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Anchor Academic Publishing
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Fundamentals of
Semiconductor Physics
Mijoe Joseph
Joseph, Mijoe: Fundamentals of Semiconductor Physics, Hamburg, Anchor Academic
Publishing 2015
PDF-eBook-ISBN: 978-3-95489-919-7
Druck/Herstellung: Anchor Academic Publishing, Hamburg, 2015
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Contents
Chapter 1.
The Band Theory of Solids
6
1.1.
Structure of Atoms
7
1.1.1.
Hydrogen Atom
7
1.1.2.
Bohr's Postulates
7
1.1.3.
Atomic Radii of Orbits and Energy Levels
8
1.1.4.
Quantum Numbers
9
1.1.5.
Quantum Numbers as the Electron Address in an Atom
12
1.2.
Many-Electron Atoms
13
1.2.1.
Pauli's Exclusion Principle
13
1.2.2.
Distribution of Electrons over the Shells
13
1.3.
Degeneracy of Energy Levels in Free Atoms, Removal of
Degeneracy by External Effect
16
1.3.1.
Degenerate state
16
1.3.2.
Degeneracy, Removed by an External Field
16
1.4.
Formation of Energy Bands in Crystals
18
1.4.1.
Splitting of Energy Levels in a Crystal
18
1.4.2.
Allowed and Forbidden Bands
19
1.5.
Filling of Energy Bands by Electron
20
1.5.1.
Filled Levels Create Filled Bands While Empty Levels
Form Empty Bands
20
1.5.2.
Overlapping of Energy Bands in a Crystal
21
1.6.
Division of Solids into Conductors, Semiconductors, and
Dielectrics
22
1.6.1.
Conductors
23
1.6.2.
Semiconductors and Dielectrics
23
1.6.3.
Energy Band Occupancy and Conductivity of Crystals
24
Chapter 2.
Electrical Conductivity of Solids
25
2.1.
Bonding Forces in a Crystal Lattice
25
2.1.1.
Crystal as a System of Atoms in Stable Equilibrium State
25
2.1.2.
Repulsive and Attractive Forces
25
2.1.3.
Covalent bond
27
2.1.4.
Not Every Two Hydrogen Atoms May Form a Molecule
28
2.1.5.
Semiconductors as Typical Covalent Crystals
29
1
2.2.
Electrical Conductivity of Metals
30
2.3.
Conductivity of Semiconductors
33
2.4.
Intrinsic Semiconductors
33
2.4.1.
Electron Conductivity
34
2.4.2.
Hole Conductivity
36
2.4.3.
The Number of Holes Equals the Number of Free Electrons 37
2.5.
Doped (Impurity) Semiconductors
38
2.5.1.
Donor Impurities
38
2.5.2.
Hole Semiconductors
41
2.6.
Effect of Temperature on the Charge Carrier Concentration
in Semiconductors
42
2.6.1.
Effect of Temperature on Conductivity of Intrinsic
Semiconductors
43
2.6.2.
Impurity Semiconductors
44
2.6.3.
Degenerate Semiconductors
47
2.7.
Temperature Dependence of Electrical Conductivity of
Semiconductors
49
2.7.1.
Scattering by Ionized Impurity Atoms
50
2.7.2.
Scattering by Thermal Vibrations
52
2.7.3.
Temperature Dependence of Conductivity of
Semiconductors
53
Chapter 3.
Non equilibrium Processes in Semiconductors
55
3.1.
Generation and Recombination of Non equilibrium Charge
Carriers
55
3.1.1.
Free Carrier Generation
55
3.1.2.
Free Carrier Recombination
55
3.1.3.
Equilibrium Carriers
56
3.1.4.
Non equilibrium Carriers
56
3.1.5.
Electra-neutrality Condition
57
3.1.6.
Recombination Rate
58
3.1.7.
The Concept of Trapping Cross Section
59
3.1.8.
Types of Recombination
60
3.1.9.
Radiative Recombination
60
3.1.10.
Non radiative Recombination
61
3.1.11.
Trapping Sites and Trapping Centers
61
3.1.12.
Surface Recombination
62
3.2.
Diffusion Phenomena in Semiconductors
62
3.2.1.
Diffusion Current
62
3.2.2.
The Einstein Relation
64
3.2.3.
Holes Pursue Electrons
64
3.2.4.
Diffusion and Recombination
65
2
3.3.
Photo-conduction and Absorption of Light
66
3.3.1.
Photo-induced Processes
66
3.3.2.
Laws Governing Photo-conductivity
67
3.3.3.
Quantum Efficiency
69
3.3.4.
Photo-current Spectral Distribution Curve
70
3.3.5.
Intrinsic Photo-conductivity
71
3.3.6.
Impurity Photo-conductivity
72
3.3.7.
Absorption Spectrum
73
3.3.8.
Exciton Absorption
73
3.4.
Luminescence
74
3.4.1.
Photoluminescence
75
3.4.2.
Fluorescence
76
3.4.3.
Phosphorescence
77
Chapter 4.
Contact Phenomena
80
4.1.
Work Function of Metals
80
4.1.1.
Double Electric Layer
80
4.1.2.
Electrical Image Force
81
4.1.3.
Total Work Function
81
4.2.
The Fermi Level in Metals and the Fermi-Dirac Distribution
Function
83
4.2.1.
Fermi Level
83
4.2.2.
Fermi-Dirac Distribution Function
85
4.2.3.
Effect of Temperature
86
4.2.4.
Physical Meaning of the Fermi Level
87
4.3.
The Fermi Level in Semiconductors
88
4.3.1.
Intrinsic Semiconductor
89
4.3.2.
Impurity Semiconductors
89
4.3.3.
Effect of Temperature on the Position of the Fermi Level
90
4.3.4.
The Fermi Level in Degenerate Semiconductors
92
4.3.5.
The Fermi Level and Work Function in Semiconductors
93
4.4.
The Contact Potential Difference
95
4.5.
Metal-to-Semiconductor Contact
98
4.5.1.
State of Equilibrium
98
4.5.2.
Barrier Layer
100
4.5.3.
Bending of Bands
100
4.5.4.
Anti-barrier Layers
101
4.6.
Rectifier Properties of the Metal-Semiconductor Junction
101
4.6.1.
External Potential Difference Applied in the Forward
Direction
103
4.6.2.
External Potential Difference Acting in the Cut-Off
Direction
105
3
4.6.3.
Current-Voltage Characteristic of the Metal -
Semiconductor Junction
106
4.6.4.
Ohmic Contact
106
4.7.
p
- n Junction
107
4.7.1.
Methods of Obtaining p
- n Junctions
107
4.7.2.
p
- n Junction at Equilibrium
110
4.7.3.
Diffusion Current
111
4.7.4.
Contact potential Difference
112
4.7.5.
