High-Temperature Optical Spectral Weight and Fermi-liquid Renormalization in Bi-Based Cuprate Superconductors

The optical conductivity $\sigma (\omega )$ and the spectral weight $W(T)$ of two superconducting cuprates at optimum doping, ${\mathrm{Bi}}_2{\mathrm{Sr}}_{2-x}{\mathrm{La}}_x{\mathrm{CuO}}_6$ and ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$, have been first measured up to 500 K. Above 300 K, $W(T)$ deviates from the usual $T^2$ behavior in both compounds, even though $\sigma (\omega \to 0)$ remains larger than the Ioffe-Regel limit. The deviation is surprisingly well described by the $T^4$ term of the Sommerfeld expansion, but its coefficients are enhanced by strong correlation, as shown by the good agreement with dynamical mean field calculations.

Figure 1

(a) Normal-state $ab$-plane conductivity of ${\mathrm{Bi}}_2{\mathrm{Sr}}_{1.6}{\mathrm{La}}_{0.4}{\mathrm{CuO}}_6$ and (in the inset) resistivity of both samples. (b) Normal-state $ab$-plane conductivity of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$ and (in the inset) reflectivity at the two extreme temperatures of both samples, with arrows marking their screened plasma frequencies ${\omega }_p$. In both main panels, the diamonds at $\omega =0$ indicate the ${\sigma }_{\mathrm{dc}}$ of the sample at the same $T$ (at low $T$ both of them are out of the scale). The horizontal arrows mark the Ioffe-Regel limits to ${\sigma }_{\mathrm{dc}}$ (see text).

Figure 2

Temperature-dependent optical spectral weight $W({\omega }_p,T)$ of optimally doped (a) Bi-2201 and (b) Bi-2212, normalized to the (extrapolated) value at $T=0$. The I-R data (red circles) are compared with DMFT results for the restricted sum rule (blue diamonds) of the single-band Hubbard model. Also shown are theoretical calculations for the noninteracting system ($U=0$) and the lowest-order Sommerfeld expansion, where the coefficient $B$ is simply rescaled by the QP DMFT weight ($Z$ scaled). In panel (a) DMFT results for the total sum rule are displayed for comparison (green squares). In the inset the dotted (dashed) line indicates the fit performed on $W({\omega }_p,T)$ data using Eq. (2) up to the second (fourth) order.

Figure 3

Normalized coefficients $b(\Omega )=B/W_0$, $c(\Omega )=C/W_0$, and $c(\Omega )/b(\Omega )$ for ${\mathrm{Bi}}_2{\mathrm{Sr}}_{1.6}{\mathrm{La}}_{0.4}{\mathrm{CuO}}_6$ and ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$, as obtained from the fits to $W(T,\Omega )$. The lines are guides to the eye.