Wiedemann-Franz Law and Abrupt Change in Conductivity across the Pseudogap Critical Point of a Cuprate Superconductor

The thermal conductivity $\kappa$ of the cuprate superconductor ${\mathrm{La}}_{1.6-x}{\mathrm{Nd}}_{0.4}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ was measured down to 50 mK in seven crystals with doping from $p=0.12$ to $p=0.24$, both in the superconducting state and in the magnetic field-induced normal state. We obtain the electronic residual linear term ${\kappa }_0/T$ as $T\to 0$ across the pseudogap critical point $p^{\star }=0.23$. In the normal state, we observe an abrupt drop in ${\kappa }_0/T$ upon crossing below $p^{\star }$, consistent with a drop in carrier density $n$ from $1+p$ to $p$, the signature of the pseudogap phase inferred from the Hall coefficient. A similar drop in ${\kappa }_0/T$ is observed at $H=0$, showing that the pseudogap critical point and its signatures are unaffected by the magnetic field. In the normal state, the Wiedemann-Franz law, ${\kappa }_0/T=L_0/\rho (0)$, is obeyed at all dopings, including at the critical point where the electrical resistivity $\rho (T)$ is $T$ linear down to $T\to 0$. We conclude that the nonsuperconducting ground state of the pseudogap phase at $T=0$ is a metal whose fermionic excitations carry heat and charge as conventional electrons do.

Popular Summary

Cuprates are copper oxide materials that superconduct at record high temperatures. Aside from high-temperature superconductivity itself, the chief mystery of cuprates is their pseudogap phase, a state that coexists and competes with superconductivity but whose fundamental nature is still unclear. Here, we provide clear and sharp signatures of the pseudogap phase in the vicinity of its critical point, data that will be key to informing future theories.

Our study is based on low-temperature (down to 0.05 K) thermal conductivity measurements, which probe how electrons behave upon entering the pseudogap phase at the critical point. When superconductivity is suppressed by applying a magnetic field, we find that the Wiedemann-Franz law, a fundamental property of metals that relates thermal and electrical transport, is obeyed at all dopings. This is true at and below the pseudogap critical point itself, showing that the ground state of the pseudogap phase is metallic and not insulating.

Upon crossing below the critical point, the thermal conductivity exhibits a sharp and pronounced drop, implying that the carrier density must decrease abruptly upon entering the pseudogap phase. Importantly, thermal conductivity allows us to probe the superconducting state in the absence of a magnetic field, where this loss of carrier density at the critical point is still observed, showing that it is not a field-induced effect.

These findings provide new experimental constraints for future theories of the pseudogap phase of cuprates.

Figure 1

(a) Temperature-doping phase diagram of Nd-LSCO, showing the superconducting $T_c$ (gray dome) and the pseudogap temperature $T^{\star }$ extracted from the electrical resistivity (red dots, Ref. [19]) and from angle-resolved photoemission spectroscopy (red square, Ref. [20]). The red diamond marks the position of $p^{\star }=0.23$, the doping for the onset of the pseudogap phase in Nd-LSCO. The red dashed line is a guide to the eye. (b) Electrical resistivity vs temperature for Nd-LSCO at $p=0.22$ and 0.24, at $H=0$ (gray data) and in the normal state at $H=33\mathrm{T}$ (colored). The pseudogap temperature $T^{\star }$ (arrow) is defined as the temperature below which $\rho (T)$ deviates from its $T$-linear behavior at high temperature (black line). Here, $T^{\star }=50\mathrm{K}$ at $p=0.22$, and $T^{\star }=0$ at $p=0.24$. (c) Electrical resistivity of Nd-LSCO at $p=0.24$ and $H=16\mathrm{T}$ (blue) with a linear fit (black line). The red dot is $L_0/({\kappa }_0/T)$, with ${\kappa }_0/T$ measured in the same sample at $H=15\mathrm{T}$ (Fig. 2), showing that the Wiedemann-Franz law is perfectly satisfied.

Figure 2

(a) Thermal conductivity $\kappa$ vs temperature plotted as $\kappa /T$ vs $T$, for Nd-LSCO at $p=0.22$ (red) and 0.24 (blue), in $H=0$ (open symbols) and 15 T (dots). The lines are linear fits to the data over the temperature range shown. The $y$ intercept of the fit is the residual electronic term ${\kappa }_0/T$. The horizontal dashed lines are calculated from the Wiedemann-Franz law $L_0/\rho (0)$ using the measured $\rho (0)$ (see text). (b) ${\kappa }_0/T$ as a function of applied magnetic field for $p=0.22$ (red) and 0.24 (blue). At both dopings, ${\kappa }_0/T$ saturates at high field, showing that the normal state has been reached. The error bars reflect the uncertainty on the fits shown in (a), which comes from varying the temperature range. For $p=0.24$, the error bars are smaller than the symbols.

Figure 3

$\kappa /T$ vs temperature for Nd-LSCO at $H=15\mathrm{T}$, for (a) $p=0.22$, 0.23, and 0.24, (b) $p=0.20$, 0.21, and 0.22, and (c) $p=0.12$ and 0.15. In panels (a) and (b), the lines are linear fits to the data over the entire range shown. In panel (c), the lines are power-law fits to the data over the entire range of the data.

Figure 4

(a) ${\kappa }_0/T$ in Nd-LSCO at $H=0$ (open red circles) and 15 T (red dots), and $L_0/\rho (0)$ at $H=15\mathrm{T}$ (blue squares), as a function of doping. The error bars on $L_0/\rho (0)$ come from the geometric factor error, $\pm 10\%$, and the uncertainty on estimating $\rho (0)$, $\pm 5\mu \mathrm{\Omega }\mathrm{cm}$ ($\pm 0.5\mu \mathrm{\Omega }\mathrm{cm}$ for $p=0.24$). The error bar on ${\kappa }_0/T$ is $\pm 0.01\mathrm{mW}/{\mathrm{K}}^2\mathrm{cm}$, which is smaller than the symbols. (b) Ratio $({\kappa }_N/T)/[L_0/\rho (0)]$ as a function of doping, where ${\kappa }_N/T$ is the normal state ${\kappa }_0/T$, measured at $H=15\mathrm{T}$. (c) Ratio $({\kappa }_N/T)/(L_0/{\rho }_0)$ as a function of doping, where ${\rho }_0$ is proportional to the level of disorder in each sample (see text). In panels (b) and (c), the error bars come from the uncertainty on $\rho (0)$, estimated to be $\pm 5\mu \mathrm{\Omega }\mathrm{cm}$ ($\pm 0.5\mu \mathrm{\Omega }\mathrm{cm}$ for $p=0.24$), on ${\rho }_0$, estimated to be $\pm 2\mu \mathrm{\Omega }\mathrm{cm}$ ($\pm 0.5\mu \mathrm{\Omega }\mathrm{cm}$ for $p=0.24$), and the error on ${\kappa }_0/T$ as in (a). The gray vertical band in all panels gives the position of $p^{\star }$. The lower black line in (c) is equal to $p$, showing that the normal-state conductivity of the pseudogap phase (PG) at $T=0$ is only a fraction $\sim p$ of its value just above $p^{\star }$.