Band Structure of the p
- n Junction at Equilibrium
112
4.7.6.
p
- n Junction as a Barrier layer
115
4.8.
Rectifying Effect of the p
- n Junction
117
4.8.1.
Reverse Current
117
4.8.2.
Forward Current
119
4.8.3.
Injection of Carriers
120
4.8.4.
Current-Voltage Characteristic of the p
- n Junction
121
4.8.5.
Effect of Temperature on the Rectifier Properties of the
p
- n Junction
122
4.9.
Breakdown of the p
- n Junction
123
4.9.1.
Avalanche Breakdown
124
4.9.2.
Tunnel Breakdown
124
4.9.3.
Thermal Breakdown
125
4.9.4.
Surface Breakdown
126
4.10.
The Electric Capacitance of the p
- n Junction
126
Chapter 5.
Semiconductor Devices
128
5.1.
Hall Effect and Hall Pickups
128
5.1.1.
Lorentz Force
128
5.1.2.
Hall Effect in the Extrinsic Semiconductors
129
5.1.3.
Practical Application of the Hall Effect
131
5.1.4.
The Hall Effect in Semiconductors with Mixed
Conductivity
132
5.1.5.
Hall Effect in Intrinsic Semiconductors
133
5.1.6.
Hall Pickups-and Their Applications
134
5.1.7.
Hall Magnetometers
134
5.1.8.
Heavy-Current Ammeters
134
5.1.9.
Hall Pickups as Signal Transducers
135
5.1.10.
Hall Microphone
136
5.2.
Semiconductor Diodes
136
5.2.1.
Rectifier Diodes
136
5.2.2.
Stabilizer Diodes (Stabilitrons)
137
5.2.3.
Varicaps
139
5.3.
Tunnel Diodes
140
4
5.3.1.
Manufacturing of the Tunnel Diodes
140
5.3.2.
The p
- n Junction between Degenerate Semiconductors 141
5.3.3.
Tunnel Transitions of Electrons in Equilibrium State
141
5.3.4.
Operation of the Tunnel Diode at Forward Bias Voltage
143
5.3.5.
Operation of the Tunnel Diode at the Reverse Bias
145
5.3.6.
Generation of Continuous Oscillations with the Help of
the Tunnel Diode
145
5.3.7.
Backward Diode
147
5.4.
Transistors
149
5.4.1.
p
- n - p Transistor
149
5.4.2.
Injection of Holes to the Base
151
5.4.3.
Collector Junction
151
5.4.4.
How Does the Transistor Amplify?
153
5.4.5.
Methods of Connecting a Transistor
153
5.4.6.
Common Emitter Connection
153
5.5.
Semiconductor Injection (Diode) Lasers
155
5.5.1.
Photons Create Photons
155
5.5.2.
Population Inversion
156
5.5.3.
Creation of Population Inversion at the Junction Between
Degenerate Semiconductors
158
5.6.
Semiconductor at Present and in Future
160
5
CHAPTER 1
The Band Theory of Solids
When researchers first come across semiconductors, there was a
clear-cut division of all solids into two large groups, viz. conductors (in-
cluding all metals) and insulators (or dielectrics) and these differed in
principle in their properties. These new semiconductor materials could
not be included in either of these groups. On the one hand, they con-
ducted electric current, although to a much lesser extent than metallic
conductors, and on the other, they did not always conduct. Nevertheless,
they did conduct electricity and so were named semiconductors (or half
conductors).
Later, it was discovered that semiconductors differ from metals both
in the way they conduct and in the way external factors influence their
conduction. For example, the effect of temperature on conductivity of
metallic conductors and semiconductors is quite opposite. In metals, an
increase in temperature causes a gradual decrease in conductivity, while
the heating of semiconductors results in a sharp increase in conductivity.
The introduction of impurities also has different effects on conductivity
of metallic conductors and semiconductors. In metals, as a rule, impuri-
ties worsen conductivity, while in semiconductors the introduction of a
negligibly small amount of certain impurities can raise the conductivity
by tens or even hundreds of thousands of times.
Finally, if we send a beam of light of a flux of some particles onto a
conductor, it will have practically no effect on its conductivity. On the
other hand, irradiation or bombardment of a semiconductor causes a
drastic increase in its conductivity.
It is interesting to note that these properties of semiconductors are,
to a considerable extent, typical of dielectrics, hence it would be much
more correct to call semiconductors semi-insulators or semi dielectrics.
In order to explain the behavior of semiconductors in various condi-
tions, to account for their properties and to predict new effects, we must
consider their structural peculiarities. That is why we shall start with the
discussion of the atomic structure of matter.
6
1.1. Structure of Atoms
1.1.1. Hydrogen Atom. From the course of physics you should
know that an atom consists of a nucleus and electrons rotating around it.
This model of an atom was proposed by the English physicist Rutherford.
In l913, the Danish physicist Niels Bohr, one of the founders of quantum
mechanics, used this model for the first correct calculations of hydrogen
atom that agreed well with experimental data. His theory of the hydro-
gen atom has played an extremely important role in the development of
quantum mechanics, though it underwent considerable changes later.
1.1.2. Bohr's Postulates. According to the Rutherford-Bohr model,
hydrogen atom consists of a singly charged positive nucleus and one elec-
tron rotating around it. To a first approximation, it can be assumed that
the electron moves along the trajectory which is a circle with the fixed
nucleus at its center. According to the laws of classical electrodynamics,
any accelerated motion of a charged body (including the electron) must
be accompanied by the emission of electromagnetic waves. In the model
under consideration, the electron moves with a tremendous centripetal
acceleration, and therefore it should continuously emit light. Should it
do so, its energy would gradually decrease and the electron would come
closer and closer to the nucleus. Finally, the electron would unite with
the nucleus ("fall" on it). Nothing of this kind occurs in reality, and atoms
do not emit light in their unexcited state. In order to explain this fact,
Bohr formulated two postulates.
According to Bohr's first postulate, an electron can only be in an orbit
for which its angular momentum (i.e. the product of the electron mo-
mentum mv by the radius r of the orbit) is a multiple of h
/2 (where h
is Planck's constant). While the electron is in one of these orbits, it does
not emit energy. Each allowed electron orbit corresponds to a certain
energy, or certain energy state of the atom, which is called a stationary
state. Atoms do not emit light in a stationary state. The analytic expres-
sion of Bohr's first postulate is
mv r
= n
h
2
where n
= 1, 2, 3... is an integer called the principal quantum
number.
Bohr's second postulate states that absorption or emission of light by
an atom occurs during transitions of the atom from one stationary state
to another. The energy is absorbed or emitted upon transition in certain
amounts, called quanta, whose value hv is determined by the difference
in energies corresponding to the initial and final stationary states of the
7
F
IGURE
1
atom:
hv
= W
m
- W
n
where W
m
is the energy of the initial state of the atom, W
n
the energy
of its final state, and v the frequency of light emitted or absorbed by the
atom.
