High-field studies of superconducting fluctuations in high-$T_c$ cuprates: Evidence for a small gap distinct from the large pseudogap

We have used large pulsed magnetic fields up to 60 T to suppress the contribution of superconducting fluctuations (SCFs) to the $ab$-plane conductivity above $T_c$ in a series of YBa${}_2$Cu${}_3$O${}_{6+x}$ from the deep pseudogapped state to slight overdoping. Accurate determinations of the SCF contribution to the conductivity versus temperature and magnetic field have been achieved. Their joint quantitative analyses with respect to Nernst data allow us to establish that thermal fluctuations following the Ginzburg-Landau scheme are dominant for nearly optimally doped samples. The deduced coherence length $\xi \left(T\right)$ is in perfect agreement with a Gaussian (Aslamazov-Larkin) contribution for $1.01T_c\lesssim T\lesssim 1.2T_c$. A phase-fluctuation contribution might be invoked for the most underdoped samples in a $T$ range which increases when controlled disorder is introduced by electron irradiation. For all dopings we evidence that the fluctuations are highly damped when increasing $T$ or $H$. This behavior does not follow the Ginzburg-Landau approach, which should be independent of the microscopic specificities of the superconducting state. The data permits us to define a field $H_c^{\prime }\left(T\right)$ and a temperature $T_c^{\prime }$ above which the SCFs are fully suppressed. The analysis of the fluctuation magnetoconductance in the Ginzburg-Landau approach allows us to determine the critical field $H_{c2}\left(0\right)$. The actual values of $H_c^{\prime }\left(0\right)$ and $H_{c2}\left(0\right)$ are found to be quite similar and both increase with hole doping. These depairing fields, which are directly connected to the magnitude of the superconducting gap, do therefore follow the $T_c$ variation which is at odds with the sharp decrease of the pseudogap $T^{*}$ with increasing hole doping. This is on line with our previous evidence that $T^{*}$ is not the onset of pairing. So the large gap seen by spectroscopic experiments in the underdoped regime has to be associated with the pseudogap. We finally propose here a three-dimensional phase diagram including a disorder axis, which makes it possible to explain most peculiar observations done so far on the diverse cuprate families.

Figure 1

Temperature dependences of the resistivites of the four different pure YBCO${}_{6+x}$ samples studied.

Figure 2

Resistivity increase normalized to its zero-field value plotted versus (a) $H$ and (b) $H^2$ for temperatures above $T_c$ in the optimally doped sample OPT93.6. For $T$ $⩾140$ K, a $H^2$ dependence of the MR, represented as dashed lines in (b), is observed for all values of field. For lower $T$, it is only seen above a given magnetic field $H_c^{\prime }$ (arrows), which is taken as the threshold field necessary to completely restore the normal state.

Figure 3

Enlarged view of the variation of the MR in optimally doped sample OPT93.6 which makes it possible to better visualize the deviations from the $H^2$ normal-state dependence for $T⩽130$ K.

Figure 4

The MR coefficient $a_{\mathrm{trans}}$ measured at 55 T (solid symbols) is plotted versus $T$ in logarithmic scales and compared to that obtained at low field and higher $T$ (open symbols), for both optimally doped OPT93.6 and underdoped UD57. Low-field data in this latter case are taken from Ref. 39. In both cases the continuity of the data emphasizes that the magnitude of $H$ has been sufficient to restore the normal state.

Figure 5

(a) Field variation of the resistivity for decreasing temperatures down to 60 K in UD57. (b) The MR is plotted versus $H^2$ for $80\text{K}⩽T⩽117$ K. The full lines are fitting curves using Eq. (2) in the high-field range.

Figure 6

Decomposition of the magnetoconductivity $-\Delta \sigma \left(H\right)$ measured at 85 K for the UD57 sample in a normal-state contribution and a superconducting contribution. The zero-field value of $\Delta {\sigma }_{SF}\left(H\right)$ gives the value of the paraconductivity at 85 K.

Figure 7

SC fluctuation contribution to the zero-field conductivity $\Delta {\sigma }_{SF}(T,0)$ for the four YBCO samples studied here (Ref. 45). The enlargement of the high-$T$ range shown in the inset gives an estimate of the accuracy on the determination of $T_c^{\prime }$, the onset of SC fluctuations. Lines are guides for the eyes.

Figure 8

The values of $T_c^{\prime }$ (squares) and $T^{☆}$ (circles) are plotted versus the hole doping for the four samples studied. The solid line indicates the superconducting dome. Contrary to $T_c^{\prime }$ that is rather insensitive to hole doping, $T^{☆}$ is found to increase with decreasing doping and crosses the $T_c^{\prime }$ line near optimal doping.[16]

Figure 9

SC fluctuation contribution to the conductivity $\Delta {\sigma }_{SF}(T,H)$ in UD85 plotted versus $H^2$. The initial linear decays displayed as dashed lines visualize the quadratic field dependence observed for low magnetic fields. The arrows indicate the threshold fields $H_c^{\prime }\left(T\right)$ taken at $\Delta {\sigma }_{SF}(T,H)=1\times {10}^3{\left(\Omega \text{m}\right)}^{-1}$. The plotted curves correspond to increasing temperatures from top to bottom.

