Magnetic and Superconducting Materials

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Contents Unit - 1: Conducting Materials 1.1 - 1.42 1.1 Introduction 1.1 1.1.1 Classical free electron theory 1.1 1.1.2 Quantum free electron theory 1.1 1.1.3 Zone theory (or) Band theory 1.2 1.2 Assumptions (postulates) of Classical free electron theory 1.2 1.3 Basic terms involved in the free electron theory 1.3 1.4 Success or uses of Classical free electron theory 1.4 1.5 Drawbacks of Classical free electron theory 1.4 1.6 Expression for Electrical Conductivity () 1.5 1.6.1 Expression for electrical conductivity 1.5 1.6.2 Correct expression for electrical conductivity of conductors 1.7 1.7 Thermal conductivity (K) 1.7 1.7.1 Expression for thermal conductivity (K) of an electron 1.7 1.8 Wiedemann-franz Law 1.10 1.8.1 Derivation 1.10 1.8.2 By Quantum theory 1.11 1.9 Quantum Free Electron Theory 1.12 1.9.1 Assumptions (Postulates) of Quantum free electron theory 1.13 1.9.2 Advantages of Quantum free electron theory 1.13 1.9.3 Drawbacks of Quantum free electron theory 1.13 1.10 Fermi – Dirac Distribution Function 1.14 1.10.1 Effect of Temperature on Fermi Function 1.14 1.10.2 Fermi level, Fermi Energy and their importance 1.15 1.11 Density of States 1.16 1.11.1 Carrier concentration in metals 1.19 1.11.2 Average energy of an electron at 0K 1.20 1.12 Work Function 1.21 1.12.1 Explanation 1.21 Solved Problems 1.23 Short Questions with Answers 1.35 Part – B Questions 1.41 Assignment Problems 1.42 www.annauniversityplus.comUnit - 2: Semiconducting Materials 2.1 - 2.54 2.1 Introduction 2.1 2.1.1 Properties of semiconductor 2.2 2.2 Classification of Semiconductors 2.2 2.2.1 Intrinsic semiconductors 2.3 2.2.2 Compound Semiconductors 2.3 2.2.3 Difference between N-type and P-type semiconductor 2.3 2.2.4 Difference between Elemental and Compound Semiconductors 2.4 2.3 Classification of Conductors, Insulators and Semiconductors Based on Band Theory 2.4 2.3.1 Conductors 2.4 2.3.2 Insulators 2.5 2.3.3 Semiconductors 2.6 2.3.4 Mobility and Conductivity in Semiconductors 2.6 2.4 Carrier Concentration in Intrinsic Semi-conductors 2.7 2.4.1 Density of electrons in conduction band 2.8 2.4.2 Density of Holes in Valence band 2.10 2.4.3 Intrinsic Carrier Concentration 2.13 2.5 Fermi level and variation of fermi level with temperature in an intrinsic semiconductor 2.14 2.6 Density of Electrons and Holes In Terms of E 2.15 g 2.7 Variation of Fermi level in Intrinsic semiconductor 2.16 2.8 Electrical Conductivity in Intrinsic Semi-conductor 2.17 2.9 Determination of Band Gap Energy of a Semiconductor 2.18 2.10 Extrinsic Semiconductor 2.19 2.10.1 N-type Semiconductor (Donor impurity) 2.20 2.10.2 P – type Semiconductor (Acceptor Impurities) 2.21 2.11 Charge Densities in a Semiconductor 2.22 2.12 Carrier Concentration in P-type Semi-conductor 2.23 2.12.1 Expression for the density of holes in valence band in termsof N 2.25 A 2.13 Carrier Concentration in N-type Semi Conductor 2.26 2.13.1 Expression for the density of electrons in conduction band in terms of N 2.28 D 2.14 Variation of Fermi Level with Temperature and Concentration of Impurities in P-type Semiconductor 2.30 www.annauniversityplus.com2.15 Variation of Fermi Level with Temperature and Concentration of Impurities in N-type Semiconductor 2.31 2.16 Hall Effect 2.31 2.16.1 Hall Effect 2.32 2.16.2 Hall Effect in n –type Semiconductor 2.32 2.16.3 Hall Effect in p-type Semiconductor 2.34 2.16.4 Hall Coefficient Interms of Hall Voltage 2.35 2.16.5 Experimental Determination of Hall Effect 2.36 2.16.6 Application of Hall Effect 2.37 Solved Problems 2.38 Short Questions with Answer 2.46 Part - B Questions 2.53 Assignment Problems 2.54 Unit - 3: