SEMICONDUCTOR COMPONENTS OF ELECTRONIC CIRCUITS
ELECTRICAL CONDUCTIVITY OF SEMICONDUCTORS
Semiconductors include materials that at room temperature have a specific electrical resistance from 10 -5 to 10 10 Ohm cm (in semiconductor technology it is customary to measure the resistance of 1 cm 3 of a material). The number of semiconductors exceeds the number of metals and dielectrics. The most commonly used are silicon, gallium arsenide, selenium, germanium, tellurium, various oxides, sulfides, nitrides and carbides.
Basic principles of the theory of electrical conductivity.
An atom consists of a nucleus surrounded by a cloud of electrons, which are in motion at some distance from the nucleus within layers (shells) determined by their energy. The farther a spinning electron is from the nucleus, the higher its energy level. Free atoms have a discrete energy spectrum. When an electron transitions from one allowed level to another, more distant one, energy is absorbed, and during the reverse transition, it is released. The absorption and release of energy can only occur in strictly defined portions - quanta. Each energy level can contain no more than two electrons. The distance between energy levels decreases with increasing energy. The “ceiling” of the energy spectrum is the ionization level at which an electron acquires energy that allows it to become free and leave the atom.
If we consider the structure of atoms of various elements, we can distinguish shells that are completely filled with electrons (internal) and unfilled shells (external). The latter are weaker connected to the nucleus and interact more easily with other atoms. Therefore, electrons located on the outer unfinished shell are called valence electrons.
Fig.2.1. The structure of bonds of germanium atoms in the crystal lattice and symbols of forbidden and allowed zones.
When molecules are formed, different types of bonds operate between individual atoms. For semiconductors, the most common are covalent bonds formed by sharing valence electrons with neighboring ones. For example, in silicon, an atom of which has four valence electrons, covalent bonds arise in the molecules between four neighboring atoms (Fig. 2.1a).
If atoms are in a bound state, then the valence electrons are acted upon by the fields of electrons and nuclei of neighboring atoms, as a result of which each individual allowed energy level of the atom is split into a number of new energy levels, the energies of which are close to each other. Each of these levels can also only contain two electrons. The set of levels, each of which can contain electrons, is called the allowed band (1; 3 in Fig. 2.1, b). The gaps between permitted zones are called prohibited zones (2 in Fig. 2.1, b). The lower energy levels of atoms usually do not form bands, since the internal electron shells in a solid interact weakly with neighboring atoms, being, as it were, “shielded” by the outer shells. In the energy spectrum of a solid, three types of bands can be distinguished: allowed (fully filled) bands, forbidden bands and conduction bands.
Allowed The zone is characterized by the fact that all its levels at a temperature of 0 K are filled with electrons. The upper filled band is called the valence band.
Prohibited The zone is characterized by the fact that within its boundaries there are no energy levels at which electrons could be located.
The conduction band is characterized by the fact that the electrons located in it have energies that allow them to free themselves from bonds with atoms and move inside a solid, for example, under the influence of an electric field.
The separation of substances into metals, semiconductors and dielectrics is carried out based on the band structure of the body at absolute zero temperature.
In metals, the valence band and conduction band mutually overlap, so at 0 K the metal has electrical conductivity.
For semiconductors and dielectrics, the conduction band at 0 K is empty and there is no electrical conductivity. The differences between them are purely quantitative - in the band gap ΔE. For the most common semiconductors ΔE=0.1÷3 eV (for semiconductors, on the basis of which they hope to create high-temperature devices in the future, ΔE=3÷6 eV), for dielectrics ΔE>6 eV.
In semiconductors, at a certain temperature value different from zero, some of the electrons will have energy sufficient to move into the conduction band. These electrons become free, and the semiconductor becomes electrically conductive.
The departure of an electron from the valence band leads to the formation of an unfilled energy level in it. The vacant energy state is called a hole. In the presence of an electric field, valence electrons from neighboring atoms can move to these free levels, creating holes elsewhere. This movement of electrons can be considered as the movement of positively charged fictitious charges—holes.
Electrical conductivity due to the movement of free electrons is called electronic, and electrical conductivity due to the movement of holes is called hole conductivity.
