum dot lasers, QD memory devices, QD photo-detectors and even
quantum cryptography. The emission wavelength of a quantum dot is a function of its size. Obviously, by
making quantum dots of different sizes, we can create light of different colours (Eisler et al., Appl. Phys. Lett.;
June 2002).
The semiconductor technology is based on the number of charges and their energy. The electronic devices
such as transistors work due to flow of charge. The electron can be assumed as tiny rotating bar magnet with
two possible orientations: spin-up or spin-down. An applied magnetic field can flip electrons from one state to
another. In this way, spin can be measured and manipulated to represent the 0’s 1’s of digital programming,
analogous to the “current on and current off” states in a conventional silicon chip. The study of electron spin in
materials is called spintronics. Spintronics is based on the direction of spin and spin coupling (Nature, April
2002).
A revolutionary new class of semiconductor electronics based on the spin degree of freedom could be
created. The performance of conventional devices is limited in speed and dissipation whereas spintronics
devices are capable of much higher speed at very low power. Spintronics transistors may work at a faster
speed, are also smaller in size and will consume less power.
Electon spins can be oriented in one direction or the other, called spin-up or spin-down. When electrons
spins are aligned in one direction, these create a net magnetic moment as observed in magnetic materials like,
iron and cobalt. Magnetism is an intrinsic physical property associated with the spins of electrons in a material.
The electron spin may exist not only in the up or down state but also in infinitely many intermediate states
because of its quantum nature depending on the energy of the system. This property may lead to highly parallel
computation which could make a quantum computer work much faster for certain types of calculations. In
quantum mechanics, an electron can be in both spin-up and spin-down states, at the same time. The mixed
state could form the base of a computer, built around not binary bits but the quantum bits or qubit. It is any
combination of a 1 or a 0. The simplest device using spin-dependent effect is a sandwich with two magnetic
layers surrounding a non-magnetic metal or insulator. If the two magnetic layers are different, then the magnetization direction of one can be rotated with respect to the other. This leads to the utilization of these structures
as sensor elements and for memory elements. Scientists and engineers are now trying to use the property of the
electron-like spin rather than charge to develop new generation of microelectronic devices which may be more
versatile and robust than silicon chips and circuit elements. Spins appears to be remarkably robust and move
easily between semiconductors. In case of electron transport from one material to another, the spins do not lose
its orientation or scatter from impurities or structural effects.
All spintronics devices work accordingly to simple principle: information is stored into the spins as a
particular spin orientation (up or down); the spins being attached to mobile electrons, carry the information
along the wire and the information is read at the terminal. Two recent discoveries:
(i) Optically induced long-lived coherent spin-state in semiconductors, and
(ii) Ferromagnetism in semiconducting GaMnAs will lead to revolutionary advances in photonics and
electronics, such as, very fast, very dense memory and logic at extremely low power, spin quantum
devices like Spin-Fets, Spin-LEDs, and Spin-RTDs and quantum computing in conventional semiconductors at room temperature. The emerging technology of spintronics may soon make it possible to
store movies on a palmpilot or build a new computer.
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Electronics—Theory and Applications
To understand the operation of semiconductor devices, it is necessary to study the semiconductor materials
in some detail.
1.2 Atomic Structure and Energy Level
To understand how semiconductors work, one must have a good knowledge of atomic structure. We know that
matter is composed of compounds and elements. The elements are the basic materials found in nature. When
elements are combined to form a new material, we have a compound. The smallest particle that an element can
be reduced to and still retain its properties is called an atom.
Although the atoms of different elements have different properties, they all contain the same subatomic
particles. There are a number of different subatomic particles, but only three of these are of interest in basic
electronics—the proton, the neutron and the electron.
The proton and the neutrons are contained in the nucleus of the atom, and the electrons revolve around the
nucleus along specific orbits. The electrons and the protons are the particles that have the electrical properties.
Neutrons have no electrical charge. Usually, atoms have the same number of electrons and protons, and so
they are electrically neutral. If an atom does have more electrons, it is called a negative ion. If it has more protons, it is called a positive ion.
Fig. 1.3 shows the representations of the atomic structures of H, B, Si, P and Ge atoms. Fig. 1.3 (a) represents the hydrogen atom. It contains one electron revolving around one proton which is the nucleus. The nucleus of H atom contains no neutrons.
Fig. 1.3 (b) represents the structure of a Boron atom. Its nucleus contains 5 protons (P) and 5 neutrons (N).
There are 5 electrons revolving around the nucleus in different orbit. There are two electrons in the first orbit, 2
electrons in the second orbit and only 1 electron in the outermost or valence orbit.
Fig. 1.3 (c) represents the structure of a Silicon atom. It contains 14 protons and 14 neutrons in the nucleus.
There are 14 electrons revolving around the nucleus in different orbits. There are 2 electrons in the first orbit, 8
electrons in the second orbit and 4 electrons in the valence orbit.
Figs. 1.3 (d) and 1.3 (e) represent the structure of Phosphorus and Germanium atoms respectively. We note
that Phosphorus contains 5 electrons in the outermost orbit called valence electrons whereas Ge atom contains
4 electrons in the outermost orbit.
The electrons in the inner orbits of an atom do not normally leave the atom. But the electrons which are in
the outermost orbit, so called valence orbit do not always remain confined to the same atom. Some of these valence electrons in certain materials called metals move in a random manner and may travel from one atom to
another in a crystal lattice. These electrons are called as free electrons. It is due to the presence of these free
electrons in a material, conduction is possible and it constitutes the current. The electrons in the inner orbits of
the atom remain bound to the nucleus and are, therefore, called bound electrons.
The tendency of an atom to give up its valence electrons depends on chemical stability. When an atom is
stable, it resists giving up electrons, and when it is unstable, it tends to give up electrons. The level of stability
is determined by the number of valence electrons, because the atom strives to have its outermost or valence
shell completely filled.
If an atom’s valence orbit is more than half filled, then atom tends to fill its orbit. So, since 8 is the maximum electrons that can be held in the valence orbit, elements with 5 or more valence electrons make good insulators, since they tend to accept rather than give up electrons. On the other hand, atoms with less than 4
valence electrons tend to give up their electrons, thereby the valence shell is empty, this would allow the next
shell, which is already filled, to be the outermost shell. These atoms make the best electrical conductors. The,
elements Si(14) and Ge(32) have 4 valence electrons, and are neither good conductors and nor good insulators.
These are called semiconductors.
Semiconductors and Junction Diodes
5
Fig. 1.3 Atomic Structure of a Few Atoms.
Most characteristics of semiconductors can be easily explained by means of an energy level diagram. We
are familiar that each isolated atom has only a certain number of orbits available. These available orbits represent energy levels for the electrons in the atom. According to Bohr’s theory of atomic structure only discrete
values of electron energies are possible. An electron energy is usually expressed in electron volt (l eV = 1.6 ×
10–19 J = 1.6 × 10–12 erg). An electron can have only certain permissible values, i.e., no electron can exist at an
energy level other than a permissible one’. Energy level diagram for hydrogen atom is shown in Fig. 1.4. The
permissible energy levels for hydrogen atom are numbered n = 1, 2, 3 ... in increasing order of energy.
In any atom, an electron orbiting very close to the nucleus in the first orbit is tightly bound to the nucleus
and possesses only a small amount of energy. The greater the distance of an electron from the nucleus, the
greater is its total energy. The total energy of an electron includes Kinetic and Potential energies. Obviously,
an electron orbiting far from the nucleus would have a greater energy, and hence it can be easily knocked out
of its orbit. This makes it clear that why the valence electrons having maximum energy take part in chemical
reactions and in bonding the atoms together to form solids.
When radiations impinge on an atom, the energy of the electrons increases. As a result, electrons are excited to higher energy levels. The excited state does not last long and very soon, the electron after emitting out
energy in the form of heat, light or other radiations, fall back to the original energy level.
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Electronics—Theory and Applications
1.3 Energy Band Diagrams in Solids
The simple energy level diagram of Fig. 1.4 for electron energies is no longer applicable when one discusses a
solid. A solid is formed when atoms bond together. In a solid, the orbit of an electron is influenced not only by the
charges in its own atom but by electrons and nuclei of every atom in the solid. Each electron in a solid occupies a
different position inside the solid and hence no two electrons can have exactly the same pattern of surrounding
charges. Obviously, the orbits of electrons in a solid are different.
Fig. 1.4 Energy Levels of an Isolated Hydrogen Atom.
When one is considering a solid in bulk, then the simple energy level diagram in Fig. 1.4 modifies to that
shown in Fig. 1.5. All electrons belonging to the first orbits have slightly different energy levels because no
two electrons see exactly the same charge environment. Since there are billions of first-orbit
electrons, the slightly different energy levels
form a group or band. Similarly, the billions of
second orbit electrons, all with slightly different
energy levels, form the second energy band. And
all third orbit electrons form the third band.
Silicon is a commonly used semiconductor
having atomic number 14. Obviously, it has 4
electrons in its outermost or valence orbit.
Clearly, the third band becomes the valence band
(Fig. 1.5). In Fig. 1.5, all these three bands are
shown completely filled. Although the third
shell of an isolated atom of silicon is not completely filled as it has only 4 electrons whereas it
could accommodate a maximum of 8 electrons,
the third energy band or valence band of a bulk
silicon material behaves as if completely filled.
Fig. 1.5 Energy Bands of Silicon at Absolute Zero.
It is so, because in solid silicon each atom
Semiconductors and Junction Diodes
7
positions itself between four other silicon atoms, and each of these neighbours share an electron with the central atom. In this manner, each atom now has 8 electrons associated with it thereby filling the valence band
completely, i.e., all the permissible energy levels in the band are occupied by electrons. Obviously, no electron
in a filled band can move and hence an electron in a completely filled band cannot contribute to electric current. At absolute zero temperature, electrons cannot move through the solid Silicon material and hence it acts
like a perfect insulator. Beyond the valence band there is a conduction band. At absolute zero temperature, the
conduction band is empty, i.e., no electron has enough energy to go into a conduction band. Definite amount of
energy is needed to shift the electron from the valence band to the conduction band. This amount differs from
one substance to another substance and helps to classify them as conductors, insulators and semiconductors.
1.4 Conductors, Semiconductors and Insulators
The band structure in a solid determines whether the solid is an insulator or a conductor or a semiconductor.
The bands are filled upto a certain level by the electrons within each atom. The highest band in which electrons
are still predominantly attached to their atoms are found is called the valence band. This is the band in which
the valence (outermost) electrons from each atom will be located. These are the electrons that are the possible
carriers of electricity. However, in order for an electron to conduct, it must get up to a slightly higher energy so
that it is free of the grip of its atom.
Insulators. Let us consider the case when the valence band in full, i.e., when there are no more available
energy levels. Then, a valence electron must jump (increase its energy) into the next higher band to be free. If
the energy gap between the valence band and the conduction band is too large, then the electron will not be
able to make that jump. Such a material will not be a good conductor of electricity, and is called an insulator.
Forbidden energy gap in an insulator is about 5 eV or even more (Fig. 1.6 (a)). The band theory of solids tells
us that an insulator is a material in which the valence bands are filled and the forbidden energy gap between
valence band and conduction band is too great for the valence electrons cannot jump at normal temperatures
from VB to the CB. An insulator does not conduct at room temperature because there are no conduction
electrons in it. However, an insulator may conduct if its temperature is very high or if a high voltage is applied
across it. This is known as the breakdown of the insulator.
Conductors. There are actually two possibilities for a substance to be a conductor. One is that the valence band
is not completely filled. Then an electron in the valence band can get free of its atom by simply jumping to a
Conduction
Band
Conduction
Band
Conduction
Band
Energy (eV)
Eg ≈ 5 eV or more
Conduction
Band
Eg ≈ 1 eV
Valence
Band
Valence
Band
Valence
Band
Valence
Band
(a)
(b)
(c)
(d)
Fig. 1.6 Energy Band Diagram for (a) Insulators, (b) and (c) Conductors, and (d) Semiconductors.
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Electronics—Theory and Applications
higher energy level within the same band. This jump requires a small amount of energy, and many electrons
can, therefore, make that jump (Fig. 1.6 (b)).
Another situation in which a conductor results stems from the fact that the size of the energy gaps between
the valence band and the conduction band are very small and different for different materials. It can even disappear, when the valence band and conduction band overlap (Fig. 1.6(c)). Therefore, the band theory tells us
that we have a conductor when
(i) The valence band is not filled, so electrons can move to higher states in the valence band and be free,
or
(ii) When there is no energy gap between the valence band and the conduction band, so electrons can easily make the transitions from the valence to the conduction band e.g., metals.
Semiconductors. There is a case in between ‘conductor’ and ‘insulator’. A material with intermediate
properties is called a semiconductor. In a semiconductor,
(i) The valence band is filled, and
(ii) Although there is an energy gap between the valence band and the conduction band yet the energy gap
is not very large.
The energy-band diagram for a semiconductor is shown in Fig. 1.6 (d). In this case, the forbidden energy
gap is of the order of 1 eV (for Silicon, Eg= 1.12 eV and for Ge, Eg = 0.72 eV). Silicon and Germanium are
very good examples of semiconductors. The gaps between their valence and conduction bands are sufficiently
small that the normal thermal energy of the solid at room temperature is enough to knock a few electrons into
the conduction band. These electrons conduct a current, but, as the name ‘Semiconductor’ implies, a semiconductor does not conduct as best as a real conductor. We will discuss it later on. The conductivity of these substances has a number of peculiarities:
(i) The dependence of conductivity on temperature is opposite to that of metals. The conductivity of
semiconductors, in contradiction to that of conductors (metals), may decrease rapidly with temperature.
At low temperatures, a semiconductor may become an insulator. Hence, the distinction between
semiconductors and insulators is purely quantitative and, to a large extent, conventional. The resistance
of most semiconductors is considerably more sensitive to changes in temperature than metals. Compact
temperature meters of high sensitivity may be constructed using semiconducting thermal resistors
(thermistors).
(ii) In a number of cases semiconductors may possess positive as well as negative temperature coefficient.
1.5 Fermi Level
In the preceding article, we have seen that as far as solid-state theory is concerned only the upper energy bands
(valence bands) are of interest, since electrons at lower levels practically do not take part in interactions among
atoms. How can the behaviour of upper band electrons be described? Since we are dealing with a very large
number of electrons, it is natural to use statistical physics methods and consider an aggregate of such electrons
as a kind of gas, usually called as electron gas.
The state of each electron of such a gas may be represented by a point (px, py, pz) in momentum space. The
direction of motion of an electron is parallel to its radius vector P and the energy of an electron depends on its
momentum. Let us consider as a crude approximation that the electrons in solid behave like free particles, i.e.,
we neglect the potential energy of the field in which the electrons move and the interaction between electrons.
If the electrons are free, the relationship between their energy and momentum is given by E = (1/2m)p2.
This means that in momentum space a surface of total energy is a sphere. Such a sphere is usually called a
Semiconductors and Junction Diodes
9
Fermi sphere. One may call the Fermi surface a surface of maximum energy as the states of an electron gas are
contained in a sphere of radius
p max = 2m E max
It is most important to determine how the electrons may distribute themselves in a band among the energies
from zero upto Emax. The number of electrons per unit volume then can be accommodated up to an energy E is
given by
n=
16π( 2me3 )1 / 2
3h
3
E 3/ 2
(1.1)
If the metal is in its ground state, which occurs at absolute zero, all electrons occupy the lowest possible
energy levels compatible with the Pauli exclusion principle (i.e., each level can accomodate two electrons: one
with spin up and one with spin down), as indicated in Fig. 1.7(b). If the total number of electrons per unit volume n0 is less than the total number of energy levels available in the band, the electrons will then occupy all
energy states up to a maximum energy, designated by EF, and called the Fermi energy. Thus the Fermi energy
EF is defined as the energy of the topmost filled level in the ground state of n electrons. If we set E = EF in
Eq. (1.1), we must have n = n0. Therefore for the Fermi energy we obtain the value
2
EF
h 2  3n 0  3
=


8 me  π 
1.1(a)
The energy distribution of electrons in the metal ground state corresponds to the shaded area in Fig. 1.7(a).
When the Fermi energy is equal to the energy band width, the band is fully occupied.
Fig. 1.7 (a) Density of energy states of free electrons in a solid (b) and (c). Distribution of free electrons among energy states in the
conduction band. (d) Occupation of energy states at a temperature different from absolute zero.
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Electronics—Theory and Applications
When band is not completely full, a small amount of energy is enough to excite the uppermost electrons to
nearby energy levels as indicated in Fig. 1.7(c). However, only the uppermost electrons can be thermally about
0.025 eV, which is very small compared with EF , and the exclusion principle makes it impossible for the
low-energy electrons to be excited into nearby occupied states. The distribution of electrons among the energy
level in a thermally excited state of the lattice corresponds to the shaded area in Fig. 1.7(d). The electrons
which have been thermally excited are those with an energy greater than EF. The states occupied by the excited
electrons fall in an energy region of the order of 20 kT about EF.
From the above discussion one can easily conclude that the concept of Fermi level serves the reference energy level from which all other energies are conveniently measured. The probability F(E) of a state corresponding to energy E being occupied by an electron at temperature T°K is given by
1
1.1(b)
F (E ) =
 E − EF 
1+ exp 

 kT 
Here k is Boltzmann constant. Three cases of interest are:
(i) At T = 0°K, if E > EF then F(E) = 0, i.e., energy state is empty.
(ii) At T = 0°K, if E < EF then F(E) = 1, i.e., energy state is occupied by an electron,
(iii) At T ≠ 0°K, and E = EF then F(E) = 1/2, i.e., energy state is 50% occupied.
Obviously, one finds that at T = 0°K, all the energy states above EF are empty, whereas all those below
Fermi energy (EF) are filled with electrons. With the rise in temperature, states above Fermi energy level (EF)
no more remain empty. They are then occupied by the electrons to some extent. Fig. 1.8 shows the position of
Fermi level in energy band diagram in the case of pure or intrinsic semiconductors. It depicts that Fermi level
for an intrinsic semiconductor is situated in the middle of forbidden gap, [See eq. (1.9a)] i.e., between conduction and valence band, and the position of Fermi level is independent of temperature. This reveals that when
the temperature is raised there is a greater possibility of electrons being found above the Fermi level with an
equal possibility of finding an electron vacancy so called ‘hole’ below Fermi level. We will discuss it in later
sections.
Fig. 1.8 Fermi Level in Intrinsic Semiconductor at 0°K.
1.6 Intrinsic Semiconductors
Semiconductor devices, e.g., diodes and transistors, are made from a single crystal for semiconductor material,
e.g. germanium or silicon. To make a semiconductor device, a sample of semiconductor must be in its purest
Semiconductors and Junction Diodes
11
form. A semiconductor in its purest form is called an intrinsic semiconductor. A semiconductor is not truly
intrinsic unless its impurity content is less than 1 part impurity in 100 million parts of semiconductor. As stated
earlier that at 0°K temperature, the Fermi level (EF) in intrinsic semiconductors, lies nearly mid way between the
valence and conduction band.
As the temperature is raised, some electrons near the top of the valence band are thermally ionised into
the lowest levels of the conduction band. The conduction band now contains a few electrons, and the
valence band has a few missing electrons or holes*. Thermal ionization, therefore, is said to create
hole-electron pairs, since there is one hole for each electron. As both electrons and holes transport charge
through the semi-conductor, they are collectively termed charged carriers or simply carriers. In applied
electric and magnetic fields a hole acts as if it is a positive electron. To understand the phenomenon of
conduction of current in a semiconductor, it is essential to study its crystal structure.
1.6 (a) Crystal Structure of Semiconductors
It is the nature of some solid materials to form themselves into bodies called crystals which have characteristic
geometric shapes. These are the crystalline substances. Exact opposites are those other solids which form into
shapeless masses and are said to be plastic, non crystalline, or amorphous. Quartz is a familiar example of crystalline substances where as a block of rubber is amorphous. Elements and compounds both can be crystalline. So
can metals and non metals. Virtually all minerals are crystalline.
Externally, a crystal has several flat faces which are arranged symmetrically w.r.t. each other. Internally, it
has a certain orderly arrangement of atoms in a repeating system called a lattice. Both externally and internally, a
single crystal of a given true crystalline material looks like all other crystals of that material. A single crystal may
be large to the point of hugeness or it may be so small as to be visible only with the aid of magnifier.
Inside the crystal lattice, certain loosely bound electrons (called valence electrons) in the outer orbit of one
atom align themselves with similar electrons in adjacent atoms to form covalent bonds which hold the atoms
together in the orderly structure of the lattice. Thus, in any covalent bond there are shared electrons, so called
because they are shared by neighbouring atoms.
Germanium (Ge) and Silicon (Si) are two very typical semiconductors. In Germanium the atom has 32 electrons distributed in four orbits whereas in silicon there are 14 electrons distributed in three orbits. The outermost
orbit called the valence orbit contain 4 electrons in each case (Fig. 1.3).
The crystals are tetrahedral in structure and each atom shares one of its electrons with its neighbour. Such a
sharing of its electrons between two neighbouring atoms is called a covalent bond. A simplified representation
of the crystalline structure of a semiconductor (Ge) at absolute zero is shown in Fig. 1.9.
At very low temperature near absolute zero all the electrons in the atoms are tied up strongly by these
bonds, but with the rise of temperature, a few electrons break away from some of the covalent bonds and get
themselves freed creating vacant spaces, deficient of electrons known as holes. A hole is equivalent to a net
positive charge equal to that of electron. Whenever a free electron is generated, a hole is created simultaneously,
i.e., free electrons and holes are always generated in pairs. Obviously, the concentration of free electrons and
holes will always be equal in an intrinsic semiconductor. This type of generation of free-electron hole pairs in
semiconductors is called as thermal generation.
Owing to the thermal vibrations the free electrons in a semiconductor crystal jump from one bound position
to another. This is equivalent to the motion of a hole relatively in opposite direction and thus gives an electric
current in the direction of motion of the hole. The conductivity of semiconductor is, therefore, due to the
*
When an electron moves from the valence band it leaves behind a vacant energy state with a positive charge. Moreover, as the electron
moving from the valence band is located at the top edge of the band has a negative effective mass and its absence in the valence band is
associated with a positive mass. The vacant energy state in the valence band therefore has a positive charge and a positive mass. The
vacant energy state is called a ‘hole’.
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Electronics—Theory and Applications
Fig. 1.9 A Simplified Representation of the Crystalline Structure of a Semiconductor (Ge) at Absolute zero.
motion of the holes and of the electrons. [Fig. 1.10(b)]. Such intrinsic semiconductors produce very weak electric current not adequate for useful work.
Fig. 1.10
(a) Crystalline Structure of Ge at room temperature; (b) Generation of Electron-hole Pair in an
Intrinsic Semiconductor (Ge).
1.6 (b) Carrier (or Electron and Hole) Concentrations in Intrinsic Semiconductors
In order to calculate the intrinsic carrier concentration, we first calculate the number of electrons excited into
the conduction band at any temperature T°K and which in turn are free to migrate in the crystal. In carrying out
these calculations it will be assumed that the electrons in the conduction band behave as if they are free with
effective mass m*e and energy will be measured from the top of the valence band. We further take the electron
density of states in the conduction band is equal to that for free electrons i.e.,
Semiconductors and Junction Diodes
13
1
 2m*e  2
N (E ) =
 ( E − E g )1 / 2

