ELECTRICAL MATERIAL SCIENCE

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1 TOMSK POLYTECHNIC UNIVERSITY ELECTRICAL MATERIAL SCIENCE Recommended for publishing as a study aid by the Editorial Board of Tomsk Polytechnic University Draftsmen V.S. Kim, O.A.Anisimova Tomsk Polytechnic University Publishing House

2 УДК (031) Д79 Д79 Vladimir S. Kim, Olga A. Anisimova The text book on the course Electrical Material Science This textbook covers the fundamentals of processes in magnetic, conducting, semiconducting and dielectric materials. The behavior of materials under the influence of magnetic and electrical fields as well as under the action of temperature, mechanical stresses, radiation, and environment is described. A classification of electrical engineering materials, their sphere of application, and the requirements to the materials are discussed also. The textbook was prepared at the Department of Electromechanical Plants and Materials TPU and is intended for students learning on the program Electrical power engineering and Electrical engineering. УДК (031) Readers Ph.D, associate professor A.N. Dudkin; Ph.D, vice director of Institute of strength and materials P.P. Kaminskij 2

3 CONTENT PREFACE GENERAL PROVISIONS Part 1. Magnetic Materials.15 Chapter 1. Classification of substances on the magnetic properties General Concepts Ferromagnetism and ferrimagnetism...16 Chapter 2. The basic properties and characteristics of magnetic materials The process of magnetization and the magnetic permeability The magnetization of ferro- and ferrimagnetic in alternating magnetic field Magnetically soft materials (soft magnets, soft ferrites) Magnetically hard materials (hard ferrites, hard magnets) Magnetic materials for special purposes Test questions to the Part 1: Magnetic Materials Part 2. Conductive Materials Chapter 3. General Concepts Chapter 4. The basic properties of conductors The conductivity and resistivity of the conductors Thermal conductivity of metals Thermoelectric power Temperature coefficient of linear expansion Mechanical properties of conductors Superconductivity Skin effect Chapter 5. Classification of conductors Metals and alloys with high conductivity Metals and alloys with high resistivity Metals and alloys for special purposes...56 Test questions to the Part 2: Conductive Materials...59 Part 3. Semiconductors...60 Chapter 6. Electrical conduction of semiconductors...60 Chapter 7. Thermoelectric phenomena and Hall effect Thermoelectric effects Chapter 8. Optical and photoelectric phenomena in semiconductors 80 Chapter 9. Electron-hole or n-p junction Chapter 10. Semiconductor materials and processes for their preparation 87 Test questions to the Part 3: semiconductor materials 98 Part 4. Dielectric materials Chapter 11. Electrical processes in dielectrics.101 3

4 11.1. Electrical conduction of dielectrics Polarization of dielectrics The dielectric losses Dielectric breakdown Chapter 12. Mechanical, thermal and chemical properties of dielectrics.168 Chapter 13. Classification of dielectrics Test questions to the Part 4: Dielectric materials..188 References to chapter References to chapter References to General provisions.190 References to chapter References to chapter

5 PREFACE New materials as well as new combination or modification of traditional materials are used today in all electrical, electromechanical and power devices to meet the increasing requirements of electrical and power engineering in quality of the materials used. The study of physical, mechanical and chemical properties of substances that could be used as electrical materials is needed to assure a long life and efficient work of that plant and equipment. The understanding of material properties only allows professionals in the fields of electrical, electromechanical and power to create new equipment and competently apply high diversity of nowadays electrical materials for manufacturing and operation of various electrical and power devices. The course of Electrical Materials Science has the following problems: 1. Study of the theoretical background needed for studying the properties of electrical materials. 2. Classification of electrical materials for their application, composition and properties. 3. Definition of the main characteristics required to evaluate the applicability of electrical materials in electrical engineering and power generation devices. 4. Discussion of the most technically and economically substantiated applications of the electrical engineering materials. 5

6 GENERAL PROVISIONS The materials used in various fields of electrical engineering can be divided into engineering structural materials and electrical engineering materials. Engineering structural materials used to manufacture auxiliary parts and components of electrical engineering devices. In other words the engineering structural materials have to stand the mechanical stress, generally. Electrical materials are characterized by certain properties with respect to electromagnetic fields and they are applied in technique with respect of these properties. Electric and magnetic fields (separately or together) can affect the electrical materials during the operation of the electrical engineering equipment. The electrical materials may have extremely different behavior under the influence of magnetic field. All electrical materials are divided into strongly magnetic materials and low-magnetic (nonmagnetic). All materials are divided into conductors, semiconductors and dielectrics according to the electrical materials behavior in the electric field. One of the most important properties of a substance with respect to the electric field is its electrical conductance. Conductance is the ability of a substance to conduct the electric current under the influence of electric field. Electrical current is an ordered movement of electrical charges that is any electrically charged particles (i.e., charge carriers: protons, electrons, ions or others). So ability of a substance to carry electricity depends on presence, type and amount of free charge carriers in it. When the sample of substance is placed in an electric field of intensity E, then the force F = qe acts on every free charge carrier q in the direction of electric field E. As a result a positively charged particle +q moves in the direction of the vector E, and a negatively charged particle q moves in the opposite direction. So the action of electric field on the substance results in the ordered motion (as opposed to random thermal motion of particles) of free charge carriers that is the electric current. The current density j is the total electric charge which is carried by the field per unit time through the unit area perpendicular to E. If the substance has free charge carriers of one type only, the current density j can be written as j = qnv d, (0.1) where n is the concentration of free charge carriers being in a volume of the substance [m 3 ]; v d is the average speed of ordered motion of free charge carriers (charge-drift velocity) because of acting the electric field E: 6

7 v d =Е, (0.2) where is a charge carrier mobility [m 2 /V s]. As it can be seen from (0.2), the mobility is the drift velocity of free charge carriers in the electric field of E=1 V/m. In view of (0.2), expression (0.1) takes the following form: j = qnе = Е. (0.3) The proportionality coefficient = qn in (0.3) is called the electrical conductivity [S/m, 1 S (Siemens) = 1 Ohm 1 ]. If there are N types of free charge carriers q i in the substance with concentrations of n i and mobilities of i then the specific electric conductivity of that substance is γ N qi ni μi. (0.4) i1 Equation (0.3) is a differential form of Ohm's law. Normally it is convenient to use the reciprocal of conductivity, that is 1/ = rather than conductivity itself. The reciprocal of conductivity is called the electrical resistivity of the substance [Ohm m]. Well-known formulation of Ohm's law is the current I through a sample is in direct proportion to the applied voltage U: I=G U, where the factor G is the conductance of sample. The conductance (the measuring unit in SI is Siemens, symbol S) shows the ability of the sample to conduct electricity. In practice, it is more convenient to use the reciprocal value 1/G=R which is known as the resistance (the measuring unit of R is Ohm, symbol Ω=1/S). The resistance R (or conductance G) depends not only on properties of the substance from which it is made but on the sample shape also. That is why, the resistance R (or conductance G) cannot be characteristic of the substance in opposite to the resistivity or conductivity γ. If a homogeneous isotropic sample of arbitrary shape is in the electric field E which does not change over time, then G = 1/R = = /, (0.5) where G and R are the conductivity and resistivity, respectively, of the substance; is a parameter which depends on the sample geometry. This parameter is = A/l for the sample with a constant cross section A along the length l (for example, wire or a cable core, the dielectric of parallel-plate capacitor). It is = 2πl/ln(D/d) for a cylinder with outer diameter D, inner diameter d, and the axial length l (for example, dielectric of cylindrical capacitor, insulation of coaxial cable). The resistivity (or conductivity ) of various materials substantially differ from each other. The resistivity of materials in the superconducting state is practically zero, while the resistivity of very low-density gas tends to 7

8 infinity. Even for solids only that are in normal conditions, the values cover 25 orders of magnitude: from ~10 8 Ohmm for the best metallic conductors up to Ohmm for the best insulators. Usually the conductors are substances with an electrical resistivity less than 10 5 Ohmm whereas the dielectrics are substances with more than 10 7 Ohmm. The electrical resistivity of semiconductors may vary in very wide range: Ohmm. However, the classification of substances on the electrical properties is not based on values of only but it also must take into account the nature of a substance electrical conductivity as well as dependences of on temperature and other factors. The basic electrical property of conductors is the high electrical conductivity in comparison with other materials. So, conductive materials offer very little resistance to the electrical current. The application of conductors is due to the property to conduct the electrical current very well mainly. There are two types of conductive materials: solutions of salts or acids (electrolytic conductors), and metals (pure metals and alloys or their melts). Electrolytic conductors have high conductivity (low resistance) because of presence of ions in the solution. Ions are formed in a liquid when a salt or an acid is dissolved in it; additionally, ionization occurs when a metal is immersed in an alkaline or an acid solution. The resulting liquid is a fairly good conductor of electricity and is known as an electrolyte. Ions (free charge carriers) carry out electrical current easily but the amount of ions become smaller and smaller with time until electrolytic conductor would exhaust. In other words, the electrical properties of electrolytes are unstable. Metals are good conductors because the outer orbits of metal atoms (the valence shells) in adjacent atoms overlap one another, allowing electrons to move freely between the atoms. This type of chemical bonding is called the metal bond. Thanks to the metal bond, the concentration of free charge carriers (electrons) in metals remains constant and that s why the metals keep their electrical properties changeless long time. A semiconductor is a material whose resistance is midway between typical resistances of a good conductor and a good insulator. The main feature of a semiconductor is an extremely strong dependence of the conductivity on the concentration and type of impurities, as well as on external actions such as the heat, electrical field, radiation, and others. Commonly used semiconductor materials include silicon and germanium (in diodes, transistors and integrated circuits), cadmium sulphide (in photoconductive cells), gallium arsenide (in lasers, and light-emitting diodes), etc. Silicon is the most widely 8

