Origin of positive out-of-plane magnetoconductivity in overdoped ${\mathbf{Bi}}_{1.6}{\mathbf{Pb}}_{0.4}{\mathbf{Sr}}_2{\mathbf{CaCu}}_{1.96}{\mathbf{Fe}}_{0.04}{\mathbf{O}}_{8+\delta }$

To elucidate the pseudogap phase diagram including the overdoped state of high transition temperature (high-$T_c$) cuprates, we must understand the origin of the positive out-of-plane magnetoconductivity observed in these compounds. For this purpose, the out-of-plane resistivity ${\rho }_c(T,H)$ of an overdoped ${\mathrm{Bi}}_{1.6}{\mathrm{Pb}}_{0.4}{\mathrm{Sr}}_2{\mathrm{CaCu}}_{1.96}{\mathrm{Fe}}_{0.04}{\mathrm{O}}_{8+\delta }$ (Bi-2212) single crystal is measured under pulsed magnetic fields up to 60 T. We show that the superconducting density-of-states depletion effect, in addition to the pseudogap effect, clearly appears below the superconducting fluctuation regime, and the contribution becomes dominant in the superconducting state.

Figure 1

(a) Out-of-plane resistivity ${\rho }_c\left(T\right)$ for ${\mathrm{Bi}}_{1.6}{\mathrm{Pb}}_{0.4}{\mathrm{Sr}}_2{\mathrm{CaCu}}_{1.96}{\mathrm{Fe}}_{0.04}{\mathrm{O}}_{8+\delta }$ with and without a 17.5-T magnetic field. Here, ${\rho }_c^n$ is a linear extrapolation of ${\rho }_c\left(T\right)$ at higher temperatures. The arrow indicates the pseudogap opening temperature $T^{*}\approx 166$ K. The inset shows the negative magnetoresistance (MR) on ${\rho }_c$ under 17.5 T. (b) Temperature dependence of the out-of-plane MR under several fields. (c) In-plane resistivity ${\rho }_{ab}\left(T\right)$ with and without a 17.5-T magnetic field. The inset shows the temperature dependence of $d{\rho }_{ab}/dT$. The arrow indicates the onset temperature of the superconducting fluctuation $T_{\mathrm{scf}}\approx 70$ K.

Figure 2

(a) Magnetic field dependence of ${\rho }_c$ for ${\mathrm{Bi}}_{1.6}{\mathrm{Pb}}_{0.4}{\mathrm{Sr}}_2{\mathrm{CaCu}}_{1.96}{\mathrm{Fe}}_{0.04}{\mathrm{O}}_{8+\delta }$ below and above $T_c$. The arrows indicate the peak field $H_{\mathrm{peak}}$, and the vortex melting field $H_m$ ($H_m$ is defined as a field in which ${\rho }_c=0.01{\rho }_c^n$ [16]). (b), (c) Out-of-plane magnetoconductivity (MC), ${\sigma }_c\left(H\right)–{\sigma }_c\left(0\mathrm{T}\right)$, at 50 and 60 K, respectively. In (b), the normal state ${\sigma }_c\left(0\mathrm{T}\right)$ was assumed as an extrapolated value to the zero field of the high-field data. (d) Out-of-plane MC above 65 K for steady magnetic fields. (b)–(d) The solid lines are linear extrapolations of MC at higher fields. Here, $H_{c2}$ is defined as the field at which MC deviates $1\%$ as an average from these lines. (e) Schematic representation for the two component analysis used above $T_c$.

Figure 3

(a) Temperature dependence of $a$. Black and orange circles represent the estimation using data of a pulsed magnet [Figs. 2 and 2] and of a steady magnet [Fig. 2], respectively (for some data points, the raw MC data were not shown). The dashed line represents a linear extrapolation of the $a$ values. (b), (c) Magnetic field dependences of the raw $d{\sigma }_c/dH$ data and the pseudogap contribution ($2aH$)-subtracted data for 40 K and 30 K, respectively. The broken lines represent the pseudogap contribution. The solid lines represent linear extrapolations of the subtracted data. (d)–(f) Magnetic field dependence of raw ${\sigma }_c$ data and the pseudogap contribution $\mathrm{\Delta }{\sigma }_c^{\mathrm{PG}}$-subtracted data at 40, 30, and 20 K, respectively. Here, $\mathrm{\Delta }{\sigma }_c^{\mathrm{SC}-\mathrm{DOS}}+{\sigma }_c^{qp}\left(0\mathrm{T}\right)$ corresponds to the subtracted data.

Figure 4

Magnetic field dependence of ${\rho }_{ab}$ for ${\mathrm{Bi}}_{1.6}{\mathrm{Pb}}_{0.4}{\mathrm{Sr}}_2{\mathrm{CaCu}}_{1.96}{\mathrm{Fe}}_{0.04}{\mathrm{O}}_{8+\delta }$ at several temperatures above and below $T_c$. The arrows indicate the upper critical field $H_{c2}$ ($H_{c2}$ is defined as the field at which ${\rho }_{ab}$ decreases $1\%$ from the high field constant value), and the vortex melting field $H_m$ ($H_m$ is defined as a field in which ${\rho }_{ab}=0.01{\rho }_{ab}^n$).

Figure 5

Characteristic fields versus temperature. The melting field $H_m$, peak field $H_{\mathrm{peak}}$, and upper critical field $H_{c2}$ are shown as solid squares, solid triangles, and circles, respectively. The $H_{c2}$ data points represented by black solid circles, blue solid circles, orange solid circles, and black open circles have been determined in Figs. 2, 2, 4, 2, 3, and 3, respectively. Here, an open symbol indicates that the data were obtained by extrapolation. The dashed lines are numerical fits for each field (see text).