$c$-axis optical spectra and charge dynamics in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$

Out-of-plane ($c$-axis-polarized) optical reflectivity spectra and $c$-axis charge transport properties are studied for single crystals of a high-$T_c$ system ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ over a wide compositional range $0<~x<~0.30$. The measurements are made at various temperatures in the normal and superconducting states over an energy range from 0.003 to 40 eV. The present study focuses on the evolution of the $c$-axis spectrum with doping and provides a full set of the optical and transport data on this single-layer system together with the previously published data of the in-plane spectra [S. Uchida et al., Phys. Rev. B 43, 7942 (1991)]. As in the case of the in-plane spectrum, the spectral weight is transferred upon doping from the high- to low-energy region. Different from the in-plane spectrum the transferred weight forms a band which is centered at relatively high energy (∼ 2 eV) and does not appreciably move with increasing $x$. As a consequence, the $c$-axis optical conductivity [${\sigma }_c\left(\omega \right)$] is extremely small in the lowest-energy region (<0.3 eV) until the compound is overdoped. Particularly, in the underdoped regime ($x<\mathrm{}0.13\mathrm{}$) the low-energy ${\sigma }_c\left(\omega \right)$ is too small to form a Drude peak, and is further suppressed with reducing temperature. This is in common with the pseudogap effect, observed for the bilayer system $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$, and is connected to the semiconducting $c$-axis resistivity. A Drude peak in ${\sigma }_c\left(\omega \right)$ develops only in the highly doped compounds ($x>~0.18$). Only in the overdoped regime is ${\sigma }_c\left(\omega \right)$ dominated by a sharp Drude term and the anisotropic resistivity $\frac{{\rho }_c}{{\rho }_{\mathrm{ab}}}$ almost constant over a wide temperature range, giving strong evidence for a three-dimensional metallic state. In the superconducting state the $c$-axis infrared optical response is quite anomalous as it is characterized by a sharp plasma edge. In the underdoped regime, the strongly suppressed spectral weight in the normal state gives rise to a sharp plasma edge within a gap region, which can be identified as a Josephson plasma in the weakly Josephson-coupled layered superconductor. In the highly doped superconducting regime, an appreciable Drude-like component remains in the spectrum even at temperatures well below $T_c$, leading to a substantial damping of the Josephson plasma. Such a gapless spectrum is perhaps associated with a crossover from the underdoped to the nonsuperconducting overdoped regime.