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The classical theory of the electrical conductivity of metals originated at the beginning of the 20th century. Its founder was the German physicist Carl Rikke. He empirically established that the passage of a charge through a metal is not associated with the transfer of conductor atoms, in contrast to liquid electrolytes. However, this discovery did not explain what exactly is the carrier of electrical impulses in the metal structure.
The experiments of the scientists Stuart and Tolman, conducted in 1916, allowed to answer this question. They managed to establish that the smallest charged particles - electrons - are responsible for the transfer of electricity in metals. This discovery formed the basis of the classical electronic theory of the electrical conductivity of metals. From that moment, a new era of research on metallic conductors began. Thanks to the results obtained, we now have the opportunity to use household appliances, production equipment, machines and many other devices.
How does the electrical conductivity of different metals differ?
The electronic theory of the electrical conductivity of metals was developed in the studies of Paul Drude. He was able to discover such a property as resistance, which is observed when an electric current passes through a conductor. In the future, this will allow us to classify different substances in terms of conductivity. From the results obtained, it is easy to understand which metal is suitable for the manufacture of a particular cable. This is very important point, since improperly selected material can cause a fire as a result of overheating from the passage of excess voltage current.
Silver has the highest electrical conductivity. At a temperature of +20 degrees Celsius, it is 63.3 * 104 centimeters-1. But it is very expensive to make wiring from silver, as it is quite rare metal, which is mainly used for the production of jewelry and decorative ornaments or investment coins.
The metal with the highest electrical conductivity among all elements of the non-noble group is copper. Its indicator is 57 * 104 centimeters-1 at a temperature of +20 degrees Celsius. Copper is one of the most common conductors used in domestic and industrial applications. It withstands constant electrical loads well, is durable and reliable. The high melting point allows you to work without problems for a long time in a heated state.
In terms of prevalence, only aluminum can compete with copper, which ranks fourth in electrical conductivity after gold. It is used in networks with low voltage, as it has almost half the melting point of copper, and is not able to withstand extreme loads. Further distribution of places can be found by looking at the table of electrical conductivity of metals.
It is worth noting that any alloy has a much lower conductivity than a pure substance. This is due to the merging of the structural network and, as a consequence, the disruption of the normal functioning of electrons. For example, in the production copper wire a material with an impurity content of no more than 0.1% is used, and for some types of cable this figure is even stricter - no more than 0.05%. All of the above indicators are the electrical conductivity of metals, which is calculated as the ratio between the current density and the magnitude of the electric field in the conductor.
Classical theory of electrical conductivity of metals
The main provisions of the theory of electrical conductivity of metals contain six points. First: a high level of electrical conductivity is associated with the presence a large number free electrons. Second: electricity arises by external action on the metal, in which electrons from random motion pass into an ordered one.
Third: the strength of the current passing through the metal conductor is calculated according to Ohm's law. Fourth: different number elementary particles in the crystal lattice leads to unequal resistance of metals. Fifth: the electric current in the circuit occurs instantly after the start of the impact on the electrons. Sixth: with an increase in the internal temperature of the metal, the level of its resistance also increases.
The nature of the electrical conductivity of metals is explained by the second paragraph of the provisions. At rest, all free electrons revolve randomly around the nucleus. At this point, the metal is not able to reproduce on its own. electric charges. But one has only to connect an external source of influence, as electrons instantly line up in a structured sequence and become carriers of electric current. As the temperature rises, the electrical conductivity of metals decreases.
This is due to the weakening molecular bonds in the crystal lattice elementary particles begin to rotate in an even more chaotic manner, so the construction of electrons in a chain becomes more complicated. Therefore, it is necessary to take measures to prevent overheating of conductors, as this negatively affects their performance properties. The mechanism of electrical conductivity of metals cannot be changed due to existing laws physics. But it is possible to neutralize the negative external and internal influences that interfere with the normal course of the process.
Metals with high electrical conductivity
The electrical conductivity of alkali metals is high level, since their electrons are weakly attached to the nucleus and easily line up in the desired sequence. But this group is distinguished by low melting points and enormous chemical activity, which in most cases does not allow them to be used for the manufacture of wires.
Metals with high electrical conductivity open form very dangerous to humans. Touching a bare wire will cause an electrical burn and a powerful discharge to all internal organs. This often results in instant death. Therefore, for the safety of people, special insulating materials are used.
Depending on the application, they can be solid, liquid or gaseous. But all types are designed for one function - to isolate the electric current inside the circuit so that it cannot affect external world. The electrical conductivity of metals is used in almost all areas modern life people, so safety is a top priority.
The most complete and accurate phenomenon of electrical conductivity of metals is described by the quantum theory of solids. However, in order to find out the most general issues one can confine oneself to consideration on the basis of the classical electron theory. According to this theory, the set of electrons in a crystal can, with a certain approximation, be likened to an ideal gas, considering the motion of electrons to obey the laws classical mechanics. In this case, the interaction of electrons with each other is not considered at all, and the interaction of electrons with ions crystal lattice reduces to ordinary elastic collisions.
