proportional to T : n ~T. Therefore, the thermal conductivity coefficient should be inversely proportional to temperature, which is in qualitative agreement with experiment. At temperatures below the Debye temperature, l is practically independent of T, and the thermal conductivity is entirely determined by the dependence on T of the heat capacity of the crystal C V ~ T 3 . Therefore, at low temperaturesλ ~T 3 . The characteristic dependence of thermal conductivity on temperature is shown in Figure 9.

In metals, in addition to lattice thermal conductivity, it is also necessary to take into account thermal conductivity due to heat transfer by free electrons. It explains the high thermal conductivity of metals in comparison with non-metals.

3. Electronic structure of crystals.

3.1 Movement of electrons in a periodic field. Band structure of the energy spectrum of electrons in a crystal. Bloch functions. dispersion curves. effective mass.

In a solid, the distances between atoms are comparable to their sizes. Therefore, the electron shells of neighboring atoms partially overlap each other, and at least the valence electrons of each atom are in a sufficiently strong field of neighboring atoms. An exact description of the motion of all electrons, taking into account the Coulomb interaction of electrons with each other and with atomic nuclei, is an extremely difficult task, even for a single atom. Therefore, the self-consistent field method is usually used, in which the problem is reduced to describing the motion of each individual electron in the effective potential field created by atomic nuclei and the averaged field of other electrons.

Let us first consider the structure of the energy levels of a crystal, based on the tight-binding approximation, in which it is assumed that the binding energy of an electron with its atom significantly exceeds the kinetic energy of its movement from atom to atom. At large distances between atoms, each of them has a system of narrow energy levels corresponding to the bound states of an electron with an ion. As the atoms approach each other, the width and height of the potential barriers between them decrease, and due to the tunneling effect, electrons get the opportunity to move from

one atom to another, which is accompanied by the expansion of energy levels and their transformation into energy zones.(Fig. 10). This is especially true of weakly bound valence electrons, which are able to easily move through the crystal from atom to atom, and to a certain extent become similar to free electrons. The electrons of the deeper energy levels are each much more strongly bound to their own atom. They form narrow energy bands with wide ranges of forbidden energies. On fig. 10 conventionally shows the potential curves and energy levels for the Na crystal. The general nature of the energy spectrum of electrons depending on the internuclear distance, d, is shown in Figure 11. In some cases, the upper levels are broadened so strongly that neighboring energy bands overlap. On fig. 11 this is the case for d = d1 .

Based on the Heisenberg-Bohr uncertainty relation, the width of the energy band, ∆ε , is related to the residence time τ of an electron at a certain lattice site by the relation: ∆ε τ > h. Due to the tunneling effect, the electron can seep through the potential barrier. According to the estimate, at an interatomic distance d ~ 1Aτ ~ 10 -15 s, and therefore ∆ε ~ h/τ ~ 10 -19 J ~ 1 eV, i.e. the band gap is on the order of one or several eV. If a crystal consists of N atoms, then each energy band consists of N sublevels. A 1 cm3 crystal contains N ~ 1022 atoms. Consequently, at a band width of ~ 1 eV, the distance between sublevels is ~ 10 -22 eV, which is much less than the energy of thermal motion under normal conditions. This distance is so negligible that in most cases the zones can be considered practically continuous.

In an ideal crystal, the nuclei of atoms are located at the nodes of the crystal lattice, forming a strictly periodic structure. In accordance with this, the potential energy of an electron, V(r ) , also periodically depends on the spatial coordinates, i.e. has translational symmetry:

lattices, a i (i = 1,2,3,…) are vectors of basic translations.

Wave functions and energy levels in a periodic field (1) are determined by solving the Schrödinger equation

which are the product of the equation of a plane traveling wave, ei kr, and a periodic factor, uk (r) = uk (r + a n ), with a lattice period. Functions (3) are called Bloch functions.

For V(r) = 0, Eq. (2) has a solution in the form of a plane wave:

where m is the particle mass. The dependence of the energy E on the wavenumber is shown dispersion curve. According to (5), in the case of a free electron, this is a parabola. By analogy with free motion, the vector k in equation (3) is called the wave vector, and p = h k is called the quasi-momentum.