Magnetic and Superconducting Materials 3.1 - 3.76 3.1 Introduction 3.1 3.1.1 Basic Definitions 3.1 3.2 Origin of Magnetic Moments 3.4 3.3 Classification of Magnetic Materials 3.6 3.3.1 Diamagnetic materials 3.7 3.3.2 Paramagnetic Materials 3.8 3.3.3 Ferromagnetic materials 3.9 3.3.4 Dia, Para and Ferro magnetic materials – Comparison 3.11 3.4 Domain Theory of Ferromagnetism 3.12 3.4.1 Energies involved in the domain growth (or) Origin of Domain theory of Ferromagnetism 3.13 3.5 Antiferromagnetic Materials 3.16 3.6 Ferrimagnetic Materials 3.17 3.7 Hysteresis 3.17 3.7.1 Explanation of hysteresis on the basis of Domains 3.19 3.8 Hard and Soft Magentic Material 3.20 3.8.1 Hard Magnetic Materials 3.20 3.8.2 Soft Magnetic Materials 3.22 3.8.3 Difference between Hard and Soft magnetic materials 3.23 www.annauniversityplus.com3.9 Ferrites 3.23 3.9.1 Properties 3.23 3.9.2 Structures of Ferrites 3.24 3.9.3 Regular spinal 3.24 3.9.4 Inverse spinal 3.25 3.9.5 Types of interaction present in the ferrites 3.25 3.9.6 Properties of ferromagnetic materials 3.26 3.9.7 Application of Ferrites 3.26 3.10 Magnetic Recording and Readout Memory 3.27 3.10.1 Magnetic parameters for Recording 3.27 3.10.2 Storage of Magnetic Data 3.28 3.10.3 Magnetic Tape 3.29 3.10.4 Magnetic Disc Drivers 3.29 3.10.5 Floppy Disk 3.31 3.10.6 Magnetic bubble Materials 3.32 SUPER CONDUCTORS 3.11 Introduction to Superconductivity 3.34 3.12 Properties of Superconductors 3.35 3.12.1 Critical magnetic field (Magnetic Property) 3.35 3.12.2 Diamagnetic property (Meissener effect) 3.36 3.12.3 SQUID (Superconducting Quantum Interference Device) 3.37 3.12.4 Effect of heavy Current 3.38 3.12.5 Persistence of Current 3.38 3.12.6 Effect of pressure 3.38 3.12.7 Isotope effect 3.39 3.12.8 General properties 3.39 3.13 Types of Super Conductors 3.39 3.13.1 Difference between Type I and II superconductors 3.42 3.13.2 Difference between High T and Low T superconductors 3.43 C C 3.14 High Temperature (High-T ) Superconductors 3.43 c 3.15 Bcs Theory of Superconductivity 3.44 3.16 Applications of Superconductors 3.47 www.annauniversityplus.com3.17 Engineering Applications 3.48 3.17.1 Cryotron 3.48 3.17.2 MAGLEV (MAGnetic LEVitation) 3.49 3.17.3 Josephson Devices 3.50 Solved Problems 3.51 Short Questions with Answers 3.61 Part B – Questions 3.74 Assignment Problems 3.76 Unit - 4: Dielectric Materials 4.1 - 4.41 4.1 Introduction 4.1 4.2 Basic Definitions 4.1 4.2.1 Electric flux density (D) 4.1 4.2.2 Permittivity  4.1 4.2.3 Dipole moment () 4.2 4.2.4 Polarization 4.2 4.2.5 Polarization vector 4.2 4.2.6 Polar and Non-polar Molecules 4.2 4.2.7 Dielectric Constance (or) Relative Permittivity ( ) 4.3 r  4.2.8 Electric Susceptibility ( ) 4.3 e 4.2.9 Different between Polar and Non - Polar molecules 4.4 4.3 Polarization Mechanisms Involved in a Dielectric Material 4.4 4.3.1 Electronic Polarization 4.4 4.3.2 Calculation of electronic polarization  4.5 e 4.3.3 Electronic polarization in terms of and  4.8 r 4.3.4 Ionic Polarization 4.8 4.3.5 Orientation Polarization 4.10 4.3.6 Space Charge Polarization 4.11 4.3.7 Total Polarization 4.12 4.4 Frequency and Temperature Dependence of Polarization Mechanism 4.13 4.4.1 Frequency dependence 4.13 4.4.2 Temperature dependence 4.14 4.5 Comparision of Types of Polarisation 4.15 4.6 Internal Field (or) Local Field 4.15 4.6.1 Clausius Mosotti Equations 4.20 www.annauniversityplus.com4.7 Dielectric Loss 4.21 4.8 Dielectric Breakdown 4.23 4.8.1 Types of dielectric breakdown 4.23 4.9 Ferro – Electricity and Its Applications 4.26 4.9.1 Properties of Ferroelectric Materials 4.26 4.9.2 Hysteresis of Ferroelectric Materials 4.27 4.9.3 Application of Ferroelectric Materials 4.28 4.10 Applications of Dielectric