Defects in crystals. Presentation on the topic "defects in crystal lattices" Description of the presentation Presentation Defects in crystals on slides

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Ideal crystals, in which all atoms would be in positions with minimal energy, practically do not exist. Deviations from the ideal lattice can be temporary or permanent. Temporary deviations arise when the crystal is exposed to mechanical, thermal and electromagnetic vibrations, when a stream of fast particles passes through the crystal, etc. Permanent imperfections include:

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point defects (interstitial atoms, vacancies, impurities). Point defects are small in all three dimensions, their sizes in all directions are no more than several atomic diameters;

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linear defects (dislocations, chains of vacancies and interstitial atoms). Linear defects have atomic sizes in two dimensions, and in the third they are significantly larger in size, which can be commensurate with the length of the crystal;

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flat, or surface, defects (grain boundaries, boundaries of the crystal itself). Surface defects are small in only one dimension;

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volumetric defects, or macroscopic disturbances (closed and open pores, cracks, inclusions of foreign matter). Volume defects have relatively large sizes, incommensurate with the atomic diameter, in all three dimensions.

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Both interstitial atoms and vacancies are thermodynamic equilibrium defects: at each temperature there is a very certain number of defects in the crystalline body. There are always impurities in lattices, since modern methods of crystal purification do not yet allow obtaining crystals with a content of impurity atoms of less than 10 cm-3. If an impurity atom replaces an atom of the main substance at a lattice site, it is called a substitutional impurity. If an impurity atom is introduced into an interstitial site, it is called an interstitial impurity.

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A vacancy is the absence of atoms at the sites of a crystal lattice, “holes” that were formed as a result of various reasons. It is formed during the transition of atoms from the surface to the environment or from lattice nodes to the surface (grain boundaries, voids, cracks, etc.), as a result of plastic deformation, when the body is bombarded with atoms or high-energy particles. The concentration of vacancies is largely determined by body temperature. Single vacancies can meet and combine into divacancies. The accumulation of many vacancies can lead to the formation of pores and voids.

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Description of the presentation Presentation Defects in crystals on slides

Energy changes that occur during the formation of defects in a perfect crystal. The gain in entropy associated with the presence of a choice of positions is called configuration entropy and is determined by the Boltzmann formula S = k ln. W, where W is the probability of the formation of a single vacancy, proportional to the number of regular atoms forming the lattice (10 23 per 1 mole of substance).

Various types of defects in crystals: a) vacancy; b) interstitial atom; c) a small replacement defect; d) large replacement defect; e) Frenkel defect; f) Schottky defect (a pair of vacancies in the cation and anion sublattices)

The energy of displacement of an atom from its position in the lattice. Energy barrier. To move an atom from its position, activation energy is required. ΔE – defect formation energy; E * - activation energy. 1 / 1 1 E k. T sn C N e , 2/ 2 2 E k. T mn C N e Equilibrium will be established if n 1 = n 2: under equilibrium conditions, there are vacancies and interstitial atoms in the metal lattice! //Ek. T m s. N N Ce

Dislocations. Mechanical properties and reactivity of solids. 1) - metals usually turn out to be much more ductile than can be expected based on calculations. The calculated value of shear stress in metals is 10 5 - 10 6 N/cm 2, while experimentally found values ​​for many metals do not exceed 10 - 100 N/cm 2. This indicates that there are some “weak links” in the structure of metals , thanks to which metals are deformed so easily; 2) - on the surfaces of many well-cut crystals, under a microscope or even with the naked eye, spirals along which the crystal grew are visible. Such spirals cannot form in perfect crystals; 3) - without ideas about the existence of dislocations, it would be difficult to explain such properties of metals as plasticity and fluidity. Plates of magnesium metal, for example, can be stretched, almost like rubber, to several times their original length; 4) - hardening in metals could not be explained without invoking ideas about dislocations.

Arrangement of atoms around an edge dislocation An edge dislocation is an “extra” atomic half-plane that does not pass through the entire crystal, but only through part of it. Edge dislocation projection.

Movement of an edge dislocation under the action of shear stress. If you connect points A and B, then this will be a projection of the slip plane along which dislocations move. Dislocations are characterized by the Burgers vector b. To find the magnitude and direction of b, it is necessary to describe a contour around the dislocation, mentally drawing it from atom to atom (Fig. e). In a defect-free region of the crystal, such a contour ABCD, constructed from translations to one interatomic distance in each direction, is closed: its beginning and end coincide at point A. On the contrary, contour 12345 surrounding the dislocation is not closed, since points 1 and 5 do not coincide. The magnitude of the Burgers vector is equal to the distance 1 - 5, and the direction is identical to the direction 1 - 5 (or 5 - 1). The Burgers vector of an edge dislocation is perpendicular to the dislocation line and parallel to the direction of motion of the dislocation line (or direction of shear) under the action of an applied stress.

Screw dislocation With continued shear stress, indicated by the arrows, the SS ' line and slip marks reach the back face of the crystal. To find the Burgers vector of a screw dislocation, let us again imagine contour 12345 (Fig. a) “circling” around it. Vector b is determined by the magnitude and direction of the segment 1 - 5. For a screw dislocation, it is parallel to the dislocation line SS ' (in the case of an edge dislocation, it is perpendicular) and perpendicular to the direction of movement of the dislocation, coinciding, as in the case of an edge dislocation, with the direction of shear or slip.

A dislocation line that changes the nature of the dislocation from screw to edge. Origin and movement of a dislocation loop The nature of dislocations is such that they cannot end inside the crystal: if in some place on the crystal surface a dislocation enters the crystal, this means that somewhere on another part of the surface it leaves the crystal.