In the weak coupling approximation, we consider the motion of almost free electrons, which are affected by the perturbing field of the periodic potential of the ionic cores. In contrast to free motion, in a periodic field V(r) Eq. (2) does not have solutions for all values ​​of E . Regions of allowed energies alternate with zones of forbidden energies. In the weak coupling model, this is explained by the Bragg reflection of electron waves in the crystal.

Let's consider this question in more detail. The condition of maximum reflection of electron waves in a crystal (Wulff-Bragg condition) is determined by formula (17) part I. Considering that G = n g, from here we get:

Consider a system of finite intervals that do not contain values ​​of k satisfying relation (7):

( - n g /2

The area of ​​change k in three-dimensional k - space, given by the formula

(8) for all possible directions, defines the boundaries of the n-th Brillouin zone. Within each Brillouin zone (n= 1,2,3,…) the energy of an electron is a continuous functionk, and at the boundaries of the zones it suffers a discontinuity. Indeed, when condition (7) is satisfied, the amplitude of the incident,

ψ k (r ) = uk (r) ei kr

and reflected

ψ -k (r) = u - k (r) e -i kr

waves will be the same, u k (r) = u -k (r). These waves give two solutions to the Schrödinger equation:

This function describes the accumulation of negative charge on positive ions, where the potential energy is the smallest. Similarly, from formula (9b) we obtain:

ρ 2 (r) = |ψ 2 (r)|2 =4 u g/2 2 (r)sin 2 (gr/2)

This function describes such a distribution of electrons, in which they are located mainly in the regions corresponding to the midpoints of the distances between the ions. In this case, the potential energy will be greater. The function ψ 2 will correspond to the energy E2 > E1 .

forbidden bands of width Eg . The energy Е`1 determines the upper boundary of the first zone, and the energy Е2 defines the lower boundary of the second zone. This means that when electron waves propagate in crystals, energy ranges arise for which there are no solutions of the Schrödinger equation that have a wave character.

Since the nature of the energy dependence on the wave vector significantly affects the dynamics of electrons in a crystal, it is of interest to consider, for example, the simplest case of a linear chain of atoms located at a distance a from one another along the x axis. In this case, g = 2π /a. Figure 12 shows the dispersion curves for the first three one-dimensional Brillouin zones: (-

π/ a< k <π /a), (-2π /a < k < -π /a; π/ a < k < 2π /a), (-3π/ a < k < -2π /a; 2π /a < k < 3π /a). К запрещенным зонам относятся области энергии Е`1 < E < E2 , E`2 <

E< E3 и т.д.

On fig. 12 presented extended zone scheme, in which different energy zones are located in VK - space in different Brillouin zones. However, it is always possible, and often convenient, to choose the wave vector in such a way that its end lies inside the first Brillouin zone. We write the Bloch function in the form:

lie in the first Brillouin zone. Substituting to in formula (11), we obtain:

has the form of the Bloch function with Bloch multiplier (13). The index n now indicates the number of the energy zone to which the given function belongs. The procedure for reducing an arbitrary wave vector to the first Brillouin zone is called reduced zone diagrams. In this scheme, vectork takes the values ​​-g/2< k < g/2 , но одному и тому же значениюк будут отвечать различные значения энергии, каждое из которых будет соответствовать одной из зон. На рисунке 13 представлена схема приведенных зон для одномерной решетки, соответствующая расширенной зонной схеме на рисунке 12.

Thus, the existence of energy band gaps is due to the Bragg reflection of de Broglie electron waves from crystalline planes. The discontinuity points are determined by the conditions of maximum wave reflection.

According to the laws of quantum mechanics, the translational motion of an electron is considered as the motion of a wave packet with wave vectors close to the vector k. The group velocity of the wave packet, v , is given by

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Band structure of the electronic energy spectrum in solids. Models of free and strongly bound electrons

3.2. Band Structure of the Energy Spectrum in the Tight-Coupling Model

3.2.1. Formation of the band structure of the energy spectrum.

So, when a bond is formed between two atoms, two molecular orbitals are formed from two atomic orbitals: bonding and loosening with different energies.

