Conducting Materials I PDF - EE-AC 317, September 2024

Introduction Electron Theory of Conductivity Electrical Resistivity Module 2: Conducting Materials I EE-AC 317...

Introduction Electron Theory of Conductivity Electrical Resistivity Module 2: Conducting Materials I EE-AC 317 Engr. Xavier Julian P. Fernandez Department of Electrical Engineering Cebu Technological University - Main Campus September 2024 EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Topics Covered 1 Introduction 2 Electron Theory of Conductivity 3 Electrical Resistivity of Metals EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Contents 1 Introduction 2 Electron Theory of Conductivity 3 Electrical Resistivity of Metals EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Conducting Materials The conductivity of a material depends on the presence of free electrons. The materials which conduct electricity due to free electrons when an electric potential difference is applied across them are known as conducting materials. Elements with only one electron on their valence shell are the best conductor because this allows the freest flow of electrical current because the opposition of an atom of these elements from taking on or loosing electrons is low. Conducting materials are good conductors of electricity and heat. Au, Ag, Cu, Ag are the examples of conducting materials. EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Conductivity Chart EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Conductivity of Materials Electrical conductivity Electrical conductivity (σ) is defined as the rate of charge flow across a unit area in a conductor per unit potential (voltage) gradient. Mathematically, J σ= (1) E where J is the current density and E is the electric field. Conductivity is the reciprocal of resistivity and has a unit of S/m. EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Contents 1 Introduction 2 Electron Theory of Conductivity 3 Electrical Resistivity of Metals EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Classical Free Electron Theory I The free electron gas model assumes that: 1 A metal contains a large number of free electrons which are free to move about in entire volume of the metal like the molecules of a gas in a container. 2 The free electrons move in random directions and collide with either positive ions fixed in the lattice or other free electrons. All the electrons are elastic and there is no loss of energy. 3 The velocity and the energy distribution of free electrons obey the classical Maxwell-Boltzmann statistics. 4 The free electrons are moving in a completely uniform potential field due to the ions fixed in the lattice. EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Classical Free Electron Theory II 5 In the absence of electric field, the random motion of free electrons is equally probable in all directions so that the current density vector is zero. 6 When the external electric field is applied across the ends of a metal, the electrons drift slowly with some average velocity known as drift velocity in the direction opposite to that of electric field. This drift velocity is superimposed over the random velocity. This drift velocity is responsible for the flow of electric current in a metal. EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Classical Free Electron Theory III 7 During electron movement, the number of electrons flowing through the materials is governed by the equation NA D n= Wa where NA is Avogadro’s number, D is the material density, and Wa is the atomic weight. EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Electron Movement Drift velocity Suppose that in a conductor, the number of free electrons available per m3 of the conductor material is n and let their axial drift velocity be v meters/second. In time dt, distance travelled would be v × dt. If A is area of cross-section of the conductor, then the volume is A × v × dt. Obviously, all these electrons will cross the conductor cross-section in time dt. If qe is the charge of each electron, then the charge which crosses the section in time dt is dq = nqe Avdt. Hence, we can also define current to be dq nqe Av dt i= = = i = nqe Av (2) dt dt where v is the drift velocity, which is the average velocity of the free electrons to which they move towards the positive terminal under the influence of an electrical field. EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Electron Movement Carrier mobility Carrier mobility refers to how easily charge carriers (like electrons or holes) can move through a material when an electric field is applied. The mobility (µ) of a charge carrier is mathematically defined as the velocity of the carriers per unit of electric field applied, or vdrift µ= (3) E where E is the electric field applied. Factors affecting carrier mobility Carrier mobility is influenced by factors such as (1) temperature, (2) material impurities, and (3) material type. EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Electron Movement Collision time When an electron collides with another particle, its direction and/or speed can change (we call this the scattering event). The average time taken by a free electron between two successive collisions is called collision time. Mathematically, σme τ= (4) nqe2 where σ is the electrical conductivity of the material and me is the electron mass. Factors affecting collision time Collision time is influenced by factors such as (1) temperature, (2) density, (3) impurities, and (4) electric field. EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Electron Movement Mean free path The average distance travelled by a free electron between two successive collisions is called mean free path. The mean free path is the product of the average speed of the electrons and the collision time. λ = vave τ (5) Relaxation time It is defined as the time taken by a free electron to reach its equilibrium position from the disturbed position in the presence of an electric field. Relaxation time is also denoted by τ , which is essentially the same with collision time. EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Example Problems Problem 1 A uniform silver wire has a resistivity of 1.54 × 10−8 Ω·m at room temperature. For an electric field along the wire of 1 V/cm, compute the average drift velocity of electron assuming that there are 5.8 × 1028 conduction electrons per m3. Also calculate the mobility. Answer: µ = 6.9973 × 10−3 m2 V−1 s−1 , vdrift = 0.69973 m/s Problem 2 The following data is given for a copper material: D = 8.92 × 103 kg · m−3 , ρ = 1.73 × 108 Ω·m, Wa = 63.5 kg. Calculate the mobility and the average time collision of electrons in copper obeying classical laws. Answer: τ = 2.380 × 10−27 s, µ = 4.27 × 10−16 m2 V−1 s−1 EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Problem 3 Calculate the drift velocity of electrons in copper and current density in a wire of diameter 0.16 cm which carries a steady current of 10 A, given that n = 8.46 × 1028 m–3. Answer: vdrift = 3.67 × 10−4 m/s Problem 4 A conducting rod contains 8.5 × 1028 electrons per cubic meter. Calculate its resistivity at room temperature and also the mobility of electrons if the collision time for electron scattering is 2 × 10–14 sec. Answer: ρ = 2.09 × 10−28 Ω · m, µ = 3.512 × 10−3 m2 V−1 s−1 EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Contents 1 Introduction 2 Electron Theory of Conductivity 3 Electrical Resistivity of Metals EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Resistivity Electrical resistivity is the reciprocal of electrical conductivity and is the measure of the ability of a material to oppose the flow of current. Total Resistivity It has been observed experimentally that the total resistivity of a metal is the sum of the contributions from thermal vibrations ρt , impurities ρi , and plastic deformation ρd ; that is, the scattering mechanisms act independently of one another. ρtotal = ρt + ρi + ρd (6) The equation above is sometimes known as Mathiessen’s rule. EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Influence of Temperature to Resistivity For the pure metal and all the copper–nickel alloys shown in the figure, the resistivity rises linearly with temperature above about -2200◦ C. Thus, ρt = ρ0 + aT (7) where r0 and a are constants for each particular metal. This dependence of the thermal resistivity component on temperature is due to the increase with temperature in thermal vibrations and other lattice irregularities (e.g., vacancies), which serve as electron-scattering centers. EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Influence of Impurities to Resistivity For additions of a single impurity that forms a solid solution, the impurity resistivity ρi is related to the impurity concentration ci in terms of the atom fraction (at %/100) as follows: ρi = Aci (1 − ci ) (8) where A is a composition-independent constant that is a function of both the impurity and host metals. EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity Influence of Plastic Deformation to Resistivity Plastic deformation also raises the electrical resistivity as a result of increased numbers of electron-scattering dislocations. The movement of dislocations can cause local distortions in the crystal lattice. These distortions can also scatter electrons, contributing to increased resistivity. EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I Introduction Electron Theory of Conductivity Electrical Resistivity... Thank you for your attention! EE-AC 317 CTU-MC · Department of Electrical Engineering Module 2: Conducting Materials I