( 2π ) 2  h 2 
1
(1.2)
In addition to the density of states function we also require the probability function for calculating the electron
density in the conduction band. The appropriate probability function is the Fermi function f (E) given by equation (1.1). As stated earlier that Fermi level Ep is in the middle of the band gap in the case of intrinsic semiconductors. If (E – EF) < 4 kT, i.e., the lower edge of the conduction band is about 4kBT above EF, one can neglect
1 in the denominator of equation (1.1) and Fermi function f(E) reduces to
F(E) = exp (EF – E)/kT
(1.3)
The number of electrons per unit volume having their energy in the range dE in the conduction band can be
obtained from
ne(E) dE = Ne(E) f(E) dE
(1.4)
∞
n e = ∫ n e ( E )dE.
or
Fg
This leads to
ne = Ne exp [(EF – Eg)/kT]
(1.5)
3
Here
 2πm*e kT  2
Ne = 2

2
 h

is called as the effective density of states in the conduction band. Equation (1.5) gives us the electron concentration in the conduction band.
In order to calculate the hole density nh where h refers to hole, we assume that the holes near the top of the
valence band behave as if they are free particles with an effective mass m*h . One can express the density of hole
states in the valence band as
3
1  2m*  2
N h ( E ) dE = 2  2h  ( − E )1 / 2 dE per unit volume
2π  h 
Here, one must remember that the energy is measured positive upwards from the top of the valence band. If
fh(E) represents the probability of occupation of the states by holes in the valence band then it must be equal to
the probability of the electron states being unoccupied in the valence band, i.e.,
1
exp[( E − E F )kT ] + 1
exp [( E − E F ) / kT ]
=
exp [( E − E F ) / kT ] +1
f h ( E ) = 1− f e ( E ) = 1−
(1.6)
Since E< EF, being in the valence band, so exp [(E – EF )/kT] <<1 and hence we can always neglect exp [(E –
Ep)/kT] in comparison to 1 in the denominator and one obtains
fh(E) = exp [(E – EF)/kT]
(1.7)
Using the relation nh(E)dE = Nh(E)fh(E)dE, one obtains the hole concentration in the valence band as
nh = Nv exp [– EF/kT]
(1.8)
14
Electronics—Theory and Applications
 2πm*h kT 
N v = 2

 h2 
where
is called the effective density of states in the valence band.
Since for intrinsic semiconductors, we have ne = nh and, therefore, from (1.5) and (1.8), one obtains
m*e
3/ 2
exp[( E F − E g )/kT ] = m*h
or
EF =
Eg
+
2
3/ 2
exp[ − E F /kT ]
 m* 
3
kT log  *h 
4
 me 
(1.9)
At T = 0°K, equation (1.9) reduces to
EF =
Eg
2
(1.9a)
1.6 (c) Fermi Level in Intrinsic Semiconductor
From (1.9) it follows .that at T = 0°K, m*e = m*h and Fermi level in the intrinsic semiconductor lies exactly
mid-way in the forbidden gap, i.e.,
EF = Eg/2
Obviously, the Fermi level EF which can be the highest occupied energy level at T = 0 K lies near the middle
of the band gap for an intrinsic semiconductor (Fig. 1.10(c)). The Fermi-Dirac (FD) distribution function f (E)
at T = 0 K is plotted in Fig. 1.10(d).
Fig. 1.10 (c) Energy band diagram; (d) FD distribution function f (E) for an intrinsic semiconductor
In general m*h > m*e , i.e., m*h is slightly greater than m*e and so the Fermi level is raised slightly as the temperature T increases. But for all practical purposes the Fermi level in intrinsic semiconductors can be assumed to
be constant for a wide range of temperatures. It is worthwhile to mention that the Fermi level in semiconductors is not a static level, but a dynamic one, since it changes appreciably with change of temperature and impurities. Variation of Fermi level with impurity concentration enables one, the operation of the various
semiconductor devices.
1.6 (d) Law of Mass Action
Multiplying Equations (1.5) and (1.8), one obtains
3
 Eg 
 2πkT 
np = n e n h = 4  2  ( m*e m*e ) 3 / 2 exp  −

 h 
 kT 
Semiconductors and Junction Diodes
15
where ne = n and nh, = p and Eg is the band gap energy. Since band gap energy Eg is constant and hence the
product np is also constant, i.e.,
np = n i2 = const [T 3 exp ( − E g /kT )]
(1.10)
Here ni: is called the intrinsic density of either carrier. From eqn. (1.10) it is evident that n i2 or np is a constant
depending on the temperature and the width .of the forbidden gap. It does not depend on the impurities introduced as long as the impurities do not change the width of the forbidden energy gap. Equation (1.10) is called
the law of mass action. The result also holds good in the presence of impurities as well.
Because np is a constant independent of impurity concentration at a given temperature, the introduction of a
small proportion of a suitable impurity to increase n, say must decrease p. This result is important in practice,
as one can reduce the total carrier concentration (n + p) in an impurity crystal, sometimes enormously, by the
controlled introduction of suitable imparities. Such a reduction is called compensation of one impurity type by
adding another.
1.6 (e) Electrical Conductivity
The electrical conductivity of intrinsic semiconductors called intrinsic conductivity in the very low temperature range (~ 0 K) is due to intrinsic charge carriers, i.e., due to electrons and holes. Such conductivity is sometimes termed as intrinsic conductivity.
Since there are two types of carriers in the intrinsic semiconductor, electrons and holes, its specific conductance is the sum of the conductivity σe = | e | nµe due to free electrons, with the concentration n and mobility µe,
and of the conductivity σh = | e | pµh due to the presence of holes, with the concentration p and mobility µh. The
mobility is defined as the magnitude of the drift velocity per unit electric field
µ = ν/E
(1.11)
The mobility is defined to be positive for both electrons and holes, although their drift velocities are opposite.
The electrical conductivity of an intrinsic semiconductor is given by
σ = | e | (nµe + pµh)
(1.12)
Since for an intrinsic semiconductor n = p, we have
σi = | e | n (µe + µh)
(1.13)
Here σi, denotes the intrinsic conductivity. Substituting the value of n, one obtains
3
 2πkT  2
σ i = 2 | e |  2  ( m*e m*h ) 3 / 4 exp [ − E g /2kT ] (µ e + µ h )
 h 
(1.14)
It is worthwhile to mention that the value of Eg is more in the case of silicon (Eg = 1.12 eV) than in the case
of germanium (Eg = 0.72 eV). Obviously, less number of electron-hole pairs will be generated in silicon than
in germanium at room temperature. This means that the conductivity of silicon will be less than that of germanium at room temperature.
When a battery is connected across a semiconductor (Fig. 1.11) the electrons experience a force towards
the positive terminal of the battery; and holes towards the negative terminal. The random motion of electrons
and holes gets modified. Over and above the random motion, there also occurs a net movement, called drift.
The random motion does not contribute to any electric current. The free electrons drift towards the positive
terminal of the battery, whereas the holes towards the negative terminal. The electric current flows through the
semiconductor in the same direction as in which the holes are moving. Since the electrons are negatively
16
Electronics—Theory and Applications
charged, the direction of conventional electric current is opposite to the direction of their motion as shown in
Fig. (1.11). Although, the two types of charge carriers move in opposite directions, the two currents are in the
same direction.
Fig. 1.11 Electric Current in an Intrinsic Semiconductor.
When the flow of charge carriers in a semiconductor is due to an applied voltage the resultant current is called
a drift current. A second type of current called as diffusion current also exists in a semiconductor. The diffusion
current flows as a result of a gradient of carrier concentration, i.e., the difference of carrier concentration from
one region to another. The diffusion current is also due to the motion of both electrons and holes.
Let us now investigate the effect of temperature on intrinsic semiconductor. From eqn. (1.14), it is evident
that the exponential term exp [– Eg /2kT] dominates all other temperature dependence. Writing eqn. (1.14) as
3


 2πkT  2 * * 3 / 4

+ log 2 | e |  2  ( me mh ) (µ e + µ h )
log σ i = −


 h 
2kT


Eg
or
log ρ = − log σ i
3


 2πkT  2 * * 3 / 4

=
− log 2 | e |  2  ( me mh ) (µ e + µ h ) (1.15)


 h 
2kT


Eg
where ρ is the resistivity.
The intrinsic semiconductor has a small conductivity. In a sample of
germanium at room .temperature the intrinsic carrier concentration is 2.5 ×
1019/m3. With the rise in temperature, the conductivity increases, i.e., resistivity decreases. Fig. (1.12) shows plot of log ρ vs. 1/T for some intrinsic
semiconductors.
1.7 Extrinsic Semiconductors
We have seen that the conductivity of intrinsic semiconductors is very
small and hence they are not suitable for any useful work except as a heat or
Fig. 1.12 Plot of log ρ vs. 1/T for
Some Intrinsic Semiconductors.
Semiconductors and Junction Diodes
17
light sensitive resistances. The conductivity can, however, be enormously increased by addition of suitable
impurity in a very small proportion, i.e., nearly 1 in 106 parts of the semiconductor. This process is called
doping. Doping is done after the semiconductor material has been refined to a high degree of purity. A doped
or impurity semiconductor is known as an extrinsic semiconductor.
1.7 (a) N-Type Semiconductors
Germanium and Silicon are tetravalent. The impurity atoms may be either pentavalent or trivalent, i.e., from
group V and III of the periodic table. If a small quantity of a pentavalent impurity (having 5-electrons in the
outermost orbit) like Arsenic (As), Antimony (Sb) or Phosphorus (P) is introduced in Germanium, it replaces
equal number of Germanium atoms without changing the physical state of the crystal. Each of the four out of five valency electrons of impurity say of Arsenic enters into covalent bonds with
Germanium, while the fifth valency electron is set free to move
from one atom to the other as shown in Fig. (1.13). The impurity
is called donor impurity as it donates electron and the crystal is
called N-type semiconductor. A small amount of Arsenic (impurity) injects billions of free electrons into Germanium thus increasing its conductivity enormously. In N-type semiconductors
the majority carriers of charge are the electrons and holes are
minority carriers. This is because when donor atoms are added to
a semiconductor, the extra free electrons give the semiconductor
a greater number of free electrons than it would normally have.
And, unlike, the electrons that are freed because of thermal agitation, donor electrons do not produce holes. As a result, the current
carriers in a semiconductor doped with pentavalent impurities are
Fig. 1.13 N-type Semiconductor.
primarily negative electrons.
The impurity atom has five valence electrons. After
donating one electron, it is left with + 1 excess charge.
It then becomes a positively charged immobile ion. It
is immobile because it is held tightly in the crystal by
the four covalent bonds as shown in Fig. 1.13.
It is important to understand that in N-type semiconductors, although electrons (negative charges) are the
majority carriers, but the semiconductor doped with
impurity remains electrically neutral. Free electrons
and holes are generated in pairs due to thermal energy
and negative charge of electrons donated by impurity
atoms is exactly balanced by positive charge of the immobile ions. Representation of an N-type semiconducFig. 1.14 Representation of an N-type Semiconductor.
tor is shown in Fig. 1.14, we have not shown Silicon or
Germanium atoms in this figure. One should assume
them as a continuous structure over the whole background. The fixed or immobile ions are regularly distributed in the crystal structure. The electrons and the holes, being free to move, are shown randomly distributed at
any moment.
Since N-type semiconductors have extra free electrons, and pure semiconductors do not, the energy band
diagram for a doped semiconductor is slightly different from that of a pure semiconductor. In effect, another
18
Electronics—Theory and Applications
energy level exists; a level for the donor electron,
which is closer to the conduction band. The forbidden band for the donor electron is much narrower than the forbidden band for the valence
electron; so one can see that it is much easier to
cause electron flow in an N-type semiconductor
(Fig. 1.15).
1.7 (b) P-type Semiconductor
Fig. 1.15 Excess Free Elections in N-type Semiconductors Produce
a Donor Energy Level Close to the Conduction Band.
When a trivalent impurity (having 3 electrons in
outermost orbit) like Indium (In), Boron (B) or Gallium (Ga) is added in a Germanium intrinsic semiconductor, the impurity atoms will displace some of the Germanium atoms in the crystal during its formation as
shown in Fig. 1.16. In this case only three out of the four possible covalent bonds are filled while the fourth
bond is vacant and the vacancy acts as a hole. Hence a hole moves relative to the electron in a direction opposite to the direction of electron, when an electron moves from one bond to the other. This trivalent impurity
known as the acceptor or P-type impurity injects into the crystal billions of holes and the majority carriers of
the charge are the holes responsible for the conductivity of the crystal. For this reason such crystals are called
P-type semiconductors or P-type crystals. P-type semiconductor can be represented as shown in Fig. 1.17.
Fig. 1.16 P-type Semiconductor.
The energy band diagram of P-type semiconductor also differs from that of the pure superconductor.
Since there is an extra number of holes, which tend
to attract electrons, they aid in starting current flow.
As a result, the acceptor energy level is also somewhat higher than that of the valence band. However,
it is not as high as the donor level (Fig. 1.18a).
P-type semiconductors will conduct easily than pure
semiconductors, but not quite as easy as n-type
semiconductors.
Fig. 1.17 Representation of a P-type Semiconductor.
Fig. 1.18 (a) Excess Holes in P-type Semiconductors Introduce
an Acceptor Energy Level Close to Valence Level.
Semiconductors and Junction Diodes
19
1.8 Important Properties of Extrinsic Semiconductors
We have seen that the introduction of impurities in pure semiconductors increases the density of one type or
another type of charge carriers. The product of holes and electrons in a semiconductor is constant depending
on the width of energy gap and temperature and hence the introduction of the impurities results in an increase
in the density of one type of carrier and a reduction in the density of the other type of carrier. In an extrinsic
semiconductor, the carriers introduced by the impurities are called majority carriers and the other type are
called minority carriers. It is important to note that the low value for minority carrier density is due to added recombination.
Let Nd be the donor impurity density, Na the acceptor impurity density, p the density of holes and n the
density of electrons in an extrinsic semiconducting material. From the condition of charge neutrality, we have
Nd + p = Na + n
or
n = (N d − N a ) + p = (N d − N a ) +
n i2
n
(Q np = n i2 )
n 2 − ( N d − N a ) n − n i2 = 0
or
(1.16)
Solving the quadratic equation in n, one obtains
n=
or
(N d − N a ) ±
( N d − N a ) 2 + 4n i2
2
n ≈ ( N d − N a ) when ( N d − N a ) >> n i
(1.17)
Obviously, the electron density (n) in the N-type semiconductor equals the difference in the donor and acceptor impurity densities when they are large compared to the intrinsic density, ni,. Similarly the hole density
(p) in a P-type semiconductor is given by
p = (Na – Nd)
when
(Na – Nd) >> ni
1.8 (a) Donor and Acceptor States
When an impurity atom from group V of the periodic table, say Phosphorous is added to a pure Ge or Si crystal
as a pentavalent impurity, these impurity atoms enter the lattice by substitution for normal atoms, and not in
interstitial positions. These impurity atoms contribute five electrons per atom to the valence band, i.e. we have
an extra electron per impurity atom. These additional electrons (which cannot be accommodated in the
valence band of the original lattice) occupy some discrete energy levels just below the conduction band; the
separation may be a few tenths of an electron volt. These excess electrons are released by the impurity atoms
and excited into the conduction band. The excited electrons then contribute to the electrical conductivity of the
semiconductor. Conversely, the impurity may consist of atoms having fewer electrons than of a semiconductor [Fig. 1.18(b)]. For the cases in which Si and Ge are the host substances, the impurity atoms could be boron
or aluminium, each of which contributes only three electrons. In this situation the impurity introduces vacant
discrete energy levels, very close to the top of the valence band. Therefore it is easy to excite some of the more
energetic electrons in the valence band into the impurity levels. This process produces vacant states, or holes,
in the valence band. These holes then act as positive electrons [Fig. 1.18(c)]. We must note that to produce significant changes in the conductivity of a semiconductor, it is sufficient to have about one impurity atom per
million semiconductor atoms.
We have already seen that the crystal as a whole remains neutral because the electron remains in the crystal. We have further seen that the band diagram of N-type or P-type semiconductor differs from that of the pure
20
Electronics—Theory and Applications
Fig. 1.18 Impurities in a semiconductor (b) Donors, or N-Type (c) Acceptors, or P-Type
semiconductor. The band diagram for N-type or P-type semiconductor explains clearly why the conduction
becomes possible in impure semiconductors at comparatively low temperatures.
1.8 (b) Fermi Level in Extrinsic Semiconductor
We have read that Fermi level is situated in the middle of the band gap in an intrinsic semiconductor as the
electron and hole densities are equal. When the intrinsic semiconductor is doped, the carrier densities change,
consequently the position of the Fermi level also changes. The shift in the position of the Fermi level can easily
be related to the majority carrier density in an extrinsic semiconductor if it is assumed that the addition of impurities do not affect the densities of energy states in the conduction and valence bands.
Let Nc and Nv denote the density of states in the conduction band and density of states in the valence band,
respectively. We have for an intrinsic semiconductor
and
 E fi − E c  
n = N c exp 
 
 kT  

 E c − E fi  
p = N v exp 
 
 kT  
(1.18)
Here Efi is the energy associated with the Fermi level in an intrinsic semiconductor. For an intrinsic semiconductor, we have n = p and therefore from (1.18), we have
 E c − E v − 2E fi 
Nc
= exp 

Nv
kT


Let Efn be the energy associated with the Fermi level in an N-type semiconductor having an electron density
n, we have
 E fn − E c 
n = N c exp 

 kT 
and
∴
or
 E v − E fn 
p = N c exp 

 kT 
 2E fn − E c − E v
n n2 N c
=
=
exp 
p n i2 N v
kT


 2( E fn − E fi ) 
 = exp 

kT



 E fn − E fi 
n = n i exp 


 kT
(1.19)
Semiconductors and Junction Diodes
21
Similarly, one obtains for a p-type semiconductor
 E fi − E fi 
p = n i exp 

 kT 
(1.20)
Thus the shift in the Fermi level in the n
and p type of semiconductor can be expressed as
n
ni
p
= kT ln
pi
E fn − E fi = kT ln
E fi − E fp





