$c$-Axis Optical Sum Rule and a Possible New Collective Mode in ${\mathrm{L}\mathrm{a}}_{2-x}{\mathrm{S}\mathrm{r}}_x{\mathrm{C}\mathrm{u}\mathrm{O}}_4$

We present the $c$-axis optical conductivity ${\sigma }_{1c}(\omega ,T)$ of underdoped ($x=0.12$) and optimally doped ($x=0.15$) ${\mathrm{L}\mathrm{a}}_{2-x}{\mathrm{S}\mathrm{r}}_x{\mathrm{C}\mathrm{u}\mathrm{O}}_4$ from 4 meV to 1.8 eV obtained by a combination of reflectivity and transmission spectra. In addition to the opening of the superconducting gap, we observe an increase of conductivity above the gap up to 270 meV with a maximal effect at about 120 meV. This may indicate a new collective mode at a surprisingly large energy scale. The Ferrell-Glover-Tinkham sum rule is violated for both doping levels. Although the relative value of the violation is much larger for the underdoped sample, the absolute increase of the low-frequency spectral weight, including that of the condensate, is higher in the optimally doped regime. Our results resemble in many respects the observations in ${\mathrm{Y}\mathrm{B}\mathrm{a}}_2{\mathrm{C}\mathrm{u}}_3{\mathrm{O}}_{7-\delta }$.

Figure 1

The $c$-axis reflectivity (a), transmission (b), and optical conductivity (c) of the UD ($x=0.12$) and the OD ($x=0.15$) LSCO samples. The inset compares ${\sigma }_1(\omega ,300\mathrm{K})$ (solid line) with the previous results based on reflectivity data only (Ref. 12: dashed line; Ref. 13: dotted line).

Figure 2

The temperature dependence of (a) conductivity at selected energies, (b) a Fabry-Pérot maximum position at $\approx 2150{\mathrm{c}\mathrm{m}}^{-1}$, (c) the condensate SW, and (d) the transmission of the OD sample at $1000{\mathrm{c}\mathrm{m}}^{-1}$ with a fine temperature resolution. The dotted lines denote the curve slopes above and below $T_c$, which is marked by the vertical line.

Figure 3

(a) ${\Delta }_s{\sigma }_1(\omega )$ (the hatched area corresponds to the slope change of SW at $\omega =0$, ${\Delta }_{s}D$), the error bars represent experimental uncertainty of ${\Delta }_s{\sigma }_1(\omega )$ below $200{\mathrm{c}\mathrm{m}}^{-1}$; the inset expands the higher frequency region, where transmission allows a very accurate determination of ${\sigma }_1(\omega )$. (b) ${\Delta }_{s}R(\omega )$ in the phonon range. (c) ${\Delta }_s{ϵ}_1(\omega )$ at high frequencies, as derived from the shift of Fabry-Pérot fringes. The solid curves represent the fit as described in the text.

Figure 4

The demonstration of the FGT sum rule violation in LSCO: (a) the relative finite-frequency SW, $-{\Delta }_{s}A(\omega )/{\Delta }_{s}D$; (b) absolute SW, $-{\Delta }_{s}A(\omega )-{\Delta }_{s}D$; (c) the temperature dependence of $A({\Omega }_c)$ (open circles) and $A({\Omega }_c)+D$ (solid circles). $T_c$ is marked by the vertical line.