Scheme of the appearance of a dislocation loop (ring) Scheme of the appearance of vacancies (b) by the annihilation of two dislocations of the opposite sign (a). In reality, direct application of an external deforming force is not necessary for the formation of dislocations. This force can be thermal stresses arising during crystallization, or, for example, similar stresses in the area of ​​foreign inclusions in a solidifying metal ingot during cooling of the melt, etc. In real crystals, excess extraplanes can arise simultaneously in different parts of the crystal. The extraplane, and therefore the dislocations, are mobile in the crystal. This is their first important feature. The second feature of dislocations is their interaction with the formation of new dislocations, dislocation loops similar to those shown in the figures below, and even the formation of vacancies due to the annihilation of two dislocations of opposite sign.

Mechanical strength of metals. Frenkel's model. The destructive force is usually called stress and denoted by σ. According to this model, the resistance σ first increases as the shift along the x axis increases and then drops to zero as soon as the atomic planes shift by one interatomic distance a. When x>a the value of σ increases again and again falls to zero at x = 2a, etc., i.e. σ(x) is a periodic function that can be represented as σ = A sin (2 π x/a ) , for the region of small x A = G /(2π), where G is Young’s modulus. A more rigorous theory subsequently gave a refined expression σ m ax = G /30. Diagram of the shift of atomic planes (a) and the dependence of voltage on distance in the crystal (b).

Experimental and theoretical values ​​of the shear strength of some metals. Roller model of shift of atomic planes of a crystal | F 1 + F 2 |=| F 4 + F 5 | the entire roller system is in balance. One has only to slightly change the balance of forces with a weak external influence, and the top row of rollers will move. Therefore, the movement of a dislocation, i.e., a collection of defective atoms, occurs at low loads. The theory gives σ m ax, which shifts a dislocation, in the form σ m ax = exp ( - 2 π a / [ d (1- ν) ]), where ν is Poisson's ratio (transverse elasticity), d is the distance between slip planes, and - period of the crystal lattice. Assuming a = d, ν = 0.3, we obtain the values ​​of σ m ax in the last column of the table, from which it can be seen that they are much closer to the experimental ones.

Scheme of caterpillar movement Schemes of dislocation-type movement: a - tensile dislocation, b - compressive dislocation, c - carpet movement. “First, let’s try to drag the caterpillar along the ground. It turns out that this is not easy to do; it requires significant effort. They are due to the fact that we are trying to simultaneously lift all pairs of caterpillar legs off the ground. The caterpillar itself moves in a different mode: it tears off only one pair of legs from the surface, carries them through the air, lowers them to the ground, then repeats the same with the next pair of legs, etc., etc. After doing this all pairs of legs will be transported through the air, the entire caterpillar as a whole will move the distance by which each pair of legs alternately shifted. The caterpillar does not drag any pair of legs along the ground. That’s why it crawls easily.”

Ways to control dislocation defects. Fixation by impurities. An impurity atom interacts with a dislocation and the movement of such a dislocation, burdened with impurity atoms, turns out to be difficult. Therefore, the efficiency of dislocation pinning by impurity atoms will be determined by the interaction energy E, which in turn consists of two components: E 1 and E 2. The first component (E 1) is the energy of elastic interaction, and the second (E 2) is the energy of electrical interaction. Fixation by foreign particles. Foreign particles are microscopic inclusions of a substance different from the base metal. These particles are introduced into the metal melt and remain in the metal after it solidifies when the melt cools. In some cases, these particles enter into a chemical interaction with the base metal, and then these particles already represent an alloy. The mechanism of dislocation pinning by such particles is based on different speeds of movement of dislocations in the metal matrix and in the material of foreign particles. Fixation with inclusions of the second phase. The second phase is understood as the release (precipitates) of an excess concentration of an impurity from a metal-impurity solution compared to the equilibrium one. The separation process is called solid solution decomposition. Intertwining of dislocations. When the density of dislocations in a metal is high, they become intertwined. This is due to the fact that some dislocations begin to move along intersecting slip planes, preventing the movement of others.

Qualitative view of the solubility curve. If the crystal contained a concentration of C m at a temperature T m and was quickly cooled, then it will have a concentration of C m at low temperatures, for example, at T 1, although the equilibrium concentration should be C 1. The excess concentration ΔC = C m – C 1 should be at sufficiently long heating will drop out of the solution, because only then will the solution assume a stable equilibrium state corresponding to the minimum energy of the system A 1- x B x.

Methods for detecting dislocations a) Micrograph (obtained in a transmission electron microscope, TEM) of a Sr crystal. Ti. O 3 containing two edge dislocations (100) (marked in the figure). b) Schematic representation of an edge dislocation. c) Micrograph of the surface of a Ga crystal. As (obtained in a scanning tunneling microscope). At point C there is a screw dislocation. d) Scheme of a screw dislocation.

Visualization of dislocations using a transmission electron microscope. a) Dark lines on a bright background are dislocation lines in aluminum after 1% stretching. b) The reason for the contrast of the dislocation region - and the curvature of crystallographic planes leads to electron diffraction, which weakens the transmitted electron beam

a) Etching pits on the surface (111) of bent copper; b) on the surface (100) c) (110) recrystallized Al -0.5% Mn. Dislocations can also be made visible in a conventional optical microscope. Since the areas around the point where dislocations reach the surface are more susceptible to chemical etching, so-called etch pits are formed on the surface, which are clearly visible in an optical microscope. Their shape depends on the Miller indices of the surface.