(1.21)
Fig. 1.19 represents the shift in the
Fermi level in the N- and P-types of semiconductors.
Fig. 1.19 Shift in the Fermi level in the N- and P-type of Semiconductors.
1.8 (c) Thermal Ionization of Extrinsic Semiconductors
When the temperature of an extrinsic semiconductor is raised above 0°K, the impurity atoms get ionized. Due
to ionization the donor impurity atoms give rise to electrons in the conduction band and the acceptor impurity
atoms give rise to holes in the valence band. These electrons or holes alongwith those generated by intrinsic
action, then serve as the current carriers at a temperature. Two cases of interest are
(i) Conduction electron concentration is equal to
n = n0 exp [(EF – Eg)/kT] ≈ Nd
(1.22)
This shows that under the present conditions the concentration of conduction electrons is approximately equal to the concentration of donors. This means that all the donors are ionized. Equation
(1.22) suggests that at room temperature the impurity concentration for Si and Ge upto 1014 to 1016 donors per cc suits this range, i.e., if we dope an intrinsic semiconductor crystal with this concentration
of donors, one can certainly predict that one will have ~1016 electrons/cc.
(ii) At higher temperature, the carrier concentration is proportional to N d .
1.8 (d) Charge Densities in Extrinsic Semiconductors
The density of impurity atoms in N- and P-materials is so low compared to the density of semiconductor atoms
that the rate of thermal pair generation is not affected appreciably by the presence of impurity atoms. In the
case of intrinsic semiconductors we have seen that the concentration product
np = n i2
[1.22(a)]
was a constant at a given temperature.
Assuming all impurity atoms in extrinsic semiconductors to be ionized at the usual operating temperatures,
the free charge densities in impurity material can be based upon ND, the donor atom density in N-material, or
NA, the acceptor atom density in P-material.
The electrical neutrality of the material demands
p = ND = n + N A
[1.22(b)]
22
Electronics—Theory and Applications
The L.H.S. of equation 1.22(b) gives the total positive charge as the sum of holes in the valence bonds and
the positive charge associated with the donor atoms that have given up electrons and become positive ions.
The R.H.S. of equation 1.22(b) sums the negative charge of free electron density n and the negative charge due
to electrons held by the ionized acceptor atoms.
Using 1.22(a), one can write
n=
n i2
p
and
p=
n i2
[1.22(c)]
n
only donor impurities are introduced in N-type material, so NA = 0. The donor density will be made much
larger than the density of intrinsic holes, or
ND >> p
and in N-type material the electron density is written from equation 1.22(b) as
[1.22(d)]
n ≅ ND
on the similar reasoning for P-type material, we have ND = 0 and NA >> n. One obtains density relations in
P-type material as
n≅
n i2
[1.22(e)]
NA
[1.22(f)]
p ≅ NA
From the above results we can conclude that the density of majority carrier approximates the impurity atom
density at usual ambient temperatures, and the density of minority carriers is reduced below the intrinsic level.
This means that the increased electron density in N-type material raises the probability that an electron will
meet and recombine with a hole, and so the hole density is decreased to maintain n2 constant.
1.9 Semiconductor Devices
A semiconductor device can be defined as a unit which consists, partly or wholly, of semiconducting materials
and which can perform useful functions in electronic apparatus and solid state research. Examples of semiconductor devices are semiconductor diodes (P-N junction), transistors, integrated circuits (Ic) etc. Si, Ge and
GaAs are most commonly used materials for the fabrication of semiconductor devices. For convenience the
properties of these semiconductors are summarized in Table 1.1.
Table 1.1 Properties of Si and Ge Semiconductors
Property
Symbol
Unit
Value
Germanium (Ge)
Atomic Numbers
32
14
72.6
28.08
144.63
kg/m3
5.32 × 103
2.33 × 103
5.32 × 103
atoms/m3
4.4 × 1028
5 × 1028
2.21 × 1028
16
11.8
10.9
eV
0.785
1.21
1.43
(Contd...)
Atomic Weight
Density
Atom Concentration
Relative Dielectric Constant
εr
Band gap at 0°K
Ego
GaAs
Silicon (Si)
Semiconductors and Junction Diodes
Property
Band gap at 300 K
Intrinsic Carrier
(generation) at 300 K
Symbol
Unit
Eg
eV
Germanium (Ge)
Silicon (Si)
0.72
1.1
1.32
3
2.5 × 1019
1.5 × 1016
9.0 × 1012
ni
m
Diamond
Diamond
Zincblende
a
Å
5.65748
5.43086
5.6534
°C
936
1420
1250
Melting Point
Minority Carrier life time
second (s)
Breakdown Field
Diffusion Constant
GaAs
Carriers
Crystal Structure
Lattice Constant
Value
Dn
(electrons)
~10
–3
7
~2.5 × 10
~ 3 × 10
–3
~ 4 × 105
~10
m2/s
0.009842
0.003367
0.001036
0.004662
0.001295
0.000906
4.7 × 1022
Effective Density of states in
the conduction band
Nc
m–3
1.04 × 1025
2.8 × 1025
Effective Density of states in
the valence band
Nv
m–3
6.4 × 1024
1.02 × 1025
Mobility (Drift) at 300 K
Work function
Raman Phonon energy
σi
~ 10–8
7
V/m
Dn (holes)
Intrinsic conductivity
23
Sm
–1
2.2428
0.4325 × 10
–3
7.0 × 1022
1.2832 × 10–6
2
µn
(electron)
m
V −s
µp
(holes)
m
V −s
W
Volt (V)
4.4
4.8
4.7
eV
0.037
0.063
0.035
0.38
0.13
0.85
0.18
0.05
0.04
2
Almost all semiconductor devices are comprised of a single crystal semiconductor incorporating two or
more semiconducting regions of different impurity density. The difference in the electric fields and carrier
densities associated with differently doped regions, called junctions, permit a wide range of essentially nonlinear conductivity effects in devices incorporating two, three or more distinct regions. Most semiconductor
devices can be understood through the simplest of such junctions, called the P-N junction, which is a system of
two semiconductors in physical contact, one with excess of electrons (N-type) and other with excess of holes
(P-type.)
1.10 P-N Junction
When a P-type semiconductor is brougth into contact with N-type semiconductor as the process of
crystallisation is taking place, the resulting combination is called a P-N junction. This junction has important
properties and is, in effect, the basis of modern semiconductor theory and practice. Most semiconductor
devices contain one or more P-N junction. The most important characteristic of a P-N junction is its ability to
conduct current in one direction only. In the reverse direction it offers very high resistance.
Formation of a P-N Junction: Fig. 1.20 shows three types of such junctions. In Fig. 1.20 (a), P and N regions
have been grown into the germanium block by mixing acceptor and donor impurities, respectively, into the
single crystal during its formation. This is known as a grown junction. It is worthwhile to mention that the
grown type of P-N Junction is not a sandwich made by attaching a P block to an N block, but actually consists
of P and N layers in a single piece of germanium.
24
Electronics—Theory and Applications
Fig. 1.20 Types of P-N Junction.
The diffused junction in Fig. 1.20 (b) is made by placing a pellet of acceptor impurity, such as indium, on
one face of a wafer of N-type germanium and then heating the combination to melt the impurity. Under proper
conditions of temperature and time, a portion of the impurity metal will diffuse a short distance into the wafer,
thereby creating a region of P-type germanium in intimate association with W-type bulk. This is also called an
alloyed junction or fusion-alloy junction from the fact that a small amount of pellet material alloys with the
germanium.
Fig. 1.20 (c) shows a point contact type. Here, a fine, pointed wire (catwhisker) makes pressure contact
with the face of an N-type germanium wafer. After assembly, the device is electroformed by passing a
high-current surge momentarily across the junction of Wafer and Whisker. The heat generated during the short
interval drives a few electrons from the atoms in the region of the point contact, leaving holes and thus converting into p-type a small volume of germanium immediately under and around the point.
Silicon P-N junction is produced in a similar manner. In most instances, the Silicon has been processed in
such a way as to make it P-type. To create the junction, an n-type material is either inserted at the proper point in
the crystal process or (in the diffused, junction process) an n-type material is later inserted into the body of P-type
wafer. Like germanium diodes, silicon diodes also are produced both in junction and point contact types.
P-N Junction with no external voltage: Fig. 1.21 shows a P-N junction just immediately after it is formed.
There is no external voltage connected to the P-N junction. Since N-type material has a high concentration of
free electrons while P-type material has a high concentration of holes, the following processes are initiated:
(i) At the junction, holes from the P region diffuse into the N region and free electrons from the N region
diffuse into the P region. This process is called diffusion. Holes combine with the free electrons in the
N-region whereas electrons combine with the holes in the P-region.
Fig. 1.21 A P-N Junction When Just formed.
(ii) The diffusion of holes from P region to N region and electrons from N region to P region across the
junction takes place because they move haphazardly due to thermal energy and also because there is a
Semiconductors and Junction Diodes
25
difference in their concentrations (The P region has more
holes whereas N region has more free electrons) in the two
regions.
(iii) As the free electrons move across the junction from N type
to P type, positive donor ions are uncovered, i.e., they are
robbed off free electrons. Hence a positive charge is built
on the N-side of the junction. At the same time, the free
electrons cross the junction and uncover the negative
acceptor ions by filling in the holes. Therefore, a net negative charge is established on P-side of the junction. When
a sufficient number of donor and acceptor ions is uncovered, further diffusion is stopped. It. is because now, a barrier is set up against further movement of charge carriers.
This is called potential barrier or junction barrier. The
potential barrier is of the order of 0.1 to 0.3 volt. Fig. 1.22
shows the electrostatic potential difference across the P-N
junction. How this potential barrier is developed? When a
sufficient number of donor and acceptor ions is uncovered, further diffusion is prevented. It is because now positive charge (ions) on N-side repels holes to across from P
type to N type and negative charge (ions) on N-side repels
free electrons to enter from N type to P type. Because of
this a difference in potential exists between the two sections, which inhibits further electron-hole combinations at
the junction, and the Fermi level of the two sides is in the
Fig. 1.22 Potential barrier across the P-N juncsame level as shown in Fig. 1.22 (c).
tion.
(iv) The region across the P-N junction in which the potential
changes from positive to negative is called the depletion region. The width of this region is of the order
of 6 × 10–8 m. Since this region has immobile (fixed) ions which are electrically charged, it is also
called as the space-charge region. Outside this region on each side of the junction, the material is still
neutral.
(v) The potential barrier for a silicon P-N junction is about 0.7 V, whereas for a germanium P-N junction
it is approximately 0.3 V.
The potential barrier discourages the diffusion of majority carriers across the junction. However, the potential
barrier helps minority carriers (few free electrons in the P region and few holes in the N region) to drift across
the junction. The minority carriers are constantly generated due to thermal energy. But electric current cannot
flow since no circuit has been connected to the P-N junction.
Width of Depletion Layer and Height of Potential Barrier: Let x1 and x2 be the width of depletion layer in P
and N sides respectively of a P-N diode junction, then total width of depletion layer
x = x 1 + x2
Similarly if V1 and V2 are the potential barriers in P and N sides of P-N junction, then the net junction potential
barrier (Fig. 1.22b).
V0 = V1 + V2
26
Electronics—Theory and Applications
The Poisson’s equation in one dimension is
d 2V
ρ
=− 1
ε
dx
2
(i)
where ε is the permittivity of medium. The charge density in P-side of depletion layer is given by
ρ = – eNa
(ii)
where Na represent the density of acceptor ions.
∴ Equation (i) for P-side of depletion layer is given by
d 2V
dx
2
=
ρNa
ε
(iii)
on integration, one obtains
dV eN a
x + C1
=
dx
ε
(iv)
where C1 is constant of integration. Using boundary condition, i.e., at
x = − x1 ,
dV
=0
dx
(since there is, no variation of potential in region exclusing depletion layer). Equation (iv) gives
C1 =
∴
eN a
x1
ε
dV eN a
eN a
eN a
x+
x1 =
=
( x + x1 )
dx
ε
ε
ε
(v)
on integration, one obtains
V=

eN a  x 2
+ x1 x + C 2 ,

ε  2

where C2 is another constant of integration. Now at x = 0, V = 0 (at junction)
∴
c2 = 0
Hence
V=

eN a  x 2
+ x1 x

ε  2

At x = – x1, V = V1
∴
∴
V1 =
eN a
ε

 x2
eN a x12
 1 − x2  = −
1 
 2
2ε


|V1 | =
eN a 2
x
2ε 1
(vi)
Semiconductors and Junction Diodes
27
If Nd is density of donor ions in N-side of depletion layer, then Poission’s equation for this becomes
d 2V
dx
2
=−
ρ
eN d
=−
ε
ε
(vii)
(Q ρ = eN d )
Proceeding as above and applying boundary conditions that at x = 0,
V = 0 and at x = x 2 ,
dV
= 0,V = V2 , one obtains
dx
V2 =
eN d x 22
(viii)
2ε
∴ The height of the potential barrier across the junction is obtained as
 N a x2 + N d x2 
1
2
V0 = |V1 | + |V2 | = e 

2ε


(ix)
Since the crystal as a whole is electrically neutral, the number of both sides of charge carriers must be equal, i.e.
N a x1 A = N d x 2 A
where A is the area of crystal
x1 N d
=
x2 N a
∴
or
x2 =
(x)
Na
x1
Nd
Substituting the value of x2 in (ix), one obtains
V0 =
εN a 2  N a 
x  1+

2ε 1  N d 
1
1
1
1
This gives
 2  2εV0 ( N a /N d )  2

2εV0
x1 = 

 =
 e (N a + N d )
 ε N a {1+ ( N 0 /N d )}
and
 2  2εV0 ( N a /N d )  2

2εV0
x2 = 
 =

 eN d {1+ ( N d /N a )}
 e (N a + N d )
1 
1
1

 2εV0  2   ( N d /N a )  2  ( N a /N d )  2 
x = x1 + x 2 = 
= 
+
 
 (N a + N d ) 
 e 


 (N a + N d )  


or
1 
1 
 2εV0  2   N a + N d  2 
x=

 e    N a N d  


(xi)
28
Electronics—Theory and Applications
Width of the depletion region changes as square root of the voltage. In order to exemplify the order of thickness commonly met with, let us consider the example of Germanium PN diode in which Nd = 1021/m3 and
Na = 1023/m3. For these concentrations across the junction V0 = 0.31 volt, and since for Germanium εr = 16
(Here ε = εr ε0), one finds that the width of the unbiased junction is x = 0.75 × 10–6 m or 0.75 micron. We must
note that in this numerical example Na >> Nd. Obviously, Eq. (xi) reduces to
1
 2 εV0  2
x=

 eNd 
(xii)
This shows that heavier doping on the side of junction extends the depletion layer to the other side. Moreover, the depth of penetration or the width of the depletion region varies inversely as the square root of doping
density on the opposite side.
Because of the presence of space charges of opposite kinds across the junction there is a voltage gradient
confined within the depletion region. Obviously, the transition region thus acts like a parallel plate capacitor in
which plates are separated by the distance x. The junction capacitance is thus
CJ =
dQ ε
= A Farad
dV x
(xiii)
where A is the area of the junction. Substituting (x) from (xi), one obtains
1
C J  εe N a N d  2
=

A  2V0 N a + N d 
Farad
m2
(xiv)
For a special case, i.e. for Na >> Nd, Eq. (xiv) reduces to
1
2
C J  εe
=
Nd 
A  2V0

(xv)
For the junction of 0.75 µ in the above numerical example the junction capacitance from Eq. (xiii) or by substituting Eq. (xii) in Eq. (xiii), one obtains
1
2
C J  εe
=
Nd 
A  2V0

(xvi)
For the junction width of 0.75 µ in the above numerical problem the junction capacitance from Eq. (xiii)
works out as CJ/A = 189 µF/m2 or 18.9 m µF/cm2. In a typical case if the junction area is A = 10–3 cm2, then we
obtain its capacitance CJ = 18 pF.
Effect of Temperature on Barrier Voltage: The barrier voltage depends on doping density and temperature.
For a given junction the doping density remains constant; therefore barrier voltage depends on temperature.
With the rise in temperature, more minority charge carrier’s are produced, leading to their increased drift
across the junction. Consequently the equilibrium occurs at a slightly lower barrier potential. Obviously, the
barrier potential decreases with rise of temperature. It is found that for both germanium and silicon
semiconductors
∆ V0 • – 2t m V
Here t°C denote the change in temperature, i.e. barrier potential decreases by about 2 mV per degree C.
Semiconductors and Junction Diodes
29
1.11 Forward and Reverse Biasing
(i) Forward Biasing
We have seen that the natural tendency of the majority carriers (free electrons in the N-section and holes in the
P-section) was to combine at the junction. This is how the depletion region and potential barrier were formed.
Actually, the combination of electrons and holes at the junction allows electrons to move in the same direction
in both the P and N sections. In the N-section, free electrons move toward the junction; in the P section, for the
holes to move toward the junction, valence electrons move away from the junction. Therefore, electron flow in
both the sections is in the same direction. This, of course, would be the basis of current flow.
With the P-N junction alone, the action stops because there is no external circuit and because of the potential barrier that builds up. So, for current to flow, a battery can be connected to the diode to overcome the
potential barrier. And the polarity of the battery should be such that the majority carriers in both sections are
driven toward the junction. When the battery is connected in this way, it provides forward bias, causing forward or high current to flow, because it allows the majority carriers to provide the current flow.
To apply forward bias, positive terminal of the battery is connected to P type and negative terminal to N
type as shown in Fig. 1.23. The applied forward potential establishes an electric field which acts against the
potential barrier field. Obviously, the resultant field is weakened and the barrier height is reduced at the P-N
Fig. 1.23 P-N Junction Showing Forward Bias.
30
Electronics—Theory and Applications
junction, Fig. 1.23 (b) and (c). Since potential barrier height is very small (~ 0.2 V) and hence a small forward
voltage is sufficient to completely eliminate the barrier. Obviously, at some forward voltage the potential barrier at the P-N junction can be eliminated altogether. Then the junction resistance will become almost zero,
and a low resistance path is established for the entire circuit. Thus a large current is generated in the circuit
even for the small potential difference applied. Such a circuit is called forward bias circuit and the current is
called forward current. The salient features of the forward bias circuit are summarized below:
(i) At some forward voltage, the potential barrier is eliminated altogether.
(ii) The P-N junction offers low forward resistance (rf) to current flow.
(iii) The magnitude of current in the circuit due to the establishment of low resistance path depends upon
the applied forward voltage, and it reduces as the voltage is increased there by forward current increases (Fig. 1.28). It is given by
I = I 0 ( e v / ηVT − 1)
where
(i)
V = applied voltage
T
here T = Temperature in Kelvin
VT =
11600
η = 1 for Ge and 2 for Si
The mechanism of current flow in a forward biased P-N junction is as follows:
(i) The free electrons from the negative battery terminal continue to arrive into the N-region while the
free electrons in the N-region move towards the P-N junction.
(ii) The electrons travel through the N-region as free electrons. Obviously, current in N region of the
P-N junction is by free electrons.
(iii) When these free electrons reach the P-N junction, they combine with holes and become valence electrons. Since a hole is in the covalent bond and hence when a free electron combines with a hole, it becomes a valence electron.
(iv) Current in the P region is by holes. The
electrons travel through P-region as valence electrons.
(v) These valence electrons after leaving the
crystal, flow into the positive terminal of
the battery.
The current flow in a forward biased P-N junction is illustrated in Fig. 1.24.
Fig. 1.24 Current Flow under Forward Bias.
(ii) Reverse Biasing
We have seen that for forward current flow, the battery must be connected to drive the majority carriers
towards the junction, where they combine to allow electrons to enter and leave the P-N junction. If the battery
connections are reversed, the potential at the N side will draw the free electrons away from the junction, and
the negative potential at the P side will attract the holes away from the junction. With this battery connection,
then, the majority carriers cannot combine at the junction, and majority current cannot flow. For this reason,
when a voltage is applied in this way, it is called reverse bias.
However, reverse bias can cause a reverse current to flow, because minority carriers are present in the
semiconductor sections. Remember, that although the P section is doped to have excess holes, yet some
electrons are freed because of thermal agitation. Also, although the N section is doped to have excess free
Semiconductors and Junction Diodes
31
electrons, some electrons are freed to produce holes in the N section. The free electrons in the P section, and
the holes in the N section are the minority carriers. Now, with reverse bias, one can see that the battery
potentials repel the minority carriers toward the junction. As a result, these minority carriers cross the P-N
junction in exactly the same way that the majority carriers did with forward bias. However, since there are
much fewer minority carriers then there are majority carriers, this minority current, or reverse current as it is
usually called, is much less and is the order of µA, with the same voltage than majority or forward, current
would be. Reverse bias P-N junction is shown in Fig. 1.25. The salient features of reverse biased P-N junction
are following:
(i) The height of the potential barrier is increased and width of Depletion region also increases, Fig. 1.25
(b) and (c).
(ii) The reverse bias P-N junction offers very high resistance to current flow. This resistance is called reverse resistance (Rr).
(iii) Due to the establishment of high resistance path, very small current flows in the circuit. This current is
usually called as the leakage current or reverse saturation current.
The current in the above situation is given by
I = I 0 ( e −V / ηVT − 1)
(ii)
Since applied voltage V is negative and as it increases the first term in R.H.S. reduces very fast and current
from equation (ii) reduces to I = (–) I0 which is quite low and constant.
Fig. 1.25 P-N Junction Showing Reverse Bias.
32
Electronics—Theory and Applications
1.12 Volt-Ampere Characteristics of P-N Junction
The P-N junction can be represented by a symbol of arrow and dash as
shown in Fig. 1.26. The arrow head represents the P-section of the crystal
and shows the direction of flow of holes or conventional current. Since the
Fig. 1.26 Symbol of the P-N JuncP-N junction diode’s resistance changes according to the direction of current
tion Diode.
flow and hence it is called a nonlinear device. Basically, its nonlinearity is
dependent on the polarity of the applied voltage. For current in the forward
direction, it has a resistance of only a few hundred ohms. In the reverse direction, its resistance is often close to 100,000 ohms.
Fig. 1.27 shows the circuit arrangement for drawing the V-I characteristics of a P-N junction diode.
A graph between the potential difference across the P-N junction and the
current through the junction is called the V-I characteristic of the P-N junction diode and is shown in Fig. 1.28.
In the forward direction, though considerably more current flows and the
current for the most part, increases linearly as the bias voltage is increased.
In the forward direction, then, the P-N junction can be considered a linear device over a large portion of its operating curve. The small portion of the
curve that is just above zero bias is nonlinear. This results because both maFig. 1.27 Circuit Arrangement for
jority and minority current actually comprise the overall current. Since the
Drawing V-I Characterminority carriers are low energy carriers, majority current starts first, and
istics of a P-N Junction.
then as the voltage is raised, minority current joins in,
causing a nonlinear rise in current. But as the voltage is
increased further, minority current becomes saturated
since there are only few minority carriers. The curve
then follows the majority current increase which is
linear.
Because of the non-linearity of the curve, if a very
small signal voltage is applied to the diode so that it
only operates around the knee, the signal will be distorted. The signals must be large enough so that they
operate mostly over the linear part of the curve.
When reverse bias is applied, a slight reverse current flows. This reverse current increases only negligibly as the bias voltage is increased a lot (20 to 25
volts). At this stage current suddenly rises in the reverse direction due to the breakdown of the crystal,
Fig. 1.28 V-I Characteristics of a P-N Junction Diode.
i.e., covalent bonds of the crystal are broken in very
large number. This breakdown reverse bias is called the Breakdown or zener voltage. Diodes are also designed to produce a useful wide range of zener breakdown region, and are used in special voltage-regulating circuits. We will discuss zener diodes later on.
1.13 Static and Dynamic Resistance of a Diode
One of the most important properties of the diode is its resistance in the forward and reverse biasing. An ideal
diode must offer zero resistance in forward bias and infinitely large resistance in reverse bias. Truly speaking,
Semiconductors and Junction Diodes
33
no diode can act as an ideal diode, i.e., an actual diode does not behave as a perfect conductor when forward
biased and as a perfect insulator when reverse biased. We consider the two resistances of the diode
(i) d.c. or static resistance
(ii) a.c. or dynamic resistance
(i) d.c. or static resistance: When P-N junction diode is forward biased, it offers a definite resistance in the
circuit. This resistance is called as the dc or static resistance (RF) of the diode, dc resistance of diode is simply
the ratio of the dc voltage across the diode to the dc current flowing through it at a particular instant.
RF =
V
I
(ii) a.c. or dynamic resistance: The ac or dynamic resistance of a diode, at a particular dc voltage, is equal to
the reciprocal of the slope of the characteristic at that point, i.e.,
rf =
change in voltage
∆V
=
resulting change in current ∆I
It is not a constant quantity but depends on operating voltage. For forward bias greater than the internal
junction voltage, the dynamic resistance varies inversely with the current, but for reverse bias its value is much
larger.
Under forward bias for large applied voltage, the diode current at junction temperature T is given as

