EFFECTS OF DEFECTS AND ANHARMONICITIES ON DYNAMICAL PROPERTIES OF HIGH TC SUPERCONDUCTORS

Abstract

The historical landmark discovery of superconductivity by H. K. Onnes (1911), now has developed a rich heritage in experimental as well as in theoretical community of condensed matter physics and material science. The journey of superconductivity has received several milestones during this long period and can be classified into two sections, namely; Conventional superconductivity (1911-1985) and High TC superconductivity (1986 and onwards). The conventional era of superconductivity includes various remarkable developments; namely; Meissner effect by Meissner and Ochsenfeld (1933); Two fluid model of Gorter and Casimir (1933); London Theory (1934); phenomenological ‘macroscopic theory’ of superconductivity of Ginzburg and Landau (1950); Isotope effect by Maxwell and Reynolds et al. (1950); Investigation of type-I and type-II superconductors and Abrikosov (1957) Vortices ; The microscopic BCS theory of superconductivity by Bardeen, Cooper, and Schrieffer (1957); correspondence between GL theory and BCS theory by Gor’kov (1959); Josephson effect by Josephson (1962) and wide technological applications of superconductors, e.g., magnetic levitation, magnetic resonance imaging (MRI), nuclear magnetic resonance (NMR), and superconducting transformers, etc. The BCS theory appeared as a most successful theory which explained all essential aspects of the superconductivity phenomenon and heart of theory resides in electron-phonon interaction. i ii The era of High TC superconductivity commences with the breakthrough discovery of superconductivity in Ba-La-Cu-O first high-TC superconductor (cuprate superconductor) by Bednorz and M¨uller (1986). The most striking feature, common to all cuprate superconductors is the layered crystal structure and dimensionality plays an decisive role, which drawn the attention of experimental as well as of theoretical physicists to study the high temperature superconducting systems. The basic building block of cuprate superconductors is the one or more CuO2 layers. The critical temperature and superconductivity in these systems depend upon the doping concentration and presence of number of CuO2 layers. The Y Ba2Cu3O7−δ is crowned as representative of cuprate family with unique dynamical properties of high temperature superconductors like thermal-, acoustical-, optical-, and electronic properties. All derivatives of Y Ba2Cu3O7−δ superconductor have defects and vibrations of apical oxygen along c-direction show influenced effects of defects and anharmonicity. The long pending anharmonic electron-phonon problem in high TC superconductors attracted the physicist with the verity that anharmonicity is responsible for many different dynamical properties of the solids. Several proposed mechanisms of high TC superconductivity considered that it is the phonon that helps to join the two electrons into superconducting pairs (cooper pairs, bipolarons, and pairons). Various remarkable results show the importance of the electron-phonon interactions in high TC superconductivity and it is always suspected that electron-phonon interaction directly or indirectly is responsible for superconductivity. The exact treatment of lattice anharmonicity is complicated from the theoretical point of view because an anharmonic perturbation is never small the phonon wave functions are highly influenced by the anharmonic potential. The inclusion of defects and disorder in a crystal are well-known which drastically modify the energy scenario of the crystal and may elucidate the physical properties of these systems in wider perspective. Further, defects play an inevitable role in the understanding of physics of high temperature superconductivity and their effects in these systems are never small. The presence of both defects and anharmonicities in a crystal give rise to the ‘localilized’, ‘anharmonic’ iii and ‘localized-anharmonicity modes’. Many features of the real crystal can only be explained by considering the anharmonic terms in the expansion of the potential energy, that give rise to the coupling of normal modes in terms of phonon-phonon interactions and this makes the study of thermodynamic properties essentially as a many body problem. The study of lattice vibrations plays a decisive role in characterizing the mechanism and various dynamical properties of high temperature superconductors which is not studied to a satisfactory extent. The density of states govern many physical properties of solids and plays central role in the investigation of the lattice energy, heat capacity of a crystal, which are the most important properties of crystal and to investigate the signature of the electron-phonon interaction in high temperature superconductors. The theoretical investigations taking the effects of impurity and anharmonicity on thermodynamical properties of high temperature superconducting system are important in understanding of underlying physics of high temperature superconductors. In