In an absolutely pure and homogeneous semiconductor at a temperature other than 0 K, free electrons and holes are formed in pairs, i.e. the number of electrons is equal to the number of holes. The electrical conductivity of such a semiconductor (intrinsic), due to paired carriers of thermal origin, is called intrinsic.
The process of formation of an electron-hole pair is called pair generation. In this case, the generation of a pair can be a consequence not only of the influence of thermal energy (thermal generation), but also of the kinetic energy of moving particles (impact generation), electric field energy, light irradiation energy (light generation), etc.
The electron and hole formed as a result of the rupture of the valence bond undergo chaotic motion in the volume of the semiconductor until the electron is “captured” by the hole, and the energy level of the hole is “occupied” by an electron from the conduction band. In this case, the broken valence bonds are restored, and the charge carriers—electron and hole—disappear. This process of restoring broken valence bonds is called recombination.
The period of time that elapses from the moment of generation of a particle that is a charge carrier until its recombination is called the lifetime, and the distance traveled by the particle during its lifetime is called the diffusion length. Since the lifetime of each carrier is different, to unambiguously characterize a semiconductor, the lifetime is most often understood as the average (statistical average) lifetime of charge carriers, and the diffusion length is the average distance that a charge carrier travels during the average lifetime. The diffusion length and lifetime of electrons and holes are related to each other by the relations
; 
(2,1)
where , is the diffusion length of electrons and holes;
, – lifetime of electrons and holes;
– diffusion coefficients of electrons and holes (density of charge carrier fluxes at a unit gradient of their concentrations).
The average lifetime of charge carriers is numerically defined as the period of time during which the concentration of charge carriers introduced in one way or another into the semiconductor decreases by e once ( e≈2,7).
If an electric field of intensity E is created in a semiconductor, then the chaotic movement of charge carriers will be ordered, i.e. holes and electrons will begin to move in mutually opposite directions, with holes in a direction coinciding with the direction of the electric field. Two counter-directed flows of charge carriers will arise, creating currents whose densities are equal
Jn dr = qnμ n E; Jp dr = qpμ p E,(2,2)
Where q– charge of the charge carrier (electron);
n, p– the number of electrons and holes per unit volume of a substance (concentration);
μ n , μ p – mobility of charge carriers.
The mobility of charge carriers is a physical quantity characterized by their average directional velocity in an electric field with a strength of 1 V/cm; μ =v/E, Where v– average carrier speed.
Since charge carriers of opposite sign move in opposite directions, the resulting current density in the semiconductor
J dr = Jn dr + Jp dr =( qnμ n +qpμ p)E (2.3)
The movement of charge carriers in a semiconductor, caused by the presence of an electric field and potential gradient, is called drift, and the current created by these charges is called drift current.
Movement under the influence of a concentration gradient is called diffusion.
The specific conductivity of a semiconductor σ can be found as the ratio of the specific current density to the electric field strength
σ =1/ρ= J/E=qnμ n +qpμ p,
where ρ is the resistivity of the semiconductor.
Impurity electrical conductivity. The electrical properties of semiconductors depend on the content of impurity atoms in them, as well as on various defects of the crystal lattice: empty lattice sites, atoms or ions located between lattice sites, etc. Impurities are acceptor and donor.
Acceptor impurities. Atoms of acceptor impurities are capable of accepting one or more electrons from the outside, turning into a negative ion.
If, for example, a trivalent boron atom is introduced into silicon, a covalent bond is formed between boron and four neighboring silicon atoms and a stable eight-electron shell is obtained due to the additional electron taken from one of the silicon atoms. This electron, being “bound,” turns the boron atom into a stationary negative ion (Figure 2.2, a). In place of the departed electron, a hole is formed, which is added to its own holes generated by heating (thermal generation). In this case, the concentration of holes in the semiconductor will exceed the concentration of free electrons of its own conductivity (p>n). Therefore in a semiconductor
Fig.2.2. Structure (a) and band diagram (b) of a semiconductor with acceptor impurities.
hole electrical conductivity will predominate. Such a semiconductor is called a p-type semiconductor.