 eV  
I = I 0 exp 
 − 1
 ηkT  

where η = 1 for Ge and 2 for Si. Differentiating the above w.r.t. V, one obtains
 e  eV / ηkT
 e 
dI
= I0 
=I
e

dV
 ηkT 
 ηkT 
∴
rf =
dV ηkT
 k T
=
= η 
 e I
dI
Ie
 1.38 × 10–23  T
= η

 1.6 × 10−19  I
or
rf = 8.625 × 10–5 η (T/I).
1.14 Space Charge (or Depletion Region) Capacitance
When no external voltage is applied, the width (d) of the depletion region of a P-N junction diode comes out to
be about 5 × 10–7 m and is mostly in the N-type material. The movement of majority carriers i.e. holes and electrons across the junction causes opposite charges to be stored at this distance ‘d’ apart. This is effectively a parallel plate capacitor whose capacitance CT (often called space-charge capacitance) is approximately 20 pF
with no external bias.
As forward bias is applied the depletion region decreases and the capacitance (CT) increases. Under
reverse-bias conditions, the depletion region increases and CT decreases. This property of voltage variable
capacitance is made use in varactors, varicaps or voltacaps.
34
Electronics—Theory and Applications
Space Charge or Transition Capacitance: The thickness of depletion layer changes with voltage applied
across the junction. Reverse biasing of a P-N junction, causes majority carriers to move away from the
junction, therefore uncovering more immobile charges. This increases the thickness of depletion layer.
Similarly the forward biasing of a P-N junction decreases the amount of uncovered charge and hence the
thickness of depletion layer. This change in the uncovered charge with the applied voltage may be
considered as a capacitive effect. It gives rise to transition capacitance or junction capacitance CT, given
by
CT =
dQ εA
=
dV
x
Here x is thickness of depletion layer and A is junction cross-sectional area.
One can show that the width x of depletion layer is given by
1
 2εV0 N a + N d  2
x=
N a N d 
 e
1
∴
 εe N a N d  2
CT = 
 A
 2V0 N a + N d 
C T ∝ V0–1 / 2
or
Obviously, the transition or junction capacitance is not constant but depends on applied voltage and hence it is
also known as voltage-variance capacitance.
Diffusion or Storage Capacitance CD: When forward biased the P-N junction develops a dominating diffusion
capacitance CD. This arises due to injected charge stored near the junction outside the transition region and is
defined as the rate of change of injected charge with applied voltage, i.e.,
CD =
dQ
dV
CD is found to be proportional to current (I) and is much larger than the CT CD has larger value in forward direction and smaller value in reverse direction. The value of CD in forward direction varies from 500 pF to 200 pF.
Obviously, CD predominates under forward bias and arises due to minority carriers, while CT predominates
under reverse bias and arises due to majority carriers.
1.15 Effect of Temperature on P-N Junction Diodes
We know that temperature rise boosts the generation of electron-hole pairs in semiconductors and increases
their conductivity. From a consideration of the energies of the carriers crossing the depletion region in a P-N
junction diode, an involved calculation yields the following relation between diode current, voltage and temperature: (see Appendix D)
I = I0 exp [V/η VT – 1]
Here
I = the diode current (forward if positive, reverse if negative)
I0 = the diode reverse current, also called the reverse saturation current, at temperature T.
V = the diode voltage, positive for forward bias, negative for reverse bias; in volts.
(1.23)
Semiconductors and Junction Diodes
η
VT
T
35
= 1 for germanium and 2 for silicon.
= T/11600, a quantity in volts, dependent upon temperature and is known as volt temperature equivalent.
= Temperature of the diode junction (°K)
On increasing the temperature the forward characteristic (Fig. 1.28) shifts to left, showing increase in current
for same voltage and it shifts to right when temperature is decreased showing biased current.
1.16 Zener Diodes
We have seen that in the breakdown region, large changes in diode current produce only small changes in diode
voltage. So a semiconductor P-N diode designed to operate in the breakdown, region may be employed as a
constant voltage device. The diodes used in such a manner are called avalanche breakdown or zener diodes.
They are used as a voltage regulator in the manner shown in Fig. 1.29 (b). The voltage source V and the
resistance R are so selected that the diode operates in the breakdown region. The diode voltage in this region
which is also the voltage across the load RL is called zener voltage (Vz) and the diode current is called zener current (IZ). As the load current (IL) or the supply voltage changes, the diode accommodates itself to these changes
and maintains nearly constant load voltage (Vz).
Fig. 1.29 (a) Symbol of Zener Diode, (b) Zener Diode as a Voltage Regulator.
The diode will continue to regulate the voltage until the diode current falls to knee current Izk (Fig. 1.29)(c).
Depending upon the nature of the semiconductor and its doping, the breakdown voltage in diode ranges from about
3 volt to several hundred volts. The breakdown phenomenon is reversible and harmless so long as the safe operating temperature is maintained.
The mechanism of diode breakdown at reverse voltage is explained below:
(i) Avalanche Breakdown. In this mechanism, the minority charge carriers (electrons in P type and holes in N
type) acquire sufficient energy from the applied reverse voltage to produce new charge carriers by removing
valence electrons from the covalent bonds. The new carriers in turn produce additional charge carriers and the
process multiplies to give large reverse currents. The diode is then said to be in the region of avalanche
breakdown, usually, a junction with a broad depletion layer (therefore a low field intensity) breaks down by
this mechanism. With the increase of temperature, the vibrations of the atoms in the crystal increase which
increases the possibility of collisions of the charge carriers with the lattice atoms and reduces the possibility
for the carriers to gain sufficient energy to start avalanche process. Thus at high voltages, the avalanche
process is prominent and does take place to cause diode breakdown. The operating voltages in such diodes
range from several volts to a few hundred volts.
36
Electronics—Theory and Applications
(ii) Zener Breakdown. In this mechanism, the breakdown is initiated through a direct rupture of covalent bonds
rather than avalenche process due to the existence of strong electric field across depletion layer. A junction
V 

having a narrow depletion layer  and hence high field intensity E = r  will break down by this mechanism.

