Module 3.1. Electrical Properties of Materials PDF

MODULE – 3.1: ELECTRICAL PROPERTIES OF MATERIALS Electrical Properties: Review of classical free electron theory, limitations of classical free electron theory. Postulates of quantum free electron theory, Fermi energy, Fermi velocity, Fermi temperature. Fermi factor and its dependence on energy a...

MODULE – 3.1: ELECTRICAL PROPERTIES OF MATERIALS Electrical Properties: Review of classical free electron theory, limitations of classical free electron theory. Postulates of quantum free electron theory, Fermi energy, Fermi velocity, Fermi temperature. Fermi factor and its dependence on energy and temperature. Electrical conductivity (qualitative expression using effective mass and Fermi velocity). Merits of quantum free electron theory. Problems. Practical Topics: 1. Fermi energy of copper Self-study: Classical free electron theory Classical free electron theory (CFET) The first theory was developed by Drude and Lorentz in 1900. According to this theory, metal contains free electrons which are responsible for the electrical conductivity and electrons obey the laws of classical mechanics. Conductivity by phonon vibrations Assumptions/Postulates of the CFET  A metal is imagined as the structure of three dimensional arrays of atoms consisting of valence electrons.  Collection of valence electrons from all the atoms is free to move about the whole volume of the metals like the molecules of a perfect gas in a container.  The free electrons are treated as equivalent to gas molecules and they are assumed to obey the laws of kinetic theory of gases.  In metals, the positive ion cores are at fixed positions and the free electrons move in random directions and collide with either positive ions or other free electrons. All such collisions are considered elastic i.e., there is no loss of energy.  In the absence of the electric field, the energy associated with each electron at a temperature T is given by (3/2) kT where k is Boltzmann constant (1.38 x 10-23 J/K). Dr. Pratibha. S, Assistant Professor, BMSCE Page 1  It is related to the KE through the relation (3/2)kT = (1/2)m(v th)2, where vth = thermal velocity.  The electric field produced due to the ionic cores is considered to be uniform throughout the body of the metal and it is neglected.  Force of repulsion between the electrons and force of attraction between electrons and lattice ions is neglected.  When an electric field is applied to the metal, the free electrons are accelerated in the direction opposite to the direction of applied electric field. Failures of classical free electron theory: Classical free electron theory failed to account for many experimental fact among which the notable once are specific heat, temperature dependence of electric conductivity (𝜎), and the dependence of electrical conductivity on electron concentration. 1) Temperature dependence of electrical conductivity: It has been experimentally observed that for metals, the electrical conductivity 𝜎 is inversely proportional to temperature T, 1 𝜎𝑒𝑥𝑝 = 𝑇………………(1) 1 Whereas according to CFET, 𝜎𝐶𝐹𝐸𝑇 = ………………(2) √𝑇 From (1) and (2) it is clear that the prediction of classical free electron theory is not agreeing with experimental observation. 