Metals contain a huge number of free electrons moving in the interstitial space of the crystal. In 1 cm 3 there are about 10 23 atoms. Therefore, with a metal valence Z, the concentration n of free electrons, also called conduction electrons, is equal to. All of them are in random chaotic thermal motion, moving in the space of the crystal at an enormous speed, the average value of which is about 10 8 cm/s. Due to the randomness of thermal motion, the number of electrons moving in any direction is always on average equal to the number of electrons moving in the opposite direction, due to which, in the absence of an external electric field, the charge carried by electrons through any section of the crystal is equal to zero. Under the action of an electric field, each electron acquires an additional speed, due to which the entire collective of electrons in the metal begins to move in the direction opposite to the direction of the applied field strength. The appearance of a directed movement of electrons determines the occurrence of an electric current in the conductor.
For every electron electric field intensity E acts with force F = eE. Under the action of this force, the electron acquires an acceleration
where e is the electron charge and m is its mass.
According to the laws of classical mechanics in free space the speed of the electrons would increase indefinitely; the same would be observed when they move in a strictly periodic field (for example, in an ideal crystal with atoms at rest at the nodes).
In reality, due to violations of the periodicity in the potential field of the lattice, the directed movement of electrons in the crystal turns out to be quite insignificant. These disturbances are primarily associated with thermal vibrations of atoms (in the case of metals, atomic residues) at the nodes of the crystal lattice (in this case, the amplitude of vibrations is greater, the higher the temperature of the crystal). In addition, a crystal always contains various defects caused by the presence of impurity atoms, empty spaces at nodes, atoms at interstices, and dislocations. The boundaries of crystal blocks, cracks, cavities, etc., also affect.
Under these conditions, electrons constantly experience collisions and waste the energy acquired in the electric field. Therefore, in reality, the speed of electrons under the action of the force of an external field increases only in the area between two collisions. The average length of this section is called the mean free path of an electron and is denoted by λ.
So, accelerating on the length of the free path, the electron acquires an additional speed of directed motion
where τ is the free path time, or the average time between two successive collisions of an electron with defects. Knowing the free path length λ, one can calculate the free path time τ by the formula
where υ 0 is the speed of the chaotic thermal motion of an electron. The mean free path of an electron λ is usually very small and does not exceed 10 -5 cm. Therefore, both the mean free path τ and the velocity addition itself Δυ turn out to be small. Since then
Assuming that when colliding with a defect, an electron almost completely loses the speed of directed motion, we can express the average speed of directed motion, called the drift velocity, as follows:
Proportionality factor
between the average drift velocity and the field strength E is called the electron mobility.
The name of this quantity accurately reflects its physical meaning: mobility is the drift velocity acquired by electrons in an electric field of unit intensity. A more rigorous calculation, taking into account the fact that even during chaotic thermal motion, electrons move not with constant speedυ 0 , and have different speeds, leads to twice greater value for electron mobility:
Accordingly, for the drift velocity, the expression is more accurate
Let us now find an expression for the current density in metals. Since under the action of an external electric field, electrons acquire an additional drift velocity, then in a unit of time through any area perpendicular to the field strength, all electrons that are separated from this area at a distance not exceeding in the volume of a parallelepiped long (Fig. 15). If the concentration of free electrons in the metal is n, then their number in the volume of this parallelepiped will be equal to . The current density, determined by the charge transferred by these electrons through a unit area, will be expressed as follows.
The electronic conductivity of metals was first experimentally proven by the German physicist E. Rikke in 1901. Through three polished cylinders tightly pressed against each other - copper, aluminum and again copper - long time(within a year) passed an electric current. The total charge that passed during this time was equal to 3.5·10 6 C. Since the masses of copper and aluminum atoms differ significantly from each other, the masses of the cylinders would have to change noticeably if the charge carriers were ions. The results of the experiments showed that the mass of each of the cylinders remained unchanged. Only insignificant traces of mutual penetration of metals were found in the contacting surfaces, which did not exceed the results of the usual diffusion of atoms in solids. Consequently, free charge carriers in metals are not ions, but particles that are the same in both copper and aluminum. Only electrons could be such particles.
Direct and convincing proof of the validity of this assumption was obtained in the experiments set up in 1913 by L. I. Mandelstam and N. D. Papaleksi and in 1916 by T. Stuart and R. Tolman.
A wire is wound on the coil, the ends of which are soldered to two metal disks isolated from each other (Fig. 1). A galvanometer is attached to the ends of the disks using sliding contacts.
The coil is brought into rapid rotation, and then abruptly stopped. After a sharp stop of the coil, free charged particles will move along the conductor by inertia for some time, and, consequently, an electric current will appear in the coil. current will exist a short time, because due to the resistance of the conductor, the charged particles slow down and the ordered movement of the particles stops.
