Metal-insulator transition and pseudogap in ${\text{Bi}}_{1.76}{\text{Pb}}_{0.35}{\text{Sr}}_{1.89}{\text{CuO}}_{6+\delta }$ high-$T_c$ cuprates

It is inferred from bulk-sensitive muon Knight shift measurement for a ${\mathrm{Bi}}_{1.76}{\mathrm{Pb}}_{0.35}{\mathrm{Sr}}_{1.89}{\mathrm{CuO}}_{6+\delta }$ single-layer cuprate that the metal-insulator (MI) transition (in the low-temperature limit, $T\to 0$) occurs at the critical hole concentration $p=p_{\mathrm{MI}}=0.09\left(1\right)$, where the electronic density of states (DOS) at the Fermi level is reduced to zero by the pseudogap irrespective of the Néel order or spin glass magnetism. Superconductivity also appears for $p>p_{\mathrm{MI}}$, suggesting that this feature is controlled by the MI transition. More interestingly, the magnitude of the DOS reduction induced by the pseudogap remains unchanged over a wide doping range ($0.1\le p\le 0.2$), indicating that the pseudogap remains as a hallmark of the MI transition for $p>p_{\mathrm{MI}}$.

Figure 1

Examples of fast-Fourier-transformed $\mu \mathrm{SR}$ spectra observed at $\sim 250$ K in (a) LD and (c) OPT, with the temperature dependence of the muon Knight shift $K_i$ ($i=1,2$, corresponding to the signal with amplitude $A_i$) in the respective samples being shown in (b) and (d). The small downturn below $\sim 50$ K in (d) is commonly observed for $K_1$ and $K_2$ and is due to the diamagnetism of the impurities (see the text).

Figure 2

Temperature dependences of the muon Knight shift ${\overline{K}}_{\mu }$ (at 6 T) in (Bi,Pb)2201. A magnified view is shown for $T<40$ K (left). The solid curves are fits obtained using Eq. (5). The onset temperature of the pseudogap ($T^{*}$) is indicated by black arrows. Inset: $T_c$ vs $p$ for measured samples.

Figure 3

(a) Hole concentration ($p$) dependence of $T_c, T^{*}$, and ${\mathrm{\Delta }}_1$ in (Bi,Pb)2201. The $c$-axis resistivity data (${\rho }_c$) are quoted from Refs. [21, 22, 23], the NMR data are from Refs. [19, 20], and the ARPES data are from Ref. [3]. (b) Superfluid density $[n_s$(0), brown rhombuses] and $K_0$ (inverted green triangles) vs $p$, where the latter corresponds to the residual DOS for $T\to 0$ in the normal state. (c) $K_{\max }$ (red squares, corresponding to the DOS at $T>T^{*}$) and $K_{\mathrm{pg}}$ (blue triangles, corresponding to the DOS depleted by the pseudogap) vs $p$. (d) Definitions of $K_{\max }, K_{\mathrm{pg}}$, and $K_0$ for ${\overline{K}}_{\mu }$ in OPT (quoted from Fig. 2).

Figure 4

(a) Schematic illustration of correspondence between $K_{\max }, K_{\mathrm{pg}}, K_0$, and the DOS at the Fermi level in momentum space. (b) $K_{\max }\simeq K_{\mathrm{pg}}$ suggests a pointlike residual DOS ($K_0\simeq 0$) at $p=p_{\mathrm{MI}}=0.09\left(1\right)$. (c) The increase of $K_0$ with $p$ suggests the development of the DOS around the nodes. (d) Continuous increase of the DOS with $p$ as $T_c$ is reduced to zero. The magnitude of the pseudogap (${\mathrm{\Delta }}_1$) decreases monotonously with increasing $p$.