When a voltage is applied to this semiconductor, the hole component of the current will dominate, i.e. J n
If the impurity content is small, which is most often the case, then their atoms can be considered isolated. Their energy levels are not split into zones. In the band diagram (Fig. 2.2b), impurity levels are depicted by dashes. The valence levels of the acceptor impurity are located in the lower part of the band gap, therefore, with a small additional energy (0.01 - 0.05 eV), electrons from the valence band can move to this level, forming holes. At low temperatures, the probability of electrons passing through the band gap is many times less than the probability of their transition from the valence band to the level of the acceptor impurity.
If the concentration of impurities in a semiconductor is high enough, then the acceptor impurity levels split, forming a band that can merge with the valence band. Such a semiconductor is called degenerate. In a degenerate semiconductor, the concentration of charge carriers with intrinsic electrical conductivity is significantly lower than in a non-degenerate one. Therefore, their qualitative feature is the low dependence of the semiconductor characteristics on the ambient temperature. In this case, the share of thermal charge carriers with intrinsic electrical conductivity compared to impurity carriers will be small.
Donor impurities. Atoms of donor impurities have valence electrons weakly associated with their nucleus (Fig. 2.3, a). These electrons, without participating in interatomic bonds, can easily move into the conduction band of the material into which the impurity was introduced. In this case, a positively charged ion remains in the lattice, and the electron is added to the free electrons
Fig.2.3. Structure (a) and band diagram (b) of a semiconductor with donor impurities.
own electrical conductivity. The donor level is located in the upper part of the band gap (Fig. 2.3, b). The transition of an electron from the donor level to the conduction band occurs when it receives a small additional energy. In this case, the concentration of free electrons in the semiconductor exceeds the concentration of holes and the semiconductor has electronic conductivity. Such semiconductors are called n-type semiconductors. If, for example, an atom of pentavalent phosphorus is introduced into silicon, then its four valence electrons will enter into a covalent bond with four electrons of silicon and will find themselves in a bound state (Fig. 2.3, a). The remaining phosphorus electron becomes free. In this case, the concentration of free electrons is higher than the concentration of holes, i.e. electronic conductivity predominates. As the impurity concentration increases, the donor levels split, forming a zone that can merge with the conduction band. The semiconductor becomes degenerate.
Charge carriers whose concentration predominates in the semiconductor are called major, and charge carriers whose concentration in the semiconductor is less than the main ones are called minority.
In an impurity semiconductor at low temperatures, impurity electrical conductivity predominates. However, as the temperature rises, the intrinsic electrical conductivity continuously increases, while the impurity conductivity has a limit corresponding to the ionization of all impurity atoms. Therefore, at sufficiently high temperatures, electrical conductivity is always intrinsic.
Drude's theory was developed in 1900, three years after the discovery of the electron. The theory was then refined by Lorentz and is now the classical and current theory of conductivity of metals.
Electronic Drude-Lorentz theory
According to the theory, current carriers in metals are free electrons.
Drude suggested that electrons in a metal obey and can be described by the equations of molecular kinetic theory. In other words, free electrons in a metal obey the laws of MKT and form an “electron gas.”
Moving in the metal, electrons collide with each other and with the crystal lattice (this is a manifestation of the electrical resistance of the conductor). Between collisions, electrons, by analogy with the mean free path of ideal gas molecules, manage to overcome the average path λ.
Without the action of an electric field accelerating electrons, the crystal lattice and electron gas tend to a state of thermal equilibrium.
Here are the main provisions of Drude's theory:
- The interaction of an electron with other electrons and ions is not taken into account between collisions.
- Collisions are instantaneous events that suddenly change the electron's speed.
- The probability for an electron to experience a collision per unit time is 1 τ.
- The state of thermodynamic equilibrium is achieved through collisions.
Despite many assumptions, the Drude-Loretzn theory explains well the Hall effect, the phenomenon of conductivity and the thermal conductivity of metals. That is why it is relevant to this day, although only the quantum theory of solids could provide answers to many questions (for example, why there are free ions and electrons in a metal).
Drude's theory explains the resistance of metals. It is caused by collisions of electrons with nodes of the crystal lattice.
Heat release, according to the Joule-Lenz law, also occurs due to the collision of electrons with lattice ions.
Heat transfer in metals is also carried out by electrons, and not by a crystal lattice.
Teria Drude does not explain many phenomena, such as superconductivity, and is not applicable in strong magnetic fields; in weak magnetic fields it may lose applicability due to quantum phenomena.