The direction of the current indicates that it is created by the movement of negatively charged particles. The charge transferred in this case is proportional to the ratio of the charge of the particles that create the current to their mass, i.e. \(~\Delta q = \frac(q_0)(m)\). Therefore, by measuring the charge passing through the galvanometer for the entire time of the existence of the current in the circuit, it was possible to determine the ratio \(~\frac(q_0)(m)\). It turned out to be equal to 1.8·10 11 C/kg. This value coincides with the ratio of the electron charge to its mass found earlier from other experiments.
Thus, an electric current in metals is created by the movement of negatively charged electron particles. According to the classical electronic theory of the conductivity of metals (P. Drude, 1900, H. Lorenz, 1904), a metal conductor can be considered as a physical system of a combination of two subsystems:
- free electrons with a concentration of ~ 10 28 m -3 and
- positively charged ions vibrating around the equilibrium position.
The appearance of free electrons in a crystal can be explained as follows.
When atoms combine into a metal crystal, the outer electrons most weakly bound to the atomic nucleus are detached from the atoms (Fig. 2). Therefore, positive ions are located at the nodes of the crystal lattice of the metal, and electrons that are not connected with the nuclei of their atoms move in the space between them. These electrons are called free or conduction electrons. They perform a chaotic movement, similar to the movement of gas molecules. Therefore, the totality of free electrons in metals is called electron gas.
If an external electric field is applied to the conductor, then a directed movement is superimposed on the random chaotic movement of free electrons under the action of the forces of the electric field, which generates an electric current. The speed of movement of the electrons themselves in the conductor is a few fractions of a millimeter per second, however, the electric field arising in the conductor propagates along the entire length of the conductor at a speed close to the speed of light in vacuum (3 10 8 m / s).
Since the electric current in metals is formed by free electrons, the conductivity of metal conductors is called electronic conductivity.
Electrons under the influence of a constant force acting from the electric field acquire a certain speed of ordered movement (it is called drift). This speed does not further increase with time, since when colliding with ions of the crystal lattice, electrons transfer the kinetic energy acquired in the electric field to the crystal lattice. As a first approximation, we can assume that on the mean free path λ (this is the distance that an electron travels between two successive collisions with ions) the electron moves with acceleration \(~a = \frac(eE)(m)\) and its drift velocity increases linearly with time\[~\upsilon = at = \ frac(eEt)(m)\]. At the moment of collision, the electron transfers kinetic energy to the crystal lattice. Then it accelerates again, and the process repeats. As a result, the average speed of the ordered motion of electrons is proportional to the electric field strength in the conductor \(~\mathcal h \upsilon \mathcal i \sim E\) and, consequently, the potential difference at the ends of the conductor, since \(~E = \frac Ul\ ), where l- conductor length.
It is known that the current strength in the conductor is proportional to the speed of the ordered motion of particles\[~I = en \mathcal h \upsilon \mathcal i S\], which means, according to the previous one, the current strength is proportional to the potential difference at the ends of the conductor: I ~ U. This is the qualitative explanation of Ohm's law on the basis of the classical electronic theory of the conductivity of metals.
However, there are difficulties with this theory. It followed from the theory that the resistivity should be proportional to the square root of the temperature (\(~\rho \sim \sqrt T\)), meanwhile, according to experience, ρ ~ T. In addition, the heat capacity of metals, according to this theory, should be much greater than the heat capacity of monatomic crystals. In reality, the heat capacity of metals differs little from the heat capacity of non-metallic crystals. These difficulties were only overcome in quantum theory.
In 1911, the Dutch physicist G. Kamerling-Onnes, studying the change in the electrical resistance of mercury at low temperatures, found that at a temperature of about 4 K (i.e. at -269 ° C), the resistivity abruptly decreases (Fig. 1) almost down to zero. This phenomenon of turning electrical resistance to zero G. Kamerling-Onnes called superconductivity.
Subsequently, it was found that more than 25 chemical elements- metals at very low temperatures become superconductors. Each of them has its own critical transition temperature to a state with zero resistance. Its lowest value for tungsten is 0.012 K, the highest for niobium is 9 K.
Superconductivity is observed not only in pure metals, but also in many chemical compounds and alloys. In this case, the elements themselves, which are part of the superconducting compound, may not be superconductors. For example, NiBi , Au 2 Bi , PdTe , PtSb and others.
Substances in the superconducting state have unusual properties:
- electric current in a superconductor can exist for a long time without a current source;
- inside a substance in a superconducting state, it is impossible to create a magnetic field:
- the magnetic field destroys the state of superconductivity. Superconductivity is a phenomenon explained from the point of view of quantum theory. A rather complicated description of it is beyond the scope school course physics.
Until recently, the widespread use of superconductivity was hindered by the difficulties associated with the need for cooling to ultralow temperatures, for which liquid helium was used. Nevertheless, despite the complexity of the equipment, the scarcity and high cost of helium, since the 60s of the XX century, superconducting magnets have been created without thermal losses in their windings, which has practically possible to obtain strong magnetic fields in relatively large volumes. It is these magnets that are required to create controlled installations. thermonuclear fusion with magnetic plasma confinement, for powerful particle accelerators. Superconductors are used in various measuring instruments, especially in instruments for measuring very weak magnetic fields with the highest precision.