If W
m
> W
n
, the atom emits energy, and if W
m
< W
n
the energy is
absorbed. Quanta of light are called photons.
Thus, according to Bohr's theory, the electron in an atom cannot
change its trajectory gradually (continuously) but can only "jump" from
one stationary orbit to another. Light is emitted just when the electron
goes from a more distant stationary orbit to a nearer stationary orbit.
1.1.3. Atomic Radii of Orbits and Energy Levels. The radii of al-
lowed electron orbits can be found by using Coulomb's law, the relations
of classical mechanics, and Bohr's first postulate. They are given by the
following expression:
r
= n
2
h
2
4
2
me
2
The nearest to the nucleus allowed orbit is characterized by n
= 1.
Using the experimentally obtained values of m, e and h, we find for the
radius of this orbit
r
1
= 0.53 × 10
-8
cm
This value is taken for the radius of the hydrogen atom. Any other
orbit with a quantum number n has the radius
r
n
= n
2
r
1
Hence, the radii of successive electron orbits increase as n
2
(Fig. 1).
The total energy of an atom with an electron in the n
th
orbit is given
by the formula
W
n
= -
2
2
me
4
n
2
h
2
8
F
IGURE
2
These energy values are called atomic energy levels. If we plot the
possible values of energy of an atom along the vertical axis, we shall
obtain the energy spectrum of the allowed states of an atom (Fig. 2)
It can be seen that with increasing n, the separation between suc-
cessive energy levels rapidly decreases. This can be easily explained:
an increase in the energy of an atom (due to the energy absorbed by
the atom from outside) is accompanied by a transition of the electron to
more remote orbits where the interaction between the nucleus and the
electron becomes weaker. For this reason, a transition between neigh-
boring far orbits is associated with a very small change in energy. The
energy levels corresponding to remote orbits are so close that the spec-
trum becomes practically continuous. In the upper part the continuous
spectrum is bounded by the ionization level of the atom
( n = ) which
corresponds to the complete separation of the electron from the nucleus
(the electron becomes free).
The minus sign in the expression for the total energy of an atom in-
dicates that atomic energy is the lower the closer is the electron to the
nucleus. In order to remove the electron from the nucleus, we must ex-
pend a certain amount of energy, i.e. supply a definite amount of energy
to the atom from outside. For n
= , i.e. when the atom is ionized, the
energy of an atom is taken equal to zero. This is why negative values of
energy correspond to n
= . The level with n = 1 is characterized by
the minimum energy of the atom and the minimum radius of the allowed
electron orbit. This level is called the ground, or unexcited level. Levels
with n
= 2, 3, 4... are called excitation levels.
1.1.4. Quantum Numbers. According to Bohr's theory electrons
move in circular orbits. This theory provided good results only for the
simplest atom, viz. the hydrogen atom. But it could not provide quan-
titatively correct results even for the helium atom. The next step was
9
the planetary model of an atom. It was assumed that electrons, like the
planets of the solar system, move in elliptical orbits with the nucleus at
one of the foci. However, this model was also soon exhausted since it
failed to answer many questions.
This is connected with the fact that it is impossible in principle to de-
termine the nature of the motion of an electron in an atom. There are no
analogues of this motion in the macro-world accessible for observation.
We are not only unable to trace the motion of an electron but we cannot
even determine exactly its location at a particular instant of time. The
very concept of an orbit or the trajectory of the motion of an electron
in an atom has no physical meaning. It is impossible to establish any
regularity in the appearance of an electron at different points of space.
The electron is "smeared" in a certain region usually called the electron
cloud. For an unexcited atom, for example, this cloud has a spherical
shape, but its density is not uniform. The probability of detecting the
electron is highest near the spherical surface of radius r
1
, corresponding
to the radius of the first Bohr orbit. Henceforth, we shall assume that the
electron orbit is a locus of points which are characterized by the high-
est probability of detecting the electron or, in other words, the region of
space with the highest electron cloud density.
The electron cloud will be spherical only for the unexcited state of
the hydrogen atom for which the principal quantum number is n
= 1
(Fig. 3 a) .When n
= 2, the electron, in addition to a spherical cloud
whose size is now four times greater, may also form a dumb-bell-shaped
cloud (Fig. 3 b). The nonsphericity of the region of predominant electron
localization (electron cloud) is taken into account by introducing a sec-
ond quantum number l, called the orbital quantum number. Each value
of the principal quantum number n has corresponding positive integral
values of the quantum number l from zero to
(n - 1):
l
= 0, 1, 2, ...(n - 1)
For example, when n
= 1, l has a single value equal to zero. If n = 3,
l may assume the values 0, 1 and 2. For n
= 1 the only orbit is spherical,
therefore l
= 0. When n = 2, both the spherical and the dumb-bell-
shaped orbits are possible, hence l may be equal either to zero or unity.
For n
= 3, l = 0, 1, 2. The electron cloud corresponding to the value l = 2
has quite a complicated shape. However, we are not interested in the
shape of the electron cloud but in the energy of the atom corresponding
to it.
The energy of the hydrogen atom is only determined by the value of
the principal quantum number n and does not depend on the value of
the orbital number l. In other words, if n
= 3, the atom will have the
10
(
A
)
(
B
)
F
IGURE
3
energy W
3
, regardless of the shape of the electron orbit corresponding
to the given value of n and various possible values of l. This means that
upon a transition from the excitation level to the ground level, the atom
will emit photons whose energies are independent of the value of l.
While considering the spatial model of an atom, we must bear in
mind that electron clouds have definite orientations in it. The position
of an electron cloud in space relative to a certain selected direction is
defined by the magnetic quantum number m, which may assume integral
values from
-l to +l, including 0. For a given shape (a given value of
l), the electron cloud may have several different spatial orientations. For
l
= 1, there will be three, corresponding to the -1, 0 and +1 values of the
magnetic quantum number m. When l
= 2, there will be five different
orientations of the electron cloud corresponding to m
= -2, -1, 0, 1 and
+2. Since the shape of the electron cloud in a free hydrogen atom does
not influence the energy of the atom, the more so it applies to the spatial
orientation.
Finally, a more detailed analysis of experimental results revealed that
electrons in the orbits may themselves be in two different states deter-
mined by the direction of the electron spin. But what is the electron
spin?
In 1925, English physicists G. Uhlenbeck and S. Goudsmit put for-
ward a hypothesis to explain the fine structure of the optical spectra of
some elements. They suggested that each electron rotates about its axis
like a top or a spin. In this rotation the electron acquires an angular mo-
mentum called the spin. Since the rotation can be either clockwise or
anticlockwise, the spin (in other words, the angular momentum vector)
may have two directions. In h
/2 units, the spin is equal to 1/2 and
has either "
+ " or " - " sign depending on the direction. Thus, the elec-
tron orientation in the orbit is determined by the spin quantum number
equal to ±1/2. It should be noted that the spin orientation like the
11
orientation of the electron orbit, does not affect the energy of hydrogen
atom in a free state.
