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Файл:Steady electric current. Tutorial
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Dielectrics have free levels in the empty upper band, but the electron energy
is insufficient to move to them.
An electric field acting on an electron should increase its kinetic energy and thereby “raise” it to a nearby higher energy level. But there are no
unoccupied free levels within the valence band. And the electric field energy
is insufficient to transfer electrons from the valence band to empty levels of
the upper band (fig. 5.3).
Fig. 5.3. Energy band diagram in dielectrics. The valence band
is completely filled and separated from the empty upper band
by a wide forbidden band
The specific electrical conductivity of dielectrics σ ~ 10
−11
S∙m−1 is al-
most twenty orders of magnitude less than that of metals.
5.4. I n t r i n s i c s e m i c o n d u c t o r s i n b a n d th e o r y
Semiconductors have a relatively narrow band gap, smaller than that
of dielectrics, separating the filled valence band from the upper empty band.
Typical values of the band gap width in semiconductors are 0.1–4.0 eV. Semiconductors with a band gap width of less than ~0.3 eV are usually called narrow-gap semiconductors, semiconductors with a band gap width of more than
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~3 eV are called wide-gap semiconductors. In typical semiconductors Ge, Si,
Se, the width of the band gap at T = 300 K is ΔΕfb = 0.67; 1.11; 1.74 eV,
respectively.
At temperature T = 0 K the conductivity of intrinsic semiconductors
is zero (fig. 5.4a). The valence band of semiconductors is completely filled,
and the conduction band is empty, there are no electrons in it. In this case,
the semiconductor behaves as a dielectric.
As the temperature increases, due to thermal energy, electrons transition from the valence to the upper empty band. In this case, free electrons are
formed in the conduction band (fig. 5.4b) and vacant free levels in the valence band (holes).
а b
Fig. 5.4. Energy band diagram in intrinsic semiconductors: a is T = 0 K,
there are no free charge carriers; b is T ≠ 0 K, electrons appear
in the empty upper band and holes in the valence band
With increasing temperature, the intrinsic conductivity of semiconductors increases due to the increase in the number of current carriers
. (5.2)
The conductivity of semiconductors is excited conductivity, for example, as a result of heating, irradiation with light, or fast particles.
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5.5. E x t r i n s i c s e m i c o n d u c t o r s i n b a n d t h e o r y
Extrinsic semiconductors are obtained by introducing impurity atoms
with a valence greater or lesser than that of the atoms of the original intrinsic
semiconductor into the crystal lattice of the semiconductor. In impurity semiconductors, in addition to intrinsic conductivity, additional extrinsic conductivity arises.
Impurity atoms distort the electric field of the crystal and the system
of energy levels. Impurity energy levels appear, located in the forbidden
zone. Impurities affect conductivity because electrons from additional levels
can more easily move into the conduction band than from the valence band.
The conductivity of the semiconductor increases. Donor and acceptor impurities are distinguished.
Silicon and germanium are elements of the 4th group of the periodic
table; for them, the donor impurities are elements of the 5th group, i. e.,
P, As, Sb. The impurity donors provide additional electrons that easily pass
into the conduction band. The energy levels of the impurity electrons (donor
levels) are located near the bottom of the conduction band (fig. 5.5).
Fig. 5.5. Energy band diagram in extrinsic semiconductorswith donor
impurities. Energy levels of impurity electrons (donor levels) are located
near the bottom of the conduction band, ΔΕ* ≈ 0,05–0,01 eV
Elements of 3th group of the periodic table B, Al, Ga, In for silicon
and germanium are acceptor impurities. Acceptor impurities form empty energy levels in the forbidden zone, not occupied by electrons – acceptor levels.
Acceptor levels are located slightly above the top of the valence band
(fig. 5.6). The transition of electrons from the valence band to acceptor levels
leads to the appearance of vacancies in the valence band.
Fig. 5.6. Energy band diagram in extrinsic semiconductors with donor
impurities. Energy levels of impurity electrons (acceptor levels) are located
near the top of the valence band: ΔΕ* ≈ 0.05–0.01 eV
Since additional empty acceptor levels are located near the top of
the valence band, it is easier for electrons to move to them than to the
empty upper band. Vacancies (holes) appear in the valence band, creating
hole (n-type) conductivity. The conductivity of the semiconductor increases.
In general, both intrinsic and extrinsic conductivity are realized in
a semiconductor. Electron (n-type) or hole (p-type) extrinsic conductivity is
determined by the concentration of impurity atoms and does not depend on
temperature. With increasing temperature, the contribution of the intrinsic
conductivity of the semiconductor increases.
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Self-check questions
1. What are the difficulties and shortcomings of the classical electron
theory of Drude–Lorentz?
2. Name the main assumptions used in the band theory of solids.
3. Describe the process of formation of energy bands in solids.
4. Why do outer electrons form the widest energy bands, while the
inner electron bands are narrow?
5. How many energy levels are contained in the allowed band?
6. What is forbidden energy band or band gap? Can electrons have
energy within the forbidden band?
7. What is the spin of an electron? Formulate the Pauli principle.
8. What is the energy interval between adjacent levels of the allowed
band in a solid?
9. Specify the possible structure of energy bands so that the solid
would be a conductor.