At present, 10-15% of energy is spent in power transmission lines to overcome the resistance of wires. Superconducting lines, or at least inputs to large cities, will bring enormous savings. Another field of application of superconductivity is transport.
On the basis of superconducting films, a number of high-speed logical and memory elements for computing devices have been created. In space research, it is promising to use superconducting solenoids for radiation protection of astronauts, docking of ships, their deceleration and orientation, and for plasma rocket engines.
Currently, ceramic materials have been created that have superconductivity at a higher temperature - over 100 K, that is, at a temperature above the boiling point of nitrogen. The ability to cool superconductors with liquid nitrogen, which has an order of magnitude higher heat of vaporization, greatly simplifies and reduces the cost of all cryogenic equipment, and promises a huge economic effect.
Literature
Aksenovich L. A. Physics in high school: Theory. Tasks. Tests: Proc. allowance for institutions providing general. environments, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsia i vykhavanne, 2004. - C. 279-282.
Topic of the lesson: Fundamentals of the electronic theory of conductivity of metals.
Electric current in electrolytes.
The type of activity is mixed.
Combined class type.
Learning objectives of the lesson: to form students' understanding of electronic
conductivity of metals; consider the experiments of Mandelstam and Papaleksi;
to formulate Ohm's law in the light of the electronic theory of conductivity of metals.
Lesson objectives:
Educational:
Establish differences in the conditions for the existence of an electric current in solid,
liquid, gaseous bodies. To form a concept of the nature of electric current in
metals.
Developing. Develop the ability to observe, form an idea of the process
scientific knowledge.
Educational. Develop cognitive interest in the subject, develop the ability to
listen and be heard.
Planned educational results:
contribute to strengthening
practical orientation in teaching physics, the formation of skills
apply the acquired knowledge in different situations.
Personal: promote emotional perception of physical objects,
ability to listen, clearly and accurately express their thoughts, develop initiative and
activity in solving physical problems, to form the ability to work in groups.
Meta-subject: develop the ability to understand and use visual aids
(drawings, models, diagrams). Development of understanding of the essence of algorithmic
instructions and skills to act in accordance with the proposed algorithm.
Subject: master the physical language, the ability to recognize compounds
parallel and sequential, the ability to navigate in wiring diagram,
collect diagrams. Ability to generalize and draw conclusions.
Lesson progress:
1. Organization of the beginning of the lesson (marking absentees, checking the readiness of students
to the lesson, answers to students' questions on homework) - 25 min.
The teacher informs the students about the topic of the lesson, formulates the objectives of the lesson and introduces
students with a lesson plan. Students write the topic of the lesson in their notebooks. Teacher
creates conditions for motivation of educational activity.
Mastering new material:
Electric current in various environments.
Electric current in metals is the ordered movement of electrons
electric current in solutions (melts) of electrolytes is a directional
movement of ions of both signs in opposite directions
electric current in gases is the ordered movement of ions and electrons under
by the action of an electric field.
A body in which there are free charge carriers, that is
an electric current in a vacuum can be created by the orderly movement of any
charged particles (electrons, ions).
Conductors, dielectrics, semiconductors, electrolytes.
Wired cues
charged particles that can move freely inside this body.
Dielectric (insulator) - a substance that practically does not conduct electric current.
The concentration of free charge carriers in the dielectric does not exceed 108 cm–3.
The main property of a dielectric is the ability to polarize in an external
electric field. From the point of view of the zone theory solid body dielectric -
substance with a bandgap greater than 3 eV.
Semiconductor cues
intermediate place between conductors and dielectrics and differs
from conductors by a strong dependence of conductivity on concentration
impurities, temperature and exposure various kinds radiation. Main
property of a semiconductor is an increase in electrical conductivity with
rising temperature
electrolytics
- a substance whose melt or solution conducts electricity
current due to dissociation into ions, however, the substance itself does not have an electric current
conducts. Examples of electrolytes are
solutions of acids, salts and bases. Electrolytes are conductors of the second kind,
substances that in solution (or melt) consist wholly or partly of ions
and therefore have ionic conductivity.
- a material that, by its specific conductivity, occupies
The nature of electric current in metals.
Metals have electronic conductivity. experimental
proof of:
K. Rikke's experience: passed a current of hundreds of amperes for a long time
me. I expected: copper will appear in aluminum. Result: negative
e. the current is not a directed movement of ions.
Stuart Tolman Experience:
1913 - Mandelstam - Papaleksi offered,
1916 - Stewart - Tolman carried out experimentally.
Length l wire = 500 m (in the coil). The coil rotated with v \u003d 500 m / s: at cut
under deceleration, the free particles moved by inertia. By
the deviation of the galvanometer needle was determined by the specific charge, according to
the direction of deflection is the charge sign.
Electronic theory of metals (P. Drude, G.A. Lorenz)
1. Free electrons in metals behave like molecules of an ideal
gas. but val>> vtherm.
2. The movement of free electrons in metals obeys the laws
Newton.
