Electrical Conductivity in Metals & Semiconductors Dr. S. Gowda Professor of Physics P.E.S. College of Engineering Mandya – 571 401, Karnataka (An Autonomous Institution affiliated to VTU, Belagavi) DEPARTMENT OF PHYSICS Course Title : Engineering Physics Course Code : P18PH12/22 Unit - III ELECTRICAL CONDUCTIVITY and SEMICONDUCTORS Notes Academic Year : 2019 1 Electrical Conductivity in Metals & Semiconductors Dr. S. Gowda Professor of Physics Electrical Conductivity in Metals Syllabus: Electrical Conductivity in Metals: Free electron concept. Classical free-electron theory – Assumptions. Failure of classical free-electron theory. Quantum free-electron theory – Assumptions. Fermi-Dirac Statistics. Fermi-energy – Fermi factor, Fermi Velocity, Fermi Temperature. Calculation of Fermi Energy at T = 0 K and T > 0 K. Density of states (with derivation). Merits of quantum free-electron theory. Problems. Introduction: Materials can be classified into three types based on the conductivity of heat and electricity. They are; 1. Conductors (Example : Metals – Copper, Aluminum, Silver, Gold) 2. Semiconductors (Example : Germanium, Silicon) 3. Insulators ( Example : Wood, Mica, Glass) Electron Theory of Metals The electron theory of metals explains the following : • Structural, electrical and thermal properties of materials. • Elasticity, cohesive force and binding in solids. • Behaviour of conductors, semiconductors, insulators etc., In solids, electrons in the outermost orbit of atoms are called valance electrons, which determine the properties of the materials. The electron theory is applicable to all solids (both metals and non-metals). This theory explains the electrical, thermal and magnetic properties of solids There are three stages for the development of electron theory of metals; namely, Classical free electron theory, Quantum free electron theory and Zone theory. 1. A classical free electron theory is a macroscopic theory proposed by Paul Drude in 1900. After the discovery of electron by JJ Thomson, this theory was elaborated by Lorentz in 1909. Hence this theory is also known as Drude & Lorentz. According to this theory metals contains free electrons which are responsible for the electrical conductivity in metals and obeys the laws of classical mechanics (Maxwell-Boltzmann distribution). 2. Quantum free electron theory is a microscopic theory developed by Sommerfeld in 1928. According to this theory, the free electrons move with a constant potential obeys quantum laws (Fermi-Dirac statistics). 3. Zone theory was developed by Bloch in 1928. According to this theory, free electrons move in periodic potential provided by lattice. This theory is also known as band theory of solids. Classical Free Electron Theory (Drude-Lorentz Theory): Free Electron Concept: All metal atoms consists of valance electrons and they are responsible for physical properties of metals such as electrical & thermal conductivities, thermoelectricity, thermionic & Photoelectric effect ete., 2 Electrical Conductivity in Metals & Semiconductors Dr. S. Gowda Professor of Physics According to Drude-Lorentz theory, when a large number of atoms arranged in three dimensional lattice points to form a metal, the boundaries of the neighbouring atoms slightly overlap with each other as shown in figure. Due to this overlapping, the valance electrons of all the atoms are free to move within the metal lattice. These electrons are called free electrons. These free electrons are move randomly in all directions through the conductor with average speed of the order of 106 m/s. This is similar to the motion of gas molecules confined in a vessel. Since the free electrons are responsible for electrical and thermal conduction in metals, they are also called as conduction electrons. All metals contain free electrons which act just as a gas molecules moving in every direction throughout the lattice. The average velocity due to the thermal energy is zero since the electrons are going in every direction. There is a way of affecting this free motion of electrons, which is by use of an electric field. This process is known as electrical conduction and theory is called Drude-Lorentz theory. The assumptions of the Drude-Lorentz classical theory of free- electrons are the following. Assumptions of Classical free-electron theory: 1. All metals contain large number of free electrons which move freely through the positive ionic core of the metals. Since these free electrons causes conduction in metal under the application of electrical field, they are called as conduction electrons. 2. The free electrons are treated as equivalent to gas molecules; the laws of classical kinetic theory of gases can be applied to them. Therefore these electrons have mean free path (λ), mean collision time (T), average speed (v). 3. In the absence of the electric field, the kinetic energy associated with an electron at a temperature T is given by 2 2 ½ mvth = 3/2 kT = ½ m c Where vth is the thermal velocity of the electron and which is equal to root mean square velocity c . 