# Optical conductivity in hole doped cuprates

An example of in-plane optical conductivity, measured in La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) as a function of doping is shown in Fig. 1 ^{[1]}. Initial doping appears to introduce states within the charge transfer gap. Eventually, a Drude peak develops; that is, a peak at zero frequency corresponding to metallic conductivity.

Measurements of the in-plane optical conductivity at low temperature allow one to distinguish a Drude peak from "mid-IR" states, as illustrated for underdoped LSCO and YBCO in Fig. 2 ^{[2]} (For the samples at lower doping, the shading indicates a distinction between a Drude peak, corresponding to mobile carriers, and a mid-infrared peak). The mid-IR weight is large compared to the Drude weight. To quantify this effect, the effective carrier density within a spectral band from zero up to frequency $\omega$ is given by

\begin{equation}
N_{\rm eff}(\omega) = \int_0^\omega d\omega'\, \sigma_1(\omega')
\end{equation}

Evaluating $N_{\rm eff}$ in the Drude peak, one obtains the lower set of open squares in the lower panels of Fig. 1 in Carrier Concentration . These results show a trend very similar to the carrier density $n_{\rm H}$ determined from the Hall coefficient, $R_{\rm H}$, measured below room temperature (lower filled circles). In contrast, integrating the optical conductivity through the mid-IR range yields the upper sets of open squares, which match the $n_{\rm H}$ obtained from $R_{\rm H}$ at high temperature (upper filled circles).