2) Dependence of 𝜎 on electron concentration: 𝑛𝑒 2 𝜏 According to classical free electron theory, the electrical conductivity 𝜎 is given by 𝜎 = where 𝑚 n is the electron concentration. i.e. free electron theory suggest that 𝜎 is proportional to electron concentration n. Experimental conductivity values for different metals are as follow Metal Electron concentration(n)per m3 Conductivity (𝜎) Siemens per meter Ag 5.85*1028 6.30*107 Cu 8.45*1028 5.88*107 Cd 9.28*1028 0.15*107 Zn 13.10*1028 1.09*107 Ga 15.30*1028 0.67*107 Al 18.06*1028 3.65*107 Practically it is observed from above table that 𝜎 is not strictly proportional to electron concentration. Dr. Pratibha. S, Assistant Professor, BMSCE Page 2  3) Dependence of 𝜎 on specific heat: 𝟑 The molar specific heat of a gas at constant volume is given by 𝑪𝒗 = 𝟐 𝑹………..(1) According to experiments, Specific heat of metal 𝐶𝑣 = 10−4𝑅𝑇 ---------- (2 ) From (1) and (2) it is clear that the prediction of classical free electron theory is not agreeing with experimental observation. Drift velocity (Vd): “The average velocity with which free electrons move in a steady state opposite to the direction of the electric field in a metal is called drift velocity”. In the absence of applied field, the free electrons move rapidly in all direction. i,e the net current moving across any given plane at any instant is zero. If a constant electric field is applied, the electron will experience a force eE and get accelerated. This causes the electrons to drift in a direction opposite to that of E. The accelerated electrons collide with ions of the metal. During collision their velocities decreases. Due to repeated collisions the average acceleration of the electrons is reduced to zero. The electrons acquire a constant average velocity opposite to the field. This velocity is called drift velocity (Vd). The drift motion is directional and causes current flow, called drift current. Resistivity and mobility: Electrons tend to accelerate in the direction of the field, they are deflected in random directions by vibrating lattice points, which is scattering. The overall effect of scattering of electrons by vibrating lattice manifests as resistance to electric current. Resistance is the physical effect brought about the vibrating lattice in a material by virtue of which the accelerating effects of an applied field on the conduction electrons is annulled so that the electrons settle into state of constant velocity which is proportional to the strength of the applied field. For a material of uniform cross section, the resistance R is directly proportional to length of wire (L) and inversely proportional to cross sectional area (A) 𝝆𝑳 𝑹= Where 𝜌 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑟𝑒𝑠𝑖𝑠𝑡𝑖𝑣𝑖𝑡𝑦 𝑨 Resistivity: "It is the property of the material measuring opposition offered by material to flow of current” 𝟏 𝟏 𝐜𝐨𝐧𝐝𝐮𝐜𝐭𝐢𝐯𝐢𝐭𝐲 = ; 𝝈= 𝒓𝒆𝒔𝒊𝒔𝒕𝒊𝒗𝒊𝒕𝒚 𝝆 Mobility of electrons: “The mobility of electrons is defined as the magnitude of drift velocity acquired by the electrons in unit electric field” 𝑽𝒅 Mobility of electrons is given by 𝝁 = Where vd is drift velocity, E is electric field applied 𝑬 Quantum free electron theory After the development of quantum mechanics, a new free electron theory was proposed by Dr. Pratibha. S, Assistant Professor, BMSCE Page 3 Sommerfeld in 1928.He succeeded in overcoming many of the drawbacks of classical free electron theory while retaining all its essential features. He treated electrons quantum mechanically and included the effects of Pauli’s exclusion principle. This new theory proposed by Sommerfeld is known as Quantum free electron theory. Assumptions of Quantum free electron theory: 1) The energy values of the conduction electrons are quantized. The allowed energy values are realized in terms of a set of energy levels. 