9 used material, and it is found in many rocks and stones (e.g., sand is the silicon dioxide) The substance with very low conductivity and the ability to be polarized (i.e., an electric field can exist in the substance bulk) is called the dielectric material. Description of property variety of dielectrics is one of the main purposes of this book. A dielectric material (such as glass or plastic) has a very high resistance to current flow (good insulators) because of the covalent bond. This type of chemical bonding means that the outer orbits of the atoms do not overlap one another and that material almost has no free charge carriers, making it very difficult for electrical current to flow through the material. Almost all substances including dielectrics physically can be solid, liquid or gaseous. It is known that there is the fourth state of matter plasma. Plasma is a highly ionized gas that occurs when a gas is exposed to strong electric field and plasma is good conductor. Solid dielectrics may have crystalline, amorphous or mixed structure. The crystalline structure can be realized as a single crystal (e.g., diamond or salt) or a polycrystal (e.g., almost all metals and alloys). A single crystal consists of repetitive identical crystalline cells. Every crystalline cell has the same volume and contains the same number of atoms in the same positions. The atomic arrangement of a typical balk centered crystal is shown in Fig. 1. A single crystal has the perfect order in the arrangement of atoms in all directions over its volume. Because of the crystalline structure all single crystals are anisotropic. A polycrystal consists of a large number of fine single crystals (crystal grains or crystallites). Crystallites may have the same or different crystal structure bat these always randomly oriented relative to one another. That is why a polycrystal is isotropic substance normally as distinguished from a single crystal. However, if the crystallite orientation is made more ordered (e.g., by means of machining), the polycrystalline becomes an isotropic material. Such polycrystalline substance with artificial anisotropy is called the textured material. Figure 1. NaCl crystal lattice structure. 9

10 An amorphous material (e.g., glass, ceramics) is characterized by the absence of strictly ordered arrangement of atoms. The atom arrangement in an amorphous substance is unordered like in a liquid. That atom arrangement can be created during a solidification of liquid at too fast lowering of temperature when there is the rapid increase in viscosity. In this case the atomic movement necessary to form the crystal structure is hampered. The absence of ordered atomic structure makes an amorphous substance typically isotropic. An amorphous-crystalline (some metals and all polymers) substance is a partially crystallized amorphous substance. For example, crystallites are growing in most of polymers and some glass compositions. The higher is the temperature of the crystallization process the higher is the rate of crystallization. Due to the formation of small crystals in the glass it loses its transparency and it becomes an amorphous-crystalline substance known as a glass ceramics or pyroceramics. Figure 2. Molecules of cross-linked polyethylene. Atomic structure. All substances gases, liquids and solids consist of atoms (chemical elements). Each atom comprises several much smaller particles, the principle ones being electrons, protons and neutrons. The difference between the smaller particles lies not only in their difference in mass (a proton is 1840 times more 'massive' than an electron), but also in the electrical charge associated with them. For example, a proton has a positive electrical charge whilst an electron has a negative electrical charge. The charge on the proton is equal to but of opposite polarity to that on the electron. The mass of the neutron is equal to that of the proton, but it has no electrical charge. The more is a chemical element number in Mendeleev's Periodic Table the more protons and neutrons its nucleus contains. Therefore it becomes heavier and 10

11 has bigger positive electrical charge. Electrons fill the shells of an atom, compensating the positive charge of the nucleus. Outer electrons are the weakest bonded electrons in the atom and called valence electrons. The other electrons together with nucleus form the atomic core and called core electrons. The simple model of atom is shown in Fig. 3. Figure 3. A crude model of atom. The nature of matter ensures that each atom is electrically balanced, that it has as many electrons as it has protons. Under certain circumstances an atom, or a molecule, or a group of atoms can acquire an electrical charge; the atom or group of atoms is then known as an ion. A negative ion (an anion) contains more electrons than are necessary for electrical neutrality and a positive ion (a cation) contains fewer electrons than necessary for neutrality. The destruction of molecules results in appearing of anion and cation at the same time. The ionization of atoms or molecules leads to the appearance of only negative or positive ions. Negative ions can appear under the influence of external factors (e.g., radiation) when the atom (or molecule) loss the valence electron. Some atoms are able to catch additional electron under certain conditions and it leads to appearing of the positive ions. In any case both molecule distraction and ionization results in appearing of free charge carriers in a substance. Types of chemical bonds. Structure and properties of all substances depend on chemical elements from which it consists of and on a type of chemical bonds between atoms. Usually it makes distinction between a chemical bond and an intermolecular interaction according to the energy of an interaction. The chemical bond is much stronger then the intermolecular interaction because it has much 11

12 higher energy. The energy of a chemical bond is of 100 kj/mole but energy of an intermolecular interaction is of 530 kj/mole for the hydrogen bond and 0,11 kj/mole for the van der Waals interactions. It is the chemical bonds that guarantee a stability of substance properties under external influence. Metallic bond is the bond between atoms with similar low electronegativity. Metallic atoms share weakly bonded valence electrons, which form an electron gas. Electron gas has a cementing effect on metal crystalline structure and causes high electrical conductivity, thermal conductivity and plasticity of a metal. Figure 4. A crude model of a metallic bond. Covalent bond. Covalent bonding is characterized by the sharing of one or more pairs of electrons between atoms, in order to produce a mutual attraction, which holds the resultant molecule together. Atoms tend to share electrons in such a way that their outer electron shells are filled. The pair of electrons is stable because of the exchange interaction between opposite oriented spins and orbital magnetic moments of electrons. If bonded atoms have the same electronegativity then covalent bond is non-polar and the molecule has electric (dipole) moment of zero. The more electronegativity of the atoms differ the more polar molecule is created. Usually a pair of nonmetallic atoms creates the covalent polar bond (HCl, H 2 O, etc.). The simple model of non-polar and polar covalent bonds is shown in Fig

13 Figure 5. A crude model of a polar and non-polar covalent bond Ionic bond. An ionic bond is a type of covalent bonding between atoms which strongly differ in their electronegativity. It can be formed after two or more atoms give up (or gain) electrons, so as to become ions. Oppositely charged ions are bound by the Coulomb attraction. This type of bonding is most typical for inorganic dielectrics, and occurs between metal and non-metal atoms (Na + -Cl, Li + -F, Ca + -Cl, etc.). Intermolecular interactions. The van der Waals forces appear due to the electrostatic dipole-dipole interaction between molecules with nonuniform charge distribution of electrons. The hydrogen bond (H bond) is the attractive interaction of a hydrogen atom with an electronegative atom, such as nitrogen, oxygen, fluorine, or chlorine (N, О, F, Cl ), that comes from another molecule or chemical group. The hydrogen must be covalently bonded to another electronegative atom to create the bond. These bonds can occur between molecules (intermolecularly), or within different parts of a single molecule (intramolecularly). The hydrogen bond is stronger than a van der Waals interaction, but weaker than covalent or ionic bonds. This type of bond occurs in both inorganic molecules such as water (H 2 O) or crystal potassium dihydrogen phosphate (КН 2 РО 4 ) and organic molecules such as DNA (deoxyribonucleic acid). Classification of substances from the band theory of solids. It is known that the emission spectrum of gas is a linear spectrum. Atoms in a gas separated from each other at very large distances. The linear spectrum means that each atom has a set of well-defined energy states (energy levels). These energy levels are energies of the electron shells. The farther the electron shell is away from the nucleus, the higher is its energy level. Most energy levels of the atom are filled with electrons in the ground state. If the atom receives energy from any outer source (the light, heat, or radiation) then one or more valence electrons could pass to the higher energy 13

14 levels (the excitation energy level). The excitation levels are not sustainable. The electron loses energy W returning to the ground state. This energy W is equal to difference between excitation energy level W 2 and ground level W 1 and the electron gives it as a photon: W = W 2 W 1 = h, (0.6) where h is Planck's constant 6, J c; is the photon frequency. 3 5 W 4 Figure 6. Energy levels of a single atom (left) and a non-metallic solid (right): 1 the ground energy levels; 2 valence band (filled electron band); 3 levels of the atomic excited state; 4 band gap; 5 conduction band (zone of free energy levels) 1 atom solids 2 If the substance is in a condensed (i.e., liquid or solid) state then the interaction among atoms is strong. The interaction of atoms leads to the degeneracy of the each linear atomic energy level. As a result, the valence energy levels form a continuous range of energies which is called the valence band, and the excitation energy levels form the conduction band. In nonmetallic materials, where atoms are connected each other by covalent or ionic bonds, the valence and conduction bands are separated by the forbidden energy gap (band gap). The band gap is the interval of energies that are prohibited for electrons. In metals there is no band gap because of metallic type of bonding (Figure 6). Valence electrons of a non-metallic substance fill the valence band energy levels (2 at Fig. 6). The valence band is filled with valence electrons in the ground state. It means that the electrons form chemical bonds and there are not free charge carriers in the substance. The substance cannot carry the electrical current. Only if the electron gets the energy enough to overcome the band gap W (4) it may escape from the local covalent bond. In this case it goes from the valence band (2) to the conduction band (5). The electron becomes the free charge carrier and it is able to take part in the electrical conductivity. The energy W (the band-gap energy, the energygap width) is required to appear the free charge carriers in the substance and 14

15 it is called the activation energy of conductivity. Substances with a high activation energy (W > 3 ev) have weak conductivity in normal conditions. Those substances are called dielectrics (insulators). The semiconductors have activation energy W < 3 ev. That is why even a small external power influences (heat, irradiation, electric field, etc.) is enough to transfer electrons from the valence band to the conduction band and the semiconductor is capable to conduct current. The valence band overlaps the conduction band in metals and alloys according to the metallic bond. It means that the valence electrons of a metal are not localized near the lattice ions and can move freely throughout the volume. Therefore, the activation energy of metal W = 0, and the conductivity of metals and alloys is very high. The band structure of the material depends on the structure of its constituent atoms and the type of bonding between them. The crystal structure also has an impact. For example, the carbon in the diamond structure is a very good insulator (W = 5,2 ev) but in the graphite structure it has metallic properties. Thus, there is a qualitative difference between substances with metallic and covalent bonding (metals and non-metallic materials). This difference is expressed in the presence or absence of the gap in the band diagrams. In practice, this leads to the fact that metals always have the conductivity but the activation of conductivity in non-metallic materials needed to make energy from the outside. The difference between dielectrics and semiconductors is conventional and it is rather conditional. This difference is expressed in the value of activation energy W only. 15

16 Part 1. Magnetic Materials Chapter 1. Classification of substances on the magnetic properties 1.1. General Concepts A substance using in electrical engineering devices with regard to their magnetic properties and characterized by the ability to collect, store and convert magnetic energy is called a magnetic material. Discovery of the magnetic field around a wire with electric current at the beginning of last century allowed to Ampere to put forward the assumption that the cause of the magnetism is micro electric currents in molecules of a substance. This idea has been developed theoretically and experimentally in the second quarter of the XX century with creating of the quantum mechanics and the modern theory of atom. Magnetic properties of a material are determined by its atomic structure and depend primarily on the magnitude of atomic magnetic moments. The magnet poles do not exist separately in contrast to positive and negative electric charges. The elementary source of magnetism is an elementary magnet which is characterized by a certain magnetic moment. Experimental and theoretical studies have shown that the magnetism of an atom (elementary magnetic dipole) is generated by three micro electric currents : 1) the presence of the electron spin magnetic moment, which is associated with a corresponding mechanical moment of the electron; 2) the orbital motion of electrons in the atom, creating the orbital magnetic moment; 3) the magnetic moment of the atomic nucleus, which is created by the spin moments of protons and neutrons. The spin magnetic moment of the nucleus is at least 10 3 times less than the one of electrons. Therefore, it can be assumed that it is the electrons which are the origins of the elementary magnetic moments in a substance. The resulting spin magnetic moment of the atom is determined as the sum of the spin magnetic moments of individual electrons. According to the Pauli principle two electrons with opposite spins only may stay in the same quantum state. The resultant spin momentum of these two electrons is zero. Only if the atom has odd number of electrons, then there are electron shells with uncompensated electrons and the atom has a permanent magnetic moment. Completely filled shells do not give a resultant spin angular momentum because in this case each spin in one direction of the atomic shell corresponds 16

