Chapter 10. The concept of the zone theory of solids

The idea of \u200b\u200bthe valence as an atom ability to form chemical bonds with a certain number of other atoms in the application to a solid loses its meaning, since the possibility of collective interaction is realized here. So in the molecule of the valence of atoms and are equal to one, and in the crystal, each atom is surrounded by 6th atoms and vice versa.

The energy spectrum of an isolated atom is determined by the interaction of electrons with the nucleus and is discrete. The energy states of electrons in the solid body are determined by its interaction both with their core and with the nuclei of other atoms. In the core crystal atoms are located periodically along any direction (Fig. 56). Therefore, the electron is moving in a periodic electric field (near the nuclei the potential energy of an electron is less than in the interval between the kernels). This leads to the fact that instead of a discrete atomic energy level in a solid containing N.atoms, occur N.closely located energy levels located apart from each other, which form the energy zone. In this sense, they are talking about the splitting of the energy level into the energy zone. Neighboring energy levels in the zone are detached from each other by 10 -23 eV. For comparison, we indicate that the average thermal energy electrons at temperature T. \u003d 300 K is ~ 10 -2 eV. As a result, the electron spectrum can be considered within the quasi-drier zone.

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

Zones of permitted energies are separated by zones of prohibited energies. Their width is comparable to the width of the zones of allowed energies. With an increase in energy, the width of the allowed zones increases, and prohibited - decreases (Fig. 57).

§2. Metals, semiconductors, dielectrics

Differences in the electrical properties of solid bodies are explained by various filling of the electrons of the allowed energy zones and the width of the prohibited zones. In order for the body to conduct electricity It is necessary to have free energy levels in the permitted zones that the electrons could be proceeded under the action of an electric field.

Metals

Consider the sodium crystal. His electronic formula. Sodium energy diagram is shown in Fig. 58.

An isolated atom has a discrete energy spectrum. Under the rapprochement of atoms, starting from a certain interatomic distance, the energy levels are cleavage into the zone. First of all, external levels are split: vacant 3 rthen half filled level 3 s.. With a decrease in distance r.before r. 1 takes off 3 r-and 3. s.-Realized zone. On distance r \u003d R. 0 (r. 0 is an equilibrium interatomic distance in a crystal) rapprochement of atoms stops. Valentines 3. s.electrons can occupy any condition within this zone. Levels 1. s.and 2 s.can split only when r.< r 0 and in chemical bond do not participate. 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 zone of permitted energies formed by valence levels, there will be 8 N. States (Number s.Cost 2 N.; number rCost 6. N.). Atom has one valence electron, so in this zone will be N.electrons, occupying conditions in accordance with the principle of Pauli and the principle of least energy. Consequently, part of the states in the free zone.

Crystals in which the zone formed by the levels of valence electrons is partially filled, belong to Metals. This zone is called the conduction zone.

Semiconductors and dielectrics

Consider the energy structure of semiconductors and dielectrics at a typical semiconductor - crystalline silicon (Z \u003d 14), whose electronic formula. In education crystal latticestarting from some interatomic distance r. 1 \u003e R. 0 (r. 0 - the equilibrium interatomic distance in the crystal) occurs sp. 3-hybridization of silicon electronic states, which leads not just to overlapping 3 s.and 3. rzones, and to their merger and formation of one 3 sp. 3 hybrid valence zone (Fig. 59), in which the maximum possible number of electrons 8 N.. In crystalline silicon, every atom forms 4 tetrahedral ties, completing its valence shell to eight electrons. As a result, all 8 in the valence zone N.states are busy. Thus, in semiconductors and dielectrics zone formed by levels of valence electrons - Valence zone (PT) - fully filled. The next vacant 4. s.-zone does not overlap with the valence zone during the interatomic distance r. 0, and separated from it the zone of prohibited energies (ZZ) . Electrons that are in the valence zone can not participate in conductivity, since all states in the zone are occupied. In order to appear in the crystal, it is necessary to translate electrons from the valence zone to the next free zone of the allowed energies. The first free allowable zone located above valence zone calledconductivity zone (zP). The energy gap between the bottom of the conduction zone and the ceiling of the valence zone is called The width of the prohibited zone W G..

Depending on the width of the prohibited zone, all crystal bodies divided into three classes:

1. Metals - 0.1 eV;

2. Semiconductors -;

3. Dielectrics - ‰ 4 eV.

Accordingly, the body has such specific resistance values:

1. Metals - ρ \u003d 10 -8 10 -6 Ohm · m;

2. Semiconductors - ρ \u003d 10 -6 10 8 Ohm · m;

3. Dielectrics - ρ\u003e 10 8 Ohm · m.

At a temperature T. \u003d 0 Semiconductors are dielectrics, but with increasing temperature, their resistance decreases sharply. Dielectrics during heating earlier melting occurs than an electronic conductivity arises.

