Classical electronic theory of Drude-Lorentz conductivity. Electronic theory of conductivity of metals 36 main provisions of the classical electronic theory of conductivity

Estimation of the mean free path of electrons in metals. In order to obtain, according to the formula (9.5.2.), coinciding with the experimental values, it is necessary to take< >much more than the true ones, in other words, to assume that the electron passes hundreds of interstitial distances without collisions with lattice ions, which is inconsistent 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 mol) heat capacity of the metal must be much greater than the atomic heat capacity of dielectrics, which do not have free electrons. According to the Dulong and Petit 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 . 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 Dulong and Petit law is well fulfilled. Consequently, the presence of conduction electrons has practically no effect on the value of heat capacity, which is not explained by the classical electron theory.

These discrepancies between theory and experience can be explained by the fact that the motion of electrons in metals is subject not to the laws of classical mechanics, but to the laws of quantum mechanics and, therefore, the behavior of conduction electrons must be described not by Maxwell-Boltzmann statistics, but by quantum statistics. Therefore, the difficulties of the elementary theory of the electrical conductivity of metals can be explained only by quantum theory, which will be considered later. It should be noted, however, that the classical electron theory has not lost its significance even up to the present time, 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 clear in comparison with quantum theory.

Lecture plan

5.1. Classical theory of electrical conductivity of metals.

5.2. Derivation of Ohm's law and the Joule-Lenz law.

5.3. Disadvantages of the classical theory of electrical conductivity of metals.

Classical theory of electrical conductivity of metals

Any theory is considered complete only if it traces the path from the elementary mechanism of the phenomenon to the macro-relations found in it, which are used in technical practice. In this case, it was irresistible to connect the features of the ordered movement of free charges in a conductor (electrical conductivity) with the basic laws of electric current. First of all, it was necessary to elucidate the nature of current carriers in metals. Fundamental in this sense were the experiments of Rikke 1 , in which for a long time ( ≈ year) the current was passed through three metal cylinders connected in series ( Cu, A1, Cu) of the same section with carefully polished ground ends. A huge charge flowed through this circuit (≈ 3.5 10 6 C). Despite this, no (even microscopic) traces of the transfer of matter from cylinder to cylinder were found (which was confirmed by careful weighing). From this it was concluded that in metals, in the process of transfer of electric charge, some particles are involved that are common (identical) for all metals.

The nature of such particles could be determined by the sign and magnitude of the specific charge (the ratio of the charge of the carrier to its mass) - an individual parameter for any of the microparticles known today. The idea of ​​such an experiment is as follows: during a sharp deceleration of a metal conductor, current carriers weakly connected to the grating should move forward by inertia. The result of such a shift is a current pulse, and the sign of the carriers can be determined from the direction of the current, and, knowing the dimensions and resistance of the conductor, it is possible to calculate the specific charge of the carriers. Such experiments gave values ​​of the ratio , which coincided with the specific charge of electrons. Thus, it was finally proved that the carriers of electric current in metals are free electrons. When a crystal lattice of a metal is formed (when isolated atoms approach each other), valence electrons weakly bound to the nuclei break away from the metal atoms, become “free” and can move throughout the volume. Thus, metal ions are located at the nodes of the crystal lattice, and free electrons move randomly between them.

The founders of the classical theory of electrical conductivity of metals, Drude 2 and Lorentz 3, showed for the first time that any set of noninteracting microparticles

Rikke Carl Victor Eduard (1845 - 1915), German physicist

2 Drude Paul Karl Ludwig (1863 - 1906), German physicist

3 Lorenz Hendrik Anton (1853 - 1928), Dutch theoretical physicist

particles (including free electrons in a metal) can be considered as an ideal gas, that is, all the conclusions of the molecular kinetic theory are applicable to free electrons in a metal.

The conduction electrons collide with the ions of the lattice during their movement, as a result of which a thermodynamic equilibrium is established between the ideal gas of free electrons and the lattice. The average speed of free electrons can be found in accordance with the expression for the arithmetic average speed of the chaotic thermal motion of ideal gas molecules (see formula (8.26) in lecture 8, part I):

which at room temperature (T ≈ 300 K) gives<u> = 1.1 10 5 m/s.

When an external electric field is applied to a conductor, in addition to the thermal motion of electrons, their ordered motion also arises, that is, an electric current. The average speed of the ordered movement of electrons -<v> can be determined according to (4.4). At the maximum allowable current density in real conductors (≈ 10 7 A / m 2), a quantitative estimate gives<v> ≈ 10 3 -10 4 m/s. Thus, even in limiting cases, the average speed of the ordered motion of electrons (causing the electric current) is much less than their speed of chaotic thermal motion (<v> << <u>). Therefore, when calculating the resulting velocity, we can assume that (<v> + <u>) ≈ <u>. It has already been noted above that the ultimate goal of the classical theory of electrical conductivity of metals is the derivation of the main laws of electric current, based on the considered elementary mechanism of the movement of current carriers. As an example, consider how this was done when deriving Ohm's law in differential form.