17 to the spin directed antiparallel and the total magnetic moment produced by a the pair of electrons is zero. For example in metals, the magnetic moment of filled shells is zero and the outer valence electrons are shared all over the metal. Therefore, the spin magnetic moments of electrons are not zero in atoms having not completely filled (unfinished) core (inner) electron shells only. That is the atoms of transition metals. According to the classical physics, the orbital magnetic moment is due to an elementary ring current caused by the orbital motion of electron in the atomic shell. From quantum theory it is known that the non-zero resulting orbital magnetic moment can be observed only in the non-circular orbits. The orbital and spin magnetic moments of individual electrons are added to the resulting orbital spin moment of the atom. The measure unit of atomic magnetic moments is the Bohr magneton B : e 24 â 9, 2710 J/Ò, h/2π, (1.1) 2m where h = 6, [J s] is the Planck's constant; е = 1, [C] (Coulomb) is the electron charge; m = 9, [kg] is the mass of the electron. Determination of the total magnetic moment of an atom is greatly facilitated by the fact that the filled shells as the orbital and spin magnetic moments are compensated, so only incompletely filled electron shells are taken into account Ferromagnetism and ferrimagnetism The spin magnetic moment is the main point in the creation of the atomic magnetic moment for ferromagnets of the group of iron. In the first approximation, the magnetic moment of atom is determined by the algebraic sum of spin magnetic moments of unfilled electron shells. For example, in the unfilled 3d-shell of the iron atom there are only six electrons instead of ten. The spins of the five of them are parallel to each other and one is antiparallel. Consequently, the total magnetic moment of the iron atom should be equal to 4 B. In fact, the magnetic moment of the iron atom is equal to 2,218 B. The discrepancy in the values of the magnetic moment of the iron atom is due to a violation of spin orientation in the solid state which is associated with the interatomic interactions and the overlap of the electronic levels of atoms. If a piece of iron is put in an external magnetic field, then it will be magnetized because of orientation of atomic magnetic moment interacting with the field. 17

18 The main magnetic characteristics If there is a solenoid of length l [m] with the number of turns N and a current I [A] flows through it, then it creates the magnetic field with the intensity of H=N I/l [A/m]. A substance in the magnetic field is magnetized and the magnetization of the substance М [A/m] is in direct proportion to the magnetic field intensity H [A/m]: M H, where dimensionless coefficient is known as the magnetic susceptibility. The magnetic susceptibility shows how strong orientation of atomic magnetic moments (i.e., magnetization) of the substance can be achieved in the magnetic field. Magnetic induction B [T] (Tesla) characterizes the total magnetic field inside the material. This total field is the sum of the external H and inner (induced) M fields: B 0 H M 0 1 H 0 H 0 H, (1.2) where 0 = H/m is the magnetic constant, which characterizes the magnetic permeability of vacuum (the standard magnetic substance); is the relative magnetic permeability of the substance, which shows how many times the permeability of the substance is more than the permeability of vacuum (dimensionless); а = 0 is the absolute magnetic permeability of the material [H/m]. Thus the magnetic constant is the absolute magnetic permeability of the vacuum. According to the magnetic properties all substances can be divided into five gropes: diamagnetic, paramagnetic, ferromagnetic, ferrimagnetic, and antiferromagnetic materials. The susceptibility of different magnetics varies in magnitude and sign as well as has different dependences on temperature, magnetic field intensity and so on. Diamagnet is a substance with the atomic magnetic moment close to zero because of compensation of both the orbital and spin magnetic moments. The magnetic susceptibility of a diamagnetic is negative and very small 10 5 and the magnetic permeability is independent on the external magnetic field. It has the value a little less than 1: 0, The diamagnetism is originated by the persistent microscopic eddy currents in the entire volume of the diamagnetic material which are induced by the external magnetic field. According to the classical physics, the external magnetic field H seeks to turn the electron orbit perpendicular to H direction. This action of the magnetic field causes the precession of the electron orbit around the H direction. It looks like the precession of a gyroscope when the axis of the rapidly rotating gyroscope does not coincide with the vertical and the 18

19 gyroscope axis rotates around the vertical. The precession of the electron orbit is equivalent to an additional motion of an electron and gives rise to a current that induces a magnetic moment directed opposite to the direction of the external field. This induced moment causes the diamagnetism. Since the electron orbit of any atom has the precession in magnetic field the diamagnetism is inherent in all substances, but the diamagnetism is noticeable only if it not overridden by stronger paramagnetism and ferromagnetism. It is typical for a diamagnet that it is pushed out by the inhomogeneous magnetic field. Diamagnets are such substances as hydrogen, rare gases, nitrogen, chlorine, water and most organic compounds, a number of noble metals as Cu, Ag, Au, Be, Zn, Cd, Hg, Pb, In, Ga, Sb, as well as graphite, glass and etc. Other substances whose atoms have permanent magnetic moments can be paramagnetic, ferromagnetic, antiferromagnetic, or ferrimagnetic materials depending on the interaction force between the magnetic moments of atoms. Paramagnet is a substance which atomic magnetic moments are too weak and almost do not react upon each other. The thermal motion of molecules destroys the order in atomic magnetic moments direction even at room temperature. Hence, all directions of the atomic magnetic moments have statistically equal probability (Fig. 1.1 a). As a result the total magnetic moment of a paramagnet is zero. An external magnetic field can create the preferential direction of elementary magnetic moments arrangement and the paramagnetic can be magnetized. However, the magnetization of the paramagnetic is weak at normal temperature and field H. Magnetic susceptibility is positive and has a value of ~ The permeability of paramagnets is 1,001 and it is almost independent on the external magnetic field. If the temperature increases at the constant field intensity, the thermal motion disrupts ordering process and therefore the magnetization decreases. The Curie's law describes the temperature dependence of paramagnetic susceptibility: C/ T (1.3) where C is the Curie constant and T is the absolute temperature, K. It was established experimentally in 1895 for paramagnetic gases and rare-earth elements If the interaction between the elementary magnetic moments cannot be ignored (as for transition metals) the more general Curie-Weiss law is valid: C/ T, (1.4) 19

20 where is the Weiss constant which is may be more or less than zero for different substances. a b c d Figure 1.1. Schematic representation of the arrangement of the magnetic moments of atoms in the paramagnetic (a), ferromagnetic (b), antiferromagnetic (c), and ferrimagnetic (d) substances In Fig. 1.2,a the dependence of the magnetization M (H) is shown for the diamagnetic (1) and paramagnetic (2) materials in weak field H. In both cases, the dependence of M on H is linear and therefore is independent on H. The external field of intensity about A/m at room temperature, and at T = 1 K it has to be of A/m is required to magnetize paramagnetic up to the saturation (Fig. 1.2b). The saturation is the point where all the elementary magnetic moments are parallel. It is typical for a paramagnet that it is drawn into by the inhomogeneous magnetic field. Paramagnetic substances include the oxygen, nitrous oxide, salt, iron, cobalt and nickel, alkali metals, as well as Mg, Ca, A1, Cr, Mo, Mn, Pt, Pd, etc. Ferromagnet is a substance the interaction between atomic magnetic moments of which is in such a way that they are arranged parallel to each other (Fig. 1.1, b). Most ferromagnets have a crystalline structure and they are characterized by large positive values of (up to hundreds of thousands or millions) as well as by complex non-linear dependence of on temperature and external magnetic field. A characteristic feature of ferromagnets is a capacity at normal temperatures to be strongly magnetized even in weak fields. Ferromagnets have very important characteristic called the Curie point T C. At temperatures above T C thermal disordering causes a sharp drop in permeability and degeneration of the ferromagnetic into 20

21 paramagnetism (see Fig. 1.3). For example the Curie point of pure iron is 1043 K, for nickel T C = 631 K, and for cobalt T C = 1404 K. M М 2 M m 0 Н 1 Н m Н a b Fig The dependence of the magnetization M on magnetic field intensity H: (a)- for the diamagnetic (1) and paramagnetic (2); (b) - for the paramagnet at low temperatures or in the very strong field Ferromagnet is a substance the interaction between atomic magnetic moments of which is in such a way that they are arranged parallel to each other (Fig. 1.1, b). Most ferromagnets have a crystalline structure and they are characterized by large positive values of (up to hundreds of thousands or millions) as well as by complex non-linear dependence of on temperature and external magnetic field. A characteristic feature of ferromagnets is a capacity at normal temperatures to be strongly magnetized even in weak fields. Ferromagnets have very important characteristic called the Curie point T C. At temperatures above T C thermal disordering causes a sharp drop in permeability and degeneration of the ferromagnetic into paramagnetism (see Fig. 1.3). For example the Curie point of pure iron is 1043 K, for nickel T C = 631 K, and for cobalt T C = 1404 K. Ferromagnetics are also characterized by the presence of hysteresis loop. The induction B is not determined only by the magnetic field H but it also depends on the previous history of the sample. It means that the induction of a ferromagnetic sample B depends on the magnitude and direction of the magnetic field H acting on it before. For example, the induction B in field H is more if the material before has been magnetized up to saturation by the field of the same direction, otherwise the induction B is less if the saturation field had before the opposite direction. Hysteresis is due to the irreversibility of the magnetization process. It leads to energy dissipation and reduce the quality of the magnetizing devices in which the ferromagnetic material is used as a core (magnetic core). On the other hand a 21

