Raman study of the anharmonicity in YBa2_2Cu3_3Ox_x

A systematic Raman study in the visible carried out on the YBa2Cu316,18Ox (x=6-7) compounds, with isotopic substitution of 18O for 16O, has detected a doping dependent deviation from harmonic behavior for the frequency shift of the in-phase mode, a smaller amount of anharmonicity for the apex mode, and almost no effect for the out-of-phase B1g-symmetry phonon. It appears that the amount of anharmonicity depends strongly on the oxygen concentration; it diminishes close to the tetragonal to orthorhombic structural phase transition and close to optimal doping, while it reaches its maximum value for the ortho-II and a tetragonal phase. The almost zero anharmonicity at optimal doping persists even at 77K. The data in the overdoped oxygen concentration, where a softening of the in-phase phonon frequency occurs, indicate that the anharmonicity is not enhanced by the sudden increase in the CuO2 buckling. The results fully agree with recent studies of the ortho-II phase but they do not comply with a static double-well potential of the apical oxygen atom at optimal doping.Comment: Dedicated to Prof. K. A. M\"uller on the Occasion of his 90th Birthda

The Hubbard model within the equations of motion approach

The Hubbard model has a special role in Condensed Matter Theory as it is considered as the simplest Hamiltonian model one can write in order to describe anomalous physical properties of some class of real materials. Unfortunately, this model is not exactly solved except for some limits and therefore one should resort to analytical methods, like the Equations of Motion Approach, or to numerical techniques in order to attain a description of its relevant features in the whole range of physical parameters (interaction, filling and temperature). In this manuscript, the Composite Operator Method, which exploits the above mentioned analytical technique, is presented and systematically applied in order to get information about the behavior of all relevant properties of the model (local, thermodynamic, single- and two- particle ones) in comparison with many other analytical techniques, the above cited known limits and numerical simulations. Within this approach, the Hubbard model is shown to be also capable to describe some anomalous behaviors of the cuprate superconductors.Comment: 232 pages, more than 300 figures, more than 500 reference