Download Physical Properties of Elements and Semiconductors

CHAPTER
1
Physical Properties of Elements
and Semiconductors
1.1 Introduction
Resistance (R)
Semiconductors constitute a large class of substances which
have resistivities lying between those of insulators and
conductors. The resistivity of semiconductors varies in 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
Temperature (T)
properties have been well investigated, are Germanium (Ge),
Silicon (Si) and Tellurium (Te). The study of their electrical Fig. 1.1 The temperature dependence
of the resistance in semiconductors.
properties reveals that
(i) Semiconductors have negative temperature Coefficient of resistance, i.e., the resistance
of semiconductor decreases with increase in temperature and vice versa as shown
Group
III
IV
B
C
V
VI
VII
Period
II
III
Si
P
S
IV
Ge
As
Se
V
Sn
Sb
Te
Fig. 1.2 Position of semiconductor elements in the Mendeleev Table.
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in Fig. 1.1. For example, Germanium is actually an insulator at low temperatures
but it becomes good conductor at high temperature.
(ii) The resistivity of semiconductors lies between that of a good insulator and of a metal
conductor, i.e., lying in the range 10–4 to 104 Ω-m.
(iii) The electrical conductivity of a semiconductor is very much affected when a suitable
impurity, e.g., Arsenic, Gallium, Indium etc. is added to it. This property of
semiconductors is most important.
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 etc. is fabricated on a single chip is
called an Integrated Circuit (IC). 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.
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 semiconductor device it may be one microwatt.
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.
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.
Physical Properties of Elements and Semiconductors
!
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
orbit around the nucleus. The electrons and the protons are the particles that have the electrical
properties. Neutrons have no electrical charge. An, atom has the same number of electrons
and protons, and so it is 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.
To understand the location and energy of each electron in an atom, one must have the
knowledge of following four quantum numbers:
(i) Principal Quantum Number (n). This characterises the average distance of an
electron from the nucleus and corresponds to the principal energy level in which electron
resides. Obviously, n gives some idea about the position of the electron around the nucleus. n
can have positive integer values starting from 1, i.e. n = 1, 2, 3,...... . The principal energy levels
or shells having different values of n are designated by the letters K, L, M, N and so on. The
maximum numbers of the electrons that can be accomodated in a shell corresponding to n is
equal to 2n2.
(ii) Azimuthal Quantum Number (l). This is also called as orbital angular
momentum quantum number and gives a measure of the angular momentum of an electron
in the orbit. Physically, l indicates the shape of the classical orbit. For a given value of n, l can
take all positive integer values from o to (n – l). The particular l value defines the subshell, and
the sub shells with l = 0, 1, 2, 3,...... are designated as s, p, d, f, g,...... respectively.
For a given principal quantum number, the energies of various subshells are in the
order of s < p < d < f ...... . Obviously, an electron in the s-sub shell has lower energy than in the
p sub shell with same n. The magnitude of this angular momentum if D = l( l + 1) .
(iii) Magnetic Quantum Number (ml). This is also called as orbital magnetic
number and this determines the preferred orientation of the orbitals in space with respect to
an applied magnetic field. We know that the magnetic moment of an electron due to its orbital
motion gives rise to a magnetic field which can interact with an external magnetic field. The
electrons orient themselves in certain preferred region of space around the nucleus under the
influence of the external field.
For a given value of l, ml can take integer values between – 1 to + 1 including 0, i.e. total
allowed values of ml are (2l + 1). The magnitude of the component of angular momentum along
the direction of the magnetic field is ml D .
(iv) Spin Quantum Number (ms). Electron is spinning about its own axis in the atom.
The spin of the electron produces a spin magnetic moment which can either parallel or anti
parallel to the surrounding magnetic field. For an electron there are two spin states. Spin
quantum number is concerned with the spinning of the electron about its own axis. Thus ms
can take only two possible states, +½ or – ½.
Pauli Exclusion Principle. This states that in an atom no two electrons can exist in
the same quantum state, i.e. in an atom there cannot be two electrons with the same value of
all the four quantum numbers. With the help of this principle, one can write the configuration
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of electrons. We may note that all the electrons with the same value of n constitute a shell and
a shell can have a maximum of 2n2 electrons.
The energy state capacity of different shells and subshells are as follows:
K-shell. This corresponds to n = 1 and this has one subshell, namely, 1s. This corresponds
to n = 1, l = 0, ml = 0, ms ± ½ and has 2 states.
L-shell. This corresponds to n = 2 and this has two subshells, namely, 2s and 2p.2s sub
shell corresponds to n = 2, l = 0, ml = 0 and has 2 states and 2p subshell correspond to n = 2,
l = 1, ml = 0, ± 1 and has 6 states.
