Classical electronic theory of Drude-Lorentz conduction. Electronic theory of conductivity Basic principles of the theory of electrical conductivity

From the standpoint of classical electronic theory, the high electrical conductivity of metals is due to the presence of a huge number of free electrons, the movement of which obeys the laws of classical Newtonian mechanics. In this theory, the interaction of electrons with each other is neglected, and their interaction with positive ions is reduced only to collisions. In other words, conduction electrons are considered as an electron gas, similar to a monatomic, ideal gas. Such an electron gas must obey all the laws of an ideal gas. Consequently, the average kinetic energy of the thermal motion of an electron will be equal to , where is the mass of the electron, is its root-mean-square velocity, k is Boltzmann’s constant, T is the thermodynamic temperature. Hence, at T = 300 K, the root-mean-square speed of thermal motion of electrons is »10 5 m/s.

The chaotic thermal movement of electrons cannot lead to the emergence of an electric current, but under the influence of an external electric field, an ordered movement of electrons occurs in a conductor at a speed of . The value can be estimated from the previously derived relationship, where j is the current density, is the electron concentration, e is the electron charge. As the calculation shows, "8×10 -4 m/s. The extremely small value of the value compared to the value is explained by the very frequent collisions of electrons with lattice ions. It would seem that the result obtained for contradicts the fact that the transmission of an electrical signal over very long distances occurs almost instantly. But the fact is that the closure of an electrical circuit entails the propagation of an electric field at a speed of 3 × 10 8 m/s (the speed of light). Therefore, the ordered movement of electrons at speed under the influence of the field will occur almost immediately along the entire length of the circuit, which ensures instantaneous signal transmission.

On the basis of classical electronic theory, the basic laws of electric current discussed above were derived - Ohm's and Joule-Lenz's laws in differential form and. In addition, the classical theory provided a qualitative explanation of the Wiedemann-Franz law. In 1853, I. Wiedemann and F. Franz established that at a certain temperature the ratio of the thermal conductivity coefficient l to the specific conductivity g is the same for all metals. Wiedemann-Franz law has the form , where b is a constant independent of the nature of the metal. The classical electron theory explains this pattern as well. Conduction electrons, moving in a metal, carry with them not only an electric charge, but also the kinetic energy of random thermal motion. Therefore, those metals that conduct electricity well are good conductors of heat. The classical electronic theory qualitatively explained the nature of the electrical resistance of metals. In an external field, the ordered movement of electrons is disrupted by their collisions with positive ions of the lattice. Between two collisions, the electron moves at an accelerated rate and acquires energy, which it gives back to the ion during a subsequent collision. We can assume that the movement of an electron in a metal occurs with friction similar to internal friction in gases. This friction creates resistance in the metal.

However, the classical theory encountered significant difficulties. Let's list some of them:

1. A discrepancy between theory and experiment arose when calculating the heat capacity of metals. According to kinetic theory, the molar heat capacity of metals should be the sum of the heat capacity of atoms and the heat capacity of free electrons. Since atoms in a solid body perform only vibrational movements, their molar heat capacity is equal to C=3R (R=8.31 J/(mol×K) - molar gas constant); free electrons move only translationally and their molar heat capacity is equal to C=3/2R. The total heat capacity should be C»4.5R, but according to experimental data C=3R.

They also knew that the carriers of electric current in metals are negatively charged electrons. All that remained was to create a description of electrical resistance at the atomic level. The first attempt of this kind was made in 1900 by the German physicist Paul Drude (1863-1906).

The meaning of the electronic theory of conductivity comes down to the fact that each metal atom gives up a valence electron from the outer shell, and these free electrons spread throughout the metal, forming a kind of negatively charged gas. In this case, the metal atoms are combined into a three-dimensional crystal lattice, which practically does not interfere with the movement of free electrons inside it ( cm. Chemical bonds). As soon as an electrical potential difference is applied to a conductor (for example, by shorting two terminals of a battery at its two ends), free electrons begin to move in an orderly manner. At first they move uniformly accelerated, but this does not last long, since very soon the electrons stop accelerating, colliding with lattice atoms, which, in turn, begin to oscillate with increasing amplitude relative to the conditional rest point, and we observe the thermoelectric effect of heating the conductor.