Let us now see what happens during the formation of a crystal. Here are possible two different options: when a metallic state arises when atoms approach each other and when a semiconductor or dielectric state arises.

metal condition can arise only as a result of the overlap of atomic orbitals and the formation of multicenter orbitals, leading to the complete or partial collectivization of valence electrons. Thus, a metal, based on the concept of initially bound atomic electron orbitals, can be represented as a system of positively charged ions combined into one giant molecule with a single system of multicenter molecular orbitals.

In transition and rare earth metals, in addition to the metallic bond that arises during the collectivization of electrons, there can also exist covalent directed bonds between neighboring atoms with completely filled bonding orbitals.

The collectivization of electrons, which ensures the bonding of all atoms in the lattice, leads to a 2N-fold (including spin) splitting of atomic energy levels and the formation of a band structure of the electronic energy spectrum when the atoms approach each other.

A qualitative illustration of the change in discrete energy levels of isolated atoms () with a decrease in the interatomic distance is shown in Figure 30a, which shows the splitting of energy levels with the formation of narrow energy zones containing 2N (including spin) different energy states (Fig. 30a).

Rice. thirty.

The width of the energy bands (), as will be shown below, depends on the degree of overlap of the wave functions of the electrons of neighboring atoms or, in other words, on the probability of the transition of an electron to a neighboring atom. In general, the energy bands are separated by forbidden energy intervals, called forbidden zones(Fig. 30a).

When the s- and p-states overlap, several "bonding" and "loosening" zones are formed. From this point of view, the metallic state arises if there are zones not completely filled with electrons. However, in contrast to the weak coupling (nearly free electron model), in this case, the electronic wave functions cannot be considered as plane waves, which greatly complicates the procedure for constructing isoenergetic surfaces. The nature of the transformation of the wave functions of localized electrons into Bloch-type wave functions describing collectivized electrons is illustrated in Fig. 30b,c.

Here it should be emphasized once again that it is the collectivization of electrons, that is, their ability to move in the crystal lattice, that leads to the splitting of the energy levels of bound states and the formation of energy bands (Fig. 30c).

Semiconductor ( and dielectric) state provided by directed covalent bonds. Almost all atomic semiconductors have a diamond-type lattice, in which each pair of atoms has a covalent bond formed as a result of sp 3 hybridization [NE Kuzmenko et al., 2000]. There are two electrons on each sp 3 orbital that binds neighboring atoms, so that all bonding orbitals are completely filled.

Note that in the model of localized bonds between pairs of neighboring atoms, the formation of a crystal lattice should not lead to a splitting of the energy levels of the bonding orbitals. In fact, a single system of overlapping sp 3 -orbitals is formed in the crystal lattice, since the electron density of a pair of electrons on -bonds is concentrated not only in the region of space between atoms, but is also different from zero outside these regions. As a result of the overlap of wave functions, the energy levels of the bonding and antibonding orbitals in the crystal are split into narrow non-overlapping zones: a completely filled binding zone and a free antibonding zone located higher in energy. These zones are separated by an energy gap.

At temperatures other than zero, under the action of the energy of the thermal motion of atoms, covalent bonds can be broken, and the released electrons are transferred to the upper band on antibonding orbitals, on which the electronic states are not localized. Thus, it happens delocalization bound electrons and the formation of a certain number, depending on the temperature and band gap, itinerant electrons. Collectivized electrons can move in the crystal lattice, forming a conduction band with the corresponding dispersion law. However, now, just as in the case of transition metals, the motion of these electrons in the lattice is described not by plane traveling waves, but by more complex wave functions that take into account the wave functions of the bound electronic states.

When an electron is excited with one of the covalent bonds, hole - an empty electronic state to which a charge is attributed+q . As a result of the transition of an electron from neighboring bonds to this state, the hole disappears, but at the same time an unoccupied state appears on the neighboring bond. So the hole can move through the crystal. Just like electrons, delocalized holes form their own band spectrum with the corresponding dispersion law. In an external electric field, transitions of electrons to a free bond prevail in the direction against the field, so that holes move along the field, creating an electric current. Thus, during thermal excitation, two types of current carriers appear in semiconductors - electrons and holes. Their concentration depends on temperature, which is typical for the semiconductor type of conductivity.