22 ferromagnetic material becomes a permanent magnet (that is material remains magnetized after the removal of the magnetizing field) because of the irreversibility of the magnetization process only. The parallel arrangement of the atomic magnetic moments of neighboring atoms is called atomic ferromagnetic order. It is characterized by the fact that the ferromagnet is in a state of spontaneous magnetization up to the saturation even in the absence of an external field. This magnetization depends on temperature. It has the maximum (true magnetization) at T = 0 K and it becomes lower with the temperature increasing. T C Figure 1.3. A typical dependence of the magnetic permeability μ of a ferromagnetic on the temperature T. T C is the Curie point. The explanation of the ferromagnetic properties of solids was given by Russian physicist Rosing and French physicist Weiss. They suggested that the spontaneous magnetization in macroscopic areas called domains is due to the internal molecular field (atomic ferromagnetic order). The sample of a ferromagnetic substance is not a permanent magnet without a magnetic field because the spontaneous magnetization of different domains has different direction. The magnetic moments of the atoms are arranged parallel to each other in each domain, that is the domain is magnetized up to saturation. However, the sum of all magnetic moments of domains is equal to zero without a magnetic field because they have different directions. The method to observe the domain boundaries and the existence of the ferromagnetic domain structure was developed by Soviet scientist Akulov and Bitter in It was shown experimentally that the domain has dimension up to tens of micrometers and the magnetic moment of the domain is about of the atomic magnetic moment. 22 T

23 The presence of unfilled atomic core shells is the necessary condition for the ferromagnetism. But this condition is not enough. For example, atoms of all transition elements have the unfilled core shells but only three from the eight iron-group elements (Fe, Co, Ni) and six from the 14 lanthanides (Gd, Tb, Dy, Ho, Er and Tm) have the ferromagnetic properties. To explain this fact the condition under which the parallel arrangement of atomic magnetic moments within the domain has to be known. It is the condition under which the ordering energy is necessary to magnetize a domain up to saturation enough to prevent heat disorder. It was found by Soviet physicist Ya. Frenkel and independently by German theorist W. Heisenberg on the basis of quantum theory that the atomic ferromagnetic order is due to the electrostatic interaction among protons and electrons which has no analogue in classical physics. The energy of electrostatic interaction between the electrons depends on the spin orientations. The difference of energies of two electrons in systems with parallel and antiparallel spins is called the exchange energy. This energy is the quantum correction to the Coulomb interaction and it is a consequence of the identity principle of the quantum particles. The exchange energy is not zero if the electrons orbits of neighboring atoms are overlap each other. Therefore, the exchange energy is important only for electrons belonging to the same atom or to adjacent atoms at certain distances between them. The energy of the exchange interaction is proportional to the integral of the exchange energy A. The sign and magnitude of A depends on the ratio of the distance a between the atoms (lattice parameter) to the diameter d of the unfilled electron shells involved in the exchange interaction. A positive sign of exchange energy integral A is required to occur the spontaneous magnetization as for Fe, Co, Ni in Fig It is possible if а/d > 1,5. If A <0 the antiparallel arrangement of magnetic moments of neighboring lattice atoms is energetically preferable. That is why Mn and Cr having A <0 does not possess ferromagnetic properties. The dependence A = f(a/d) explains the ferromagnetic properties of some alloys of non-ferromagnetic components such as alloys of the manganese (Mn) with bismuth (Bi) or antimony (Sb), etc. The introduction atoms of certain elements into the Mn lattice cause the increase in interatomic distances in the alloy up to the value necessary to make possible the spontaneous magnetization. Thus, the conditions for ferromagnetism are: 1. the presence of internal unfilled shells (d or f) in atoms; 2. the positive value of the exchange energy integral that takes place when the size of the unfilled shell is small in comparison with the distance between atoms in the lattice (a/d > 1,5). 23

24 A Fe Co Ni Gd 1.5 a/d Mn Cr Figure 1.4. The dependence of the exchange energy integral A on the ratio of lattice parameter a to the diameter d of the inner unfilled shell Antiferromagnet is a substance the interaction among atomic magnetic moments of which leads to the antiparallel orientation (see Fig. 1.1, c). The exchange energy integral of the antiferromagnetic is A <0. It means that the antiparallel arrangement of magnetic moments is energetically profitable in the absence of an external magnetic field. There is a mutual compensation of the atomic magnetic moments because magnetic moments of neighboring atoms are equal in magnitude and antiparallel in direction. French physicist L. Neel was a significant researcher of antiferromagnets, so the theory of antiferromagnetism and ferrimagnetism is known as the theory of Neel. But the theoretical description of the antiferromagnetism was created by the Soviet physicist L. Landau in the One of the main provisions of the Neel is a following. A crystal lattice of an alloy is the structure consisting of two (or more) sublattices and the sublattices are magnetized in opposite directions. Theoretical description of the antiferromagnetic and ferromagnetic structures was proved by experiments using the methods of neutron diffraction. The intensity of scattering depends on the direction of the magnetic moments of atoms or ions. It allows determining the orientation of the magnetic moments by means of a neutron ray diffraction analysis. Antiferromagnets have the distinctive temperature dependence of magnetic susceptibility. The magnetic moments of the sublattices compensate each other and the magnetic moment of anantiferromagnet is zero at T = 0 K. The antiparallel arrangement of magnetic moments is gradually broken as the temperature increases and the magnetic susceptibility increases reaching a maximum at the Neel point T N (Fig. 1.5). At the Neel point the ordered arrangement of magnetic moments is completely broken and an 24

25 antiferromagnet becomes a paramagnetic. After the Neel point the dependence (Т) obeys the Curie-Weiss law (1.4). The magnetic susceptibility of the antiferromagnet is small ( ). In the weak magnetic field, the susceptibility of antiferromagnet is practically independent on the external field intensity. In the strong field, is usually a complicated function of the field. T N T Rare-earth metals such as Ce Pr, Nd, Figure 1.5. The dependence of the Sm, and Eu, and Cr, Mn, many oxides, magnetic susceptibility of an chlorides, fluorides, sulfides, carbonates of antiferromagnet on the temperature T transition metals, for example, based on manganese (MnO, MnCl 2, MnF 2, MnS 2 etc.) or based on Fe, Co, Ni, Cr, etc. are antiferromagnets Ferrimagnet (or uncompensated antiferromagnet) is a substance (chemical compound) atomic magnetic moments of which are antiparallel oriented like in case of the antiferromagnet (Fig. 1.1, d). However, the magnetic moment magnitudes have different values because the crystal lattice consists of different elements so that the resultant magnetization may be large enough. Compounds that have the crystal structure of the spinel type, garnet, rock salt, or hexagonal are ferrites. All ferrites are ferrimagnets. Ferrites with spinel-type structure are compounds of iron oxide Fe 2 O 3 with oxides of other metals. The structural formula of the ferrite is MeO Fe 2 O 3, where Me is a divalent metal (Fe, Ni, Mn, Zn, Co, Cu, Cd, Mg, etc.). There are also singlecomponent and multicomponent ferrites. In a constant (or low frequency) magnetic field the magnetic characteristics of ferrites (, B S ) are less than the characteristics of ferromagnets, so the practical application of ferrites were much later in connection with the development of high-frequency art. Ferrites are oxide compounds and the resistivity of a ferrite is times bigger than one of a metallic ferromagnet. It is a serious advantage for the high-frequency art. Ferrites belong to a class of semiconductors or insulators according to the resistivity. Thanks to the high resistivity of a ferrite it has the very small eddy currents under the influence of the variable magnetic field. Ferrites are used at frequencies up to hundreds of megahertz while the metallic magnetic materials are used at frequencies up to several tens of kilohertz only. 25

26 Ferrimagnets have the domain structure like ferromagnets. Magnetic properties of the ferrimagnet are closely related to the crystal structure just as ferromagnets. The resulting magnetic moment of the ferrimagnet is determined by the magnetic moments of the sublattices. Many properties of ferromagnets are qualitatively similar to the properties of ferromagnets but there are significant differences between these substances. Ferrimagnets differ from ferromagnets by the smaller value of saturation induction. Additionally ferrimagnets have a more complex temperature dependence of the saturation induction. It can be explained by the presence of two or more sublattices in crystal structure of the ferromagnet with uncompensated magnetic moments ordered in the opposite direction. If there are two sublattices only the ferrimagnet has the resulting saturation magnetization М S as the vector sum of saturation magnetizations of both sublattices М S1 and М S2. Chapter 2. The basic properties and characteristics of magnetic materials 2.1. The process of magnetization and the magnetic permeability The magnetization of ferro- and ferrimagnetics is described by the dependence of the flax density (magnetic induction) B on the magnetic field intensity H (Fig. 2.1 a). This dependence is known as the initial magnetization curve or B-H curve. The initial magnetization curve starts from B = 0 and shows how the flux density increases with increasing field strength. It's important to know that the magnetization curves for ferromagnetic materials are all strongly dependent upon purity, heat treatment and other factors. There are two types of B-H curves: 1) the initial magnetization curve which is obtained by a monotonic increase of H or very similar the normal magnetization curve which is a locus of the vertices of the hysteresis loops obtained under reversal magnetization (Fig. 2.1 a); 2) the unhysteretic magnetization curve obtained under simultaneous action of direct magnetic field and alternating magnetic field with amplitude decreasing to zero (Fig. 2.1 b). The initial curve is not very different from the normal curve but it is hardly reproduced and cannot be used for a comparison of magnetic properties of different magnetics. Unhysteretic magnetization curve is characterized by a rapid increase in the flax density up to saturation in a weak constant magnetic field. The normal magnetization curve is an essential 26

27 characteristic of magnetic materials and it is widely used to characterize the material magnetization. The absolute permeability of substance is the most important characteristic and it can be determined by using the normal magnetization curve: а = В/Н., В max В s in Four distinct regions in both dependences of B-H curve and of on Н can be identified according to the different processes of magnetization (Fig. 2.1). These regions correspond to the change in the magnetic domains structure. Region 1 is the region of the weakest magnetic field characterized by a reversible displacement of domain boundaries (reversible growth of domains) and the slow linear rise of В(Н). The atomic magnetic moments have the transition orientation М b М domain boundary domain 1 layer 2 Figure 2.2. Magnetic moment directions in a domain boundary c H a 27 Figure 2.1. (a) The normal magnetization curve B (H), (b) an anhysteretic magnetization curve, and (c) the dependence of the magnetic permeability μ (H) of a ferromagnetic material. 1 very weak field; 2 weak field; 3 intermediate field; 4 strong field. between the directions of magnetization of adjacent domains in the boundary layer (Fig. 2.2). All domains have approximately the same volume without the external magnetic field (Fig. 2.3, a). If the external field is weak the magnetic moments in a domain boundary tend to turn in the direction of the external field. It leads to the displacement of domain boundaries so that the energetically favorable domains (magnetized close to the direction of the field) rise in volume at the expense of other domains (Fig. 2.3, b). The boundary displacement is a reversible process in the weak magnetic field. If the external field decreases the boundary comes back and at H=0 it takes initial position. Magnetization is zero again. Magnetic permeability in in Region 1 is called the initial permeability. It is