Subsequent investigations and calculations have shown that it is im-
possible to explain the electron spin simply by its rotation about the axis.
When the angular velocity of the electron was calculated, it was found
that the linear, velocity of points on the electron equator (if we assume
that the electron has the spherical shape) would be higher than the veloc-
ity of light, which is impossible. The spin is an inseparable characteristic
of the electron like its mass or charge.
1.1.5. Quantum Numbers as the Electron Address in an Atom.
Thus, we have learned that ,in order to describe the motion of the elec-
tron in an atom or, as physicists say, to define the state of an electron in
an atom, we must define, a set of four quantum numbers: n, l, m, and
.
Roughly speaking, the principal quantum, number n defines the size
of the electron orbit, The larger n, the greater region of space is embraced
by the corresponding electron cloud. By setting the value of n, we define
the number of the electron shell of the atom. The number n, itself can
acquire any integral value from 1 to
n
= 1, 2, 3, ...
The orbital quantum number l define the shape of the electron cloud.
From the entire set of orbits corresponding to the same value of n., the
orbital number l selects those having the same shape. To each value of
l there corresponds its own sub shell. The number of sub shells is equal
to n, since l may acquire the values from 0 to
(n - 1):
l
= 0, 1, 2...(n - 1)
The magnetic quantum number m defines the spatial orientation of
the orbit in group of orbits with the same shape, i.e. belonging to the
same sub shell. In each sub shell, there are
(2l + 1) orbits with different
orientations, since m may assume the values from 0 to
±l:
m
= -l, -(l - 1), ... - 1, 0, +1, ...., +(l - 1), +l
Finally, the spin quantum number
defines the orientation of the
electron spin in the given orbit. Spin has only two values:
= ±1/2
While considering the hydrogen atom and using the concepts of
"shell", "sub shell" and "orbit", we spoke about the opportunities avail-
able to the single electron in this atom rather than about the atomic
structure. The electron in the hydrogen atom may go from one shell to
another and from orbit to orbit within the same shell.
12
The pattern of electron distribution in many electron atoms and their
possible transitions are much more complicated.
1.2. Many-Electron Atoms
1.2.1. Pauli's Exclusion Principle. In discussing the structure of
many-electron atoms, we must consider a very important principle for-
mulated in 1925 by the Swiss physicist W. Pauli. This principle states
that there cannot be two electrons in an atom in the same quantum state
described by the set of four quantum numbers (n, l, m, and
). In other
words, only one or two electrons may be simultaneously in any station-
ary orbit in an atom. In the latter case, the spins of the electrons must
have opposite directions, i.e. for one electron
= +1/2, while for the
other
= -1/2.
Taking into account the Pauli exclusion principle and knowing the
number of stationary orbits characterized by different quantum numbers,
we can determine the possible number of electrons in every atomic shell
and sub shell (see Table 1)
1.2.2. Distribution of Electrons over the Shells. The first shell,
whose principal quantum number is n
= 1, does not split into sub-shells,
since it has only one quantum number l associated with it and this is
equal to zero. In this case m
= 0 as well, so we may conclude that the
first shell consists of only one orbit which can be occupied, according to
Pauli's exclusion principle, by only two electrons.
The second shell
(n = 2) consists of two sub shells since l can be
either 0 or 1. In atomic physics, letter symbols instead of the numerical
values of l are used for describing sub shells. For example, regardless
of the value of the principal. quantum number n, all sub shells with
l
= 0 are denoted by s, sub shells with l = 1 are denoted by p, for l = 2
the symbol d is used, and so on. In this connection, it is said that the
second shell consists of the s
- and p-sub shells. The s-sub shell (l = 0)
consists of one circular orbit and may contain only two electrons, while
the p
-sub shell consists of three orbits (m may be equal to -1, 0, and
+1) and may contain six electrons. The total number of electrons in the
second shell is equal to eight.
Similarly, we can calculate the possible number of electrons in any
shell and sub shell. For example, there can be 10 electrons in the 3d
-
sub shell (n
= 3, 1 = 2), viz. two electrons in each of the five orbits char-
acterized by different values of the quantum number m. The maximum
number of electrons in any sub shell is equal to 2
(2l+1). In spectroscopy,
letter symbols (terms) are ascribed to different shells: the first shell is de-
noted by K, the second by L, the third by M , and so on.
13
Quantum
Numbers
Notation
for
a
sub
shell
level
No
of
orbits
in
a
sub
shell
No
of
electrons
in
a
level
T
otal
no
of
electrons
in
a
shell
n
m
l
1
0
0
±
1
2
1
s
1
2
2
2
0
0
±
1
2
2
s
1
2
6
1
-
1,
0,
+
1
±
1
2
2
p
3
6
3
0
0
±
1
2
3
s
1
2
18
1
-
1,
0,
+
1
±
1
2
3
p
3
6
2
-
2,
-
1,
0,
+
1,
+
2
±
1
2
3
d
5
10
T
ABLE
1
14
F
IGURE
4
The single electron in a hydrogen atom is in a centrally symmetric
field of the atomic nucleus; its energy is determined solely by the value
of the principal quantum number n and does not depend on the values of
the other quantum numbers. On the other hand, in many-electron atoms
each electron is in the field created both by the nucleus and by the other
electrons.
Consequently, the energy of an electron in many-electron atoms turns
out to depend both on the principal quantum number n and on the orbital
number l , though remaining independent of the values of m and
.
This feature of many-electron atoms leads to considerable differences
between their energy spectrum and the spectrum of hydrogen atom.
Fig.4 shows a part of the spectrum for many-electron atom (the energy
levels of the first three atomic shells). Dark circles on the levels indicate
the maximum number of electrons which can occupy the corresponding
sub shell.
It is well known that a system not subjected to external effect tends
to go into the state with the lowest energy. Atom is not an exception in
this respect. As the atomic shells are filled, the electrons tend to occupy
the lowest levels and would all occupy the first level if there were no
limitations imposed by Pauli's exclusion principle. The only electron in
the hydrogen atom occupies the lowest orbit belonging to the 1s- level.
In the helium atom, the same orbit contains also the second electron,
and the first atomic shell is filled. It should be noted that helium is an
inert gas, and its great stability is due to the complete outer shell.
15
In the lithium atom, there are only three electrons. Two of them
occupy the first shell, and the third is in the second shell with n
= 2 (it
cannot occupy the first shell due to Pauli's exclusion principle). Lithium is
an alkali metal whose valency equals unity. This means that the electron
in the second shell is weakly bound to the atomic core and can be easily
detached from it. This can be judged from the ionization potential which
for lithium is only equal to 5.37 V , while for helium it is equal to 24.45 V.