To obtain a metal material with increased strength, it is necessary to create a large number of dislocation pinning centers, and such centers must be evenly distributed. These requirements led to the creation of superalloys. New metal functional materials. "Designing" the structure of alloys A superalloy is at least a two-phase system in which both phases differ primarily in the degree of order in the atomic structure. The superalloy exists in the Ni - Al system. In this system, an ordinary mixture can be formed, i.e., an alloy with a chaotic distribution of Ni and Al atoms. This alloy has a cubic structure, but the nodes of the cube are replaced by Ni or Al atoms randomly. This disordered alloy is called the γ phase.

Along with the γ phase in the Ni - А l system, an intermetallic compound Ni 3 А l can also be formed, also with a cubic structure, but ordered. Cuboids Ni 3 А l are called γ ‘ -phase. In the γ '-phase, Ni and A l atoms occupy the sites of the cubic lattice according to a strict law: for one aluminum atom there are three nickel atoms. Scheme of dislocation movement in an ordered crystal

C diagram of dislocation pinning by inclusions of another phase. DD – moving dislocation. To create a superalloy, nickel is melted and mixed with aluminum. When the molten mixture is cooled, the disordered γ phase first solidifies (its crystallization temperature is high), and then small-sized cuboids of the γ '-phase are formed inside it as the temperature decreases. By varying the cooling rate, it is possible to regulate the kinetics of formation, and hence the size of inclusions of the γ ‘-phase Ni 3 А l.

The next step in the development of high-strength metallic materials was the production of pure Ni 3 Al without the γ phase. A type of fine-grained mosaic structure of metal. This material is very fragile: chipping occurs along the grain boundaries of the mosaic structure. Here other types of defects are revealed, in particular the surface. Indeed, on the surface of the crystal there is a break in chemical bonds, i.e. a violation is a break in the crystal field, and this is the main reason for the formation of a defect. Dangling chemical bonds are unsaturated, and in contact they are already deformed and therefore weakened. Scheme of breaking chemical bonds on the crystal surface.

To eliminate these defects it is necessary: ​​- either to produce a monocrystalline material that does not contain individual grains-crystallites; - or find a “buffer” in the form of impurities that would not penetrate in noticeable quantities into the volume of Ni 3 Al, but would be well adsorbed on the surface and fill vacancies. Isovalent impurities have the greatest affinity for vacancies, i.e. impurities whose atoms are in the same group of the Periodic Table as the atom removed from the crystal lattice and forming the vacancy. Superalloys Ni 3 Al and Ni 3 Al are widely used today as heat-resistant materials at temperatures up to 1000°C. Similar cobalt-based superalloys have slightly lower strength, but retain it up to a temperature of 1100°C. Further prospects are associated with the production of intermetallic compounds of Ti. Al and T i 3 A l in their pure form. Parts made from them are 40% lighter than the same parts made from nickel superalloy.

Alloys with easy deformability under load. The method for creating such metallic materials is to produce a structure with very small crystallite grains. Grains with dimensions less than 5 microns slide over each other under load without destruction. A sample consisting of such grains can withstand a relative tension Δ l / l 0 = 10 without destruction, i.e., the length of the sample increases by 1000% of the original length. This is the effect of superplasticity. It is explained by the deformation of bonds in grain contacts, i.e., a large number of surface defects. Superplastic metal can be processed almost like plasticine, giving it the desired shape, and then a part made of such material is heat treated to enlarge the grains and quickly cooled, after which the effect of superplasticity disappears, and the part is used for its intended purpose. The main difficulty in producing superplastic metals is achieving a fine grain structure.

It is convenient to obtain nickel powder by the leaching method, in which the Al - Ni alloy is crushed using Na alkali. OH leach aluminum to produce a powder with a particle diameter of about 50 nm, but these particles are so chemically active that they are used as a catalyst. The activity of the powder is explained by a large number of surface defects - broken chemical bonds that can attach electrons from adsorbed atoms and molecules. Scheme of rapid crystallization of a metal melt sprayed in a centrifuge: 1 - cooling gas; 2 - melt; 3 - melt jet; 4 - small particles; 5 - rotating disk Scheme of dynamic pressing of metal powders: 1 - projectile, 2 - powder, 3 - mold, 4 - gun barrel

Laser glazing method. The term is borrowed from porcelain (ceramic) production. Using laser radiation, a thin layer on the metal surface is melted and rapid cooling is applied at rates of the order of 10 7 K/s. The limiting case of ultra-fast hardening is the production of amorphous metals and alloys - metallic glasses.

Superconducting metals and alloys Material Al V In Nb Sn Pb Nb 3 Sn Nb 3 Ge Т с, К 1, 19 5, 4 3, 4 9, 46 3, 72 7, 18 18 21. . . 23In 1911 in Holland, Kamerlingh Onnes discovered a decrease in the resistivity of mercury at the boiling point of liquid helium (4.2 K) to zero! The transition to the superconducting state (ρ = 0) occurred abruptly at a certain critical temperature Tc. Until 1957, the phenomenon of superconductivity had no physical explanation, although the world was busy searching for more and more new superconductors. Thus, by 1987, about 500 metals and alloys with different Tc values ​​were known. Niobium compounds had the highest Tc.

