Frequency- and temperature-dependent conductivity in ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\mathit{x}}$ crystals

The results of a systematic study of the optical properties of the ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\mathit{x}}$-based insulators and superconductors are reported. Specifically, we present measurements and analysis of the optical reflectivity R of a series of ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\mathit{x}}$ crystals in the frequency range from 30 to 20 000 ${\mathrm{cm}}^{\mathrm{-}1}$ (4 meV to 2.5 eV), and temperature range from 10 to 270 K. From R we obtain the real part of the frequency-dependent optical conductivity σ(ω) by Kramers-Kronig analysis. In our discussion, we emphasize the development of structure and spectral weight in σ(ω) as the compounds change from insulators to high-${\mathit{T}}_{\mathit{c}}$ superconductors with varying O content or Al doping. We identify the free carrier, and interband components of σ(ω), and focus on the free-carrier component.

The free-carrier component is analyzed by calculating σ(ω,T) within a model in which carriers scatter from a spectrum of dispersionless oscillators parametrized by ${\mathrm{\alpha }}^2$F(ω) (where α is the coupling constant and F(ω) is the density of modes). For ω<50 meV, σ(ω,T) is well described by weak coupling (λ≃0.4) to an F(ω) which is broad on the scale of ${\mathit{k}}_{\mathit{B}}$T. From the fit we obtain the inelastic scattering rate as a function of T, and the spectral weight in the translational, or Drude mode, of the quasiparticles. Above 50 meV, σ cannot be fit by this scattering model, with any ${\mathrm{\alpha }}^2$F(ω), which suggests a two-component picture of σ(ω,T). As T is lowered, a ‘‘knee’’ in R(ω), and a threshold in the corresponding σ(ω), is resolved, which we associate with the low-frequency edge of this second component. In addition, a second threshold in the range 15–20 meV is seen at low T, although the magnitude of the change in R is close to our detection limit of 1%. We compare the properties of these thresholds with expectations for a superconducting energy gap as described in BCS theory. Finally, we discuss the implications of other experiments, which also probe the spectrum of low-energy excitations in the cuprate superconductors, on the interpretation of σ(ω,T).