As the number of electrons in an atom increases, the outer sub shells
and shells are filled. For example, starting with boron, which has 5 elec-
trons, the 2p-sub shell is filled. This process is completed in inert gas
neon which has a fully filled second shell and is thus characterized by
the great stability. The eleventh electron in the sodium atom starts pop-
ulating the third shell (3s
-sub shell), and so on.
1.3. Degeneracy of Energy Levels in Free Atoms, Removal of
Degeneracy by External Effect
1.3.1. Degenerate state. .We have already noted that in many-
electron atoms the energy of electrons is only determined by the values
of the quantum numbers n and l and does not depend on the values of m
and
. This can be illustrated by the energy spectrum shown in Fig. 4.
Indeed, all six electrons in the 3p-sub-shell, for example, have the same
energy, although they have different values of m and
. States described
by different sets of quantum numbers but having the same energy are
called degenerate. Similarly, the energy levels corresponding to these
states are also called degenerate. The levels are degenerate while the
atoms are in the free state. If however, the atoms are placed in a strong
magnetic or electric field, the degeneracy is partially or completely re-
moved. Let us illustrate this removal of degeneracy with respect to the
quantum number m.
1.3.2. Degeneracy, Removed by an External Field. Different val-
ues of the quantum number m correspond to different spatial orienta-
tions of similar electron orbits. In the absence of an external field, dif-
ferent orientations of the orbits do not affect the energy of the electrons.
If however, we place an atom in an external field, the field will act differ-
ently on the electrons in orbits oriented in different ways with respect to
the direction of this field. As a result, changes in energies of electrons in
similarly shaped but differently oriented orbits will be different both in
magnitude and in sign: energies of some. electrons will increase while
those of others will decrease. The energy levels for different electrons in
the spectrum will also change their arrangement. Moreover, instead of
one energy level corresponding to all electrons in similar orbits several
16
F
IGURE
5
sub levels appear in the spectrum, the number of sub level being equal
to the number of differently oriented similar orbits, i.e. to the number
of possible values of the quantum number m. Fig. 5 shows the result
of an external electric field acting on the 3d
-level, for which n = 3 and
l
= 2. It can be seen that splitting of the level into sub levels and the
displacement of sub levels occur simultaneously.
The process in which previously indistinguishable (from the point
of view of energy) degenerate levels become distinguishable is called
the removal of degeneracy. Let us illustrate degeneracy removal with
another example.
We consider an electron having a certain energy W
0
in a one-
dimensional space characterized by the coordinate x (Fig. 6). In the
absence of an external field, the state of this electron is described by
one energy level W
0
irrespective of the direction of its motion. In other
words, in the absence of an external field the energy level W
0
is doubly
degenerate. If we apply an external electric field, say, along the x-axis,
the energy of the electron becomes dependent on the direction of its mo-
tion. If the electron moves along the x-axis, it will be decelerated by the
external field, its energy becoming W
0
- eE x (where x is the distance
covered by the electron).
If the electron moves in the opposite direction, its energy becomes
W
0
+ eE x. Correspondingly, the appearance of two different states is
manifested in the energy spectrum by the slitting of the degenerate level
W
0
into two non degenerate levels W
0
- eE x and W
0
+ eE x. In other
words, the degeneracy is removed under the effect of the external field,
17
F
IGURE
6
1.4. Formation of Energy Bands in Crystals
1.4.1. Splitting of Energy Levels in a Crystal. Let us do the fol-
lowing mental experiment. Take N atoms of a substance and arrange
them at a sufficiently large distance from each other but in such a way
that this arrangement reproduces the crystalline structure of the mate-
rial. Since the separation between the atoms is large, we can ignore
their interaction and consider them free. In each of these atoms, there
are degenerate levels with degeneracies equal to the number of differ-
ently oriented similar orbits in corresponding sub shells. Let us now
start bringing the atoms closer, retaining their mutual arrangement. As
the atoms converge, come closer, they begin to experience the influence
of their approaching neighbors, which is similar to the influence of an ex-
ternal electric field. The smaller the separation between the atoms, the
stronger is the interaction between them. Owing to this interaction, de-
generacy of the energy levels characterizing the free atoms is removed:
each degenerate level splits into
(2l + 1) non degenerate levels. All the
atoms in a crystal generally exist under the same conditions (except for
those which form the external boundary of the crystal). It could seem
therefore that each atom should contribute the same set of non degen-
erate sub levels into the energy spectrum that characterized the crystal
as a whole viz. one 1s- sub level, three 2p-sub levels, five 3d-sub lev-
els, and so on. Each sub level may contain two electrons with opposite
spins. Although this splitting actually occurs, the corresponding sub lev-
els obtained from similar atomic levels differ from each other in energy,
some of them are higher in the energy spectrum of the crystal than the
18
F
IGURE
7
initial levels of the individual atoms, while others lie somewhat lower.
This difference can be explained by Pauli's exclusion principle general-
ized for the entire crystal as a single entity. According to this principle,
no two non degenerate sub levels in a crystal may have the same energy.
Therefore, when the crystal is formed, each energy level spreads into an
energy band consisting of N
(2l + 1) non degenerate sub levels differing
in energy. For example, the 1s-level spreads into 1s-band consisting of
N sub levels which may contain 2N electrons, the 2p-level spreads into
2p-band consisting of 3N sub levels which may contain 6N electrons,
and so on.
The formation of energy band in a crystal from discrete energy levels
of individual atoms is shown schematically in Fig. 7. The shorter the
distance r, the stronger the effect of the neighboring atoms and the more
the levels are "smeared". The energy spectrum of a crystal is determined
by the smearing of the levels corresponding to the inter atomic distance
a
0
, typical of a given crystal.
The degree of smearing of levels depends on their depth in an atom.
The inner electrons are strongly coupled to their nuclei and are screened
from external effects by the outer electron shells. Therefore the corre-
sponding energy levels are weakly smeared. Naturally, the electrons in
the outer shells are most strongly affected by the field of the crystal lat-
tice, and the energy levels corresponding to them are smeared the most.
It should be noted that smearing of levels into energy bands does not de-
pend on whether there are electrons on these levels or whether they are
empty. In the latter case, the smearing of levels indicates the broadening
of the range of possible energies which the electron may acquire in the
crystal.
1.4.2. Allowed and Forbidden Bands. From what has been said
above, it follows that there is an entire band of allowed energy values
19
corresponding to each allowed energy level in a crystal, i.e. there is
an allowed band. Allowed bands alternate with the bands of forbidden
energy, or forbidden bands. Electrons in a pure crystal cannot have an
energy lying in the forbidden bands. The higher the allowed atomic level
on the energy scale, the more the corresponding band is smeared. As the
energy increases, the forbidden bands becomes narrower.
