Low anisotropy of the upper critical field in a strongly anisotropic layered cuprate ${\mathrm{Bi}}_{2.15}{\mathrm{Sr}}_{1.9}{\mathrm{CuO}}_{6+\delta }:$ Evidence for a paramagnetically limited superconductivity

We study angular-dependent magnetoresistance in a low-$T_c$ layered cuprate ${\mathrm{Bi}}_{2.15}{\mathrm{Sr}}_{1.9}{\mathrm{CuO}}_{6+\delta }$. The low $T_c\sim 4$ K allows complete suppression of superconductivity by modest magnetic fields and facilitates accurate analysis of the upper critical field $H_{c2}$. We observe a universal exponential decay of fluctuation conductivity in a broad range of temperatures above $T_c$ and propose a method for extraction of $H_{c2}\left(T\right)$ from the scaling analysis of the fluctuation conductivity at $T>T_c$. Our main result is observation of a surprisingly low $H_{c2}$ anisotropy $\sim 2$, which is much smaller than the effective mass anisotropy of the material $\sim 300$. We show that the anisotropy is decreasing with increasing field and saturates at a small value when the field reaches the paramagnetic limit. We argue that the dramatic discrepancy of high-field and low-field anisotropies is clear evidence for paramagnetically limited superconductivity.

Figure 1

(a) Scanning electron microscopy image of the sample OP(4.3). (b) Temperature dependence of the $c$-axis resistance at $H=0$ and 14 T. The compound has a low $T_c\simeq 4$ K and the pseudogap onset temperature $T^{*}\simeq 110$ K. (c) Current-voltage characteristic of a small mesa at $T=1.8$ K and $H=0$.

Figure 2

$T$ dependencies of the in-plane resistance at magnetic fields (a) perpendicular and (b) parallel to the $ab$ planes. (c) Comparison of the data from (a) and (b) for zero and 17 T. For $H\parallel ab$ the $R_{ab}\left(T\right)$ is shifted to lower temperatures. For $H\perp ab$ the superconducting transition is completely suppressed and $R_{ab}$ is shifted upwards, indicating presence of a positive orbital magnetoresistance in the normal state. (d) MR in a perpendicular field below and just above $T_c$. Note that the saturation field $\sim H_{c2}\left(T\right)$ is decreasing with $T\to T_c$. Panels (e) and (f) show MR for both field orientations (e) below and (f) above $T_c$. The positive MR at $T>T_c$ is caused both by suppression of superconducting fluctuations and an additional orbital normal state MR.

Figure 3

Temperature dependencies of the $c$-axis resistance for (a) $H\perp ab$ and (b) $H\parallel ab$. (c) Comparison of the data from (a) and (b) for $H=0$ and 17 T. Panels (d) and (e) show $c$-axis MR for the two field orientations (d) below and (e) above $T_c$. It is seen that the normal state negative MR is present for both field orientations. (f) $R_c\left(H^{\perp }\right)$ measured up to 65 T (data from Ref. [30]). It is seen that there is both a positive MR at low fields due to suppression of the supercurrent and a negative MR at high fields due to suppression of the PG.

Figure 4

(a) and (b) Angular dependencies of in-plane resistances for rotation around two orthogonal axes in the $ab$ plane. Dashed lines in (b) represent scaled data from (a) [64]. (c) Angular dependence of the $c$-axis resistance. The peak at $\Theta =0^{\circ }$ is due to onset of Josephson flux flow. (d) Theoretical angular dependencies of flux-flow resistances for a 2D model with an anisotropy $\gamma =5$. Note that the cusp at $\Theta =0^{\circ }$ becomes sharper at $H>H_{c2}^{\perp }$ because superconductivity survives only in a narrow range of angles around $\Theta =0^{\circ }$. A similar narrowing is seen in panels (a)–(c). Panels (e) and (f) represent comparison of (e) in-plane and (f) out-of-plane angular MR (symbols) with resistances at the corresponding perpendicular $H^{\perp }=Hsin\left(\Theta \right)$ (solid lines) and parallel $H^{\parallel }=Hcos\left(\Theta \right)$ field components at $T=2$ K and $H=17$ T.

Figure 5

Angular dependence of (a) in-plane and (b) $c$-axis resistances at $H=10$ T for different temperatures. The cusp at $\Theta =0^{\circ }$ is vanishing at $T\gtrsim 2T_c$ for $R_{ab}$ and at $T>T_c$ for $R_c$. (c) Angular-dependent MR above $T_c$, normalized by $R(\Theta ={90}^{\circ })$. At $T\ge 10$ K the MR is varying in a smooth 3D manner. (d) Temperature dependencies of absolute values of angular MR amplitudes normalized by the field (in the semilogarithmic scale). Note that the in-plane MR decays at a scale $T\sim T_c$ and the out-of-plane MR at the PG temperature $T^{*}\simeq 110$ K. Panels (e) and (f) represent comparison of (e) in-plane and (f) out-of-plane angular MR at $T=7.7$ K $>T_c$ with the MR at the corresponding parallel and perpendicular field components. The in-plane angular MR at not too small angles is dominated by the perpendicular field component (solid line). The $c$-axis angular MR is given by the additive contribution of the two field components (dashed lines).

Figure 6

Fluctuation part of the in-plane conductivity $\Delta {\sigma }_{ab}\left(T\right)=1/R_{ab}\left(T\right)-1/R_n\left(T\right)$, normalized by the normal state resistance, for fields (a) perpendicular and (b), (c) parallel to the $ab$ plane. Curves in panels (b) and (c) were obtained from the same data using different $R_n\left(T\right)$: (b) $R_n=R_{ab}(H=17$ T), (c) linear extrapolation from high $T$, shown by the dashed line in Fig. 2. It is seen that fluctuation paraconductivity decays approximately exponentially with increasing $T$ with an almost field independent slope. Panels (d) and (e) show data from (a) and (c), respectively, shifted by $T_c\left(H\right)$. (f) Field dependence of paraconductivity $\Delta {\sigma }_{ab}\left(H^{\perp }\right)$ at different $T$. An approximately exponential decay is seen.

Figure 7

(a) The upper critical field perpendicular (filled squares) and parallel to layers (filled circles and rhombuses) obtained from the scaling analysis of fluctuation conductivity according to Eq. (4) at $T>T_c\left(H\right)$. For comparison we also show middle points $H_{50\%}\left(T\right)$ of in-plane (small open circles) and out-of-plane (small open squares) resistive transitions in parallel field. The dashed line represents the $\sqrt{T_c-T}$ dependence. Open triangles (right axis) show $T$ dependence of the superconducting gap, obtained by intrinsic tunneling spectroscopy [30]. (b) $T$ dependence of the anisotropy of the upper critical field ${\gamma }_H=H_{c2}^{\parallel }/H_{c2}^{\perp }$. It is seen that at low temperature it saturates at a small value ${\gamma }_H\left(0\right)\sim 2$. (c) Angular anisotropy of the in-plane resistance $R_{ab}\left({90}^{\circ }\right)/R_{ab}\left(0^{\circ }\right)$ at $T=2$ K as a function of magnetic field. The anisotropy rapidly decreases with increasing $H$ as soon as the field approaches the paramagnetic limit.