10. What is a hybrid zone and how is it formed?
11. How is the electrical conductivity of metals explained and what
does it depend on in the band theory?
12. How does the band theory explain the practical absence of electri-
cal conductivity in dielectrics?
13. How are energy bands filled in dielectrics and what is the value of
the forbidden band?
14. Which semiconductors are called intrinsic and which are extrinsic?
15. What impurities give electron (n-type) or hole (p-type) extrinsic
conductivity?
16. How does the conductivity of intrinsic and extrinsic semiconduc-
tors depend on temperature?
17. How does the energy band diagram of intrinsic semiconductors
change when the temperature increases from 0 K to room temperature and
above?
18. Write and explain energy band diagram extrinsic semiconductors
with donor impurities, where are the energy levels of impurity electrons (donor levels) located.
19. Write and explain the energy band diagram extrinsic semiconductors with acceptor impurities, where are the acceptor levels located.
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C O N C L U S I O N
“The technology of the future is, first of all,
physics in its applications”
Abram Ioffe
It is difficult to imagine the modern world without a huge variety of
electrical devices that use one or another action of electric current, from electric irons and kettles to the most complex electronics. Electric current is the
orbital movement of electrons in atoms and molecules; atmospheric currents
and currents in wires, currents in the neural networks of living organisms...
Note that some animals have learned to produce and use current for defense
and attack.
This tutorial contains only topics related to steady current. It presents
the basic concepts of electric current, the thermal effect of current, the laws
of its flow in various media, and explains the laws based on classical electron
theory and quantum band theory of solids. The author hopes to present the
ability of current to create a magnetic field and the issues of magnetism in
the tutorial “Magnetic Field”.
Readers, of course, have noticed that each section of the tutorial is
accompanied by typical tasks with detailed explanations and mandatory
questions for self-checking the material covered. The author wants them not
to limit themselves to this tutorial and provides a bibliographic list of books
for more in-depth self-education.
Make it a rule to be constantly interested in new ideas and discoveries,
and not only in the field of physics. Academician Ioffe's idea is confirmed by
modern discoveries in the field of semiconductors, alternative sources of
electricity, controlled thermonuclear fusion, robotization of pro-duction and
work on the creation of artificial intelligence.
Never stop at what you have achieved. “Who does not go forward,
goes backwards: there is no standing still”.
Good luck with your studies!
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B I B L I O G R A P H Y
1. Arkhipov, V. Electrostatics: тutorial / V. Arkhipov. – Kazan, KNRTU
Press, 2024. – 90 p.
2. Feynman, R. P. The Feynman Lectures on Physics: in 3 vol. Vol. 2.
Mainly electromagnetism and matter / R. P. Feynman, R. B. Leighton,
M. Sands. – Boston: Addison-Wesley, 2005. – 556 p.
3. Griffiths, D. Introduction to Electrodynamics / D. Griffiths. –
3rd edition. – London: Prentice-Hall International (UK) Limited, 1999. –
596 p.
4. Good, R. H. Classical Theory of Electric and Magnetic Fields /
R. H. Good, T. J. Nelson. – New York: Academic Press, 2013. – 654 p.
5. Han, F. A. Modern Course in University Physics / F. Han,
Sh. Zheng, Sh. Li. – New Delhi: World Scientific Publishing, 2024. – 600 p.
6. Irodov, I. E. Problems in General Physics / I. E. Irodov. – Moscow:
Mir, 1981. – 402 p.
7. Jackson, J. Classical Electrodynamics / J. Jackson. – 3rd edition. –
Berkeley John Wiley&Sons, Inc., 1998. – 832 p.
8. Matweev, A. N. Electricity and Magnetism / A. N. Matweev. –
Moscow: Mir, 1987. – 448 p.
9. Purcell, E. Electricity and Magnetism (Berkeley Physics Course).
Vol. 2 / E. Purcell, D. Morin. – 3rd edition. – Cambridge: Cambridge University Press, 2011. – 853 p.
10. Raj, M. G. Problems in Electrostatics / M. G. Raj. – New Delhi:
Anmol Publications, 1998. – 121 p.
11. Savelyev, I. V. Physics General Cource: in 3 vol. Vol. 2. Electricity and Magnetism, Waves, Optics / I. V. Savelyev. – Moscow: Mir,
1980. – 509 p.
12. Wolkenstein, V. S. Problems in General Physics / V. S. Wolkenstein. – Moscow: Mir, 1987. – 382 p.
13. Yavorsky, V. M. Handbook of Physics / V. M. Yavorsky,
A. A. Detlaf. – Moscow: Mir, 1972. – 1132 p.
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EDUCATIONAL EDITION
Signed in print 29.09.2025
Sheet size 60×84 1/16
Offset paper
Digital print
5,11 conv. pr. sh.
5,5 publ. sh.
Edition 400 copies
Order 50/25
Kazan National Research Technological University Press
Offset laboratory of Kazan National Research Technological University
420015, Kazan, Karl Marx street, 68
Viktor Arkhipov
STEADY ELECTRIC CURRENT
Виктор Палладиевич Архипов
ПОСТОЯННЫЙ ЭЛЕКТРИЧЕСКИЙ ТОК
Editor Е. Shevchenko
Computer layout A. Rakhmankulova
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