Optical Properties of Strongly-Coupled d-wave Superconductors with an Anisotropic Momentum Dependent Interaction

Abstract

<p>We have studied within the Eliashberg framework properties of a two-dimensional d-wave superconductor where the electron-boson interaction leading to pairing in the superconducting state is a highly anisotropic function of momentum. Adding to the momentum anisotropy is an underlying single particle band structure based on a two-dimensional tight-binding model. The convolution of anisotropic interaction and anisotropic energy dispersion leads to a self energy which is strongly k-dependent in the Brillouin zone. Additionally, the boson spectral weight was assumed to extend over a large energy range with significant weight at both high and low frequencies. The study begins with numerical calculations of the self-consistent electronic self energy. Renormalization effects are shown to play an important role in the quasiparticle properties. The effects of strong inelastic scattering on the electronic spectral density in both the normal and superconducting state are investigated. Numerical calculations of the in-plane optical conductivity and electronic Raman scattering cross-sections are made for both the normal and superconducting states. Using a simple model to extract the effective quasiparticle scattering rates and mass renormalization factors, comparison of the optical results are made to the fundamental quasiparticle properties. The effects of momentum space anisotropy and strong electron-boson coupling on the optical scattering rates are discussed. The low frequency behaviour of the various Raman scattering channels in the presence of strong inelastic scattering is investigated. It is shown that both inelastic and elastic scattering can lead to a crossover from ω³ to ω dependence of the low frequency Β₁g spectra in the superconducting state, while the B₂g response remains linear in frequency. An effective boson spectral density, g²x"(ω), is extracted from the the calculated conductivity using a simple model. It is shown that the optical conductivity data may be used to extract the signature of the electron-boson coupling. The effective spectral density is then used in an isotropic model to determine whether or not such a function extracted from the conductivity would capture all the details of the momentum dependent interaction. The effects of resonant impurity scattering on the low frequency conductivity and Raman spectra are also investigated. Numerical calculations confirm the existence of universal values for the zero frequency conductivity and zero frequency slope of the B₂g Raman response in the superconducting state. No such limit is observed in the Raman B₁g spectrum.</p>