Continuous current. If an electric current is excited in a metal ring, then at normal, for example, room temperature, it quickly dies out, since the flow of current is accompanied by heat losses. At T ≈ 0 in a superconductor, the current becomes undamped. In one of the experiments, the current circulated for 2.5 years until it was stopped. Since the current flows without resistance, and the amount of heat generated by the current is Q = 0.24 I 2 Rt, then in the case of R = 0 there are simply no heat losses. There is no radiation in the superconducting ring due to quantization. But in an atom the momentum and energy of one electron are quantized (take on discrete values), and in a ring the current, i.e. the entire set of electrons, is quantized. Thus, we have an example of a cooperative phenomenon - the movement of all electrons in a solid is strictly coordinated!

Meissner effect Discovered in 1933. Its essence lies in the fact that an external magnetic field at T< Т с не проникает в толщу сверхпроводника. Экспериментально это наблюдается при Т=Т с в виде выталкивания сверхпроводника из магнитного поля, как и полагается диамагнетику. Этот эффект объясняется тем, что в поверхностном слое толщиной 0, 1 мкм внешнее магнитное поле индуцирует постоянный ток, но тепловых и излучательных потерь нет и в результате вокруг этого тока возникает постоянное незатухающее магнитное поле. Оно противоположно по направлению внешнему полю (принцип Ле-Шателье) и экранирует толщу сверхпроводника от внешнего магнитного поля. При увеличении Н до некоторого значения Н с сверхпроводимость разрушается. Значения Н с лежат в интервале 10 -2 . . . 10 -1 Т для различных сверхпроводников. http: //www. youtube. com/watch? v=bo 5XTURGMTM

If there were no Meissner effect, the conductor without resistance would behave differently. When transitioning to a state without resistance in a magnetic field, it would maintain a magnetic field and would retain it even when the external magnetic field is removed. It would be possible to demagnetize such a magnet only by increasing the temperature. This behavior, however, has not been observed experimentally.

In addition to the superconductors considered, which were called superconductors of the first kind, superconductors of the second kind were discovered (A, V. Shubnikov, 1937; A. Abrikosov, 1957). In them, an external magnetic field, upon reaching a certain H c1, penetrates into the sample, and electrons, whose velocities are directed perpendicular to H, begin to move in a circle under the influence of the Lorentz force. Vortex filaments appear. The “trunk” of the thread turns out to be a non-superconducting metal, and superconducting electrons move around it. As a result, a mixed superconductor is formed, consisting of two phases - superconducting and normal. Only when another, higher value of Hc is reached, the 2 filaments, expanding, come closer together, and the superconducting state is completely destroyed. The values ​​of Нс2 reach 20. . . 50 T for such superconductors as Nb 3 Sn and Pb. Mo 6 O 8 respectively.

Josephson structure diagram: 1-dielectric layer; 2-superconductors The structure consists of two superconductors separated by a thin dielectric layer. This structure is located at a certain potential difference specified by the external voltage V. From the theory developed by Feynman, the expression for the current I flowing through the structure follows: I= I 0 sin [(2e. V/h)t+ φ 0 ], where I 0 = 2Kρ/ h (K is the interaction constant of both superconductors in the Josephson structure; ρ is the density of particles carrying the superconducting current). The quantity φ 0 = φ 2 - φ 1 is considered as the phase difference between the wave functions of electrons in contacting superconductors. It can be seen that even in the absence of external voltage (V = 0), a direct current flows through the contact. This is the stationary Josephson effect. If we place the Josephson structure in a magnetic field, then the magnetic flux Ф causes a change in Δ φ, and as a result we get: I= I 0 sinφ 0 cos (Ф / Ф 0), where Ф 0 is the magnetic flux quantum. The value of Ф 0 = h с/е is equal to 2.07·10 -11 T cm 2. Such a small value of Ф 0 allows the production of ultra-sensitive magnetic field meters (magnetometers) that detect weak magnetic fields from the biocurrents of the brain and heart.

The equation I= I 0 sin [(2e. V/h)t+ φ 0 ] shows that in the case of V ≠ 0 the current will oscillate with a frequency f = 2 e. V/h. Numerically, f falls into the microwave range. Thus, the Josephson contact allows you to create alternating current using a constant potential difference. This is the non-stationary Josephson effect. An alternating Josephson current, just like an ordinary current in an oscillatory circuit, will emit electromagnetic waves, and this radiation is actually observed experimentally. For high-quality Josephson S - I - S contacts, the thickness of the dielectric layer I must be extremely small - no more than a few nanometers. Otherwise, the coupling constant K, which determines the current I0, is greatly reduced. But the thin insulating layer degrades over time due to the diffusion of atoms from superconducting materials. In addition, the thin layer and the significant dielectric constant of its material lead to a large electrical capacitance of the structure, which limits its practical use.

Basic qualitative ideas about the physics of the phenomenon of superconductivity. Mechanism of formation of Cooper pairs Let us consider a pair of electrons e 1 and e 2, which are repelled by the Coulomb interaction. But there is also another interaction: for example, electron e 1 attracts one of the ions I and displaces it from the equilibrium position. The I ion creates an electric field that acts on the electrons. Therefore, its displacement will affect other electrons, for example, e 2. Thus, the interaction of electrons e 1 and e 2 occurs through the crystal lattice. An electron attracts an ion, but since Z 1 > Z 2, the electron, together with the ion “coat,” has a positive charge and attracts a second electron. At T > T c, thermal motion blurs the ion “coat”. The displacement of an ion is the excitation of lattice atoms, i.e., nothing more than the birth of a phonon. During the reverse transition, a phonon is emitted and is absorbed by another electron. This means that the interaction of electrons is the exchange of phonons. As a result, the entire collective of electrons in the solid body turns out to be bound. At any given moment, an electron is more strongly connected to one of the electrons in this collective, i.e., the entire electronic collective seems to consist of electron pairs. Within a pair, electrons are bound by a certain energy. Therefore, only those influences that overcome the binding energy can affect this pair. It turns out that ordinary collisions change the energy by a very small amount, and it does not affect the electron pair. Therefore, electron pairs move in the crystal without collisions, without scattering, i.e., the current resistance is zero.

