Numerical and experimental studies of
\nhigh-temperature superconducting systems

The objective of this thesis is to study the correlated behaviour of electron and lattice \ndynamics in the high-temperature cuprate superconductors. In particular, we \nare interested in the lattice (i.e., structural and phonon) and electronic dynamics in \nthese systems and how they may influence the driving mechanisms of high-Tc superconductivity. \nWe do this through two methods: numerical calculations on the \ntwo-dimensional Hubbard model and inelastic light scattering experiments. \nFor the numerical component of this work, we compute the static and dynamical \nlongitudinal charge and spin susceptibilities as functions of real frequency on a twodimensional \nsquare lattice with weak-coupling for the t − t' − U Hubbard model via \na perturbative, diagrammatic approach. In the static case, the spin susceptibility \nis dominated by q = (π, π) spin fluctuations that are largest near half-filling, while \nthe charge susceptibility has a clear multi-peak structure with a minimum located \nat the van Hove singularity of the non-interacting dispersion, a feature not observed \nin the commonly employed random-phase approximation. For dynamical charge and \nspin susceptibilities, we plot the dispersion for each respective collective electronic \nexcitation on the real frequency axis. These excitations were found to split into \nseparate modes surrounding the van Hove singularity, only to merge again in the longwavelength \nlimit. From an experimental standpoint, these results will prove useful \nfor those studying electronic excitations as our calculations indicate that electron \ndoped cuprates should be probed at higher energies in order to detect spin and charge \nexcitations in comparison to the hole doped cuprates. \nFor the experimental aspect of this work, we study the single-crystal Bi₂Sr₂CaCu₂O₈₊δ \n(Bi-2212) cuprate by Brillouin light scattering spectroscopy at room-temperature. We \nstudy the acoustic phonon behaviour in the normal state and gain further information \non the structural dynamics which currently remain unresolved. From spectra \nii \ncollected with an excitation source of 532 nm, we identify the Rayleigh surface mode, \na longitudinal resonance mode, and six bulk acoustic modes (four quasi-transverse and \ntwo quasi-longitudinal), contrary to the typically expected three. From bulk spectral \npeaks, the extracted frequency shift and linewidth were used to determine the optical \nextinction coefficient-to-refractive index ratios. These ratios were subsequently used \nto obtain the extinction coefficient, penetration depth, optical absorption coefficient, \nand complex dielectric function, and were found to be 5 − 7 times larger than those \nobtained from optical interference measurements. \nThe number of bulk modes observed in the Brillouin data suggests Bi-2212 is an incommensurate \ncomposite crystal consisting of two weakly interacting sublattices. Further \nanalysis of the Brillouin data allows us to assign the two independent sublattices as \nBi₂Sr₂O₄ and CaCu₂O₄, calculate the hypersound velocity of the observed surface and \nbulk acoustic phonon modes, and derive an expression relating the bulk crystals longitudinal \nacoustic phonon velocity to the bulk longitudinal acoustic phonon velocities \ncorresponding to the two sublattices. Finally, we determined the elastic constants \nC₁₁, C₂₂, C₃₃, C₄₄, C₅₅, C₁₂, and C₂₃ and related sublattice elastic constants. \nOverall, the utility of the experimental work presented in this thesis is multi-fold. \nFirstly, it has refined the acoustic phonon velocities, and subsequently the elastic \nconstants, of Bi-2212 which have received little attention. Secondly, our derived expression \nrelating the bulk and sublattice longitudinal velocities will aid in expanding \nupon the model of incommensurate composite crystals and the study incommensurate \ncrystallography in general. Lastly, and most importantly, our experimental verification \nthat Bi-2212 is comprised of two interacting sublattices alludes to a highly \nnon-trivial optic phonon structure and the existence of low-lying optic phonon modes. \nIdentification of such modes may play an important role in understanding the driving \nmechanisms surrounding high-Tc superconductivity.

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