28 the characteristic of the magnetic material and it is determined experimentally at Н 0,1 А/м. Region 1 is called the initial region of permeability or Rayleigh law region. 1 1 H 1 H M 3 a b c Figure 2.3. The scheme of magnetization processes: (a) initial state at H = 0; there are four single domain with different directions of the vectors of spontaneous magnetization (indicated by arrows); (b) the displacement of domain boundaries under the influence of an external magnetic field; the growth of a domain favorably oriented relative to the field H; (c) the rotation of the magnetization vector M with a further increase of the magnetic field Region 2 is the region of irreversible displacement of domain boundaries. It is characterized by the strongest dependence of B on H. The permeability passes through a maximum in this region. There is the remanent magnetization because the domain boundaries do not return in the initial positions when the external field is reduced to zero: B0 at H=0. Irreversible displacement of domain boundaries occurs only when the energy of the individual boundary layers depending on the magnetic field strength reaches a certain value. In this regard, the region 2 consists of continuous series of jumps in the magnetization. This part of the magnetization curve is called area of the Barkhausen effect. Region 3 is the region of the magnetization vector rotation. The magnetization vectors of domains turn in the field direction (Fig. 2.3, c). The magnetization is about to saturation in this region when spontaneous magnetization vectors are oriented parallel to the external field direction. The flax density B is close to the Figure 2.4. The dependence of the magnetic permeability on the magnetic field frequency f f maximum value B max and decreases almost up to 1. Region 4 is a region of the saturation. B is almost independent on H in this region. Further increase of H gives a very small increase in magnetization 28

29 due to the paraprocess and the magnetization is considered complete with the saturation. Paraprocess under the action of external very strong magnetic field is the orientation of elementary magnets within the domain which have been disoriented by thermal motion. The magnetization process of ferro- and ferrimagnetic materials under the influence of the magnetic field H can be divided into two subprocesses: 1. domain boundaries displacement (Fig. 2.1, region 1 and 2 and Fig. 2.3, b). 2. orientation of the domain magnetic moments in the magnetic field direction (Fig. 2.1, region 3 and Fig. 2.3 c). There is the differential permeability for a more detailed description of the dependence of (H) in addition to the initial Н and the maximum max permeability: 1 db a 0 dh. (2.1) Depending on whether is defined in the constant or a variable field it is called static or dynamic ( ~ ) permeability: 1 Bm 0 H, (2.2) m where B m and H m are the amplitude values of the flax density and magnetic field intensity correspondingly. If the pulsed field H P is used to magnetize the sample, the material is characterized by the pulse permeability P : 1 BP P 0 H, (2.3) P where B P is the maximum change of the flax density. The static and dynamic magnetic permeability are different because the magnetization in an alternating magnetic field is influenced by the eddy currents, magnetic viscosity, and resonance phenomena. The magnetic permeability in an alternating magnetic field depends on the frequency f. The permeability decreases due to the inertia of the magnetic processes with increasing the field frequency (Fig. 2.4). The magnetization at low frequencies (region I in Fig.2.4) always has enough time to reach the saturation during the field period and the permeability is equal to the static permeability. The orientation of domain magnetic moments gradually lags from the field direction with growing frequency in the region II (region of dispersion). As a result, the permeability is decreasing and has the intermediate value in this region. At very high frequencies (region III in Fig.2.4), the magnetization has no time to follow the alternating magnetic 29

30 field and the material does not feel the field. In this case, the permeability is close to The magnetization of ferro- and ferrimagnetic in alternating magnetic field The magnetization curve has a form of the hysteresis loop (Fig. 2.5) when a magnetic material is magnetized by an alternating magnetic field H. In this case H varies along the cycle: from 0 to Н m, 0, H m, and 0 again (Н m is amplitude of the magnetic field intensity). At the same time the flax density B periodically comes through the hysteresis loop characteristic points: 0, B s, B r, 0, B s, B r, 0. The phenomenon of hysteresis in ferromagnetic materials is the result of two effects: rotation of magnetization and changes in size or number of magnetic domains. The magnet has the domain structure and the magnetization does not vary across each domain. There are relatively thin domain walls between domains in which the direction of magnetization rotates from the direction of one domain to another (Fig 2.2). If the magnetic field changes, the walls move, changing the relative sizes of the domains. Inverse domains grow by displacement of domain boundaries and addition or subtraction of the initial orientated domains (called nucleation and denucleation). When Н = Н m the magnet has the reversed magnetization. The magnetic moment per unit volume is smaller than it would be in a singledomain magnet because the domains are not magnetized in the same direction but domain walls involve rotation of only a small part of the magnetization, so it is much easier to change the magnetic moment. The shape of the hysteresis loop depends on the former magnetic states of material ( magnetic history ), the material magnetic properties, the frequency and amplitude of magnetic field. The hysteresis loop takes the finalstate form (be balanced) after passing a number (normally 5-10) cycles only. This process is called magnetic preparation or magnetic accommodation. Figure 2.5. B-H curve and hysteresis loop of ferromagnetic material. The characteristic points of a symmetrical hysteresis loop are important characteristics of a magnetic material: 30

31 1) the saturation induction B s is the maximum corresponding to the magnetization maximum of magnet; 2) the remanence (residual induction) B r is the flux density remaining in zero field after a large magnetic field is applied (enough to achieve saturation); 3) the coercivity (coercive force) Н С is the magnetic field intensity required to reduce the magnetization of that material to zero after the magnetization of the sample has been driven to saturation. Thus coercivity measures the resistance of a ferromagnetic material to becoming demagnetized. The specific magnetic hysteresis loss per one cycle of magnetization is defined by the hysteresis loop area: Phl HdB [J/m 3 ], (2.4) or per a mass unit at the magnetic field frequency f [Hz]: P f / D HdB [W/m 3 ], (2.5) hl where D is a density of the magnetic substance [kg/cm 3 ]. In addition to the hysteresis loss, there is the eddy current loss in ferroand ferrimagnetic materials in the alternating magnetic field: 2 2 P 1.64 d f B / D [W/kg], (2.6) ec where d is the sample thickness [m]; В m is the flux density amplitude [T]; is the resistivity [Ωm] Magnetically soft materials (soft magnets, soft ferrites) Magnetically soft materials can be magnetized up to the saturation even in a weak magnetic field because of a high permeability μ and these magnets have a narrow hysteresis loop because of a low coercivity Н С. It means the material's magnetization can easily reverse direction without dissipating much energy (i.e. without big hysteresis losses). The division of magnets into soft and hard according to the coercive force Н С is conditional. According to the Russian standard the magnetically soft materials are ferroand ferromagnetic materials with Н С <4 ka/m. The high resistivity of a material prevents eddy currents in the core, another source of energy loss. Hence, the more is the resistivity of material, the higher frequency at which it can be used. All metals have small resistivity, so there are big eddy currents in metals at a high frequency of alternating magnetic field. Such soft magnetic metals as iron (low carbon 31 m

32 electrical steel), electrical (siliceous) steel, and permalloy (nickel-iron or Fe- Ni-Co alloy) are used at zero or low (up to hundreds of hertz or few of kilohertz) frequency only. At higher frequencies the soft magnetic materials having the resistivity like a semiconductor (or even a dielectric) are used: soft ferrites and magnetic insulators. For example, containing nickel, zinc, and/or manganese ferrites with high resistivity are extensively used in the cores of RF transformers and inductors because of their comparatively low losses at high frequencies. Thin rolled (up to several micrometers) metal materials are used sometimes at high frequencies and at pulsed mode especially. Technically pure iron contains less than 0.05% carbon and minimal amount of other impurities. Iron is a cheap material and it is well-stamped and processed by all machine tools. Iron has good magnetic properties in a permanent magnetic field. Iron is used for manufacturing electrical engineering devices intended for use in static magnetic field only due to its low electrical resistivity. Technically pure iron is the raw material for almost all ferromagnetic alloys. Electrolytic iron is manufactured by electrolysis. It has not well enough magnetic properties because of the high hydrogen concentration. The magnetic properties can be significantly improved by the remelting in a vacuum and multiple annealing. If it has been treated in that way, the electrolytic iron has the following magnetic properties: Н С = 30 A/m; max = In fact the electrolytic iron is rarely used because it is an expensive enough material. Carbonyl iron is produced by the thermal decomposition of iron pentacarbonyl Fe(CO) 5. The iron can be produced as the powder, sponge, or other form depending on the decomposition conditions. The carbonyl iron has to be thermally treated in hydrogen to obtain good magnetic properties. Carbonyl iron is widely used as the ferromagnetic phase of a magnetodielectric material. Sometimes a carbon steel or a special steel with carbon content of % is used instead of the technically pure iron. The magnetic properties of these steels are worse than properties of iron but they can be improved by annealing. Silicon electrical steel is a solid solution of silicon in iron with silicon content up to 3.2%. Manganese and aluminum can be added up to 0.5%. Silicon significantly increases the electrical resistivity of the steel, which decreases the induced eddy currents and narrows the hysteresis loop of the material, thus lowering the core loss. However, the grain structure hardens and embrittls the metal, which adversely affects the workability of the material, especially when rolling it. When alloying, the concentration levels of carbon, sulfur, oxygen and nitrogen must be kept low, as these elements indicate the presence of carbides, sulfides, oxides and nitrides. These 32

33 compounds increase the hysteresis losses and decrease the magnetic permeability even in particles as small as one micrometer in diameter. The presence of carbon has a more detrimental effect than sulfur or oxygen. Carbon also causes magnetic aging when it slowly leaves the solid solution and precipitates as carbides, thus resulting in an increase in power loss over time. For these reasons, the carbon level is kept to 0.005% or lower. The carbon level can be reduced by annealing the steel in a decarburizing atmosphere, such as hydrogen. Textured by the cold rolling steel has anisotropy of magnetic properties. Permalloy is a nickel-iron magnetic alloy with about 20% iron and 80% nickel content. Permalloys typically have the face centered cubic crystal structure with a lattice constant of approximately nm in the vicinity of a nickel concentration of 80%. It is notable for its very high magnetic permeability which makes it useful as a magnetic core material for electrical and electronic equipment, as well as for the magnetic shielding against magnetic fields. Commercial permalloy alloys typically have relative permeability of around 100,000 compared to several thousand for ordinary steel. In addition to high permeability, permalloy alloys have a low coercivity, almost zero magnetostriction, and significant anisotropic magnetoresistance. The low magnetostriction is critical for industrial applications, allowing it to be used in thin films where variable stresses would otherwise cause a ruinously large variation in magnetic properties. Permalloy's electrical resistivity generally varies within the range of 5% depending on the strength and the direction of an applied magnetic field. A disadvantage of permalloy is that it is not very ductile or workable, so applications requiring shapes, such as magnetic shields, are made of other high permeability alloys such as nickel-iron alloy (μ-metal). Permalloy is used in transformer laminations and magnetic recording heads. As stated previously, the magnetic materials used at high (from several to tens of MHz) and ultra-high frequencies (from hundreds to tens of thousands of MHz) must have a low electric conductivity. Ferrites and magnetic insulators meet this requirement. Ferrites are chemical compounds consisting of ceramic materials with iron (III) oxide (Fe 2 O 3 ) as their principal component and some metal oxides. Ferrites for the radio frequencies include a manganese-zinc ferrite (MnZn) and nickel-zinc ferrite (NiZn). These ferrites are two-component systems with the formulas Mn a Zn (1-a) Fe 2 O 4 and Ni a Zn (1-a) Fe 2 O 4 correspondingly. MnZn have higher permeability and saturation induction than NiZn. NiZn ferrites exhibit higher resistivity than MnZn, and are 33