2) The distribution of electrons in the various allowed energy levels occurs as per Pauli’s exclusion principle. 3) The free electrons travel in a constant potential inside the metal but stay confined within its boundaries. 4) The attraction between the free electrons and the lattice ions, and the repulsion between the electrons themselves are ignored. Density of states: We know that the permitted energy levels for electrons in a solid material will be in terms of bands. Each band is spread over an energy range of few electron volts. The number of energy levels in each band will be extremely large. Therefore, in a small energy range, the energy values appear to be virtually continuous over the band spread. However, number of energy levels per unit energy range varies with energy in band. This variation is realized through a function g(E), which is known as density of states function. Density of states g (E) can be defined as it is the number of available energy states per unit volume per unit energy centred at E. It is mathematically continuous function and the product g(E) dE gives the number of states per unit volume in an energy interval dE at E. Let us consider the case of free electrons in a material. The possible energy values for the free electrons corresponds to only the set of energy levels available vacant but adjacent to the filledenergy levels in the band. Consider such a band as shown in fig below. Let the energy band be spread in an energy interval between E1 and E2. Dr. Pratibha. S, Assistant Professor, BMSCE Page 4  Consider an infinitesimally small increment dE at arbitrary energy value E in theband. Since dE is an infinitesimally small increment in E, we can assume that g(E) remains constant between E and E+dE. Then the number of energy levels in the range E and (E+dE)is obtained by evaluating the product of g(E) and dE 3 8√2𝜋𝑚2 𝑔(𝐸). 𝑑𝐸 = [ ] √𝐸. 𝑑𝐸 ℎ3 As per this equation, the number of energylevels in an energy interval dE is proportional to √𝐸.A plot of g(E) verses E is shown in fig below. Fermi energy: The energy corresponding to the highest occupied levels at zero degree absolute is called the Fermi energy, and the energy levels is referred to as Fermi level. The Fermi energy is denoted as 𝐸𝐹. Thus at T=O⁰K all the energy levels lying above the Fermi level are empty and those lying below are completely filled as shown in fig. Fermi velocity (VF) The velocity of the electrons which are present at the Fermi level is called Fermi velocity. 𝟏 𝑬𝑭 = 𝒎𝒗𝟐𝑭 𝟐 𝟐𝑬𝑭 𝒗𝑭 = √ 𝒎 where, VF is Fermi velocity, is EF Fermi energy, and m is mass of the electron. Fermi temperature (TF) The temperature at which average kinetic energy of an electron in a solid is equal to the thermal energy at 0 K. The thermal energy of an electron is kT. 