3. Free electrons in the process of chaotic motion collide
predominantly with ions of the crystal lattice.
4. Moving until the next collision with ions, electrons
accelerated by an electric field and acquire kinetic
energy Ek.
Construct a satisfactory quantitative theory of motion
electrons in a metal based on the laws of classical mechanics
impossible. But you can roughly explain Ohm's law.
dependence of metal resistivity on
temperature, where temperature coefficient of resistance
(table value). Completely correctly explain conductivity
metals allows only quantum theory.
Superconductivity.
The phenomenon was discovered by H.KamerlingOnnes (Holland) in 1911 on mercury and
is that at ultralow temperatures the resistance
conductor can abruptly drop to 0. Ie. in such conductors
energy is consumed for heating. In 1933 W. Meisner discovered the phenomenon
consisting in the fact that the external magnetic field does not penetrate deep into
superconductor, if the value magnetic field less critical
values
Currently open
high-temperature superconductors predicted by V. Ginzburg
(temperatures above the temperature of liquid nitrogen).
(Meissner effect).
Basic provisions of the classical theory of electronic conductivity.
one). Current carriers in metals are electrons, the movement of which is subject to
the law of classical mechanics.
2). The behavior of electrons is similar to the behavior of molecules ideal gas(electronic
gas).
3). When electrons move in a crystal lattice, one can ignore
collision of electrons with each other.
four). In the elastic collision of electrons with ions, the electrons completely transfer to them
energy stored in the electric field.
Velocity of ordered motion of electrons in a metal.
Homework: Message
ELECTRICAL CONDUCTIVITY OF METALS AND SEMICONDUCTORS
Electrical conductivity of metals
The corresponding quantum mechanical calculation shows that in the case of an ideal crystal lattice, conduction electrons would not experience any resistance during their movement and the electrical conductivity of metals would be infinitely large. However, the crystal lattice is never perfect. Violations of the strict periodicity of the lattice are due to the presence of impurities or vacancies (i.e., the absence of atoms in the site), as well as thermal vibrations in the lattice. Scattering of electrons by impurity atoms and by photons leads to the appearance of electrical resistance in metals. The purer the metal and the lower the temperature, the lower this resistance.
The electrical resistivity of metals can be represented as
where count - resistance due to thermal vibrations of the lattice, approx is the resistance due to the scattering of electrons by impurity atoms. term col decreases with decreasing temperature and vanishes at T = 0K. term approx at a low concentration of impurities does not depend on temperature and forms the so-called residual resistance metal (i.e. the resistance that a metal has at 0K).
Let there be a unit volume of metal n free electrons. Let's call the average speed of these electrons drift speed
. By definition
In the absence of an external field, the drift velocity is zero and there is no electric current in the metal. When an external electric field is applied to the metal, the drift velocity becomes different from zero - an electric current arises in the metal. According to the law Ohma drift speed is finite and proportional to the force 
.
It is known from mechanics that the speed of steady motion is proportional to the external force applied to the body F when, in addition to force -
F, the resistance force of the medium acts on the body, which is proportional to the speed of the body (an example is the fall of a small ball in a viscous medium). Hence we conclude that in addition to the force 
, the "friction" force acts on the conduction electrons in the metal, the average value of which is equal to
(r- coefficient of proportionality).
The equation of motion for the "average" electron has the form
,
where m *
is the effective mass of the electron. This equation allows you to find the steady value 
.
If, after the establishment of a stationary state, the external field is turned off 
, the drift velocity begins to decrease and, upon reaching the state of equilibrium between the electrons and the lattice, vanishes. Let us find the law of the drift velocity decrease after the external field is turned off. Putting in 
, we get the equation
We are familiar with this type of equation. His solution looks like
,
where 
- the value of the drift speed at the moment the field is turned off.
It follows that during the time
the value of the drift speed decreases in e once. Thus, the value is the relaxation time characterizing the process of establishing equilibrium between electrons and the lattice, disturbed by the action of an external field 
.
The given formula can be written as follows:
.
The steady value of the drift velocity can be found by equating to zero the sum of the force 
and friction force:
.
.
We obtain the steady value of the current density by multiplying this value 
on the charge of an electron e and electron density n:
.
Proportionality factor between 
is the electrical conductivity
. In this way,
.
The classical expression for the electrical conductivity of metals has the form
,
where is the mean free path of electrons, m is the ordinary (not effective) mass of the electron.
From a comparison of formulas and it follows that the relaxation time coincides in order of magnitude with the mean free path of electrons in a metal.
Based on physical considerations, it is possible to estimate the quantities included in the expression, and thereby calculate, in order of magnitude, the conductivity
. The values obtained in this way are in good agreement with experimental data. Also, in agreement with experience, it turns out that
varies with temperature according to the law 1/ T. Recall that classical theory gives that
inversely proportional
.