Literature: [W. Harrison, 1972, ch. II, 6.7; D. G. Knorre et al., 1990; K.V. Shalimova, 1985 , 2.4; J.Ziman et al., 1972, ch.8, 1]

3.2.2. Wave function of an electron in a crystal

In the tight-binding model, the wave function of an electron in a crystal can be represented as a linear combination of atomic functions:

where r is the radius vector of the electron, r j- radius vector j th lattice atom.

Since the wave function of itinerant electrons in a crystal must have the Bloch form (2.1), the coefficient With _( j) for an atomic function on j-th node of the crystal lattice should have the form of a phase factor , that is,

Physical bases and

And technology of electronic means

Physical foundations

E.N. VIGDOROVICH

Tutorial

"Physical Foundations"

MGUPI 2008

UDC 621.382 Approved by the Academic Council

as a teaching aid

electronic media technology

Tutorial

M. Ed. MGAPI, 2008

Edited by

prof. Ryzhikova I.V.

The textbook contains brief material on the physical foundations of the processes of formation of the properties of electronic means.

The manual is intended for teachers, engineering and technical workers and students of various specialties

______________________________

@ Moscow State Academy of Instrument Engineering and Informatics, 2005

1. ENERGY SPECTRUM OF CHARGE CARRIERS

The task before us is reduced to consideration of the properties and behavior of charged particles in a crystalline solid.

From the courses of atomic physics and quantum mechanics, the behavior of electrons in a single isolated atom is known. In this case, electrons may not have any values ​​of energy E, but only some. The energy spectrum of electrons acquires a discrete character, as shown in Fig. 1.1 in. Transitions from one energy level to another are associated with the absorption or release of energy.

Rice. 1.1. Scheme of the formation of energy bands in crystals:

a - the arrangement of atoms in a one-dimensional crystal; b - distribution of intracrystalline potential field; in - arrangement of energy levels in an isolated atom; d - location of energy zones

The question arises of how the energy electronic levels in atoms will change if the atoms are brought closer to each other, that is, if they are condensed into a solid phase. A simplified picture of this one-dimensional crystal is shown in fig. 1.1 a.

It is not difficult to obtain a qualitative answer to this question. Consider what forces act in a single atom, and what - in a crystal. In an isolated atom, there is a force of attraction by the atomic nucleus of all their electrons and the repulsive force between electrons. In a crystal, due to the close distance between atoms, new forces arise. These are the forces of interaction between nuclei, between electrons belonging to different atoms, and between all nuclei and all electrons. Under the influence of these additional forces, the energy levels of the electrons in each of the atoms of the crystal must somehow change. Some levels will go down, others will go up on the energy scale. This is what first consequence approach of atoms. Second consequence due to the fact that the electron shells of atoms, especially the outer ones, can not only come into contact with each other, but can even overlap. As a result, an electron from one level in any of the atoms can go to the level in the neighboring atom without energy expenditure and, thus, move freely from one atom to another. In this regard, it cannot be argued that a given electron belongs to any one particular atom, on the contrary, an electron in such a situation belongs to all atoms of the crystal lattice simultaneously. In other words, it happens socialization electrons. Of course, complete socialization occurs only with those electrons that are on the outer electron shells. The closer the electron shell is to the nucleus, the stronger the nucleus holds the electron at this level and prevents the movement of electrons from one atom to another.

The combination of both consequences of the approach of atoms leads to the appearance on the energy scale, instead of individual levels, of entire energy zones (Fig. 1.1, d), i.e., regions of such energy values ​​that an electron can have while being within a solid body. The band width should depend on the degree of bonding of the electron with the nucleus. The greater this connection, the smaller the splitting of the level, i.e., the narrower the zone. In an isolated atom, there are forbidden values ​​of energy that an electron cannot possess. It is natural to expect that something similar will be in a solid. Between zones (now no longer levels) there can be forbidden zones. Characteristically, if the distances between the levels in an individual atom are small, then the forbidden region in the crystal can disappear due to the overlap of the resulting energy bands.

