Diamagnetism and Cooper pairing above $T_c$ in cuprates

In the cuprate superconductors, Nernst and torque magnetization experiments have provided evidence that the disappearance of the Meissner effect at $T_c$ is caused by the loss of long-range phase coherence, rather than the vanishing of the pair condensate. Here we report a series of torque magnetization measurements on single crystals of ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$ (LSCO), ${\text{Bi}}_2{\text{Sr}}_{2-y}{\text{La}}_y{\text{CuO}}_6$ (Bi 2201), ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$, and optimal ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_7$. Some of the measurements were taken to fields as high as 45 T. Focusing on the magnetization above $T_c$, we show that the diamagnetic term $M_d$ appears at an onset temperature $T_{onset}^M$ high above $T_c$. We construct the phase diagram of both LSCO and Bi 2201 and show that $T_{onset}^M$ agrees with the onset temperature of the vortex Nernst signal $T_{onset}^{\nu }$. Our results provide thermodynamic evidence against a recent proposal that the high-temperature Nernst signal in LSCO arises from a quasiparticle contribution in a charge-ordered state.

Figure 1

(Color online) Magnetization curves of sample LSCO 09 with Sr content $x=0.09$ and transition temperature $T_c=24\text{K}$ measured in magnetic fields $H$ up to 14 T. (a) The (total) effective magnetization $M_{eff}$ vs $H$ at temperatures $4.2\le T\le 200\text{K}$. Below $T_c$, the $M_{eff}\text{−}H$ curves are hysteretic when $H$ lies below the melting field $H_m\left(T\right)$, which is experimentally defined as the field at which the up-sweep branch deviates from the down-sweep branch ($H_m$ is indicated by an arrow for the curve at 7 K). Above $H_m$, the curves become reversible. (b) The diamagnetic magnetization $M_d$ vs $H$ at temperatures $4.2\le T\le 30\text{K}$. (c) Curves of $M_d$ vs $H$ at $22\le T\le 80\text{K}$ displayed in expanded scale. In LSCO 09, the diamagnetic signal persists to more than 60 K above $T_c$. In (b) and (c), the bold curve is measured at the separatrix temperature $T_s=22\text{K}$.

Figure 2

(Color online) Magnetization curves $M_d$ vs $H$ in LSCO 09 measured in intense fields up to 33 T. $T_c$ lies between the curves at 22 K (bold) and 25 K.

Figure 3

(Color online) Comparison of magnetization curves $M_d$ vs $H$ in LSC0 17 and LSCO 12 measured in intense fields up to 45 T. Panel (a) shows curves for sample LSCO 17 ($x=0.17$ and $T_c=38\text{K}$). Panel (b) displays curves of sample LSCO 12 ($x=0.12$ and $T_c=27\text{K}$).

Figure 4

(Color online) Curves of magnetization $M_d$ vs $H$ in Bi 2201 measured in intense fields. Panel (a) displays results on UD sample Bi 2201 (La content $y=0.7$ and $T_c=12\text{K}$). Note that most of the results shown are at $T$ above $T_c=12\text{K}$. They approach zero at the same nominal field scale $\sim 42\text{T}$. Panel (b) plots curves of $M_d$ vs $H$ measured in another UD Bi 2201 (La content $y=0.6$ and $T_c=18\text{K}$). Panels (c) and (d) display $M_d$ curves measured in optimally doped Bi 2201 (La content $y=0.44$ and $T_c=30\text{K}$) and OV Bi 2201 (La content $y=0.2$ and $T_c=20\text{K}$). In all these Bi 2201 samples, the curves of $M_d$ vs $H$ remain strikingly nonlinear high above $T_c$. In each panel, the curve closest to $T_c$ is shown in bold. A preliminary version of panel (a) was published in Ref. 12.