Practical application of low-temperature superconductors. Superconducting magnets, made of Nb 3 Sn superconducting alloy wire. At present, superconducting solenoids with a field of 20 T have already been built. Materials corresponding to the formula M x Mo 6 O 8, where the metal atoms M are Pb, Sn, Cu, Ag, etc., are considered promising. The highest magnetic field (approximately 4 0 T) obtained in Pb solenoid. Mo 6 O 8. The colossal sensitivity of Josephson junctions to a magnetic field served as the basis for their use in instrument making, medical equipment and electronics. SQUID is a superconducting quantum interference sensor used for magnetoencephalography. Using the Meissner effect, a number of research centers in different countries are conducting work on magnetic levitation - “floating” above the surface to create high-speed magnetic levitation trains. Induction energy storage devices in the form of a circuit with undamped current and electric power transmission lines (EPL) without losses through superconducting wires. Magnetohydrodynamic (MHD) generators with superconducting windings. They have an efficiency of converting thermal energy into electrical energy of 50%, while for all other power plants it does not exceed 35%.

Defects in crystals are divided into:

Zero-dimensional

One-dimensional

Two-dimensional

Point defects (zero-dimensional) - violation of periodicity at lattice points isolated from each other; in all three dimensions they do not exceed one or more interatomic distances (lattice parameters). Point defects are vacancies, atoms in interstices, atoms in sites of a “foreign” sublattice, impurity atoms in sites or interstices.

Vacancies– absence of an atom or ion in a crystal lattice site; Implemented or interstitial atoms or ions can be both intrinsic and impurity atoms or ions that differ from the main atoms in size or valency. Substitutional impurities replace particles of the main substance at lattice nodes.

Linear(one-dimensional) defects – The main linear defects are dislocations. The a priori concept of dislocations was first used in 1934 by Orowan and Theiler in their study of plastic deformation of crystalline materials, to explain the large difference between the practical and theoretical strength of a metal. Dislocation– these are defects in the crystal structure, which are lines along and near which the correct arrangement of atomic planes characteristic of the crystal is disrupted.

Surface defects of the crystal lattice. Surface lattice defects include stacking faults and grain boundaries.

Conclusion: All types of defects, regardless of the cause of their occurrence, lead to a violation of the equilibrium state of the lattice and increase its internal energy.

Defects in crystals

Any real crystal does not have a perfect structure and has a number of violations of the ideal spatial lattice, which are called defects in crystals.

Defects in crystals are divided into zero-dimensional, one-dimensional and two-dimensional. Zero-dimensional (point) defects can be divided into energy, electronic and atomic.

The most common energy defects are phonons - temporary distortions in the regularity of the crystal lattice caused by thermal motion. Energy defects in crystals also include temporary lattice imperfections (excited states) caused by exposure to various radiations: light, X-ray or γ-radiation, α-radiation, neutron flux.

Electronic defects include excess electrons, electron deficiencies (unfilled valence bonds in the crystal - holes) and excitons. The latter are paired defects consisting of an electron and a hole, which are connected by Coulomb forces.

Atomic defects appear in the form of vacant sites (Schottky defects, Fig. 1.37), in the form of displacement of an atom from a site to an interstitial site (Frenkel defects, Fig. 1.38), in the form of the introduction of a foreign atom or ion into the lattice (Fig. 1.39). In ionic crystals, to maintain the electrical neutrality of the crystal, the concentrations of Schottky and Frenkel defects must be the same for both cations and anions.

Linear (one-dimensional) defects in the crystal lattice include dislocations (translated into Russian, the word “dislocation” means “displacement”). The simplest types of dislocations are edge and screw dislocations. Their nature can be judged from Fig. 1.40-1.42.

In Fig. 1.40, and the structure of an ideal crystal is depicted in the form of a family of atomic planes parallel to each other. If one of these planes breaks inside the crystal (Fig. 1.40, b), then the place where it breaks forms an edge dislocation. In the case of a screw dislocation (Fig. 1.40, c), the nature of the displacement of atomic planes is different. There is no break inside the crystal of any of the atomic planes, but the atomic planes themselves represent a system similar to a spiral staircase. Essentially, this is one atomic plane twisted along a helical line. If we walk along this plane around the axis of the screw dislocation (dashed line in Fig. 1.40, c), then with each revolution we will rise or fall by one pitch of the screw equal to the interplanar distance.

A detailed study of the structure of crystals (using an electron microscope and other methods) showed that a single crystal consists of a large number of small blocks, slightly disoriented relative to each other. The spatial lattice inside each block can be considered quite perfect, but the dimensions of these areas of ideal order inside the crystal are very small: it is believed that the linear dimensions of the blocks range from 10-6 to 10 -4 cm.

Any given dislocation can be represented as a combination of an edge and a screw dislocation.

Two-dimensional (planar) defects include boundaries between crystal grains and rows of linear dislocations. The crystal surface itself can also be considered as a two-dimensional defect.