d
An increase in temperature increases the energy of the valence electrons and makes it easier for these electrons
to escape from these covalent bonds. Smaller applied voltage is, therefore, required to pull these electrons
from the crystal lattice. The zener effect plays an important role only in diodes with breakdown voltage below
about 6 volts. The zener diode is always used in reverse biased condition (Fig. 1.29 (b)).
The range of voltages about the breakdown voltage in which a zener diode conducts in reverse direction is
called tolerance.
We must note that during manufacturing, it is very difficult to have exact doping for every zener diode of
the same number, i.e. type. Obviously, breakdown voltages of zeners of same number also differ slightly. This
range of breakdown voltages for the same type of zener diodes is referred as tolerance. Let us consider, for example, a particular type of zeners marked 9V, 10% tolerance. These zener diodes may have breakdown voltage from 8.1V (9 – 0.9) to 9.9 V (9 + 0.9) instead of sharp 9V for all.
The breakdown voltage of zener diode depends upon operating temperature. It is found to decrease
with increase in junction temperature. This is due to the increased reverse current, (i.e. increase in minority carriers) that flows with increasing temperature. The decrease in breakdown voltage is about 2 mV per
degree centigrade rise in temperature.
The maximum power which a zener diode can dissipate (or handle) without damage is referred as its power
rating and denoted by PZM. Zener diodes for the commercial purposes available in the market have power rating from 1/4 W to more than 50 W.
Power rating (PZM) is the product of maximum current IZM which a zener diode can handle and the rated or
operating voltage of a zener diode (VZ), i.e.
PZM = I ZM VZ
A data sheet, sometimes includes the maximum current rating of a zener diode. The maximum value of current which a zener diode can handle at its rated voltage without damage is referred as its maximum current rating (IZM).
When a Zener diode is operated in the breakdown region, as increase in current produces a slight increase in
voltage. This means that a zener diode has a small a.c. resistance. This a.c. resistance is called zener resistance
(often referred as zener impedance ZZT). Sometimes, we find the specification about this resistance on the data
sheets supplied by the manufactures.
The opposition offered to the current flowing through the zener diode in the operating region is known as
zener resistance (RZ) or Zener impedance (ZZ).
Zener diodes find wide commercial and industrial applications, e.g. voltage stablizer, meter protection,
wave shaping, etc.
1.16(a) Constant Current Diode
These are the diodes that work exactly opposite to Zener diode. These diodes keep the current constant flowing through them when the voltage changes, i.e. instead of holding the voltage constant, these diodes hold the
current constant. The range of voltage over which a diode can keep the current constant is known as voltage
compliance. These diodes are optimised for a particular voltage compliance.
1.17 Tunnel Diode
It is a device just like a P-N junction, which offers negative resistance under certain bias conditions. The negative resistance of the diode is due to tunneling and it is called a tunnel diode or Esaki diode. It is made very
Semiconductors and Junction Diodes
37
Fig. 1.30(a)
much like an ordinary P-N junction diode, except that both the P and N regions are heavily doped (1 : 104 )
thousand times more than ordinary diode. It is used as an active device in electronic circuits in the frequency
range of few megahertz. The semiconductors with very high impurity concentration are referred to as degenerate semiconductors. Typical tunnel diode characteristic is shown in Fig. 1.29 (a).
As a result of high impurity levels, the contact potential is
high, the depletion layer is very narrow and the Fermi levels
lie in the conduction band for N-side and in the valence band
of the P side (Fig. 1.29 (b)). Under these circumstances, a
very abrupt transition from P to N type material is achieved
within the crystal. Since depletion region is very narrow; this
gives rise to extremely large electric fields.
Tunnel Effect. When a P-N semiconductor is heavily doped,
(it has many majority carriers and ions), and under forward
biased the hole and valence electron random drift is heavy.
As a result, it is not uncommon for a large number of
electrons to fill holes and release energy to only a few other
valence electrons. These few valence electrons, then, have
their energy levels raised considerably so that they can
cross from the N to the P section and current increases as
shown by OA [F ig. 1.30a], even with little or no applied
voltage. This action which seems to allow a valence
electron to cross a potential barrier without enough applied
external energy is called the tunnel effect because it seems
as though the valence electron ‘tunnels’ through the
forbidden band as shown in Fig. 1.30 (c). On further
increasing voltage the barrier height decreases (Fig. 1.30
(d)) and Fermi level EFN is much raised so current decreases
as shown by AB (Fig. 1.30 (a)). This region is called
negative resistance region. On further increasing voltage
the conduction band of N type is in level with C B of P type
Fig. 1.30 (b–e) Tunnel Diode—Characteristics and
Energy Levels.
38
Electronics—Theory and Applications
and free electrons of N type easily cross to
P type thereby current BC again increases
like ordinary diode as shown by BC (Fig
1.30a).
The negative resistance region AB (Fig.
1.30) allows the diode to be used as an
oscillator. It can also be used as an electronic switch since it has a good response
in negative resistance region. By its nature,
the tunnel diode has a rather high reverse
Fig. 1.31 Equivalent Circuit of a Tunnel Diode.
current, but operation under this condition
is not generally used.
The equivalent circuit of tunnel diode is as shown in Fig. 1.31. The series resistance RS is the ohmic resistance. The series inductance LS depends upon the lead length and geometry of the diode package. The junction
capacitance C depends upon the bias and is usually measured at the valley point. The negative resistance [– Rn]
has a minimum at the point of inflection between peak current Ip and valley current Iv. Typical values for these
parameters for a tunnel diode having peak current value Ip = 10 mA are Rn = (–) 30 Ω, Rs = 1 Ω, Ls = 5 nH and
C = 20 pF.
It is a voltage controllable device (like tetrode). Since between Ip and Iv we have three values of voltage for
a given current and on account of its multivalued feature its is used in pulse and digital circuiting.
Our main interest in the tunnel diode is its application as a very high speed switch. Since tunneling takes
place at the speed of light, the transient response is limited only by total shunt capacitance (junction plus stray
wiring capacitance) and peak driving current. Switching times of the order of a nanosecond are reasonable and
times as low as 50 p.secs have been obtained.
The advantages of a tunnel diode are (i) low cost,
(ii) low noise, (iii) high speed, (iv) environmental
immunity and (v) low power.
The disadvantages of the diode are its low output
voltage swing and the fact that it is a two terminal
device unlike ordinary diode in which current flows
only when forward biased. Because of this latter feature, there is no isolation between the input and output and it leads to serious circuit design problems.
However we may have a special type of tunnel
diode whose peak value current is of the order of
valley current. Such a diode is called Backward
Diode. V.I Characterstic of a backward diode is
shown in Fig. 1.32. We may use it under reverse biased condition as ordinary diode with the advantage
that now the diode current is available just beyond
zero volt whereas in ordinary diode, current is not
Fig. 1.32 V-I Characteristic of Backward Diode.
available till applied voltage is more than (break
over voltage) 0.7 volt in case of silicon and so in case a back diode it is conveniently used as rectifier for V
small (mV) input signals (For details of backward diode see section 1.23).
1.18 Photodiode
A photo-diode is a P-N junction diode packed into a transparent plastic packet working under reverse biased
condition.
Semiconductors and Junction Diodes
39
Principle. A very small current flows through a P-N junction diode when it is reverse biased. It is because,
minority charge carrier takes part in conduction. The number of minority carrier depend upon the working
temperature. However when photons (light) of suitable frequency is incident on the junction, the number of
charge carrier increases. But this happens only when light radiations such the junction which is not the case in
ordinary diode.
Working. In a photo diode visible light is focussed on the junction through a lens (Fig. 1.33). Light on being
incident produces free electrons and holes. Thus number of charge carriers increases and current increases.
Generally a photodiode is optimized for its sensitivity to light. As the light intensity increases more and more
charge carriers are generated and reverse bias current increases Fig. (1.34).
Fig. 1.34 V-I Characteristic of Photo-diode.
Fig. 1.33 Equivalent Circuit of a Photo-diode.
In this respect a photodiode acts as a photo detector, a device which converts incoming light signal into
electrical signal.
Volt-Ampere characteristic: Under normal reverse bias the current in a P-N junction diode is given by
I = I 0 ( e −V / ηkT − 1) ≈ − I 0
since applied voltage V is negative. So at higher values of V the first term goes to zero.
On illuminating the diode under reverse biased conditions the number of hole-electrons pairs increases.
Which is proportional to the intensity of incident light (number of photons). Hence the total current flowing
through the diode,
I = I 0 + IS
where IS is the short circuit current generated on account of photoelectric effect. In general the volt-ampere
characteristic is given by
40
Electronics—Theory and Applications
I = I S + I 0 ( e −V / ηkT − 1)
(1.24)
This characteristic is drawn in Fig. 1.34.
Small Signal Model: Under ideal conditions a photo-diode under reverse biased condition is shown in Fig.
1.35 (a). In Fig. 1.35 (b) parasitic elements C, R and r are taken into consideration, where R represents reverse
biased resistance. C represents the transition capacitance and ‘r’ represents the bulk ohmic resistance. The
barrier capacitance C is of the order of 10 pF, R ≈ 50 MΩ and r = 100Ω. In Fig. 1.34 the dark current I0
corresponds to the reverse current due to thermally generated electron- hole pairs. From equation (1.24) we
see that if diode terminals are open circuited then the total current I has be zero. Under such conditions, a
voltage known as photovoltaic potential V0C will appear across diode. It is given by
 I 
Voc = ηVT ln  1+ s 
 Io 
(1.25)
where VT is the volt equivalent of temperature and η is a constant equivalent to unity for Ge and 2 for Si.
Fig. 1.35 Photo-diode under (a) Reverse Biased Condition, (b) Parasitic Elements.
Thus V0C increases logarithmically with Is or with intensity of incident light. Now if a load (RL) is connected
across the diode, the voltage drop across it will correspond to incident light intensity. The output voltage generated is of the order of few hundred millivolts across RL of few kilo-ohms. Large area photovoltaic P-N junction banks have been used for conversion of solar energy into electrical energy which is the basis of solar cells.
1.18(a) Solar Cells
Solar cells, which convert the sunlight directly into electricity, at present furnish the most important long duration power supply for satellites and space vehicles. These cells have also been successfully employed in
small-scale terrestrial applications. Today, the solar cell is considered a major device for obtaining energy
from the sun. Since it can covert sunlight directly to electricity with high conversion efficiency, and can provide nearly permanent power at low operating cost, and is virtually free of pollution. Recently, research and
pollution development of low-cost, flat panel solar cells, thin film devices, concentrator systems, and many
innovative concepts have increased. We can expect that in near future, the costs of small solar power modular
units and solar power plants will be economically feasible for large-scale use of solar energy.
The first solar cell were developed by chapin et al. in 1954 using a diffused silicon p-n function. Subsequently, the cadmium-sulphide solar cell was developed by Raynolds et al. in 1954. To date, solar cells have
been made of many other semiconductors, using various device configurations and employing single-crystal,
polycrystal, and amorphous thin film structures.
Semiconductors and Junction Diodes
41
Photo-voltaic Effect and Solar Cells
The photo-voltaic effect can be observed in nature in a variety of materials, but the materials that have shown
the best performance in sunlight are the semiconductors. In photovoltaic conversion, the solar radiation falls
on devices called solar cells which convert the sunlight directly into electricity. The principal advantages associated with solar cells are that they have no moving parts, require little maintenance, and work quite satisfactorily with beam or diffuse radiation.
A typical schematic representative of a conventional silicon solar cell is shown in Fig. 1.35(a). Silicon cells
are thin wafers about 300 µm in thickness and 3 to 4 cm in diameter sliced from a single crystal of an type or
p-type doped silicon. A shallow p-n function is formed at one end by diffusion. Metal electrodes made from a
Ti-Ag solder are attached to the front and back side of the solar cell. On the front and back side of the cell are
attached metal electrodes made from a Ti-Ag solder. On the front side, the electrode is in the form of a metal
grid with fingers which permit the sunlight to go through, while on the backside, the electrode completely covers the surface. An antireflection contain of SiO having a thickness of about 0.1µm is also put on the top surface of the cell.
Fig. 1.35 (a) Schematic representation of a Silicon Solar Cell
When a monochromatic radiation of wavelength λ is incident on the front surface of a cell, it is absorbed
and pairs of positive and negative charges, called electron-hole pairs are created. The generation rate of
electron-hole pairs at a distance x from the semiconductor surface is given by
(1.26)
G ( λ , x ) = α ( λ )F ( λ )[ 1− R ( λ )] exp [ −α ( λ )x]
2
where α(λ) is the absorption coefficient F(λ) the number of incident photons/cm /s per unit bandwidth, and
R(λ) the fraction of these photons reflected from the surface. The positive and negative charges are separated
because of the p-n function. The direct current thereby produced is collected by the metal electrodes and flows
to the external load. The total photocurrent at a given wavelength is given by
J ( λ ) = J p ( λ ) + J n ( λ ) + J dr ( λ )
(1.27)
where Jp and Jn are photo current densities due to holes and electrons respectively and Jdr is the photocurrent
per unit bandwidth, representing photocurrent generation taking place within the depletion region.
42
Electronics—Theory and Applications
The conversion efficiency the can be derived from the device is given by
η=
I mVm
Pin
(1.28)
where Vm and Im are the voltage and current at maximum power point and Pin is the power density of the sunlight. For Silicon cells, n range between 10 and 15 per cent. An important property of the material affecting the
conversion efficiency is the hand gap energy (Eg). Photons of solar radiations having energy (E) less than the
band gap energy are not usually absorbed. On the other hand, photons of sunlight having E > Eg have their
energy partially utilized with the quantity (E – Eg) being wasted as heat. The value of Eg for Silicon is 1.20 eV.
This value is quite close to the optimum value desirable from the stand point of obtaining maximum
efficiency.
Apart from the band gap energy, the current voltage characteristic of a solar cell also influences its efficiency. Fig. 1.35(b) show a
typical characteristic for a solar cell. The current Isc obtained at zero
voltage is called the short circuit current, while the voltage Voc
obtained with an open circuit is called the open circuit voltage. The
open-circuit voltage Voc and the short-circuit current Isc are determined for a given light level by the cell properties. The product of
these quantities is the ideal power of the cell. The maximum useful
power is the area of the largest rectangle that can be formed under
the I-V curve. Calling these values of voltage and current Vm and Im
We can see that the maximum delivered power illustrated by the
rectangle in the Figure 1.35(b) is less than Voc Isc product. The ratio
Im Vm/Isc Voc is called the fill factor, and is a Figure of merit for solar
cell design. Typical values of the above quantities for a silicon cell
Fig. 1.35 (b) A typical current-voltage charare as follows: Voc = – 450 to 600 mV, Isc= –30 to 50 mA/cm2 and
acteristic for an ideal solar cell
Fill factor = 0.65 to 0.80.
A well-made Si cell can have about 10 per cent efficiency for solar energy conversion, providing approximately 100W/m2 of electric power under full illumination. This is modest amount of power per unit solar cell
area, considering the effort in fabricating a large area of Si cells. one approach to obtaining more power per
cell is to focus considerable light onto the cell using mirrors. Although Si cells lose efficiency at the resulting
at the resulting high temperatures, Ga As and related compounds can be used at 100°C or higher. In such solar
concentrator system more effort and expense can be put into the solar cell fabrication. Since fewer cells are required. For example, a Ga As — Al Ga As hetrofunction cell provides good conversion efficiency and operates at the elevated temperatures common in solar concentrator systems.
Applications: Some of the important applications of photodiodes are (i) Photo detection (both visible and
invisible), (ii) Demodulation, (iii) Logic circuits, (iv) Switching, (v) Optical communication system.
1.19 Light-emitting Diode (LED)
LED is a solid state (P-N junction diode) light source which has replaced incandascent lamps in many day to
day applications.
ED is just not an ordinary P-N junction diode where silicon is used. Here we use compounds having
elements like gallium, arsenic and phosphorus which are semi-transparent unlike silicon which is opaque.
(Gallium-arsenide gives infrared radiations and gallium-arsenide-phosphide gives visible light either red or
yellow.)
Semiconductors and Junction Diodes
43
Principle. Whenever a P-N junction is forward biased, free electrons and holes recombine at the junction.
Here we may understand the situation by assuming that electrons fall into a hole. While jumping the electron
releases (radiates) energy since electron jumps from a higher energy level (N-type) to a hole (P-type) in lower
energy level (P-type). The energy radiated is in the form of heat and light (Fig. 1.36) but none of the light is
visible since silicon is opaque and so no light can come out. Instead of silicon if we use a semitransparent
material (like gallium, Arsenic compounds etc.) then the light emitted due to recombination of free electrons
and holes at the junction will escape to the surroundings and we shall perceive it. The colour of the radiations
coming out will depend upon the material used. Mostly we find red, green or amber colour when the material
used is gallium arsenide phosphide. We do get infrared radiations also when the compound used is gallium
arsenide.
Fig. 1.36 (a) Schematic Construction of LED (b) symbol.
Working: An LED is forward biased with a small resistance in series (Fig. 1.37). The construction of LED is
such that they have typical voltage drop of 1.5 to 2.5 V for currents between 10 and 50 mA. When forward
biased.
The exact voltage drop depends on the colour, tolerance, and other factors depending on the nature of
construction.
Incidentally LED’s have low reverse voltage ratings. A typical value is 3V for TIL 221 (red LED). So if an
LED is reversed biased then voltage greater than 3V may destroy it. One way to protect is to connect another
ordinary diode in parallel with the given LED [Fig. 1.37(b)].
Applications: LED’s which produce visible radiations are used in instrument displays, calculators, digital
clocks etc.
Here we have LED arrays. An LED array is a group of LED’s that display numbers, letters or other symbols. The most common LED array is the seven segment display shown in Fig. 1.38 (a). The display contains
seven rectangular LED’s (A to G). Each LED is called a segment because it forms part of the character being
displayed. Fig. 1.38 (b) shows the schematic arrangement of the seven LED’s constituting a display. Their
44
Electronics—Theory and Applications
Fig. 1.37 LED Circuit.
Fig. 1.38 LED Array.
anodes have been joined to common terminal (3). So a positive voltage when applied to it drives all anodes. By
grounding one or more cathodes. We can forward bias and of the diodes which in turn form any digit from 0 to
9. For instance by grounding the cathodes of A, B and C. We display the number ‘7’ in Fig. 1.38 (a) or by
grounding the cathodes of A, B, C, D and G we display the number ‘3’.
A seven segment array can also display capital letters ‘A’, ‘E’, ‘F’, plus lower case letters ‘b’ and ‘d’ which
are commonly used in microwave ovens, stereo tuners and microprocessor trainer kits. A typical display TIL
312 shown in Fig. 1.39 (b) represents the schematic diagram of the same, showing left and right decimal points
also. Fig. 1.39 (a) shows the pin-out of it.
The advantages of an LED are:
(i) Low voltage of operation
(ii) Long life (more than 20 yrs)
(iii) Fast ON-OFF switching (10–9 sec)
Semiconductors and Junction Diodes
45
Fig. 1.39 A Typical Display TIL 312.
The other uses of LED are:
1.
2.
3.
4.
For indicating power ON/OFF (Power level indicators)
Optical switching applications
Solid state video displays
Optical communication—Energy coupling circuits.
All natural colours are composed from three primary colours namely red, green and blue. LED in red and
green have been available—but the blue one has been missing. Blue light emitters have several potential applications. They can be used for full colour displays of large area. They can be used in traffic lights as replacement for
ordinary bulbs resulting in huge power savings as well as cost. Use of blue lasers can also result in higher density
storage of information in optical CD-ROMs. Blue light emitters have been fabricated using several materials including ZnO, ZnSe and SiC. These attempts have been reasonably successful. However maximum success in
last few years has been achieved with GaN. This material has proved to be very useful as it is also capable of
operating at high power density, high temperature and unfriendly enviornment. GaN having a band gap of 3.4 eV
can give continuously varying band gap by combining with AIN to get upto 6.2 eV and with In N to get a band
gap down to 1.9 eV. The high thermal conductivity and superior stability of this material makes it ideal for several applications over other competing materials.
1.19(a) Liquid Crystal Display (LCD)
Just like LED we have another type of display that uses seven segment called Liquid Crystal Display also
known as electroluminescent display. It consists of a thin layer of normally transparent liquid crystal material
between two electrodes.
When an electric field is applied, the liquid crystal material between the two electrodes becomes turbulent,
reflecting and scattering ambient light. It provides excellent brightness under high ambient light conditions
and requires only 50 µW of power per segment, there by total power for one complete display of 7 segment is
350 µW. This power is much less than that of an LED display, but the life expectancy is not as high as that for
LED which is 10,000 hours minimum.
These displays are used in Watches, Pocket Calculators, Pocket televisions and portable instrument displays.
46
Electronics—Theory and Applications
1.20 Varactor Diode
An ordinary PN junction diode under reverse biased condition causes the majority charge carriers to move
away from the junction, thereby uncovering more immobile charges, i.e. ions. Hence the thickness of the
space charge layer at the junction increases with reverse voltage. This increase in uncovered charge with
applied voltage, may be considered a capacitive effect. We may define an incremental capacitance CT by
CT =
dQ
dV
(1.34)
where dQ is the increase in covered charge caused
by an increase dV in voltage. The quantity CT is
generally referred to as transition region or space
charge or Barrier Capacitance.
This quantity is not constant but depends upon
the magnitude of reverse voltage. A variation of CT
with applied reverse voltage is shown in Fig. 1.40.
The larger the reverse voltage, the larger is the
space charge width and smaller the capacitance CT.
Similarly for increase in voltage in the forward biased condition the space charge width will decrease and ct will increase.
Fig. 1.40 Variation of CT with reverse voltage for a typical silicon
Diodes which are based on voltage variable
diode at 25°C
capacitance are called varactors. A varactor is also
called varicap, voltacap, epicap or turning diode. A circuit model for a varactor
diode under reverse bias is shown in Fig. 1.41(a). A schematic symbol of a
varactor is shown in Fig. 1.41 (b).
The resistance RS represents the body (ohmic) series resistance of the diode.
Typical values of which is 8.5Ω at 4V. The reverse diode resistance Rr ≈ 1 MΩ
and that’s why it may be treated as open circuit across CT.
The capacitance of a varactor is given by
Fig. 1.41(a) Varactor diode
C=
εA
W
(1.34a)
where, ε is the permittivity of depleted region of semiconductor diode, A is the area of junction and W is the
width of the depletion region. Since the width of the space charge (or depleted region) is approximately proportional to the square-root of the reverse bias voltage, V, whereas the area A and permittivity ε being constant;
we have
C=
K
V
,
where K = εA.
Applications
Such diodes are useful in a number of circuits:
(i)
(ii)
(iii)
(iv)
Voltage tuning of an LC resonant circuit (Electronic tuning circuit in radio, television etc.)
Self balancing bridge circuits
Parametric Amplifiers
Automatic frequency control device
Semiconductors and Junction Diodes
47
Fig. 1.42
In voltage tuning circuits on changing applied reverse voltage. The capacitance of the circuit (including
Varactor Diode) will change i.e. increasing voltage shall reduce the capacitance. Hence the resonant frequency of the L-C circuit will increase. This frequency can measured by using a variable oscillator. Thus,
there is linear relationship between applied reverse voltage and resonant frequency upto a certain limit.
1.21 Schottky Diode
While discussing rectification, we have seen that an ordinary diode is turned off when the bias changes from
forward to reverse. It happens only in case of lower frequency signals. However, as the frequency increases, a
point reaches where an ordinary diode could not turn off fast enough to prevent noticeable current during part
of the reverse half cycle (Fig. 1.43(c)). One finds that this happens due to storage of charges around the junction during forward biasing.
We have seen that when a diode is forward biased, the charges are stored in different energy band near the
junction because of the lifetime of the minority carriers. This effect is called as charge storage. When we
suddently reverse-biased a forward-biased diode, a large reverse current exists for a while because of this storage
charge. Obviously, the charge storage prelongs the reverse recovery time (trr). The reverse recovery time is the
48
Electronics—Theory and Applications
Fig. 1.43
time taken to turn off a forward biased time. More preciesly, it is the time taken by a diode to drop the reverse current to 10% of forward current.
For ordinary diodes, trr is approximately 4ns. At low frequencies, i.e. below 10 MHz, its effect is not even
noticed [Fig. 1.43(b)]. However, at higher frequencies, i.e. more than 10 MHz, its effect is noticeable, since
the output signal begins to deviate from its normal shape [Fig. 1.43(c)]. This problem is eliminated in Schottky
diode.
When a semiconductor is brought into contact with a metal, there is formed in the semiconductor a barrier
layer from which charge carriers are severely depleted. The barrier layer is also called a depletion layer or exhaustion layer. Schottky diode is based on this effect. For the construction of a Schottky diode, a metal like gold,
silver or platinum is used on one side of the junction and doped silicon (N-type) is used on the other side [Fig.
1.44(a)]. When Schottky diode is unbiased, the free electrons on n-side are in smaller orbit than the free electrons
on the metal side. This difference in the size of the orbit creates a barrier potential (0.25 V only). This barrier potential is called Schottky barrier.
Fig. 1.44(b) shows a schematic symbol of a Schottky diode. We may note that the bar (line) looks almost a
rectangular S.
When a Schottky diode is forward biased, free electrons on N-side gain sufficient energy to travel to larger
orbit. Due to this, the free electrons just cross the junction and enter the metal, producing a large forward current. We know that metal has no holes and hence there is no charge storage around the junction and the reverse
recovery time (trr) is zero. This means the Schottky diode is switched off at once when it is reverse-biased.
Fig. 1.44
This is why, Schottky diodes can easily rectify signals of frequency above 30 MHz without distortion. The
output of half wave-rectifier using Schottky diode is shown in Fig. 1.44 (d) when the input signal frequency is
100 MHz.
To maximize power output and efficiency, IMPATT diodes are usually designed. The word IMPATT
stands for “impact ionization avalanche transit time”. IMPATT diodes employ impact-ionization and transit-time properties of semiconductor structures to produce negative resistance at microwave frequencies. The
negative resistance arises from two delays which cause the current to lag behind the voltage. One is the “
Semiconductors and Junction Diodes
49
avalanche delay” caused by finite buildup time of the avalanche current; the other is the “transit-time delay”
from the finite time for the carriers to cross the drift region. When these two delays add up to half-cycle time,
the diode resistance is negative at the corresponding frequency.
Schottky diodes find several applications. We mention below few of them.
(i) We know that the speed of the digital computers depends upon the switching time of their diodes and
transistors. Obviously, Schottky diodes find their best application in digital computers due to their fast
switching operation.
(ii) Schottky diodes are best suited for low voltage rectification due to their small barrier potential
(0.25 V).
1.22 Step Recovery Diodes or Charge Storage Diode or Snapback diode
In contrast to fast-recovery diodes, a charge storage diode is designed to store charge while conducting in the
forward direction and upon switching to conduct for a short period in the reverse direction. A particularly
interesting charge-storage diode is the step-recovery diode or snapback diode which conducts in a reverse
direction for a short period then abruptly cuts off the current as the stored charges have been dispelled. This
cutoff occurs in the range of picoseconds and results in a fast-rising wavefront rich in harmonics.
The symbol of a step recovery diode is shown in Fig. 1.45(a). Step recovery diode has unusual doping profile in which the density of carriers decreases near the junction. This unusual distribution of carriers causes a
phenomenon called reverse snap-off and that is why it is also called snapback diode.
When a high frequency (say 20 MHz) signal is applied in the circuit (Fig. 1.45(b)), during positive half
cycle, the diode conducts like an ordinary silicon diode. However, during the negative half cycle the reverse
current exists for a while because of the stored charges, then suddenly the reverse current drops to zero [Fig.
1.45(c)]. It is as if the diode is snapped off and hence it is also known as snapback diode.
Fig. 1.45
We must note that output of this diode is non-sinusoidal periodic wave which contains a sharp spike in the
negative half cycle. From the Fourier series theorem, we know that any non-sinusoidal periodic wave is equivalent to the superposition of sinusoidal components called harmonics. These harmonics have the frequencies f,
2f, 3f, ... nf, where f is the fundamental frequency of the waveform.
The output of this diode [Fig. 1.45(c)] is rich of harmonics and can be filtered to produce a sine wave of
required higher frequency. Because of these characterstics, these diodes are used as harmonic generators and
pulse formers most step-recovery diodes are made from Si with relatively long minority carrier life times ranging
from 0.5 to 0.5 µs. We must note that the lifetimes are about 1000 times longer than for fast-recovery diodes.
50
Electronics—Theory and Applications
1.23 Backward Diode
It is just a zener diode in which a doping level is increased to such an extent that its zener effect, i.e. reverse
breakdown occurs near to zero.
Usually, zener diodes have breakdown voltages more than 2V. The backward diods are prepared to have reverse breakdown voltage near to zero (~ – 0.1 V). Obviously, in backward diodes forward conduction still occurs around + 0.7 V but the reverse conduction starts approximately at –0.1V. This means that it conducts
better in the reverse direction than in the forward direction.
Fig. 1.46(a) shows V-I characterstics of backward diode. Its symbol is similar to zener diode and is shown
in Fig. 1.46 (b).
The backward diode can be used for rectification of small signals, microwave detection, and mixing. Similar to the tunnel diode, the backward diode has a good frequency response because there is no minority carrier
storage effect. In addition, the current-voltage characterstic is insensitive to temperature and to radiation
effect, and the backward diode has very low 1/f noise.
Fig. 1.46
1.24 Thermistors and Barretters
We have read that the electrical conductivity of a semiconductor changes significantly with temperature and
has a negative temperature coefficient of resistivity. This property is utilised in a device called thermistor,
whose resistance is temperature sensitive, usually decreasing with temperature. Conventional wire-wound
metallic resistors have positive temperature coefficients of resistivity. Commercial thermistors are usually
made of sintered mixtures of Mn2O3, NiO2 and CO2O3. A thermistor consists of a semiconductor bead of approximately 0.04 cm in diameter. Two thin wires are attached to the bead to provide for the two terminals. Diameter of the wire is approximately 0.25 µm.
Thermistors finds use in control systems operated by temperature changes, in the measurement of microwave power, in thermometry, and as a thermal relay. In electronic circuits, thermistors have been used
to compensate for the change in resistance with temperature of ordinary components where variation of
component values cannot be tolerated.
The I-V characterstic of a thermistor has a negative slope. Devices that exhibit, in some region, a negative
slope in their I-V characterstics are useful for making oscillators, amplifiers and switching circuits. However,
thermistors are not suitable in these applications because their response characterstics are too slow. There are
certain bulk semiconducting compounds which have negative resistance characterstics over a limited range of
operating parameters utilizing mechanism unrelated to the temperature sensitivity of the resistivity. These materials have been used to obtain devices based on Gunn effect.
A heavily doped semiconductor shows metallic properties.It has a positive temperature coefficient of resistivity owing to the decrease of the carrier mobility with increasing temperature. Such a device is called
sensistor. Thermistors also find extensive use as sensing elements in microwave power measuring equipments
Semiconductors and Junction Diodes
51
and as temperature sensors of electronic thermometers. Thermistors are capable of yielding power information over the power range of 10–5 mW to 20 mW with a typical burn out level of 400 mW.
A barretter has a positive temperature coefficient of resistance, consists of an approximately mounted
piece of “Wollaston” platinum wire having diameter of approximately 1.25 µm. Barretters are capable of
yielding power information over the range of 10–5 mW to 20 mW. A typical burnout level is 20 mW. For low
level rf power application below 103 MHz, 0.001A, “Littlefuse” may be used as a barretter.
1.25 Photoconductor
It consists simply of a slab of semiconductor (in bulk or
thin film form) with ohmic contacts affixed to oppose ends
as shown in Fig. 1.47. When light is incident on a semiconductor, electrons are excited from the valence band into the
conduction band. Obviously, electron-hole pairs in excess
of those generated thermally are produced. The conductivity of the material increases owing to the increased number
of carriers. Under the influence of light on the surface of a
photo conductor, carriers are generated either by
band-to-band transitions (intrinsic) or by transitions involving forbidden gap energy levels (extrinsic), resulting
in an increase in conductivity.
For the intrinsic photoconductor, the conductivity is
given by σ = e (µn n + µp p), and the increase of conductivity under illumination is mainly due to the increase in the
number of carriers. The corresponding long- wavelength
cut off of the photon of frequency vc in this case is given
by
λc =
hc
1.24
=
(µm )
E g E g (eV)
Fig. 1.47
(1.35)
where λc is the wavelength corresponding to the semiconductor band gap Eg, c is the speed of light. For wavelengths (λ) shorter than λc, the incident radiation is absorbed by the semiconductor, and hole-electron pairs are
generated. For the extrinsic case, photoexcitation may occur between a band edge and an energy level in the
energy gap. Photoconductivity can take place by the absorption of photons of energy equal to or greater than
the energy separation of the band gap levels and the conduction or valence band. In this case the long wavelength cut off is determined by the depth of the forbidden-gap energy level.
Fig. 1.48 shows the photo response of an intrinsic as well as
extrinsic (a semiconductor with deep impurity level, e.g., thin film
deposition on insulating substrate which introduce many materials
that find application as sensitive photoconductors are CdS and Cd Te.
The performance of a photo-conductor is measured in terms of
(i) the photo-conductivity gain, and (ii) the response time of the
detector. Typical gain and response time for a photo-conductor are
105 and 10–3 seconds respectively. Photoconductivity devices are
used as relays in digital or control circuits, as light meters, and to
record a modulating light intensity such as in a sound track.
Fig. 1.48 Photo response of a photoconductor
52
Electronics—Theory and Applications
1.26 Gunn Effect and Gunn Diode
Ridley-Watkins and independently Hilsum predicted that semiconductor materials under certain conditions
can offer differential negative resistance. This differential negative resistance is a bulk effect and due to transfer of electrons from one valley to another in the conduction band. Gunn while experimenting on a sample of
N-type GaAs and some other III-V compounds, found that the current through the sample increased linearly
with voltage till a certain threshold voltage. Beyond the threshold voltage a number of current pulses appeared
with a time interval proportional to the length of the sample. This threshold field is high (~ 400 V/mm). The
oscillations lie in the microwave range and set in the negative differential conductance (NDC) region where
the current decreases with increase in electric field E (Fig.
1.49). This behaviour is a consequence of the band structure
of these materials. In the region of velocity field curve, where
v decreases with increasing E, the differential mobility
(dv/dE) of the electrons becomes negative. The reason is
following.
n-GaAs has a direct energy band gap. The band gap is 1.4
eV. The free electrons in n-GaAs normally occupy the lowest
energy states in the conduction band. The effective mass of
electrons in this situation is and their mobility µ1 is high
Fig. 1.49 A possible characterstic of electron drift ve(≈ 0.02 m2/v-s). So as a result of the transfer of the electrons
locity vs. field for a semiconductor exhibitcurrent begins to decrease with increase in field because
ing the transferred electron mechanism
T = n1eµ1 E + n2eµ2E, where n1 and n2 are the concentration of
electrons having the mobility µ1 and µ2 respectively. The average drift velocity of an electron is
 n µ + n2µ 2 
V = 1 1
E
 n1 + n 2 
(1.36)
(n1 + n2) being the total electron concentration.
If the transition of the electrons from the high-mobility state to the low-mobility state occurs rapidly over a
range of field E, one observes that v diminishes with field E beyond a certain thershold field Eth, shown by region AB in Fig. 1.49, when all the electrons move to the low-mobility state, the drift velocity is v = µ2E. One
finds that v continues to increase slowly with E. This builds up the successive oscillations.
Owing to the occurrance of the NDM in the velocity-field curve, Gunn diodes are used as sources of microwaves especially where high power is not a requirement. Gunn diodes are also used as local oscillators for
mixers in microwave receivers over the frequency range 1 to 100 GHz. A Gunn device can carry just before
the threshold voltage one of two possible currents, depending on the presence or absence of a domain. This
can, therefore, be used as a high-speed binary logic.
Gunn diode is a versatile semiconductor device. These are commercially available for pulsed operation
yielding a power output of 5W in the frequency range 5.0 to 12.0 GHz.
1.27 IMPATT, TRAPATT AND QWITT Diodes
IMPATT (Impact Ionization Avalanche Transit Time) diodes employ impact-ionization and transit-time
properties of semiconductor structures to produce negative resistance at microwave frequencies. TRAPATT
(Trapped Plasma Avalanche Triggered Transit) diode is an IMPATT related device. These are quite new devices and used as oscillators in the microwave region. These devices work well in the breakdown region.
These devices work well in the breakdown regions. Although IMPATT operation can be obtained in simpler
structures, the Read diode is best suited for illustration of basic principles.
Semiconductors and Junction Diodes
53
The Read diode consists essentially of two regions: (i) the
n+ -p region at which avalanche multiplication occurs and
(ii) the i (essentially intrinsic) region through which generated holes must drift in moving to the p+ contact. One can
built similar devices in the p+-n-i-n+ configuration, in which
electrons resulting from avalanche multiplication drift
through the i-region taking advantage of the higher mobility
of electrons compared to holes.
The device operates in a negative conductance mode
when the a.c. component of current is negative over a portion
Fig. 1.50 The Read diode (a) The basic device conof cycle during which the a.c. voltage is positive, and
figure (b) Electric field distribution in the
device under reverse bias.
vice-versa. The negative conductance occurs because of two
processes, causing the current to lag behind the voltage in
time: (i) a delay due to avalanche process and (ii) a further delay due to the transit time of the carriers across
the drift region. When these two delay times combine to produce a net phase-shift between 90° and 270°, the
diode resistance is negative at the corresponding frequency. Consequently, negative conductance occurs and
the device can be used for oscillation and amplification.
The structures that have been successfully employed for IMPATT devices are the PN, PIN, P+ NN+, P+
NIN+ and N+ PIP+ (called read diodes). These devices are usually constructed from silicon and gallium arsenide. At present, IMPATT diode can generate the highest CW (Continuous Wave) power output at millimeter-wave frequencies, i.e. microwave power above 30 GHz. A major drawback of the IMPATT diode for very
high frequency operation is the fact that the avalanche process, which depends on random impact ionization
events, is inherently noisy.
In TRAPATT diode, the TRAPATT mode of operation, i.e. the operating frequency is substantially lower
than the transit time frequency and the efficiency is considerably higher. Under large-signal conditions the
periodic avalanching of the diode begins at the high-field side and sweeps rapidly across the diode, leaving it
substantially filled by a highly conducting plasma of holes and electrons whose space-charge depresses the
voltage to very low values. Since the plasma cannot rapidly escape, this mode is called TRAPATT mode. This
diode has been tried experimentally in pulsed transmitter of phased array radar systems.
A variety of approaches have been investigated to find alternative methods for injecting
carriers into the drift region without relying on
the avalanche mechanism. A particularly interesting device is QWITT (Quantum Well Infection Transit Time) diode, which employes
resonating tunneling through a quantum well to
inject electrons into the drift region. The device
consists of a single GaAs quantum well between
two Alx Ga1-x As barriers, in series with a drift
region of undoped GaAs. This structure is then
placed between two n+-GaAs regions to form
two contacts (Fig. 1.51). In this device, one can
achieve maximum resonant tunneling of electrons through the well if the dc bias is properly
adjusted.
QWITT diode is a low-noise injection mechanism with superior high frequency characterstics.
Fig. 1.51 Structure of a QWITT diode
54
Electronics—Theory and Applications
The length of the transit time as well as the shape of the current pulse can be optimized to obtain the best
power-frequency performance from the QWITT diode. This diode should extend the normal frequency limit
associated with transit-time devices, while providing higher output power than simple quantum-well RTD
oscillations.
PIN Diodes
This is a P-N junction with greatly improved switching times. Obviously this is a PN junction with a doping
profile tailored so that an intrinsic layer the “I-region” is sandwitched between a player and an N-layer as
shown in Fig. 1.52.
In practice, however, the idealized I (intrinsic)-region is approximated by either
a high resistivity P layer (referred to as π-layer) or a high resistivity N-layer (referred to as v layer). The resistivity of I layer is typically 103 Ω-m.
When no external voltage is applied between the terminals of PIN diode, the
Fig. 1.52 PIN Diodes
concentration gradient across the function cause diffusion of carriers. The width of
the depletion region in the I-layer is relatively large. Now, if a reverse bias is applied and increased gradually, the depletion region becomes thicker. At swept out voltage (a particular value of
the reverse bias), all the free carriers are swept out of the intrinsic layer. With further increase of reverse bias,
the depletion region widens in the highly doped semiconductor regions. PIN diode offers a very high resistance under reverse bias. We must note that the breakdown voltage of the diode under reverse bias is also very
large.
The PIN diode when used as a switch operates between the ON and OFF states. In ON state the diode is forward biased when minority carriers are infected into the high resistivity region between the highly doped
regions. The increase in the densities of carriers reduces the resistance and the diode impedance is low. This
causes a current flow. The forward resistance of the diode varies with the forward bias. In the OFF state the
high resistivity region is completely swept out and the diode impedance is very high as the capacitance is very
low.
The PIN diode has found wide applications in microwave circuits. It can be used as a microwave
switch with essentially constant depletion layer capacitance and high power handling capability. When
PIN diodes are used as microwave switches and when they are biased in the OFF condition the bias is usually beyond the swept-out voltage (usually – 2V). PIN diodes can also be used as a variolosser (variable
attenuator) by controlling the device resistance which varies approximately linearly with the forward current. PIN diodes can also modulate signals upto the GHz range. We must note that the forward
characterstics of a thyristor (chapter 3) in its ON state closely resemble those of PIN diode.
SOLVED PROBLEMS
Example 1.1. Find the conductivity and resistivity of a pure silicon crystal at temperature 300°K. The density of electron hole pair per cc at 300°K for a pure silicon crystal is 1.072 × 1010 and the mobility of electron µn = 1350 cm2/volt-sec and hole mobility µh = 480 cm2/volt-sec.
Solution: Conductivity of pure silicon crystal is given by
ni = 1.072 × 1010
σ = n i e(µ e + µ h )
σi = 1.072 × 1010 × 1.6 × 10–19 (1350 + 480) = 3.14 × 10–6 mho/cm
µn = 1350 cm2/Volt-sec
µh = 480 cm2/Volt-sec
e = 1.6 × 10–19 Coulomb
Semiconductors and Junction Diodes
55
Resistivity of silicon crystal is given by
ρi =
1
1
=
= 3.18 × 105 Ohm -cm
σ i 3.14 × 10–6
= 3.18 × 10–3 Ohm - m
Example 1.2. A silicon wafer is doped with phosphorus of concentration 1013 atoms/cm3. If all the donor
atoms are active, what is its resistivity at room temperature? The electron mobility is 1200 cm2 /Volt-sec
charge on the electron is 1.6 × 10–19 Coulomb.
Solution:
µ = 1200 cm/volt-sec
σ =µ en
σ = 1200 × 1.6 × 10–19 × 1013 = 19.2 × 110–4 mho/cm
e = 1.6 × 10–19C
n = 10–13 = Np
Resistivity
ρ=
1
1
=
= 5.2 × 102 Ohm -cm
σ 19.2 × 10–4
Example 1.3. In a semi-conductor, it is found that three quarters of current is being carried by electrons and
one quarter by holes. If at this temperature the drift speed of the electrons is two and half times that of
holes, determine the ratio of electrons to holes present.
Solution: Current I = In + Ip
Here
In → current due to electrons, Ip → current due to holes
Let drift speed due to holes be Vp, then
I = In + Ip = nn eVn + np eVp
According to the problem,
In =
and
3
5
I = ne e V p
4
2
(i)
1
I = n p eV p
4
(ii)
Ip =
Dividing (i) by (ii), we obtain
nn
2 6
= 3 × = = 1.2
np
5 5
Example 1.4. Find the resistance of an intrinsic germanium rod 1 cm long, 1 mm wide and 1 mm thick at
temperature of 300°K. For germanium ni = 2.5 × 1013, µn = 3900 cm2/Volt-sec at 300°K.
Solution:
σ = n i e (µ e + µ h )
56
Electronics—Theory and Applications
= 2.5 × 1013 × 16
. × 10−19 ( 0.39 + 019
. )
or
Now
∴
= 2.32 mho/m
1
ρ=
Ohm × m
2.32


