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12 J. Roth - Solid State Theory0.3 OverviewLet us now give a brief overview on what we will discuss in this lecture.The starting point, Hamiltonian (1), and the Born-Oppenheimer approximationare very general. We can use it to describe gases, liquids, or solids, as we do in thislecture. Moreover, we will only consider solids in the crystalline form, where theyare perfectly ordered in a periodic fashion. Actually, almost all materials becomesolid crystals at zero temperature. The notable exception is 4 He which, due tostrong quantum fluctuations of the Bose-Einstein condensated 4 He bosons, remainsa liquid. As we will see in this lecture, periodicity is a symmetry which, if cleverlyexploited, is of tremendous help. Hence, in the following chapter• Crystal structure.we will study this periodicity and the corresponding symmetry. We can also envisagethat we deal in this Section with the Hamiltonian (14) for the nuclei, wherewe neglect the movement of the nuclei, i.e., determine the equilibrium positions.However, since the determination of E0 el is such a formidable task, which we wouldneed to carry out for many nuclei positions, we will focus on the general symmetries.In the next chapter• Lattice vibrations.we will then include the motion of the nuclei around their equilibrium position.This way, we will discover elementary excitations of a solid: phonons. We will callelementary excitations, those modes of the system that at low enough energies (andhence, temperatures), can be considered as eigenmodes of the solid. These modeswill be real eigenmodes (i.e. modes with an infinite lifetime) only within certainapproximations (here disregarding the electron-phonon coupling). However, in thisway we will be able to understand the experiments and look for corrections byconsidering interactions among the elementary excitations, that in general will leadto a finite lifetime, as also seen in experiments. We will be more precise aboutthis in the coming chapters. Here we will have also the first encounter with secondquantization, much in the same way as in elementary quantum mechanics, whendiscussing the harmonic oscillator (creation and annihilation operators).Next we will focus on the electronic degrees of freedom described by Hamiltonian(6) within the Born-Oppenheimer approximation. First, we will consider theelectrons as non-interacting, and discuss• Electrons in a periodic potential.Since we know that electrons are charged, this may appear at a first sight as agross simplification. However, at the beginning of quantum mechanics, Sommerfeldconsidered electrons as non interacting particles and took only into account thefact that they obey the Fermi-Dirac statistics. In such a way he could explain thetemperature dependence of the specific heat in metals as well as their magnetic
J. Roth - Solid State Theory 13susceptibility. We will come to understand this apparent contradiction only later,when we discuss the effects of interaction. For the moment we will introduce theconcept of a band-structure, that is very useful to understand simple metals andsemiconductors.In order to complete the picture developed, we will come then, to consider• Interacting electrons.Here we will go deeper into the concepts of second quantization and develop a theoreticalframe that should allow us to deal in principle with fully interacting electrons.Unfortunately this goal was still not reached and different approaches are the subjectof current research. Nevertheless, the theoretical language developed here shouldbe useful to make contact with undergoing research work. On the other hand, itwill be useful to understand the solution of the contradiction mentioned above. Wewill see that in this case, the elementary excitation is the so-called quasiparticle,namely a fermionic excitation with spin 1/2 and charge −e that interacts weaklywith the other ones. This concept, introduced by Landau, explains the behavior ofmost metals. There are however, exceptions like one-dimensional systems or hightemperature superconductors, but they are the subject of more advanced lecturesand current research.The subjects above constitute the first half of these lectures. Students from theDiplomstudiengang may decide here, whether they continue, and make a full lecture,or they stop, and can combine this half-lecture with another elective one.The second half is devoted to different subjects that are important in the physicalunderstanding of solids. The first one is• Collective excitations.By them we denote those elementary excitations, that arise from the cooperativearrangement of electrons in long-lived modes like plasmons and excitons. We willdiscuss these in the general frame developed before as well as with simple picturesthat are helpful to understand their experimental manifestation.One of the most dramatic manifestations of quantum mechanics is the realizationof a state of matter, where transport of electric current takes place without losses,that is• Superconductivity.We will review here the phenomena, concepts, and theory of conventional superconductivity,a knowledge that is indispensable if we would like to understand a newform of superconductivity that is present in the so-called high temperature superconductors,where a different, new state of matter is suspected due to the evidencegiven by an increasing number of experiments. These new materials that were discoveredin 1986 by J. Bednorz and K.A. Müller, who received the Nobel Prize in
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Solid State Theory - Institut fÃ¼r Theoretische und Angewandte Physik ...

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