The separation of sub levels in an allowed band is very small. In real
crystals ranging from 1 to 100 cm
3
in size, the sub levels are separated
by 10
-22
to 10
-24
eV . This difference in energy is so small that the bands
are considered to be continuous. Nevertheless, the fact that sub levels in
the bands are discrete and the number of sub levels in the band is always
finite plays a decisive role in crystal physics, since depending on the fill-
ing of the bands by electrons, all solids can be divided into conductors,
semiconductors, and dielectrics.
1.5. Filling of Energy Bands by Electron
1.5.1. Filled Levels Create Filled Bands While Empty Levels Form
Empty Bands. Since the energy bands in solids are formed from the
levels of individual atoms, it is quite obvious that their filling by elec-
trons will be determined above all by the occupancy of the corresponding
atomic levels by electrons.
Let us consider by way of an example the lithium crystal. In the free
state, the lithium atom has three electrons. Two of these are in the 1s-
shell, which is thus completed. The third electron belongs to the 2s
-sub
shell, which is half-filled. Consequently, when a crystal is formed, the1s-
band turns out to be filled completely, the 2s- band is half-filled, while
the 2p-, 3s-, 3p-, etc. bands in an unexcited lithium crystal are empty,
since the levels from which they are formed are unoccupied.
The same is true for all alkali metals. For example, when a sodium
crystal is formed, the 1s
-, 2s-, and 2p- bands are completely filled,
since the corresponding levels in sodium atoms are completely packed by
electrons (two electrons in the 1s
-level, two electrons in the 2s- level,
and six electrons in the 2p
- level). The eleventh electron in the sodium
atom only half-fills the 3s
- level, hence the 3s- band too is half-filled
with electrons.
When crystals are formed by atoms with completely filled levels, the
created bands in general are also filled completely. For example, if we
constructed a crystal from neon atoms, the 1s
- , 2s- , and 2p- bands in
the energy spectrum of such a crystal would be completely filled (each
neon atom has 10 electrons which fill the corresponding energy levels).
20
F
IGURE
8
The remaining upper-lying bands (3s, 3p, etc.) would turn out to be
empty.
1.5.2. Overlapping of Energy Bands in a Crystal. In some cases
the problem of filling the energy bands by electrons is more complicated.
This refers to crystals of rare-earth elements and those with a diamond-
type lattice, among which the most interesting for us are the crystals of
typical semiconductors, viz. germanium and silicon.
At a first glance, the crystals of rare-earth elements must only have
completely filled and empty bands in their energy spectrum. Indeed, the
beryllium atoms, for example, which have four electrons each, are char-
acterized by two completely filled levels, 1s and 2s levels. In magnesium
atom, which has 12 electrons, the levels 1s, 2s, 2p and 3s are also com-
pleted. However, the upper energy bands in crystals of the rare-earth
elements, which are created by completely filled atomic levels, are in
fact only partially filled. This can be explained by the fact that the en-
ergy bands corresponding to the upper levels are smeared so much in the
process of crystal formation that the bands overlap. As a result of this
overlapping, hybrid bands are formed, which incorporate both filled and
empty levels. For example, a hybrid band in a beryllium crystal is formed
by the completed 2s-levels and the empty 2p-levels (Fig. 8), while in the
magnesium crystal, by the filled 3s-levels and empty 3p-levels. It is due
to the overlapping that the upper energy bands in rare-earth crystals are
filled only partially.
21
In semiconductor crystals with diamond-type lattices band overlap-
ping leads to quite the opposite result. In silicon atoms, for example,
the 3p-level (3p-sub-shell) contains only two electrons, though this level
may be occupied by six electrons. It is natural to expect that during the
formation of a silicon crystal, the upper energy band (the 3p-band) will
only be filled partially, while the preceding band (the 3s-band) will be
filled completely (since it is formed by the completely filled 3s-level). Ac-
tually the overlapping during the formation of a silicon crystal not only
leads to the appearance of a hybrid bands composed of the 3s- and 3p-
sub-levels, but also to a further splitting of the hybrid band into two sub
bands separated by the forbidden energy gap W
g
(Fig. 9). In all, the
3s
+ 3p hybrid band must have 8 electron vacancies per atom (2 vacan-
cies in the 3s-sub-shell and 6 in the 3p-sub-shell). After the splitting of
the hybrid band,4 vacancies per atom are found to be in each sub band.
Trying to occupy the lower energy levels, electrons of the third shells of
silicon atoms (there are four of them-two in the 3s-sub-shell and two in
the 3p-sub-shell) just fill the lower sub band, leaving the upper sub band
empty.
1.6. Division of Solids into Conductors, Semiconductors, and
Dielectrics
Physical properties of solids, and first of all their electric properties,
are determined by the degree of filling of the energy bands rather than
by the process of their formation. From this point of view all crystalline
bodies can be divided into two quite different groups.
F
IGURE
9
22
(
A
)
(
B
)
(
C
)
F
IGURE
10
1.6.1. Conductors. The first group includes substances having a
partially-filled band in their energy spectrum above the completely filled
energy bands (Fig. 10 a). As was mentioned above, a partially filled
band is observed in alkali metals whose upper band is formed by un-
filled atomic levels, and in alkali-earth crystals with a hybrid upper band
formed as a result of the overlapping of filled and empty bands. All sub-
stances belonging to the first group are conductors.
1.6.2. Semiconductors and Dielectrics. The second group com-
prises substances with absolutely empty bands above completely filled
bands (Fig. 10 b,c). This group also includes crystals with diamond-
type structures, such as silicon, germanium, gray tin, and diamond it-
self. Many chemical compounds also belong to this group, for example,
metal oxides, carbides, metal nitrides, corundum (Al
2
So
3
) and others.
The second group of solids includes semiconductors and dielectrics.
The uppermost filled band in this group of crystals is called the va-
lence band and the first empty band above it, the conduction band. The
upper level of the valence band is called the top of the valence band and
denoted by W
v
. The lowest level of the conduction band is called the
bottom of the conduction band and denoted by W
c
.
In principle, there is no difference between semiconductors and di-
electrics. The division in the second group into semiconductors and di-
electrics is quite arbitrary and is determined by the width W
g
of the for-
bidden energy gap separating the completely filled band from the empty
band. Substances with forbidden band widths W
g
2eV belong to the
semiconductor subgroup.
23
Germanium (W
g
0.7eV ), silicon (W
g
1.2eV ) gallium arsenide
GaAs (W
g
1.5eV ), and indium antimonide I nS b (W
g
0.2eV ) are typ-
ical semiconductors.