length
10−6
Resistance R = ρ i 
 = 2.32 × −2
10
 area of cross section 
R=
1
2.32 × 10−4
ohm = 4.31 k Ω
Example 1.5. A sample of Germanium is made of P material by adding acceptor atoms at a rate of one atom
per 4 × 108 Germanium atoms. The acceptor density is assumed to be zero and ni = 2.5 × 1019 per m3 at
300°K. There are 4.4 × 1028 Germanium atoms/m3. The acceptor density is found to be 1.1 × 1020
atoms/m3.
Solution:
n i2 = np = n p N a
∴
np
−~
n i2
Na
np
∴
ni
=
=
6.25 × 1038
1.1 × 10
20
5.6 × 1018
2.5 × 1019
= 5.6 × 1018
= 0.22
Example 1.6. Find the value of the applied forward voltage for a P-N junction diode if IS = 50 micro
amp/cm, I = 2 amp/cm2 and e/kT= 40/volt.
Solution:
I = Is ( e eV / kT − 1)
e 40V =
∴
∴
V=
6
J
~ 2 × 10
−1 −
Js
30
2 × 105
2.303
= 0.277 volt.
log 10
40
3
Example 1.7. The approximate value of P-N junction current under forward bias is given by I • Is eeV/kT.
Show that the incremental resistance Re, defined by ∆V/∆I is equal to 1/40I at the room temperature
(e/kT= 40/Volt).
Solution:
I = Is exp [eV/kT] = Is e40V
Semiconductors and Junction Diodes
Q
Re =
57
∆V
1
=
∆I 40I
Example 1.8. Determine the number density of donor atoms which have to be added to an intrinsic germanium semi-conductor to produce an N-type semi-conductor of conductivity 5 mho/cm. The mobility of
electrons in the N-type semi-conductor is 3850 cm2 /Volt-sec.
Solution:
σ n = nq µ n = N D q µ n
σn = 5 mho/cm
5
∴ n = ND = σn/q µn =
. × 10−19 × 3850
16
2
µn = 3850 cm /volt-sec
= 0.8 × 1018 per cm 3
q = 1.6 × 10–19 C
Example 1.9. The saturation current density of a PN junction Germanium diode is 250 mA/m2 at 300°K.
Find the voltage that would have to be applied across the junction to cause a forward current density of 105
A/m2 to flow.
Solution:

 eV  
I = I s exp   − 1
 kT  

Dividing throughout by volume, one obtains the equation in the form of current density as

 eV  
J = J s exp   − 1
 kT  

 eV 
exp   − 1 = J /J s
 kT 
∴
∴
5
10
 eV 
exp   − 1 =
= 4 × 105
−3
 kT 
250 × 10
J s = 250 mA/m 3
J = 105 A/m 2
eV ~
− log e ( 4 × 105 ) = 12.9
kT
or
∴
V=
12.9 × 1.38 × 10−23 × 300
1.6 × 10−19
= 0.33 V
Example 1.10. A PN function in series with a resistance of 5 × 103 Ω is connected across a 50 and 1 mΩ
respectively. Show that the forward and reverse bias current in the circuit are 9.9 mA and 4.975 ×10-2 mA
respectively.
Solution: (i) Forward bias
Current =
Applied voltage
50
=
= 9.9 mA
Junction resistance + external resistance 5000 + 50
58
Electronics—Theory and Applications
(ii) Reverse bias
Current =
50
5000 × 106
= 4.975 × 10−2 mA
Example 1.11. Assume that the silicon diode in the following circuit requires a minimum current of 1 mA to
be above the knee point (0.7 V) of its I-V characteristics. Also assume that the voltage across the diodes is
independent of current above the knee point.
(i) If VB = 5 V, what should be the maximum value of R so that the voltage is above the knee point ?
(ii) If VB = 5 V, what should be the value of R to establish a current of 5 mA in the circuit?
(iii) What is the power dissipated in the resistance R and in the diode, when a current of 5 mA flows in
the circuit at VB = 6 V.
(iv) If R = 1 kΩ, what is the minimum voltage VB required to keep the diode above the knee point?
Solution:
(i) The minimum voltage across the diode is = 0.7 V.
Obviously, this is above the knee point of the characteristic
curve. Thus the voltage drop across the resistance R = V′ =
(5 – 0.7)V = 4.3 V. The minimum current i = 1 mA = 10–3 A.
∴
R=
4.3
V′
= −3 Ω = 4.3 k Ω
i 10
(ii) The current through the resistance R = 5 mA = 5 ×10–3A and
also the voltage drop across R = (5 – 0.7)V = 4.3 V
∴
R=
4.3
5 × 10
−3
= 0.86 × 10−3 Ω = 860 Ω
Fig. 1.53
(iii) We have VB = 6V, and hence the voltage V′ across the resistance is V´= 6 – 0.7 = 5.3V
and current i = 5 mA = 5 × 10–3A
∴
The power (P) dissipated through the resistance R is
P = iV´ = 5 × 10–3 × 5.3 W
= 26.5 ×10–3 W = 26.5 mW
The power P′, dissipated in the diode is
P ' = i × 0.7 W = 5 × 10−3 × 7 × 10−1 W
= 3.5 × 10−3 W = 3.5 mW
(iv) To keep the diode above the knee point, the minimum current required is 1mA.
∴
VR = voltage drop across R (= 1 kΩ )
= 1 × 103 × 10–3 = 1V
We have the minimum voltage drop across the diode = 0.7 V
∴
VB = (1 + 0.7)V = 1.7 V.
Semiconductors and Junction Diodes
59
Example 1.12. A Zener diode (BC-147) has a Zener voltage of 9V with 10% tolerance at 25°C. Find the
range of voltage ratings at which it will breakdown? If the temperature is raised to 75°C, what will be the
new breakdown voltage when decrease in Zener voltage is about 2mV per degree centigrade?
Solution: Given, average breakdown voltage Vz = 9 V
Tolerance = 10% =
10
× 9 = 0.9 V
100
∴ Range of breakdown voltage = (9 – 0.9) to (9 + 0.9) V = 8.1 V to 9.9 V.
In the second case, the increase in temperature = 75 – 25 = 50°C
Fall in breakdown voltage = 2 × 50 = 100 mV = 0.1 V
∴ New breakdown voltage = 9 – 0.1 = 8.9 V.
Obviously, the range of new breakdown voltage = (8.1 – 0.1) to (9.9 – 0.1) V = 8.0 V to 9.8 V
Example 1.13. A varactor diode has a capacitance of 15 pF when the reverse bias voltage across it is 5 V. If
the diode bias voltage is increased to 10 V, then show the capacitance will be 10.6 pF.
Solution: We know that the capacitance of a varactor diode is inversely proportional to the square root of the
1
bias voltage, i.e. C ∝
.
V
C2
V
= 1
C1
V2
∴
or
C 2 = C1
V1
5
= 15 ×
= 10.6 pF
V2
10
Example 1.14. Will Silicon and Germanium semiconductors at 1000°K ambient temperature still be semiconductors?
Solution: We know the forbidden energy gap between conduction and valence bands for a semiconductor is
nearly 1 CV. For Ge and Si the energy gap is 0.785 eV and 1.21 eV respectively at 0°K. The energy gap
decreases with increase in temperature which is represented by the expression
E g (T ) = 121
. − 3.60 × 10 − 4T (for Si)
. – 3.60 × 10– 4 (1000) = 0.85 eV
E g (1000) = 121
∴
Obviously, Si and Ge will remain semiconductors at 1000°K ambient temperature.
Example 1.15. An abrupt function diode has transition temperature of 20 pF when reverse biased at 5 V. If
the voltage is increased by 1 V, show that the capacitance will decrease by 1.7 pF.
Solution: C =
∴
K
V
( K = eA )
20 =
K
5
∴ K = 20 5 = 44.8
60
Electronics—Theory and Applications
44.8
∴ Capacitance for 1V increase =
5 +1
= 18.3 pF
∴ Decrease in Capacitance = 20 – 18.3 = 1.7 pF.
Example 1.16. Fig. 1.54 shows the plot of log of resistivity versus
reciprocal of temperature for two different semiconductors A and B.
Assume that mobility is proportional to T–3/2, find (a) which material has wider band gap? (b) which material will require light of
shorter wavelength for generation of a electron-hole pair?
Solution:
(a) Resistivity, ρ =
∴
1
n i (µ n + µ p ) e
Fig. 1.54
µ p = µ p 0 × T −3 / 2
1
ρ=
A0 T
1
3 / 2 − E go / 2 kT
e
[µ n 0 T
−3 / 2
+ µ p 0 T −3 / 2 ] e
1
=
∴
µ n = µ n 0 × T −3 / 2
A 0 (µ no + µ p 0 ) e exp ( − E g 0 /2kT )
log ρ = − log [ A 0 ( µ n 0 + µ p 0 ) e] +
Eg 0
2kT
=C+
m
T
[where C and D are constants: C = − log[ A 0 ( µ n 0 + µ p 0 ) e; m = E g 0 / 2 k ]
This is a equation of straight line whose slope, m = Eg0 / 2k. From Fig. 1.54, we note that the graph of material
B has higher slope and hence material B has wider band gap.
(b) We know that an electron-hole pair will be created if the energy of the incident photon is equal to or
higher than the band gap. The critical wavelength (λc) is given by
λc =
hc
6.6 × 10−34 × 3 × 108 1.24 × 10−6
=
=
m
Eg e
Eg
E g × 1.6 × 10−19
Obviously, the material B with higher value of Eg will require a shorter wavelength of light electron-hole pair
creation.
Example 1.17. Two Germanium diodes D1 and D2 are connected in series
across 5V battery as shown in Fig. 1.55. (a) Find voltage across each
diode assuming breakdown voltage of diodes is greater than 5V. What
is the effect of temperature? (b) Find current if Vz = 4.9 V and I0 = 5 µA.
Solution: (a) The two diodes are connected in series and hence the same
current I flows in D1 and D2. Obviously, it is in forward direction through
D2 and in reverse direction through D1. Since D2 diode is forward biased, V2
will be very small and hence V1 (= 5 – V2) will be very much larger than VT
Fig. 1.55
Semiconductors and Junction Diodes
61
(= 0.026 V). This means the current will be equal to reverse saturation current I0. Now we consider diode D2.
We have