𝑬𝑭 = 𝒌𝑻𝑭 𝑬𝑭 𝑻𝑭 = 𝒌 where, 𝑻𝑭 is Fermi temperature, 𝑬𝑭 is Fermi energy, and k is Boltzmann constant Dr. Pratibha. S, Assistant Professor, BMSCE Page 5 Fermi factor and its temperature dependence: Fermi factor is the probability of occupation of a given energy state for a material in thermal equilibrium. The probability f(E) that a given energy state with energy E is occupied at a steady temperature T, is given by 1 𝑓(𝐸) = (𝐸−𝐸𝐹 ) 𝑒 𝐾𝑇 +1 Where EF is the Fermi energy and K is the Boltzmann constant. The dependence of Fermi factor on temperature, and the effect on occupancy of energy level is as shown in figure. Let us consider different cases of distribution as follows. (i) Probability of occupation for E < EF at T=O When T= O and E < EF, we have for the probability, 1 1 𝑓(𝐸) = = =1 𝑒 −∞ + 1 0 + 1 Therefore f (E) =1, for E < EF Here f(E ) = 1 or 100% means the energy level is completely occupied, and E < EF applies to all the energy levels below EF. Therefore, at T=O, all the energy levels below Fermi level are completely filled. (ii) Probability of occupation for E > EF at T=O When T= O and E > EF, we have for the probability, 1 1 1 𝑓(𝐸) = = = =0 𝑒∞ +1 ∞+1 ∞ Therefore f (E) = 0, for E >EF. i.e at T=O, f (E) = 0% means all the energy levels above Fermi levels are completely empty. (iii) Probability of occupation at ordinary temperature: At ordinary temperature the value of probability remains 1 for E EF the probability value falls off to zero rapidly. “Hence, as the temperature increases, the electrons close to the Fermi level shifts to higher energy levels which were empty at 0 K” Dr. Pratibha. S, Assistant Professor, BMSCE Page 6 From the discussion we may say that the Fermi energy is the most probable or the average energy of the electrons across which the energy transition occurs at temperature above zero degree absolute. This may be considered as the physical basis for the concept of Fermi energy. Variation of 𝒇(𝑬) at different temperatures (T0 < T1 < T2 < T3) Electron density of states N(E) N(E) is given by the product of the number of energy levels in the same range and the Fermi factor. N (E) dE= f (E) X g (E) dE Electron density (n) The number of electrons per unit volume of the material ‘n’ can be evaluated by integrating the above expression from E=0 to E=Emax , where Emax is the maximum energy possessed by the electrons. E=Emax 𝑛= ∫ N (E) dE E=0 E=Emax 𝑛= ∫ f (E) X g (E) dE E=0 Relation between electron density, n and Fermi energy, EF at 0 K We know that, at T=0K, f (E)=1 and Emax =EF We also know that, 3 8√2𝜋𝑚2 𝑔(𝐸). 𝑑𝐸 = ( ) √𝐸. 𝑑𝐸 ℎ3 E=Emax 𝑛= ∫ f (E) X g (E) dE E=0 E=𝐸𝐹 3 8√2𝜋𝑚2 𝑛 = ∫ 1X ( ) √𝐸. 𝑑𝐸 ℎ3 E=0 Dr. Pratibha. S, Assistant Professor, BMSCE Page 7  3 E=𝐸𝐹 8√2𝜋𝑚2 𝑛=( ) ∫ 1 X √𝐸. 𝑑𝐸 ℎ3 E=0 3 16√2𝜋𝑚2 3 𝑛=( ) (𝐸𝐹 ) 2 3ℎ3 3 3𝑛ℎ3 (𝐸𝐹 )2 =( 3) 16√2𝜋𝑚2 2 3 3 3𝑛ℎ 𝐸𝐹 = ( 3) 16√2𝜋𝑚2 2 ℎ2 3 3 2 𝐸𝐹 = ( ) ( ) (𝑛)3 8𝑚 𝜋 2 𝐸𝐹 = 𝐵(𝑛)3 2 ℎ2 3 3 Where, 𝐵 = (8𝑚) (𝜋) = 5.85 × 10−38 𝐽 Quantum expression for electrical conductivity 𝑛𝑒 2 𝜏 𝑛𝑒 2 𝜆 𝜎= = ∗ 𝑚∗ 𝑚 𝑣𝐹 where, 𝑚∗ is effective mass of electron, 𝜆 is mean free path, and 𝑣𝐹 is Fermi velocity Successes of QFET 1. Specific heat at constant volume (Cv) 2. Dependence of electrical conductivity on temperature 3. Dependence of electrical conductivity on electron concentration 1. Specific heat constant volume (Cv) According to QFET, the expression for specific heat is given by 2𝑘 𝐶𝑣 = ( ) 𝑅𝑇 𝐸𝐹 Taking 𝐸𝐹 ≅ 5𝑒𝑉 for metals, we get 2𝑘 ( ) ≈ 10−4 𝐸𝐹 𝐶𝑣 = 10−4 𝑅𝑇 This value is in excellent agreement with the experimental value. “Hence, the QFET successfully explain the specific heat property of metals” Dr. Pratibha. S, Assistant Professor, BMSCE Page 8  2. Dependence of electrical conductivity (𝝈) on temperature (T) According to the QFET, 1 𝜎∝ 𝑇 “Hence the QFET successfully explain the dependence of σ on T” 3. Dependence of electrical conductivity (σ) on electron concentration (n) According to QFET, 𝑛𝑒 2 𝜆 𝜎= 𝑚∗ 𝑣𝐹 𝜆 From the above equation, it is clear that, depends both on the n and. 