34 therefore more suitable for frequencies above 1 MHz. There are also lithiumzinc, lead-zinc, and some other types of ferrites. These ferrites are extensively used in the cores of RF transformers and inductors, magnetic antenna, stators and rotors of high-frequency micromotors, etc. For ferrites used in alternating fields apart from the initial magnetic permeability measured at high frequency usually needed to know the magnetic dissipation factor tg m (or the relative magnetic dissipation factor tg m / in ) and critical frequency f cr. The frequency at which starts the sharp increase in loss is called the critical frequency. Ferrites for microwave devices operating in the frequency range from hundreds to tens of thousands of MHz are lithium, magnesium, nickel, magnesium ferro-aluminates, and nickel or magnesium ferro-chromates. There are also ferrites with the structure of the mineral garnet. These ferrites have the structural formula (Me 2 O 3 ) 3 (Fe 2 O 3 ) 5 or Me 3 Fe 5 O 12, where Me is the yttrium trivalent ion or a rare earth element (lanthanide). Magnetic insulators (magnetodielectrics) are conglomerates where the particles of a milled ferromagnetic material are separated by an electric insulating covering of a nonmagnetic material which is both a mechanical binder. Magnetic properties of a magnetic insulator are determined by the ferromagnetic particles size and shape, their spatial distribution in the binder, as well the ratio between the volume contents of the ferromagnetic filler and the insulator. The magnetic properties of the magnetic insulator weakly depend on the characteristics of the initial ferromagnetic material. Like the ferrite, the magnetic insulator has high resistivity and it is a high-frequency magnetic material. Magnetic insulators have some advantages over the ferrites, especially the higher stability of the properties. In addition, the plastics processing using for a magnetic insulator manufacture makes it possible to get a product much higher class of accuracy and clarity than the ceramic processing which is used for ferrites. But a number of electromagnetic parameters of magnetic insulators are worse than of ferrites. The most widely used magnetic insulators have the alsifer (Al-Si-Fe alloy) or the carbonyl iron as filler Magnetically hard materials (hard ferrites, hard magnets) Magnetically hard materials have a high coercivity (Н С >4 ka/m according to Russian standard) and high remanence after magnetization. The high coercivity means the materials are very resistant to becoming demagnetized, an essential characteristic for a permanent magnet. They also conduct magnetic flux well and have a high magnetic permeability. This enables a magnetically hard material to store stronger magnetic fields than 34

35 iron itself. They are cheap, and are widely used in household products such as refrigerator magnets. They are used for the manufacture of permanent magnets, recording devices and storage media, acoustic equipment, etc. The main parameters of a hard magnet are: B H maximum specific magnetic energy W m [J/m 3 ], 2 where B and H are flax density and magnetic field intensity from the second quadrant of the hysteresis; the coercive force Н С ; the residual induction B r. For example, the hard ferrites (so-called ceramic magnets) are composed of iron and barium or strontium oxides. Ceramic, or ferrite, magnets are made of a sintered composite of powdered iron oxide and barium/strontium carbonate ceramic. Given the low cost of the materials and manufacturing methods, inexpensive magnets (or non-magnetized ferromagnetic cores, for use in electronic components such as radio antennas, for example) of various shapes can be easily mass-produced. The resulting magnets are non-corroding but brittle and must be treated like other ceramics. The most common hard ferrites are: strontium ferrite, SrFe 12 O 19 (SrO 6Fe 2 O 3 ), a common material for permanent magnet applications; barium ferrite, BaFe 12 O 19 (BaO 6Fe 2 O 3 ), a common material for permanent magnet applications. Barium ferrites are robust ceramics that are generally stable to moisture and corrosion-resistant. They are used in e.g. subwoofer magnets and as a medium for magnetic recording, e.g. on magnetic stripe cards; cobalt ferrite, CoFe 2 O 4 (CoO Fe 2 O 3 ), used in some media for magnetic recording. The following magnetic alloys are used as magnetically hard materials. 1) Hard-to-deform alloys Al + Ni + Fe (Al-Ni alloy; alni) and Al + Ni + Co + Fe (alnico). The magnetic alloys are widely used in industrial and consumer applications where strong permanent magnets are needed; examples are electric motors, electric guitar pickups, microphones, sensors, loudspeakers, traveling-wave tubes, and cow magnets. In many applications they are being superseded by rare earth magnets, whose stronger fields (B r ) and larger energy products (BH max ) allow smaller size magnets to be used for a given application. 2) Form-in-place alloys Co+Mo+Fe (comol), Cr+Co+Fe, V+Co+Fe (vicalloy), Cu+Ni+Co (cunico), Cu+Ni+Fe (cunife). Used for the manufacture of complex configuration magnets, compass needles, magnetic 35

36 spring gauges, and magnetic systems of hysteresis motor, recording devices or storage media. 3) Intermetallic compounds such as REM + Co (REM is a rare earth element such as samarium (Sm), praseodymium (Pr)) have the best magnetic properties and they are used in magnetron magnetic systems and other electronic equipment. A magnetic powder of gamma ferric oxide -Fe 2 O 3 with particles of needle-like shape is used for manufacturing magnetic tapes and disks. Chromium dioxide CrO 2 is also widely applied Magnetic materials for special purposes If the magnetic material application depends not only on magnetic properties but on some other characteristics that material is called a magnetic material for special purposes. Magnetic materials for special purposes include: magnetic materials with rectangular hysteresis loop (magnetic materials with RHL), magnetostrictive materials, magnetic films with cylindrical magnetic domains (CMD) and several others. RHL magnet has the hysteresis loop with the squareness ratio = B r /B s of at least The value of the squareness ratio is very close to 1 and it means that there are two well-defined magnetic states of RHL magnet with opposite directions of magnetization: states with +B r or B r. RHL magnets widely used in automation, computing, multi-channel communication systems, etc. as an element allows to distinguish two stable states: 0 and 1. Magnetic materials with RHL can be divided into three groups: ferrites textured ferromagnetic alloys used in the form of tapes with a thickness of 0.5 mm to a few micrometers, and thin ferromagnetic films. The most widely used ferrites are systems Li-Mg-Mn, Mg-Mn, Li-Na, and Mg-Mn-Zn-Ca. The squareness of the ferrite hysteresis loop is due to the chemical composition, sintering and subsequent cooling, but not due to a special treatment. RHL ferrites have not the texture and their properties are isotropic. Permalloys with crystallographic or magnetic texture are used as textured ferromagnetic alloys with the RHL. Magnetostriction is a property of ferromagnetic materials that causes them to change their shape or dimensions during the process of magnetization. The variation of material's magnetization due to the applied magnetic field changes the magnetostrictive strain until reaching its saturation value. The effect was first identified in 1842 by James Joule when observing a sample of nickel. Magnetostriction causes losses due to frictional heating in susceptible ferromagnetic cores. The effect is also responsible for the high pitch buzzing 36

37 sound that can be heard near transformers on alternating current carrying pylons. Internally, ferromagnetic materials have a structure that is divided into domains, each of which is a region of uniform magnetic polarization. When a magnetic field is applied, the boundaries between the domains shift and the domains rotate, both of these effects cause a change in the material's dimensions. The change in the material's dimensions is anisotropic and it is called the linear magnetostriction because it leads to changes in the magnet shape primarily without changing its volume. Linear magnetostriction is described by the magnetostrictive coefficient L which is the fractional change in length (the relative deformation) as the magnetization of the material increases from zero to the saturation value elongation of the sample: L = l/l 0 where l = l l 0 is the absolute elongation of a sample from its initial size l 0 to final length l. L is often expressed in parts per million or microstrains. Magnetostrictive materials can convert magnetic energy into kinetic energy, or the reverse, and are used to build actuators and sensors. Magnetostriction is a direct application in magnetostrictive generators of sonic and ultrasonic frequencies as well as some radio circuits and devices (the replacement of quartz for frequency stabilization in electro-mechanical filters, etc.). Cobalt exhibits the largest room temperature magnetostriction of a pure element at 60 microstrains. Among alloys, the highest known magnetostriction is exhibited by Terfenol-D, (Ter for terbium, Fe for iron, NOL for Naval Ordnance Laboratory, and D for dysprosium). Terfenol-D, Tb x Dy 1-x Fe 2, exhibits about 2000 microstrains in a field of 160 ka/m at room temperature and is the most commonly used engineering magnetostrictive material. Another very common magnetostrictive composite is the amorphous alloy Fe 81 Si 3.5 B 13.5 C 2 with its trade name Metglas 2605SC. Favourable properties of this material are its high saturation magnetostriction constant of about 20 microstrains and more, coupled with a low magnetic anisotropy field strength, H A, of less than 1 ka/m (to reach magnetic saturation). As a magnetostrictive material also used nickel, permalloy, alfer, a number of other alloys and some ferrites. Magnetic films are layers of magnetic material with thickness ranging from fractions of micrometers to a few micrometers deposited on a nonmagnetic sublayer by means of a vacuum evaporation, a cathodic magnetron sputtering, or an electrolytic deposition. A sublayer for a planar magnetic film can be made from glass, glass-ceramic, quartz plates, not 37