M-shell. This corresponds to n = 3 and has 3s, 3p and 3d subshells. The 3s subshell
corresponds to n = 3, l = 1, and ml = 0, ± 1 and has 6 states. The 3d subshell corresponds to
n = 3, l = 2, ml = 0, ± 1, ± 2 and has 10 states.
Similarly, one can find subshells in other shells, e.g., N, etc. According to Pauli exclusion
principle, the electronic distribution in first four shells are shown in Table 1.1. We may note
that in all seven shells are required for all the chemical elements in the periodic table.
Table 1.1. Electron distribution in subshells
Shell
K
L
M
N
n
1
2
3
4
l
0
0
1
0
1
2
0
1
2
3
Subshell
s
s
p
s
p
d
s
p
d
f
No. of electrons
2
2
6
2
6
10
2
6
10
14
The electrons in the inner most shells are very strongly attached to the atom and hence
they cannot be easily separated. In other words, the electrons that are very near to the nucleus
are most tightly bound and hence have the lowest energy. Clearly, the atoms that have electrons
in the closed shells will have very stable configurations.
The atomic number Z gives the numbers of electrons orbiting about the nucleus and the
superscripts are used to designate the number of electrons in a particular subshell, e.g. Z for
silicon is 14 and its electronic configuration is given as 1s22s22p63s23p2. We may note that
silicon has only two electrons in the outer most subshell. Electronic configurations of C, Si, Ge,
Sn and Pb belonging to same subgroup (IV A) in the periodic table is given in Table 1.2.
Table 1.2. Electronic configuration of some elements of group IV A
Element
Atomic number
Electronic configuration
C
6
1s22s22p2
Si
14
1s22s22p63s 23p2
Ge
32
1s22s22p63s23p63d104s24p2
Sn
50
1s22s22p63s23p63d104s 24p64d105s25p2
Pb
82
1s22s22p63s23p63d104s24p64d104f 145s25p65d106s26p2
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Physical Properties of Elements and Semiconductors
Fig. 1.3 shows the representation 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 shells.
There are two electrons in the first shell, 2 electrons in the second shell and only 1 electron in
the third outermost or valence shell.
3rd shell
Proton
Electron
2nd shell
1st shell
Nucleus
1P
(b) Boron atom
(a) Hydrogen atom
H
(c) Silicon atom
B
Si
(5 )
(1 )
(d) Phosphorus atom
P
(15 )
(14 )
(e) Germanium atom
Ge
(32 )
Fig. 1.3 Atomic structure of a few atoms.
Fig. 1.3 (c) represents the structure of a Silicon atom. It contains 14 protons and 14
neutrons the nucleus. There are 14 electrons revolving around the nucleus in different shells.
There are 2 electrons in the first shell, 8 electrons in the second shell and 4 electrons in the
outermost or valence shell.
Fig. 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 shell called valence
electrons whereas Ge atom contains 4 electrons in the outermost shell.
The electrons in the inner shells of an atom do not normally leave the atom. But the
electrons which are in the outermost shell, so called valence shell do not always remain confined
to the same atom. Some of these valence electrons 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, that electrical conduction is possible.
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
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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 shell is more than half filled, then atom tends to fill its shell. So,
since 8 is the maximum electrons that can be held in the valence shell, 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.
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 (1 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.
En (eV)
0
n=
n=4
Brackett series
n=3
Paschen
series
n=2
n=1
Balmer
series
–0.85
–1.5
–3.4
–13.6
Lyman
series
Fig. 1.4 Energy level diagram of hydrogen.
In any atom, an electron orbiting very close to the nucleus in the first shell 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.
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Physical Properties of Elements and Semiconductors
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. Very soon,
the electron after emitting out energy in the form of heat, light or other radiations, fall back to
the original energy level.
1.3 Energy Band Theory
Most of the metals and semiconductors have crystalline structure. A crystal is made of a space
array of atoms of molecules in regular repetition in three dimensions of some fundamental
building block. The electronic energy levels for a gaseous element are the same as for a single
free atom because the individual atoms in a gas are well apart and has negligible influence on
each other. However, in a crystal the individual atoms are so closely packed that the resulting
energy levels gets modified due to interaction between the atoms in a crystal lattice. We may
note that when atoms form crystals, the energy, levels of the inner shell electrons are not
affected appreciably. On the otherhand, the energy levels of the outer-shell electrons are
considerably altered as these electrons are shared by the adjacent atoms in the crystal. As a
consequence of this interaction between the electrons of outer-shell, the energy levels spread
up to form a band of energy (Fig. 1.5).
Isolated
atoms
Energy
2N electrons
6N states
2N electrons
6N p states
2N electrons
2N s states
2N electrons
2N states
Lower energy levels remain
unaffected by crystal formation
a
Interatomic spacing
Fig. 1.5 Energy band formation in a crystal.