4. Since the motion of the electrons is random, the net current is zero in the absence of electric field. But when an electric field is applied, current is produced due to the drift velocity of the electrons. 5. The electric field (or Potential) due to positive ionic cores is considered to be uniform throughout the metal and hence neglected. But the force of attraction between the electrons & lattice ions and the force of repulsion between the electrons themselves are considered to be negligible. 3 Electrical Conductivity in Metals & Semiconductors Dr. S. Gowda Professor of Physics Drift velocity, mean free path, Mean collision time, Relaxation time, conductivity, Mobility and resistivity. 1) Drift velocity (Vd) : It is the average velocity acquired by the electrons in a direction opposite to the direction of the applied electric field. eE = Drift velocity is given by m τ Where E is the strength of the applied electric field τand is the relaxation time. 2) Mean Free Path (λ): The average distance travelled by the conduction electrons between two successive collisions with lattice ions is known as mean free path. i.e., λ = vτ, where τ is the mean collision time & v = vd+vth -4 v is the combined velocity drift and thermal velocities. Since vd =10 m/s is very 6 small compared to thermal velocity vth = 10 m/s, hence λ = vth τ 3) Mean Collision Time (τc) : It is the average time that elapses between two successive collisions of an electron with lattice points. If v is the total velocity of the electrons due to combined effect of thermal & drift velocities, then τc = λ/v 4) Relaxation Time (τr) : When an electric field is applied, the free electrons drift slowly in a direction opposite to that of the applied electric field with an average velocity v′av vav = vav′ in the presence of electric field. If the electric field is turned off suddenly, the average velocity vav reduces exponentially to zero. − τ ′ t r 1 ′ Therefore, vav = vave , If t = τr , then vav = e vav Therefore, the relaxation time (τr) is time during which the average velocity vav decreases 1 to ( e ) times its value of time when the field is turned off. The relaxation time is a measure of the rate at which relaxation takes place. For metals the values of τc & τr are equal. Electrical conductivity in metals : Electrical conductivity of a metal is the ability of the metal to allow electrons to flow through it. The expression for electrical conductivity of a metal is given by ne2 σ = τ m Where n is the number of free electrons/unit volume & τ is the relaxation time. Mobility of electrons : Mobility is defined as the magnitude of the drift velocity acquired by the electrons in unit electric field. The expression for the mobility is v e µ = d = τ E m Electrical resistivity : It is the property of the metal and defined as the reciprocal of electrical conductivity. 1 m i.e., ρ = = σ ne2τ 4 Electrical Conductivity in Metals & Semiconductors Dr. S. Gowda Professor of Physics In metals, the resistivity is due to the scattering of conduction electrons. Scattering of electrons may be due to lattice vibrations or due to impurities. a) Scattering due to lattice vibrations (Phonons) – When the temperature of metal is increased, due to vibrations of lattice ions, the scattering of electrons may take place. The resistivity due to this type of scattering is called ‘ideal resistivity’ denoted by ρph. This is temperature dependent. b) Scattering due to impurities – The impurities in metals may also scatter electrons and the resistivity due to this is denoted by ρi. This is temperature independent. Thus in any metal, the total resistivity is ρ = ρ ph + ρ i This is known as Mattthienssen’s rule Success of Classical Free Electron Theory: .. The successes of this theory are:- 1. It verifies Ohm’s law i.e., V = IR σ T 2. It derives Wiedemann-Frenz i.e., ∝ T σ E 3. It explains Electrical (σE) and Thermal conductivity (σT) of metals 4. It explains optical properties of metals. Failures/Drawbacks of Classical Free-Electron Theory: 1. Specific Heat (CV) : The specific heat of a gas at constant volume is given by 3 CV = 2 R , where R is the universal gas constant But experimentally it was observed that the specific heat of a metal by its conduction electrons is given by − 4 CV = 10 RT Thus, the experimental value of Cv is very much lesser than the expected value of Cv. According to classical free electron theory Cv is independent of temperature, but the experimental value of Cv is directly proportional to temperature. Hence classical free electron theory fails to explain Cv. 2. Mean Free Path (λ) : According to classical free electron theory, the mean free path . λ = cτ , where c is the root mean square velocity & τ is the relaxation time of m m conduction electrons, we also know that ρ = or τ = ne2τ ne2ρ By substituting the values of e, m, n & ρ for a metal, τ can be calculated.