The average speed of electrons can be calculated using the formula for an ideal gas:
Here k is Boltzmann's constant, T is the metal temperature, m is the electron mass.
When an external electric field is turned on, the chaotic movement of particles of the “electron gas” is superimposed by the ordered movement of electrons under the influence of field forces, when the electrons begin to move in an orderly manner with an average speed u. The magnitude of this speed can be estimated from the relationship:
where j is the current density, n is the concentration of free electrons, q is the charge of the electron.
At high current densities, calculations give the following result: the average speed of chaotic motion of electrons is many times (≈ 10 8) greater than the speed of ordered motion under the influence of a field. When calculating the total speed, it is assumed that
u → + v → ≈ v →
Drude's formula
The Drude formula is derived from the Boltzmann kinetic equation and has the form:
σ = n q 2 τ m *
Here m * is the effective mass of the electron, τ is the relaxation time, that is, the time during which the electron “forgets” in which direction it moved after the collision.
Drude derived Ohm's law for currents in metal:
Tolman and Stewart Experience
In 1916, the experiment of Tolman and Stewart provided direct evidence that electrons are current carriers in metals.
The essence of the experience was as follows.
Tolman and Stewart Experience
A conducting coil with a wire of length L rotated around its axis at high speed, and its ends were closed to a galvanometer. When the coil was sharply slowed down, the free electrons in the metal continued to move by inertia, and the galvanometer recorded a current pulse.
Assuming that free electrons obey Newton's laws of mechanics, we can write that when the conductor stops, the electron acquires acceleration v " (directed along the wires in the coil). In this case, the electron is acted upon by a force directed opposite to the acceleration.
Under the influence of this force, the electron behaves as if it were acted upon by the field E = - m v " q. The emf arising in the coil during braking can be written as:
ε = ∫ L E d l = - m v " q ∫ L d l = - m v " q L
Assuming that the acceleration is the same in each turn, we can write Ohm’s law for the coil, and then calculate the charge passing through it during the time d t:
I R = - m v " q L
d q = I d t = - m L d v q R d t d t = - m L d v q R
The charge passed from the moment of braking to stopping:
q = - m L q R ∫ v 0 0 d v = - m L v 0 q R
The experiment of Tolman and Stewart was in good agreement with the theory; the experimentally obtained ratio q m corresponded to the ratio of the electron charge to its mass.
Example
At T = 300 K, calculate the average speed of thermal motion of free electrons.
Let's calculate the average speed using the formula for an ideal gas:
k = 1.38 10 - 23 J K
m = 9.31 10 - 31 k g
We substitute the values and calculate:
v = 8 1, 38 10 - 23 3 10 2 3, 14 9, 31 10 - 31 ≈ 10 5 m s
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Classical electronic theory of electrical conductivity of metals and its experimental justification. Wiedemann-Franz law.
Electric current in metals is the ordered movement of electrons under the influence of an electric field.
This assumption was experimentally confirmed in the experiment of K. Riecke (1911).
An electric current was passed through a chain of three successive cylinders - copper, aluminum and copper again - for a long time (about a year) - a total of 3.5 μC of charge passed through the cylinders. However, no traces of transfer of substances (copper or aluminum) were found. It followed that the electrical conductivity of metals was due to free charges common to all metals - only electrons were suitable for this role.
Another convincing proof of the electronic nature of current in metals was obtained in experiments with the inertia of electrons (the experiment of Tolman and Stewart) (1916).
A coil with a large number of turns of thin wire was driven into rapid rotation around its axis. The ends of the coil were connected using flexible wires to a sensitive ballistic galvanometer. The untwisted reel was sharply slowed down, and
a short-term current arose in the circuit due to the inertia of charge carriers. The total charge flowing through the circuit was measured with a galvanometer.
When braking a rotating coil, each charge carrier e with mass m is acted upon by a braking force, which plays the role of an external force, i.e. a force of non-electric origin:
The external force per unit charge is, by definition, the field strength of the external forces:
Consequently, in the circuit when the coil is braking, a electromotive force:
During the braking time of the coil charge q will flow into the circuit, equal:
Where is the length of the coil wire, I– instantaneous current value in the coil, R– total resistance of the circuit, – initial linear speed of the wire.