Materials 4.28 4.10.1 Dielectrics in Capacitors 4.29 4.10.2 Insulating materials in transformers 4.29 Solved Problems 4.31 Short Questions with Answers 4.36 Part - B Questions 4.41 Assignment Problems 4.41 Unit - 5 : Modern Engineering Materials 5.1 - 5.30 5.1 Introduction 5.1 5.2 Metallic Glasses 5.1 5.2.1 Glass transition temperature 5.1 5.2.2 Methods of production of Metallic Glasses 5.2 5.2.3 Types of Metallic Glasses 5.3 5.2.4 Properties of Metallic glasses 5.3 5.2.5 Applications of Metallic glasses 5.5 5.3 Shape Memory Alloys 5.6 5.3.1 Definition 5.6 5.3.2 Working Principle of SMA 5.7 5.3.3 Characteristics of SMA 5.8 5.3.4 Properties of Ni – Ti alloy 5.9 5.3.5 Advantages of SMA’s 5.9 5.3.6 Disadvantages of SMA’s 5.9 5.3.7 Applications of SMA’s 5.9 5.4 Nano Materials 5.10 5.4.1 Introduction 5.10 5.4.2 Definitions 5.11 5.4.3 Synthesis of Nanomaterials 5.11 5.4.4 Chemical Vapour Deposition (CVD) 5.13 www.annauniversityplus.com5.5 Properties of Nanoparticles 5.14 5.6 Applications of Nanoparticles 5.15 5.7 Non linear materials (NLO materials) 5.17 5.7.1 Higher Harmonic Generation 5.17 5.7.2 Experimental Proof 5.19 5.7.3 Optical mixing 5.20 5.8 Biomaterials 5.20 5.8.1 Biomaterials Classifications 5.21 5.8.2 Conventional implant devices 5.21 5.8.3 Biomaterials Properties 5.23 5.8.4 Modern Engineering MaterialsBiomaterials Applications 5.24 Short Questions with Answers 5.25 Part B - Questions 5.30 Index I.1 - I.2 www.annauniversityplus.com1 Conducting Materials 1.1 INTRODUCTION The electron theory of solids explains the structures and properties of solids through their electronic structure. This theory is applicable to all solids both metals and non metals. This theory also explains the bending in solids behavior of conductors and insulators, electrical and thermal conductivities of solids, elasticity and repulsive forces in solids etc,.. The theory has been developed in three main stages. 1.1.1 Classical free electron theory This theory was developed by Drude and Lorentz. According to this theory, a metal consists of electrons which are free to move about in the crystal molecules of a gas it contains mutual repulsion between electrons is ignored and hence potential energy is taken as zero. Therefore the total energy of the electron is equal to its kinetic energy. 1.1.2 Quantum free electron theory Classical free electron theory could not explain many physical properties. In classical free electron theory, we use Maxwell-Boltzman statics which permits all free electrons to gain energy. In Somerfield developed a new theory, in which he retained some of the features of classical free electron theory included quantum mechanical concepts and Fermi-Dirac statistics to the free electrons in the metals. This theory is called quantum free electron theory. Quantum free electron theory permits only a few electrons to gain energy. www.annauniversityplus.com1.2 ENGINEERING PHYSICS - II 1.1.3 Zone theory (or) Band theory Bloch developed the theory in which the electrons move in a periodic field provided by the Lattice concept of holes, origin of Band gap and effective mass of electrons are the special features of this theory of solids. This theory also explains the mechanism of super conductivity based on band theory. 1.2 ASSUMPTIONS (POSTULATES) OF CLASSICAL FREE ELECTRON THEORY 1. A Solid metal has nucleus with revolving electrons. The electrons move freely like molecules in a gas. 2. The free electrons move in a uniform potential field due to the ions fixed in the lattice. 3. In the absence of electric field (E=0), the free electrons move in random directions and collide with each other. During this collision no loss of energy is observes since the collisions are elastic as shown in figure. 