Substances for which W g
> 3eV belong to dielectrics. Well-known
dielectrics include corundum (W
g
7eV ), diamond (W
g
> 5eV ), boron
nitride (W
g
4.5eV ), and others.
The arbitrary nature of the division of second group solids into di-
electrics and semiconductors is illustrated by the fact that many generally
known dielectrics are now used as semiconductors. For example, silicon
carbide with its forbidden band width of about 3eV is now used in semi-
conductor devices. Even such a classical dielectric as diamond is being
investigated for a possible application in semiconductor technology.
1.6.3. Energy Band Occupancy and Conductivity of Crystals. Let
us consider the properties of a crystal with the partially filled upper band
at absolute zero
(T = 0). Under these conditions and in the absence of
an external electric field, all the electrons will occupy the lowest energy
levels in the band, with two electrons in a level, in accordance with Pauli's
exclusion principle.
Let us now place the crystal in an external electric field with intensity
E. The field acts on each electron with a force F
= -eE and accelerates
it. As a result, the electron's energy increases, and it will be able to go to
higher energy levels. These transitions are quite possible, since there are
many free energy levels in the partially filled band. The separation be-
tween energy levels is very small, therefore even extremely weak electric
fields can cause electron transitions to upper-lying levels. Consequently,
an external field in solids with a partially filled band accelerates the elec-
trons in the direction of the field, which means that an electric current
appears. Such solids are called conductors.
Unlike conductors, substances with only completely filled or empty
bands cannot conduct electric current at absolute zero. In such solids,
an external field cannot create a directional motion of electrons. An ad-
ditional energy acquired by an electron due to the field would mean its
transition to a higher energy level. However, all the levels in the va-
lence band are filled. On the other hand, there are many vacancies in
the empty conduction band but there are no electrons. Common elec-
tric fields cannot impart sufficient energy for electron to transfer from
the valence band to the conduction band (here we do not consider fields
which cause dielectric breakdown). For all these reasons an external
field at absolute zero cannot induce an electric current even in semicon-
ductors. Thus, at this temperature a semiconductor does not differ at all
from a dielectric with respect to electrical conductivity.
24
CHAPTER 2
Electrical Conductivity of Solids
2.1. Bonding Forces in a Crystal Lattice
2.1.1. Crystal as a System of Atoms in Stable Equilibrium State.
How is a strictly ordered crystal lattice formed from individual atoms?
Why cannot atoms approach one another indefinitely in the process of
formation of a crystal? What determines a crystal's strength?
In order to answer these questions, we must assume that there are
forces of attraction F
at
and repulsive forces F
r ep
which act between atoms
and which attain equilibrium during the formation of a crystal structure.
Irrespective of the nature of these forces, their dependence on the inter-
atomic distance turns out to be the same (Fig. 11 a). At the distance r
>
a
0
attractive forces prevail; while for r
< a
0
, repulsive forces dominate.
At a certain distance r
= a
0
which is quite definite for a given crystal,
the attractive and repulsive forces balance each other, and the resultant
force F
r
(which is depicted by curve 3) becomes zero. In this case, the
energy of interaction between particles attains the minimum value W
0r
(Fig. 11 b ). Since the interaction energy is at its minimum at r
= a
0
atoms remain in this position (in the absence of external excitation),
because removal from each other, as well as any further approach, leads
to an increase in the energy of interaction. This means that at r
= a
0
,
the system of atoms under consideration is in stable equilibrium. This
is the state which corresponds to the formation of a solid with a strictly
definite structure, viz. a crystal.
2.1.2. Repulsive and Attractive Forces. Curve 2 in ( Fig. 11 a)
shows that repulsive forces rapidly increase with decreasing distance r
between the atoms. Large amounts of energy are required in order to
overcome these forces. For example, when the distance between a pro-
ton and a hydrogen atom is decreased from r
= 2a to r = a/2 (where
a is the radius of the first Bohr orbit), the energy of repulsion increases
300 times. For light atoms whose nuclei are weakly screened by elec-
tron shells, the repulsion is primarily caused by the interaction between
nuclei. On the other hand, when many-electron atoms get closer, the re-
pulsion is explained by the interaction of the inner, filled electron shells.
25
(
A
)
(
B
)
F
IGURE
11
26
The repulsion in this case is not only due to the similar charge of the
electron shells but also due to rearrangement of the electron shells. At
very small distances, the electron shells should overlap, and orbits com-
mon to two atoms will appear. However, since the inner, filled orbits
have no vacancies, and extra electrons cannot appear in them due to
the Pauli exclusion principle, some of these electrons must go to higher
shells. Such a transition is associated with an increase in the total energy
of the system, which explains the appearance of repulsive forces.
Obviously, the nature of repulsive forces is the same for all atoms
and does not depend on the structure of outer, unfilled shells. On the
contrary, forces of attraction which act between atoms are much more
diverse in nature, which is determined by the structure and degree of
filling of the outer electron shells. Bonding forces acting between atoms
are determined by the nature of attractive forces. When considering the
structure of crystals, the most important bonds are the ionic, covalent,
and metallic, and these should be well known to you from the course
of chemistry. Here, we shall only consider the covalent bond, which
determines the basic properties of semiconductor crystals.
2.1.3. Covalent bond. Covalent bond is the main one in the forma-
tion of molecules or crystals from identical or similar atoms. Naturally,
during the interaction of identical atoms, neither electron transfer from
one atom to another nor the formation of ions takes place. The redistri-
bution of electrons, however, is very important in this case as well. The
process is completed not by the transfer of an electron from one atom
to another but by the collectivization of some electrons: these electrons
simultaneously belong to several atoms.
Let us see how the covalent bond is formed in the molecule of hy-
drogen, H
2
. Whilst the two hydrogen atoms are far apart, each of them
" possesses" its own electron, and the probability of detecting "foreign"
electrons within the limits of a given atom is negligibly small. For exam-
ple, when the distance between the atoms is r
= 5nm, an electron may
appear in the neighboring atom once in 10
12
y ears. As the atoms come
closer, the probability of "foreign" electron appearing sharply increases.
For r
= 0.2nm, the transition frequency reaches 10
12
sec
-1
, and at a fur-
ther approaching the frequency of electron exchange becomes so high
that "own" and "foreign" electrons appear at the same frequency near
the two nuclei. In this case, the electrons are said to be collectivized.
The bond based on the "joint possession" of two electrons by two atoms
is called the covalent bond.
The formation of the covalent bond must naturally be advantageous
from the point of view of energy. Curve 1 in Fig. 12 shows the total
27
energy of two hydrogen atoms as a function of their separation (the total
energy of two infinitely remote hydrogen atoms not interacting with one
another is taken as the zero level). It can be seen that as atoms approach
one another, their total energy decreases and attains its minimum value
at r
= a
0
which corresponds to the separation of atoms in a hydrogen
molecule.
