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Syllabus & Lecture Notes 1 I. SYLLABUS AND INTRODUCTION The course is taught on basis of lecture notes which are supplemented by the textbook Ashcroft and Mermin. Some practical information: Professor: Michel van Veenendaal • Office: Faraday West 223 • Tel: 815-753-0667 (NIU), 630-252-4533 (Argonne) • e-mail: [email protected] • web page: www.niu.edu/ veenendaal • ∼ Office hours: I am at NIU Tu/Th. Feel free to walk into my office at any time. You can always make an • appointment if you are worried that I might not be there. Official office hours will be established if you feel that the “open door” policy does not work. Prerequisites: there are no official prerequisites for the course. However, a knowledge of quantum mechanics at • the 560/561 level is recommended. Mathematical concepts that will be used are calculus, vector algebra, Fourier transforms, differential equations, linear algebra (in particular matrices and eigenvalues problems). Homework: several homework sets will be given. They will be posted on the web site. • Midterm: one midterm will be given. • Attendence: There is no required attendence. • Additional further reading • F. Wooten, Optical Properties of Solids (Academic Press, New York, 1972). CONTENTS Background: A sort background and history of solid-state physics is given. • The electronic structure: tight-binding method (1D). First, we study a diatomic molecule starting from hydrogen • wavefunctions. We create an understanding why two atoms prefer to from a molecule. The molecule is then made longer until an infinitely long one-dimensional molecule is formed. The eigenenergies of the chain are calculated analytically. As an additional example we consider the benzene molecule. The electronic structure: nearly free-electron model (1D). In this section, we start from the opposite limit and • consider free electrons moving in one dimension. A periodic potential representing the presence of nuclei is then added. Comparison of results for tight-binding and nearly-free electron model. The results of the two opposite limits • are compared and their connections are shon. Other ways of keeping atoms together. Different forms of crystal binding are discussed: covalent bonds, ionic • crystals, and van der Waals forces. Formalization: Bloch theorem. The Bloch theorem and its connection to the periodicity of the lattice is discussed. • Phonons in one dimension. In this section, nuclear vibrations are introduced. Collective nuclear motion leads • to phonons. We first consider a monoatomic linear chain. Periodicity and basis. A linear chain is introduced consisting of alternating atoms of a different kind. This is • known as a lattice plus a basis. Effect of a basis on the electronic structure. The effects of introducing two different types of atoms on the • electronic structure is demonstrated. It is shown that a change in the periodicity can change the conductive properties from metallic to semiconducting. 2 Effect of a basis on the phonon dispersions. For the phonon dispersion, the effect of introducing a basis is the • creation of optical phonon modes in addition to the acoustical modes. Crystal structures. After introducing several concepts in one dimension, we have a closer look at crystal struc- • tures in two and three dimension. We look at the symmetry of crystals and their effects on the material properties. Measuring crystal structure: Diffraction. This section describes how neutrons, electron, and, in particular, X- • rays are scattered from a crystal lattice. The focus on the conceptual understanding of diffraction. Bragg’s law is derived. The reciprocal lattice. The reciprocal lattice (the Fourier transform of the lattice in real space) is introduced. • The concepts of diffraction and Brillouin zone are formalized. Free electron in two and three dimensions. The free electron model is studies in two dimensions. Special • attention is paid to the Brillouin zones, the Fermi surface for different electron fillings, the density of states, Nearly-free electron in two dimensions. A periodic potential is introduced in the free-electron model in two • dimensions. The effects on the electronic structure and the Fermi surface are studied. Tight-binding in two dimensions. As in one dimension, the nearly free-electron model is compared to the • tight-binding model and its differences and, in particular, the similarities are discussed. The periodic table. We now look at more realistic systems and see what electronic levels are relevant for the • understanding of the properties of materials. The filling of the different atomic levels is discussed. Band structure of selected materials: simple metals and noble metals. We apply the concepts that we have • learned in one and two dimensions to the band structure of real materials. First, aluminium is