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Physical Chemistry B Physical Chemistry B Marco Montalbano 08/08/2018 1 This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. Le immagini sono esenti da CC, in quanto reperite dal materiale didattico fornito dal professor Martinazzo Contents 1 Drude model of metals 3 1.1 Basic assumption of the Drude model . 3 1.2 Hall effect and magnetoresistance . 11 1.2.1 Quantum Hall effect . 17 1.3 Optical properties . 19 1.4 Thermal properties . 25 1.4.1 Thermoelectric effect . 28 2 Drude-Sommerfeld model of metals 32 2.1 Basic assumptions of the Drude-Sommerfeld model . 33 2.2 Density of states . 39 2.3 Fermi-Dirac distribution . 42 2.4 Drude vs Drude-Sommerfeld model . 56 3 The structure of crystalline solids 59 3.1 Reciprocal lattice . 64 3.2 X-Ray diffraction . 66 4 Bloch model of solids 75 4.1 Band structure of solids . 84 5 Electronic band structure of simple systems 93 5.1 Solid system with very low potential . 93 5.2 Tight binding method . 94 5.2.1 Monodimensional monoatomic solid . 94 5.2.2 Bidimensional monoatomic solid . 99 5.2.3 Hexagonal lattice . 101 5.2.4 Monodimensional monoatomic solid: two orbitals for each atom . 104 5.2.5 Square monoatomic lattice: four orbitals for each atom . 109 5.2.6 Two different atoms in cell, one base function for each atom111 5.2.7 Biatomic linear chain . 113 5.3 Graphene . 116 5.4 Resume of the main features of the Tight Binding method . 128 2 CONTENTS 3 5.5 Getting the potential . 129 6 Semiclassical model of electrons in bands 132 6.1 Drude-Boltzmann conductivity . 144 A Group Theory 148 Chapter 1 Drude model of metals Metals are a very peculiar class of solid materials since they show unique prop- erties that many other classes do not behave such as high electrical and thermal conductivity, ductility, malleability and some peculiar surface properties (such as a very high surface tension). Metals are very important chemically wise since they occupy 2/3 of the periodic table of elements. Nevertheless, they can be described using some simple qualitative and quantitative models which are still used by physics and engineers nowadays 1.1 Basic assumption of the Drude model In metals, at least two particles of opposite charge have to be present since electrons, which are responsible for the transport properties of metals, are neg- atively charged. Drude imagined that the negative charges were compensated by much heavier positive charges which are considered to be immobile. He also thought that valence electrons of an atom in a metal are not bond to the atom anymore but are instead free of moving along all the metal, while the posi- tively charged ions are immobile. Core electrons remain bound to the metal ion while valence electrons are free to move. Valence electrons in a metal are called conduction electrons (figure 1.1). Drude applied his kinetic theory of the electron gas to the conduction elec- trons which are assumed to have a mass m which freely move in a background of heavy immobile positive ions. The force imposed by all of the positively charged ions on the conduction electrons is zero just for symmetry reasons, and in the end the effect of the presence of the positive charges is that of imposing the electrons to stay in the metal. If we consider the case of a monovalent metal such as Li with ions arranged at a certain distance a of 2-3 A˚2 in a lattice configuration, we can calculate the density of this electron gas considering the presence of just one ion per cell. − − 1 e 1 e 23 −3 − ρe− ' = = 10 cm · e (10 A)˚ 3 10 · (10−8) cm3 4 CHAPTER 1. DRUDE MODEL OF METALS 5 Figure 1.1: (a) Schematic picture of an isolated atom. (b) In a metal the nucleus and the ion core retain their configuration in the free atom, but the valence electrons leave the atom to form the electron gas. so 1023 electron per every cm3 of solid metal are present. Another important quantity for the description of a solid metal is the number of electrons per unit volume N · Zρ n = a m A where Z is the number of valence electrons of an element, ρm is the nominal density of the metal and A is the atomic mass of the element. the ratio ρm=A is the number of moles per unit volume. The average number of electrons per unit volume is much larger than the normal conditions for an electron gas, and so the classical theory of the electrons gas won't be a satisfying description of these systems anymore. The typical distance between electrons can be