F
IGURE
12
2.1.4. Not Every Two Hydrogen Atoms May Form a Molecule.
The above process of the formation of the hydrogen molecule is pos-
sible only when the electrons in atoms being combined have opposite
spins. Only in this case may the electrons occupy the same electron or-
bit, which is a common orbit for the combined atoms. If, however, the
electrons of the approaching hydrogen atoms have parallel spins, i.e. are
in the same state determined by the same set of four quantum numbers
(n, 1, m, and
), then according to Pauli's exclusion principle they cannot
occupy the same orbit. Such atoms will be repelled rather than attracted
to one another, and their total energy will increase and not decrease with
decreasing r (curve 2 in Fig. 12). Obviously, a molecule is not formed in
this case.
An important feature of covalent bonds is their saturation. This prop-
erty is also one of the manifestations of Pauli's exclusion principle and
consists in the impossibility of a third electron to take part in the for-
mation of the covalent bond. Having combined into the covalent bond,
electrons are unable to form another bond and at the same time they
"do not allow" other electrons to penetrate the combined orbit. For this
28
reason hydrogen molecule, for example, consists of two atoms, and the
H
3
molecule cannot be formed.
2.1.5. Semiconductors
as
Typical
Covalent
Crystals. The
diamond-type crystals exhibit the most typical features of covalent
bonds. Typical representatives of this group are semiconductor crystals
and among them the well known germanium and silicon. Atoms of these
elements have four electrons in the outer shell, each of which forms a
covalent bond with four nearest neighbors (Fig. 13). In this process,
each atom gives its neighbor one of its valence electrons for "partial
possession" and simultaneously gets an electron from the neighbour on
the same basis.
F
IGURE
13
As a result, every atom forming the crystal "fills up" its outer shell to
complete the population (8 electrons), thus forming a stable structure,
which is similar to that of the inert gases (in Fig. 14, these 8 electrons are
conventionally placed on the circular orbit shown by the dashed curve).
Since the electrons are indistinguishable, and the atoms can exchange
electrons, all the valence electrons belong to all the atoms of the crystal
to the same extent. A semiconductor crystal thus can be treated as a
single giant molecule with the atoms joined together by covalent bonds.
Conventionally, these crystals are depicted by a plane structure (Fig. 14).
where each double line between atoms shows a covalent bond formed
by two electrons.
29
F
IGURE
14
2.2. Electrical Conductivity of Metals
The best account of this phenomenon is given by the quantum theory
of solids. But to elucidate the general aspects, we can limit ourselves to
a consideration based on the classical electron theory. According to this
theory, electrons in a crystal can, to a certain approximation, be identified
with an ideal gas by assuming that the motion of electrons obeys the
laws of classical mechanics. The interaction between electrons is thus
completely ignored, while the interaction between electrons and ions of
the crystal lattice is reduced to ordinary elastic collisions.
Metals contain a tremendous number of free electrons moving in the
interstitial space of a crystal. There are about 10
23
atoms in 1 cm
3
of a
crystal. Hence, if the valence of a metal is Z, the concentration (number
density) n of free electrons (also called conduction electrons) is equal to
Z
× 10
23
cm
-3
.They are all in random thermal motion and travel through
the crystal at a very high velocity whose mean value amounts to 10
8
cm
/sec. Due to the random nature of this thermal motion, the number
of electrons moving in any direction is on the average always equal to
the number of electrons moving in the opposite direction, hence in the
absence of an external field the electric charge carried by electrons is
zero. Under the action of an external field, each electron acquires an
additional velocity and so all the free electrons in the metal move in
the direction opposite to the direction of the applied field intensity. The
30
directional motion of electrons means that an electric current appears in
the conductor.
In an electric field of intensity E, each electron experiences a force
F
= eE. Under the action of this force, the electron acquires the acceler-
ation
a
=
F
m
=
eE
m
where e is the charge of an electron and m is its mass.
According to the laws of classical mechanics, the velocity of electrons
in free space would increase indefinitely. The same would be observed
during their motion in a strictly periodic field (for example, in an ideal
crystal with the atoms fixed at the lattice sites).
Actually, however, the directional motion of electrons in a crystal is
quite insignificant due to imperfections in the lattice's potential field.
These imperfections are mostly associated with thermal vibrations of the
atoms (in the case of metals, atomic cores) at the lattice sites, the vi-
brational amplitude being the larger the higher the temperature of the
crystal. Moreover, there are always various defects in crystals caused
by impurity atoms, vacancies at the lattice sites, interstitial atoms, and
dislocations. Crystal block boundaries, cracks, cavities and other macro
defects also affect the electric current.
In these conditions, electrons are continuously colliding and lose the
energy acquired in the electric field. Therefore, the electron velocity
increases under the effect of the external field only on a segment between
two collisions. The mean length of this segment is called the mean free
path of the electron and is denoted by
.
Thus, being accelerated over the mean free path, the electron ac-
quires the additional velocity of directional motion
v
= a
where
. is the mean free time, or the mean time between two suc-
cessive collisions of the electron with defects. If we know the mean free
path
, the mean free time can be calculated by the formula
=
v
0
+ v
where v
0
is the velocity of random thermal motion of the electron.
Usually, the mean free path
of the electron is very short and does not
exceed 10
-5
cm. Consequently, the mean free time
. and the increment
of velocity
v are also small. Since
v
v
0
, we have
31
v
0
Assuming that upon collision with a defect the electron loses practi-
cally the velocity of directional motion, we can express the mean velocity,
called the drift velocity, as follows:
-
v
=
v
2
=
eE
2m
.
=
e
2mv
0
E
= uE
The proportionality factor
u
=
e
2m
v
0
between the drift velocity v and the field intensity E is called the elec-
tron mobility.
The name of this quantity reflects its physical meaning: the mobil-
ity is the drift velocity acquired by electrons in an electric field of unit
intensity. A more rigorous calculation taking into account the fact that
in random thermal motion electrons have different velocities rather than
the constant velocity v
0
gives a double value for the electron mobility:
u
=
e
m
v
0
Accordingly, a more correct expression for the drift velocity is given
by the formula
-
v
=
e
mv
0
E
Let us now find the expression for the current density in metals. Since
electrons acquire an additional drift velocity ¯
v under the action of an ex-
ternal electric field, all the electrons that are a t a distance not exceeding
¯
v from a certain area element normal to the direction of the field inten-
sity will pass through i t in a unit of time. If the area of this element is
S ,all the electrons contained in the parelleopide of length ¯
v; will pass
through it in a unit of time (Fig. 15) . If the concentration of free elec-
trons in the metal is n, the number of electrons in this volume will be
n¯
vS The current density, which is determined by the charge carried by
these electrons through unit area, can be expressed as follows:
j
=
en¯
vS
S
=
ne
2
mv
0
E
32
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