V  
I = I 0 exp   − 1
 VT  

Putting I = I0 and V = V2, we have

V  
I 0 = I 0 exp  2  − 1
 VT  

V 
exp  2  − 1 = 1
 VT 
∴
or
V2 = VT ln 2 = 0.026 × 0.693 = 0.018 V
∴
V1 = 5 − 0.018 = 4.892 V.
Effect of temperature: V2 = VT ln 2 = kT ln 2.
So V2 will increase with temperature.
(b) If Vz is 4.9 V then D1 will breakdown. This means V1 = 4.9 V.
∴
V2 = 5 – 4.9 = 0.1 V.
Now using I0 = 5 µA, V2 = 0.1 V, one obtains

 0.1  
I = 5 exp 
 − 1 = 229 µA.
 0.026  

Example 1.18. A LED has a greater forward voltage drop than does a common signal diode. A typical LED
can be modeled as a constant forward voltage drop VD = 1.6V. Its luminuous intensity Iv varies directly
with forward current and is described by Iv = 40 iD ≈ millicandela (mcd). A series circuit consists of such
an LED, a current-limiting resistance R, and a 5V dc source Vs. Find the value of LED current iD such that
the luminuous intensity is 1 mcd.
Solution: We have
I v = 40 i D
We must have
iD =
IV
1
=
= 25 mA.
40 40
Example 1.19. Light doping of P-type semiconductor material is defined as the case for which p >> n is not
valid (n >> p is not valid for an N-type semiconductor). Derive a procedure to determine the number of
mobile carriers for the case of light doping consider ND and NA are immobile donor and acceptor ions respectively.
Solution: For P-type doping, ND = 0. Whence, by charge neutrality and mass-action law
62
Electronics—Theory and Applications
p+ND = p = n+N A =
n 2 + N A n − n i2 = 0
or
n i2
n
Solving the quadratic equation for n and discarding the extraneous negative root, one obtains
n=
1
 − N A + N A2 + 4n i2 

2
Knowing n, one obtains from mass-action law
p=
n i2
n
For N-type doping, NA = 0. By analogous procedure,
p=
1
 − N D + N D2 + 4N i2 

2
n=
n i2
p
SUMMARY
1. Semiconductors constitute a large class of substances characterized by negative temperature coefficient
of resistance.
2. Semiconductors have half filled outer shells, and are neither good insulators nor good conductors.
3. Semiconductors have a forbidden band that represents the amount of energy needed to move a valence
electron to conduction band to cause conduction.
4. Crystal structure of semiconductors, e.g., germanium and silicon reveals that a pair of electrons is shared
between atoms to complete the valence shells of the individual atoms. Such bonding between atoms is
termed as covalent bonding.
5. In a pure semiconductor the number of holes and electrons are equal. Such a semiconductor is termed
intrinsic.
6. Conduction in a pure semiconductor consists of electron flow in the conduction band and hole flow in the
valence band. Hole conductivity is the result of motion of bound electrons along the bonds.
7. Impurities are added to pure semiconductors, in a process called doping, to increase the number of
current carriers. Trivalent or acceptor, impurities have only 3 valence electrons and thereby form holes in
the bonds, which accept electrons from the semiconductor material, making it positive or P-type.
Pentavalent, or donor impurities have 5 valence electrons, and produce excess electrons that make the
semiconductor material a negative or N-type.
8. P-type and N-type refer to the majority current carriers produced by doping; the overall semiconductor
remains electrically neutral Minority current carriers, produced by thermal energy, are opposite to the
majority carriers.
9. The majority current carriers in an N-type material are electrons, while the minority carriers are holes.
The reverse is true for a P-type material. The doped semiconductor as a whole remains electrically
neutral.
10. A doped semiconductor acts like a resistor. The resistance of doped semiconductor is called bulk
resistance. Greater is doping, lesser is bulk resistance.
Semiconductors and Junction Diodes
63
11. A P-N junction is formed in a process which may be of the grown, alloyed, or diffused type.
12. A P-N junction has a built-in potential hill or barrier that exists across the depletion region and may be
increased or decreased by the application of an external voltage.
13. A forward-biased P-N junction has the positive terminal applied to the P side and the negative to the
N-side. The reverse is true for a reverse-biased junction.
14. The avalanche breakdown and zener breakdown are two different mechanisms by which a P-N junction
breaks.
15. Avalanche breakdown occurs in ordinary diodes, i.e. thicker functions. After avalanche breakdown, the
P-N junction is destroyed for ever and the diode is said to be burnt-off.
16. Zener breakdown occurs in thin junctions (zener diodes) and it is the runaway increase in minority
carriers during reverse current flow. It is also caused by the release of high energy valence electrons. It is
used in voltage-regulating diodes. After zener breakdown, the junction is not destroyed till the current
flowing through it exceeds the rated value.
17. The tunnel effect is the movement of valence electrons from the valence energy band to the conduction
band with little or no applied energy. Valence electrons seem to tunnel through the forbidden energy
band. The tunnel effect provides a negative resistance region in the tunnel diode where increasing voltage
results in decreasing current. Tunnel diodes are used in switching and oscillatory circuits.
18. A germanium device is seldom used at a temperature higher than 75°C, a silicon device seldom higher
than 175°C.
19. A P-N junction exhibits a transition capacitive effect of several picrofarad under reverse bias condition,
but may have of diffusion capacitance of many microfarads in the forward-biased state.
20. Photo diode is a two element semiconductor light sensative device. They convert light energy into
electrical energy directly. Such types of diodes are optimised for their sensitivity to light.
21. An LED is semiconductor diode which emits visible light from its P-N junction when it is forward biased.
They are used as readouts in alphanumeric displays. They are also used as pilot lamps in most of the
electronic and electrical devices.
22. An SCR is a three element, four layer (NPNP) solid state device which permits current to flow through it is
one direction only (i.e. when forward biased) and bias is equal to or greater than forward breakover voltage.
The third element called gate, controls the breakover voltage. Higher the gate current, the lower the breakover
voltage. Once the SCR is ON the gate loses control over the conduction of SCR. It continues to conduct as
long as it is forward biased and as long as the current through it is equal to greater than holding current.
23. Varactor diode is a special type of solid state diode whose capacitance varies with applied reverse
k
, where k = EA. The capacitors of varactors is
voltage. The capacitance of a varactor is given by C =
V
controlled by the applied voltage, therefore, they have replaced mechanically tuned capacitors in many
applications, e.g., television receivers, FM receivers, automobile radius, communication equipment, etc.
24. The diodes that keep the current constant flowing through them when the voltage changes are known as
constant current diodes.
25. A backward diode is just a zener diode in which a doping level increases to such an extent that its zener
effect (reverse breakdown) occurs near to zero (say –0.1 V).
A. Review Questions
1. What are semiconductors? How do they differ from conductors and insulators? Why an increase in
temperature increases conductivity of a semiconductor?
64
Electronics—Theory and Applications
2. Explain with suitable diagrams the conduction band, valence band and forbidden band and hence explain
the behaviour of conductor, semiconductor and insulator. Explain electrons and holes contribution to
electrical conduction.
3. Derive the expression for the conductivity of an intrinsic semi- conductor.
4. What are intrinsic and extrinsic semiconductors? Discuss the loca- tions of Perm levels under suitable
limiting conditions and give the necessary theory.
5. What is impurity conduction in semiconductors? Explain how the presence of a small impurity in a
semiconductor modifies its conduction properties.
6. Explain the movement of electrons and holes in a semiconductor. In what respect N-type and P-type
semiconductors differ from each other.
7. What is drift current and diffusion current in a semiconductor? Write an expression for the total electron
current density in a semiconductor.
8. How will you draw the characteristics of a semiconductor diode?
9. How an abrupt junction formed?
10. What happens to the depletion region in a p-n diode under forward bias and reverse bias conditions?
11. Explain the working of a P-N junction. Discuss forward and reverse biasing of P-N junction diode.
12. What happens to the depletion region in a p-n diode under forward bias and reverse bias conditions?
13. Explain with diagrams forward and reverse biasing of a P-N junction. What is meant by avalanche
breakdown?
14. Explain with reference to Zener diode characteristic curve the following:
(i) IZK (ii) IZT (iii) ZZ
where symbols have usual meaning.
15. What do you understand by zener breakdown? Draw a typical characteristic of a zener diode. Identify the
breakdown region and explain it.
16. A zener diode is a p-n function yet it is different from an ordinary p-n function. Explain.
17. Describe an experiment for drawing the characteristic of a zener diode.
18. Draw the (V-I) characteristics of a tunnel diode and discuss its applications.
19. Explain the mechanism of electrical conduction in a typical semiconductor like Germanium or Silicon.
How the conductivity of a pure semiconductor is affected by adding traces of trivalent and pentavalent
impurities?
20. What is meant by doping a semiconductor? What basic properties must these doping elements possess to
make a pure semiconductor a P-type or N-type? Explain the physical principles involved.
21. Write short notes on the following:
(i) Energy bands in solids, (ii) Avalanche breakdown, (iii) Zener breakdown, (iv) P type and N type
semiconductors, (v) Tunnel diode, (vi) Varactor diode, (vii) Light emittinmg diode, (viii) Solar cell, (ix)
Schotty diode, (x) Backward diode
22. Give the structure of light emitting diode. Explain its working and give some of its uses.
23. Explain, why Zener breakdown voltage has a negative temperature coefficient whereas the avalanche
breakdown voltage has a positive temperature coefficient? Why dynamic resistance of an ideal Zener
diode is zero but the dc resistance is not zero.
24. What is a breakdown diode? Explain the origin of breakdown of a function.
25. Draw the volt-ampere characterstic of a tunnel diode. Explain the occurrence of the negative differential
resistance in the characterstic of a tunnel diode. Write some of the uses of tunnel diode.
Semiconductors and Junction Diodes
65
B. Numerical Problems
1. Mobilities of electrons and holes in a sample of intrinsic germanium at room temperature are 3600 cm2
/volt-sec, and 1700 cm2 /volt-sec, respectively. If the electron and hole densities are each equal to 2.5 ×
[Ans. 2.12 mho/m]
1013 per cc, calculate its conductivity.
2. The concentration of the acceptor atoms in P-type germanium crystal is 4 × 1015 per cc. Find the
conductivity of the crystal at 300°K. Hole mobility in germanium at 300°K is 1900 cm2 per volt-sec. All
the acceptor atoms are ionized at this temperature.
[Ans. 121.6 mho/m]
3. Calculate the current produced in a small germanium plate of area 1 cm2 and of thickness 0.3 mm. When
a potential difference of 2 V is applied across the faces. Given concentration of free electrons in
germanium is 2 × 1019 m3 and the mobility of electrons and holes are 0.36 m2/volt-sec, and 0.17
[Ans. 1.13 amp]
m2/volt-sec, respectively.
4. A specimen of pure germanium at 300°K has a density of charge carriers of 2.5 × 1019 per m3. It is doped
with a donor impurity atoms at the rate of one impurity atom for every 106 atoms of germanium. All
impurity atoms may be supposed to be ionised. The density of germanium atom is 4.2 × 1028 atoms/m3.
Find the resistivity of the doped germanium if electron mobility is 0.36 m2/volt-sec.
[Ans. 0.41 × 10–3 Ω-m]
5. What concentration of donor atoms per cc is required in a germanium crystal so that the conductivity of
the crystal is 2 mho/cm at 300°K. For germanium µn, = 3900.
[Ans. 320.5 × 1019/m3]
6. Find the conductivity of germanium which is doped with donor atoms of concentration Nd= 1017/cm3 and
compare it to that of intrinsic case. Given ni = 2.5 × 1013/cm3, µn = 3800 cm2/volt-second.
[Ans. 6.25 × 1015/m3]
7. Compare the number of electron hole pairs per cm in a pure silicon crystal at the temperature of (i) 27°C
and (ii) 57°C. For silicon Eg – 1.13 eV.
[Ans. 8.39]
8. Find the value of P-N junction diode current in the forward bias condition, when the applied voltage
across it is 0.05 volt and IS = 50 µA. Given e/kT= 40/volt.
[Ans. 0.319 milli amp.]
3
9. Determine the donor concentration in N-germanium having a resistivity of 0.015 Ω-m. µc = 0.36
m2/volt-sec. Repeat for P-germanium of equal resistivity, µn, = 0.17 m2/volt-sec.
10. In the circuit (Fig. 1.50) using a zener voltage regulator, calculate the limit of the Vin for getting regulated
voltage without damaging the diode. Take the dynamic resistance of the diode as negligible and
temperature coefficient negligible; (Assume Iz min = 5 mA).
11. An N-type Si bar is 2 cm long and has a cross-section of 2 mm × 2 mm. When a one volt battery is
connected across it, a current of 8 mA flows. Determine: (i) doping level (ii) drift velocity.
[Ans. (i) 192 × 1015 / cm3 (ii) 650 cm/s]
12. (a) Show that the conductivity of a semiconductor is minimum when it is lightly doped with P-type
impurity such that
P = ni
µn
µp
(b) Show that the minimum conductivity is 2n i µ n µ p q.
(c) With the help of the above result determine the minimum conductivity of silicon.
[Ans. (c) σmin = 3.87 × 10–6 (Ohm-cm)–1]
66
Electronics—Theory and Applications
13. A Silicon diode operates at a fixed bias of 0.4 V. Determine the factor by which the current will get
multiplied when its temperature is raised from 25°C to 150°C.
[Ans. 638]
14. When a diode is reverse biased with 8V it has a junction capacitance of 15 pF. When the reverse biased is
increased to 12V, the capacitance drops to 13.05 pF. Find whether it is abrupt or graded junction?
[Ans. graded junction type]
C. Short-Question-Answers
1. What is the order of energy for forbidden band in (i) diamond (ii) silicon (iii) germanium and (iv)
aluminium?
[Ans. (i) 9eV (ii) 1.2eV (iii) 0.74eV (iv) Zero.]
2. What happens to the resistance and conductance of a semiconductor on heating?
[Ans. Resistance decreases but conductance increases.]
3. What is the order of resistively of a metal, and insulator and a semiconductor?
[Ans. Resistively of (i) Metal •10–6 Ω-cm, (ii) Insulator •1013 Ω-cm and (iii) semiconductor lies
between 1013 Ω and 10–6 Ω cm]
4. What are the important characteristics of a semiconductor?
[Ans. (i) it has covalent bonding (ii) it is crystalline (iii) it has a negative temperature coefficient of
resistance (iv) its conductivity increases with the addition of impurities.]
5. What is mobility? Write its units?
[Ans. The drift velocity per unit electric field is called mobility (µ). Its unit is N–1S–1mC or N–1mA].
6. Why metallic bodies are always opaque?
[Ans. We know that metallic solids have partially filled conduction bands. The energy of photons in the
visible light region varies between 1eV and 3eV. As light is incident on metallic solids, free electrons of
the conduction band absorb the energy of the incident photons. Obviously, no photons are allowed to pass
through, the metallic solids behave as though opaque.]
7. A small portion of indium (In) is incorporated in germanium (Ge). Is the crystal N type or P type?
[Ans. Obviously, the crystal is P type as indium is trivalent and one of the covalent bonds will remain
without electrons giving excess of holes.]
8. What are the charge carriers in N-type and P-type semiconductors?
[Ans. The charge carriers in N-type semiconductors are the electrons while in P-type the charge carriers
are holes. Why a semiconductor is damaged by a strong current?]
9. Why a semiconductor is damaged by a strong current?
[Ans. The safety limits of temperature for germanium and silicon are about 80°C and 200°C respectively. A
strong current passing through a semiconductor heats it up beyond these temperatures. At these
temperatures a large number of charge carriers are available for conduction and the specific resistance of
the crystal become very low and thus they are damaged, i.e., loose the property of semiconductors.]
10. What is the difference between hole-current and electron flow?
[Ans. The hole current is in the direction of conventional current which is due to the flow of positive
charge while the electron flow is opposite to the conventional current as it is due to the flow of negative
charges.]
11. What is the Germanium diode zener voltage?
[Ans. When a reverse bias of about 25 volts is applied to the crystal, the excessively high temperature
destroys the covalent structure of germanium and the reverse current rises sharply. This breakdown
voltage is called zener voltage (Vz).]
Semiconductors and Junction Diodes
67
12. What do you understand by ‘tolerance’ of a zener diode?
[Ans. The range of voltages about the breakdown voltage in which a zener diode conducts in reverse
direction is called tolerance?]
13. What do you understand by power rating of a zener diode?
[Ans. The maximum power which zener diode can handle without damage is known as its power rating.]
14. What do you understand by maximum current rating of a zener diode?
[Ans. The maximum value of current which a zener diode can handle at its rated voltage without damage
is known as its maximum current rating.]
15. Which type of diode used for the rectification of very high frequency (MHz)?
[Ans. Schottky diode.]
16. Which diode is best suited for the rectification of very small peak voltage (~ 0.5 V)?
[Ans. Backward diode.]
17. Which diode has a negative resistance?
[Ans. Tunnel diode.]
D. Objective Questions
1. N-type germanium is obtained on doping intrinsic germanium by
[a] Phosphorous
[b] Aluminium
[c] Boron
[d] Gold
2. Depletion region is a zone which contains
[a] holes only
[b] electrons only
[c] both electrons and holes
[d] neither electrons nor holes
3. In a semiconductor diode arrow represents
[a] N type material
[b] P type material
[c] both P and N type materials
[d] none of the above
4. Zener diode is used for
[a] rectification
[b] amplification
[c] stabilization
[d] none of the above
5. For a tunnel diode a decrease in current causes
[a] voltage constancy
[b] decrease in voltage
[c] Increase in voltage
[d] none of the above
6. PN junction is formed when P type semiconductor and N type semiconductor are joined
[a] together
[b] physically
[c] to get homogeneous material chemically
[d] in such a manner that electrons and holes diffuse to give depletion layer.
7. The depletion region of a junction diode is formed
[a] when forward bias is applied to it
[b] when the temperature of the junction is reduced
[c] under reverse bias
[d] during the manufacturing process.
8. The width of the depletion layer of a junction
[a] is independent of applied voltage
[b] is increased under reverse bias
[c] decreases with light doping
[d] increases with heavy doping
68
Electronics—Theory and Applications
9. The LED or the light emitting diode
[a] is made from one of the two basic semiconducting materials, silicon or germanium.
[b] is made from the semiconducting compound gallium arsenide phosphide.
[c] emits light when forward biased.
[d] emits light when reverse biased.
10. The p-side of a junction diode is earthed and the n-side is given a potential of –2V. The diode will
[a] not conduct
[b] conduct partially
[c] break down
[d] conduct
11. For detecting light intensity we use a/an
[a] photodiode in reverse bias
[b] photodiode in forward bias
[c] LED is a reverse bias
[d] ED in forward bias
12. When a p-n function diode is forward biased, the flow of current across the function is mainly due to
[a] diffusion of charges
[b] drift of charges
[c] depends on the nature of the material
[d] both drift and diffusion of charges
13. A p-n function diode cannot be used
[a] as a rectifier
[b] for increasing the amplitude of an ac signal
[c] for getting light radiation
[d] for converting light energy into electrical energy
14. A strong electric field across a P-N junction that causes covalent bonds to break apart is called
[a] reverse breakdown
[b] avalanche breakdown
[c] lever breakdown
[d] low voltage breakdown
15. A light emitting diode produces light when
[a] forward biased
[b] reverse biases
[c] unbiased
[d] none of the above
16. A solar cell is an example of
[a] photo emissive cell
[b] photo radiation cell
[c] photo voltaic cell
[d] photo conductive cell
17. When holes leave the P-material to fill electrons in the N-material to fill electrons in the N-material the
process is called
[a] diffusion
[b] depletion
[c] avalanche breakdown
[d] zener breakdown
18. A varacter diode is optimised for
[a] high output current
[b] high output
[c] its variable inductance
[d] its variable
19. A diode which has zero breakdown voltage is known as
[a] tunnel diode
[b] Zener diode
[c] Schottky diode
[d] backward diode
20. A zener diode is used as
[a] a coupler
[b] a rectifier
[c] an amplifier
[d] a voltage regular
Semiconductors and Junction Diodes
69
E. Mark which of the following statements is true or false
1. Breaking a covalent bond produces a free electron, which moves about the lattice in a random manner
[1] True
[2] False
2. A semiconductor is damaged by a strong current
[1] True
[2] False
3. Charge carriers in N type semiconductor are holes
[1] True
[2] False
4. Minority current carriers, produced in a semiconductor by thermal energy, are opposite to majority
carriers
[1] True
[2] False
5. The arrow in a PN diode represents the P section, and corresponds to the anode in an electron tube
[1] True
[2] False
6. Light emitting diode emits light when it is forward biased
[1] True
[2] False
7. The material used for the construction of LED is Ga As P
[1] True
[2] False
8. A zener diode is operated in forward characterstic region
[1] True
[2] False
9. A tunnel diode has zero breakdown voltage
[1] True
[2] False
10. The barrier potential of a Schottky diode is 0.25 v
[1] True
[2] False
11. A varactor diode is optimised for high output current.
[1] True
[2] False
12. Schottky diodes are made of n-type semiconductors and metal
[1] True
[2] False
13. Current in a 1.2 W, 6V zener diode should be limited to a maximum of 0.2 A
[1] True
[2] False
14. When a PN junction is heavily doped, its breakdown voltage will increase
[1] True
[2] False
15. The light emitting diode (LED) is usually made from metal oxide
[1] True
[2] False
F. Fill in the Blanks
1.
2.
3.
4.
5.
Free electrons are .......... current carriers and holes are .......... current carriers.
The potential barrier increases with .......... bias and decreases with .......... bias.
Forward bias utilizes .......... carriers to carry the current flow, and reverse bias utilizes .......... carriers.
Excessive forward bias causes .......... breakdown, and excessive reverse bias causes .......... breakdown.
The tunnel effect is the movement of valence electrons from the valence energy band to the .......... band
with little or no applied energy.
70
Electronics—Theory and Applications
6. The tunnel effect provides a .......... resistance region in the tunnel diode where increasing voltage results
in .......... current.
7. Under forward biased conditions an LED emits light when free .......... and .......... recombine at the
junction.
8. The most common LED arrary is the .......... indicator.
9. A photodiode is optimized for its sensitivity to .......... and it should be .......... biased.
Answers
Section D:
1(a), 2(d), 3(b), 4(c), 5(c), 6(d), 7(d), 8(b), 9(b,c), 10(d), 11(a), 12(a), 13(b), 14(c), 15(b), 16(c),
17(a), 18(d), 19(a), 20(d).
Section E: 1(True), 2(True), 3(False), 4(True), 5(True), 6(False), 7(True), 8(False), 9(True), 10(True),
11(False), 12 (True), 13(True), 14(False), 15(False).
Section F: 1(Negative, Positive), 2(Reverse, Forward), 3(Majority, Minority), 4(Avalanche, Zener),
5(Condition), 6(Negative, Decreasing), 7(Electrons, Holes), 8(Seven Segment),
8(Light Reverse).
Appendix A
Hall Effect
An effect whereby a conductor carrying an electric current perpendicular to an applied strong magnetic field
develops a voltage gradient which is transverse to both the current and the magnetic field. It was discovered by
E. H. Hall in 1879. One can obtain the important information about the nature of conduction process in semiconductors and metals through analysis of this effect. Whether a given sample of semiconductor material is n
or p-type can be determined by observing the Hall effect. If the electric current is caused to flow through a
sample of semiconductor material and the magnetic field is applied in a direction perpendicular to the current,
the charge carriers are crowded to one side of the sample, giving rise to an electric field perpendicular to both
the current and the magnetc field. This development of a transverse electric field is known as one Hall effect.
The effect is also used to in the Hall probe for the measurements of magnetic fields, and in magnetically operated switching devices.
The strength of the electric field EH produced is given by the relationship EH = RHjB, where j is the current
density, B is the magnetic flux density, and RH is a constant called the Hall coefficient. The value of RH can be
shown to be 1/ne, where n is the number of charge carriers per unit volume and e is the electronic charge.
Hall Effect in Semiconductors
Hall effect in semiconductors originates from motion of both electrons and holes. This provides the
best experimental evidence of the motion of holes,
i.e. the production of a transverse electric field by
the motion of charge carriers in a magnetic field.
With an applied electric field and a transverse component of magnetic field, both electrons and holes
are deflected transversely toward the same side of
the sample as shown in Fig. A.1.
The drifting carriers experience Lorentz forces
in the magnetic field Hz, and thereby suffer deflecFig. A.I Hall Effect in Semiconductors.
tions. Both electrons and holes are deflected towards the same side of the sample as shown in Fig.
A.1. The situation in all sections is similar. The two types of carriers in semiconductors under this arrangement, so called Hall geometry, therefore, tend to combine and cancel each other at the front surface. However,
this cancellation is incomplete, and thus there is a net charge which accumulates on the front surface of the
sample. Because of these surface charges on the front and the rear surfaces of the sample, an electric field is
produced in Y direction, known as Hall effect in semiconductors. εy may be calculated as follows.
The electron and hole current densities along x axis can be expressed as
J xe = n e µ e ε x 