4. When the presence of electric field ( ) the free electrons are accelerated E 0 in the direction opposite to the direction of applied electric field, as shown in figure. Fig1.1 Absence of electric field (E= 0) Presence of electric field ( ) E 0 5. Since the electrons are assumed to be perfect gas, they obey the laws of classical theory of gases. 6. Classical free electrons in the metal obey Maxwell-Boltzmann statistics. www.annauniversityplus.comCONDUCTING MATERIALS 1.3 1.3 BASIC TERMS INVOLVED IN THE FREE ELECTRON THEORY 1. Drift Velocity (V ) d The drift velocity is defined as the average velocity acquired by the free electron in particular direction, due to the applied electric field. Average distance travelled by the electron Drift Velocity = Time taken  1 V ms d t 2. Mobility () The mobility is defined as the drift velocity (V ) acquired by the electron per d unit electric field (E). V 211 d  m V s E 3. Mean free path () The average distance travelled by a electron between two successive collision is called mean free path. 4. Mean collision time ( ) (or) Collision time c It is the time taken by the free electron between two successive collision.   sec c V d 5. Relaxation time () It is the time taken by the electron to reach equilibrium position from disturbed position in the presence of electric field. l  sec V d Where l is the distance travelled by the electron. The value of relaxation time is –14 of the order of 10 sec. 6. Band gap (E ) g Band gap is the energy difference between the minimum energy of conduction band and the maximum energy of valence band. www.annauniversityplus.com1.4 ENGINEERING PHYSICS - II 7. Current density (J) It is defined as the current per unit area of cross section of an imaginary plane holded normal to the direction of the flow of current in a current carrying conductor. I –2 J A m A 1.4 SUCCESS OR USES OF CLASSICAL FREE ELECTRON THEORY 1. It is used to verify Ohm’s law. 2. It is used to explain electrical conductivity () and thermal conductivity of (K) of metals. 3. It is used to derive Widemann-Franz law. 4. It is used to explain the optical properties of metal. 1.5 DRAWBACKS OF CLASSICAL FREE ELECTRON THEORY 1. It is a macroscopic theory. 2. According to classical free electron theory, all the free electrons will absorb energy, but the quantum free electron theory states that only few electrons will absorb energy. 3. This theory cannot explain the Compton effect, Photo-electric effect, para- magnetism and ferromagnetism, etc., 4. This theory cannot explain the electrical conductivity of semiconductors and insulators. 5. Dual nature of light radiation cannot be explained. 6. The theoretical and experimental values of specific heat and electronic specific heat are not matched. K 7. By classical theory = T is constant for all temperature, but by quantum  K theory = T is not a constant for all temperatures.  8. The Lorentz number obtained by classical theory does not have good agreement with experimental value and theoritical value, it is rectified by quantum theory. www.annauniversityplus.comCONDUCTING MATERIALS 1.5 1.6 EXPRESSION FOR ELECTRICAL CONDUCTIVITY ( )  Definition The electrical conductivity is defined as the quantity of electricity flowing per unit area per unit time at a constant potential gradient. 