We note that the calculations that led to the formula are equally suitable both for the classical interpretation of the motion of conduction electrons in a metal and for the quantum mechanical interpretation. The difference between these two interpretations is as follows. In the classical consideration, it is assumed that all electrons are perturbed by an external electric field, according to which each term in the formula receives an addition in the direction
opposite 
. In the quantum mechanical interpretation, one has to take into account that only electrons occupying states near the Fermi level are perturbed by the field and change their velocity. Electrons located at deeper levels are not perturbed by the field, and their contribution to the sum does not change. In addition, in the classical interpretation, the denominator of the formula should be the usual mass of the electron m, in the quantum mechanical interpretation, instead of the usual mass, the effective mass of the electron should be taken m *
. This circumstance is a manifestation of the general rule, according to which the relations obtained in the approximation of free electrons turn out to be valid for electrons moving in the periodic field of the lattice, if we replace the true mass of the electron in them m effective mass m *
.
Superconductivity
At a temperature of the order of several kelvins, the electrical resistance of a number of metals and alloys abruptly turns into zero-substance, passes into superconducting state. The temperature at which this transition occurs is called critical temperature and denoted T k . Highest observed value T k is 20 K.
Experimentally, superconductivity can be observed in two ways:
1) including in the general electrical circuit superconductor link. At the moment of transition to the superconducting state, the potential difference at the ends of this link vanishes;
2) by placing a superconductor ring in a magnetic field perpendicular to it. After cooling the ring below, turn off the field. As a result, a continuous electric current is induced in the ring. The current in such a ring circulates indefinitely.
The Dutch scientist G. Kamerling-Onnes, who discovered the phenomenon of superconductivity, demonstrated this by transporting a superconducting ring with a current flowing through it from Leiden to Cambridge. In a number of experiments, the absence of current decay in the superconducting ring was observed for about a year. In 1959, Collins reported that he observed no decrease in current for two and a half years.
In addition to the absence of electrical resistance, the superconducting state is characterized by the fact that the magnetic field does not penetrate into the bulk of the superconductor. This phenomenon is called Meissner effect. If a superconducting sample is cooled by being placed in a magnetic field, at the moment of transition to the superconducting state, the field is pushed out of the sample, and the magnetic induction in the sample vanishes. Formally, we can say that a superconductor has zero magnetic permeability ( = 0). Substances with < 1 are called diamagnets. Thus, a superconductor is an ideal diamagnet.
A sufficiently strong external magnetic field destroys the superconducting state. The value of magnetic induction at which this happens is called critical field and denoted B k . Meaning B k depends on the sample temperature. At critical temperature B k = 0, with decreasing temperature value B k increases tending to 
- the value of the critical field at zero temperature. An approximate view of this dependence is shown in Fig. 1
If we amplify the current flowing through the superconductor included in the common circuit, then at the value of the current strength I k the superconducting state is destroyed. This value of current is called critical current. Meaning I k depends on the temperature. The form of this dependence is similar to the dependence B k from T(see fig. 1).
Superconductivity is a phenomenon in which quantum mechanical effects are found not on microscopic, but on large, macroscopic scales. The theory of superconductivity was created in 1957 by J. Bardeen, L. Cooper and J. Schrieffer. It is called briefly the BCS theory. This theory is very complex. Therefore, we are forced to confine ourselves to presenting it at the level of popular science books, which, apparently, will not be able to fully satisfy the demanding reader.
The key to superconductivity lies in the fact that, in addition to Coulomb repulsion, electrons in a metal experience a special kind of mutual attraction, which in the superconducting state prevails over repulsion. As a result, conduction electrons are combined into so-called cooper couples. The electrons in such a pair have oppositely directed spins. Therefore, the spin of the pair is zero, and it is a boson. Bosons tend to accumulate mostly energy state, from which it is relatively difficult to transfer them to an excited state. Consequently, Cooper pairs, having come into coordinated motion, remain in this state for an indefinitely long time. Such a coordinated movement of pairs is the superconductivity current.
Let us explain what has been said in more detail. An electron moving in a metal deforms (polarizes) a crystal lattice consisting of positive ions. As a result of this deformation, the electron is surrounded by a "cloud" of positive charge, which moves along the lattice along with the electron. The electron and the cloud surrounding it are a positively charged system, to which another electron will be attracted. Thus, the ionic lattice plays the role of an intermediate medium, the presence of which leads to attraction between electrons.
In quantum mechanical language, the attraction between electrons is explained as a result of the exchange between electrons of lattice excitation quanta - phonons. An electron moving in a metal violates the regime of lattice vibrations - it excites phonons. The excitation energy is transferred to another electron, which absorbs the phonon. As a result of such an exchange of phonons, an additional interaction between electrons arises, which has the character of attraction. At low temperatures, this attraction for substances that are superconductors exceeds the Coulomb repulsion.
The interaction due to the exchange of phonons is most pronounced for electrons with opposite momenta and spins. As a result, two such electrons combine into a Cooper pair. This pair should not be thought of as two stuck together electrons. On the contrary, the distance between the electrons of the pair is very large, it is approximately 10 -4 cm, i.e. exceeds the interatomic distances in the crystal by four orders of magnitude. Approximately 10 6 Cooper pairs overlap noticeably; occupy the total space.