J xh = p e µ h ε x 
(1)
72
Electronics—Theory and Applications
The total current density Jx is
J x = J xe + J xh = e( nµ e + pµ h ) ε x
(2)
(Small changes in mobility due to magnetic resistance are neglected.)
Let us now obtain the electron current density (Jye) along Y direction due to Lorentz force. Lorentz force
acting on an electron is – e(Ve × B) = eVeBz, where Ve is the drift velocity of the electron. This Lorentz force is
equivalent to an electric field – VeBz Since Ve = – µeεx and therefore this field may be expressed as Bz µeεx.
Using (1), one obtains
Jye = neµe(Bz µeεx)
Replacing Bz by Hz we have
J ye = neµ 2e H z ε x
and hole current density is given by
J yh = peµ n2 H z ε x
The total transverse current density is the sum of these:
(
)
J y = J ye + J yh = e nµ e2 − pµ h2 H z εx
(3)
A Hall field εy is required to make the net Y current zero. The ratio εy/εx is thus the same as Jy/Jx:
(
)
e ( nµ e + pµ h ) ε y = e nµ e2 − pµ h2 H z ε x
∴
εy =
pµ h2 − nµ e2
nµe + pµh
(4)
H z Ex
The Hall coefficient RH is given by
RH =
εy
HzJx
=
pµ h2 − nµ e2
e ( nµ e + pµ h )
2
(5)
Equation (5) is like that for a metal if the semiconductor is extrinsic (n >> p or p >> n). Measurements of a
and RH for strongly p-type and n-type extrinsic semiconductors permit determination of Hall mobilities from
(5). Since µn, µp, n and p are all temperature dependent, Hall effect in a semiconductor exhibits wide temperature variations, corresponding to the increase of intrinsic carriers at higher temperatures.
From equation (5), it is obvious that the magnitude of Hall coefficient is inversely proportional to the carrier concentration. The coefficent of proportionality involves a factor which depends on the energy distribution of the carriers and the way in which the carriers are scattered in their motion. However, the value of this
factor normally does not differ from unity by more than a factor of two. The situation is more complicated
when more than one type of carrier is important for the conduction. The Hall coefficient then depends on the
concentrations of the various types of carriers and their relative mobilities.
The product of the Hall coefficient and the conductivity is proportional to the mobility of the carriers when
one type of carrier is dominant. The proportionality involves the same factor which is contained in the relationship between the Hall coefficient and the carrier concentration. The value obtained by taking this factor to
be unity is referred to as the Hall mobility.
Semiconductors and Junction Diodes
73
Experimental Determination of the Hall Coefficient
To measure the carrier concentration directly, the most common
method is the use of the Hall effect.
Fig. A.2 shows the basic set up,
where a rectangular specimen of the
semiconductor of width b and thickness t is placed between the pole
pieces of an electromagnet such that
Fig. A.2 Basic set up for the determination of Hall Coefficient
a magnetic field B is applied along
the z-direction. With the help of a battery, the current I is allowed to pass through the sample in the x-direction
(Fig. A.2). One can measure the Hall voltage VH with the help of two probes placed at the centres of the top and
bottom surfaces of the sample. We have
(6)
VH = E H t
But EH = RH J B and hence Eq. (6) takes the form
(7)
VH = R H J t
We have the current density
J=
I
bt
(8)
where bt is the cross-sectional area of the sample. Making use of Eq. (8), Eq. (7) becomes
R IB
VH = H
b
VH b
or
RH =
IB
(9)
one can measure the quantities appearing on the r.h.s. of Eq. (9) and hence RH can be readily determined. In
S.I. system, RH is obtained in m3/Coulomb. We must note that the polarity of VH is opposite for N-type and
P-type of materials and hence RH has opposite signs for the two types of semiconductors.
Uses of the Hall Effect
(i) Determination of semiconductor type. The Hall voltage VH is measured by placing the two probes at the
centres of the top and bottom face of the sample. If the magnetic flux density is B Wb/m2, then we have, n = 1/e
RH where RH = 1/ne.
Since RH is positive for P-type and negative for N-type semiconductor, the sign of RH is used to determine
the type of semiconductor specimen.
(ii) Determination of carrier concentration. We have n = 1/ | eRH|, where RH = 1/ne. Obviously, by
measuring RH, one can find the electron concentration n and hole concentration p.
(iii) Determination of mobility of carriers. If the conduction is due to one type of carriers, e.g. electrons, we have
σ = neµ n
i.e.
σ
= σR H
ne
V b 
µn = σ  H 
 IB 
µn =
[using Eq. 9. ]
74
Electronics—Theory and Applications
Obviously, by knowing σ, one can determine the mobility µn. This mobility, i.e. given by σ |RH | is referred to
as Hall mobility.
(iv) Measurement of Magnetic Flux Density. We have seen that Hall voltage VH is proportional to the
magnetic flux density B for a given current I passing through a sample. Thus knowing the sample dimensions
and RH , one can determine the magnetic field by measuring I and VH. One can also use it as the basis for the
design of a magnetic flux density meter.
(v) Measurement of Power in an Electromagnetic wave. We know that in an electromagnetic wave in free
space the magnetic field H and the electric field E are at right angles to each other. Thus, if a semiconductor
sample is placed parallel to E it will derive a current I in the semiconductor. The semiconductor sample is
subjected simultaneously to a transverse magnetic field H producing a Hall voltage across the semiconductor
sample. The Hall voltage will be proportional to the product E and H across the sample, i.e. to the magnitude of
the poynting vector of the electromagnetic wave. Obviously, the Hall effect can be used to determine the
power flow in an electromagnetic wave.
(vi) Hall Effect Multiplier. If the magnetic field B is produced by passing a current I′ through an air-core coil,
B will be proportional to current I′ . The Hall voltage VH is thus proportional to the product II′ . This forms the
basis for the design of a multiplier.
Example 1. A sample of Si is doped with 1017 phosphorus atoms/cm3. What would you expect to measure for
its resistivity? What Hall voltage would you expect in a sample 100 µm thick if Ix = 1 mA and Bz = 1 kG
(= 10–5 Wb/cm2)?
Solution. Mobility for silicon 700 cm2/V-s. Obviously, the conductivity is σ = e µn n = 1.6 × 10–19 × 700 ×
1017 = 11.2 (Ω-cm)–1.
ρ = σ −1 = 0.0893 Ω -cm.
The resistivity is
R H = − ( en ) −1 = −62.5 cm 3 / C
The Hall coefficient is
The Hall voltage is
VH =
Ix Bz
10−3 × 10−5
RH =
× ( −62.5) = −62.5µV
x
10−2
REVIEW QUESTIONS
1. What is Hall effect? Briefly discuss the physical origin of the Hall effect. Mention uses of Hall effect.
2. What is Hall coefficient? Show that for a P-type semiconductor the Hall coefficient RH is given by, RH =
1/eP. Describe an experimental set up for the measurement of Hall Voltage, VH.
3. For an semiconductor show that the Hall Coefficient is given by
RH = −
1 pµp 2 − nµ n2
e ( pµ p + nµ n ) 2
where µP and µN are the mobilities of holes and electrons respectively.
Show that for an intrinsic semiconductor the above expression reduces to
RH = −
1 µ n −µ p 


ni e  µ n + µ p 
[Hint. n = p = ni for an intrinsic semiconductor]
Appendix B
(a) Conduction in Semiconductors by Charge Drift
(b) Conduction in Semiconductors by Diffusion of Charge
(c) Barrier Potential and Volt-current Equation for the P-N Diode
(a) Conduction in semiconductors by Charge Drift. The free electrons and holes of the semiconductor move in
random paths due to the fluctuation of thermal energies. When an electric field is applied along the
semiconductor material, a component of velocity in the direction of the electric field is added to the random
thermal velocities. This directed motion of charge in the semiconductor is called drift. The drift of the charge can
be described in terms of the field intensity and the conduction properties.
A current density J is defined as the charge passing through a unit area per second. In terms of a charge density, ρ, moving with a velocity v through the plane of the observation, one obtains
 amp 
J = ρv  2 
m 
(1)
Also ρ = – ne so Je = – neve, where ve is the mean electron velocity. The random thermal components of velocity will cancel across the plane and hence leaving the directed velocity component due to the electric field.
Similarly, for holes, one obtains
Jh = pevh
where vh, is the mean hole velocity. Since ve is oppositely directed to vh, the total current density due to both
electrons and holes is
v 
 v
J = e ( nv e + pv h ) (amp / m 2 ) = eE  n e + p h 
 E
E
(2)
where E is the field intensity.
The conductivity of the material is defined as
σ=
J
= e ( nµ e + pµ h )
E
(3)
where µ is called the charge mobility and it is defined as the velocity achieved per unit field-intensity.
In extrinsic semiconductors the number of mobile charges is relatively fixed at n = ND or p = NA, and the
conductivity varies with impurity density. In the case of intrinsic semiconductors the number of mobile
charges created by thermal pair generation increases faster than the mobility falls with temperature, and the
conductivity rises with temperature. In an intrinsic semiconductor the current is largely due to electrons, only
small fraction is due to holes. However, the situation in the case of metals is quite different. Metals have large
and relatively fixed numbers of mobile charges. As temperature rises, the increased agitation of the atomic nuclei reduces the mobility of the charges and the conductivity of metal falls with rising temperature.
Resistivity ρ =
1
1
=
σ e( nµ e + pµ n )
76
Electronics—Theory and Applications
The mobility of electrons in a semiconductor is considerably greater than that of holes.
(b) Conduction in semiconductors by diffusion of
charge. Fig. B.1 illustrates a charge density
gradient in non-homogeneous material. It will
produce a net movement of charge by diffusion.
The thermal agitation velocities remain random and due to this there will be more free electrons in high density n-region with velocity
components directed toward the p-region then
there will be electrons in the p-region with velocity component directed toward the n-region. The
result is a net movement or diffusion of electrons
from n-region to p-region. This represents a diffusion current. Similarly, we have net movement
or diffusion of holes from p to n region.
Fig. B.1 Charge Density Gradient in Extrinsic (Non-homogeneous)
Semiconductors.
J h = − e Dh
dp
dx
(4)
1
∆ x ∆v where
2
∆v is the mean thermal velocity reached by the holes between collision with atoms. The unit of Dh is m2/s (e is
in Coulomb, dp/dx is in m–4 and Jh is in A/m2). Eq. (4) is sometimes referred as Fick’s law. – dp/dx is the density gradient of holes in the + x direction. Similarly we can show that the electron density gradient, a diffusion
current density due to electrons moving to the left is given by
dn
(5)
J e = eDe
dx
where Dh is a diffusion constant, diffusion coefficient or diffusivity for holes, defined as Dh =
Being of electrons, this is a conventional current to the right. Obviously, both currents move to the right and
are additive.
Diffusion current exists because of a space gradient of charge density. Drift current is created by an electric
field, whereas diffusion can occur in a region free of electric fields, a condition that is present in many semiconductor devices.
Since drift and diffusion currents may occur simultaneously, we can use Eqs. (4) and (5) and the current
densities to obtain the total current densities as
dp
(6)
J h = eµ h pE − eDh
dx
dn
(7)
J e = eµ e nE + eDe
dx
The net current density
r
dp 
dn

(7(a))
J = J n + J e = e [ nµ e + pµ n ] E + e  De
− Dh
dx
dx 

(c) Barrier potential and volt-current equation for the P-N diode. To use a PN junction diode as a circuit
element we need to know its volt-ampere relationship. We now develop the voltage-current equation for the
PN diode from the barrier potential and the N and P charge densities.
Semiconductors and Junction Diodes
77
We have the diffusion constant D and the charge mobility µ as measures of the thermodynamic actions of
charges. The Einstein relation is
De Dh kT
(8)
=
=
e
µe µh
where kT/e represents the mean thermal energy of the particles and is often called the voltage equivalent of
temperature, at room temperature 300 K, kT/e = 0.026 V and it is dimensionally equal to work per unit charge.
It could be stated in Volts and it is often called the voltage equivalent of temperature, VT, i.e.
VT =
kT
T
=
e 11,600
At room temperature ( ~ 300 K), one finds that kT/e = 0.026 V.
The current must be zero in the open circuited PN junction. Considering the electron component, we set
Je = 0 in Eq. (7) and we obtain
µ
dn
= − e E dx
n
De
Using Eq. (8), we can write as
dn
e
=−
E dx
n
kT
(9)
Integrating the above expression across the junction, [Fig. B.1] from x1 to x2, we obtain
nn
dn
e
∫ n = kT
np
x2
∫ ( − E ) dx
x1
where the subscript p indicates a p material and subscript n indicates an n material.
The integration of – E from x1 to x2 leads to the value of the barrier potential VB with N positive to P. We obtain the result of integration as
nn
(10)
= exp[VB / ( kT / e )]
np
Eq. (10) is the relation between the electron density at the junction face in the n-region to the electron density at the junction face in the p-region. The exponent is a measure of the average ability of charges to transit
the barrier or represents the ratio of barrier height to average energy of the charges.
Similarly by putting Jf, = 0 in eq. (6), one obtains
pp
pn
= exp[(VB / ( kT / e )]
(11)
Eqs. (10) and (11) are called as Boltzmann equations. Since the barrier potential is a function of the equilibrium densities of the mobile charges at the junction faces and hence on the junction faces the electron
charge densities are nn • ND and np •n i2 /N A . Using these results, we obtain from eq. (10) as
VB =
kT N A N D
ln
e
n i2
Eq. (12) is the relation between the barrier potential and the impurity densities that create it.
(12)
78
Electronics—Theory and Applications
In Fig. 1.24, a small potential V is applied to the junction, with the positive terminal to p as a forward
voltage. A forward current appears as i to the p terminal. The depletion region is supplied with mobile charge
from the source V and, with electrons added to n and holes added to p, the junction barrier potential is reduced
to VB – V.
The PN-Diode Equation. One can derive a quantitative expression for the diode current from the barrier
potential and the n and p charge densities.
From equation (8), we have
De Dn kT
=
=
µe µn
e
(13)
In the open circuited p-n junction the current must be zero. Thus the expression for current flow, i.e.
Eq. (7) gives
µ
dn
e
E dx
= − e E dx = −
n
De
kT
(14)
One can integrate the above expression across the junction from x1 to x2, the transition region, as
nn
dn
e
∫ n = kT
np
x2
∫ ( − E ) dx
(15)
x1
The integration on the right hand side leads to the value of the barrier potential VB with n positive to p. The
Eq. (15) thus reduces to
nn
 eV 
= exp  B 
 kT 
np
(16)
Eq. (16) gives the ratio of electron densities at the junction (depletion region face) in the n region to that in the
p-region. Similarly, one obtains
pp
pn
 eV 
= exp  B 
 kT 
(17)
Equations (16) and (17) are Boltzmann’s equations. If a small potential V is applied to the junction with the
positive terminal to the p-type, i.e. as a forward bias, the junction barrier potential reduces to VB – V. The mobile charge from the source moves to the depletion region (electrons added to n and holes to p). The current far
from the junction is due to majority carriers. The diffusion at the junction is by minority carriers predominantly in the n-region. Thus under the forward bias condition, Eq. (17) for the hole density on the right face of
the junction (n-region) becomes
p n + ∆p n = p p e − e (V B −V ) / kT = ( p p e − eV B / kT )

 eV  
∆p n = p p e − eV B / kT exp   − 1
 kT  

(18)
Similarly, one can obtain the electron density at the left face of the junction (p-region) as
n p + ∆n p = n n e − e (V B −V ) / kT = [ n n e − eV B / kT ]e eV / kT
∴
∆n p = n n e − eV B / kT [ e eV / kT − 1]
(19)
Semiconductors and Junction Diodes
79
Using (2) and assuming the junction area A, one obtains the total current across the junction as
I = JA = Ae [ ∆n pVe + ∆p nVn ]
= Ae [ n nVe + p p v n ] e − eV B / kT × [ e eV / kT − 1]
(20)
= I s [ e eV / kT − 1]
(21)
Eq. (21) is known as the diode equation (plotted in Fig. 1.28). Eq. (21) shows that for V positive, i.e. forward bias, the current increases rapidly and for V negative, i.e. reverse bias, the current decreases to a limiting
value IS. This current is called the reverse bias saturation current. Is is small due to the low rate of pair generation at ambient temperature, but it is strongly temperature dependent. The ratio of forward to reverse current at
a given applied voltage is called rectification ratio.
REVIEW QUESTIONS
1. Explain the conduction in semiconductors by charge drift.
2. Explain the phenomenon of diffusion of current carriers in semiconductors. Define diffusion constant
and write its unit. write Einstein’s relation between mobility and diffusivity.
3. Write Fick’s law equation and define the diffusion constant.
4. Derive the equation
I = I S [ e eV / kT − 1]
for a PN diode. The symbols have usual meaning.