2 ne 11 = Ohm m  m 1.6.1 Expression for electrical conductivity Fig 1.2 Moment of Electron When an electric field (E) is applied to a conductor the free electrons are accelerated and give rise to current (I) which flows in the direction of electric filed flows of charges is given in terms of current density. Let ‘n’ be the number of electrons per unit volume and ‘e’ be the charge of the electrons. The current flowing through a conductor per unit area in unit time (current density) is given by J = nV (e) d nV (e) J = – ... (1) d The negative sign indicates that the direction of current is in opposite direction to the movement of electron. Due to the applied electric field, the electrons acquire an acceleration ‘a’ can be given by Drift Velocity (V ) d Accelaration (a) = Relaxation time () www.annauniversityplus.com1.6 ENGINEERING PHYSICS - II V d a =  V = a ... (2) d When an electric field of strength (E) is applied to the conductor, the force experienced by the free electrons in given by F = – eE ... (3) From Newton’s second Law of motion, the force acquired by the electrons can be written as F = ma ... (4) Comparing equation (3) & (4) –eE = ma e E a = ... (5) m Now, substituting the value of ‘a’ from the equation (2),we get eE V = ... (6) d m Substitute equation (6) in (1) eE  n (e) J =  m  2 ne E J = ... (7) m J The electrical conductivity = E 2 ne  = m 2 ne  The electrical conductivity m The electrical conductivity of a material is directly proportional to the free electron concentration in the material. www.annauniversityplus.comCONDUCTING MATERIALS 1.7 1.6.2 Correct expression for electrical conductivity of conductors By using the classical free electron theory, quantum free electron theory and band theory of solids we can get, 2 ne The electrical conductivity, = m Where m- effective mass of free electron  - Electrical conductivity  - Relaxation time n - Number of electrons 1.7 THERMAL CONDUCTIVITY (K) Definition The thermal conductivity is defined as the amount of heat flowing through an unit area per unit temperature gradient. Q 11 Wm   = K dT  A  dx  The negative sign indicates that heat flows hot end to cold end. Where K is the thermal conductivity of metal. Q is the amount of heat energy. dT is the temperature gradient. dx In general, the thermal conductivity of a material is due to the presence of lattice vibrations (ie., photons and electrons). Hence the total thermal conduction can be written as. K K K = electron photons total 1.7.1 Expression for thermal conductivity (K) of an electron Consider a metal bar with two planes A and B separated by a distance ‘’ T T T T from C. Here is hot end and is cold end. ie., 1 2 1 2 www.annauniversityplus.com1.8 ENGINEERING PHYSICS - II  A C B Direction of T T 1 2 flow of heat  Fig.1.3 Thermal Conductivity Let ‘n’ be the number of conduction electrons and ‘v’ be the velocity of the K electrons. is the Boltzmann constant B From kinetic theory of gases 1 2 mv Energy of an electron at A = 2 3K T B 1 = ... (1) 2 The kinetic energy of an electron at 1 2 mv B = 2 3K T B 2 = ... (2) 2 The net energy 3K (T T )  B 1 2 ... (3)   transferred from A to B 2  Fig.1.4 moment of electron field Let as assume that there is equal probability for the electrons to move in all the six directions. Each electrons travels with thermal velocity ‘V’ and ‘n’ is the free electron density then on average of 1/6 nv electron will travel in any one direction. www.annauniversityplus.comCONDUCTING MATERIALS 1.9 No. of electrons crossing per unit area in unit time at C 1 = nv ... (4) 6  (Average energy transfer from A to B)  The energy carried by Q   the electrons from A to B (No of electron crossin g per unit area)   3K (T T ) 1  B 1 2 nv Q =  2 6  1 K (T T )nv Q = ... (5) B 1 2 4 We know that the thermal conductivity, Q K = dT  A  dx  The heat energy transferred per unit sec per unit area dT K Q = A=1 unit area dx K(T T ) 1 2 Q = ... (6) 2  dT  T T , dx 2 1 2 Comparing equations (5) and (6), K(T T ) 1 1 2 K (T T )nv = B 1 2 4 2 1 K nv Thermal conductivity K =  B 2 K nv B K Thermal conductivity K 2 www.annauniversityplus.com1.10 ENGINEERING PHYSICS - II 1.8 WIEDEMANN-FRANZ LAW Statement The ratio between the thermal conductivity (K) and electrical conductivity (σ) of a metal is directly proportional to the absolute temperature of the metal. K K  T or  LT  –8 –2 Where L is called Lorentz number, the value of L is 2.44 × 10 WK (as per Quantum Mechanical value). 