38 coated or covered with a dielectric film of SiO, SiO 2, A1 2 O 3 non-magnetic metals, and other material. Table 2.1 Values of the magnetostrictive strain during the longitudinal magnetostriction at the magnetic saturation and normal temperature are given for some materials. magnetostrictive material nickel (Ni) permalloy (45 % Ni, 55 % Fe) permendur (49 % Со, 2 % V, 49 % Fe) alloy 65K (65 % Со, 35 % Fe) ferrous-ferric oxide (magnetite) FeOFe 2 O magnetostrictive material ferrite vibrox-1 (Ni-Cu-Co-ferrite) Cobaltic ferrite (СоО-Fe 2 O 3 ) ТbFe 2 : polycrystal monocrystal The most widely used film made from alloys Fe-Ni, Fe-Ni-Co, Mn-Bi, etc. The film thickness range is limited by the fact that properties of a large thickness film are similar of the bulk magnet, and the ferromagnetic properties are gradually disappeared at much lower thicknesses. Test questions to the Part 1: Magnetic Materials 1) What magnetic material types are there? How can you describe them? 2) What is a way of the magnetization of a diamagnetic and paramagnetic? 3) What is the main difference between a ferrimagnet and an antiferromagnet? 4) Give the definition for the magnetic susceptibility χ. How is it connected with the relative magnetic permeability? 5) Give the definitions for the relative and absolute magnetic permeabilities. What properties are they describe? What is the magnetic constant? 6) What are the Neel point and the Curie point? 7) Describe the normal magnetization curve of ferro-and ferrimagnetic materials. 8) Describe the hysteresis loop according to the magnetization process. What are characteristic points of it? 9) What is the hysteresis loss in magnets? 38

39 10) What is the eddy currents loss in magnets? How is it possible to reduce them? 11) Give an explanation of the dependence of (Т) of magnets. 12) Give an explanation of the dependence of (Н) of magnets. 13) Give an explanation of the dependence of (f) of magnets. 14) What is the criterion of magnets division into soft and hard? 15) What magnetically soft materials do you know? For what are they used? 16) What magnetically hard materials do you know? For what are they used? 17) What is the magnetostriction? How is it used? What is the main characteristic of a magnetostrictive material? 18) What magnetic materials can be used at high and ultrahigh frequencies? Explain, why. 19) Describe the features and methods of textured magnetic material producing. 20) What is the magnetodielectric? How are their used? 21) Is it possible to make a magnetic core of the transformer from a hard ferrit? Prove your answer. 22) What is a ferrite? Where are they used? 23) What magnetic materials can be used in devices of sound frequencies? Prove your answer. 24) Why magnetic cores of electrical machines and transformers are made laminated (consisting of individual plates)? 39

40 Part 2. Conductive Materials Chapter 3. General Concepts There are solid and liquid conductive materials. Solid conductors are the metals and alloys as well as some modifications of carbon. Liquid conductors include molten metals and alloys as well as water solutions of salts, acids, or alkalis (electrolytes). All gases and vapors including a pair of metals at relatively low temperatures and under not so high voltage are good insulators. However, at very high temperature in strong electric field the ionization processes occur in a gas. In this case, the gas becomes a conductor with the electronic and ionic conductivity. Highly ionized gas with equal number of free electrons and positively charged ions per unit volume is very good conductive medium called plasma. The electrical current passing through a solid or liquid metal is the directional movement (drift) of free electrons. In other words, solid and liquid metals have electronic conductivity and are called electronic conductors or first-class conductors. The band theory of solids (see General Provisions ) explains the very good conductivity of metals by the fact the valence band is directly adjacent to (or even overlaps) the conduction band. It means that all valence electrons of a metal can go the conduction band easily and therefore the number of free charge carriers in the metal is always constant and very large. The current passing through an electrolyte is the directional movement of positively and negatively charged ions. So, electrolytes have the ionic type of conductivity and are called ionic conductors or second-class conductors. The current flow through an electrolyte is related to the phenomenon of electrolysis. The electrolyte composition and its properties are changing over time during the electric current flow through it in accordance with the Faraday's laws. In this regard, electronic (metal) conductors are of the greatest interest in terms of applicability in the electrical engineering. According to the classical electron theory of metals, the solid metal conductor is described as the crystal lattice of positively charged ions placed in the nearly free electron gas. Electron gas is formed by collectivized (almost free) valence electrons of metal atoms. Each metal atom gives at least one valence electron in the collectivized electron gas, thus the concentration of free charge carriers in the metal conductor is not less than per mole. Some experimentally found laws such as the Ohm's law or the law of electric power losses (Joule-Lenz low) can be mathematically described by means of classical concepts applying the statistics laws of the gas theory to the free electron gas of metal. Moreover, this classical approach allows 40

41 describe the link between electrical and thermal conductivity of metals (Wiedemann-Franz-Lorentz law) as well. But the classical theory is not able to solve a number of important problems: the discrepancy between theoretical and experimental dependences of electrical conductivity on temperature, for example, or the fact that theoretically calculated values of the metal specific heat is higher than the experimental data. These difficulties have been resolved in frame of the quantum mechanics only. According to the quantum theory the valence electrons of a metal are in so-called degeneracy state. In contrast to the classical theory, the energy of degenerated electron gas is nearly independent on temperature. That is the thermal motion does not change the energy of the electron. The electron gas becomes similar to the normal gas at higher temperatures of about 1000 K only. Nevertheless, both the classical and quantum theories agree that metals have good electrical and thermal conductivity due to the metallic bond. Chapter 4. The basic properties of conductors 4.1. The conductivity and resistivity of the conductors Since each atom of the conductor gives at least one electron to the electron gas, the concentration of free charge carriers is extremely high. For example, the concentration of free electrons n is ~ 5, m -3 in Ag, ~ 8, m -3 in Cu, and 8, m -3 in Al. The concentration of free electrons in the metal is essentially independent on temperature which sharply distinguishes between conductors and non-metallic substances (semiconductors and dielectrics). The values of metal resistivity at normal temperature covering only three orders of magnitude and lays in the range from for Ag and up to 10 μω m (micro Ohm meter or 10 6 Ohm meter) for some alloys. Some characteristic of metals are given in Table The conductivity is reciprocal of the resistivity and in case of metals it usually is measured in MS/m (mega Siemens per meter or 10 6 S/m). According to the classical theory of metals (Drude-Lorentz theory) free electrons obtain the drift rate vd under the action of electric field E in addition to the velocity of the thermal motion vt. The drift velocity has the opposite direction to the vector E whereas the thermal velocity has no the definite direction and the ordered motion of the free charge carriers (electric current) is defined by the value and direction of vd. 41

42 Tm, С Table 4.1 D, mg/m 3 C J/(kgK), W /(mk) TCl, 10 6 K 1, μωm α, 10 4 K 1 ϕ, ev METALL mercury Hg gallium Ga sodium Na indium In tin Sn cadmium Cd lead Pb zinc Zn magnesium Mg aluminum Аl silver Ag gold Au copper Cu beryllium Be nickel Ni cobalt Co iron Fe palladium Pa titanium Ti platinum Pt thorium Th zirconium Zr niobium Nb molybdenum Mo tantalum Ta rhenium Re tungsten W 38,9 29,7 97, ,55 5,91 0,97 7,28 7,31 8,65 11,4 7,14 1,74 2,70 10,5 19,3 8,94 1,83 8,90 8,71 7,87 12,2 4,5 21,4 11,5 6,4 8,6 10,2 16,6 20,5 19, ,4 7,2 5,1 6,5 4,7 4,4 0,958 0,56 0,046 0,09 0,12 0,076 0,21 0,059 0,045 0,028 0,016 0,024 0,0172 0,04 0,073 0,062 0,098 0,11 0,42 0,105 0,186 0,41 0,18 0,057 0,135 0,21 0,055 Notice. Metals are arranged in order of increasing melting temperature Tm. D is the density, C is the specific heat, is the thermal conductivity coefficient, TCl is the temperature coefficient of linear expansion, α is the temperature coefficient of resistivity, ϕ is the electron work function. Data are given at normal temperature and pressure. The electron drift velocity vd increases from zero with constant acceleration a = F/m between two collisions of the electron with ions of the metal crystal lattice (here m is a mass of the electron; F is the force acting on the electron F = E e; e is the electron charge). The mean distance coming by an electron without collisions is called the mean free path l. For metals under the normal conditions vt >> vd (vt is about 10 5 m/s and vd is only of ,5 2,3 4,4 4,0 4,4 3,6 4,3 4,4 4,8 4,3 3,9 5,0-4,5 4,8 4,1 5,3 3,3 3,8 4,0 4,2 4,1 4,8 4,5

43 10 3 m/s in a field of E = 1 V/m). Therefore, the mean free time between two collisions of an electron with ions is defined as =l/vt. Assuming the kinetic energy of the electron thermal motion mv 2 т /2 is the same of an ideal gas molecule (the energy is (3/2)kT, where k is the Boltzmann constant and T is the absolute temperature) and using equations (0.1) and (0.3), the dependence of the resistivity on temperature is following: 1/ mkT n e l. (4.1) According to (4.1), if n and l are independent on temperature, the resistivity of a metal has to be directly proportional to T. This is the mistake of classical approach and the only quantum physics gives more correct dependence (Т). Thus, the resistivity of a metal is determined by the mean free path l that is by the crystal structure of a metal. Pure metals with the perfect crystal lattice have the lowest resistivity. Any lattice distortions (because of impurities, for example) make the free path l shorter and lead to the increase in. The temperature dependence of a metal resistivity Let us introduce the common concept of a temperature coefficient which will be used as here as below. The temperature coefficient of some material characteristic z (TCz) is the logarithmic derivative of this characteristic on temperature: 1 dz d TCz ln z. z dt dt (4.2) As it is seen from (4.2), the temperature coefficient of any characteristic is measured in K 1. The positive sign of TCz means the z increases with increasing T and vice versa. If the characteristic z is a function of several variables x, y which in turn are depend on T and this dependence has form z Ax m y n..., where A, m, n... are constants, then TCz mtcx ntcy... (4.3) As a rule the temperature coefficient of resistivity TCρ is denoted by symbol α: d 1 (4.4) dt 43

44 Normally α is positive for metals because the resistance of metals increases with increasing temperature, and according to classical theory α of pure solid metals should be close to the TC of an ideal gas: 1/273 = K 1. The amplitude of the thermal vibrations of metal ions increases with increasing temperature and the collision probability of drifting electrons with ions increases too. As a result, the electron mean free path and its mobility decreases. Therefore the metal resistivity grows with the temperature. If the, μωm metal has the phase transition from solid to liquid state or from liquid to evaporation, its resistance 0,2 increases abruptly at the temperatures of the phase transitions (Fig. 4.1). 0, Т, С Figure 4.1. This is the dependence of the copper resistivity on temperature. The sharp growth of ρ is at the copper melting point of 1083 C If the resistivity of metal 0 at temperature T 0 and the temperature coefficient of metal resistivity α are known, then the conductor resistivity at temperature T can be calculated by the formula: 1 TK T T 0 0 The change of resistivity of metals at reversible deformations The change in ρ of metals at elastic deformations is caused by changes in the electron mean free path and mobility. The volume per ion in a metal crystal lattice changes at an elastic deformation and it changes the oscillation amplitude of ions. The volume per ion becomes bigger at the strain deformation. It leads to the increase in the amplitude of the ion thermal oscillations. The increase in the ion oscillation amplitude in turn causes the decrease in the mobility of electrons and grows. On the contrary, the compression reduces the amplitude of the ion oscillation at given temperature and decreases. Therefore, the reversible (elastic) strain of conductor leads to increase in and the reversible compression results in the decrease in the conductor resistivity. The dependence of ρ on the elastic deformation can be approximately described by 0 1 s, (4.5) where is the resistivity of the metal under mechanical stress ; 0 is the resistivity of the metal is not exposed to mechanical stress; s is the pressure 44