Thus, the energy spectrum of electrons in a crystal has a band structure . . A quantitative solution of the problem of the spectrum of electrons in a crystal using the Schrödinger equation also leads to the conclusion that the energy spectrum of electrons in a crystal has a band structure. Intuitively, one can imagine that the difference in the properties of different crystalline substances is uniquely associated with the different structure of the electron energy spectrum (different widths of allowed and forbidden zones)

Quantum mechanics, in order to explain a number of properties of matter, considers elementary particles, including the electron, both as a particle and as a kind of wave. That is, an electron can be simultaneously characterized by the energy values E and momentum p, as well as the wavelength λ, frequency ν, and wave vector k = p/h. Wherein, Е=hν and p = h/λ. Then the movement of free electrons can be described by a plane wave, called the de Broglie wave, with a constant amplitude.

Chapter 10

The idea of ​​valency as the ability of an atom to form chemical bonds with a certain number of other atoms in application to a solid body loses its meaning, since the possibility of collective interaction is realized here. So in a molecule, the valencies of atoms and are equal to one, and in a crystal, each atom is surrounded by 6 atoms and vice versa.

The energy spectrum of an isolated atom is determined by the interaction of electrons with the nucleus and has a discrete character. The energy states of electrons in a solid are determined by its interaction both with its own nucleus and with the nuclei of other atoms. In a crystal, the nuclei of atoms are located periodically along any direction (Fig. 56). Therefore, the electron moves in a periodic electric field (near the nuclei, the potential energy of the electron is less than in the gap between the nuclei). This leads to the fact that instead of a discrete atomic energy level in a solid containing N atoms, arises N closely spaced energy levels that form an energy band. In this sense, one speaks of the splitting of an energy level into an energy zone. Neighboring energy levels in the band are separated from each other by 10 -23 eV. For comparison, we point out that the average thermal energy of electrons at a temperature T= 300 K is ~ 10 -2 eV. As a result, the electron spectrum inside the band can be considered quasi-continuous.

The number of states in the band is equal to the product of the number of atoms in the crystal and the multiplicity of the atomic energy level from which the band was formed. The multiplicity of an energy level is understood as the number of electrons that can be at this level in compliance with the Pauli principle.

Zones of allowed energies are separated by zones of forbidden energies. Their width is comparable to the width of the allowed energy zones. With an increase in energy, the width of the allowed bands increases, while the width of the forbidden bands decreases (Fig. 57).

§2. Metals, semiconductors, dielectrics

Differences in the electrical properties of solids are explained by the different filling of the allowed energy bands with electrons and the width of the band gaps. In order for a body to be able to conduct an electric current, it is necessary to have free energy levels in the allowed zones, to which electrons could go under the influence of an electric field.

Metals

Consider a sodium crystal. Its electronic formula is . The energy diagram of sodium is shown in fig. 58.

An isolated atom has a discrete energy spectrum. When the atoms approach each other, starting from a certain interatomic distance, the energy levels split into zones. First of all, the outer levels are split: vacant 3 R, then half-filled level 3 s. As the distance decreases r before r 1 overlap occurs 3 R- and 3 s-zones of allowed energies. On distance r = r 0 (r 0 is the equilibrium interatomic distance in the crystal), the approach of atoms stops. Valence 3 s electrons can occupy any state within this band. Levels 1 s and 2 s can only split r< r 0 and do not participate in chemical bonding. Communication is carried out by a collective of valence electrons, the energy states of which form a common zone obtained as a result of overlapping.

In the allowed energy zone formed by valence levels, there will be 8 N states (number s- states 2 N; number R- states 6 N). An atom has one valence electron, so this zone will contain N electrons occupying states in accordance with the Pauli principle and the principle of least energy. Consequently, some of the states in the zone are free.

Crystals in which the band formed by the levels of valence electrons is partially filled belong to metals. This band is called the conduction band.

Semiconductors and dielectrics

Let us consider the energy structure of semiconductors and dielectrics using the example of a typical semiconductor - crystalline silicon (Z = 14), whose electronic formula is . During the formation of a crystal lattice, starting from a certain interatomic distance r 1 >r 0 (r 0 is the equilibrium interatomic distance in the crystal) occurs sp 3 -hybridization of electronic states of silicon, which leads not only to overlapping 3 s and 3 R zones, but to their merger and formation of a single 3 sp 3 hybrid valence band (Fig. 59), in which the maximum possible number of electrons is 8 N. In crystalline silicon, each atom forms 4 tetrahedral bonds, completing its valence shell to eight electrons. As a result, in the valence band all 8 N states are busy. Thus, for semiconductors and dielectrics band formed by valence electron levels- valence band (VZ) - completely filled. Next vacant 4 s- band does not overlap with the valence band at interatomic distance r 0 , and is separated from it by the band of forbidden energies (ZZ) . Electrons located in the valence band cannot participate in conduction, since all states in the band are occupied. In order for a current to appear in the crystal, it is necessary to transfer electrons from the valence band to the next free band of allowed energies. First free zone above valence band is called conduction band (CB). The energy gap between the bottom of the conduction band and the top of the valence band is called band gap Wg.