Point defects such as vacancies are present in every crystal, no matter how carefully it is grown. Moreover, in a real crystal, vacancies are constantly generated and disappeared under the influence of thermal fluctuations. According to the Boltzmann formula, the equilibrium concentration of PV vacancies in a crystal at a given temperature (T) is determined as follows:

where n is the number of atoms per unit volume of the crystal, e is the base of natural logarithms, k is Boltzmann’s constant, Ev is the energy of vacancy formation.

For most crystals, the energy of vacancy formation is approximately 1 eV, at room temperature kT » 0.025 eV,

hence,

With increasing temperature, the relative concentration of vacancies increases quite quickly: at T = 600° K it reaches 10-5, and at 900° K-10-2.

Similar reasoning can be made regarding the concentration of defects according to Frenkel, taking into account the fact that the energy of formation of interstitials is much higher (about 3-5 eV).

Although the relative concentration of atomic defects may be small, the changes in the physical properties of the crystal caused by them can be enormous. Atomic defects can affect the mechanical, electrical, magnetic and optical properties of crystals. To illustrate, we will give just one example: thousandths of an atomic percent of some impurities in pure semiconductor crystals change their electrical resistance by 105-106 times.

Dislocations, being extended crystal defects, cover with their elastic field of a distorted lattice a much larger number of nodes than atomic defects. The width of the dislocation core is only a few lattice periods, and its length reaches many thousands of periods. The energy of dislocations is estimated to be on the order of 4 10 -19 J per 1 m of dislocation length. The dislocation energy, calculated for one interatomic distance along the dislocation length, for different crystals lies in the range from 3 to 30 eV. Such a large energy required to create dislocations is the reason that their number is practically independent of temperature (athermicity of dislocations). Unlike vacancies [see formula (1.1), the probability of the occurrence of dislocations due to fluctuations of thermal motion is vanishingly small for the entire temperature range in which the crystalline state is possible.

The most important property of dislocations is their easy mobility and active interaction with each other and with any other lattice defects. Without considering the mechanism of dislocation motion, we point out that in order to cause dislocation motion, it is enough to create a small shear stress in the crystal of the order of 0.1 kG/mm2. Already under the influence of such a voltage, the dislocation will move in the crystal until it encounters any obstacle, which may be a grain boundary, another dislocation, an interstitial atom, etc. When it encounters an obstacle, the dislocation bends, bends around the obstacle, forming an expanding dislocation loop, which then becomes detached and forms a separate dislocation loop, and in the area of ​​the separate expanding loop there remains a segment of linear dislocation (between two obstacles), which, under the influence of sufficient external stress, will bend again, and the whole process will repeat again. Thus, it is clear that when moving dislocations interact with obstacles, the number of dislocations increases (their multiplication).

In undeformed metal crystals, 106-108 dislocations pass through an area of ​​1 cm2; during plastic deformation, the dislocation density increases by thousands and sometimes millions of times.

Let's consider what effect crystal defects have on its strength.

The strength of an ideal crystal can be calculated as the force necessary to tear atoms (ions, molecules) away from each other, or to move them, overcoming the forces of interatomic adhesion, i.e. the ideal strength of a crystal should be determined by the product of the magnitude of the interatomic bond forces by the number of atoms , per unit area of ​​the corresponding section of the crystal. The shear strength of real crystals is usually three to four orders of magnitude lower than the calculated ideal strength. Such a large decrease in the strength of the crystal cannot be explained by a decrease in the working cross-sectional area of ​​the sample due to pores, cavities and microcracks, since if the strength was weakened by a factor of 1000, the cavities would have to occupy 99.9% of the cross-sectional area of ​​the crystal.

On the other hand, the strength of single-crystalline samples, in the entire volume of which approximately the same orientation of the crystallographic axes is maintained, is significantly lower than the strength of a polycrystalline material. It is also known that in some cases crystals with a large number of defects have higher strength than crystals with fewer defects. Steel, for example, which is iron “spoiled” by carbon and other additives, has significantly higher mechanical properties than pure iron.

Imperfection of crystals

So far we have considered ideal crystals. This allowed us to explain a number of characteristics of the crystals. In fact, crystals are not ideal. They may contain a large number of various defects. Some properties of crystals, in particular electrical and others, also depend on the degree of perfection of these crystals. Such properties are called structure-sensitive properties. There are 4 main types of imperfections in a crystal and a number of non-main ones.

The main imperfections include:

1) Point defects. They include empty lattice sites (vacancies), interstitial extra atoms, and impurity defects (substitutional impurities and interstitial impurities).

2) Linear defects.(dislocations).

3) Planar defects. They include: surfaces of various other inclusions, cracks, outer surface.

4) Volumetric defects. They include the inclusions themselves and foreign impurities.

Non-major imperfections include:

1) Electrons and holes are electronic defects.

2) Phonons, photons and other quasiparticles that exist in a crystal for a limited time

Electrons and holes

In fact, they did not affect the energy spectrum of the crystal in an unexcited state. However, in real conditions, at T¹0 (absolute temperature), electrons and holes can be excited in the lattice itself, on the one hand, and on the other hand, they can be injected (introduced) into it from the outside. Such electrons and holes can lead, on the one hand, to deformation of the lattice itself, and on the other hand, due to interaction with other defects, disrupt the energy spectrum of the crystal.

Photons

They cannot be seen as true imperfection. Although photons have a certain energy and momentum, if this energy is not enough to generate electron-hole pairs, then in this case the crystal will be transparent to the photon, that is, it will freely pass through it without interacting with the material. They are included in the classification because they can influence the energy spectrum of the crystal due to interaction with other imperfections, in particular with electrons and holes.

