Temperature range of superconducting fluctuations above $T_c$ in YBa${}_2$Cu${}_3$O${}_{7-\delta }$ single crystals

Microwave absorption measurements in magnetic fields from 0 up to 16 T were used to determine the temperature range of superconducting fluctuations above the superconducting critical temperature $T_c$ in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\mathrm{\delta }}$. Measurements were performed on deeply underdoped, slightly underdoped, and overdoped single crystals. The temperature range of the superconducting fluctuations above $T_c$ is determined by an experimental method which is free from arbitrary assumptions about subtracting the nonsuperconducting contributions to the total measured signal and/or theoretical models to extract the unknown parameters. The superconducting fluctuations are detected in the $\mathit{ab}$ plane, and $c$-axis conductivity, by identifying the onset temperature $T^{\prime }$. Within the sensitivity of the method, this fluctuation regime is found only within a fairly narrow region above $T_c$. Its width increases from 7 K in the overdoped sample ($T_c=89\hspace{0.5em}\mathrm{K}$) to, at most, 23 K in the deeply underdoped sample ($T_c=57\hspace{0.5em}\mathrm{K}$), so that $T^{\prime }$ falls well below the pseudogap temperature $T^{*}$. Implications of these findings are discussed in the context of other experimental probes of superconducting fluctuations in the cuprates.

Figure 1

Microwave absorption in slightly underdoped YBCO ($T_c=87\mathrm{\hspace{0.5em}}\mathrm{K}$). External dc magnetic field is applied parallel to the $c$ axis (black: $B_{\mathit{dc}}=0$; red: $B_{\mathit{dc}}=8\mathrm{\hspace{0.5em}}\mathrm{T}$; green: $B_{\mathit{dc}}=16\mathrm{\hspace{0.5em}}\mathrm{T}$). (a) $H_{\mathrm{mw}}$ is in the $c$ direction so that the induced microwave currents flow in the $\mathit{ab}$ plane. (b) $H_{\mathrm{mw}}$ lies in the $\mathit{ab}$ plane so that the induced microwave currents flow partly in the $c$-direction and partly in the $\mathit{ab}$ plane. Insets show the difference in absorption between $B_{\mathit{dc}}=16\mathrm{\hspace{0.5em}}\mathrm{T}$ and zero field for their respective configurations.

Figure 2

Microwave absorption in deeply underdoped YBCO ($T_c=57\mathrm{\hspace{0.5em}}\mathrm{K}$). External dc magnetic field is applied parallel to the $c$ axis (black: $B_{\mathit{dc}}=0$; red: $B_{\mathit{dc}}=8\mathrm{\hspace{0.5em}}\mathrm{T}$; green: $B_{\mathit{dc}}=16\mathrm{\hspace{0.5em}}\mathrm{T}$). (a) $H_{\mathrm{mw}}$ is in the $c$ direction so that the induced microwave currents flow in the $\mathit{ab}$ plane; (b) $H_{\mathrm{mw}}$ lies in the $\mathit{ab}$ plane so that the induced microwave currents flow partly in the $c$ direction and partly in the $\mathit{ab}$ plane. Insets show the difference in absorption between $B_{\mathit{dc}}=16\mathrm{\hspace{0.5em}}\mathrm{T}$ and zero field for their respective configurations.

Figure 3

Microwave absorption in the deeply underdoped UD57 YBCO sample before (dotted lines) and after aging at room temperature in air (solid lines). The latter data show that the intake of oxygen takes place during aging and a small fraction of the sample becomes optimally doped. External dc magnetic field is applied parallel to the $c$ axis (black: $B_{\mathit{dc}}=0$; green: $B_{\mathit{dc}}=16\mathrm{\hspace{0.5em}}\mathrm{T}$). (a) $H_{\mathrm{mw}}$ is in the $c$ direction so that the induced microwave currents flow in the $\mathit{ab}$ plane; (b) $H_{\mathrm{mw}}$ lies in the $\mathit{ab}$ plane so that the induced microwave currents flow partly in the $c$ direction and partly in the $\mathit{ab}$ plane. Insets show the difference in absorption between $B_{\mathit{dc}}=16\mathrm{\hspace{0.5em}}\mathrm{T}$ and zero field for their respective configurations.