Figure 5

(Color online) Comparison of high-field magnetization curves $M_d$ vs $H$ in OPT Bi 2212 and UD Bi 2212. Panel (a) (Ref. 12) displays the $M_d$ curves in OPT Bi 2212 $\left(T_c=88\text{K}\right)$ in fields up to 45 T. Panel (b) plots magnetization curves in UD Bi 2212 $\left(T_c=50\text{K}\right)$ in fields up to 33 T. In each panel, the curve closest to $T_c$ is shown in bold. Panel (a) is taken from Ref. 12.

Figure 6

(Color online) The low-field magnetization curves of OPT Bi 2201 (La content $y=0.44$). In (a), the striking weak-field nonlinearity is highlighted by displaying the variation in $M_d$ from $H=-2$ to $+2\text{T}$. Although $T_c=30\text{K}$ (bold curve), diamagnetism is observed up to the onset temperature $T_{onset}^M\simeq 70\text{K}$. Panel (b) shows the low-field curvature in expanded scale. The measurements were performed using a very soft cantilever beam. The divergent curvature of $M_d$ at zero $H$ is consistent with fragile London rigidity (Refs. 11, 14).

Figure 7

(Color online) Contour plot of the diamagnetic magnetization $\left|M_d\left(T,H\right)\right|$ of UD Bi 2201 with La content $y=0.7$ and $T_c=12\text{K}$ (arrow). The spacing between adjacent contour lines is 10 A/m for $T<T_c$. The upper critical field $H_{c2}$ (defined by extrapolating $M_d\to 0$) is plotted as open circles.

Figure 8

(Color online) Magnetization curves of OPT ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{7-\delta }$. Panel (a) displays curves of $M_d$ vs $H$ from $T=89$ to 150 K. Some curves were taken (at slower sweep rate) only to 1 T. Above $T_c=92\text{K}$ (bold curve), a sizeable diamagnetic signal persists to $T_{onset}^M\sim 130\text{K}$. Panel (b) plots the $T$ dependence of the observed $M_{eff}$ with $H=14\text{T}$. As in the other cuprates, $M_{eff}$ may be fit to a straight line above the onset of the diamagnetic signal $T_{onset}^M$. The dashed curve is the Meissner signal $M_{sq}$ measured by SQUID magnetometry with $H=10\text{Oe}$.

Figure 9

(Color online) Plots of the temperature dependence of $M_{eff}\left(T\right)$ in Bi 2201 (a) and in LSCO (b), showing the onset of diamagnetism as $T$ is decreased. In both panels, the value of $M_{eff}$ measured at $H=14\text{T}$ is plotted vs $T$ in samples with various doping levels $x$. In general, $M_{eff}$ at high $T$ varies weakly vs $T$, as shown by the straight lines which are of the form $A+BT$. Relative to this linear background, $M_{eff}$ shows a strong downwards deviation starting at the onset temperature $T_{onset}^M$ (indicated by arrows).

Figure 10

(Color online) Expanded view of the total observed magnetization $M_{eff}$ vs $H$ in LSCO 09 (a) and LSCO 12 (b). Above $T_{onset}^M$, the $T$ dependence of the curves is only from the paramagnetic Van Vleck term. Below $T_{onset}^M$, however, the diamagnetic term $M_d$ grows rapidly to dominate the $T$ dependence. The stability of the torque cantilever and the resolution in $\tau$ are sufficient to allow closely spaced curves to be resolved. The uncertainty in measuring $M_{eff}$ makes the largest contribution to the error bars in $T_{onset}^M$.

Figure 11

(Color online) Phase diagram comparing the onset temperatures for the Nernst and diamagnetism signals vs doping $x$ in ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$ (a) and in ${\text{Bi}}_2{\text{Sr}}_{2-y}{\text{La}}_y{\text{CuO}}_6$ (b). The superconducting transition temperature $T_c$ (solid circles) is plotted with the onset temperature $T_{onset}^{\nu }$ determined by the Nernst effect (solid diamonds) and $T_{onset}^M$ determined by torque magnetometry (open squares). In panel (b) for ${\text{Bi}}_2{\text{Sr}}_{2-y}{\text{La}}_y{\text{CuO}}_6$, a large La content $y$ implies small hole carrier concentration (UD regime).