1.8.1 Derivation By Classical theory, we can drive Widemann-Franz law using the expressions for electrical and thermal conductivity of metals. The expression for thermal conductivity K nv B K = 2 The expression for electrical conductivity 2 ne  = m 1/ 2K nv K B = 2  ne / m 2 m K v K 1 B   v =  2  2 e K K 1 2 B mv = 2  2 e We know that kinetic energy of an electron 3 1 2 K T  mV = B 2 2 www.annauniversityplus.comCONDUCTING MATERIALS 1.11 K K 3 B K T = B 2  2 e 2 K K T 3 B = 2  2 e 2 K K 3 B = 2 T 2 e K  L  L is called Lorentz number T Thus, it is proved that the ratio of thermal conductivity and electrical conductivity of a metal is directly propotional to the absolute temperature of the metal. 2 K 3 B Where Lorentz number L = 2 2 e 2 23 3 1.38 10  L = 2 19 2 1.6 10  82 L = 1.12 10 WK It is found that the classical value of Lorentz number is only one half of the –8 –2 experimental value (2.44 × 10 WK ). The discrepancy of L value is the failure of the classical theory (Experimental and Theoretical). This can be rectified by quantum theory. 1.8.2 By Quantum theory By Quantum theory the mass ‘m’ is replaced by effective mass m 2 ne The electrical conductivity  =  m www.annauniversityplus.com1.12 ENGINEERING PHYSICS - II According to Quantum theory, the expression for thermal conductivity is modified by considering the electron specific heat as 2 2 nK T  B K =  3 m 2 2  nK T  B   K 3 m  = 2  ne  m  2 2  K T  K B  = T 2 3 e   K LT =  2 2  K  B  Where L = 2  3 e  223 2 (3.14) (1.38 10 ) L = 19 2 3 (1.6 10 ) –8 –2 L = 2.44 × 10 WK This is gives the correct value of Lorentz number and it in good agreement with the experiment value. 1.9 QUANTUM FREE ELECTRON THEORY The failure of classical free electron theory paved this way for Quantum free electron theory. It was introduced by Sommer field in 1928. This theory is based on making small concepts. This theory was proposed by making small changes in the classical free electron theory and by retaining most of the postulates of the classical free electron theory. www.annauniversityplus.comCONDUCTING MATERIALS 1.13 1.9.1 Assumptions (Postulates) of Quantum free electron theory 1. In a metal the available free electrons are fully responsible for electrical conduction. 2. The electrons move in a constant potential inside the metal. They cannot come out from the metal surface have very high potential barrier. 3. Electrons have wave nature, the velocity and energy distribution of the electron is given by Fermi-Dirac distribution function. 4. The loss of energy due to interaction of the free electron with the other free electron. 5. Electron’s distributed in various energy levels according to Pauli Exclusion Principle. 1.9.2 Advantages of Quantum free electron theory 1. This theory explains the specific heat capacity of materials. 2. This theory explains photo electric effect, Compton Effect and block body radiation. etc. 3. This theory gives the correct mathematical expression for the thermal conductivity of metals. 1.9.3 Drawbacks of Quantum free electron theory 1. This theory fails to distinguish between metal, semiconductor and Insulator. 2. It also fails to explain the positive value of Hall Co-efficient. 3. According to this theory, only two electrons are present in the Fermi level and they are responsible for conduction which is not true. www.annauniversityplus.com

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