studied, which is a good example of a nearly free-electron model. We then compare the noble metals, such as copper and gold. Here, the s and p like electrons behave as nearly-free electrons. However, the d electrons behave more like tightly bound electrons. Thermal properties. In this section, we study the thermal properties of solids. In particular, we discuss the • temperature dependence of the specific heat due to electrons and phonons. For phonons, we distinguish between acoustical and optical phonons. Optical spectroscopy. The optical properties of solids are discussed in a semi-classical model. Using the dielectric • properties of materials, it is explained why metals reflect optical light, whereas insulators do not. The different colors of aluminium, gold, and silver are discussed. Quantum-mechanical treatment of optical spectroscopy. The relationship between the semiclassical approach of • the optical properties of solids and their electronic structure is discussed. It is shown that the classical oscillators correspond to interband transitions. Relation to absorption. A relation is made between the dielectric function and absorption (Fermi’s Golden Rule) • as derived in quantum mechanics. Thomas-Fermi screening. Using the concepts of dielectric properties introduced in the previous section, it is • explained why many systems behave like metals, despite the presence of strong Coulomb interactions. The idea is known as screening. Many-particle wavefunctions. It is shown how to construct many-particle wavefunctions. Many-particle effects • become important when electrons and phonons can no longer be treated as independent particles due to electron- electron and electron-phonon interactions. Magnetism. Different types of magnetism are discussed. In diamagnetism, a local moment is created by the • magnetic field. For paramagnetism, local moments are already present due to the presence of electron-electron interactions. The effect of the magnetic field is the alignment of the local moments. Ferromagnetism. In ferromagnetism, the magnetic moments in a materials align parallel spontaneously below a • certain temperature. Antiferromagnetism. Antiferromagnetism is similar to ferromagnetism. However, the magnetic moments align • in the opposite direction. 3 Phonons. Phonons in three dimensions are formally derived. The phonons are quantized making them effective • particles. Electron-phonon interaction. The interactions between the electrons and the phonons is derived. • Attractive potential. It is shown how the interaction between the electrons and the phonons can lead to an • attractive potential between the electrons. Superconductivity and the BCS Hamiltonian. Using an effective Hamiltonian including the attractive interaction • between the electron, it is shown how a superconducting ground state can be made. This theory is known as BCS theory after Bardeen, Cooper, and Schrieffer. BCS ground state wavefunction and energy gap. The ground-state wavefunction and the energy gap of the BCS • model are described. Transition temperature. The temperature where a solid becomes superconducting is calculated and its relation • to the superconducting gap is derived. Ginzburg-Landau theory. This section describes a phenomenological model, known as Ginzburg-Landau theory, • to describe superconductivity. Flux quantization and the Josephson effect. It is shown how magnetic flux is quantized due to the supercon- • ducting current. This quantization leads to peculiar effects in the current across an insulating barrier. This is known as the Josephson effect. 4 II. BACKGROUND From Hoddeson, Braun, Teichamnn, Weart, Out of the crystal maze, (Oxford University Press, 1992). Obviously, people have been interested in the proporties of solids since the old ages. The ancient Greeks (and essentially all other cultures) classified the essential elements as earth, wind, air, and fire or in modern terms: solids, liquids, gases, and combustion (or chemical reactions in general). Whole periods have been classified by the ability to master certain solids: the stone age, the bronze age, the iron age. And even our information age is based for a very large part on our ability to manipulate silicon. Many attempts have been made to understand solids: from Greek philosophers via medieval alchemists to cartesian natural philosophers. Some macroscopic properties could be framed into classical mechanics. Optical conductivity in solids was worked out in the theories of Thomas Young and Jean Fresnel in the late eighteenth and early nineteenth century. Elastical phenomena in solids had a long history. Macroscopic theories for electrical conductivity were developed by, e.g. Georg Ohm and Ludwig Kirchhoff. Another good example of a mechanical model for a solid are the theories of heat conductivity
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