These collisions have a decelerating effect on electrons, similar to how, say, it is difficult for a person to move at a sufficiently high speed in a dense crowd of people. As a result, the speed of electrons is set at a certain average level, which is called migration speed, and this speed, in fact, is by no means high. For example, in ordinary household electrical wiring, the average speed of electron migration is only a few millimeters per second, that is, electrons do not fly along the wires, but rather crawl along them at a pace worthy of a snail. The light in a light bulb comes on almost instantly only because all these slow electrons start moving. simultaneously, as soon as you press the switch button, the electrons in the coil of the light bulb also begin to move immediately. That is, by pressing the switch button, you produce an effect in the wires similar to what would happen if you turned on a pump connected to a watering hose filled to capacity with water - a stream at the end opposite to the pump will rush out of the hose immediately.

Drude took the description of free electrons very seriously. He assumed that inside a metal they behave like an ideal gas, and applied to them the ideal gas equation of state, quite fairly drawing an analogy between the collisions of electrons and the thermal collisions of molecules of an ideal gas. This allowed him to formulate the formula for electrical resistance as a function of the average time between collisions of free electrons with atoms of the crystal lattice. Like many simple theories, the electronic theory of conductivity is good at describing some basic phenomena in the field of electrical conductivity, but is powerless to describe many of the nuances of this phenomenon. In particular, it not only does not explain the phenomenon of superconductivity at ultra-low temperatures ( cm. The theory of superconductivity, on the contrary, predicts an unlimited increase in the electrical resistance of any substance as its temperature tends to absolute zero. Therefore, today the electrically conductive properties of matter are usually interpreted within the framework of quantum mechanics ( cm.

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.

When molecules are formed, different types of bonds operate between individual atoms. For semiconductors, the most common are covalent bonds formed by sharing the valence electrons of neighboring atoms. For example, in germanium, the atom of which has four valence electrons, covalent bonds arise in molecules between four neighboring atoms (Fig. 2.1, a).

Rice. 2.1. Structure of bonds of the germanium atom in the crystal lattice (a) and symbols of forbidden and allowed (b)

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 in Fig. . The gaps between permitted zones are called prohibited zones (2 in Fig.). 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.

The allowed band 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.

The forbidden band is characterized by the fact that within its limits 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 of the AE. The most common semiconductors (semiconductors on the basis of which they hope to create high-temperature devices in the future) in dielectrics.

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.

Valence electrons from neighboring atoms, in the presence of an electric field, 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 forming 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 charge carrier is different, for an unambiguous characteristic of a semiconductor, the lifetime is most often understood as the average (statistical average) lifetime of the charge carriers, and the diffusion length is the average distance that the charge carrier travels during the average lifetime. The diffusion length and lifetime of electrons and holes are related to each other by the relations

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 a semiconductor decreases by a factor of ().

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, and the holes will move in the direction coinciding with the direction of the electric field. Two counter-directed flows of charge carriers will arise, creating currents whose densities are equal

where q is the charge of the charge carrier (electron); - the number of electrons and holes per unit volume of the substance; , - mobility of charge carriers.

The mobility of charge carriers is a physical quantity characterized by their average directional speed in an electric field with intensity , where v is the average speed of the carrier.

Since charge carriers of opposite sign move in the opposite direction, the resulting current density in the semiconductor

The movement of charge carriers in a semiconductor, caused by the presence of an electric field and a 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 a can be found as the ratio of the specific current density to the electric field strength:

where is the resistivity of the semiconductor.

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.

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. 2.1, a).

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 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.

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 forming 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)

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. Jn

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 of intrinsic electrical conductivity is significantly less 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.