5.2. Derivation of Ohm's law and Joule–Lenz law

Let there be an electric field in a metal conductor with intensity . From the side of the field, the electron experiences the action of the Coulomb force F = eE and gains momentum. According to Drude theory at the end of the mean free path<l> an electron collides with an ion of the lattice, gives off the energy accumulated during movement in the field (the speed of its ordered movement becomes equal to zero). Moving with uniform acceleration, an electron acquires a speed by the end of its free path , Where is the average time between two successive collisions of an electron with lattice ions. The average speed of the directed motion of an electron is equal to

Because (<v> + <u>) ≈ <u>, then and (5.1) takes the form . Thus, the current density, according to (4.4), can be represented as

. (5.2)

Comparing this expression with Ohm's law in differential form, it can be seen that these expressions are identical, provided that the conductivity

Thus, within the framework of the classical theory of the electrical conductivity of metals, Ohm's law was derived in differential form.

Similarly, the Joule-Lenz law was derived, a quantitative relationship was obtained between specific conductivity and thermal conductivity, taking into account the fact that in metals the transfer of electricity and heat is carried out by the same particles (free electrons) and a number of other relationships.

SEMICONDUCTOR COMPONENTS OF ELECTRONIC CIRCUITS

ELECTRICAL CONDUCTIVITY OF SEMICONDUCTORS

Semiconductors include materials that at room temperature have electrical resistivity 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 provisions of the theory of electrical conductivity.

An atom consists of a nucleus surrounded by a cloud of electrons that are in motion at some distance from the nucleus within layers (shells) determined by their energy. The farther from the nucleus is a rotating electron, the higher its energy level. Free atoms have a discrete energy spectrum. During the transition of an electron from one allowed level to another, more distant, energy is absorbed, and during the reverse transition, it is released. Absorption and release of energy can occur only 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 level of ionization at which the electron acquires the energy that allows it to become free and leave the atom.

If we consider the structure of atoms of various elements, then we can distinguish shells that are completely filled with electrons (internal) and unfilled shells (external). The latter are weaker bound 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 the bonds of germanium atoms in the crystal lattice and symbols of forbidden and allowed zones.

During the formation of molecules, various types of bonds act between individual atoms. For semiconductors, the most common are covalent bonds formed due to the socialization of neighboring valence electrons. For example, in silicon, an atom of which has four valence electrons, covalent bonds arise in molecules between four neighboring atoms (Fig. 2.1, a).

If the atoms are in a bound state, then the fields of electrons and nuclei of neighboring atoms act on valence electrons, 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 contain only two electrons. The set of levels, each of which can contain electrons, is called the allowed zone (1; 3 in Fig. 2.1, b). The gaps between the allowed zones are called forbidden zones (2 in Fig. 2.1, b). The lower energy levels of atoms usually do not form zones, since the internal electron shells in a solid interact weakly with neighboring atoms, being, as it were, "shielded" by the outer shells. Three types of bands can be distinguished in the energy spectrum of a solid body: allowed (completely 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 top filled band is called the valence band.

Forbidden the zone is characterized by the fact that within its limits there are no energy levels on which electrons could be located.

The conduction band is characterized by the fact that the electrons in it have energies that allow them to be released from bonds with atoms and move inside a solid body, for example, under the influence of an electric field.

The separation of substances into metals, semiconductors and dielectrics is performed based on the band structure of the body at absolute zero temperature.

In metals, the valence band and the conduction band mutually overlap, so at 0 K the metal has electrical conductivity.

In 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. The most common semiconductors have 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 other than zero, some of the electrons will have enough energy to pass 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. The valence electrons of neighboring atoms in the presence of an electric field can move to these free levels, creating holes elsewhere. Such movement of electrons can be considered as the movement of positively charged fictitious charges-holes.

The electrical conductivity due to the movement of free electrons is called electronic, and the electrical conductivity due to the movement of holes is called hole.

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 of not only the impact of thermal energy (thermal generation), but also the kinetic energy of moving particles (impact generation), the energy of the electric field, the energy of light irradiation (light generation), etc.

The electron and hole formed as a result of breaking the valence bond perform chaotic motion in the bulk 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, the electron and the hole, disappear. This process of repairing broken valence bonds is called recombination.

The time interval that has elapsed from the moment of generation of a particle, which is a charge carrier, to 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 of the carriers is different, for an unambiguous characteristic of a semiconductor, the lifetime is most often understood as the average (statistically 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 by the relations

; (2,1)

where , is the diffusion length of electrons and holes;

, is the lifetime of electrons and holes;

are the diffusion coefficients of electrons and holes (density of charge carrier fluxes at a unit concentration gradient).

The average lifetime of charge carriers is numerically defined as the time interval during which the concentration of charge carriers introduced into a semiconductor in one way or another decreases by e once ( e≈2,7).

If an electric field of intensity E is created in a semiconductor, then the chaotic motion of charge carriers will be ordered, i.e. holes and electrons will begin to move in mutually opposite directions, and holes - in the direction coinciding with the direction of the electric field. Two oppositely directed flows of charge carriers will arise, creating currents whose densities are equal to

J n dr = qnμ nE; Jp dr = qpμ p E,(2,2)

Where q– the charge of the charge carrier (electron);

n,p is 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 directed 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 = J n 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 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 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 in 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 outside, turning into a negative ion.

If, for example, a trivalent boron atom is introduced into silicon, then a covalent bond is formed between boron and four neighboring silicon atoms, and a stable eight-electron shell is obtained due to an additional electron taken from one of the silicon atoms. This electron, being “bound”, turns the boron atom into a fixed 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 intrinsic conduction (p>n). Therefore, in a semiconductor

Fig.2.2. Structure (a) and band diagram (b) of a semiconductor with acceptor impurities.

hole conductivity will prevail. Such a semiconductor is called a p-type semiconductor.

When a voltage is applied to this semiconductor, the hole component of the current will prevail, i.e. J n