Figure 10

The field $H_c^{\prime }$ at which SC fluctuations disappear and normal state is fully restored is reported versus $T$ for the four pure samples studied. Dashed lines represent the fitting curves to Eq. (8) using data with solid symbols. When $H_c^{\prime }\left(T\right)\gtrsim 40T$ (open symbols), the data are somewhat underestimated as the maximum applied field is not sufficient.

Figure 11

Magnetoresistivity $\delta \rho /{\rho }_0$ plotted versus $H$ (a) or $H^2$ (b) for a UD57 irradiated sample with $T_c=5$ K. Dotted lines give the normal-state behavior restored in high fields. They parallel each other, which points out that the MR and the scattering time $\tau$ are nearly $T$ independent for strong disorder.

Figure 12

Comparison of the SCF conductivity $\Delta {\sigma }_{SF}(T,0)$ in (a) and onset field $H_c^{\prime }\left(T\right)$ in (b) for pure (solid symbols) and disordered (open symbols) samples of OPT93.6 (diamonds, triangles) and UD57 (squares), with the reduced $T_c$ values as indicated in (a). In (b) dashed lines are the fitting curves using Eq. (8). Data corresponding to an irradiated OPT sample with $T_c=30$ K (open triangles) have been added (Ref. 46).

Figure 13

Variations of (a) $T_c^{\prime }$ and (b) $H_c^{\prime }\left(0\right)$ for OPT (diamonds) and UD (squares) pure and disordered samples versus $T_c$. Here the $T_c$ value is used to monitor the disorder. The full line in (a) corresponds to a slope unity which parallels the $T_c$ variation. Data from Ref. 46 corresponding to irradiated OPT samples with $T_c=30$ K and $1.9$ K have been added.

Figure 14

Comparison of the onset temperature for SCFs extracted from Nernst measurements ($T_{\nu }$) and from this work ($T_c^{\prime }$) in the same OPT93.6 and UD57 samples. The solid symbols are for the off-diagonal Peltier conductivity deduced from the analysis of the Nernst coefficient (Ref. 7) while the open ones are the values of $\Delta {\sigma }_{SF}(T,0)$ obtained in this work. The values of $T_{\nu }$ (vertical arrows) had been estimated at temperatures corresponding to the minimal values of ${\alpha }_{xy}/B$.

Figure 15

The maximum fields required to completely suppress SC at $T=0$, as inferred from Nernst, diamagnetism or transport measurements, are plotted versus the onset of superconducting fluctuations for LSCO samples [open diamonds (Ref. 6), open squares (Ref. 51)], La-Bi2201 [open circles (Ref. 51)], and YBCO pure or irradiated crystals (this work). While the results for LSCO, La-Bi2201, and irradiated YBCO samples follow more or less the same linear dependence (dashed line), the $H_c^{\prime }\left(0\right)$ values for the pure UD85, OPT93.6, and OD92.7 are much larger with respect to their $T_c^{\prime }$.

Figure 16

(a) $T$ variations of the zero-field resistivity (solid line) and the normal-state values (symbols) deduced from high-field data for OPT93.6. The colored area corresponds to the range where superconducting fluctuations are effectively present while the hatched area is the extra range found by neglecting the decay of normal-state resistivity due to the pseudogap. (b) The variation $\Delta {\sigma }_{SF}\left(T\right)$ deduced assuming a linear fit would appear more accurate though it overestimates the actual value by at least a factor 3.

Figure 17

Superconducting fluctuation conductivity $\Delta {\sigma }_{SF}$ for the four pure samples considered here plotted versus $\epsilon =\text{ln}(T/T_c)$. Values of $T_c$ have been taken here at the midpoint of the resistive transition, and error bars for $\epsilon$ using the onset and offset values of $T_c$ are indicated. The dashed line represents the expression of Eq. (10) with $s=11.7$ Å. and ${\xi }_c\left(0\right)\simeq 0.9$ Å. Full lines are guides for the eye.

Figure 18

Same as Fig. 17 for the pure and irradiated optimally doped and underdoped YBCO${}_{6.6}$ crystals. Solid symbols are for the pure samples while empty ones are for the irradiated ones. The shift to the right observed for the UD57 samples with increasing disorder is shown in Sec. 6c to result from the reduction in $T_c$ with respect to the mean-field temperature $T_{c0}$.

Figure 19

The values of ${\alpha }_{xy}/B$ taken from Ref. 7 are plotted versus the values of $\Delta {\sigma }_{SF}\left(T\right)$ determined in this work for (a) the pure OPT sample and (b) two UD57 samples pure and irradiated. The values of $\Delta {\sigma }_{SF}\left(T\right)$ reported here have been obtained by interpolation between the data reported in Fig. 17. Solid lines are linear fits for the high $T$ values while the dashed line is guide for the eye for the data near $T_c$.