It should be remembered that if there are n nodes in a branched chain, then independent equations according to the first rule can be written for (n – 1) nodes. When applying the second rule, each subsequent circuit must be chosen so that it contains at least one section of the circuit that was not included in the previously considered circuits. Thus, using formulas (3.145) and (3.146), we obtain a system of equations, which should be solved to find the parameters of the branched chain that are unknown according to the problem conditions.

3.11 Classical electronic theory of electrical conductivity of metals

Current carriers in metals, as has been experimentally established, are electrons. Based on the idea of the presence of free electrons in metals, Drude and Lorentz created the classical electronic theory of conductivity of metals.

The existence of free electrons in metals can be explained by the fact that during the formation of a crystal lattice as a result of the approach of atoms and the interaction between them, valence electrons relatively weakly associated with the nucleus are detached from the metal atoms, become free and can move throughout the entire volume of the metal. Thus, metal ions are located at the nodes of the crystal lattice, and free electrons move chaotically between them. In the classical Drude-Lorentz electronic theory, conduction electrons behave like ideal gas molecules, however, unlike ideal gas molecules, electrons collide primarily not with each other, but with ions of the crystal lattice. These collisions lead to the establishment of thermal equilibrium between the electron gas and the crystal lattice, and, therefore, the electron gas has the same temperature as the entire metal. Extending the results of the kinetic theory of gases to electron gas, the average speed of thermal motion of electrons can be estimated using the formula:

where m e 9, 1 10 31 kg is the mass of the electron. For room temperature

(T ~ 300 K) calculation using formula (3.147) gives a value of 10 5 m/s.

When an electric field is turned on, the chaotic thermal motion of electrons is superimposed on the ordered motion of electrons

(an electric current arises) with an average speed u, which can be estimated based on the formula:

j en0 u .

The maximum permissible current density, for example, for copper wires is of the order of 10 7 A/m 2, and the concentration of valence electrons for copper n 0 ~ 10 29 m – 3. This gives for u 10 3 m/s. So

thus, u.

Drude believed that when an electron collides with a node of a crystal lattice, the energy acquired by the electron along the mean free path

conductor is homogeneous and under its action the electron moves with acceleration after a collision

and by the end of the free run it will acquire an average speed

where is the average time between two consecutive

collisions.

Drude did not take into account the Maxwellian velocity distribution of electrons

attributed to everyone

electrons

the same

speed equal

Hence

Average

free length

electron. Speed

changes linearly during the free path at a const, therefore

u max

2m e

Substituting this expression into (3.148), we get:

ne2 E

and, recalling Ohm’s law in differential form, we obtain for electrical conductivity:

Note that, in accordance with the classical theory of electrical conductivity, the resistance of metals is due to collisions of electrons with ion nodes of the crystal lattice. For the Joule–Lenz law in differential form, Drude obtained

using the fact that at the mean free path the electron acquires additional kinetic energy

e 2 2

2m2

which he completely transfers to the crystal lattice, and since

collisions, then energy should be released per unit time in a unit volume

mu max2

E2,

Lorentz subsequently improved Drude's theory by applying Maxwell-Boltzmann statistics, and showed that the same results can be reached by considering the collisions of electrons with lattice points absolutely

elastic, and received the expression:

n2 e2

The classical Drude-Lorentz theory could not explain a number of experimentally observed phenomena. So from experience it follows that ~ T, and from

(3.154) it follows that ~ T. When estimating the mean free path using formulas (3.154) and (3.158), substituting experimental

greater than the interatomic distance, i.e. we have to assume that the electron travels hundreds of interstitial distances without collisions with lattice ions. Finally, for electron gas the classical theory

predicted the electronic contribution to the molar heat capacity 3 2 R. However,

It follows from the experiment that this contribution to the heat capacity of metals turns out to be negligible. The listed disadvantages were overcome only in the quantum mechanical theory of electrical conductivity.




and by the end of the free run it will acquire an average speed




where is the average time between two consecutive					collisions.