Interlayer magnetotransport in the overdoped cuprate ${\text{Tl}}_2{\text{Ba}}_2{\text{CuO}}_{6+x}$: Quantum critical point and its downslide in an applied magnetic field

Fundamental explanations of high-temperature (high-$T_c$) superconductivity must account for the profound differences in the properties of the “normal” (nonsuperconducting) state at the two extremes of charge doping: heavy and light. On the light doping side, its properties clearly violate the standard Fermi-liquid theory of metals. The key to the nature of superconducting pairing lies in understanding the transition to a conventional behavior on the overdoped side. We report a convergence of the pseudogap energy scale and the boundary that separates unconventional from a conventional metal in the zero-temperature limit, both boundaries framing a V-shaped area of “strange metal” state in the temperature-doping phase space. By accessing the low-temperature regions of the phase diagram via a high-field interlayer magnetotransport in heavily doped ${\text{Tl}}_2{\text{Ba}}_2{\text{CuO}}_{6+x}$, we show that the pseudogap boundary has the hallmarks of a quantum phase transition with a zero entropy jump. The critical doping (linkage) point consistently downshifts with magnetic field in unison with the suppression of $T_c$, suggesting that quantum critical fluctuations that destabilize the pseudogap are connected to the superconductivity with high-$T_c$.

Figure 1

(Color online) Experimental phase diagram of the cuprates vs the number of doped holes per Cu ion from charge transport. The in-plane resistivity ${\rho }_{ab}\propto T^n$ in Tl-2201 shows a smooth evolution with doping from the non-Fermi-liquidlike power law with $n\sim 1$ to a very standard $T$-squared dependence where $T_c\left(p\right)$ becomes zero (Ref. 20). $T_c$’s of the crystals used in this work are indicated by the arrows.

Figure 2

(Color online) ${\rho }_c$ vs magnetic field for the overdoped ${\text{Tl}}_2{\text{Ba}}_2{\text{CuO}}_{6+x}$ with $T_c=35\text{K}$. Here the peak at $H_{sc}$ and the upturn (or negative magnetoresistance) in the ${\rho }_c\left(H\right)$ only becomes visible above $T=10\text{K}$, when the superconductivity is sufficiently suppressed by magnetic field, see also Figs. 3a, 3b. Eventually, when the pseudogap closes, the upturn disappears.

Figure 3

(Color online) $c$-axis resistivity as a function of field and temperature for overdoped Tl-2201. [(a) and (b)] ${\rho }_c$ vs magnetic field $H$ at fixed temperatures exhibits a peak in the superconducting state at $H_{sc}$. The core feature in ${\rho }_c\left(H\right)$ that changes with doping is the upturn above $H_{sc}$—the negative MR associated with the excess resistivity due to the pseudogap. A detailed analysis of ${\rho }_c$, e.g., in Bi-2212 is in Refs. 22, 23. For the $T_c\sim 35\text{K}$ crystal in (a) this upturn is uncovered at moderate temperatures and disappears above $T_c$. (b) The upturn is reduced with overdoping and is absent for $T_c\lesssim 18\text{K}$. (c) $\delta {\rho }_c\left(H\right)$ obtained by subtracting the $H$-linear part from ${\rho }_c\left(H\right)$ at fixed $T$. Each curve is shifted vertically for clarity. $H_{\text{FL}}\left(T\right)$, marked by arrows, are the deviation points from $H$-linear MR. (d) ${\rho }_c$ as a function of $T^2$ (for $T_c=15\text{K}$) with the $T^2$ contribution subtracted, for the fields 35, 30, 25, 20, 15, and 11.5 T. Arrows mark $T_{\text{FL}}\left(H\right)$ at the deviation from $T^2$.

Figure 4

(Color online) Field-temperature diagram for overdoped Tl-2201 obtained from the high-field transport measurements. (a) $T_c=35\text{K}$. (b) $T_c=18\text{K}$. (c) $T_c=15\text{K}$. Sky blue squares, $T_{\text{FL}}\left(H\right)$, and dark blue squares, $H_{\text{FL}}\left(T\right)$, separate FL and n-FL states. Red open squares represent the superconducting limiting field $H_{sc}\left(T\right)$, which in cuprates varies exponentially (Ref. 22) with $T$. The pseudogap field $H^{\ast }$ (brown open circles) is in evidence only for the highest $T_c$. The Fermi-liquid boundary is strictly linear. It terminates at $H_{\text{QC}}\sim H_{c2}$, where the Fermi-liquid coefficient $A\propto {\gamma }^2$ in the FL [where ${\rho }_c\left(T\right)={\rho }_c\left(0\right)+A{T}^2$] diverges (Ref. 21). $\gamma$ is the electronic coefficient of specific heat and a measure of the effective mass $m^{\ast }$ of a Landau quasiparticle.

Figure 5

(Color online) Field vs hole concentration diagram for overdoped Tl-2201 obtained from the high-field transport measurements. (a) At zero magnetic field and high temperatures, the Fermi liquid emerges beyond a nearly vertical boundary $T_{\text{FL}}\left(0\right)$ at $p\approx 0.265$, close to the edge of the superconducting dome. $T^{\ast }$ vanishes nearby, at $p_c\sim 0.262$, but how it links with $T_{\text{FL}}$ as $T\to 0$ is masked by the $T_c$ dome. [Note that $T^{\ast }$ in LSCO also vanishes well beyond optimal doping (Ref. 19).] High magnetic field, by shrinking the $T_c$ dome, exposes the concave shape of the $T_{\text{FL}}\left(p,H\ne 0\right)$ boundary (see text), connecting with the $T^{\ast }$ in the $T\to 0$ limit. Note that the nearly vertical onset of the $n\simeq 2$ temperature dependence seen in the in-plane transport (Fig. 1) follows a nearly vertical boundary at the edge of the dome, in close correspondence with our results. (b) Critical doping $p_c\left(H\right)$ shifts linearly from $p_{c0}$ with field (inset), dragging the entire non-Fermi-liquid (n-FL wedge) region to lower doping. At heavy doping, it is near the edge of the superconducting state. (c) The slope of the Fermi-liquid boundary $T_{\text{FL}}\left(H\right)$ (at fixed hole concentration) diverges at $p_c\sim 0.262$. It scales with the slope of $T_{\text{FL}}\left(p\right)$ (at fixed field), see text.

Figure 6

(Color online) Scaling relations for the pseudogap temperature $T^{\ast }$ observed in two families of cuprates. (a) Linear scaling relation between $T^{\ast }$ and $H^{\ast }$. (b) Field dependence of $T^{\ast }\left(H\right)$ approaches $H^{\ast }$ with a vertical slope. The slope $\partial H/\partial T$ across a phase transition is proportional to the difference of the entropy at constant $H$ across the transition (divided by $T$ times the change in the difference of the temperature derivative of $\chi$). This is zero for a continuous transition at $T\to 0$.