The energy spectrum of electrons in the solid body differs significantly from the energy spectrum of free electrons (which is continuous) or the electron spectrum belonging to the individual isolated atoms (discrete with a certain set of available levels) - it consists of separate allowed energy zones separated by the zones of prohibited energies.

According to the quantum-mechanical postulates of boron, in an isolated atom, the electron energy can take strictly discrete values \u200b\u200b(the electron is on one of the orbital). In the case of a system of several atoms, combined with chemical bond, electronic orbitals are split off in an amount proportional to the number of atoms, forming the so-called molecular orbitals. With a further increase in the system to the macroscopic level, the number of orbital becomes very large, and the difference in electrons in neighboring orbital, respectively, very small - energy levels are split up to two almost continuous discrete sets - energy zones.

The highest of the allowed energy zones in semiconductors and dielectrics, in which at a temperature of 0 to all energy states are occupied by electrons, is called the valence, following it - the conduction zone. In conductors, the conduction zone is called the highest permitted zone in which electrons are located at a temperature of 0 K. mutual location These zones are all solids and divided into three large groups (see Fig.):

• conductors - materials in which the conduction zone and the valence zone overlap (no energy gap), forming one zone called the conduction zone (thus, the electron can move freely between them, having received any permissible low energy);
• dielectrics - materials in which zones do not overlap and the distance between them is more than 3 eV (in order to translate the electron from the valence zone in the conductivity zone requires significant energy, so the dielectric current is practically not carried out);
• semiconductors - materials in which zones do not overlap and the distance between them (the width of the forbidden zone) lies in the range of 0.1-3 eV (in order to translate an electron from the valence zone to the conduction zone, the energy is required smaller than for a dielectric, so clean Semiconductors weakly skip the current).

Zone theory is the basis modern theory solid. She made it possible to understand nature and explain the most important properties of metals, semiconductors and dielectrics. The magnitude of the prohibited zone (the energy gap between the zones of valence and conductivity) is a key value in the zone theory and determines the optical and electrical properties of the material. For example, in semiconductors, conductivity can be increased by creating a permitted energy level in the prohibited zone by doping - addition to the composition of the initial base material impurities to change its physical and chemical properties. In this case, they say that the semiconductor is impurity. It is thus that all semiconductor devices are created: solar cells, diodes, solid-state and others. The electron transition from the valence zone to the conduction zone is called the process of generating charge carriers (negative - electron, and positive - holes), and the reverse transition is the recombination process.

The zone theory has the boundaries of applicability, which proceed from the three main assumptions: a) the potential of the crystal lattice is strictly periodic; b) the interaction between the free electrons can be reduced to one-electron self-consistent potential (and the remainder is considered by the method of perturbation theory); c) interaction with phonons is weak (and can be considered on perturbation theory).

Illustrations

Author

• Razumovsky Alexey Sergeevich

Changes applied

• Nimushina Daria Anatolyevna

Sources

1. Physical encyclopedic Dictionary. T. 2. - M.: Big Russian encyclopedia, 1995. - 89 p.
2. Gurov V. A. Solid-state electronics. - M.: Technosphere, 2008. - 19 p.

Zone structure of the electronic energy spectrum in solids. Models of free and highly related electrons

3.2. Zone structure of the energy spectrum in a strong communication model

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

So, in the formation of the relationship between two atoms of two atomic orbitals, two molecular: binding and tearing with different energies are formed.

Let's see now what is happening in the formation of a crystal. Here are possible two different options: When the rapprochet of atoms occurs a metal state and when a semiconductor or dielectric state occurs.

Metal state It may occur only as a result of overlapping atomic orbitals and the formation of multicenter orbitals, leading to a complete or partial collectivization of valence electrons. Thus, metal, if proceeding from the concept of originally connected atomic electronic 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 transitional and rare earth metals, in addition to the collectivization of electrons of metallic communications arising from the collectivization of electrons, can also exist covalent directional connections Between adjacent atoms with completely filled binding orbital.

The electron collectivization, which ensures the connection of all atoms in the lattice, leads to the approach of atoms to 2N-multiple (taking into account the spin) splitting of atomic energy levels and the formation of the zone structure of the electronic energy spectrum.