Figure 20

The SCF conductivity normalized to the value in the normal state is plotted versus $T$ for the UD57 samples either pure (circles) or irradiated with $T_c$ values decreased down to 26.8 K (squares) and 11 K (triangles). The vertical bars indicate the estimated values of $T_{KT}$ which are slightly lower than the values of $T_c$ (taken at the midpoint of the superconducting transition). The arrows are for the mean-field temperature $T_c^{MF}$ estimated here from the deviation to the fitting curves using Eq. (17) (solid lines).

Figure 21

Evolution of the fluctuation magnetoconductivity $-\Delta {\sigma }_H(T,H)=\Delta {\sigma }_{SF}(T,0)-\Delta {\sigma }_{SF}(T,H)$ as a function of $H$ for the OPT93.6 sample at different temperatures, 94.4, 97, 100, and 103.4 K. The dotted lines represent the computed results from Eq. (19) with $H_{c2}\left(0\right)=180\left(10\right)$ T. They deviate from the data beyond the $H^{☆}$ field values shown by arrows and reported in Fig. 22.

Figure 22

Values of $H^{*}$ at which Eq. (19) deviates from the experimental results. Those are compared to the expected linear dependence of Eq. (21) using the determined value of $H_{c2}\left(0\right)=180\left(10\right)$ T.

Figure 23

Evolution of the fluctuation magnetoconductivity $-\Delta {\sigma }_H(T,H)=\Delta {\sigma }_{SF}(T,0)-\Delta {\sigma }_{SF}(T,H)$ as a function of $H$ for the UD57 sample at different temperatures, 82.4, 84.2, and 91.8 K. The dotted lines represent the computed results using Eq. (19) with $T_{c0}=72$ K and $H_{c2}\left(0\right)=90\pm 10$ T.

Figure 24

Comparison of the temperature dependence of $\Delta {\sigma }_{SF}$ for the OD92.5, OPT93.6, and UD85 samples with Eq. (23) (solid line) that takes into account short-wavelength cutoff in the fluctuation spectrum.[75, 76] The dashed line is for the LD expression [Eq. (10)].

Figure 25

The ratio $\Delta {\sigma }_{SF}\left(T\right)/\Delta {\sigma }_{LD}\left(T\right)$ is plotted versus $T-T_c$ in a semilogarithmic scale for all the pure samples studied. $\Delta {\sigma }_{LD}\left(T\right)$ is calculated from Eq. (10) using the value of ${\xi }_c=0.89$ Å determined in Sec. 6a. $T_{GL}$ indicates the upper bound for the GL regime. In the case of the most underdoped UD57 sample, we have assumed that the phase-fluctuation regime disappears at $T_{c0}=72$K (see Sec. 6). The full lines are exponential fits of the data above $T_{GL}$.

Figure 26

SC fluctuation contribution to the conductivity $\Delta {\sigma }_{SF}(T,H)$ plotted versus $H^2$ for the UD85 sample. In (a), the data plotted in a linear scale for the high-$T$ regime can be fitted by the exponential relationship of Eq. (26) with $H_0\simeq 25$ T. In (b), the data are plotted in a semilogarithmic scale for all the temperatures investigated. The dotted lines are curves using Eq. (19) with $H_{c2}\left(0\right)=125\left(5\right)$ T, which deviate from the experimental data for $H>H^{☆}$ indicated by arrows. The straight lines are exponential fits of the high-field data with $H_0=25\pm 3$ T whatever $T$. The plotted curves correspond to increasing temperatures from top to bottom.

Figure 27

SC fluctuation contribution to the conductivity $\Delta {\sigma }_{SF}(T,H)$ plotted versus $H^2$ in a semilog scale for (a) the OPT93.6 sample, (b) the UD57 sample, and (c) the UD57 irradiated sample with $T_c=25$ K. The full lines are exponential fits according to Eq. (26) which do not take into account the low-field data at low $T$. For all these samples, we find $H_0=25\pm 5$ T at all temperatures. For the pure UD57 sample, we have also indicated the matching curves taken from Eq. (19) with $H_{c2}\left(0\right)=90\left(10\right)$ T in order to better visualize the deviations at larger fields. Here again it is seen that Eq. (19) does not fit the data for $T\gtrsim 93.3$ K, even at low fields, if one keeps the same value of $H_{c2}\left(0\right)$ (see discussion in Sec. 8). The plotted curves correspond to increasing temperatures from top to bottom.

Figure 28

Phase diagram constructed on the data points obtained here, showing the evolution of $T_c^{\prime }$ and the onset of SCFs, with doping and disorder. The fact that the pseudogap and the SCF surfaces intersect near optimum doping in the clean limit is apparent. These surfaces have been limited to experimental ranges where they have been determined experimentally. In the overdoped regime, data taken on Tl 2201 indicates that disorder suppresses SC without any anomalous extension of the SCFs (Ref. 36).