Point imperfections (defect)

At T¹0 it may turn out that the energy of particles at the nodes of the crystal lattice will be sufficient to transfer the particle from a node to an interstitial site. At which each specific temperature will have its own specific concentration of such point defects. Some defects will be formed due to the transfer of particles from nodes to interstitial sites, and some of them will recombine (decrease in concentration) due to the transition from interstitial sites to nodes. Due to the equality of flows, each temperature will have its own concentration of point defects. Such a defect, which is a combination of an interstitial atom and the remaining free site), cancia) is a defect according to Frenkel. A particle from the near-surface layer, due to temperature, can reach the surface), the surface is an endless sink for these particles). Then one free node (vacancy) is formed in the near-surface layer. This free site can be occupied by a deeper-lying atom, which is equivalent to the movement of vacancies deeper into the crystal. Such defects are called Schottky defects. One can imagine the following mechanism for the formation of defects. A particle from the surface moves deep into the crystal and extra interstitial atoms without vacancies appear in the thickness of the crystal. Such defects are called anti-Shottky defects.

Formation of point defects

There are three main mechanisms for the formation of point defects in a crystal.

Hardening. The crystal is heated to a significant temperature (elevated), and each temperature corresponds to a very specific concentration of point defects (equilibrium concentration). At each temperature, an equilibrium concentration of point defects is established. The higher the temperature, the higher the concentration of point defects. If the heated material is cooled sharply in this way, then in this case this excess point defects will turn out to be frozen, not corresponding to this low temperature. Thus, an excess concentration of point defects is obtained in relation to the equilibrium one.

Impact on the crystal by external forces (fields). In this case, energy sufficient to form point defects is supplied to the crystal.

Irradiation of a crystal with high-energy particles. Due to external irradiation, three main effects are possible in the crystal:

1) Elastic interaction of particles with the lattice.

2) Inelastic interaction (ionization of electrons in the lattice) of particles with the lattice.

3) All possible nuclear transmutations (transformations).

In the 2nd and 3rd effects, the first effect is always present. These elastic interactions have a dual effect: on the one hand, they manifest themselves in the form of elastic vibrations of the lattice, leading to the formation of structural defects, on the other hand. In this case, the energy of the incident radiation must exceed the threshold energy for the formation of structural defects. This threshold energy is usually 2–3 times higher than the energy required for the formation of such a structural defect under adiabatic conditions. Under adiabatic conditions for silicon (Si), the adiabatic formation energy is 10 eV, threshold energy = 25 eV. For the formation of a vacancy in silicon, it is necessary that the energy of external radiation be at least greater than 25 eV, and not 10 eV as for the adiabatic process. It is possible that at significant energies of incident radiation, one particle (1 quantum) leads to the formation of not one, but several defects. The process can be cascading.

Point defect concentration

Let's find the concentration of defects according to Frenkel.

Let us assume that there are N particles at the nodes of the crystal lattice. Of these, n particles moved from nodes to interstices. Let the energy of defect formation according to Fresnel be Eph. Then the probability that another particle will move from a node to an interstice will be proportional to the number of particles still sitting at the nodes (N-n), and the Boltzmann multiplier, that is ~. And the total number of particles moving from nodes to interstice ~. Let's find the number of particles moving from interstices to nodes (recombines). This number is proportional to n, and is proportional to the number of empty places in the nodes, or more precisely the probability that the particle will stumble upon an empty node (that is, ~). ~. Then the total change in the number of particles will be equal to the difference of these values:

Over time, the flows of particles from nodes to interstices and in the opposite direction will become equal to each other, that is, a stationary state is established. Since the number of particles in interstices is much less than the total number of nodes, n can be neglected and. From here we will find

– concentration of defects according to Frenkel, where a and b are unknown coefficients. Using a statistical approach to the concentration of defects according to Frenkel and taking into account that N’ is the number of interstices, we can find the concentration of defects according to Frenkel: , where N is the number of particles, N’ is the number of interstices.

The process of formation of defects according to Frenkel is a bimolecular process (2-part process). At the same time, the process of formation of Schottky defects is a monomolecular process.

A Schottky defect represents one vacancy. Carrying out similar reasoning as for the concentration of defects according to Frenkel, we obtain the concentration of defects according to Schottky in the following form: , where nsh is the concentration of defects according to Schottky, Esh is the energy of formation of defects according to Schottky. Since the Schottky formation process is monomolecular, then, unlike Frenkel defects, there is no 2 in the denominator of the exponent. The formation process, for example, Frenkel defects, is characteristic of atomic crystals. For ionic crystals, defects, for example Schottky, can only form in pairs. This occurs because in order to maintain the electrical neutrality of an ionic crystal, it is necessary that pairs of ions of opposite signs simultaneously emerge to the surface. That is, the concentration of such paired defects can be represented as a bimolecular process: . Now we can find the ratio of the Frenkel defect concentration to the Schottky defect concentration: ~. The energy of formation of paired defects according to Schottky Er and the energy of formation of defects according to Frenkel Ef are on the order of 1 eV and can differ from each other on the order of several tenths of eV. KT for room temperatures is on the order of 0.03 eV. Then~. It follows that for a particular crystal one specific type of point defects will predominate.

Speed ​​of defect movement across the crystal

Diffusion is the process of moving particles in a crystal lattice over macroscopic distances due to fluctuations (changes) in thermal energy. If the moving particles are particles of the lattice itself, then we are talking about self-diffusion. If the movement involves particles that are foreign, then we are talking about heterodiffusion. The movement of these particles in the lattice can be carried out by several mechanisms:

Due to the movement of interstitial atoms.