that respect the investigation of the heat capacity can provide valuable information about lattice vibrations, electron density of states near the Fermi level, energy gaps, and low temperature electronic characteristics. In the present work, we have taken up the impurity-induced anharmonic electronphonon problem in high temperature superconductors with a different concept considering the effects of defects, anharmonicities, electrons and interactions thereof in the crystal and presents an elegant theoretical approach to understanding underlying physics of phenomenon of high temperature superconductivity. Therefore, it is imperative to study the effects of defects and anharmonicities on dynamical properties of high temperature superconductors. The organization of this Thesis work is as follows: Chapter I : In this chapter, we start by introducing the historical developments of superconductivity and brief survey of recent developments in the field of superconductivity. After covering the introduction of superconductivity phenomenon, motivation for present research work with the state of art of the problem and the aspects iv of the theory that are essential for the understanding of this thesis are outlined. A brief introduction of the state of art (methodology) is also presented pertaining to the theories of many-body physics. Chapter II : This chapter deals with the description of different kind of potentials and their role in the understanding of the physical properties of the crystal. The concept of choosing best suitable potential is investigated here in the new frame work. The lattice dynamics of Y Ba2Cu3O7−δ HTS has been investigated in the frame work of harmonic approximation using the Born-Mayer potential. Chapter III : This chapter is devoted to the detailed description of the state of art that we have used in the thesis. The novel electron-phonon problem of the many body theory have been theoretically dealt with the help of the method of double time temperature dependent retarded Green’s function using the quantum dynamic approach of electrons and phonons. In order to investigate the quantum dynamics of electron and phonon, we have taken almost complete Hamiltonian (without considering BCS type) that consists the contributions of the (i) unperturbed electron (ii) unperturbed phonon (iii) electron-phonon interaction term (iv) anharmonicity, and (v) defects. It is noteworthy that the main features of present work which makes different from earlier theories include non BCS Hamiltonian which automatically predicts the appearance of pairons. Adopting the equation of motion method of Zubarev, the expressions of frequency line width and frequency shift for phonon and electron have been derived. Further, using these expressions in Lehmann’s representation, the expressions of phonon density of states and electron density of states have been developed in a new frame work (without considering BCS Hamiltonian). The impurity induced anharmonic electron-phonon problem is taken up as state of art and the ab-initio formulation of phonon density of states and electron density of states via approximation free approach. The Phonon density of states can be separated into diagonal and non-diagonal contributions. The Phonon density of states have been analyzed for the model crystal of Y Ba2Cu3O7−δ superconductor for the various contributions like diagonal, non-diagonal and total contribution for phonon v density of states. The electron density of states for defect, cubic anharmonicity and electron-phonon contributions have been investigated for Y Ba2Cu3O7−δ cuprate superconductor. Chapter IV : This chapter is devoted to the one of the most important properties of the crystal the phonon heat capacity along with the lattice energy using the density of states approach taking the effects of defects, anharmonicities and electron phonon interaction. The expressions for phonon heat capacity have been derived, which can be divided in two contributions diagonal and non-diagonal contributions. The non-diagonal contribution of phonon heat capacity mainly depends on the mass change parameters. This contribution is significant only in impure crystals and vanishes for pure crystals. Chapter V : This chapter aims to investigate the electronic heat capacity with the help of the change in total energy of the system. The present theory of electronic heat capacity reveal that it is not a simple quantity but depends on a large number of factors, namely; (i) defects, (ii) cubic and quartic anharmonicity and (iii) electronphonon interactions which on a very careful inspection exhibit that the electronic heat capacity depends on defect concentration, temperature, electron-phonon and anharmonic coupling constants, electron and phonon energies. Chapter VI : Following our quest for superconductivity, a succinct summary of this thesis work is presented in this chapter along with the outlook of this work, highlighting the most important conclusions and show some possible paths for stimulating future aspects.