Far-infrared $c$-axis response of ${\mathrm{La}}_{1.87}$${\mathrm{Sr}}_{0.13}$Cu${\mathrm{O}}_4$ determined by ellipsometry

We have measured the $c$-axis optical response of a large, high-quality single crystal of ${\mathrm{La}}_{1.87}$ ${\mathrm{Sr}}_{0.13}$Cu${\mathrm{O}}_4$ between 30 and 700 ${\mathrm{cm}}^{-1\mathrm{}}$ in the 10-300 K temperature range with a far-infrared Fourier ellipsometer. Since ellipsometry provides both amplitude and phase information, we directly obtain the complex dielectric function, thereby avoiding the errors introduced by the Kramers-Kronig transformation or the reference determination. A purely algebraic treatment of our data within the two-fluid Gorter-Casimir model reveals no indication of a reduction of the normal carrier scattering rate $\Gamma$ below $T_c$. We obtain $\Gamma \approx 130\pm 70$ ${\mathrm{cm}}^{-1\mathrm{}}$ and a plasma frequency for the superconducting carriers of ${\omega }_{\mathrm{ps}}\approx 260\pm 40$ ${\mathrm{cm}}^{-1\mathrm{}}$. Thus, assuming a weak-coupling BCS gap ($3.5k_B{T}_c\approx 77$ ${\mathrm{cm}}^{-1\mathrm{}}$, with $T_c=31.5\mathrm{}$ K) we conclude that the $c$-axis optical response of ${\mathrm{La}}_{1.87}$ ${\mathrm{Sr}}_{0.13}$Cu${\mathrm{O}}_4$ can neither be modeled within the clean nor with the dirty limit.