Figure 4

Microwave absorption in overdoped YBCO ($T_c=89\mathrm{\hspace{0.5em}}\mathrm{K}$). External dc magnetic field is applied parallel to the $c$ axis (black: $B_{\mathit{dc}}=0$; red: $B_{\mathit{dc}}=8\mathrm{\hspace{0.5em}}\mathrm{T}$; green: $B_{\mathit{dc}}=16\mathrm{\hspace{0.5em}}\mathrm{T}$). (a) $H_{\mathrm{mw}}$ is in the $c$ direction so that the induced microwave currents flow in the $\mathit{ab}$ plane; (b) $H_{\mathrm{mw}}$ is parallel to the $a$ axis so that the induced microwave currents flow partly along the $c$ axis and partly along the $b$ axis. Insets show the difference in absorption between $B_{\mathit{dc}}=16\mathrm{\hspace{0.5em}}\mathrm{T}$ and zero field for their respective configurations.

Figure 5

(a) The real part of the $\mathit{ab}$-plane resistivity of the three YBCO samples, OD89 (green), UD87 (black), and UD57 (red) as determined from the microwave measurements in the mode in which the current flows only in the $\mathit{ab}$ plane [Figs. 1a, 2a, and 4a]. Metallic behavior is observed in overdoped and underdoped samples in the normal state. The thicker and thinner lines pertain to the results obtained in zero field and 16 T fields, respectively. (b) The real part of the $c$-axis resistivity determined by the decomposition of the total signal measured in the mode where the current flows partly in the $\mathit{ab}$ plane and partly along the $c$ axis as shown in the insets of Figs. 1b, 2b, and 4b. The decomposition procedure is described in the text.

Figure 6

(a) $\mathit{ab}$-plane complex conductivity of the UD87 sample. The data points of the real part of the conductivity (${\sigma }_{1\mathit{ab}}$) are represented by black symbols while red symbols are used for the imaginary part (${\sigma }_{2\mathit{ab}}$). Full circles, crosses, and open triangles stand for the conductivities in zero magnetic field, 8 and 16 T fields, respectively. The inset in (a) shows on an expanded scale the difference between the conductivities in zero field and in the field of 16 T ($\Delta {\sigma }_{1\mathit{ab}}$ blue circles, $\Delta {\sigma }_{2\mathit{ab}}$ red triangles) at temperatures above the zero-field $T_{c0}$. (b) The complex conductivity along the $c$ axis in the UD87 sample. The presentation is analogous to that of (a).

Figure 7

Presentation of the real and imaginary parts of the complex superconducting fluctuation conductivity above $T_c$ as a function of reduced temperature $t=\ln (T/T_c)$ on a log-log plot. Data for the slightly underdoped sample UD87 is shown by full symbols and the dashed lines are theoretical curves as described in the text. (a) In-plane conductivity ${\sigma }_{1\mathit{ab}}$ and ${\sigma }_{2\mathit{ab}}$. (b) Conductivity along the $c$ axis ${\sigma }_{1c}$ and ${\sigma }_{2c}$.

Figure 8

Real and imaginary parts of the complex superconducting fluctuation conductivity above $T_c$ in deeply underdoped sample UD57. Experimental values are shown by full symbols and the dashed lines are theoretical curves as described in the text. (a) In-plane conductivity ${\sigma }_{1\mathit{ab}}$ and ${\sigma }_{2\mathit{ab}}$. (b) Conductivity along the $c$ axis ${\sigma }_{1c}$ and ${\sigma }_{2c}$.

Figure 9

Real and imaginary parts of the complex superconducting fluctuation conductivity above $T_c$ in overdoped sample OD89. Experimental values are shown by full symbols and the dashed lines are theoretical curves as described in the text. (a) In-plane conductivity ${\sigma }_{1\mathit{ab}}$ and ${\sigma }_{2\mathit{ab}}$. (b) Conductivity along the $c$ axis ${\sigma }_{1c}$ and ${\sigma }_{2c}$.