Not all conduction electrons combine into Cooper pairs. At a temperature T, different from absolute zero, there is some probability that the pair will be destroyed. Therefore, along with the pairs, there are always "normal" electrons moving through the crystal in the usual way. The closer T and T k , the larger the fraction of normal electrons becomes, turning into 1 at T = T k . . Therefore, at temperatures above T k the superconducting state is possible.
The formation of Cooper pairs leads to a rearrangement of the energy spectrum of the metal. To excite an electronic system that is in a superconducting state, it is necessary to destroy at least one pair, which requires energy equal to the binding energy E the number of electrons in a pair. This energy is the minimum amount of energy that the system of electrons in a superconductor can absorb. Consequently, in the energy spectrum of electrons in the superconducting state, there is a gap of width E St, located in the region of the Fermi level. Energy values belonging to this gap are prohibited. The existence of a gap has been proven experimentally.
Thus, the excited state of an electronic system in the superconducting state is separated from the ground state by an energy gap of width E St. Therefore, quantum transitions of this system will not always be possible. At low speeds of its movement (corresponding to a current strength less than I k) its electronic system will be excited, and this means movement without friction, i.e. without electrical resistance.
Energy gap width E sv decreases with increasing temperature and vanishes at the critical temperature T k . Accordingly, all Cooper pairs are destroyed, and the substance passes into a normal (non-superconducting) state.
It follows from the theory of superconductivity that the magnetic flux Ф associated with the superconducting ring (or cylinder) through which the current circulates must be an integer multiple of 
, where q
- current carrier charge
.
Value
represents flux quantum.
Magnetic flux quantization was discovered experimentally in 1961 by Deaver and Fairbank and independently by Doll and Nebauer. In the experiments of Deaver and Fairbank, the sample was a band of tin deposited on a copper wire with a diameter of about 10 -3 cm. The wire played the role of a frame and did not pass into the superconducting state. The measured values of the magnetic flux in these experiments, as well as in the experiments of Doll and Nebauer, turned out to be integer multiples of the value in which, as q take twice the charge of an electron q = - 2e) . This serves as an additional confirmation of the correctness of the BCS theory, according to which the current carriers in a superconductor are Cooper pairs, the charge of which is equal to the total charge of two electrons, i.e. - 2e.
Semiconductors
Semiconductors are crystalline substances, in which the valence band is completely filled with electrons, and the band gap is small (for intrinsic semiconductors, no more than 1 eV). Semiconductors owe their name to the fact that in terms of electrical conductivity they occupy an intermediate position between metals and dielectrics. However, their characteristic is not the magnitude of the conductivity, but the fact that their conductivity increases with increasing temperature (recall that in metals it decreases).
Distinguish own and impurity semiconductors. Among the intrinsic ones are chemically pure semiconductors. The electrical properties of impurity semiconductors are determined by the artificial impurities present in them.
When considering the electrical properties of semiconductors, the concept of "holes" plays an important role. Let us dwell on the elucidation of the physical meaning of this concept.
In an intrinsic semiconductor at absolute zero, all levels of the valence band are completely filled with electrons, and there are no electrons in the conduction band (Fig. 2a). The electric field cannot transfer electrons from the valence band to the conduction band. Therefore, intrinsic semiconductors behave at absolute zero as dielectrics. At temperatures other than 0 K, part of the electrons from the upper levels of the valence band pass as a result of thermal excitation to the lower levels of the conduction band (Fig. 2b). Under these conditions, the electric field is able to change the state of the electrons in the conduction band. In addition, due to the formation of vacant levels in the valence band, the electrons of this band can also change their velocity under the action of an external field. As a result, the electrical conductivity of the semiconductor becomes nonzero.
It turns out that in the presence of vacant levels, the behavior of the electrons in the valence band can be represented as the movement of positively charged quasiparticles, called "holes". Since the conductivity of a completely filled valence band is equal to zero, it follows that the sum of the velocities of all electrons in such a band is equal to zero
Let us extract from this sum the speed k th electron
It follows from this relation that if k th electron in the valence band is absent, then the sum of the velocities of the remaining electrons turns out to be equal to 
. Therefore, all these electrons will create a current equal to 
. Thus, the resulting current turns out to be equivalent to the current that would be created by a particle with a charge + e, which has the speed of the missing electron. This imaginary particle is a hole.
The concept of holes can also be arrived at in the following way. Vacant levels are formed at the top of the valence band. As shown, the effective mass of an electron at the top of the energy band is negative. The absence of a particle with a negative charge (- e) and negative mass m * is equivalent to the presence of a particle with a positive charge (+ e) and positive mass | m * | those. holes.
So, in terms of its electrical properties, a valence band with a small number of vacant states is equivalent to an empty band containing a small number of positively charged quasiparticles called holes.
We emphasize that the motion of a hole is not the displacement of some real positively charged particle. The concept of holes reflects the nature of the motion of the entire multielectron system in a semiconductor.