45 coefficient characterizing the metal. Plus sign in (4.5) corresponds to the extension, and minus corresponds to the compression. α,k 1, μωm % Cu Figure 4.2. Dependences of (a) ρ and (b) α of alloys Cu-Ni on the composition (percentage by weight) Plastic deformation always increases of metals due to the lattice distortions. The resistivity can be reduced back to its initial value by recrystallization during the heat treatment (annealing). Sometimes there is a decrease in the resistivity at the compression deformation due to some secondary phenomena like metal seal, the destruction of oxide films, etc. The resistivity of metals can change in a different ways under the high hydrostatic pressure. The may increases, decreases, and has abrupt changes in value due to polymorphic transitions (changes in the crystal structure of the substance). Such abrupt changes in (in bismuth, barium, thallium, lead, etc.) under changing hydrostatic pressure are used as reference points for high pressure measurements. Resistivity of alloys The formation of the solid solution by alloyage of two metals leads to a significant increase in resistance of the alloy compare with of the initial metals. Atoms of one metal penetrate to the crystal lattice of another metal during the crystallization. As a result, the alloy has more lattice imperfections than pure metals. As it was mentioned above, the impurities and the irregularities in the crystal structure of metal lead to the increase in its resistivity. The dependence of the solid solution resistivity on the percentage of two alloying elements is shown in Fig. 4.2 (curve a). The curve has a maximum at the certain content of the alloy components. The alloy resistivity decreases up to the resistivity of one pure metal with decreasing content of another component. At the same time, usually pure metals have the temperature coefficient of resistivity α higher compare with α of alloy. The dependence of α of the solid solution on the alloying elements content has a minimum (Fig. 4.2, curve b). In some alloys the resistivity changes not only due to the 45

46 change in the charge carrier mobility but also because of simultaneous increasing of the free charge concentration with increasing temperature. In this case (not so often in fact) α of the alloy even may be less than zero like in semiconductors (see part III). If the alloying components do not form the solid solution and alloy has a polycrystalline structure including crystallite of each component with small lattice distortion then the alloy resistivity is approximately determined as the arithmetic weighted mean Thermal conductivity of metals The heat is transferred through the metal by the same free electrons mainly which determine the conductivity of metals. The number of electrons per unit volume of metal is very large. That is why the thermal conductivity of metals much more than of non-metals usually. It is clear that the greater is the metal conductivity, the greater is its thermal conductivity at other parameters being equal. The metal conductivity decreases with increasing temperature and the ratio of the thermal conductivity to electrical conductivity / must increase. It is known as the Wiedemann-Franz-Lorentz law: L 0 T, (4.6) where T is the absolute temperature [T] and L 0 is the Lorentz number: 2 2 k L 0. (4.7) 2 3 e Substitution of the Boltzmann constant k = J/K and the electron charge е = C in (4.7) gives L 0 = V 2 /K 2. The Wiedemann- Franz-Lorentz law works good for most of metals at temperatures close to normal or slightly elevated. Let us to verify this law in case of copper at room temperature. Substituting the parameters of copper = S/m and = 390 W/(mK) into (4.6) we obtain (at T = 293 K) L 0 = V 2 /K 2 which is very close to the theoretical value of L 0. In the same way for aluminum L 0 = V 2 /K 2, for lead and tin L 0 = V 2 /K 2, and for iron L 0 = V 2 /K 2. However, in the low temperature region L 0 becomes a variable: for copper, for example, it passes through a minimum and it is again close to the theoretical value of L 0 near the absolute zero of temperature. 46

47 4.3. Thermoelectric power If two different metals are in contact, the potential difference arises due to the difference in the electron work functions (see Table. 4.1) and the difference in the free electron concentration. If the temperature of contact points ( junctions of a thermocouple) are the same, the sum of potential differences is zero in a closed circuit of (two or more) metals. If one of the junctions of two metals A and B has the temperature of T 1 and another junction has the temperature of T 2 (Т 1 Т 2 ), then the thermoelectromotive force (thermo-emf or thermopower) U appears between junctions (Fig. 4.3): k na U ( T1 T2 ) ln, (4.8) e n where n A and n B are the concentrations of free electrons in metals A and B, respectively; k is the Boltzmann's constant, and e is the electron charge. B Figure 4.3. The thermocouple It can be seen from (4.8) that the thermoelectric power is proportional to the temperature difference between junctions and formula (4.8) can be written as: U K T T, (4.9) 1 2 where K is the coefficient of thermal electromotive force definite for given pair of conductors. The thermocouples are used to measure temperature and made from conductors with large and stable coefficient of thermal electromotive force. In other cases it is important to avoid the parasitic thermal EMF which could cause errors during exact dimensions in measuring circuits. So, the metals and alloys with the smallest coefficient of thermoelectric power with respect to copper are used in the windings of measuring instruments and in standard reference resistors. 47

48 4.4. Temperature coefficient of linear expansion Electrical engineering devices are made from different materials: metals, plastics, glass, ceramics, etc. There is the possibility of mechanical cracking or breaking of the devise operating at changing temperature because of the thermal expansion of materials in contact. It is important to know the behavior of different materials under changeable temperature to assure the device long life. According to (4.2), the temperature coefficient of linear expansion is TCl dl l dt 1 [K 1 ], (4.10) where l is an arbitrary linear size of the sample. The TCl of metals increase when the temperature comes close the melting point. So, the low-melting metals have relatively high TCl and the high-melting metals have relatively low TCl at normal temperature. The temperature coefficient of linear expansion TCl is used also to calculate the temperature coefficient of wire resistance TCR: TCR TC TCl. (4.11) For pure metals typically ТCl<<ТC and approximately ТCR = ТC. The formula (4.11) is of considerable practical importance for alloys with low TC Mechanical properties of conductors In a very large extent the mechanical properties of metallic conductors (tensile strength, elongation at break l/l, brittleness, hardness, etc.) depend on the mechanical and thermal treatment, the presence of impurities, etc. For example, the annealing of copper leads to the decrease in strength in times and the growth in elongation at break l/l in times Superconductivity The electrical resistivity of a metallic conductor decreases gradually as temperature is lowered. It is due to the growth in electron mobility and mean free path when the thermal vibration amplitude of ions becomes less. So, the resistivity of the metal becomes very low near the absolute zero of temperature (cryoconductivity). For example, copper, aluminum, and beryllium are good cryoconductors below 100 K. The small non-zero residual resistivity ρ res of ordinary conductors (such as copper or silver) is due to the presence of impurities and other imperfections of the crystal 48

49 lattice. Superconductors are materials which resistance drops abruptly to zero when the material is cooled below the critical temperature T cr. This transition to the superconducting state is reversible. The electric current in a loop of superconducting wire can persist indefinitely with no power source. The superconductivity was first discovered by the Dutch scientist Heike Kamerlingh Onnes in 1911 for mercury loop frozen up to the temperature of 4.2 K. The phenomenon of superconductivity has been found later in other materials in addition to mercury. 27 element-superconductors (pure metals) and over a thousand complex superconductors (alloys and compounds) are known now. The parameters of some superconductors are shown in Table Table 4.2 elemental superconductors compound superconductors substance Т cr, K В cr, T substance Т cr, K В cr, T aluminum (Al) 1,2 0,010 alloy 44 % Nb + 56 % Ti 8,7 12 mercury (Hg) 4,2 0,041 alloy 50 % Nb + 50 % Zr 9,5 11 lead (Pb) 7,2 0,080 vanadium gallide V 3 Ga niobium (Nb) 9,4 0,195 niobium stannide Nb 3 Sn The superconductive state is destroyed by the magnetic field of intensity higher than the critical Н cr and superconductor becomes an ordinary conductor. It can also be caused by the magnetic field induced by the critical current I cr passing through the superconductor. When a superconductor is cooled below its transition temperature and placed in the weak external magnetic field H, the magnetic field is ejected from the superconductor bulk. The Meissner effect does not cause the field to be completely ejected. The field penetrates the superconductor but only to a very small distance decaying exponentially to zero within the bulk of the material. It is characterized by a parameter λ which is called the London penetration depth. The Meissner effect is a defining characteristic of superconductivity. For most superconductors, the London penetration depth is on the order of 100 nm. A superconductor with little or no magnetic field within it is said to be in the Meissner state. The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value H cr. In Type II superconductors, raising the applied field past a critical value H cr1 leads to a mixed state (also known as 49

50 the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At second critical field strength H cr2, superconductivity is destroyed. Most pure elemental superconductors, except niobium, technetium, vanadium and carbon nanotubes, are Type I while almost all impure and compound superconductors are Type II. During the 1950s, theoretical condensed matter physicists arrived at a solid understanding of conventional superconductivity, through a pair of remarkable and important theories: the phenomenological Ginzburg-Landau theory (1950) and the microscopic BCS theory (Bardeen, Cooper, and Schrieffer, 1957). The high-temperature superconductors. Until 1986, physicists had believed that BCS theory forbade superconductivity at temperatures above about 30 K. In that year, Bednorz and Müller discovered superconductivity in a lanthanum-based cuprate perovskite material, which had a transition temperature of 35 K (Nobel Prize in Physics, 1987). It was soon found that replacing the lanthanum with yttrium (i.e., making YBCO) raised the critical temperature to 92 K, which was important because liquid nitrogen could then be used as a refrigerant (the boiling point of nitrogen is 77 K at atmospheric pressure). This is important commercially because liquid nitrogen can be produced cheaply on-site from air, and is not prone to some of the problems (for instance solid air plugs) of helium in piping. Many other cuprate superconductors have since been discovered, and the theory of superconductivity in these materials is one of the major outstanding challenges of theoretical condensed matter physics. From about 1993, the highest temperature superconductor was a ceramic material consisting of thallium, mercury, copper, barium, calcium and oxygen (HgBa 2 Ca 2 Cu 3 O 8+δ ) with T c = 138 K. Figure 4.4 Timeline of superconducting materials. 50

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