666 (1) - 1 I SYLLABUS AND INTRODUCTION The course is taught on basis of lecture notes which are supplemented by the textbook Ashcroft and Mermin Some

666 (1) - 1 I SYLLABUS AND INTRODUCTION The course is...

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Unformatted text preview: 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: ∼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 structures 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, Xrays 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 electronelectron 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 superconducting 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 developed in the early nineteenth century by Joseph Fourier. Franz Ernst Neumann developed a mechanical theory for the interaction between elasticity and optics, allowing him to understand anisotropy in materials such as the birefringence that occurs when an isotropic medium is subjected to pressure or uneven heating. Another important ingredient to understanding solids is symmetry. Already in 1690, it was suggested by Huygens that the regular form of solids in intricately related to their physics. The first scientific theory of crystal symmetry was set up by Ren´e-Just Ha¨ uy in 1801-2, based on atomistic principles. This was extended upon by mineralogist Christian Samuel Weiss, who, however, completely rejected the atomistic approach. He introduced the concepts of crystallographic axis. The crowning achievement is the classification by Auguste Bravais that the are only 14 ways to order a set of points so that the neighborhood of any individual lattice point looks the same. In 1901, Woldemar Voight published his book on crystallography classifying the 230 different space groups (and was shocked by the large number). Another important nineteenth century development is the reemergence of the atomistic theories. The success of continuum theories (classical mechanics, electricity and magnetism, thermodynamics) had pushed atomistic thought to the background, in particular in physics (remember the trials and tribulation Ludwig Boltzmann went through in the acceptance of his statistical approach to thermodynamics which was often considered a nice thought experiment, but hopelessly complicated). However, chemical reactions clearly indicated the discrete nature of matter. A microscopic theory proved elusive until the advent of quantum mechanics in the early twentieth century. Quantum mechanics is crucial in understanding condensed matter. For example, gases can be reasonably well described in a classical framework due to their low density. This means that we do not require quantum (Fermi-Dirac) statistics and we do not have to deal with atomic levels. Theories started accelerating after the discovery of the electron by J. J. Thompson and Hendrik Lorentz (and mind you, Lorentz and Zeeman determined the e/m ratio before Thompson). The classical theory of solids essentially treats a solid as a gas of electrons in the same fashion as Boltzmann’s theory of gases. This model was developed by Paul Drude (who, inexplicably, committed suicide at each 42, only a few days after his inauguration as professor at the University of Berlin). The theory was elaborated by Hendrik Lorentz. Although a useful attempt, the use of Maxwell-Boltzmann statistics as opposed to Fermi-Dirac statistics leads to gross errors with the thermal conductivity κ and the conducivity σ off by several orders of magnitude. The theory gets a lucky break with the Wiedemann-Franz law that the ratio κ/σ is proportional to the temperature T . (It is not quite clear to me why AM wants to start out with an essentially incorrect theory, apart from historic reasons. . . Then again, we also do not teach the Ptolemean system anymore). However, it was very important in that it was one of the first microscopic theories of a solid. Early attempts to apply quantum mechanics to solids focused on specific heat as measured by Nernst. Specific heat was studied theoretically by Einstein and Debye. The difference in their approaches lies in their assumption on the momentum dependence of the oscillators. Einstein assumed one particular oscillator frequency representing optical modes. Debye assume sound waves where the energy is proportional to the momentum. Phonon dispersion curves were calculated in 1912 by Max Born and Theodore von K´ arm´ an. However, again, this can again be done from Newton’s equations. However, further progess had to weight the development of quantum statistics which were not developed untill 1926. In the meantime, the number of experimental techniques to study condensed matter was increasing. In 1908, Kamerlingh-Onnes and his low-temperature laboratory managed to produce liquid Helium. This led to the discovery of superconductivity in mercury (and in some ways the onset of “big” physics). After discussions with Peter Paul Ewald in 1912, who was working on his dissertation with Sommerfeld on optical birefringence, Max von Laue successfully measured the diffraction of X-rays from a solid. The experiments were interpreted by William Henry Bragg and his son William Lawrence opening up the way for x-ray crystal analysis. In 1926-7, Enrico Fermi and Paul Dirac established independently a statistical theory that included Wolfgang 5 Pauli’s principle that quantum states can be occupied by only one electron. This property is inherently related to the antisymmetry of the wavefunctions for fermions. In 1927, Arnold Sommerfeld attempted a first