The value of the specific charge of current carriers in the metal obtained in the experiments turned out to be close to the specific charge of the electron 
The good electrical conductivity of metals is explained by high concentration of free electrons , equal in order of magnitude number of atoms per unit volume.
The assumption that electrons are responsible for the electric current in metals arose much earlier than the experiments of Tolman and Stewart. Back in 1900, the German scientist P. Drude, based on the hypothesis of the existence of free electrons in metals, created the electronic theory of metal conductivity. This theory was developed in the works of the Dutch physicist H. Lorentz and is called classical electron theory . According to this theory, electrons in metals behave like an electron gas, much like an ideal gas.
Electron gas fills the space between the ions that form the metal's crystal lattice. Due to interaction with ions, electrons can leave the metal only by overcoming the so-called potential barrier . The height of this barrier is called work function .
At ordinary (room) temperatures, electrons do not have enough energy to overcome the potential barrier. According to the Drude–Lorentz theory, electrons have the same average energy of thermal motion as the molecules of a monatomic ideal gas. This allows us to estimate the average speed of thermal motion of electrons using the formulas of molecular kinetic theory:
When an external electric field is applied to a metal conductor, in addition to the thermal movement of electrons, their ordered movement (drift), that is, an electric current, occurs. The value of the electron drift velocity lies within the range of 0.6 – 6 mm/s. Thus, the average speed of the ordered movement of electrons in metal conductors is many orders of magnitude less than the average speed of their thermal movement.
The low drift speed does not contradict the experimental fact that the current in the entire DC circuit is established almost instantly. Closing the circuit causes the electric field to propagate at a speed c= 3·10 8 m/s. After a while ( l– length of the chain) a stationary distribution of the electric field is established along the chain and the ordered movement of electrons begins in it.
In the classical electronic theory of metals, it is assumed that the movement of electrons obeys Newton's laws of mechanics. In this theory, the interaction of electrons with each other is neglected, and their interaction with positive ions is reduced only to collisions. It is also assumed that with each collision the electron transfers to the lattice all the energy accumulated in the electric field and therefore after the collision it begins to move with zero drift velocity.
Despite the fact that all these assumptions are very approximate, the classical electronic theory qualitatively explains the laws of electric current in metal conductors: Ohm's law, the Joule-Lenz law and explains the existence of electrical resistance of metals.
Ohm's law:
Electrical resistance of a conductor.
The classical theory of electrical conductivity of metals originated at the beginning of the twentieth century. Its founder was the German physicist Karl Rikke. He experimentally established that the passage of a charge through a metal does not involve the transfer of conductor atoms, unlike liquid electrolytes. However, this discovery did not explain what exactly is the carrier of electrical impulses in the metal structure.
The experiments of scientists Stewart and Tolman, conducted in 1916, allowed us to answer this question. They were able 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 electrical conductivity of metals. From this moment on, a new era of research into metal conductors began. Thanks to the results obtained, today we 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 electrical conductivity of metals was developed in the research 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 make it possible to classify different substances according to their conductivity level. From the results obtained, it is easy to understand which metal is suitable for making a particular cable. This is a very important point, since incorrectly selected material can cause a fire as a result of overheating from the passage of excess voltage current.
Silver metal has the highest electrical conductivity. At a temperature of +20 degrees Celsius, it is 63.3 * 104 centimeters-1. But making wiring from silver is very expensive, since it is a rather rare metal, which is used mainly for the production of jewelry and decorative items or bullion coins.
The metal with the highest electrical conductivity among all elements of the base 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 for household and industrial purposes. It withstands constant electrical loads well, is durable and reliable. The high melting point allows you to work for a long time in a heated state without problems.
In terms of abundance, 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. The 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 much lower conductivity than the pure substance. This is due to the merging of the structural network and, as a consequence, disruption of the normal functioning of electrons. For example, in the production of copper wire, a material with an impurity content of no more than 0.1% is used, and for some types of cable this indicator is even stricter - no more than 0.05%. All given 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 basic principles of the theory of electrical conductivity of metals contain six points. First: a high level of electrical conductivity is associated with the presence of a large number of free electrons. Second: electric current arises through external influence on the metal, during which electrons move from random motion to ordered motion.