𝑣𝐹 𝜆 Therefore, though the value ‘n’of for divalent metals is higher than monovalent metals, the will 𝑣𝐹 be material specific value which affects the overall conductivity of the given metal. “Hence the QFET successfully explain the dependence of σ on n” …………………………………………………………………………………………………………………………………………….. Problems: 1. Calculate the Fermi energy of sodium assuming that the metal has one free electron per atom. Given the density of sodium = 970 kg/m3 and atomic weight of sodium = 22.99. 2 2 2 ℎ2 3 3 Soln: 𝐸𝐹 = (8𝑚) (𝜋) (𝑛)3 or 𝐸𝐹 = (5.85 × 10−38 )(𝑛)3 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑒𝑛𝑐𝑒 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠×𝑁𝐴 ×𝑑𝑒𝑛𝑠𝑖𝑡𝑦 1×6.023×1026 ×970 And 𝑛 = = = 2.54 × 1028 𝑎𝑡𝑜𝑚𝑖𝑐 𝑤𝑒𝑖𝑔ℎ𝑡 22.99 2 𝐸𝐹 = (5.85 × 10−38 )(2.54 × 1028 )3 = 3.92𝑒𝑉 2. Determine the drift velocity of electrons in a copper conductor with cross sectional area of 10-6 m2 carrying a current of 4 A. The atomic weight of Cu is 63.6 and density is 8.9 g/cm3. 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑒𝑛𝑐𝑒 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠×𝑁𝐴 ×𝑑𝑒𝑛𝑠𝑖𝑡𝑦 Soln: 𝑛= = 8.424 × 1028 𝑎𝑡𝑜𝑚𝑖𝑐 𝑤𝑒𝑖𝑔ℎ𝑡 𝐼 = 𝑛𝑒𝐴𝑣𝑑 𝐼 𝑣𝑑 = = 3 × 10−4 𝑚/𝑠 𝑛𝑒𝐴 3. Determine the relaxation time in a current carrying metallic wire. The metal has 5.8 × 1028 conduction electrons /m3 and its resistivity is. 𝟓𝟒 × 10−8 Ω𝑚. 𝑛𝑒 2 𝜏 Soln: 𝜎= 𝑚∗ 𝜎𝑚∗ 𝑚∗ 1 𝜏= = = 3.97 × 10−14 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 𝑛𝑒 2 𝑛𝑒 2 𝜌 Dr. Pratibha. S, Assistant Professor, BMSCE Page 9 4. Calculate the probability of an electron occupying an energy level 0.02 eV above and below Fermi level at 200 K in a material. 1 Soln: We know that , 𝑓(𝐸) = (𝐸−𝐸𝐹 ) 𝑒 𝐾𝑇 +1 i. At 200K, above Fermi level (E-EF) is a +ve quantity. (E-EF)=0.02x1.6x10-19 J=3.2x10-21 J 1 1 ∴ 𝑓(𝐸) = 3.2×10−21 = = 0.24 𝑜𝑟 24% 𝑒 1.16 +1 𝑒 1.38×10−23 ×200 +1 ii. At 200K, below Fermi level (E-EF) is a -ve quantity. (E-EF)=-0.02x1.6x10-19 J=-3.2x10-21 J 1 1 ∴ 𝑓(𝐸) = −3.2×10−21 = = 0.76 𝑜𝑟 76% 𝑒 −1.16 + 1 𝑒 1.38×10−23 ×200 +1 5. Find the temperature at which there is 1% probability that a state with an energy 0.5 eV above Fermi energy is occupied. Soln: Given f (E) = 1%=0.01 Energy level 0.5 eV above Fermi energy means (E-EF) is a +ve quantity. (E-EF)=0.5x1.6x10-19 J=8x10-20 J We know that, 1 𝑓(𝐸) = (𝐸−𝐸𝐹 ) 𝑒 𝐾𝑇 +1 1 0.01 = 8×10−20 𝑒 1.38×10−23 ×𝑇 +1 1 0.01 = 3 𝑒 5.797×10 +1 5.797×103 𝑒 𝑇 + 1 = 100 5.797×103 𝑒 = 99 𝑇 3 5.797 × 10 = ln( 99) 𝑇 5.797 × 103 𝑇= = 1261.58 𝐾 ln( 99) 6. Consider the Fermi energy of copper at 8 eV. Calculate the Fermi velocity of free electrons in copper. 𝟐𝑬𝑭 𝒗𝑭 = √ = 1.676 x 106 m/s 𝒎 Important questions: 1. Mention any four assumptions of classical free electron theory. 2. Discuss the failures of classical free electron theory. 3. Explain in detail, any three merits of quantum free electron theory. 4. Define Fermi function. Explain the variation of Fermi function with energy and temperature. 5. Mention the postulates of quantum free electron theory and explain any two of its merits. 6. Derive the expression for Fermi energy at T=0K. Dr. Pratibha. S, Assistant Professor, BMSCE Page 10