In-plane optical conductivity of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$: Reduced superconducting condensate and residual Drude-like response

Temperature dependencies of the optical spectra of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ with $x=0.12$ and 0.15 were carefully examined for a polarization parallel to the $\mathrm{Cu}{\mathrm{O}}_2$-plane over a wide frequency range down to $8{\mathrm{cm}}^{-1}$. Selection of well-characterized crystals enabled us to measure purely in-plane polarized spectra without any additional peak. The weight of superconducting (SC) condensate estimated from the missing area in ${\sigma }_1\left(\omega \right)$ well agrees with the estimate from the slope of ${\sigma }_2\left(\omega \right)$ vs $1∕\omega$ plot, showing no evidence that the Ferrell-Glover-Tinkham sum-rule is violated in the optical spectrum. We demonstrate that the optically estimated SC condensate is much smaller than the value obtained from the $\mu \mathrm{SR}$ measurement of magnetic penetration depth. We also find an anomalous increase of conductivity in submillimeter region towards $\omega =0$ below $T_c$, which suggests the microscopic inhomogeneity in the superconducting state. Both observations are discussed in relation with the inhomogeneous electronic state that might be inherent to high-$T_c$ cuprates.

Figure 1

Comparison of the in-plane reflectivity spectra of two ${\mathrm{La}}_{1.85}{\mathrm{Sr}}_{0.15}\mathrm{Cu}{\mathrm{O}}_4$ crystals at $300\mathrm{K}$. Both of LSC#1 and LSC#2 were measured on the $ac$-faces with polarization $E\Vert a$. The spectra of LSC#2 shows a substantial contribution from the $c$-axis spectrum. The $c$-axis spectrum of LSCO is indicated by a gray curve as a reference. We can reproduce a similar profile to LSC#2, using the pure in-plane spectrum (LSC#1) and the $c$-axis spectrum, as described in the text. The calculated reflectivity assuming a 15° rotation around the $ac\text{−}∕bc$-plane or equivalently the 6.7%-admixture of the $c$-axis component is indicated by a short-dashed curve.

Figure 2

(a) Comparison of the in-plane conductivity ${\sigma }_1\left(\omega \right)$ of ${\mathrm{La}}_{1.85}{\mathrm{Sr}}_{0.15}\mathrm{Cu}{\mathrm{O}}_4$ (LSC#1) and the $c$-axis component mixed data for $\theta =15°$. (b) Comparison of the scattering rate $1∕\tau \left(\omega \right)$ for LSC#1 and the $c$-axis component admixture case. It is clear that the double peak is a result of the $c$-component admixture. It is also demonstrated that the kink at $\sim 700{\mathrm{cm}}^{-1}$ even at $300\mathrm{K}$ and is independent of the superconducting response below $200{\mathrm{cm}}^{-1}$.

Figure 3

(Color online) Temperature dependence of in-plane reflectivity spectra of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ with $x=0.15$ and $x=0.12$ with $E\Vert a$ or $b$. The inset of the upper panel shows the spectra for $x=0.15$ in submillimeter (SMM) and far-infrared (FIR) regions separately. The inset of the lower panel shows the room temperature spectra in a wider range of frequency.

Figure 4

(Color online) Temperature dependence of in-plane conductivity spectra of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ with $x=0.15$ and $x=0.12$ with $E\Vert a$ or $b$. Inset figures show the low-$\omega$ spectra and the dc conductivity. The optical data can be smoothly extrapolated to the dc values.

Figure 5

(a) Optical conductivity of ${\mathrm{La}}_{1.85}{\mathrm{Sr}}_{0.15}\mathrm{Cu}{\mathrm{O}}_4$ $(T_c=36\mathrm{K})$ with $E\Vert a$ or $b$ at $(T=40\mathrm{K})$ and $5\mathrm{K}$. The missing area $A_{\mu \mathrm{SR}}$ expected from ${\lambda }_L=280\mathrm{nm}$ obtained in the $\mu \mathrm{SR}$ study is shown by a hatched area. The solid circle indicates the dc conductivity at $40\mathrm{K}$. (b) The ratio of the missing area obtained from the present optical study $A\left({\omega }_0\right)$ to $A_{\mu \mathrm{SR}}$. MA#1 represents the ratio for $A\left({\omega }_0\right)$ calculated with the lower limit of integral of $8{\mathrm{cm}}^{-1}$. MA#2 is for the data of $A\left({\omega }_0\right)$ calculated by the dashed curve extrapolation in (a).

Figure 6

Imaginary part of condutivity ${\sigma }_2\left(\omega \right)$ of ${\mathrm{La}}_{1-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ for $x=0.12$ and 0.15 at low frequencies. $E\Vert a\left(b\right)$ and $T=5\mathrm{K}$. ${\sigma }_2^{\mathrm{res}}$ is calculated from the residual ${\sigma }_1\left(\omega \right)$ at $5\mathrm{K}$ using the KK-relation. When ${\sigma }_2^{\mathrm{res}}$ is substracted, ${\sigma }_2$ scales well with $1∕\omega$, which is the signature of the superconducting carrier response.