A high-quality illustration of the change in discrete energy levels of isolated atoms () with a decrease in the interatomic distance is shown in Figure 30A, where the splitting of energy levels is shown to form narrow energy zonescontaining 2n (taking into account the spin) of various energy states (Fig. 30a).

Fig. thirty.

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

At the overlap of S- and Pologists, several "binding" and "baking" zones are formed. The metallic state from this point of view occurs if there are zones not completely filled with electrons. However, in contrast to a weak bond (models of almost free electrons), in this case, electronic wave functions are not considered as flat waves, which greatly complicates the procedure for the construction of isoenergy surfaces. The nature of the conversion of wave functions of localized electrons into the wave functions of the Bloch type describing collectivized electrons is illustrated in Figure 30b, in.

It should be emphasized again that it is the collectivization of electrons, that is, the possibility of moving in a crystal lattice, leads to the splitting of energy levels of the states and the formation of energy zones (Fig. 30B).

Semiconductor (and Dielectric) condition Provided by directional covalent bonds. Almost all atomic semiconductors They have a diamond type grille, in which each pair of atoms has a covalent-β-flasher formed as a result of SP 3-hybridization [N.E.Kuzmenhenko et al., 2000]. On each SP 3, there are two electrons in each SP 3, there are two electrons, so that all binding orbitals are fully filled.

Note that in the model of localized bonds between pairs of neighboring atoms, the formation of a crystal lattice should not lead to the splitting of energy levels of binding orbitals. In fact, a single system of overlapping SP 3 is formed in the crystal lattice, since the electron density of the electron pair on-means concentrates not only in the area of \u200b\u200bspace between atoms, but is different from zero and outside these areas. As a result of overlapping the wave functions, the energy levels of binding and baking orbitals in a crystal are split into narrow non-overlapping zones: a fully filled binding zone and an energy located above - free loose. These zones are separated by the energy gap.

With non-zero temperatures under the action of the thermal motion of atoms atoms, covalent bonds can be breaking, and the released electrons are transferred to the upper zone on tearful orbitals, on which electronic states are not localized. So happens delocalization associated electrons and the formation of a certain number, depending on the temperature and width of the prohibited zone, collectivized electrons. Collectivized electrons can move in a crystal lattice, forming a conduction zone with the corresponding dispersion law. However, now, as well as in the case of transition metalsThe movement of these electrons in the lattice is described by non-flat running waves, but with more complex wave functions that take into account the wave functions of the bound electronic states.

When an electron is excited from one of the covalent bonds is formed hole - empty electronic condition attributed to charge+q. As a result of the transition of any electron from neighboring links to this state of the hole disappears, but at the same time there appears on the neighboring communication. So the hole can move on the crystal. As well as electrons delocalized holes form their zone spectrum with the relevant dispersion law. In the outer electric field, the transitions of electrons on the free bond is prevalent in the direction against the field, so that the holes are moved along the field, creating an electric current. Thus, with thermal excitation in semiconductors, there are two types of current carriers - electrons and holes. Their concentration depends on the temperature, which is characteristic of the semiconductor type of conductivity.

Literature: [U. Kharison, 1972, ch. II, 6.7; DG Bnorra et al., 1990; K.V.Shalimova, 1985, 2.4; J. Zaiman et al., 1972, GL.8, 1]

3.2.2. Wave Electron Function in Crystal

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

where r. - radius-vector electron, r. j. - radius-vector j.Lathe atoms.

Since the wave function of collectivized electrons in a crystal should have a Bloch form (2.1), then the coefficient FROM _ (j) with atomic function on j.the node of the crystal lattice should have the form of a phase factor, that is

proportional to: n ~ t. Consequently, the thermal conductivity coefficient should be inversely proportional to the temperature, which is qualitatively consistent with the experience. At temperatures below, the Debaevskayal practically does not depend on the OTT, and the thermal conductivity is entirely determined by the dependence of the heat capacity of the crystal of V ~ T 3. Therefore, at low temperaturesλ ~ T 3. The characteristic dependence of thermal conductivity on temperature is shown in Figure 9.

In addition to lattice thermal conductivity, it is necessary to take into account the thermal conductivity due to the transfer of heat with free electrons. It is precisely her that explains the high thermal conductivity of metals compared to non-metals.

3. Electronic structure of crystals.

3.1. Electron generation in the periodic field. The zone structure of the electron energy spectrum in the crystal. Functions of flea. Dispersion curves. Effective mass.

