Vortex correlations in the liquid states of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+y}$ with tilted columnar defects

The interlayer and intralayer vortex correlations are studied in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+y}$ with tilted and nontilted columnar defects (CDs) by the Josephson plasma-resonance experiments under various field angles. The interlayer correlation is enhanced at the recoupling field, near which we observe clear triple-resonance peaks. The recoupling field is determined only by the density of CDs in the superconducting layer, irrespective of CD angles from the $c$ axis. In the low-field decoupled liquid state, we observe an increase of the interlayer correlations due to increased pinning by enlarged defects. By contrast, scaling analysis shows little effect of the pinning strength on the interlayer and intralayer correlations in the high-field recoupled liquid state.

Figure 1

(Color online) Microwave loss as a function of $c$-axis component of the magnetic field $H_z$ in the ${\theta }_{\mathit{CD}}=80°$ sample (a) at various temperatures and (b) at various angles. Dashed lines indicate $H_z=B_{\Phi }^{*}∕5$ and $B_{\Phi }^{*}∕3$ positions.

Figure 2

(Color online) Angular dependence of $H_{pz}$ in BSCCO with ${\theta }_{\mathit{CD}}=80°$ at $73.5\mathrm{K}$. The horizontal dashed lines show $B_{\Phi }^{*}∕5$ and $B_{\Phi }^{*}∕3$. The inset shows schematic field dependence of IPC in BSCCO with tilted CDs for three magnetic-field directions.

Figure 3

(Color online) Angular dependence of (a) $H_p$ and (b) $H_{pz}$ for BSCCO with various ${\theta }_{\mathit{CD}}$ at low temperatures in the recoupled liquid state. Vertical dashed lines show ${\theta }_{\mathit{CD}}$ for each sample. (c) Calculated results of $H_{pz}$ normalized by the maximum values after scaling given by Eq. (2). The dashed line in (c) is for $\gamma =1$ and solid lines are for $\gamma =400$.

Figure 4

(Color online) (a) ${\theta }_H$ dependence of $H_{pz}$ for BSCCO with ${\theta }_{\mathit{CD}}=0°$ (open circles) and 70° (closed circles) with the same $B_{\Phi }\cos {\theta }_{\mathit{CD}}=6.84\mathrm{kG}$. Vertical dashed lines indicate ${\theta }_{\mathit{CD}}$ for each sample at $60\mathrm{K}$ in the recoupled liquid state. Angular dependence of $H_p$ for BSCCO with ${\theta }_{\mathit{CD}}=70°$ scaled back to the case of ${\theta }_{\mathit{CD}}=0°$ using Eq. (2) is shown in open squares with $\gamma =400$. (b) ${\theta }_H$ dependence of $H_{pz}$ at higher temperatures in the decoupled liquid state.

Figure 5

(Color online) Concept of angular transformation in an anisotropic superconductor with CDs between ${\theta }_{\mathit{CD}}>0°$ and ${\theta }_{\mathit{CD}}=0°$. (a) An example with $\gamma =5$ and ${\theta }_{\mathit{CD}}=70°$. (b) After application of the angular transformation to ${\theta }_H$ and ${\theta }_{\mathit{CD}}$. ${\tilde{\theta }}_{H-\mathit{CD}}$ is relative field direction from CDs. (c) Rotational transformation of (b) by an angle $-{\tilde{\theta }}_{\mathit{CD}}$. (d) Scaling back to an anisotropic system with ${\theta }_{\mathit{CD}}=0°$.

Figure 6

(Color online) $f\left(b\right)=[H_{pz}\left({\theta }_H\right)∕H_{pz}(0°)]$ as a function of $b=(2\pi s{H}_{px}∕\sqrt{H_{pz}{\Phi }_0})$ in BSCCO with $B_{\Phi }\cos {\theta }_{\mathit{CD}}=6.84\mathrm{kG}$ and ${\theta }_{\mathit{CD}}=0°$ (open squares), and 70° (open circles). Three lines show parabolic curves as $f\left(b\right)=1-{(R∕a)}^2{b}^2$. Inset shows the temperature dependence of $R∕a$ with $B_{\Phi }\cos {\theta }_{\mathit{CD}}=6.84\mathrm{kG}$ and ${\theta }_{\mathit{CD}}=0°$ (open circles), and 70° (closed circles).