Third: the strength of the current passing through a metal conductor is calculated according to Ohm's law. Fourth: different numbers of elementary particles in the crystal lattice lead to unequal resistance of metals. Fifth: electric current in the circuit arises instantly after the start of exposure to electrons. Sixth: as the internal temperature of the metal increases, the level of its resistance also increases.
The nature of the electrical conductivity of metals is explained by the second point of the provisions. In a quiet state, all free electrons rotate chaotically around the nucleus. At this moment, the metal is not able to independently reproduce electrical charges. But as soon as you connect an external source of influence, the electrons instantly line up in a structured sequence and become carriers of electric current. With increasing temperature, the electrical conductivity of metals decreases.
This is due to the fact that the molecular bonds in the crystal lattice weaken, elementary particles begin to rotate in an even more chaotic order, so the formation of electrons in a chain becomes more complicated. Therefore, it is necessary to take measures to prevent overheating of the conductors, as this negatively affects their performance properties. The mechanism of electrical conductivity of metals cannot be changed due to the current laws of physics. But it is possible to neutralize 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 at a high level, since their electrons are weakly attached to the nucleus and easily line up in the desired sequence. But this group is characterized by low melting points and enormous chemical activity, which in most cases does not allow their use for the manufacture of wires.
Metals with high electrical conductivity when opened are very dangerous for humans. Touching a bare wire will result in an electrical burn and a powerful discharge to all internal organs. This often results in instant death. Therefore, special insulating materials are used for the safety of people.
Depending on the application, they can be solid, liquid or gaseous. But all types are designed for one function - isolating the electrical current inside the circuit so that it cannot affect the outside world. The electrical conductivity of metals is used in almost all areas of modern human life, so ensuring safety is a top priority.
Current carriers in metals are free electrons, i.e. electrons weakly bound to the ions of the metal crystal lattice. This idea of the nature of current carriers in metals is based on the electronic theory of conductivity of metals, created by the German physicist P. Drude and subsequently developed by the Dutch physicist H. Lorentz, as well as on a number of classical experiments confirming the provisions of the electronic theory.
The first of these experiments - Rikke's experience(1901), in which for a year an electric current was passed through three metal cylinders (Cu, Al, Cu) of the same radius connected in series with carefully polished ends. Despite the fact that the total charge passing through these cylinders reached a huge value (C), no, even microscopic, traces of matter transfer were found. This was experimental evidence that ions in metals do not participate in the transfer of electricity, and charge transfer in metals is carried out by particles that are common to all metals. Such particles could be the discovery of electrons in 1897 by the English physicist D. Thomson.
To prove this assumption, it was necessary to determine the sign and magnitude of the specific charge of carriers (the ratio of the charge of the carrier to its mass). The idea of such experiments was as follows: if there are mobile current carriers in the metal, weakly connected to the lattice, then when the conductor is sharply decelerated, these particles should move forward by inertia. The result of the displacement of charges should be a current pulse; By the direction of the current, you can determine the sign of the current carriers, and knowing the dimensions and resistance of the conductor, you can calculate the specific charge of the carriers. These experiments were carried out in 1916 by the American physicist R. Tolman and the Scottish physicist B. Stewart. They experimentally proved that current carriers in metals are negatively charged, and their specific charge is approximately the same for all metals studied. Based on the specific charge of electric current carriers and the previously determined elementary electric charge, their mass was determined. It turned out that the values of the specific charge and mass of current carriers in metals and electrons moving in vacuum coincided. Thus, it was finally proven that the carriers of electric current in metals are free electrons.
The existence of free electrons in metals can be explained as follows: during the formation of a metal crystal lattice (as a result of the approach of isolated atoms), valence electrons, relatively weakly bound to atomic nuclei, are detached from the metal atoms, become “free” and can move throughout the entire volume. Thus, metal ions are located at the nodes of the crystal lattice, and free electrons move chaotically between them, forming a kind of electron gas, which, according to the electronic theory of metals, has the properties of an ideal gas.
During their movement, conduction electrons collide with lattice ions, as a result of which thermodynamic equilibrium is established between the electron gas and the lattice. According to the Drude-Lorentz theory, electrons have the same energy of thermal motion as the molecules of a monatomic gas.
The thermal motion of electrons, being chaotic, cannot lead to the generation of current.