In the solid body of the distance between atoms are comparable to their dimensions. Therefore, electronic shells of neighboring atoms partially overlap with each other and at least the valence electrons of each atom turn out to be in a sufficiently strong field of neighboring atoms. Exact description The movements of all electrons, taking into account the Coulomb interaction of electrons with each other and with atomic nuclei, is an extremely complex task even for a separate atom. Therefore, a method of self-consistent field is usually used, in which the task is reduced to the description of the movement of each individual electron in the field of the effective potential generated by atomic nuclei and the averaged field of the remaining electrons.

We first consider the structure of the energy levels of the crystal, based on the approximation of a strong bond, in which the electron binding energy with its atom significantly exceeds the kinetic energy of its movement from the atom to the atom. For large distances Each of them has a system of narrow energy levels corresponding to the associated electron states with an ion. Under the rapprochement of atoms, the width and height of the potential barriers between them decreases, and due to the tunnel effect, the electrons are able to move from

one atom to another, which is accompanied by the expansion of energy levels and turning them into energy zones. (Fig. 10). In particular, this applies to the weakly related valented electrons that are able to easily move on the crystal from the atom to the atom, and to a certain extent become similar to free electrons. Electrons of deeper energy levels are much stronger than associated with their atom. They form narrow energy zones with wide intervals of prohibited energies. In fig. 10 Conditionally presented potential curves and energy levels for the Na crystal. The overall nature of the energy spectrum of electrons depending on the interstitial distance, D, is shown in Figure 11. In some cases, the upper levels are embroidered so much that the neighboring energy zones overlap. In fig. 11 This takes place at d \u003d d1.

Based on the ratio of the uncertainties of Heisenberg - boron, the width of the energy zone, Δε is associated with the time of the electron's stay in a certain lattice assembly by the relation: Δε τ\u003e h. Due to the tunnel effect, the electron can leak through the potential barrier. According to the estimate, with an interatomic distance D ~ 1Aτ ~ 10 -15 C, and thereforeδε ~ H / τ ~ 10 -19 J ~ 1 eV, i.e. The width of the forbidden zone is about one or more eV. If the crystal consists of n atoms, then each energy zone consists of N sublevel. The crystal of 1 cm3 is contained by N ~ 1022 atoms. Consequently, with the width of zone ~ 1 eV, the distance between the pylons is ~ 10 -22 eV, which is significantly less than the energy of the thermal motion under normal conditions. This distance is so insignificant that in most cases the zone can be considered almost continuous.

In the perfect crystal of the atoms of atoms located in the nodes of the crystal lattice, forming a strictly periodic structure. In accordance with this, the potential energy of the electron, V (R), also periodically depends on the spatial coordinates, i.e. possessed broadcast symmetry:

lattices, a i (i \u003d 1,2,3, ...) - vectors of major broadcasts.

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

representing the product of the equation of a flat running wave, Ei Kr on a periodic factor, u k (r) \u003d u k (r + a n), with a lattice period. Functions (3) called flea functions.

When V (R) \u003d 0, equation (2) has a solution in the form of a flat wave:

where m is the mass of the particle. Energy dependence E from a wave chicken is depicted dispersion curve. According to (5), in the case of a free electron, it is a parabola. By analogy with free movement, vector in equation (3) is called a wave vector, AP \u003d H K - a quasi-pulse.

In the approximation of a weak connection, the movement of almost free electrons is considered, which acts an indignant field of the periodic potential of ionic cores. In contrast to free movement, in the periodic field V (R), equation (2) has a solution at all values. The areas of allowed energies alternate with the zones of prohibited energies. In the model of weak communication, this is explained by the Bragg reflection of electron waves in the crystal.

Consider this question more. The condition of maximum reflection of electron waves in the crystal (the condition of Wulf - Bragg) is determined by the formula (17) of Ch.I. Considering that G \u003d N G, we obtain:

Consider a system of finite intervals that do not contain values \u200b\u200bof k satisfying relation (7):

(- n g / 2

The area of \u200b\u200bchange to in three-dimensionalk - space given by the formula

(8) For all possible directions, determines the boundaries of the N - oh Brillouin zone. Within each brilluene zone (n \u003d 1,2,3, ...), the electron energy is a continuous functionk, and at the boundaries of the zones it will tear the gap. Indeed, when performing condition (7) amplitude falling,

ψ k (r) \u003d uk (r) ei kr

and reflected

ψ -k (R) \u003d U - K (R) E -i Kr

waves will be the same, u k (r) \u003d u -k (R). These waves give two solutions to the Schrödinger equation:

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

This function describes such a distribution of electrons in which they are located mainly in areas corresponding to the middle distances between ions. At the same time, the potential energy will be greater. The functions ψ 2 will correspond to the E2\u003e E1 energy.

forbidden zones EG width. Energy E`1 Determines the upper boundary of the first zone, and Energy E2 is the lower boundary of the second zone. This means that in the propagation of electron waves in crystals, there are areas of energy values \u200b\u200bfor which there are no solutions of the Schrödinger equation that have a wave nature.

