Temperature dependence of the bulk energy gap in underdoped ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$: Evidence for the mean-field superconducting transition

Understanding of the puzzling phenomenon of high-temperature superconductivity requires reliable spectroscopic information about temperature dependence of the bulk electronic density of states. Here I present a detailed analysis of the $T$ evolution of bulk electronic spectra in ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$ obtained by intrinsic tunneling spectroscopy on small mesa structures. Unambiguous spectroscopic information is obtained by obviation of self-heating problem and by improving the spectroscopic resolution. The obtained data allow accurate determination of the superconducting transition temperature and indicate that (i) the superconducting transition maintains the mean-field character down to moderate underdoping and is associated with a rapid opening of the superconducting gap, which is well described by the conventional BCS $T$ dependence. (ii) The mean-field critical temperature reaches maximum at the optimal doping and decreases with underdoping. Such behavior is inconsistent with theories assuming intimate connection between superconducting and antiferromagnetic spin gaps and supports proposals associating high-temperature superconductivity with the presence of competing ground states and a quantum critical point near optimal doping.

Figure 1

(Color online) (a) Zero-bias resistance $R_0$ vs $T$ for a mesa on a nearly optimally doped crystal SMa. Inset shows $I\text{−}V$ characteristics at $T=4.7\text{K}$ which clarifies the origin of certain parts of the resistive transition. (b) The same data are shown as a thermal activation plot $R_0/T$ (in a logarithmic scale) vs $1/T$. It is seen that, in the whole normal-state region $T>T_c$, $R_0/T$ is described by the Arrhenius law (dashed line) with a constant TA barrier $U_{\text{TA}}\simeq 25\text{meV}$.

Figure 2

(Color online) (a) $I\text{−}V$ curves at different $T$ for a small mesa on a near optimally doped SMa crystal. The kink and transition to approximately Ohmic resistance at the sum-gap voltage is clearly seen. Inset shows multiple quasiparticle branches due to one-by-one switching of intrinsic Josephson junctions into the resistive state. (b) $dI/dV$ vs voltage per junction for the same mesa. Arrows indicate the following characteristic features: the sum-gap peak, a minor double-gap dip at $T<T_c$, and a crossing point and a hump at $T>T_c$.

Figure 3

(Color online) Comparison of (a) simulated and (c) experimental IVCs for S42 at 4.9 and 40 K; (b) and (d) demonstrate collapse of the $I\text{−}V$ curves, scaled by $\Delta \left(T\right)$ and indicate that both $V$ and $I$ scales are determined by $\Delta \left(T\right)$.

Figure 4

(Color online) (a) $dI/dV\left(V\right)$ (in a semilogarithmic scale) for two mesas with different areas on a moderately underdoped S92 crystal. It is seen how heating and in-plane resistivity bend upward curves at high bias and reduce the sum-gap peak voltage for the large mesa. (b) $dI/dV\left(V\right)$ curves at different $T$ for the small mesa. In is seen that the curves maintain the linear V shape (in the semilogarithmic scale) in the whole subgap region when self-heating becomes negligible. A sudden crossover at $T=T_c$ from tunnelinglike with $T$-independent slope to thermal activation behavior with $T$-dependent slope is clearly seen. Data are from Ref. 38.

Figure 5

(Color online) (a) $dI/dV\left(V\right)$ (in a semilogarithmic scale) at different $T$ for the same mesas on near optimally doped crystal SMa before and after FIB trimming. The peak voltage is reduced in the larger mesa due to self-heating. (b) $dI/dV$ curves at different $T$ for the small mesa. The characteristic V shape in the semilogarithmic scale is observed at all $T$. As in Fig. 4, the crossover from $T$-independent to $T$-dependent slope occurs at $T_c$.

Figure 6

(Color online) Simulated TA characteristics from Eq. (3) with $U_{\text{TA}}=24\text{meV}$. They reproduce all major features of experimental curves at $T>T_c$ in Figs. 4, 5b including the V shape with the slope proportional to the reciprocal temperature and the crossing point at $eV\sim 2U_{\text{TA}}$.

Figure 7

(Color online) Improvement of spectroscopic resolution at elevated $T$ by means of $T$-differential spectroscopy for the same small mesa on S92 crystal. In (a) shown are differences between $\text{ln}\left(\sigma =dI/dV\right)$ curves from Fig. 4b at two consecutive $T$, normalized by the temperature difference. Such curves emphasize $T$-dependent spectroscopic features and collapse into the universal curve for the case of TA. In (b) the universal TA curve at high $T_1,T_2$ was subtracted to remove completely the TA background. This allows clear observation of evolution of the sum-gap peak and the double-gap dip well above the phase-coherent $T_c^{phase}$.