When an external electric field is applied to a metal conductor, in addition to the thermal motion of electrons, their ordered motion occurs, i.e. an electric current occurs.
Even at very high current densities, the average speed of the ordered movement of electrons, causing an electric current, is significantly less than their speed of thermal movement. Therefore, when calculating, the resulting speed can be replaced by the speed of thermal movement .
1. Ohm's law. Let there be an electric field of intensity E=const in a metal conductor. From the field side, charge e experiences the action of force F=eE and acquires acceleration. Thus, during the free run, the electrons move uniformly accelerated, acquiring speed at the end of the free run
,
Where
According to Drude's theory, at the end of the free path, an electron, colliding with lattice ions, gives them the energy accumulated in the field, so the speed of its ordered movement becomes equal to zero. Consequently, the average speed of directional motion of an electron
. (9.5.1.)
The classical theory of metals does not take into account the electron velocity distribution, so the average time
Substituting the value
.
Current density in a metal conductor
E,
from which it can be seen that the current density is proportional to the field strength, i.e. received Ohm's law in differential form. The proportionality coefficient between j and E is nothing more than the conductivity of the material
, (9.5.2.)
which is greater, the greater the concentration of free electrons and their average free path.
Joule-Lenz law.
By the end of the free path, the electron under the influence of the field acquires additional kinetic energy
. (9.5.3.)
When an electron collides with an ion, this energy is completely transferred to the lattice and goes to increase the internal energy of the metal, i.e. to heat it up.
Per unit time, an electron experiences an average of
If n is the electron concentration, then n
, (9.5.5.)
which goes to heating the conductor. Substituting (9.5.3.) and (9.5.4.) into (9.5.5.), we thus obtain the energy transferred to the lattice per unit volume of the conductor per unit time,
. (9.5.6.)
The quantity w is called the specific thermal power of the current. The proportionality coefficient between w and according to (9.5.2.) is specific conductivity; therefore, expression (9.5.6.) is the Joule-Lenz law in differential form.
The classical theory of electrical conductivity of metals explained the Ohm and Joule-Lenz laws, and also gave a qualitative explanation of the Wiedemann-Franz law. However, in addition to the considered contradictions in the Wiedemann-Franz law, she also encountered a number of difficulties in explaining various experimental data. Let's look at some of them.
Temperature dependence of resistance. From the formula for specific conductivity (9.5.2.) it follows that the resistance of metals, i.e. a quantity inversely proportional to , must increase proportionally (in (9.5.2.) n and< >do not depend on temperature, but ~ ). This conclusion of the electronic theory contradicts experimental data, according to which R~T.
Estimation of the mean free path of electrons in metals. In order to obtain, using formula (9.5.2.), coinciding with the experimental values, we must take< >significantly more than true, in other words, to assume that the electron travels hundreds of interstitial distances without collisions with lattice ions, which is not consistent with the Drude-Lorentz theory.
Heat capacity of metals. The heat capacity of a metal is the sum of the heat capacity of its crystal lattice and the heat capacity of the electron gas. Therefore, the atomic (i.e., calculated per 1 mole) heat capacity of the metal should be significantly greater than the atomic heat capacity of dielectrics, which do not have free electrons. According to Dulong and Petit's law, the heat capacity of a monatomic crystal is 3R. Let us take into account that the heat capacity of a monatomic electron gas is equal to . Then the atomic heat capacity of metals should be close to 4.5R. However, experience proves that it is equal to 3R, i.e. for metals, as well as for dielectrics, the law of Dulong and Petit is well satisfied. Consequently, the presence of conduction electrons has virtually no effect on the value of the heat capacity, which is not explained by classical electronic theory.
These discrepancies between theory and experiment can be explained by the fact that the movement of electrons in metals obeys not the laws of classical mechanics, but the laws of quantum mechanics and, therefore, the behavior of conduction electrons should be described not by Maxwell-Boltzmann statistics, but by quantum statistics. Therefore, the difficulties of the elementary theory of electrical conductivity of metals can be explained only by quantum theory, which will be discussed later. It should be noted, however, that the classical electronic theory has not lost its significance to this day, since in many cases (for example, at a low concentration of conduction electrons and high temperature) it gives correct qualitative results and is simple and simple in comparison with quantum theory. visual.