Since the nature of the energy dependence on the wave vector significantly affects the dynamics of electrons in the crystal, it is of interest to consider for example the simplest case of a linear chain of atoms located at a distance and one from the other along the x axis. In this case, G \u003d 2π / a. Figure 12 presents dispersion curves for the three first one-dimensional brilline 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.

In fig. 12 presented extended zone scheme, in which various energy zones are placed in space in various brilline zones. However, it is always possible, and often conveniently, choose a wave vector so that the end of it is lying inside the first brilline zone. We write the flea function in the form:

lying in the first brilline zone. Substituting to the formula (11), we get:

it has a form of flea function with a Bloch factor (13). The N index now indicates the number of the energy zone to which this function belongs. The procedure for bringing an arbitrary wave vector to the first brilline zone was named schemes of reduced zones. In this scheme, the vector accepts the -G / 2 values< 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 prohibited zones is due to the Bragg reflection of the electronic waves de Broglyl from crystalline planes. The gap points are determined by the conditions for the maximum reflection of the waves.

According to the laws of quantum mechanics, the progressive movement of the electron is considered as the movement of the wave package with wave vectors close to the vector to. The group speed of the wave packet, V is determined by the expression.

The most valuable statement in modern physics sufficient to understand all The properties of solid bodies - hypothesis about their atomic structure .

Consider on the basis of the atomic hypothesis of the idea of \u200b\u200bthe movement of electrons in solid bodies. It is excellent to try to associate the properties of a solid body with the properties of a single atom. The properties of the atom are well studied experimentally and theoretically interpreted by quantum mechanics. They can be summarized as follows.

1. Electron moving around the atomic nucleus, may be not in any condition, but only in one of so-called stationary states.

2. Status is characterized by a certain energy and electron density distribution. The aggregate of stationary states forms an electron energy spectrum in the atom. The energy spectrum is absolutely individual for each atom, this is a kind of fingerprint. The distribution of electron density shows in which areas around the electron atom is predominantly, that is, with a probability close to 1. The energy spectrum is made in the form of an energy diagram (Fig. 1.1). The state with minimal energy is called the main one. The electron is located closest to the kernel.

Fig.1.1. Energy spectrum of hydrogen atom.

the electronic properties of the crystal are defined, as well as the properties of an atom, two factors - an energy spectrum of electrons in the crystal and their statistics, that is, the distribution law by states.

The structure of the energy spectrum of the crystal is qualitatively clarified, based on the spectrum of a separate atom.

Imagine yourself N. The same atoms removed on such long distances that they do not affect each other. The energy spectrum of such an ensemble of independent atoms will consist of N. coinciding atomic spectra. Each atomic state will be simultaneously the status of the ensemble. Such states whose energies coincide are called N - multiple degenerate.

Let's start closing atoms. At a certain interatomic distance, the electrostatic forces of the electro-nuclear attraction and electron-e-repulsion will be noticeable. Total to prevail the attraction, but repulsion will lead to the fact that the previously coincided atomic levels of energy split into N. individual levels (Fig.1.4). When the interatomic distance is reached, crystal is formed. Further rapprochement impede large pushing strengths.

Fig.1.4. Education of the Energy Spectrum of the Crystal

Each atomic level turns thus in the zone of the resolved electron energies in the crystal width. If the sum of the semi-width neighboring zones is less than the distance between the corresponding atomic levels, then the allowed zones are separated forbidden zone . If the amount of semides exceeds the distance between the levels, then the adjacent allowable zones overlap, forming one, wider, permitted zone.

The described picture of the formation of the energy spectrum is applicable to metal crystals, semiconductors and dielectrics. Which type will belong to a specific crystal, determined by the number of electrons Z. in atom.

If a Z.- younger number, then Z / 2. The lowest permitted zones will be completely filled, and the rest are empty. The term "filled zone" should be understood in the sense that there is exactly in the crystal N. electrons with energies belonging to this permitted zone. The topmost of the filled zones is called a valence zone, and the following is an empty - conduction zone. Crystals with such filling of zones are called dielectrics.