Figure 8

(Color online) The same as in Fig. 7 for another underdoped S82 crystal. The collapse of curves at high $T$ into the universal TA curve is clearly seen in (a). From panel (b) it is seen that the normal-state pseudogap remains roughly $T$ independent at $T>T_c$.

Figure 9

(Color online) (a) $T$ variation of characteristic spectroscopic features for three mesas on the near optimally doped SMa crystal. Data for the same mesas are shown by symbols of the same type and color. Open symbols, the TA hump; the dashed-dotted curve, $2U_{\text{TA}}$. Downturn of $U_{\text{TA}}$ marks the $c$-axis resistive transition, $T_c^{phase}$, at which $c$-axis phase coherence is established. It is marked by the dashed vertical line. Solid symbols, the sum-gap peak; it is seen that the measured gap is distorted with increasing mesa area as a result of self-heating. Inset in (a) shows power at the peak for the same mesas. (b) The heat-compensated $T$ dependence of the bulk energy gap. Solid line shows the conventional BCS $T$ dependence with the mean-field critical temperature $T_c^{mf}=96\text{K}$. Excellent agreement with experimental data is seen. No pseudogap is observed above $T_c^{mf}$. Inset in (b) shows the effective thermal resistance for the two larger mesas.

Figure 10

(Color online) The same as in Fig. 9 for the small mesa on the moderately underdoped S92 crystal. Crosses in (a) represent the crossing point of $dI/dV\left(V\right)$ curves at two consecutive temperatures [see Fig. 4b]. Filled triangles represent half the double-gap dip. It is seen that signatures of the energy gap (the pseudogap) survive up to $T^{\ast }\simeq 130\text{K}$ (see Fig. 7). Solid and dashed lines in (b) show that $\Delta \left(T\right)$ is equally well described by the pure BCS dependence or the combined gap between the BCS gap and $T$-independent pseudogap, shown by the dashed line. However, for both fits the mean-field superconducting temperature is smaller than that for the optimally doped crystal in Fig. 9.

Figure 11

(Color online) The same as in Fig. 10 for the S43 crystal. It is seen that the superconducting transition even in moderately underdoped HTSC has a mean-field character and is accompanied by opening of the superconducting gap.

Figure 12

(Color online) The same as in Fig. 10 for the S82 crystal. Data for mesas with different areas at $T\simeq 6\text{K}$ are shown. Gray area in (b) highlights the $T$ region without $c$-axis phase coherence but with persisting gap and in-plane superconductivity.

Figure 13

(Color online) (a) In-plane $R_{ab}$ and $c$-axis $R_c$ resistive transitions in a slightly underdoped crystal, similar to SMa. Panel (b) reproduces $\Delta \left(T\right)$ from Fig. 9b along with the resistive transition for a mesa on SMa. It is seen that the onset of in-plane superconductivity coincides with the extrapolated mean-field critical temperature; the onset of $c$-axis transition is close to the middle of the in-plane resistive transition, while the temperature at which $c$-axis phase coherence is established is further reduced due to thermal fluctuations. Gray-shaded area in (b) marks the superconducting region without phase coherence in the $c$-axis direction.

Figure 14

(Color online) The IVC at $T=30\text{K}$ from Fig. 2 in a semilogarithmic scale. Thin lines are multiple integers, indicating good periodicity of quasiparticle branches. However, a small nonuniformity of the junctions is seen from the spread of critical currents $\Delta I_c$, which leads to splitting $\Delta I_{sg}$ of the sum-gap peak in $dI/dV\left(I\right)$ (red line) because different junctions reach $V_{sg}$ at slightly different currents.

Figure 15

(Color online) Simulated distortion of superconductor-insulator-superconductor (SIS) tunneling characteristics by strong self-heating. (a) Undistorted IVCs at different $T$. Simulations were made for typical parameters of our mesas, including detailed $\kappa \left(T\right)$ for Bi-2212 and for coherent directional $d$-wave tunneling. (b) Distorted IVCs at the same base $T$. (c) The mesa temperature as a function of bias. (d) $T$ dependence of the genuine superconducting gap (solid line) and the “measured” gap obtained from distorted IVCs (dashed curve). Note that even strong self-heating ($T$ reaches $T_c/2$ at $V_{sg}$ at 4.2 K) does not cause considerable distortion of the measured gap. Data are from Ref. 62.

Figure 16

(Color online) Top panel shows a sketch of current distribution in case of resistive electrodes and contact configuration for three- and four-probe measurements. The main panel shows simulated $I\text{−}V$ curves in the three-probe (dashed lines) and four-probe (solid lines) configurations for different in-plane electrode resistances. It is seen that in-plane resistance distorts the measurement of junction characteristics and that acute back bending can develop in the four-probe case for high in-plane resistance. Inset shows the in-plane IVC.