Depending on the band gap, all crystalline bodies are divided into three classes:

1. metals - ˆ0.1 eV;

2. semiconductors -;

3. dielectrics - ‰4 eV.

Accordingly, the bodies have the following resistivity values:

1. metals - ρ = 10 -8 10 -6 Ohm m;

2. semiconductors - ρ = 10 -6 10 8 Ohm m;

3. dielectrics - ρ>10 8 Ohm m.

At a temperature T= 0 semiconductors are insulators, but with increasing temperature, their resistance decreases sharply. In dielectrics, when heated, melting occurs earlier than electronic conductivity occurs.

In 1928-1931. The band theory is the basis of modern ideas about the mechanisms of various physical phenomena occurring in a solid crystalline substance when it is exposed to an electromagnetic field. This is the theory of electrons moving in a periodic potential field of a crystal lattice.

In an isolated atom, the energy spectrum of electrons is discrete, i.e., electrons can occupy only well-defined energy levels. Some of these levels are filled in the normal, unexcited state of the atom; electrons can be at other levels only when the atom is subjected to external energy influence, i.e., when it is excited. In striving for a stable state, an atom radiates an excess of energy at the moment of transition of electrons from excited states to levels at which its energy is minimal. Transitions from one energy level to another are always associated with the absorption or release of energy.

In an isolated atom, there is a force of attraction by the nucleus of the atom of all its electrons and a force of repulsion between the electrons. If there is a system of N identical atoms that are sufficiently distant from each other (for example, a gaseous substance), then there is practically no interaction between atoms, and the energy levels of electrons remain unchanged. When a gaseous substance condenses into a liquid, and then when a crystal lattice of a solid is formed, all the electronic levels available for atoms of this type (both filled with electrons and unfilled) are somewhat shifted due to the action of neighboring atoms on each other. In a crystal, due to the close distance between atoms, there are interaction forces between electrons belonging to different atoms, and between all nuclei and all electrons. Under the influence of these additional forces, the energy levels of electrons in each of the atoms of the crystal change: the energy of some levels decreases, while the energy of others increases. In this case, the outer electron shells of atoms can not only touch each other, but also overlap. In particular, the attraction of the electrons of one atom by the nucleus of the neighboring one reduces the height of the potential barrier separating the electrons of solitary atoms. That is, when the atoms approach each other, the electron shells overlap, and this, in turn, significantly changes the nature of the electron motion. As a result, an electron from one level in any of the atoms can go to the level in a neighboring atom without expending energy, and thus move freely from one atom to another. This process is called the socialization of electrons - each electron belongs to all atoms of the crystal lattice. Complete socialization occurs with the electrons of the outer electron shells. Due to the overlap of the shells, electrons can, without a change in energy, pass from one atom to another through exchange, i.e., move through the crystal. The exchange interaction has a purely quantum nature and is a consequence of the indistinguishability of electrons.

As a result of the approach of atoms on the energy scale, instead of individual levels, energy zones appear, i.e., regions of such energy values ​​that an electron can have while being within a solid body. The band width should depend on the degree of bonding of the electron with the nucleus. The greater this connection, the smaller the splitting of the level, the narrower the zone. In an isolated atom, there are forbidden energy values ​​that an electron cannot possess; in a solid, there can be forbidden zones. The energy spectrum of electrons in a crystal has a band structure. The allowed energy bands are separated by forbidden energy intervals. The width of the allowed energy bands does not depend on the size of the crystal, but is determined only by the nature of the atoms that form the solid and the symmetry of the crystal lattice. If EA is the energy of the exchange interaction between two neighboring atoms, then for crystals with a simple cubic lattice, where each atom has 6 nearest neighbors (coordination number = 6), the splitting of levels into zones will be 12EA, for a face-centered lattice (K.n. = 12 ) the width of the energy allowed zone will be 24 EA, and in the body-centered (K.n. = 8) - 16 EA.