extimated as l ' n−1=3 since n−1 is the volume occupied by a single electron, and so l represents the typical distance between electrons in the gas composing our solid. The quantity typically used is the Wigner-Seitz radius being defined as the radius of a sphere whose volume is equal to the volume occupied by each conduction electron. Obviously, these spheres must not overlap. V 1 4 = = πr3 N n 3 s and so 3 1=3 r = ' 1 − 2 A˚ s 4πn Clearly, as rs grows the density of the electron gas decreases. CHAPTER 1. DRUDE MODEL OF METALS 6 Another fundamental quantity is the thermal wavelength λβ, which repre- sents the space occupied by an electron at a certain temperature. h λβ = p 2πmkBT By inserting the numerical value of the constants 6:626 · 10−34J · s λβ = p2π · 10−30Kg·1:38 · 10−23J · (T=K) This way it is possible to build a formula in which we can insert the temperature as a variable and obtain the thermal wavelength 6:6 · 10−8m 6:6 · 102A˚ λβ = = p(T=K) p(T=K) At 100K, the thermal wavelength is about 70-80A,˚ and considering that rs is about 1-2A˚ Boltzmann statistics is not valid aymore in these conditions since the electron-electrons distance is much smaller than in a regular electron gas. In order to be described by the Boltzmann statistics the distance between atoms should be much larger than the thermal wavelength. The gas has a much higher density and is therefore defined as a degenerate gas. Drude, anyway, was not aware of this issue and still described the the Boltz- mann statistics in his description of metals. 1. First, he assumed that between a collision and the next one the electron- electron interactions and the electron-ion interactions are negligible, and so in the absence of an external field electrons can move in a straight line in a uniform motion. In the presence of an external field Newton's laws are applied but ignoring the complex effect of the presence of other ions and electrons in the lattice. While the neglect of electron-electron inter- actions is a good approximation (independent electron approximation), the neglect of electron-ion interactions is not good at all and has been abandoned even for the description of qualitative models (free-electron approximation). 2. The collisions are instantaneous events which alter sharply the velocity of an electron. Drude imagined these collisions to happen as a matter of the presence of electrons coming out of the core shells of ions. Drude, in his model, completely ignored the origin of the scattering phenomena. If an electric field is turned on its effect is felt only during the scattering phenomena since after that the velocities of the electrons are completely randomized, and so you cannot appreciate the effect of the applied ex- ternal field. The scattering phenomena are almost instantaneous and has the effect of randomizing the velocities of the electrons coming out of the collision. The velocity of an electron after the collision is that given by CHAPTER 1. DRUDE MODEL OF METALS 7 the distribution probability of equilibrium velocities, which is the one reg- ulated by the Mawxwell-Boltzmann law, and since all the values of the velocities are possible their average is zero. The absolute value of the vmax depends on the temperature since at higher temperatures it is more probable that an electron will exit from the collision with a higher veloc- ity. Another fundamental quantity in this context is the time between each collision phenomena τ . Figure 1.2: Trajectory of a conduction electron scattering off the ions, according to the naive picture of Drude. The motion of the electrons in the solid material is no balistic because of the presence of the frequent collisions. The motion of the electrons in a solid material is called drift diffusion, where the word drift represent the component of the motion relative to the presence of an electric field and the diffusion component which is related to the randomization of the velocities. The diffusive motion is at the basis of the transport phenomena in solid materials. The effect of an electric field in a metal is that of accelerating the electrons between each collision, and the increase in velocity between each collision is −|ejEτ ∆v ' F · τ = me where F is the compeling force (in this case the electric field multiplied by the charge of an electron) and τ is the time between a collision and the next one. This a sort of Newton equation.
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