Intrinsic conductivity of semiconductors
Intrinsic conduction results from the transition of electrons from the upper levels of the valence band to the conduction band. At the same time, a certain number of current carriers appear in the conduction band - electrons occupying levels near the bottom of the band, at the same time, the same number of places at the upper levels are vacated in the valence band, as a result of which holes appear
The distribution of electrons over the levels of the valence band and the conduction band are described by the Fermi-Dirac function. This distribution can be made very clear by depicting how it is done in Fig. distribution function graph together with the scheme of energy zones.
The corresponding calculation shows that for intrinsic semiconductors, the value of the Fermi level counted from the top of the valence band is equal to
,
where E is the band gap, and m d*i m e*- effective masses hole and an electron in the conduction band. Usually the second term is negligible, and we can assume 
. This means that the Fermi level lies in the middle of the band gap. Consequently, for electrons that have passed into the conduction band, the value E
-
E F differs little from half the band gap. The conduction band levels lie on the tail of the distribution curve. Therefore, the probability of filling them with electrons can be found using formula (1.23) of the previous paragraph. Putting in this formula 
, we get that
.
The number of electrons that have passed into the conduction band, and hence the number of holes formed, will be proportional to the probability. These electrons and holes are current carriers. Since the conductivity is proportional to the number of carriers, it must also be proportional to the expression. Consequently, the electrical conductivity of intrinsic semiconductors increases rapidly with temperature, changing according to the law
,
where E is the band gap, 0 - a value that changes with temperature much more slowly than the exponent, and therefore it can be considered a constant in the first approximation.
If we plot the dependency ln from T, then for intrinsic semiconductors a straight line is obtained, shown in Fig.4. The slope of this straight line can be used to determine the band gap E.
Typical semiconductors are Group IV elements periodic system Mendeleev - germanium and silicon. They form a diamond-type lattice in which each atom is bound by covalent (pair-electron) bonds with four neighboring atoms equally spaced from it. Conventionally, such a mutual arrangement of atoms can be represented as a flat structure, shown in Fig. 5. Circles with a sign 
denote positively charged atomic residues (i.e. that part of the atom that remains after the removal of valence electrons), circles with a sign 
- valence electrons, double lines- covalent bonds.
At a high enough temperature, thermal motion can break apart individual pairs, freeing one electron. The place left by the electron ceases to be neutral, an excess positive charge arises in its vicinity 
, i.e. a hole is formed (in Fig. 5 it is shown by a dotted circle). An electron from one of the neighboring pairs can jump to this place. As a result, the hole begins to wander through the crystal as well as the freed electron.
When a free electron meets a hole, they recombine(connect). This means that the electron neutralizes the excess positive charge that exists in the vicinity of the hole and loses freedom of movement until it again receives energy from the crystal lattice sufficient for its release. Recombination leads to the simultaneous disappearance of a free electron and a hole. On the level diagram, the recombination process corresponds to the transition of an electron from the conduction band to one of the free levels of the valence band.
So, two processes go on simultaneously in an intrinsic semiconductor: the birth of pairwise free electrons and holes and recombination, leading to the pairwise disappearance of electrons and holes. The probability of the first process increases rapidly with temperature. The recombination probability is proportional to both the number of free electrons and the number of holes. Therefore, each temperature corresponds to a certain equilibrium concentration of electrons and holes, which changes with temperature in proportion to the expression.
When there is no external electric field, conduction electrons and holes move randomly. When the field is turned on, an ordered motion is superimposed on the chaotic motion: electrons against the field and holes - in the direction of the field. Both motion - and holes, and electrons - leads to charge transfer along the crystal. Consequently, intrinsic electrical conductivity is determined, as it were, by charge carriers of two signs - negative electrons and positive holes.
Note that at a sufficiently high temperature, intrinsic conduction is observed in all semiconductors without exception. However, in semiconductors containing an impurity, the electrical conductivity is composed of intrinsic and impurity conductivities.
Impurity conductivity of semiconductors
Impurity conductivity arises if some atoms of a given semiconductor are replaced at the crystal lattice sites by atoms whose valency differs by one from the valency of the main atoms. Figure 6 conventionally shows the lattice of germanium with an admixture of pentavalent phosphorus atoms. A phosphorus atom needs four electrons to form covalent bonds with its neighbors. Consequently, the fifth valence electron turns out to be, as it were, superfluous and is easily split off from the atom due to the energy of thermal motion, forming a wandering free electron.
In contrast to the case considered in the previous paragraph, the formation of a free electron is not accompanied by the breaking of covalent bonds, i.e. hole formation. Although an excess positive charge arises in the vicinity of the impurity atom, it is bound to this atom and cannot move along the lattice.
Due to this charge, the impurity atom can capture an electron approaching it, but the bond of the captured electron with the atom will be fragile and easily broken again due to thermal vibrations of the lattice.
Thus, in a semiconductor with an impurity, the valency of which is one greater than the valency of the main atoms, there is only one type of current carriers - electrons. Accordingly, such a semiconductor is said to have electronic conductivity or is a semiconductor n- type (from the word negative - negative). Impurity atoms that supply conduction electrons are called donors.