Due to the movement of vacancies.

Due to the mutual exchange of places of interstitial atoms and vacancies.

Diffusion due to the movement of interstitial atoms

In fact, it is of a two-stage nature:

An interstitial atom must form in the lattice.

The interstitial atom must move in the lattice.

Example: we have a spatial lattice. Particle in an interstice.

In order for a particle to move from one interstitial site to a neighboring one, it must overcome a potential barrier of height Em. The frequency of particle jumps from one interstice to another will be proportional. Let the vibration frequency of the particles correspond to the interstices v. The number of neighboring internodes is equal to Z. Then the frequency of jumps: .

Diffusion due to vacancy movements

The diffusion process due to vacancies is also a 2-step process. On the one hand, vacancies must be formed, on the other hand, they must move. It should be noted that a free place (free node) where a particle can move also exists only for a certain fraction of time in proportion to where Ev is the energy of vacancy formation. And the frequency of jumps will have the form: , where Em is the energy of motion of vacancies, Q=Ev+Em is the activation energy of diffusion.

Moving particles over long distances

Let's consider a chain of identical atoms.

Let's assume that we have a chain of identical atoms. They are located at a distance d from each other. Particles can move to the left or to the right. The average displacement of particles is 0. Due to the equal probability of particle movement in both directions:

Let's find the root-mean-square displacement:

where n is the number of particle transitions, can be expressed. Then. The value is determined by the parameters of the given material. Therefore, let us denote: – diffusion coefficient, as a result:

In the 3-dimensional case:

Substituting the value of q here, we get:

Where D0 is the frequency factor of diffusion, Q is the activation energy of diffusion.

Macroscopic diffusion

Mentally, between planes 1 and 2, let us conditionally select plane 3 and find the number of particles crossing this half-plane from left to right and from right to left. Let the particle hopping frequency be q. Then, in a time equal to half-plane 3, half-plane 1 will intersect the particles. Similarly, during the same time, the selected half-plane from the side of half-plane 2 will intersect the particles. Then, during time t, the change in the number of particles in the selected half-plane can be represented in the following form: . Let's find the concentration of particles - impurities in half-planes 1 and 2:

The difference in volume concentrations C1 and C2 can be expressed as:

Let's consider a single selected layer (L2=1). We know that is the diffusion coefficient, then:

– Fick's 1st law of diffusion.

The formula for the 3-dimensional case is similar. Only in place of the one-dimensional diffusion coefficient, we substitute the diffusion coefficient for the 3-dimensional case. Using this analogy of reasoning for concentration, and not for the number of carriers, as in the previous case, one can find the 2nd Fickian diffusion.

– Fick's 2nd law.

Fick's 2nd law of diffusion is very convenient for calculations and practical applications. In particular for the diffusion coefficient of various materials. For example, we have some material on the surface of which an impurity is deposited, the surface concentration of which is equal to Q cm-2. By heating this material, this impurity diffuses into its volume. In this case, depending on time, a certain distribution of impurities is established throughout the thickness of the material for a given temperature. Analytically, the distribution of impurity concentration can be obtained by solving the Fick diffusion equation in the following form:

Graphically it is:

Using this principle, diffusion parameters can be found experimentally.

Experimental methods for studying diffusion

Activation method

A radioactive impurity is applied to the surface of the material, and then this impurity is diffused into the material. Then, part of the material is removed layer by layer and the activity of either the remaining material or the etched layer is examined. And thus the distribution of concentration C over the surface X(C(x)) is found. Then, using the obtained experimental value and the last formula, the diffusion coefficient is calculated.

Chemical methods

They are based on the fact that during the diffusion of an impurity, as a result of its interaction with the base material, new chemical compounds with lattice properties different from the basic ones are formed.

pn junction methods

Due to the diffusion of impurities in semiconductors, at some depth of the semiconductor, a region is formed in which the type of its conductivity changes. Next, the depth of the p-n junction is determined and the concentration of impurities at this depth is judged from it. And then they do it by analogy with the 1st and 2nd cases.

List of sources used

1. Kittel Ch. Introduction to solid state physics. / Transl. from English; Ed. A. A. Guseva. – M.: Nauka, 1978.

2. Epifanov G.I. Solid state physics: Textbook. allowance for colleges. – M.: Higher. schools, 1977.

3. Zhdanov G.S., Khundzhua F.G., Lectures on solid state physics - M: Moscow State University Publishing House, 1988.

4. Bushmanov B. N., Khromov Yu. A. Physics of Solid State: Textbook. allowance for colleges. – M.: Higher. schools, 1971.

5. Katsnelson A.A. Introduction to solid state physics - M: Moscow State University Publishing House, 1984.

Defects in the crystal structureReal metals that are used as structural
materials consist of a large number of irregularly shaped crystals. These
crystals
called
grains
or
crystals,
A
structure
polycrystalline or granular. Existing production technologies
metals do not allow obtaining them of ideal chemical purity, therefore
real metals contain impurity atoms. Impurity atoms are
one of the main sources of defects in the crystal structure. IN
Depending on their chemical purity, metals are divided into three groups:
chemically pure - content 99.9%;
high purity - content 99.99%;
ultrapure - content 99.999%.
Atoms of any impurities are sharply different in size and structure
differ from the atoms of the main component, so the force field around
such atoms are distorted. An elastic zone appears around any defects.
distortion of the crystal lattice, which is balanced by volume
crystal adjacent to a defect in the crystal structure.