To make more clear the formation of energy bands in a crystal, we consider an example
of silicon crystal made up of N atoms. Let us assume that the interatomic spacing can be
varied without affecting the fundamental crystal structure. Let, the interaction between adjacent
atoms is negligible for very large interatomic spacing, say a (Fig. 1.5). Silicon has 14 electrons
and the outermost subshells, namely, 3s and 3p contains 2 electrons each. Thus in a silicon
crystal consisting of N atoms, the outermost subshells 3s and 3p consist of 2 N electrons each.
Obviously, the 3s subshell has 2 N electrons completely occupying the available 2 N states, and
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the 3p subshell has only 2 N electrons partially occupying the available 6 N states, all at the
same energy level.
Now, we consider that the interatomic spacing is gradually decreased, i.e. we are moving
from right to left (Fig. 1.5). This means that there will be a gradual increase in the interaction
between the neighbouring atoms. The atomic wave functions overlap due to this interaction.
This makes the crystal an electronic system which obeys the Pauli exclusion principle. Clearly,
2Ns states spread out to form a band of energy. Since N is very large (~ 1023/cm3) and separation
between energy levels is small, the total spread between the minimum and maximum energy
levels become large. This spread will have several volts of energy and termed as energy band.
The lower shaded region in Fig. 1.5 shows this. In this band the 2 N states are completely filled
with 2 N electrons. Similarly, we see that at the same value of interatomic spacing at p-level,
6 Np states spread up to form a band. The upper shaded region in Fig. 1.5 shows this band we
may note that there are 6 N states available in this band but only 2 N states are occupied and
remaining 4 N states remain unoccupied.
There exists an energy gap (Eg) between two energy bands, called as forbidden energy
gap, as no electrons can occupy states in this energy gap. With the decrease in atomic spacing
this forbidden energy gap decreases and becomes zero with further reduction in the interatomic
spacing, say at ‘b’ (Fig. 1.6). When the interatomic spacing is small enough, the two energy
bands will overlap. Under these circumstances, the 6 Np states in the upper band merge with
the 2 Ns states in the lower band, giving a total of 8 N states. Half of these 8 N states, i.e. 4 N
states are occupied by the 4 N available electrons. Now, these 4 N electrons no longer belong to
either p-subshell or s-subshell but belong to the entire crystal. Clearly, at this interatomic
spacing, each atom in the crystal can contribute 4 electrons to the crystal. The band occupied
by these contributed electrons is known as valence band.
Energy
Forbidden
energy gap
4N states
0 electrons
Conduction band
0
states
Crystal lattice
spacing
Eg
4N states
4N electrons
Valence band
c
b
Interatomic spacing
Fig. 1.6 Energy band in a crystal as a function of interatomic spacing.
Now, if the interatomic spacing is further reduced, say ‘c’ (Fig. 1.6), the interaction
between the atoms becomes very large, and the energy band structure becomes as shown in
Fig. 1.6. The exact shape of the energy band structure depends on : (i) the orientation of the
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Physical Properties of Elements and Semiconductors
atoms relative to each other in space, and (ii) the atomic number of the atom, which may be
obtained from the solution of Schrödinger’s wave equation. For spacing ‘c’, 4 N states in the
valence band are completely filled by 4 N electrons. This valence band is separated from the
upper unfilled or empty band by a forbidden energy gap (Eg). However, the forbidden energy
gap contains no allowed energy states. The upper energy band has 4 N unfilled energy states
and this band is called as the conduction band.
1.4 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.
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 in a bulk mass, 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. Similarly all third orbit electrons form
the third band and so on.
Energy (ev)
Conduction
band
Energy gap
3rd band
(Valence band)
2nd band
1st band
Fig. 1.5 Energy bands of silicon at absolute zero.
Silicon is a commonly used semiconductor having atomic number 14. It has 4 electrons
in its outermost or valence orbit. Here, 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
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accommodate a maximum of 8 electrons, the third energy band or valence band of a bulk
silicon material behaves as if completely filled. It is so, because in solid Silicon each atom
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 filling the valence band
completely as shown in Fig. 1.9 (for Germanium), 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 conduction 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 (Fig. 1.6). At
absolute zero temperature, the conduction band is empty, i.e., no electron has enough energy
to travel in 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.
1.5 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 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 them
Energy (ev)
Conduction
band
Conduction
band
Conduction
band
Eg 5 eV
or more
Valence
band
(a)
Conduction
band
Eg 1 eV
Valence
band
(b)
Valence
band
Valence
band
(c)
(d)
Fig. 1.6 Energy band diagram exhibiting energy band gap in (=) Insulators, (>) and
(?) Conductors, and (@) Semiconductors.