UNIT 9 FREE ELECTRON THEORY OF METALS Structure 9.1 Introduction Objectives 9.2 Drude - Lorentz Theory Electrical Conductivity Thermal Properties 9.3 Sommerfeld Model Fermi Energy and Fermi Surface Temperature Dependence of Electrical and Thermal Properties 9.4 Summary 9.5 Terminal Questions 9.6 Solutions and Answers 9.1 INTRODUCTION In Blocks 1 and 2, you have learnt about crystal structure and crystal binding. These helped you to understand the elastic properties of solids. You have studied that elastic and to some extent, thermal properties of solids can be understood in terms of vibrations of atoms about their respective equilibrium positions. However, many physical properties of solids can be explained only when sub-atomic details such as motion of electrons are taken into consideration. In this unit, you will learn one such microscopic theory: free electron theory of metals which successfully explains some of their electrical and therinal properties. Metals are perhaps the most versatile materials from the point of view of utility. We all know that due to hardness and rigidity, metals are ideal materials for making machines, industrial equipment, household goods, automobiles, ships etc. Similarly, iron is an essential ingredient of modem structures such as ceilings, flyovers and pillars, because of its strength and stability. Further, metals like copper and aluminium are used in power transmission and distribution and so on. In view of their so many and so varied uses, it is important to understand their physical properties. In your school physics course, you learnt that electrons are responsible for electrical conduction in metals. This means that to understand the electrical properties, we have to investigate the behaviour of electrons. Efforts made to this effect led to free electron theory of metals. In (classical) free electron theory, proposed by Drude and later developed by Lorentz, it is assumed that valence electrons in a metal are analogous to a gas in a box. Using the method of kinetic theory of gases and Maxwell- Boltzmann distribution law, Drude and Lorentz explained the phenomenon of electrical conduction. You will study Drude-Lorentz theory in Sec.9.2. This theory, however, could not satisfactorily explain experimental observations on temperature dependence of resistivity and heat capacity. These limitations of the classical theory were overcome to a large extent by Sommerfeld who argued that free electrons in a metal behave as quantum mechanical particles. You will learn the details of Sommerfeld's model and related concepts in Sec. 9.3. Unfortunately, even Sommerfeld's model had limited success. In particular, it did not provide a logical theoretical basis to distinguish metals from insulators and semiconductors. To overcome this, it was proposed that free electrons in a metal are not completely free; as such they move under the influence of a periodic potential which arises due to ions situated at lattice points. This led to what is known as the hand thcon qf solidc.. You will learn it in the next unit. I Electronic Properties Objectives I After studying this unit, you should be able to: state basic assumptions of Drude-Lorentz theory of metals; explain the concept of relaxation time and its role in electrical conduction; state the limitations of Drude-Lorentz theory; describe-Sommerfeld free electron model; explain the concepts of Fermi level, Fermi energy and Fermi surface; predict the nature of electrical conduction and temperature dependence of resistivity on the basis of Sommerfeld model; derive expressions for heat capacity and thermal conductivity of metals and explain electronic contribution to them; and solve numerical problems based on these concepts. 9.2 DRUDE - LORENTZ THEORY \ We all know that metals are good conductors of electricity. When a potential difference is applied across a metallic wire, electric current begins to flow. Since electric current is the directed motion of electrons, it is only logical to think that metals possesspee electrons which are responsible for electrical conduction. You may ask: Where fromfree electrons come? Let us learn about it now. Free Electrons in Metals To understand the origin of free electrons in metals, let us consider the particular case of sodium (Na) metal. In a sodium atom, 11 electrons revolve around the nucleus in various orbits: 1~~,2~~,2p~,3~~.This is shown in Fig.9.la. The electron in the outermost orbit (3s) is responsible for most of the chemical properties of sodium and 0 is called the valence electron. The radius of the outermost orbit is nearly 1.9 A . The other 10 electrons in two inner orbits are relatively tightly bound to the nucleus. Fig.9.1: a) An isolated sodium atom; and b) sodium atoms arranged in a crystal lattice forming sodium metal When a large number ofNa atoms are brought