Locating the Missing Superconducting Electrons in the Overdoped Cuprates ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$

Overdoped high-temperature cuprate superconductors have often been understood within the standard BCS framework of superconductivity. However, measurements in a variety of overdoped cuprates indicate that the superfluid density is much smaller than expected from BCS theory and decreases smoothly to zero as the doping is increased. Here, we combine time-domain THz spectroscopy with kHz range mutual inductance measurements on the same overdoped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ films to determine the total, superfluid, and uncondensed spectral weight as a function of doping. A significant fraction of the carriers remains uncondensed in a wide Drude-like peak as $T\to 0$, while the superfluid density remains linear in temperature. These observations are seemingly inconsistent with existing, realistic theories of impurity scattering suppressing the superfluid density in a BCS-like $d$-wave superconductor. Our large measurement frequency range gives us a unique look at the low frequency spectral weight distribution, which may suggest the presence of quantum phase fluctuations as the critical doping is approached.

Figure 1

Real (a), (b) and imaginary (c), (d) parts of the THz optical conductivity as a function of the frequency ($\nu$) and temperature. Green curves in (a) and (c) indicate the conductivities at $T_c$. Vertical dashed lines in (b) and (d) denote $T_c$.

Figure 2

${\sigma }_1(\nu )$ above and below $T_c$ for overdoped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ films with (a) $T_c=27.5\mathrm{K}$, (b) $T_c=13.5\mathrm{K}$, and (c) $T_c=7\mathrm{K}$. Solid lines indicate the data. Dashed lines show a Drude fit with a single scattering rate, ${\sigma }_1(\nu )=S\tau /(1+{\nu }^2{\tau }^2)$. Shaded region represents the expected superfluid spectral weight. (d) The superfluid spectral weight $S_{\delta }$ with temperature for the film with $T_c=13.5\mathrm{K}$ as derived from the complex impedance using a two-coil MI setup ($\nu =40\mathrm{kHz}$). The dashed line represents a linear extrapolation to determine $S_{\delta }$ for $T=1.6\mathrm{K}$. Inset: The real and imaginary parts of the mutual inductance with temperature for the same film. (e) Spectral weight normalized to the normal state spectral weight $S_n$ of the superfluid ($S_{\delta }$) and of the uncondensed carriers ($S_u$) as a function of doping at $T=1.6\mathrm{K}$. $S_{\delta }$ is determined from the MI data as in (d), while $S_u$ and $S_n$ are determined from Drude fits to ${\sigma }_1(\nu )$. Yellow circles give $(S_{\delta }+S_u)/S_n$. Solid lines are guides to the eye. Error bars represent the 95% confidence interval (2 s.d.) in the fitting procedure to extract $S$.

Figure 3

(a) Scattering rate $\gamma$ in units kelvin with the temperature for all films measured. $\gamma$ is obtained with a single Drude fit to ${\sigma }_1(\nu )$ at all temperatures as shown in Figs. 2. Error bars represent the 95% confidence interval in the fitting procedure to extract the parameter $\gamma =1/\tau$. (b) $\nu {\sigma }_2$ normalized to $S_n$ versus frequency for all films at $T=1.6\mathrm{K}$. Circle and diamond symbols represent the TDTS and MI data, respectively. Dashed lines are guides to the eye. (c) The average phase uncertainty determined from the quantum Debye-Waller factor $W_Q=e^{-⟨\delta {\theta }^2⟩/2}$ as a function of doping. Here, $W_Q=S_{\delta }/S_n$. Red line is a linear guide to the eye for the data shown as yellow squares.