theory of solids using Fermi-Dirac statistics. It had some remarkable improvements over the Drude-Lorentz model. The specific heat was significantly lower due to the fact that not all electrons participate in the conductive properties, but only the ones with the highest energies (i.e. the ones close to the Fermi level). However, the model still completely ignores the effect of the nuclei leading to significant discrepancies with various experiments including resistivity. Werner Heisenberg also had an interest in metals and proposed the problem to his doctoral student Felix Bloch. The first question that he wanted Bloch to address was how to deal with the ion in metals? Bloch assumed that the potential of the nuclei was more important that the kinetic energy. A tight-binding model as opposed to Sommerfeld’s free-electron model and ended up with the theorem that the wavefunction can be written as ψ = eik·r u(r), plane wave modulated by a function that has the periodicity of the lattice. Another important development Rudolf Peierls and L´eon Brillouin was the effect of the periodic potential in the opening of gaps at particular values of the momentum k, now known as Brillouin zones. III. THE ELECTRONIC STRUCTURE: TIGHT-BINDING METHOD (1D) AM Chapter 10 A solid has many properties, such as crystal structure, optical properties (i.e. whether a crystal is transparent or opaque), magnetic properties (magnetic or antiferromagnetic), conductive properties (metallic, insulating), etc. Apart from the first, these properties are related to what is known as the electronic structure of a solid. The electronic structure essentially refers to the behavior of the electrons in the solid. Whereas the nuclei of the atoms that form the solid are fixed, the electronic properties vary wildly for different systems. Unfortunately, the electronic structure is complicated and needs to be often approximated using schemes that sometimes work and at other times fail completely. The behavior of the electrons is, as we know from quantum mechanics, described by a Hamiltonian H. In general, this Hamiltonian contains the kinetic energy of the electron, the potential energy of the nuclei, and the Coulomb interactions between the electrons. Especially, the last one is very hard to deal with. Compare this with a running competition. For the 100 m, the runners have their own lane and the interaction between the runners is minimal. One might consider each runner running for him/herself (obviously, this is not quite correct). However, for larger distances, the runners leave their lanes and start running in a group. We can no longer consider the runners as running independently, since they have to avoid each other, they might bump into each other, etc. This is a much more complicated problem. The same is true for solids. In some solids, we can treat the electrons as being more or less independent. In some solids, we have to consider the interactions between the electrons explicitly. For the moment, we will neglect the interactions between the electrons, even though for some systems, they can turn the conductive properties from metallic to insulating. Systems with strong electron-electron interactions are an important (and still largely unsolved) topic in current condensed-matter physics research. The enormous advantage is that we do not have a problem with N particles (where N is of the order of 1023 ), but N times a one-particle problem. Of course, this is what you were used to so far. You probably remember the free-electron model (to which we will return later on). First, the bands were obtained (pretty easy since εk = ~2 k 2 /2m), and in the end you just filled these eigenstates up to the Fermi level. So far, you might not have realized that this was actually an approximation. This is the type of system we will be considering at the moment. So we are left with a Hamiltonian containing the kinetic energy of the electron, the potential to the nuclei, and some average potential due to the other electrons in the solid. This is essentially an extension on the quantum-mechanical model of particles in a box with the periodic potential of the nuclei added later. We will come back to that in a while, but we start with a model that puts in the atoms explicitly (known as the tight-binding model). This allows us to start thinking about the crystal immediately, instead of putting it in as some afterthought. In general, we can write the Hamiltonian for the electron in a crystal as H = Hel = X p2 e2 1 X Zj e2 1 1 X i + + . 2m 4πε0 ij |ri − Rj | 2 4πε0 |ri − ri′ | ′ i (1) i6=i This leaves out the kinetic energy of the nuclei and their interaction. We will come back to that later. Our first assumption is that the nuclei are fixed charges in space. The electrons move so fast that that the do not notice the motion of the nuclei. This is not true and the interaction between electrons and the motion of the nuclei can lead to exciting phenomena such as superconductivity, but it is a good starting point. The first term in the equation is the kinetic energy of the electrons. The second term is the interaction between the electrons and the nuclei, where i sums over all the electrons in the solid and j sums over all the nuclei with charge Zj e. The last term is the Coulomb interaction between the electrons. The factor 1/2 avoid double counting. Including this term makes it impossible to 6 solve the problem, except for very small systems. We nee...
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