closer, inter-atomic forces come into play. As a result, Na atoms arrange themselves along a cubic lattice. Each Na atom is located at a lattice point and is separated from its nearest neighbours by a fixed distance (Fig.9.1 b). The equilibrium separation between two neighbouring atoms in Na metal is 3.7 A . The inter-atomic separation is less than the sum of radii of two nearest Na atoms. This is possible only when their outermost orbits begin to overlap. Due to this overlap, valence electron of one atom can move to the orbit of the nearby atom and is shared by both the atoms at the same time. If this argument is extended further, we may say that valence electrons of Na atoms belong to the entire specimen of metallic sodium. So we may say that valence electrons constitute free electrons I and they can participate in electrical conduction. These are, thef'efore, also called Free Electron Theory of Metals conduction electrons. The genesis- of free electrons in monovalent sodium metal holds true for all metals. Drude proposed that the physical properties of metals can be understood in terms of the free electron model. According to this model, a metal consists of positive ions and The genius of Drude lies in valence electrons which constitute a free electron gas. The free electrons move all the fact that the numerical values of the radii of atomic around the volume of the metal. On the basis of this argument, Drude attempted to orbits and interatomic explain the macroscopic properties like electrical conduction and heat capacity of separation as well as the metals in terms of the microscopic parameters such as charge, mass, and momentum concept of overlapping orbits of free electrons. Lorentz contributed significantly to the development of Drude's were not known to him when model by applying the laws of statistical mechanics (Maxwell-Boltzamann he proposed the free electron model. distribution). Therefore, the microscopic theory developed on the basis of Drude model is called the Drude-Lorentz theory. For mathematical ease, following simplifying assumptions about metals are made in Drude - Lorentz theory: 1. Valence electrons behave as free electrons and can move all around the volume of the metal. 2. Positively charged ions located at the lattice sites offer a uniform potential and do , not influence the motion of free electrons. That is, interaction between the positively charged ions and free electrons can be ignored. 3. The interaction between free electrons themselves is too small and can be ignored; therefore, the free electrons move randomely inside the metal specimen without any change in their energy, except for the occasional collisions with the ions. 4. Free electrons behave as molecules of an ideal gas and obey kinetic theory of gases and the Maxwell-Boltzmann (M-B) distribution law. - a \ @la/ a a La/- . on \ / a'. .,ai- .L (a) ~lectron (b) - Fig.9.2: a) Metal specimen consisting of fixed ions (shown by bigger dots) and free electrons; and b) metal specimen as a gas of free electrons You may now like to know: How to visualise the physical model of metal on the basis of Drude model? A metal specimen may be considered analogous to a three- dimensiolial (3-D) box containing the ions fixed at the lattice sites and the gas of free electrons. Its two-dimensional (2-D) representation is shown in Fig.9.2a. The fixed ions do not affect the motion of free electrons (Assumption 2). Thus, for analytical purposes, a metal specimen is equivalent to an empty box containing a gas of free electrons (Fig.9.2b). You will now learn how this model helped in developing an understanding of electrical conductivity. 9.2.1 Electrical Conductivity You know that electrical conduction is the flow of electric current in a solid specimen when a potential difference is maintained across its two ends. The relation between current I and potential difference V is governed by Ohm's law: 7 Electronic Properties v=IR (9.1) We can express Ohm's law in terms of parameters where R is electrical resistance offered by the material. independent of physical dimensions of the specimen by Note that Ohm's law is an empirical law; that is, it is a generalisation based on replacing I with current experimental results. You may also note that Ohm's law as expressed by Eq. (9.1) is density J and potential easier to verify experimentally. For theoretical analysis however we express Ohm's difference V by electric field &. Current density is defined as law in tem~sof current density J and electric field E (see margin remark) as electric current flowing- through unit cross-sectional area, i.e., J = IlA where A is the area of cross-section of the specimen. where o is electrical conductivity of the material. Electrical conductivity is one of the important parameters that enable us to differentiate a metal from an insulator.