Since the exchange energy of EA depends on the degree of overlap of electron shells, the energy levels of the inner shells, which are more localized near the nucleus, split less than the levels of valence electrons. Not only normal (stationary), but also excited energy levels are subject to splitting into a zone. The width of the allowed zones when moving up along the energy scale increases, and the size of the forbidden energy gaps decreases accordingly.

Each zone consists of many energy levels. Their number is determined by the number of atoms that make up the solid, ie. in a crystal of finite dimensions, the distance between levels is inversely proportional to the number of atoms. In accordance with the Pauli principle, at each energy level there can be no more than two electrons, and with opposite spins. Therefore, the number of electronic states in the band turns out to be finite and equal to the number of corresponding atomic states. The number of electrons filling a given energy band also turns out to be finite. When N atoms approach each zone, N sublevels appear. A crystal with a volume of 1 cm 3 contains 10 22 -10 23 atoms. Experimental data show that the energy extent of the valence electron band does not exceed a few electronvolts. It follows that the levels in the zone are separated from each other in energy by 10 -22 - 10 -23 eV, i.e. the levels are so close that even at low temperature this zone can be considered a zone of continuous allowed energies, such an energy zone is characterized by quasi-continuous spectrum. A negligibly small energy impact is enough to cause the transition of electrons from one level to another, if there are free states there. That is, due to the small difference in the energy of two neighboring sublevels, the orbitals of valence electrons in a crystal are perceived as a continuous zone, and not as a set of discrete energy levels.

More rigorously, we can only speak about the probability of an electron being at a particular point in space. This probability is described using wave functions x, which are obtained by solving the Schrödinger wave equation. When atoms interact and chemical bonds appear, the wave functions of valence electrons also change.

Deriving the energy spectrum of electrons in a crystal from the energy levels in isolated atoms is called the tight-binding approximation. It is more true for electrons located at deep levels and less subject to external influences. In complex atoms, the electron energy is determined by the principal quantum number n and the orbital quantum number l. Accounting for interactions in a crystal (weak coupling approximation) shows that during the formation of a crystal, the levels of atoms are split into N(2l+1) sublevels, on which 2N(2l+1) electrons can be located.

Like energy levels in isolated atoms, energy bands can be fully filled, partially filled, or empty. The inner shells in isolated atoms are filled, so the zones corresponding to them also turn out to be filled. The topmost filled band is called the valence band. This zone corresponds to the energy levels of the outer shell electrons in isolated atoms. The free, unfilled zone closest to it is called the conduction band. There is a band gap between them. The filling of the conduction band begins when the electrons in the valence band receive additional energy sufficient to overcome the energy barrier equal to the band gap.

The absence of any energy levels in the band gap is typical only for perfect crystals. Any violation of the ideality of the periodic field in a crystal entails violations of the ideality of the band structure. In a real crystal, there are always defects in the crystal lattice. If the number of defects in the crystal is small, then they will be located at considerable distances from each other, localized. Therefore, the energy state of only those electrons that are in the region of the defect will change, which will lead to the formation of local energy states that are superimposed on the ideal band structure. The number of such states is either equal to the number of defects or exceeds it if several such states are associated with a defect. The location of local states is limited by the region near the defect. Electrons located at these energy levels turn out to be associated with defects and therefore cannot participate in electrical conductivity. That is, the levels of defects on which they are located are located in the band gap of the crystal.

As the temperature rises, the amplitude of thermal vibrations of atoms increases, the degree of their interaction and the degree of splitting of energy levels increase. Therefore, the allowed zones become wider, and the forbidden ones, respectively, narrower. With a change in interatomic distances, depending on the nature of the level splitting, the band gap can either increase or decrease. This happens, for example, under the action of pressure on the crystal.

The band theory makes it possible to formulate a criterion that makes it possible to divide solid substances into two classes - metals and semiconductors (dielectrics). The band theory was originally developed for crystalline solids, but in recent years its ideas have been extended to amorphous substances as well.