Microwave measurements of the in-plane and $c$-axis conductivity in ${\text{HgBa}}_2{\text{CuO}}_{4+\delta }$: Discriminating between superconducting fluctuations and pseudogap effects

An approach to microwave measurements is used in order to determine both, the in-plane and out-of-plane conductivity of the high-$T_c$ superconductor ${\text{HgBa}}_2{\text{CuO}}_{4+\delta }$ near optimal doping. Unlike the $ab$-plane conductivity, the $c$-axis conductivity is highly sensitive to superconducting fluctuations. From a single $c$-axis data set, we can clearly discern the opening of the pseudogap at $T^{\ast }=185\left(15\right)\text{K}$, the appearance of the superconducting fluctuations at a much lower temperature $T^{\prime }=105\left(2\right)\text{K}$, and the full transition to the superconducting state at the critical temperature $T_c=94.3\text{K}$. Thus, with the present high sensitivity, we establish that the extent of the superconducting fluctuations is only about $10\text{K}$ above $T_c$.

Figure 1

(Color online) Temperature dependence of ${\rho }_{ab}$ in the single crystal of nearly optimally doped ${\text{HgBa}}_2{\text{CuO}}_{4+\delta }$ (full line) calculated from the measured microwave absorption (lower inset, with $H_{\text{mw}}\parallel c$). The dashed line represents extrapolated linear behavior above 200 K. Upper inset: difference between ${\rho }_{ab}$ and the linear behavior extrapolated to low $T$ clearly reveals $T^{\ast }$.

Figure 2

(Color online) Microwave absorption for two geometries with $H_{\text{mw}}$ parallel to the $ab$ plane. Screening currents have in-plane and out-of-plane contributions. The conductivity along the $c$ axis is more than ${10}^3$ times smaller than that parallel to the $ab$ planes and consequently the skin depth is ${10}^{3/2}$ times larger. Thus we distinguish two cases: (a) when $H_{\text{mw}}$ is aligned with the sample’s shorter edge, the perpendicular longer edge acts as the sample’s width which is then larger than $2{\delta }_{ab}$, so that the microwave field is completely screened deeply in the sample bulk, (b) when $H_{\text{mw}}$ is along the sample’s longer edge, it is the sample’s shorter edge, which turns out to be smaller than $2{\delta }_{ab}$, that acts as its width. Consequently, the microwave currents flow throughout the sample volume and do not completely screen the field, i.e., the sample is transparent to some degree. These two experimental configurations are shown in the insets of the panels (a) and (b), where profiles of the microwave magnetic field across the sample are schematically represented by the gradual change in the color intensity. The lower inset of panel (a) shows ${\rho }_c$ obtained from both measured orientations, black and light gray (red) symbols, respectively. The dark gray (blue) line is the fit to the model described in the text. Thin lines in the main panels (a) and (b) are calculated temperature dependencies of the microwave absorption as explained in the text.

Figure 3

(Color online) Microwave absorption measured with the external dc magnetic field applied in the $c$ direction, in the same geometry as in the case of Fig. 2b. The applied fields were 0, 3, and 8 T, plotted in full (black), dashed (red) and dotted (green) lines, respectively. Inset: temperature dependence of $c$-axis conductivity calculated from the microwave absorption data for zero field (full black line) and for 8 T (dotted green line).